Robust Flight Control: A Design Challenge (Lecture Notes in Control and Information Sciences)

Jean-François Magni, Samir Bennani and Jan Terlouw (Eds.)

Robust Flight Control: A Design Challenge

i

This book was rst printed by Springer-Verlag, 1997 Lexture Notes in Control and Information S ien es, 224.

Editors Jean-François Magni, Do teur ès S ien es ONERA CERT, Département d'Études et Re her hes en Automatique, BP 4025, F31055 Toulouse Cedex, Fran e. Samir Bennani, Ir. Delft University of Te hnology, Fa ulty of Aerospa e Engineering, Kluyverweg 1, 2629 HS Delft, The Netherlands. Jan Terlouw, Ir. National Aerospa e Laboratory NLR, Flight Me hani s Department, Anthony Fokkerweg 2, 1059 CM Amsterdam, The Netherlands.

ii

ROBUST FLIGHT CONTROL: A DESIGN CHALLENGE

EDITORS J.-F. Magni, S. Bennani & J. Terlouw

GARTEUR ACTION GROUP FM(AG08)

Resear h Establishments: Centro Italiano Ri er he Aerospaziali (CIRA, Italy), Deuts he Fors hungsanstalt für Luft- und Raumfahrt (DLR, Germany), Defen e Resear h Agen y (DRA, United Kingdom), Instituto Na ional de Té ni a Aeroespa ial (INTA, Spain), Laboratoire d'Automatique et d'Analyse des Systèmes (LAAS, Fran e), National Aerospa e Laboratory (NLR, The Netherlands), O e National d'Etudes et de Re her hes Aérospatiales (ONERA, Fran e).

Industry: Alenia Aeronauti a (ALN, Italy), Avro International Aerospa e (AVRO, United Kingdom), British Aerospa e, Dynami s (BAe-D, United Kingdom), British Aerospa e, Military Air raft (BAe-MA, United Kingdom), Cambridge Control Ltd (CCL, United Kingdom), Daimler-Benz Aerospa e Airbus (DASA, Germany), Fokker Air raft Company (FAC, The Netherlands), Saab Military Air raft (SMA, Sweden).

Universities: Craneld University (CUN, United Kingdom), Delft University of Te hnology (DUT, The Netherlands), Linköping University (LiTH, Sweden), Loughborough University (LUT, United Kingdom), University of Cambridge (UCAM, United Kingdom), University of Lei ester (ULES, United Kingdom), Universitá di Napoli Frederi o II (UNAP, Italy), Universitad Na ional de Edu a ión a Distan ia (UNED, Spain).

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iv

Contents Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

1

Introdu tion. Jan Terlouw and Chris Fielding . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Tutorial part 2

Multi-Obje tive Parameter Synthesis (MOPS). Georg Grübel and Hans-Dieter Joos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3

Eigenstru ture Assignment. Lester Faleiro, Jean-François Magni, Jesús M. de la Cruz and Stefano S ala . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4

Linear Quadrati Optimal Control. Fran es o Amato, Massimiliano Mattei and Stefano S ala . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5

Robust Quadrati Stabilization. Germain Gar ia, Ja ques Bernussou, Jamal Daafouz and Denis Arzelier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6 7

H1 Mixed Sensitivity. Mark R. Tu ker and Daniel J. Walker . . . . . . . . . . . 52 H1 Loop Shaping. George Papageorgiou, Keith Glover, Alex Smerlas and Ian Postlethwaite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

8 -Synthesis. 9

Samir Bennani, Gertjan Looye and Carsten S herer . . . . . . . 81

Nonlinear Dynami Inversion. Binh Dang Vu . . . . . . . . . . . . . . . . . . . . . . . . . 102

10

Robust Inverse Dynami s Estimation. Ewan Muir . . . . . . . . . . . . . . . . . . . 112

11

A Model Following Control Approa h. Holger Duda, Gerhard Bouwer, J.-Mi hael Baus hat and Klaus-Uwe Hahn . . . . . . . . . . . . . . . . . . . . . . . . . 116

12

Predi tive Control. Jan Ma iejowski and Mihai Huzmezan . . . . . . . . . . . 125

13

Fuzzy Logi Control.

Gerard S hram, Uzay Kaymak and Henk B. Ver-

bruggen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

RCAM part 14

The RCAM Design Challenge Problem Des ription.

Paul Lambre hts,

Samir Bennani, Gertjan Looye and Dieter Moormann . . . . . . . . . . . . . . 149

15

The Classi al Control Approa h. Jim E. Gautrey . . . . . . . . . . . . . . . . . . . . 180

16

Multi-Obje tive Parameter Synthesis (MOPS). Hans-Dieter Joos . . . . . 199

17

An Eigenstru ture Assignment Approa h (1).

Lester Faleiro and Roger

Pratt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

v

18

An Eigenstru ture Assignment Approa h (2). Jesús M. de la Cruz, Pablo Ruipérez and Joaquín Aranda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

19

A Modal Multi-Model Approa h. Carsten Döll, Jean-François Magni and Yann Le Gorre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

20

The Lyapunov Approa h. Jamal Daafouz, Denis Arzelier, Germain Gar ia and Ja ques Bernussou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

21

An

H1 Approa h. Mark R. Tu ker and Daniel J. Walker

-Synthesis Approa h (1). 23 A -Synthesis Approa h (2). 22

24

A

. . . . . . . . . . . . 300

Samir Bennani and Gertjan Looye . . . . . . 321 Jan S huring and Rob M.P. Goverde . . . 341

Autopilot Design based on the Model Following Control Approa h. Holger Duda, Gerhard Bouwer, J.-Mi hael Baus hat and Klaus-Uwe Hahn

25

360

Flight Management using Predi tive Control. Mihai Huzmezan and Jan M. Ma iejowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

26

A Fuzzy Control Approa h. Gerard S hram and Henk B. Verbruggen . 398

HIRM part 27

The HIRM Design Challenge Problem Des ription. Ewan Muir . . . . . . 421

28

Design via LQ Methods. Fran es o Amato, Massimiliano Mattei, Stefano S ala and Leopoldo Verde . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446

29

The

H1 Loop Shaping Approa h. George Papageorgiou, Keith Glover and

Ri k A. Hyde . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466

30

Design of Stability Augmentation System using

-Synthesis.

Karin Ståhl

Gunnarsson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486

31

Design of a Robust, S heduled Controller using

-Synthesis.

Johan An-

thonie Markerink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

32

Nonlinear Dynami Inversion and LQ Te hniques. Béatri e Es ande . . 525

33

The Robust Inverse Dynami s Estimation Approa h. Ewan Muir . . . . . 543

Con luding part 34

The Industrial View. Chris Fielding and Robert Lu kner . . . . . . . . . . . . 569

35

An Other View of the Design Challenge A hievements. Georg Grübel 605

36

Con luding Remarks. Samir Bennani, Jean-François Magni and Jan Terlouw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612

Appendix A

Used Nomen lature. Anders Helmersson and Karin Ståhl Gunnarsson

Bibliography vi

614

Author Index

Fran es o Amato . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33, 446 Joaquín Aranda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 Denis Arzelier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42, 278 J.-Mi hael Baus hat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116, 360 Samir Bennani . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81, 149, 421, 321, 612 Ja ques Bernussou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42, 278 Gerhard Bouwer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116, 360 Jesús M. de la Cruz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22, 238 Jamal Daafouz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42, 278 Binh Dang Vu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Carsten Döll . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 Holger Duda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116, 360 Béatri e Es ande . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 Pierre Fabre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Lester Faleiro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22, 218 Chris Fielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1, 569 Germain Gar ia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42, 278 Jim E. Gautrey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Keith Glover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64, 466 Rob M.P. Goverde . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 Georg Grübel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13, 605 Klaus-Uwe Hahn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116, 360 Anders Helmersson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149, 614 Mihai Huzmezan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125, 379 Ri k A. Hyde . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421, 466 Jonathan Irving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 Joseph Irvoas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Hans-Dieter Joos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13, 199 Uzay Kaymak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Paul Lambre hts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149, 421 Tony Lambre gts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Yann Le Gorre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 Gertjan Looye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81, 149, 321 Robert Lu kner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 Jan Ma iejowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125, 379 Jean-François Magni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22, 149, 258, 612 Johan Anthonie Markerink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 Alberto Martínez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Massimiliano Mattei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33, 446 Philippe Ménard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Dieter Moormann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149, 421 Ewan Muir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112, 421, 543

vii

George Papageorgiou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64, 466 Ian Postlethwaite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Roger Pratt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 Pablo Ruipérez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 Stefano S ala . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22, 33, 149, 421, 446 Carsten S herer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Gerard S hram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135, 398 Jan S huring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341, 421 Phillip Sheen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Alex Smerlas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Karin Ståhl Gunnarsson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421, 486, 614 Jan Terlouw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1, 149, 421, 612 Mark R. Tu ker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52, 300 Hans van der Vaart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Henk B. Verbruggen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135, 398 Leopoldo Verde . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 Daniel J. Walker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52, 300

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1.

Introdu tion

Jan Terlouw and Chris Fielding 1

2

1.1 The Importan e of Advan ed Control Design Methods for the European Air raft Industry European manufa turers of military and ivil air raft have rea hed a high level of expertise in designing ight ontrol laws, to a point that they an solve virtually any realisti hallenge that might be foreseen in the near future. This

apability is a result of the lessons learned by generations of engineers who have extended and passed on their skills, always driven by the ultimate requirement - that one day their ight ontrol system (FCS) had to y.

However, the

large time and eort spent to solve all problems en ountered during the design pro ess poses the question whether improvements are possible. As the s ienti ommunity sometimes laims to have invented new methods to improve urrent ways of working, there is a natural interest from industry in what the resear hers have to oer. On the other hand, s ientists are interested in realisti appli ations to justify their work and to test new on epts. It is lear that there is a strong in entive for both worlds to work together, but a tually a hieving it an be di ult.

S ientists like to develop methods

whi h have general appli ability, and this is parti ularly true for ontrol theoreti ians. On the other hand, users of design methods are, from a professional point of view, mainly interested in dedi ated methods that solve their parti ular problems. The result is that many new ideas never really break through, be ause they are simply not spe ialised and elaborated enough, or be ause there is not enough liaison between the s ienti and industrial worlds. There are now a large number of ontroller design methods that have been developed over the past twenty-ve years (some have earlier origins). In this book twelve of them are treated:

   1

Multi-Obje tive Parameter Synthesis Eigenstru ture Assignment Linear Quadrati Optimal Control

National Aerospa e Laboratory NLR, Flight Me hani s Department, Anthony Fokker-

weg 2, 1059 CM Amsterdam, The Netherlands. 2

British Aerospa e Military Air raft, Aerodynami s Department, Warton Aerodrome,

Preston PR4 1AX, UK

1

 Lyapunov Te hniques  H1 Mixed Sensitivity  H1 Loop Shaping  -Synthesis  Nonlinear Dynami Inversion  Robust Inverse Dynami s Estimation  Model Following  Predi tive Control  Fuzzy Logi Control These methods have many dierent features. A ommon feature is that ea h of them is developed to a hieve advantages over lassi al te hniques. The laimed benets range from enhan ed performan e, resulting from multi-input multioutput ontrollers, to improved e ien y and simpli ation of the design pro ess. At the same time, the most important and obvious di ulty in adopting any new method is the la k of experien e of its use in pra ti e. This book is an attempt to redu e the gap between theory and prati e, with respe t to appli ation of modern ontrol design te hniques. It deals with ight ontrol of rigid body ivil and military air raft. The twelve te hniques mentioned above will be demonstrated on the basis of two ben hmark problems [145, 177℄. But rst, some general remarks will be made about ight ontrol laws as a part of FCS design.

Flight ontrol laws design The main fun tion of the ight ontrol system (FCS) of an air raft is to ontribute to its safe and e onomi operation, su h that the intended ight missions

an be a

omplished and unexpe ted events an be handled. The heart of a modern FCS onsists of the following omponents, arranged in a logi al way to benet from the prin iple of feedba k: sensors provide a ight ontrol omputer (FCC) information on air data, inertial data and o kpit data; an FCC in whi h ight ontrol laws are implemented to determine the ommands for the a tuation systems of the air raft ontrol surfa es and throttles for engines demands. For air raft, feedba k ontrol is used to provide tight pilot ommand tra king, to attenuate external disturban es su h as gusts and turbulen e and to provide robustness against modelling errors. In the early days of ight, safety was the main on ern for FCS designers. Pilots needed signi ant eort to maintain some ight onditions under all ir umstan es. Today, safety is even more important, be ause many more people are transported, higher osts are involved in establishing safety, and the reputation of airlines and air raft manufa turers is paramount, in an in reasingly ompetitive market. Fly by Wire allows the pilot to ontrol the

2

air raft states, as an alternative to the onventional dire t ontrol of the engines and ontrol surfa es.

It gives new opportunities to in rease the overall level

of safety through the exibility oered by the ontrol laws [78℄. For example, error-tolerant ontrol laws provide ight envelope prote tion, and help the pilot to re over from unusual attitudes and su

essfully a hieve riti al manoeuvres. The use of modern FCS an be bene ial from an e onomi point of view. For ertain types of air raft, fuel onsumption an be redu ed by allowing relaxed stati stability, ountera ted by the appli ation of a tive ontrol. Another advantage related to fuel onsumption is that for large air raft the weight of Fly by Wire systems is smaller than that of onventional systems. Furthermore, the so- alled family on ept an be introdu ed. Flying dierent air raft an be made almost the same for pilots, by making appropriate adjustments in the ight ontrol laws. As a result, dierent air raft feel almost the same, therefore helping to redu e pilot training osts. Most importantly, modern FCS have ontributed to improved dynami al behaviour. Certain military air raft annot be own without a stability augmentation system.

The open loop instability, whi h is related to agility of

the air raft, is utilised to obtain better performan e and manoeuvrability of the losed-loop system.

For ivil air raft, performan e an be in reased by

appli ation of a tive systems, for example to provide gust suppression and auto-trimming, in order to a hieve improved ride quality. The performan e benets a hieved, have the penalty of tremendous osts involved in the development of an advan ed FCS. In the past, the pilot sti k was typi ally onne ted with rods or ables to the ontrol surfa es. Sin e then, the in reased safety, and e onomi al and performan e demands have for ed air raft manufa turers to extend FCS to a high level of omplexity. The danger exists that the e onomi al benets des ribed above are nullied by higher design and maintenan e osts, while omplexity an potentially have a negative ee t on safety. The large number of fun tions and requirements have in reased the number of spe ialists areas needed for the FCS design pro ess. This makes the work

hallenging from a te hni al and management point of view. People who are responsible for mode logi , redundan y design, software and hardware development, design integration and erti ation have to work losely together. In the overall pro ess, ontrol laws designers assume a modest, but entral position. They have knowledge of ight me hani s, ontrol theory, handling qualities, airframe stru ture and FCS hardware.

Their task is inuen ed by the de-

sign requirements, the ight envelope, the air raft onguration omplexity, the stores arriage and weight distribution, the required autopilot modes, the air raft stability (or instability) levels and the aerodynami nonlinearity. The work of an industrial ight ontrol laws designer who uses lassi al design te hniques (see Chapter 15) may onsist of the following simplied sequen e of a tivities. The rst step is to derive a nonlinear dynami model of the air raft to be ontrolled. Getting familiar with the dynami al behaviour by means of trimming, stability and ontrol analysis and nonlinear simulations (for stable air raft) and understanding the inuen es of the modelling assump-

3

tions is most important at this stage. Linearisation and linear simulation of the model is also performed. The next step is to dene the ontroller ar hite ture and to make a rst design whi h in ludes gain s heduling to over the air raft's ight envelope. Implementation of the ontrol law in the nonlinear model, for o-line and piloted simulation, is arried out next. This pro ess might be repeated to optimize the design. In the design pro ess, nonlinearities and model un ertainties are important issues to understand and deserve mu h attention if a robust design is to be a hieved.

Robustness of ight ontrol systems Robustness investigation deals with the dis repan y between models and reality.

It is basi ally on erned with whether a ontrolled system will work

satifa torily under the ir umstan es it will meet in pra ti e. FCS designers have always used models to in rease their knowledge about ight ontrol, and have been invloved in robustness investigations in some form, sin e the very beginning of ight. At various stages of the ight ontrol laws design pro ess, model un ertainty

an be introdu ed, for example, when linearised versions of omplex models are derived. In this ase, the term un ertainty is a tualy a misnomer, be ause the deviation between linear and nonlinear behaviour an be quantied. The same is true if known variations in, for example, the position of the entre of gravity or a time-delay in the system, are negle ted. Depending on the design te hnique used, it may be ne essary to make su h modelling assumptions (temporarily) in order to obtain a model whi h is suitable for ontroller design. Model un ertainty an also be introdu ed unintentionally due to modelling errors, unknown hara teristi s of the air raft in relation to the environment, or ina

urate information about the signals owing through the system. For example, the pre ise value of aerodynami stability derivatives and air data may not exist. A feature of several ontrol design te hniques des ribed in this book is that they deal systemati ally and sometimes expli itly with robustness. Introdu ing these systemati s into the design y le may enhan e FCS design in terms of the ee ien y of the pro ess and the performan e of the resulting ontrol systems.

Potential ontribution of modern design te hniques It seems that the European aeronauti al industry is not in the rst pla e interested in modern te hniques purely to a hieve better air raft performan e. In fa t almost any te hnique, modern or lassi al, when used to solve realisti problems with enough knowledge of the method itself, with the ne essary tools available, and based on a thorough knowledge of ight me hani s, will eventually lead to the desired results. The real interest is in the systemati approa hes behind new methods, be ause this an simplify the design y le and make it more transparent. Global ompetition for es air raft manufa turers to

ontinuously improve the e ien y of their engineering a tivities. If it an be

4

demonstrated that advan ed design te hniques lead to a design y le with better tra ability of design de isions and simpliation of the overall pro ess, the

han e that modern ontrol te hniques will be used by industry will in rease. The omplexity of the design task and the related investment made in the past in human and non-human apital, explain the areful attitude from some air raft manufa turers to repla e their well-established lassi al te hniques. Moreover, lassi al te hniques have desirable features, for example the visibility of the resulting ontroller.

At the lowest level of detail of the ontrol

law, the fun tion of every gain and dynami element an be easily understood, whi h makes designs easy to modify and a

ept. On the other hand, the visibility after integration of subsystems is partly lost at a higher level. Another advantage is that gain and phase margins are open-loop measures with a lear link to robustness. This makes them very useful for synthesis. Even though it is true that superiority is often related to simpli ity and transparan y, whi h are typi al features of lassi al ontrol te hniques, the aeronauti al industry a knowledges some disadvantages as well.

Due to his-

tori reasons, the lassi al approa h in whi h ea h mode and ight ondition is treated as a separate problem has led to mode proliferation and the need for omplex algorithms. To avoid fun tional integration at the end of the FCS design, whi h is too late, an all en ompassing and onsistent design strategy is ne essary. Throughout the design pro ess a systems approa h strategy should be applied, supported by good requirements, design tools and design models. Appli ation of advan ed te hniques promises a signi ant redu tion of design time be ause it would remove the time- onsuming lassi al one-loop-at-a-time approa h and redu e the number of design points for whi h a ontroller has to be designed.

1.2

GARTEUR A tion Group on Robust Flight Control 3

In O tober 1994, GARTEUR

A tion Group FM(AG08) was established. For

the twenty-three member organisations of this group from seven European

ountries, GARTEUR proved to be an organisation oering ex ellent onditions and support for arrying out basi , pre ompetitive resear h. GARTEUR unites resear h establishments, the aeronauti al industry and universities in A tion Groups. In FM(AG08) the following organisations parti ipated:



Resear h Establishments

 3

Centro Italiano Ri er he Aerospaziali (CIRA, Italy, Capua)

The Group for Aeronauti al Resear h and Te hnology in EuRope (GARTEUR) was

formed in 1973 and has as member ountries: Fran e, Germany, The Netherlands, Spain, Sweden and the United Kingdom.

A

ording to its Memorandum of Understanding, the

mission of GARTEUR is to mobilize, for the mutual benet of the GARTEUR member

ountries, their s ienti and te hni al skills, human resour es and fa ilities in the eld of aeronauti al resear h and te hnology. More information about GARTEUR an be found in the GARTEUR Guide [4℄.

5



Deuts he Fors hungsanstalt für Luft- und Raumfahrt (DLR, Germany, Oberpfaenhofen)

  

Defen e Resear h Agen y (DRA, United Kingdom, Bedford) Instituto Na ional de Té ni a Aeroespa ial (INTA, Spain, Madrid) Laboratoire d'Automatique et d'Analyse des Systèmes (LAAS, Fran e, Toulouse)



National Aerospa e Laboratory (NLR, The Netherlands, Amsterdam)





O e National d'Etudes et de Re her hes Aérospatiales

 

CERT-ONERA, Fran e, Toulouse ONERA-Salon, Fran e, Salon de Proven e

Industry

   

Alenia Aeronauti a (ALN, Italy, Turin) Avro International Aerospa e (AVRO, United Kingdom, Woodford) British Aerospa e, Dynami s (BAe-D, United Kingdom, Filton) British Aerospa e, Military Air raft (BAe-MA, United Kingdom, Warton)

   



Cambridge Control (CCL, United Kingdom, Cambridge) Daimler Benz Aerospa e Airbus (DASA, Germany, Hamburg) Fokker Air raft Company (FAC, The Netherlands, Amsterdam) Saab Military Air raft (SMA, Sweden, Linköping)

Universities

       

Craneld University (CUN, United Kingdom, Craneld) Delft University of Te hnology (DUT, The Netherlands, Delft) Linköping University (LiTH, Sweden, Linköping) Loughborough University (LUT, United Kingdom, Loughborough) University of Cambridge (UCAM, United Kingdom, Cambridge) University of Lei ester (ULES, United Kingdom, Lei ester) Universitá di Napoli "Fediri o II" (UNAP, Italy, Naples) Universidad Na ional de Edu a ión a Distan ia (UNED, Spain, Madrid)

The A tion Group was haired by NLR (Terlouw); CERT-ONERA (Magni) provided the vi e- hairman.

In total eight meetings were held in Amster-

dam, Madrid, Cambridge, Capua, Toulouse, Linköping, Oberpfaenhofen and (again) Amsterdam, whi h gave an extra ultural dimension to the proje t. In view of the longer term obje tive of ontributing to e ien y improvement of the ight ontrol laws design pro ess, it was de ided to follow three main streams.

6

Design Challenge The rst stream was the Design Challenge des ribed in this book. Before the start of the A tion Group it was on eived that a thorough demonstration of modern design te hniques, applied to genuine ight ontrol problems, was required in order to get the desired feedba k from industry. The aim was to present the state-of-the-art with respe t to modern (robust) ontrol in su h a way that industry ould relate to it. At the same time it was the intention to

larify what is needed for a design method to be a

epted by an industrial design o e. To a hieve this, people from industry were asked to give inputs for two ben hmark problems, whi h were subsequently developed by people from the resear h establishments and universities. The rst one, the RCAM (Resear h Civil Air raft Model) problem [145℄, is based on the automati landing of a large, modern argo air raft. The se ond, the HIRM (High In iden e Resear h Model) problem [177℄, onsiders the ontrol of a military air raft a

ross a wide design envelope. Both ben hmarks are based on six degrees of freedom mathemati al air raft dynami s models, dened in Matlab/Simulink [121, 240℄. They in lude aerodynami , engine, atmosphere and gravity models.

In addition, a tuator and

sensor hara teristi s are taken into a

ount, together with models for wind, atmospheri turbulen e and windshear. An extensive set of design requirements is given, whi h an be tested with software for frequen y and time domain evaluations.

A standard nomen lature [237℄ and a standard report lay-out were

dened at an early stage, to avoid unne essary problems later on. In order to make the ben hmarks more realisti , parameter variations (time-delay, mass and entre of gravity variations for RCAM; variations in aerodynmi derivatives and measurement errors for HIRM) were dened. hardware implementation issues are onsidered.

Furthermore, some

This puts the ben hmarks

into the ategory of robust ight ontrol problems. At the start of the proje t it was de ided to limit the s ope of the demonstration of the te hniques to design and omputer simulations. Validation of the most promising ontrol laws and design te hniques might possibly be performed in a follow-on proje t, in whi h the use of a ight simulator and a ying testbed is re ommended. The Design Challenge was not aimed at giving the answer to the question whi h method is best?, but rather to show, step by step, how modern ontrol

an be applied. The design teams were asked to highlight four main points: 1. The eort to learn, to implement and to apply the method. 2. The appli ability of the design method to ight ontrol laws design. 3. The omplexity of the resulting ontroller, its implementation and erti ation issues. 4. The robustness and performan e of the designed ontroller. A large group of ontrol engineers primarily from the European aeronauti al

7

industry has evaluated the proposed designs. This book is a summary of the results of the Design Challenge.

Computer-Aided Control System Design The se ond stream of a tivities addressed the development of a framework for

omputer-aided ontrol system design.

Several industrial members of GAR-

TEUR A tion Group FM(AG08) indi ated a need for omputer support of the design pro ess and data. A prototype was developed by NLR, based on the

ommer ial software produ ts Matlab/Simulink, SiFrame and Informix. The prototype oers fa ilities for design pro ess denition and exe ution, in luding tool integration and a entral data repository. Most important is the automati

onsisten y ontrol of all (versions of ) design information in the framework. The  lassi al design pro ess of Craneld University, des ribed in Chapter 15, is implemented in the prototype, whi h was evaluated by several FM(AG08) organisations. The results of this eort are des ribed seperately in [224℄.

Robust Flight Control Tutorial and Literature Survey Database The third stream aimed at making available a literature overview of ontrol te hniques related to ight ontrol appli ations and at providing a tutorial do ument on advan ed ontrol te hniques. CIRA has established a Robust Flight Control Literature Survey Data Base, whi h an be a

essed via the Internet [206℄. From it, referen es and summaries of arti les on robust ight ontrol an be obtained. The aim of the database is to assist designers in lassifying their ontrol problems a

ording to similar problems already treated in the literature. As su h, it an help the designer to identify the most onvenient te hnique to be used. INTA has written a do ument [53℄ in whi h tutorials of all design te hniques that are des ribed in this book and several others are presented in detail.

1.3 Outline of the book The editors would like to point out that this book is the result of a group a tivity. With respe t to the ontents, it was onsidered to be important that as many FM(AG08) organisations as possible would get the opportunity to present their results, in order to over a wide variaty of design approa hes. The ontributions were not sele ted by the editors. The book onsists of four parts.

Part I ontains tutorials of all methods

that have been applied on either RCAM or HIRM or on both problems. Parts II and III over the RCAM and HIRM ben hmark denions and the proposed designs.

With a few ex eptions, ea h design hapter has basi ally

the same stru ture.

The designs are based on the twelve methods given in

se tion 1.2. Furthermore, one RCAM design is entirely based on lassi al te hniques.

8

In Part IV, three dierent views on the Design Challenge are given. Chapter 34 presents a view from industry. A questionnaire was designed by British Aerospa e and DASA to aid evaluators in their assessment of the Design Challenge entries. Chapter 35 dis usses the Design Challenge results from the s ienti resear her's point of view. An obje tive measure of stability robustness, namely the stru tured singular value, is given for ten RCAM designs. Finally, Chapter 36 ontains some on luding remarks of the editors. One of the onditions whi h made the Design Challenge possible was the fa t that all teams have used the same nomen lature, whi h is given in Appendix A.

A knowledgements Most of the work needed for writing this book was funded by the parti ipating organisations of GARTEUR A tion Group FM(AG08).

These organisations,

whi h are listed in se tion 1.2, are given thanks for their onden e in the group and their full support until the end of the proje t. In some ases national agen ies and other resear h funding bodies have given additional nan ial help, notably the Netherlands Agen y for Aerospa e Programs (NIVR). Without their support the Design Challenge would not have been possible. FM(AG08) also wishes to express its gratitude to Aérospatiale and DRA for making available the models on whi h the RCAM and HIRM ben hmark denitions are based. Another word of thanks is to the GARTEUR organisation, in parti ular the Flight Me hani s Group of Responsables and the Exe utive Committee, for making the publi ation of this book possible. The head of the NLR Flight Division, Jan van Doorn, who has a ted as the GARTEUR Monitoring Responsable of FM(AG08), has given essential ontributions behind the s enes. He was an indispensible link between the A tion Group and the GARTEUR organisation. The editors are grateful to Chris Fielding, Derek Laidlaw, Jim Gautrey, Lester Faleiro, Daniël Walker and Jonathan Irving for he king most hapters on the use of English and proposing many suggestions for improvements. Not all results of GARTEUR A tion Group FM(AG08) related to the Design Challenge ould be presented in this book.

Several design teams joined in

at a later stage or there were other reasons why their designs ould not be in luded. In this respe t Alex Smerlas (Univ. of Lei ester) [217℄, Aldo Tonon (ALN), Jürgen A kermann (DLR), Angel Perez de Madrid (UNED) and their

olleagues are a knowledged for their valuable ontributions. This book will be presented at a GARTEUR Spe ialists' Workshop on Robust Flight Control (CERT-ONERA, Toulouse, Fran e, April 14-15, 1997). Spe ial thanks is given to CERT-ONERA for organising and hosting this workshop.

9

10

Part I

Tutorial part

11

2.

Multi-Obje tive Parameter Synthesis

(MOPS)

Georg Grübel1 and Hans-Dieter Joos

1

2.1 Theoreti al Aspe ts 2.1.1 Global Goals Any ontrol law is parametrized in some way. For example, in a P-I-D ontrol stru ture with additional shaping lters there are the P-I-D gains and the lter parameters whi h are to be tuned for losed-loop performan e. Similarly, an LQR state- or output feedba k ontrol law is parametrized by the state- and

ontrol eort weights Q and R, an eigenstru ture state feedba k ontrol law is parametrized by the eigenvalues and some eigenstru ture parameters and an

H1 ontrol law is parametrized by its input/output weighting lter parameters. Control parameter tuning for a best possible robust performan e is a time onsuming task if performed manually. This is due to the multitude of different (nonlinear) design spe i ations whi h have to be dealt with.

This is

also true if one uses advan ed analyti al ontrol synthesis te hniques su h as



synthesis. Multi-obje tive parameter synthesis (MOPS) is a general te hnique whi h

omplements a hosen ontrol law synthesis te hnique.

Having hosen an

appli ation-spe i ontrol law stru ture with parametrization, or having hosen a general ontrol synthesis te hnique with its analyti ally given parameterization, the free design parameters (e.g.

the LQR-weights) are omputed

by a min-max parameter optimization set up.

The designer formulates this

set up by spe ifying the design goals as a set of well dened omputational

riteria, whi h an be a fun tion of stability parameters (e.g.

eigenvalues),

and time- and frequen y response hara teristi s (e.g. step-response overshoot and settling time, ontrol rates, bandwidth, stability margins et ).

By this

multi- riteria formulation all the various oni ting design goals are taken are of individually, but are ompromised on urrently by a weighted min-max parameter optimization. In parti ular, robust- ontrol requirements with respe t to variations in stru tured parameter sets and operating onditions an be taken are of by a multi-model formulation whi h en ompasses the worst- ase design onditions. 1

DLR German Aerospa e Resear h Establishment, Institute for Roboti s and System

Dynami s, Control Design Engineering Group (Prof. G. Grübel), D-82234 Wessling E-mail: dieter.joosdlr.de

13

For oni ting design riteria the te hnique provides a best-possible paretooptimal ontrol tuning. Sin e the multi- riteria in ludes performan e measures whi h are dire tly used as design drivers, they provide dire t quantitative information about the design oni ts and performan e onstraints. This yields all the ne essary information on how to improve the balan e of a design tradeo within a given ontroller stru ture or a hosen ontrol synthesis method. The method allows simple (linear) ontrollers to be optimized based on omplex (nonlinear) system evaluation models, thereby linking together the steps of ontrol design and of (nonlinear) design assessment. Our experien e shows that with the same engineering eort, a dedi ated ontrol performan e improvement of about 30% an be a hieved by numeri al multi-obje tive optimization as ompared to manual design parameter tuning in a sequential manner.

2.1.2 System Model Des ription Both linear and nonlinear design models an be taken into a

ount. In a multimodel approa h linear models together with nonlinear models an be used simultaneously. There is no restri tion on the representation of system disturban es. Robustness against stru tured parameter deviations or, for example, sensor failures is a hieved by applying a ommon ontroller to a set of xed worst- ase parameter models. This model set hara terizes the worst dynami s deviations within the range of operation, e.g. the ight envelope, or a part of it. For ea h su h model the appropriate set of riteria has to be spe ied. Hen e, the multimodel problem is transformed into a multi- riteria optimization problem. In general, there exists no theory that guarantees stability or performan e robustness a ross the range of operation, if only a nite number of operating points is onsidered simultaneously. It depends on the physi al properties of the system to be designed, whether runaways an exist. If they exist, they have to be added to the set of worst- ase operating points and treated simultaneously by the multi-model approa h. Worst- ase operating points an be omputed by a dual appli ation of the parameter optimization approa h:

Find those

parameter ombinations within a given un ertainty domain whi h yield the worst performan e for the hosen multi- riteria set up [20℄. Of ourse, robustness of the ontroller about an operating point an be enfor ed in the multi-obje tive approa h by adding suitable robustness riteria (e.g. gain/phase margins) to the set of otherwise spe ied performan e riteria.

2.1.3 Controller Stru ture Both linear ontrollers and nonlinear ontrollers (e.g. fuzzy ontrollers or adaptive ontrollers) an be used. If a spe i analyti al synthesis te hnique is applied within this framework, the ontroller stru ture is bound to this synthesis te hnique.

14

2.1.4 Design Spe i ations Ea h design obje tive may be mathemati ally des ribed by a well-dened riterion

i

whose value is the smaller, the better the obje tive is a

omplished.

Examples taken from the RCAM design hallenge spe i ations are: - Steady state error, settling time and rise time for demanded steady state value

ys :

=

Z tend

t1

(y(t) ys )2 dt

- Overshoot over demanded steady state value

ys :

= max (y(t)=ys ) t - Rise time dened as the time the unit step response

y(t1 ) = 0:10 to y(t2 ) = 0:90:

= t2 - Minimal damping of the eigenvalues

y(t)

takes from

t1 evi :

= 1 min ( Re(evi )=jevi j) : i In order to get smooth riteria as a fun tion of the tuning parameters, the min- or max-fun tions involved are smoothed by an exponential approximation; see also se tion 2.5. The above eigenvalue riterion minimal damping is reformulated in su h a manner that better damping results in a smaller riterion value.

2.1.5 Analysis Information To evaluate the hosen design riteria have to be performed.

i , the respe tive analysis omputations

This usually requires eigenvalue omputations, time

response simulations and frequen y response omputations. This analysis information is used to judge the quality of a design in addition to the riteria values whi h quantify the balan e of the a hieved optimum, and whi h provide further ontroller synthesis information (see se tion 2.1.6).

2.1.6 Controller Synthesis Information To ea h riterion,

i

di is T = [Tk ℄ are omputed

an upper-bound demand value or driver value

dened by the designer. Then the tuning parameters

by solving the min-max parameter optimization problem

min max f i =di g T

i

15

subje t to performan e and tuning onstraints:

gj (T )  0; Tkmin  Tk  Tkmax: This is the MOPS synthesis formula. By iterating the demand values as a fun tion of the a hieved riteria values

i

1,

di

the resulting ompromise

trade-o solution an be driven in a desired dire tion.

2.1.7 Pra ti al Implementation Aspe ts The overall omputing time for the synthesis mainly depends on the time for

riteria evaluations. Hen e fast algorithms and software implementations [101℄ are required for the orresponding analysis omputations. It is good pra ti e to use  heap riteria where possible. The number of riteria evaluations also depends on the number of models used in a multi-model set-up. Therefore it is also bene ial to minimize the number of models by a areful sele tion of worst- ase operating points or parameter deviations.

2.1.8 Relation with other Methods Multi-obje tive parameter synthesis loses the parametri design loop with modelling-, synthesis- and analysis methods a

ording to Figure 2.1.

synthesis model

D

T

C

synthesis

plant model

P

performance/cost criteria

controller model

closed-loop model

M

simulation/ analysis

I

Figure 2.1: Design loop losed by multi-obje tive parameter synthesis. It serves to automate ontrol tuning to given performan e spe i ations. It is neutral as far as the design steps modelling, synthesis and analysis are

on erned.

2.2 Example of Appli ation The approa h has been applied for robust ight ontrol [138, 102, 100℄, a tive antenna-beam ontrol [19℄, PWM-satellite attitude ontrol [98℄, maglev vehi le

16

ontrol [190℄, (semi-)a tive ar suspension and air raft landing gear ontrol [81, 209, 254℄, robot ontrol [153℄, and others. An example appli ation is the MOPSsolution for the RCAM design hallenge [130℄.

There, for the longitudinal

ontrol, LQR PI-output feedba k is used, whereas for lateral ontrol a lassi al

ontrol stru tue [35℄ is used, thereby demonstrating the appli ation for two dierent ontroller stru tures. A nonlinear worst- ase plant analysis, also using MOPS, was performed to he k robustness within the multi-model set-up.

2.3 Computational Aspe ts The method requires the set-up of a omputation loop a

ording to Fig. 2.1 and the availability of a suitable min-max parameter optimization software. For an engineering-e ient appli ation of this te hnique it is very bene ial to have a software framework whi h supports intera tive modular problem setup and demand spe i ation as well as automated performan e evaluation (su h as ANDECS_MOPS [99℄). Multi-model/multi-obje tive performan e evaluation an be fun tionally parallelized, e.g.

by using the PVM (Parallel Virtual Ma hine) lient-server

network on ept. Thereby the omputation time an be redu ed. The multi-obje tive optimization problem an be solved by any nonlinear programming tool, sin e minimizing a set of riteria an be transformed into a

onventional s alar nonlinear programming problem; see Ÿ2.5. Using appli ation-spe i engineering riteria in pra ti e, typi ally leads to non onvex optimization problems. Thus lo al minima may exist. However, a lo al minimum solution is also a lo al best-possible pareto-optimal solution. If su h a solution is not satisfa tory, other solutions an be found by hanging the demand values, or by hanging the starting values for the tuning parameters. To avoid lo al minima, a global optimizer has to be used whi h may have the disadvantage of rather long omputing times.

2.4 Comparative Study Multi-obje tive parameter synthesis allows full exploitation of a given ontroller stru ture, as a fun tion of the ontroller parametrization.

In parti -

ular, it allows the exploitation of the a hievable trade-os between ontrol performan e and required ontrol eort. This is possible in a most detailed, appli ation-spe i way and hen e, no matter what ontroller stru ture or ontroller synthesis method is used, this te hnique, in prin iple, always yields the best possible performan e in the hosen ontext. A potential benet of multi-obje tive tuning the design parameters of an analyti synthesis method (e.g. LQR, eigenstru ture synthesis, et .) instead of dire tly tuning the parameters of a given ontroller stru ture (i.e. state- or dynami output feedba k), is primarily that usually a smaller number of parameters is to be tuned. This parti ularly holds for multi-input/multi-output

17

systems. Also, built-in performan e and robustness features of the hosen synthesis method are automati ally guaranteed. On the other hand, dire tly tuning the parameters of a spe ied ontroller stru ture allows the designer to use appli ation-proven ontroller stru tures for whi h no analyti synthesis te hniques exist, and it allows him to extend and to adapt su h stru tures during the design pro ess. If an appropriate software framework is available whi h provides a predened omputation loop and a set of standard riteria to hoose from (e.g. ANDECS), the level of required training is moderate. In this ase, no spe i mathemati al theory is required. Design spe i ations are expli itely formulated in their most natural mathemati al form and a trans ription of design spe i ations into a synthesis-spe i weighting form is not required. In omplex design-de ision problems with, say, more than 5 riteria to be handled simultaneously, an integrated data system is mandatory, to keep tra k of the de ision iterations made during the design pro ess (this holds for any design-iteration logi ). The ANDECS software provides su h an integrated data system, whi h is spe i ally-designed for multi-obje tive/multi-model design iterations.

2.5 Mathemati al Appendix 2.5.1 Preferen e order, goal oni ts and satisfa tory ompromise sets for riteria ve tors The main advantage of a multi-obje tive design is the possibility to formulate an individual riterion for ea h spe ied demand, while treating all riteria during optimization simultaneously. Here, some terms are laried by introdu ing the related on epts [128℄: - better solution in the ontext of a preferen e order for ve tor-valued

riteria, - goal oni t and  ompromise in the ontext of pareto-optimality and - satisfa tory ompromise in the ontext of demand level. (i) The individual riteria

i (T ) are ombined to give a riteria ve tor (T ).

The following preferen e order allows one to partially ompare su h ve tors:

A set of tuning parameters

T1

is said to be better than

the orresponding riteria ve tor

(T 1)

is smaller than

where smaller means

(T 1) < (T 2 ) , i (T 1) < i (T 2 ) for all i 18

T 2 , if

(T 2),

Smaller equal is dened as

(T 1)  (T 2 ) , i (T 1)  i (T 2 ) for all i 1 2 and i (T ) < i (T ) for at least one i. (ii) Trying to improve several riteria simultaneously normally leads to a goal

oni t in the sense that no riterion an be improved further without worsening another one. More pre isely:

A set of tuning parameters

T  is alled a ompromise solution,

or pareto-optimal solution, if there is no T with

(T ) < (T  ).

(iii) Usually, ompromise solutions are not unique. There exists a whole set of pareto-optimal solutions and it is up to the design engineer to de ide what trade-o is a best satisfa tory ompromise in his design ontext. The term satisfa tory an be made more pre ise by introdu ing the demand level

d referring to a riteria ve tor :

A set of tuning parameters

if T 

T  forms a satisfa tory ompromise,

belongs to the set of ompromise solutions and if

(T )  d ; where in

ve tor d

d

the demands of the designer are quantied. The

is alled the demand level.

Fig. 2.2 illustrates the above denitions for the ase of 2 riteria

2 .

Assume that that

(fT g)

1

and

fT g denotes the set of all feasible tuning parameters T and

(fT g)

is the orresponding value set. The thi k border part of

in Fig. 2.2 is the set of ompromise solutions and

Cs

marks the subset of a

satisfa tory ompromise. Note, that all solutions with riteria values smaller than the demand level

d

are satisfa tory solutions.

2.5.2 Finding a satisfa tory ompromise set by means of min-max optimization A parti ular, satisfa tory ompromise an be found by means of parameter optimization. From the riteria ve tor

(T ) and the demand level d one an form

a s alar fun tion

 = max f i (T )=di g : i

19

c2

c({T}) d* Cs c1 Figure 2.2: Demand level and satisfa tory ompromise set in two-dimensional

riteria spa e

Of ourse, we have solution if solution of

  1. 

(T )  d and therefore we have a satisfa tory

Moreover, it an be shown [205℄ that a minimum

 = min max f i (T )=di g i T

is a ompromise solution. Again, if



is less than or equal 1, the

ompromise solution is satisfa tory. Hen e the problem of nding a satisfa tory ompromise solution is redu ed to a s alar min-max optimization problem.

This is also known as goal at-

tainment with a zero ideal point [90℄. Fig. 2.3 illustrates what best possible solution is a hieved by min-max optimization in a two dimensional riteria spa e. The min-max optimization problem an be solved using standard nonlinear

 as obje tive fun tion. However, the non due to the maximum fun tion may ause problems if gradient

programming methods applied to smoothness of

based solvers are applied.

In this ase, it is preferable to reformulate the

optimization problem in one of two ways: 1. The un onstrained min-max optimization problem with smooth riteria

i (T ) is equivalent to the onstrained problem [181℄ min max f i (T )=dg , minftg ; s:t: i (T )  t : T

i

i

T;t

Solving the min-max problem in this way yields exa t solutions.

20

c2

c({T}) d* c* c1 Figure 2.3: Satisfa tory ompromise found by min-max optimization

2. Approximate solutions are found if the fun tion

 is approximated by a

smooth fun tion, as proposed in [138℄:

X  max f i (T )=di g = lim !1 1= ln ( exp( i (T )=di )) i

i

=  + lim !1 1= ln (

X

i

(( i (T )=di

exp

))) :

This approximation formulation is well suited for numeri al omputation, sin e the argument of the exponential is always less than or equal to zero. The approximated min-max problem an be solved as an un onstrained parameter optimization problem. Of ourse, the fun tion

 an be minimized dire tly if optimization methods

su h as dire t sear h methods are used, whi h do not require smooth obje tive fun tions.

21

3.

Eigenstru ture Assignment

Lester Faleiro , Jean-François Magni , Jesús M. de la Cruz and Stefano S ala 1

2

3

4

3.1 Introdu tion The theory presented here on erns the design hapters 17, 18, 19 and some aspe ts of hapter 28.

The main on epts of eigenstru ture assignment as a

design te hnique will be explored, in orporating a short explanation of how to

hoose a desired eigenstru ture based on design spe i ations. The mathemati al methods used will also be summarised, and some omments given on the use of eigenstru ture assignment.

3.2 Eigenstru ture Analysis The equations that des ribe an air raft and their relation to the time response of that air raft an be grouped together in matrix form:

x_ = Ax + Bu (3.1) y = Cx + Du where the most important of these matri es, A, des ribes the internal dynami s of the air raft. The B matrix des ribes the distribution of the a tuator inputs to the states of the air raft, and the C matrix denes how the states an be observed as outputs of the system. D is usually zero for an air raft, though non-zero matri es o

ur when air raft a

elerations are in luded in the outputs.

x

is the state ve tor,

u

y is the output measurement n states, m inputs and p outputs.

is the input ve tor and

ve tor. It will be assumed that the system has

A an be further de omposed into its onstituent eigenvalues and eigenve -

tors. The derivation of these an be found in any standard text on linear matrix algebra. Let the

n eigenvalues and eigenve tors of the system be dened by:

 = [1 : : : i : : : n ℄ and V = [v1 : : : vi : : : vn ℄ 1

(3.2)

Department of Aeronauti al and Automotive Engineering and Transport Studies, Lough-

borough University, Loughborough, Lei estershire LE11 3TU, United Kingdom. 2

CERT ONERA, Département d'études et Re her hes en Automatique, BP 4025, F31055

Toulouse Cedex, Fran e. 3

Dep. Informáti a y Atomati á. Fa ultad de Cien ias Físi as. Universidad Computense.

28040 Madrid, Spain. 4

Flight Control and Me hani s department, Centro Italiano Ri er he Aerospaziali, 81043

Capua, Italy.

22

where

AV = V 

The eigenve tor set

V

(3.3)

is a basis set for the state spa e

x; thus any ve tor in

the state spa e an be expressed as a linear ombination of the eigenve tors of the air raft system. These eigenve tors are also alled the right eigenve tors of the system. The left, or dual basis eigenve tors of the same system are given by

W , where

W T = [w 1 : : : w i : : : w n ℄ ; W A = W

(3.4)

Solving the state-spa e equations given in (3.1) yields an expression for the time response that an be found in most standard ontrol texts:

y(t) =

n X i=1

Cvi wTi ei t x0 +

n X i=1

Cv i wTi

Z t

0

ei (t  )Bu( )d

(3.5)

It is lear from this equation that there are two omponents to the time response. The rst is dependent on the initial onditions of the system, and is alled the homogeneous omponent; the se ond is dependent on an input to the system, and is alled the for ed omponent. The entire time response of a linear system thus depends on four variables: The eigenvalues of the system The eigenve tors of the system The initial onditions of the system The inputs to the system Ea h of these plays a part in the determination of the time response, and di tates the overall ee t that modes and inputs play in the output response of the system. The homogeneous omponent of equation (3.5) an be written as

y(t) = where

n X i=1

Ci ei t vi

i are the s alars wTi x0 , i = 1 : : : n.

(3.6)

This shows that the output response

is omposed of a linear ombination of eigenvalue-eigenve tor sets of the matrix

A.

Ea h of these sets is alled a mode. In every mode the eigenvalue deter-

mines the de ay/growth rate of the response and the eigenve tor determines the strength of the oupling of this mode with the outputs.

ith mode with the j th output is given by C j v i , where C j is the row of C . If C j v i = 0, then equation th mode does not ontribute to the j th output; they have (3.6) shows that the i From (3.6) we an see that the oupling of the

j th

been de oupled. As an example of how the information about the nature of eigenstru ture

an be used, let us examine a simple linear representation of the longitudinal

23

dynami s of the RCAM model, in terms of four varying states of the system.

Mathemati ally, we an determine the time response of the system to

an arbitrary initial ondition, but this does not ne essarily give us a omplete understanding of the system dynami s. Mode 1 2

Eigenvalue

0:830  1:107i 0:011  0:126i

Damping ratio

Frequen y (rad/s)

0.6

1.38

0.09

0.13

Table 3.1: Modes of the open-loop system The eigenvalues of this nominal system are shown in Table 3.1.

It an

be seen that although there are four states in the system, there are only two modes in its dynami behaviour. It is known that the Phugoid and the SPPO (Short Period Pi hing Os illation) are the two os illatory modes that hara terise air raft longitudinal motion, and that the Phugoid usually has a mu h lower frequen y and damping than the SPPO. However, if these modes were in any way un onventional, a knowledge of the eigenve tors alone would not be su ient to understand the air raft. States

q  u w

Mode 1

0:014 0:010 0:015 1

6 61:5 6 8:3 6 14:5 6 20:7

Mode 2

0:002 0:0132 0:99 0:142

6 50:2 6 34:7 6 39:5 6 41:8

Table 3.2: Eigenve tors of the open-loop system (magnitude and argument)

The only way to ensure that ea h of the modes an be attributed to parti ular air raft hara teristi s is by a subsequent examination of the right eigenve tors of the system. For this ase, these are shown in Table 3.2. The eigenve tors for a mode are read verti ally down the table. It an be seen that Mode 1 is

hara terised by a large intera tion with

w, the standard hara teristi of the u, and omparatively

SPPO. Mode 2 is hara terised by a large intera tion with little with

w,

typi al of the Phugoid. The two modes an thus be designated

as 1. SPPO and 2. Phugoid. In the time domain, the peak for ea h of these states will dier a

ording to the phase angles (arguments) of the elements of the eigenve tor, given in degrees in Table 3.2. Note that usually the magnitude, rather than the phase, in eigenstru ture assignment an be more easily visualised for the purposes of design and analysis, so only the magnitudes will be used in eigenve tor des ription from now on. Additional information about the system an be obtained by using the left eigenve tors to determine the ee t that ea h input has on ea h mode of the system.

These input oupling ve tors are given by the produ t of the left

eigenve tors and the input distribution matrix,

24

W B.

For the above example,

the input oupling is given below: Mode

Æt

Æth

SPPO

85.4

19.3

Phugoid

31.5

13.7

This shows that the SPPO will be ex ited by a taileron input to a mu h larger extent than a throttle input, and the Phugoid is the same. This qualitative eigenstru ture analysis is a tool that an be used to examine the nature of the modes of a system qui kly. Classi al te hniques usually assume that a knowledge of the system dynami s is readily available with the model.

This

is a fair assumption, but may be ome redundant if more omplex modes are involved in the open-loop system. Additionally, this te hnique of analysis is invaluable during the eigenstru ture assignment pro ess in examining the sour e of design problems.

3.3 Eigenstru ture Assignment It was shown in equation (3.6) that the output response of a air raft an be des ribed by a representation involving its eigenvalues and eigenve tors. Thus, if the eigenstru ture of the air raft an be manipulated somehow, we have a means of altering its time response. Various forms of dire t eigenstru ture assignment methodology exist, from the rst tentative steps in output feedba k by Kimura [135℄ to their further development by Andry, Shapiro and Chung [211℄ to urrent work su h as that done by Sobel, Lallman and Shapiro [219℄, [221℄ and [220℄. In essen e, all these methods are similar, and fun tion in mu h the same way. They all require the designer to spe ify a set of eigenvalues and eigenve tors for the design, and they all produ e a proportional gain matrix ontroller.

3.3.1 Determination of the desired eigenstru ture The philosophy behind dire t eigenstru ture assignment is that whilst the designer is able to spe ify a set of desired losed-loop eigenvalues

d ,

she/he

is also able to spe ify exa tly whi h elements of the desired eigenve tors

Vd

she/he would like to set to zero, where

d = [d1 : : : di : : : dp ℄ ; V d = [vd1 : : : vdi : : : vdp ℄

(3.7)

This an be illustrated by the set of eigenve tors shown in Table 3.3. We would perhaps like the SPPO mode of response to be unae ted by forward velo ity and pit h angle, and vi e-versa. We therefore spe ify these elements in the desired losed-loop eigenstru ture to be zero. We are un on erned with the values of the remaining elements, designated by an 'x'.

A similar situa-

tion o

urs with the Phugoid eigenve tor. This pro ess is ee tively assigning elements of

vi

in (3.6) to zero.

25

States

q  u w

SPPO

Phugoid

x

0

0

x

0

x

x

0

Table 3.3: Example of desired losed-loop eigenve tors

The ontrol design problem an thus be stated as follows: Given a set of

d and a orresponding set of desired eigenve tors V d , nd m  p matrix K su h that the eigenvalues of the losed-loop system matrix (A + BKC ), obtained when using the output feedba k ontrol equation desired eigenvalues

an

u = Ky; in lude

d

(3.8)

(A + BKC ) are

as a subset, and the orresponding eigenve tors of

as lose as possible to the respe tive members of the set

V d.

3.3.2 The a hievable ve tor spa e Now, from the eigenve tor equation of the losed-loop system:

(A + BKC )vi = i vi ; i = 1 : : : p

Avi 

(3.9)

i vi + BKCvi = 0

A i I B

where





(3.10)



vi = 0 zi

(3.11)

zi = KCvi

(3.12)

So, for a non-trivial solution,



vi zi





2 Ker A i I B 

n rows of the null spa e (Ker) of A i I B spa e, N i . A se ond method that an be used

and the rst able ve tor



(3.13)



form the a hiev-

to determine this

spa e an be derived from (3.10):

Dene

vi = (A i I ) 1 BKCvi

(3.14)

N i = (A i I ) 1 B

(3.15)

and now the losed-loop eigenve tors should omply with

26

vi = N i zi

(3.16)

in order to obtain the required eigenvalues. The a hievable eigenve tors lie in the subspa e spanned by the olumns of the matrix

vai must

N i . Expanding this N i is of dimension

example into more general terms, the subspa e des ribed by

m. Ni

On e the desired eigenvalues have been hosen, the range spa e of matri es

onstrains the sele tion of the losed-loop eigenve tors.

desired eigenve tors

v di

In general, the

will not reside in the a hievable eigenve tor spa e. In

order to have the resulting eigenve tor as lose as possible to the desired one, an optimum hoi e is made by proje ting the desired eigenve tor onto the a hievable spa e,

N i.

This is illustrated diagrammati ally in gure 3.1 for a simple three dimensional system. achievable vector v ia

desired vector v id

Dimension 3

This vector space, defined by the null vectors, describes the set of points over which the desired eigenvalues can be realised.

Null space vectors

Figure 3.1: Representation of de oupling in a 3-dimensional state spa e In this example, the desired ve tor an be hosen to de ouple a mode from a dimension.

As an example, say we want this mode to be de oupled from

Dimension 2. Thus, for this system, the only possible a hievable eigenve tor is given by the interse tion between the null spa e (whi h is the only pla e where the desired eigenvalue will be produ ed) and the Dimension 1/Dimension 3 plane (the lo us of points whi h does not ontain any omponent of Dimension 2).

Sin e the desired eigenve tor

vdi

ontains desired de oupling information

(i.e. a zero in the Dimension 2 row), it will lie on the Dimension 1/Dimension 3 plane. In real systems, this on ept an be expanded to de ouple modes from air raft outputs.

On e the desired eigenstru ture has been worked out, the

nal eigeve tors of the system an be produ ed.

3.3.3 Determination of the nal eigenve tors Ri an be dened su h that: A

ording to [12℄, a reordering operator f
g fvdi gRi



= dli i



and fN i 27

gRi



~i  N = D i

(3.17)

where

li

and

di

are the ve tors of spe ied and unspe ied omponents of

Ni

respe tively. The rows of the null spa e

vdi

have been reordered in the same

way. The nal eigenve tor is given by (see [12℄)

where

(
)y

y vi = N i N~i li

(3.18)

denotes the pseudo-inverse.

It is also possible to determine the nal eigenve tors without the use of proje tion. For ea h desired eigenve tor, the de oupled elements are integrated into a row ve tor

gi vi = 0,

gi

su h that if

vdi = [x x 0 x℄T , gi = [0 0 1 0℄T .

Thus,

sin e the nal eigenve tor should also have the relevant elements

de oupled. Thus, equation (3.11) an be rewritten as



A i I B gi 0





vi =0 zi

(3.19)

and for a non-trivial solution,



vi zi



2 Ker



A i I B gi 0

 (3.20)

This ve tor in the null spa e an now be suitable partitioned and its rst entries an be used to form

vai

n

3.3.4 Determination of the feedba k gain vi an now be grouped into the eigenve tor matrix V . The zi ( omputed together with vi using (3.20)) are grouped into the matrix

These eigenve tors ve tors

Z.

From (3.12) the feedba k gain satises

KCV = Z Usually, the number of olumns of

p, therefore

V

and

Z

(3.21) is equal to the number of outputs

K = Z (CV ) 1

If the number of olumns is larger than

p,

(3.22)

a dynami fedba k an be used as

detailed in Ÿ3.4.4. When

vi is omputed as in (3.18), the orresponding ve tors zi an be found

easily in order to solve (3.22). However the resulting stati feedba k gain matrix

an be determined dire tly by substituting rearranged to give:

V

into equation (3.9), whi h an be

K = B y (V  AV )(CV ) 1

(3.23)

Other ways of al ulating the gain matrix for numeri al e ien y and in the

ase of matrix non-invertibility have been des ribed in the literature ([12℄, [133℄), and an be used instead of equation (3.23) if desired.

28

3.4 Robustness to Parameter Variation Standard eigenstru ture assignment, as des ribed in previous se tions, takes performan e and de oupling into a

ount, but does not relate to any robustness requirements. Four dierent, and sometimes o-operative, ways of ta kling this problem have been pursued with the RCAM problem.

3.4.1 Open-loop ve tor proje tion It has been shown by Wilkinson in [256℄ and [160℄ that for a perturbation in the losed-loop matrix

(A + BKC ) given by
(A + BKC ), the orresponding

rst order perturbation in the relevant eigenvalue is given by:


i = wi
(A + BKC )vi where w i and v i are normalized su h that w i v i = 1.

(3.24) On the assumption that

the open-loop eigenvalues do not vary a lot with parameter variation, (3.24) shows that any variation an be related dire tly to the eigenve tors of the system. Thus, if the open-loop eigenve tors are used as the desired eigenve tors, eigenvalue sensitivity to perturbation should not be deteriorated by feedba k. This thesis is used in the RCAM design in hapter 19.

3.4.2 Iterative assignment Kautsy et al. [133℄ proposed using iterative eigenstru ture assignment to de rease the sensitivity of an eigenvalue in a state-feedba k ontrol system. An iteration is used in whi h the ve tor

vi

is repla ed by a new ve tor with maxi-

mum angle to the remainder of the urrent right eigenve tor spa e

i = 1; 2; : : : ; n in turn.

V i for ea h

The new ve tor is obtained, letting:

V i = [v1 : : : vi 1 vi+1 : : : vn ℄

(3.25)

wi (ith left eigenve tor) is orthogonal to V i , and the new v i is found by prowi (now ee tively the desired ve tor for the ith mode) into N i (whi h

je ting

ontains the a hievable right eigenve tor spa e):

vi =

N i N Ti wi k N Ti wi k2

(3.26)

thus giving a ve tor that is as orthogonal as possible to the urrent spa e whilst retaining the desired eigenvalues of the losed-loop system. This means that a perturbation in any of the elements of the remaining eigenve tors due to parameter variation will not ae t the urrent mode. The iteration is ontinued until the redu tion in the ondition number of the

V

matrix is less than some



toleran e. This is be ause the ondition number of the matrix ( ) is a measure of the overall sensitivity of the system. At the end of this iteration, a

V

matrix

for a minimum sensitivity solution remains. Ba k substitution of this matrix into equation (3.23) produ es a feedba k gain matrix.

29

Of ourse, altering the eigenve tors in this way does inevitable result in a loss of performan e.

The pro ess of de omposition and proje tion would

result in a loss of desired de oupling. However, using the null spa e des ribed in equation (3.20) an help to over ome this problem, as the null spa e itself

ontains the de oupling required. A further des ription of the use of this pro ess is given in [77℄.

3.4.3 Stability margin improvement A se ond riterion in use is a measure of loop robustness in terms of gain and phase margins. If the air raft is represented by

G(s), a variety of loop transfer

fun tions an be used to determine losed-loop system robustness. The singular values of the sensitivity fun tion plementary sensitivity fun tion fun tion

T = L(I + L) 1

S = (I + L) 1 ,

the om-

and the balan ed sensitivity

S + T , where L is the open loop gain matrix, an be used to measure

the stability margins for multiloop feedba k ontrol systems ([152℄, [50℄ and [28℄). The design pro edure in hapter 18 uses these measures, and the design in [77℄ uses similar ones. The fun tions

S

and

T

may be al ulated at the a tuator inputs or at the

sensor outputs. At the inputs,

L = KG and at the outputs, L = GK . The peak  S , T or S + T gives a robustness

value of the maximun singular value ( ) of

guarantee for all frequen ies. The formulae applied to omputing the stability margins using the sensitivity fun tion are the following:

a = 1=(S ) Gain Margin Phase Margin where gains

= [1=(1 + a); 1=(1 a)℄ = 2sin 1 (a=2)

(3.27)

a  1. The gains of the loops may thus be perturbed simultaneously by
satisfying 1=(1 + a) <
< 1=(1 a) without destabilising the losed

loop system. Similarly, the feedba k loops may be perturbed simultaneously

 satisfying j  j< 2sin 1 (a=2) without destabilising the losed loop system. The best possible gain and phase margins are obtained when  (S ) = 1, o in this ase the gain margin is [ 6 dB, +1 dB℄ and the phase margin is 60 . Similar margin equations an be devised for the T and S + T . These stability by phases

margins are known to be onservative, and a better approa h is obtained by repla ing the maximum singular value



with the stru tured singular value



[44℄, [28℄. The above des ription gives only a measure of robustness. In order to use this information in a design synthesis, an iterative loop whi h ontains the eigenstru ture assignment design pro ess, but updates the hoi e of eigenvalue and eigenve tor an be used. This pro ess produ es variable results, depending on the air raft and the initial design spe i ations, but has nonetheless been found to be useful. Previous examples of the use of these stability margins to improve robustness of air raft ontrol systems an be found in [178℄ and [76℄.

30

3.4.4 A multimodel approa h A fourth way of improving the robustness of an eigenstru ture assignment design is to use the multi-model approa h des ribed in [150℄.

The RCAM

design des ribed in Chapter 19 uses this method. It relies on produ ing a bank of linear air raft models at dierent operating points. These models are denoted

(Ai ; B i ; C i ) i = 1 : : : p.

Extra freedom to

improve robustness is introdu ed with the multi-model approa h. Instead of assigning all the available eigenstru ture to one linear model, a dierent model may be used for ea h assignment.

Thus, models with parti ularly sensitive

eigenvalues an be isolated, and the relevant eigenvalue-eigenve tor pair an be re-assigned to improve the robustness of a parti ular mode on a parti ular model. Thus, for ea h eigenvalue in turn, hoose solve for

v i , ti :



Ai

i I B i gi 0



i

and a model

(Ai ; B i ; C i ) then



vi zi = 0

(3.28)

First ase: the number of eigenve tors to be assigned is equal to the number of outputs, solve for

K

by using:

K [C 1 v 1 C 2 v 2 : : : C p v p ℄ = [z 1 z 2 : : : z p ℄

K = [z1 z2 : : : zp ℄[C 1 v1 C 2 v2 : : : C p vp ℄ 1

(3.29) (3.30)

Se ond ase: more ve tors need to be assigned. It is ne essary to use a dynami feedba k. Let

K (s) denote the transfer fun tion matrix of the feedba k. K (s)

In [150℄

is justied the fa t that we have to solve for

K (1 )C1 v1 = z1 ; K (2 )C2 v2 = z2 ; : : : Note that now, the assigned eigenvalue

i

(3.31)

appears in the equation. Finding a

solution to (3.31) is far more di ult than in the previous ase (see [150℄, [161℄ for details.)

3.5 Con lusions This hapter has shown that the main pro ess of eigenstru ture assignment an be broken up into two. The rst, and arguably most important, element is the spe i ation of eigenstru ture based on the designers requirements and experien e. The se ond is the mathemati al pro ess of eigenstru ture assignment itself. This latter pro ess onsists of nding an a hievable eigenve tor spa e whi h will produ e the desired losed-loop eigenvalues whi h have been spe ied for performan e. Ve tors an then be hosen from this spa e to give required de oupling. Additional manipulation to redu e eigenvalue sensitivity an also be employed. Robustness an best be a hieved by using eigenstru ture assignment

31

as a part of a large design strategy. Goal attainment, the use of singular values and multi-model design have been des ribed as used for the RCAM problem. Additionally, eigenstru ture an be further manipulated to give dynami

ontrollers, whi h have been des ribed for both the point design [77℄ and the multi-model design [55℄.

This is advisable in ases where additional design

freedom is required. Despite all the versatility and potential visibility of the method, eigenstru ture assignment is most useful as a tool within a fuller design environment, thus allowing the attainment of good performan e, de oupling and robustness in the resulting ontrol system.

32

4.

Linear Quadrati Optimal Control

Fran es o Amato 1, Massimiliano Mattei and Stefano S ala

1

2

4.1 Introdu tion Linear quadrati optimal ontrol is ertainly the most widely applied modern

ontrol te hnique. The fundamentals of this theory, whi h date ba k at least to the Fifties (see the germinal paper [131℄ and the bibliography therein) an be found in the Spe ial Issue on the LQG problem [1℄ whi h appeared as an IEEE Transa tion on Automati Control in 1971; sin e then, many books have been written on this subje t (see among others [10℄ and [140℄). This ontrol te hnique allows the designer to take into a

ount both requirements on the amplitude of the ontrol inputs and the settling time of the state variables; moreover, when onsidering innite horizon optimization and provided that the weighting matri es are suitably hosen, an important feature of LQ ontrol is that the resulting losed-loop system exhibits very good guaranteed multivariable stability margins.

Many appli ations of the LQ theory

have been performed in the aeronauti al eld. One of the most important is

ertainly the design of the ight ontrol system of the AFTI/F-16 air raft by General Dynami s (see [70℄). When the omplete state is not available for measurement and some or all of the measures are ae ted by noise, one an use the Kalman optimal ltering theory [1℄ (whi h turns out to be the dual of the LQ optimal ontrol theory) to design an observer of the state variables; however the robustness margins are no longer guaranteed in the presen e of an observer. If sensor noise is absent or one does not are about it, it is possible to use the degree of freedom on the design of the observer to re over the LQ robustness margins; this is the elebrated Loop

Transfer Re overy (LTR) te hnique (see [226℄), whi h, however, an be applied only when the plant under onsideration is minimum phase. Appli ations of the LTR in the aereonati al eld an be found in [64℄, [203℄, and [249℄. Finally in [231℄ some appli ations in aeronauti s of the linear quadrati optimal stati output feedba k ontrol, developed in [172℄, are provided. 1

Dipartimento di Informati a e Sistemisti a, Università degli Studi di Napoli Federi o II

via Claudio 21, 80125 Napoli, Italy, Tel.+39(81)7683172, Fax+39(81)7683686 2

Centro

Italiano

Ri er he

Aerospaziali

Via

Tel.+39(823)623949, Fax+39(823)623335

33

Maiorise,

81043

Capua

(CE),

Italy

4.2 Plant Model Requirements and Controller Stru ture Let us start by onsidering the linear time-invariant plant

x_ = Ax + Bu u x(t)

where, as usual,

x(0) = x0

(4.1)

2 IR n is the state and u(t) 2 IR m is the ontrol.

The

steady-state Linear Quadrati (LQ) optimal ontrol problem an be stated as follows:

Problem:

Q

given

[0; +1) ! IR m

 0 and R > 0, nd, if existing, the ontrol law u : t 2

whi h minimizes the ost fun tion:

J (u) = If the pair

(A; Bu )

Z

0

+1 



xT (t)Qx(t) + uT (t)Ru(t) dt :

(4.2)

is stabilizable the problem is solvable and the optimal

ontrol law turns out to be a state feedba k ontrol law in the form

u(t) = Kx(t)

(4.3)

therefore we often talk of Linear Quadrati State Feedba k (LQSF) optimal

ontrol law; the optimal gain matrix

K

is given by

K = R 1 BuT P where

P

(4.4)

is the unique positive semidenite solution of the algebrai Ri

ati

equation

AT P + P A + Q P Bu R 1 BuT P = 0 : Finally the value of J (
) orresponding to the minimum is

(4.5)

Jopt = xT0 P x0 :

(4.6)

Let us onsider the losed-loop system in Figure 4.1 given by the onne tion of (4.1) and (4.3). As shown in [152℄ and [202℄, if the weighting matrix

R

is

hosen in diagonal form, this system exhibits, at the plant input, guaranteed lower and upper multivariable gain margins of

1=2 and +1 respe tively; more60o and

over, the guaranteed lower and upper multivariable phase margins are

+60o respe tively.

weighting matrix

Therefore LQSF optimal ontrol systems, provided that the

R

is properly hosen, have good robustness properties; this

fa t has further en ouraged ontrol engineers in appli ation of this te hnique in several elds. Now we assume that not all states are available for measurement and that some or all of the measures are ae ted by white noise

x_ = Ax + Bu u + Bw w y = Cx + m 34

(4.7a) (4.7b)

u

Bu

(sI-A)

x

-1

-

K Figure 4.1: LQSF system

where

y(t) 2 IR r

and

ww ( ) = w Æ(t  ) mm ( ) = m Æ(t  )

(4.8a) (4.8b)

are the auto ovarian e fun tions of the sto hasti pro esses that

m

w and m; we assume

is stri tly positive denite.

The steady-state Linear Quadrati Estimator (LQE) problem an be stated as follows: Find a linear state estimator

x^ = L(u; y)

(4.9)

whi h minimizes the steady-state mean square re onstru tion error

where

If the pair

(A; 1w=2 )

 T  Ex (L) = t!lim E e ( t ) e ( t ) x x +1

(4.10)

ex (t) = x(t) x^(t) :

(4.11)

is stabilizable and the pair

(A; C )

is dete table, the

estimator problem is solvable; moreover the optimal estimator (whi h takes the name of Kalman Filter) is a dynami system whi h possesses a Luenberger observer stru ture

_ = A + Bu u + L(y C ) x^ =  where the optimal gain matrix

tion

(4.12b)

L is given by L = C T m1

and

(4.12a)

(4.13)

 is the unique positive semidenite solution of the algebrai Ri

ati equa-

A + AT + Bw w BwT 35

C T m1 C  = 0 :

(4.14)

Finally the value of the ost fun tion orresponding to the optimum is given by

Exopt = tr() :

(4.15)

It is readily seen that the LQ and the LQE problems are duals of ea h other. An immediate onsequen e is that, if we onsider the losed-loop system in Figure 4.2, this system exhibits at the output, the same robustness margins of the LQSF system.

(sI-A)

^ x

-1

C

-

L Figure 4.2: LQE System

u

Bu

-

(sI-A)

-1

y

x C

Bu

K ^ x

+

(sI-A)

-1

+

+

L -

C Figure 4.3: Controller-Observer Stru ture for Feedba k.

Now onsider the deterministi version of system (4.7)

x_ = Ax + Bu u y = Cx

(4.16a) (4.16b)

a well known result, the so- alled Separation Prin iple, states that, if one designs a state feedba k gain

K with A+Bu K Hurwitz, and a Luenberger observer 36

in the form (4.12) with

A + LC

Hurwitz, the losed-loop system depi ted in

Figure (4.3) and des ribed by the equations

x_ _ y u

= Ax + Bu u = A + Bu u + L(y = Cx = K

(4.17a)

C )

(4.17b) (4.17 ) (4.17d)

is asymptoti ally stable; moreover, the eigenvalues of (4.17) are those of

Bu K

and those of

A + LC .

Now assume that

K

and

L

A+

has been designed

following an LQ optimal ontrol and Kalman Filter estimator philosophy respe tively; we know from the above dis ussion that the LQ s heme without observer in Figure 4.1 is robust at the plant input and that the LQE s heme

without state feedba k in Figure 4.2 is robust at the plant output. What an we say about the robustness of the whole LQ-LQE s heme of Figure 4.3? The answer, as shown by a ounter-example in [57℄, is, in general, nothing. This last point introdu es the LTR robust ontrol te hnique, whi h is a methodology to re over, in a ontroller-observer framework, the LQ (or the LQE) robustness margins. Assume that the number of inputs is equal to the number of outputs, that is

m = r (if this hypothesis is not fullled and m < r we

an introdu e further  titious inputs), and that we are interested in obtaining good performan e in terms of amplitude of the ontrol inputs and settling time and, at the same time, good robustness properties at the plant input (a tuators) in the s heme of Figure 4.3. We pro eed in the following way: rst the matrix

K is designed following equations (4.4) and (4.5) (after suitable matri es Q and R have been hosen); then the observer gain L is hosen in su h a way that the desired LQ margins are obtained at the plant input; the last part of this se tion is devoted to detail the pro edure to design su h

L.

This pro edure is

known as LQG/LTR.

Assumption: Let

the plant (4.16) is minimum phase.

L be the solution of an optimal estimator problem with  titious input

disturban e matrix and auto ovarian e matri es given by

Bw = I w = Bu BuT m =  2 I : In this ase we have that

where

( ) is the solution of

1 L( ) = ( )C T 2 

A( ) + ( )AT + Bu BuT It is shown in [141℄ that

1 ( )C T C ( ) = 0 : 2

1 lim L( ) = Bu U 

 !0

37

(4.18a) (4.18b) (4.18 )

(4.19)

(4.20)

(4.21)

U is any orthonormal matrix. Using (4.21) and denoting G(s) = C (sI A) 1 Bu as the transfer matrix of the plant and K (s;  ) as the transfer matrix

where

of the ontroller-observer, it is readily seen that

lim K (s;  )G(s) = K (sI

 !0 Sin e

K (s;  )G(s)

A) 1 Bu :

(4.22)

is the open loop transfer matrix of the ontroller-observer

s heme in Figure 4.3, dened by opening the loop at the plant input, and

A) 1 Bu

K (sI

is the transfer matrix of the LQ ontroller in Figure 4.1, obtained

by opening the loop at the plant input, topologi al arguments lead to the

on lusion that the LQ robustness margins are asymptoti ally re overed at

 ! 0. From (4.21) follows that, when  ! 0, the observer gain L( ) goes to innity; therefore, in pra ti al situations one onsiders a given value of  , for example  = 1, and he k the degree of the plant input in Figure 4.3, when

satisfa tion of ondition (4.22) (this an be done by plotting and omparing the singular values of

K (s;  )G(s) and K (sI A) 1 Bu ).

Then the value of

 is

redu ed until the approximation of the limiting ondition (4.22) is satisfa tory and ompatible with the fa t that faster and faster observers be ome more and more transparent to sensor noise. If we desire to re over the robustness margins at the plant output we an set up the dual pro edure: rst design an optimal Kalman Filter and then design an optimal LQSF ontroller using the  titious weighting matri es

Q = CT C R = 2 I :

(4.23a) (4.23b)

In this ase, the dual relations of (4.21) and (4.22) hold

1 UC  lim G(s)K (s; ) = C (sI A) 1 L !0 lim K () =

!0

(4.24a) (4.24b)

whi h ensure the re overy of the Kalman Filter margins at the plant output. We remember, however, that this last pro edure an only be applied when

r = m or r < m (in this ase it is ne essary to introdu e  titious outputs).

When the plant (4.16) is nonminimum phase, the full re overy of the stability margins annot be obtained; however, a partial re overy may result from the modied LTR pro edure des ribed in [226℄.

4.3 Possible Design Obje tives and Design Cy le Des ription LQ optimal ontrol performs a trade-o between ontrol amplitudes and settling time; this trade-o is strongly inuen ed by the hoi e of the weighting matri es

Q

and

R.

Large values of

R

with respe t to

38

Q

will result in weak

ontrol amplitudes and a slow regulation of the state variables; onversely we have stronger ontrol amplitudes and a faster regulation. For a system in the form (4.16) with

m  r and robustness re overy at the

plant input (a tuators), the design y le is usually omposed of the following steps:

Step 1 Choose the weighting matri es

Q and R;

Step 2 Evaluate the time behaviour of states and ontrols; Step 3 If the time behaviour is satisfa tory, go to Step 4, otherwise go to Step 1; Step 4 Let

 = ;

Step 5 Evaluate

L( ) a

ording to (4.19) and (4.20);

K (s;  )G(s) and K (sI A) 1 Bu ; if the re overy is not satisfa tory, set  = = , where > 1, and go to Step 5.

Step 6 Plot the singular values of

4.4 A Simple Design Example We will now provide a numeri al example in whi h the LQ method has been used to design a Proportional plus Integral feedba k multivariable a tion. This stru ture resembles the one used in the HIRM ontrol s heme des ribed in Chapter 28. Let us onsider the linearized model of the longitudinal dynami s of the HIRM air raft in straight and level ight (Ma h=0.40, altitude=10000 feet) in the form

x_ = Ax + Bu u y = Cx where

x(0) = x0

(4.25a) (4.25b)

x = ( V  q  )T , u = ( ÆT S engineF )T

and

y = x;

we have the

following system matri es:

0

A =

B  0

Bu =

B 

9:150 10 2:717 10 3:458 10 0:00 2:482 5:855 10 1:203 0:00

2 3 3 2

6:553 6:136 10 1 9:806 1 1 1 1:166 10 9:859 10 6:091 10 7 C ; A 1 1 1:547 10 2:651 10 0:00 0:00 1:00 0:00 6:043 10 5 1 4:570 10 7 C ; C = I4 : 2:284 10 6 A 0:00

If we want to synthesize a ontroller whi h regulates velo ity and pit h rate, we have to dene an auxiliary matrix:



Cr = 01 00 01 00 39



su h that

yr = ( V q )T = Cr x :

(4.26)

We an now make referen e to the losed-loop s heme shown in Figure 4.4. Considering that the state-spa e realization of the integrator is

x_ i = e yi = xi ; where

e=r

(4.27a) (4.27b)

Cr x is the tra king error and r

is the referen e signal, we have

the following losed-loop system state equation





u Kp x^_ = A + B C r



where x ^ = xx i

 

Bu Ki x^ + 0 r 0 I

(4.28)

 ; equation (4.28) an be rewritten as





x^_ = A^ + B^ K^ x^ + B^2 r

(4.29)

where

A^ =







A 0 ; B^ = Bu 0 Cr 0



 

; B^2 = 0I

are the state-spa e matri es of an auxiliary  ti ious system and

K^ = ( Kp Ki )

(4.31)

is the state feedba k gain whi h we are going to design with the LQ method.

r

+ -

e

1 s

Ki

+

u

x

Linear plant

Cr

y

+

Kp

Figure 4.4: Closed Loop System

Now the problem is the hoi e of the weighting matri es

Q and R

appli ation of the LQ te hnique to design the PI gain matri es.

for the

Indeed our

e. This means that, ^ B^ ), the last two states, (A;

obje tive is to keep as low as possible, the tra king error in the quadrati ost fun tion dened by the system

whi h are related to the integrators, should be emphasized by in reasing the

40

relative entries of

Q.

In terms of the hoi e of

R,

a good trade-o between

performan e and ontrol energy must be found. In Figure 4.5 the time response of the system is shown under a demand

q. The results obtained by dierent hoi es of the R are ompared. Q = diag(( 10 8 10 8 10 8 10 8 10 6 10 1 )) ; 8 < R0 R0 = diag ( 25 10 5 10 13 ) : R = 100 R0 : 1000 R0

of 5deg/se on matrix

weighting

It is evident from the plots that, by in reasing the norm of the matrix

R, the

20

6

15

4

q (deg/s)

teta (deg)

ontrol energy, and onsequently the time response, des reases.

10 5 0 0

2 0

5

−2 0

10

5

30

dts (deg)

20 R=R0 R=100*R0 R=1000*R0

10 0 −10 −20 0

5

10 Figure 4.5: Simulation Results

41

10

5.

Robust Quadrati Stabilization

Germain Gar ia1;2 , Ja ques Bernussou11 , Jamal Daafouz1;2 and Denis Arzelier

5.1 Introdu tion A fundamental problem in ontrol theory is the robust stabilization problem [56℄. From a pra ti al point of view, it is ne essary to hara terize a lass of

ontrollers whi h ensures, at least, asymptoti stability for the ontrolled un ertain system.

A way to address this problem, is to extend the on ept of

Lyapunov stability to the ase of un ertain systems. The idea is to nd a single Lyapunov fun tion for the ontrolled system from whi h a single ontroller being dedu ed. When su h a Lyapunov fun tion exists, the system is said to be quadrati ally stabilizable this is why the orresponding on ept is alled quadrati stabilizability . Numerous papers deal with the quadrati stabilization problem. For norm bounded un ertain systems whi h are entral in this

hapter, a solution is given in [193℄, [86℄ and onne tions between quadrati stabilizability and

H1 ontrol are presented in [134℄.

Stability is a minimum requirement and is not su ient in pra ti e when a reasonable performan e level has to be obtained. A ommon and dire t way to a

ount for performan e is to put some onstraints on the losed-loop pole lo ations leading to robust pole lo ation design. performan e measure as

Another way is to dene a

H2 or H1 norms and, due to plant un ertainty, one

an at best minimize an upper bound on these norms. Su h approa hes are referred to as guaranteed ost designs [87℄, [91℄. It is also possible to ombine pole lo ation and guaranteed ost designs. The rst problem addressed in this hapter is to nd a linear ontrol law su h that the losed-loop system poles belong to the disk

 + j 0 and radius r.

D(r; )

with enter

The disk for pole lo ation an be hosen in su h a way

that a good ompromise between mode damping and speed is guaranteed. For

ontinuous systems, it su es to in lude it in a se tor lo ated in the left half

omplex plane. If

 is a omplex mode for the ontrolled system, !n = jj, its = Re[℄, its damping fa tor and z = !n 1 ,

undamped natural frequen y, its damping ratio, then

8 2 D(r; ) !n <  + r; < r ; z > r 1 1

LAAS-CNRS, 7 avenue du olonel Ro he, 31077 Toulouse Cedex 4

2

Also with INSA, Complexe S ientique de Rangueil, 31077 Toulouse Cedex 4

42

Another ontrol design problem whi h is dealt with in this hapter, onsiders the disk pole lo ation ombined with a guaranteed

H2 ost. When working in

the quadrati framework, two main approa hes are possible. The rst one (now very popular) is hara terized by the use of an LMI formulation (Linear Matrix Inequality) when writing the onditions for quadrati stabilizability, in luding or not performan e requirements. Being linear with respe t to the unknown matri es, the LMI formulation proposes a onvex parametrization of the robust

ontrollers. Among the good features asso iated with LMI, one an stress the fa t that there exist e ient numeri al tools (industrial pa kages) working on interior point methods.

Another interesting feature lies in the ability of the

LMIs to aggregate several onstraints, provided these are written in terms of LMIs (the ase for stru tural onstraints, integral quadrati onstraints, et . ). The se ond approa h relies on the use of Ri

ati type equations, a tool whi h it is not surprising to nd here, in the framework of linear systems with quadrati fun tions. E ient numeri al tools exist to solve parameter dependent Ri

ati equations. An advantage in expressing the onditions through Ri

ati equations is that ontrol interpretation is mu h easier. Usually in a Ri

ati equation, two weighting matri es, one for the states, and the other for the ontrol, appear. This is the ase for the Ri

ati equations arising in the quadrati stabilizability problem. Their role and ee ts on the derived ontrol are well understood and it is possible by a judi ious hoi e or by a trial and error method to sele t a

ontrol s heme satisfying some requirements. It should be noted that Ri

ati equations an be derived be ause the pole lo ation region is relatively simple (a ir le). This is the reason why in the following, the quadrati approa h will be illustrated by developing the results through the Ri

ati framework.

For

more omplex regions, no analyti al solutions in terms of matrix equations an be obtained. But for a large lass of regions named LMI regions, the problem

an be solved by LMI te hniques. For more details, see [46℄.

5.2 Preliminaries Throughout the hapter, the symbols

0; 1

respe tively denote the null matrix

and the identity matrix of appropriate dimension. of the matrix

M

M0

denotes the transpose

( omplex onjugate transpose for omplex matri es).

For

B , A < ()B means that the matrix A B is negative denite (semidenite). (M ) denotes the spe tral radius of M and  (M ) = (M 0 M )1=2 the maximum singular value. Let us onsider a ontinuous symmetri matri es

A

and

system des ribed by :

where IR

n

A

2

x_ (t) = (A +
A)x(t) + Bu(t) y(t) = Cx(t) nn , B 2 IR nm , C 2 IR pn , u(t) 2 IR m IR

(5.1)

is the input,

x(t)

2

is the state. In order to simplify the following developments, the ase of

un ertainty ae ting only the dynami matrix

A

is onsidered, noti ing that

most of the given results an be extended to un ertain

43

A and B matri es.

These

results an be found in the given referen es. There are several ways to model the un ertainty. One of the most popular is the following:

Norm bounded un ertainty where

D2

nr , IR

E

2

ln IR


A = DF E

(5.2)

dene the stru ture of the un ertainty and the

modelling parameter un ertainty

F

belongs to the set :

F = fF 2 IR rl : F 0 F  1g

(5.3)

In this way an ellipsoidal volume is dened as an un ertainty domain in the hyperspa e of the entries of of this hyperellipsoid.

A, the nominal model being dened in the enter

There exist some other ways to des ribe un ertainty.

We list below some examples whi h may be translated, after some elementary transformations, into a norm bounded un ertainty. Their pra ti al interest is dis ussed in some detail in the robust ontrol literature.

Bounded real un ertainty The un ertainty term is written as :


A = DF (1 D0 F ) 1 E nr , E 2 IR ln dene the stru ture of the un ertainty and the where D 2 IR modelling parameter un ertainty F belongs to F . D0 is a onstant matrix satisfying 1 D00 D0 > 0. Then, we have A +
A = A + D(1 D0 D0 ) 1 D0 E + D(1 D0 D0 ) 1=2
(1 D0 D0 ) 1=2 E with


0
 1.

0

0

0

0

Positive real un ertainty The un ertainty term is given by :


A = DF (1 + D0 F ) 1 E nr , E 2 IR rn dene the stru ture of the un ertainty and the where D 2 IR modelling parameter un ertainty F belongs to the set : Fp = fF 2 IR rr : F 0 + F  0g (5.4)

D0 is a onstant matrix of appropriate dimension satisfying D0 + D00 > 0. This ondition ensures that the matrix 1 + D0 F is invertible for all F 2 Fp . Then, we have A +
A = A D(D0 + D00 ) 1 E + D(D0 + D00 ) 1=2
(D0 + D00 ) 1=2 E 0 with

 1.

Moreover,

44

Stru tured un ertainty The above dened un ertainties are alled unstru tured un ertainties in the sense that they are dened through a single un ertainty matrix dened in a very global and general set.

F

whi h is

We an introdu e some stru tural

features on the un ertainty by dening multiblo k un ertainty terms, su h as


A = where


Ai

m X i=1


Ai

an be expressed by one of the following expressions


Ai = Di Fi Ei ; Fi 2 Fi

D00 i D0i > 0
Ai = Di Fi (1 + D0 i Fi ) 1 Ei ; Fi 2 Fpi ; D0 i + D0 0i > 0 with Di and Ei are onstant matri es of appropriate dimensions and the sets
Ai = Di Fi (1 D0 i Fi ) 1 Ei ; Fi 2 Fi ;

1

Fi and Fpi are dened respe tively like F and Fp .

In this way, one an take

into a

ount more pra ti al parametri un ertainty, but the onditions derived in the sequel are only su ient. In [85℄, the quadrati d stability on ept whi h is the ounterpart of quadrati stability in the ontext of pole pla ement in a disk was introdu ed. We re all below the denition this on ept.

Denition 5.1

The system

x_ (t) = (A + DF E )x(t)

is quadrati ally

if and only if there exists a positive denite symmetri matrix that :

for all

F

2 F with

(Ar + Dr F Er )0 P (Ar + Dr F Er ) P < 0

p

p

Ar = (A 1)=r; Dr = D= r; Er = E= r

P

d stable

2 IR nn su h (5.5)

(5.6)

This denition states that a system is quadrati ally d stable if there exists a single matrix

P

satisfying (5.5) for all the systems in the un ertainty domain.

Pole lo ation is meaningful in the ase of non time-varying un ertainty, i.e.

F

is a onstant matrix. It has been shown in [85℄ that equation (5.5) is in fa t a su ient ondition for quadrati stability, the matrix for the system (5.1), whatever

F

P

matrix in (5.5) is a Lyapunov

belongs to

F.

Furthermore, one may

expe t that for slowly varying un ertainty, the satisfa tion of (5.5) will ensure a good transient behaviour for the ontrolled system. It is to be noti ed that (5.6) is a dis rete Lyapunov inequality for the transformed system (5.6).

In

fa t, a system is quadrati ally d stable if and only if the transformed system is quadrati ally stable. This equivalen e allows to interpret the quadrati d stability as an

H1 norm onstraint as is done for quadrati stability in [134℄. The 45

ondition be omes: the system dened by

x_ (t) = (A +
A)x(t) is quadrati ally

d stable if and only if

kEr (s1 Ar ) 1 Dr )k1 < 1

(5.7)

In the light of this result, the quadrati d stability problem and in the sequel the quadrati d stabilization problem are equivalent to an

H1 synthesis problem

that an be solved using for example an LMI formulation or a Ri

ati equation

approa h. It is well known that the LMI te hniques are powerfull, parti ularly in the ases where multiple onstraints and obje tives have to be taken into a

ount. In the ases where analyti al solutions an be derived, for example a Ri

ati equation, the omplexity of LMI omputations remains higher than that of solving a Ri

ati equation [84℄.

5.3 Quadrati d Stabilizability by Output Feedba k In this se tion, we use the equivalen e between the quadrati d stabilization problem and a disk.

H1 ontrol synthesis to solve output feedba k pole lo ation in

The output feedba k quadrati d stabilizability is formalized in the

following denition.

Denition 5.2

The system is said to be quadrati ally d stabilizable via output

K (s) su h that the u = K (s)y) is quadrati ally d stable for all F 2 F .

feedba k if there exists a linear time-invariant ompensator

losed-loop system ( losed by

From (5.7) written for the losed-loop system, the system is quadrati ally d stabilizable via dynami linear output feedba k if and only if

kEr (s1 Ar Br K (s)Cr ) 1 Dr k1 < 1

(5.8)

As before, the problem an be solved using some standard te hniques.

The

Ri

ati equation approa h leads to

Theorem 5.3

The system (5.1) is quadrati ally d stabilizable by an output

R1 ; R2 ; Q positive denite symmetri ma > 0 and two positive denite nn ; Y 2 IR nn satisfying : symmetri matri es X 2 IR A0r (X 1 + Br (R1 ) 1 Br0 Dr Dr0 ) 1 Ar X + Er0 Er + Q = 0 (5.9) Ar (Y 1 + C 0 (R2 ) 1 Cr E 0 Er Q) 1A0 Y + Dr D0 = 0 (5.10)

ompensator if and only if, given

tri es of appropriate dimensions, there exist

r

with :

Y 1

r

r

1

Dr0 XDr > 0 Er0 Er > A0r (X 1 Dr Dr0 ) 1 Ar + Q 46

r

(5.11) (5.12)

Condition (5.12) implies that

(XY ) < 1.

ompensator is given by :

Under the previous onditions, a

p

_ = (A + BK + rDKd) + L(y C) u = K

(5.13)

where:

K = (R1 ) 1 Br0 (X 1 + Br (R1 ) 1 Br0 Dr Dr0 ) 1 Ar Kd = Dr0 (X 1 + Br (R1 ) 1 Br0 Dr Dr0 ) 1 Ar L = (1 Y X ) 1 Ar (Y 1 + Cr0 (R2 ) 1 Cr Er0 Er Q) 1 Cr0 (R2 ) 1

5.3.1 Output d stabilization algorithm The following algorithm to he k quadrati d stabilizabilty an be dedu ed from the monotoni behaviour of the solutions of the previous Ri

ati equations. Step 1 :

Step 2 :

Choose positive denite symmetri matri es

1; R2 = 1; Q = 1

and

>0

R1 ; R2 ; Q, for example R1 =

Solve the two Ri

ati equations of theorem 5.3. If the solutions are positive denite and satisfy (5.11) and (5.12), Stop. The system is quadrati ally d stabilizable by output feedba k.

Compute the ontroller with

formula (5.13). Otherwise go to step 3. Step 3 :

Take

 = =2.

If

 is less than some omputational a

ura y 0 , Stop.

The

system is not quadrati ally d stabilizable by output feedba k. Otherwise go to step 2. It is obvious that the above algorithm onverges for some number of steps.

0

0 > 0

in a nite

has to be hosen su iently small. To solve the Ri

ati

equations some standard algorithms an be used.

5.4 Quadrati d Stabilizabilty and Guaranteed Cost In this se tion, the results of robust pole lo ation in a disk are ombined with another spe i ation requirement expressed through an

H2 norm of a transfer

matrix from an external perturbation to a ontrolled output.

In fa t, this

problem an be seen as a robust pole lo ation problem with the minimization of an upper bound on a linear quadrati ost (multi obje tive ontrol design). Let the un ertain system be des ribed by :

x_ (t) = (A + DF E )x(t) + Bu(t) + B1 w(t) z (t) = C1 x(t) + D12 u(t) y(t) = C2 x(t) + D21 w(t) 47

(5.14)

where

w

is a disturban e,

2

z

IR

s

is a ontrolled output and

F

2 F.

All

matri es are onstant matri es of appropriate dimensions. We assume without

C10 D12 = 0 and B10 D21 = 0. Let us also dene : Co = f > 0 : the onditions of theorem 5.3 are satisedg

loss of generality that

and :

K = fK(s) given by (5.13)

The ontroller

:  2 Co g

K (s) an be written as : _ (t) = H (t) + Ly(t) u(t) = K(t)

where

H ; L; K are given by theorem 5.3.

(5.15)

The losed-loop system is obtained

by ombining (5.14) and (5.15).

2 

x_ _



=

z   6 6 A + 6 4 | {z } |

0 00

0B 1{z0

A

+

 |



0 {z

}

3


7 0 7 {z }5

|

E




}|

7

D F E

D z = C1 D12 K {z | Ce The transfer matrix from

B

Ae

}| 



}

x 



|

H L  K 0 {z H 

 }|

{

0 1 C2 0 {z }



x + B1 LD21  {z } | {z X B1

C



w

}

w(t) to z (t) is given by :

HF (s) = Ce [s1 Ae DF E ℄ 1 B1 If K (s) 2 K, the H2 norm of HF is expressed as : kHF k22 = Tra e(Ce L (F )Ce0 ) = Tra e(B10 Lo (F )B1 ) where L (F ) and Lo (F ) are respe tively the ontrollability and

(5.16)

(5.17) observability

gramians solutions of :



(Ae + DF E )L (F ) + L (F )(Ae + DF E )0 + B1 B10 = 0 (Ae + DF E )0 Lo (F ) + Lo (F )(Ae + DF E ) + Ce0 Ce = 0

(5.18)

The problem solved in this se tion is the following :

Find

K (s)

and

F , kHF k2 

,

0



<

1

su h that

K (s)

being as small as possible.

48

2K

and

8F 2

5.4.1 Guaranteed ost ontrollers To solve this problem, note that the existen e of a ontroller satisfying the

onditions of theorem 5.3 is independent of the hoi e of the weighting matri es

R1 ; R2

and

Q.

In fa t, an appropriate hoi e for these matri es allows us to

solve the pole lo ation with guaranteed ost design problem. Let :

0 D12 + Æ1 R1 = D12 where

0 + Æ1 R2 = D21 D21

Q = C10 C1 + Æ1

Æ > 0 is a small parameter whi h prevents singularities.

We have the

following results.

Theorem 5.4

Suppose that system (5.14) is quadrati ally d stabilizable by out-

put feedba k. Then: i)

Co 6= ;

1 > 0 and P = P 0 > 0 su h that : A0e P + PAe + 1 PDD0 P + 1 1E 0 E + Ce0 Ce + Æ1 = 0

ii) There exists

Eo1 = f1 > 0 : equation (5.19) has a solution P > 0g. Co , 1 2 Eo1 , and F 2 F , we have : P (1 )  Lo (F ); 8F 2 F

iii) Let

 2 Co , K(s) tra e (B1 B10 P (1 )).

iv) For all

(5.19)

For all

is a guaranteed ost ontroller with



2

2 (1 ) =

5.4.2 Optimization problem The previous lemma suggests solving the following optimisation problem to nd the best guaranteed ost ontroller in the sense dened above. Min

[B1B10 P (1 )℄

tra e

 2 Co 1 2 Eo1

(5.20)

We propose the following algorithm :

Algorithm.

For a representative sample of values of

Step 0 : Initialize Step 1 : Take



 (1020, for example)

2 Co

 2 Co , do

and ompute the orresponding ontroller using theorem

5.3. Step 2 : For the ontroller obtained in step 1 , solve:

2 =

rg

A

fMin tra e[B1 B10 P (1)℄g 1

2 E o1 49

(5.21)

< 

Step 3 : If

H

,

H; L

L; K 



K;

,

go to step 2.

Else go to step 1.

tra e[B1 B10 P (1 )℄ is a onvex fun tion with Eo1 and then this optimization problem an be solved by a

It an be shown that in step 2 ,

1

respe t to

over

one-line sear h algorithm.

5.5 Pra ti al Considerations  To apply the method presented in this hapter, the rst step is to derive an un ertain model for the system. Usually a nominal model is available (linearization) and the un ertainties result from parameter variations, high fre-


A an be obtained by an The D matrix disA and the E matrix over the rows

quen y phenomena or non-linear ee ts. The term

a priori knowledge of the range of parameter variations. tributes un ertainties over the olumns of

A. The size of the un ertainty is adjusted by an appropriate s aling on the D and E matri es. For the RCAM design problem, the parameters whi h vary

of

are the mass and the entre of gravity.

 The high frequen y and non linear phenomena an be minimized by appropri-

ately shaping the sensitivity fun tions. A way to do this is to sele t judi iously the weighting matri es

R1 ; R2

and

Q.

Although no systemati method to

x these matri es exists, a trial-and-error approa h allows us to adjust them. Theorem 1 hara terizes the lass of ontrollers whi h pla es the poles in a disk and the weighting matri es an be used to nd in this lass, a ontroller whi h satises other requirements. With no un ertainty, that is

D and E equal

to zero, the ontroller derived from theorem 1 is lose to an LQG ontroller dened on the triple

Q =  1,

if



(Ar ; Br ; Cr ).

Then onsidering

R1 = 1, R2 = 1

and

  and  , a ontroller with small gains K and L will be

sele ted in the lass of disk pole lo ation ontrollers. These gains have a dire t inuen e on the sensitivity fun tions. Then with shape the sensitivity fun tions.

If

D

and

E

;

and

, it is possible to

are not equal to zero, a similar

hoi e leads to the same ee ts. In pra ti e, a ompromise an be obtained by a trial-and-error approa h.

 Another degree of freedom on erns the hoi e of the parameters dening the ir le

(; r).

The values of these parameters are imposed by the settling

time and overshoot spe i ations, but there exists a ertain latitude on their sele tion.

If the radius of the ir le is too small, the problem is onstrained

and the lass of ontrollers too. In fa t, a trial-and-error approa h allows us to obtain a satisfa tory ompromise.

 The last point on erns the onservative nature of the approa h.

Consider

rst the unstru tured un ertainty. It is well known that the quadrati approa h leads to onservative results be ause a xed Lyapunov fun tion is used for the design. To alleviate this, it is possible to use parameter dependent Lyapunov fun tion approa hes developed over the last few years. If un ertainty is stru tured, the onservatism is more important. A way to redu e it is to ombine a

50

 synthesis approa h with multipliers.

5.6 Con lusion In this hapter, a robust ontrol design based on the quadrati approa h was presented. The performan e requirements are onsidered following two dierent paths. The rst one onsists of lo ating the losed-loop system poles in a disk, the parameters dening the disk ( entre and radius) being hosen in a way that ensured good transient behaviour. The se ond one onsists of dening a ost fun tion (quadrati ) and minimising a ost upper bound, leading to the well-known guaranteed ost design. In fa t, these two means to express performan e are onsidered simultaneously in this

hapter. The derived onditions expressed in terms of parameter dependent Ri

ati equations an be solved with available numeri al powerfull te hniques.

51

H1 Mixed Sensitivity

6.

Mark R. Tu ker and Daniel J. Walker 1

2

6.1 Introdu tion Classi al approa hes to feedba k design have for many years provided reliable methodologies for designing ontrollers that are robust, but these te hniques have not extended well to the multivariable ase. Modern te hniques have subsequently looked at methods for designing multivariable robust ontrollers.

H1

ontrol theory has been establishing itself

sin e the 1980's. The approa h is based on minimising over frequen y the peak values of ertain system transfer fun tions that an be hosen by the design engineer to represent design obje tives. The

H1

mixed sensitivity approa h allows the design engineer to meet

stability and performan e requirements in the presen e of modelling errors, un ertainty and perturbations arising from disturban es or noise. Input and output signals are shaped with frequen y dependent weights to meet robustness and performan e spe i ations.

H1 mixed H1 minimisation is des ribed, followed by a mixed sen-

This hapter is a tutorial hapter that will des ribe the theory of sensitivity methods.

sitivity one degree-of-freedom single input and single output design method.

Next a two degree-of-freedom multivariable mixed sensitivity design is onsidered that in ludes disturban e inputs and a mat hing model.

H1 te hniques have been applied in the design hapters 21, 22, 23, 29, 30, -synthesis methods -synthesis tutorials are given in hapters

and 31 where mixed sensitivity as well as loop shaping and have been used. Loop shaping and 7 and 8 respe tively. More extensive treatment of [215, 159, 266, 96, 61℄.

6.2

H1 theory and appli ations an be found in

H1 Minimisation

z are the output errors or r are the exogenous signals (referen e inputs and disturban es), e are the measurements and u are the ontrols. Consider the standard problem of Figure 6.1, where

osts,

1

Engineering Department, University of Lei ester, University Road, Lei ester LE1 7RH,

United Kingdom. E-mail: mrtsun.engg.le.a .uk Tel: +44 116 252 2567/2874 Fax: +44 116 252 2619 2

Engineering Department, University of Lei ester, University Road, Lei ester LE1 7RH,

United Kingdom. E-mail: wjdlei ester.a .uk Tel: +44 116 252 2529 Fax: +44 116 252 2619

52

r

z P

K

u

e

Figure 6.1: Standard Problem

A

ording to the signals, the open loop system



z e



=



P11 P12 P21 P22



P , of 6.1, is given as





r r =P u u

 (6.1)

r to z an be derived as Tzr = P11 + P12 K (I P22 K ) 1 P21

The losed loop relationship taken from

K

The standard

(6.2)

H1 optimisation problem is to nd a stabilising ontroller

whi h is proper and minimises the supremum (lowest upper bound) over

frequen y of the maximum singular value of

Tzr , the transfer fun tion from the

referen e inputs to the output errors or osts. That is, minimise

 [Tzr (s)℄ k Tzr k1 = Re(sup s) > 0

(6.3)

A stabilizing ontroller a hieving the minimum losed loop norm, k Tzr k1 =

opt , is said to be optimal. A stabilizing ontroller a hieving a losed loop norm

> opt is said to be sub-optimal. P an be represented in state spa e form as

2

3

x_ 4 z 5 = e

2

A B1 B2 4 C1 D11 D12 C2 D21 D22

32

3

x 54 r 5 u

(6.4)

It is worth noting that three spe ial ases of the standard plant A 1-blo k problem is when

D12

and

D21

P

exist.

are square and su h a problem is

mathemati ally easier to solve than a 2-blo k problem where only is square. A 4-blo k problem is when neither

D12

or

D21

D12

or

D21

is square and su h a

problem is the hardest to solve. Generally, all problems will require the solving of two algebrai Ri

ati equations, referred to as the ontrol and lter equations respe tively. In fa t the system of

P

needs to be onstru ted so that the following on-

ditions apply [92℄.

53

1.

(A; B2 ; C2 )

is stabilisable and dete table. This is required for the exis-

ten e of stabilising ontrollers. 2.

D12 has full olumn rank and D21 has full row rank.

This is su ient to

ensure that the ontroller is proper.

 3.

A jwI B2 C1 D12

 has full olumn rank for all



solution to the lter Ri

ati equation. row rank for all

w

Also

w,

enabling a stabilising

A jwI B1 C2 D21

 has full

enabling a stabilising solution to the ontrol Ri

ati

equation. The

H1 optimisation an be solved using fun tions su h as hinfopt whi h

iteratively sear hes for the optimum solution for a parti ular suboptimal

opt and using hinf  whi h produ es a

. These fun tions are available in the

Matlab Robust Control Toolbox [45℄. The ontroller produ ed will be of the same order as the system

P

used.

A high order ontroller an easily result, and so ontroller redu tion is often performed to eliminate unwanted or redundant states. A more spe i system stru ture is now onsidered.

6.3 Mixed Sensitivity - One Degree of Freedom d r

e

+

+

u G

K

y

+

-

Figure 6.2: Closed Loop Feedba k System

r,

Figure 6.2 shows a simple losed loop feedba k system with referen e input output

this

y,

output disturban e

d,

error signal

e

and ontrol signal

y=d = e=r = (I + GK ) 1 = So

u.

From

(6.5)

This is dened as the output sensitivity. To a hieve small tra king error, good transient behaviour and high bandwidth the output sensitivity needs to be small at low frequen ies whi h an be a hieved by designing

K

to have high

gain at these frequen ies. Also

u=r = u=d = K (I + GK ) 1 = KSo = (I + KG) 1 K = Si K 54

(6.6) (6.7)

Si = (I + KG) 1 is dened as the input sensitivity. (Note that in a single input single output system So = Si ). To a hieve robustness it is ne essary to where

a

ommodate disturban es and un ertainties and it is also required to limit high frequen y ontrol eort. For this

KSo

must be designed to be small at

K

high frequen ies whi h an be a hieved by designing

to have low gain at

these frequen ies. In order to meet the low and high frequen y onditions, the design will in orporate frequen y dependent weights.

z1 W1

z2

W2

d r

+

u

e

+

+

G

K

y

-

Figure 6.3: Closed Loop Feedba k System with Weights Figure 6.3 shows the system of Figure 6.2 with added weights. From this it an be written

2

z1 4 z2 e

3 5

3

2

W1 = 4 0 I

W1 G  r  W2 5 u G

whi h hen e denes the augmented plant be obtained using 6.8 in 6.2 and so the

ontroller that minimises

P.

H1

The transfer fun tion

where

> 0.

an

W1 So

k Tzr k1 W2 KSo 1 If there is a bound on the H1 norm su h that k Tzr k1 <

W1 So

W2 KSo 1

Tzr

problem is to nd a stabilising

=





(6.8)

<

(6.9)

then

(6.10)

This implies that

k W1 So k1 < k W2 KSo k1 <

(6.11)

so it an be shown that

 [So ℄ < =[W1 1 ℄  [KSo℄ < = [W2 1 ℄

W1 and W2 an be hosen to give the KSo required to a hieve the required high and In fa t W1 needs to be a low pass lter whilst W2 needs

Hen e the frequen y dependent weights bounds on the terms low frequen y gains.

(6.12)

So

and

to be a high pass lter.

55

Broadly speaking, W1 and W2 determine the performan e and robustness properties respe tively. For example, if the weights have been s aled so that gamma is about one, it follows that other words,

W1

W1 1

provides an upper bound on

So , the 1 Likewise, W2

should be hosen to mirror the desired

determined largely by performan e requirements. an upper bound on

KSo.

So .

In

latter being will provide

This an be interpreted in terms of the losed loop

KSo at s = jw, the smaller the additive model error that

system's robustness to unstru tured additive model error; the larger any given omplex frequen y

will be required to destabilize the system. (This follows from the small gain theorem). Conversely, knowledge of the likely size of the additive model error

KSo. KSo

di tates the safe upper bound on

an also be interpreted in terms

of the gain of the losed loop system from the output disturban es to a tuator useage. It should also be noted that

So

an be interpreted as determining the

system's robustness to output inverse multipli ative perturbation. larger

So

Thus the

at a given frequen y, the less robust the system is to output inverse

multipli ative perturbation at that frequen y. Given the state spa e representation of the plant and weights as

G=



AG BG CG DG



W1 =

then the state spa e form of

2 2 6 6 6 4

x_ z1 z2 e

3 7 7 7 5

=

6 6 6 6 6 6 6 4

P



A1 B1 C1 D1



W2 =



A 2 B2 C2 D2

 (6.13)

is onstru ted as

AG B1 CG 0 D1 CG 0 CG

0 A1 0 C1 0 0

0 0 0 B1 A2 0 0 D1 C2 0 0 I

BG B 1 DG B2 D1 DG D2 DG

3 72 7 7 76 74 7 7 5

3

x 7 r5 u

(6.14)

Assuming that the onstru tion of this augmented plant meets the requirements given in Se tion 6.2, then the

H1 minimisation an be performed to produ e

a robustly stabilising ontroller. Note that a 2-blo k problem is being solved as

D21

is square.

6.4 Design Pro edure The following is a simple pres riptive pro edure for designing a one degree-offreedom ontroller using mixed sensitivity. 1. Sele t the linearised plant model. 2. Sele t

W1 1

and

W2 1

to bound

So and KSo .

3. Augment the plant with the weights plant

P.

56

W1

and

W2

to form the augmented

4. Synthesise a sub-optimal ontroller or an optimal ontroller where the

H1 norm is minimised. The smaller indi ating a more robust design.

5. System analysis. The results an be tuned by adjustments to the weights and so iterations of the design loop an be performed as required. (Note: The pole-zero an ellation phenomenon an o

ur in this one degreeof-freedom mixed sensitivity te hnique. Steps to prevent this possibly undesirable situation o

urring an be found in [241℄. The subsequent two degree-offreedom approa h does not suer from this phenomenon).

6.5 Design Example The following plant model is onsidered to represent the transfer fun tion from the input voltage to the angular position of a simple motor.

G= W1

1 s(s + 1)

(6.15)

is sele ted to be an integrator, this will result in integral a tion in

the ontroller enabling good tra king and small steady state errors.

A true

integrator annot be used as this would not onform to the requirements of Se tion 6.2, so an approximate integrator is used.

The gain of this weight

determines the losed loop bandwidth. The sele ted weight is

W1 = Weight

W2

1 s + 10 6

is sele ted to be a high pass lter.

(6.16)

W2

must be proper to meet

the rank onsiderations of the augmented plant as required in Se tion 6.2. The gain and bandwidth of the weight are hosen to allow low frequen y ontrol eort but limit high frequen y ontrol eort. The sele ted weight is

W2 =

20s + 4 s + 80

(6.17)

The augmented plant is onstru ted as in 6.14 and a sub-optimal ontroller is synthesised realising a

H1 norm bound of = 1:1 opt = 1:2642. An optimal

ontroller an give better results over the whole frequen y range, but this may be a hieved through high frequen y or dire t terms in the ontroller. A suboptimal ontroller is generated without these possible unwanted terms at a slight ost to the robustness and performan e. The output sensitivity

So frequen y response of Figure 6.4 shows the desired

low gain over the operating bandwidth so reje ting low frequen y disturban es. At high frequen ies the gain is unity and around the bandwidth there is a peak in the response. The smaller this peak, the more robust the design. The magnitude of this peak determines the smallest unstru tured output inverse

57

multipli ative disturban e that will de-stabilise the system (see Appendix of [242℄). The fun tion

KSo is similarly analysed.

Figure 6.5 shows the frequen y response the step response for the losed loop system. Iterations of the design y le an now be performed to meet robustness and performan e spe i ations as required. 10

0

0

−5

−10

−10

−20 Gain − [dB]

Gain − [dB]

5

−15 −20

−30 −40

−25

−50

−30

−60

−35 −40 −2 10

−70

−1

10

0

1

2

10 10 Frequency − [rad/s]

3

10

10

−80 −2 10

−1

0

10

1

10 10 Frequency − [rad/s]

2

10

3

10

Figure 6.4: (a) Output Sensitivity Frequen y Response (b) Frequen y Response of

KSo

1.2

1

Units

0.8

0.6

0.4

0.2

0 0

1

2

3

4

5 Time − [s]

6

7

8

9

10

Figure 6.5: Step Response of the Closed Loop System

6.6 Mixed Sensitivity - Two Degree of Freedom So far it has been shown how a one degree-of-freedom ontroller an be produ ed using mixed sensitivity te hniques. The method is now extended to a two degree-of-freedom ontroller design. The system of Figure 6.6 shows a pos-

r1 ), an output disturban e input (r2 )

sible onguration, with referen e input (

58

z2

W2

r2 +

e1

r1

u

e2

+

G

K

z1

+

W1 -

M

Figure 6.6: Two Degree of Freedom Closed Loop Feedba k System

z1

M is an ideal model to mat h K is to be designed.   K an be partitioned as K = K1 K2 , su h that

and two output osts (

and

z2).

The system

the losed loop system to. Controller



u = K1 K2





e1 e2



(6.18)

K1 is a pre-lter and K2 is a feedba k ontroller. There are two methods K . Firstly, K2 ould be synthesised to robustly stabilise the loop against disturban es and un ertainty, and then K1 synthesised where

for generating the ontroller

to shape the losed loop to meet the performan e requirements. Su h a two stage approa h an oer greater exibility and may produ e better results, but

the method is ompli ated to implement needing a two step design pro edure and the resulting ontrollers are independent of ea h other and so overall are

K by

of a high order. A simpler one stage method is to generate the ontroller synthesising the feedba k ontroller

K2

and pre-lter

K1

together. Only one

synthesis is required, and the resulting ontroller is of a lower order as

K2 share the same state spa e.

K1 and

Here the one stage approa h is onsidered.

For this system the standard problem of Figure 6.1 an be formed.

The

system is represented by

2

3

2

3

z1 W1 M W1 W1 G 2 3 r1 6 z2 7 6 0 0 W2 7 6 7 = 6 7 4 r2 5 4 e1 5 4 I 0 0 5 u 0 I G e2 whi h hen e denes P . It an also be shown that   Tzr = W1 (So GK1 M ) W1 So W2 Si K1 W2 K2 So On e again the aim is to produ e a stabilising ontroller the

H1

norm of

Tzr .

K

(6.19)

(6.20)

that minimises

The four weighted fun tions to be minimised are the

dieren e between ideal and a tual systems, the output sensitivity, the ontrol eort to the referen e inputs and the ontrol eort to the outputs.

59

W1

and

W2

are frequen y dependent weights sele ted as before. So using

this design enables robustness and performan e riteria to be met in orporating performan e spe i ations in the mat hing model

P

an be formed using the denitions of

G, W1

M.

and

The state spa e form of

W2

as given in 6.13 and

using the state spa e representation of the mat hing model

M= The state spa e form of

2

2

x_ 6 6 z1 6 6 z2 6 4 e1 e2

3 7 7 7 7 7 5

Ag 6 B 1 CG 6 6 0 6 6 0 6 =6 6 D1 CG 6 6 0 6 4 0 CG

P



AM BM CM DM

 (6.21)

is

0 A1 0 0 C1 0 0 0

0 0 A2 0 0 C2 0 0

0 B1 CM 0 AM D1 CM 0 0 0

0 B 1 DM 0 BM D1 DM 0 I 0

0 Bg B1 B1 DG 0 B2 0 0 D1 D1 DG 0 D2 0 0 I DG

3 7 72 7 7 76 76 76 74 7 7 7 5

x r1 r2 u

3 7 7 7 5

(6.22)

Assuming that the onstru tion of this augmented plant with plant model and weights meets the requirements given in Se tion 6.2, then the

H1 minimi-

sation an be done to produ e a robust stabilising ontroller. It is noted that a 2-blo k problem is being solved as

D21

is square.

6.7 Design Pro edure The pro edure for designing a two degree-of-freedom ontroller using mixed sensitivity follows a similar pro edure to the one degree-of-freedom ase (6.4). The two degree-of-freedom ase requires the additional sele tion of a desired

losed loop system response model.

6.8 Design Example The following plant model has been sele ted.

3

2

(s + 5) 0 1 4 0:1(s + 5) 10(s + 1) 5 G= s(s + 1)(s + 5) s(s + 5) 0

(6.23)

The rst and se ond outputs are to be mat hed to the desired model

 1 0 M = s2 +2s+1 1 0 2 s +2s+1 60

 (6.24)

The ross oupling terms here are zero, so dening the requirement for the

losed loop system to be de oupled. Next, the weights

W1 and W2 are sele ted. These will be multivariable and W1 is sele ted as an integrator to provide good

frequen y dependent. Weight

tra king and small steady state errors. The gain on this term will determine

the error bound on the dieren e between the a tual and the ideal system, and will also determine the bandwidth for output disturban e reje tion.

W1_1 = W1_2 =

2 s + 10 6

(6.25)

The third output is the rate of the rst output and will be fed ba k to enhan e the ontrol and robustness.

Low frequen y a tivity of the signal is

required to enable the tra king of the rst output.

The weight sele ted is a

bandpass lter, sele ted to reje t disturban es around the bandwidth frequen y.

s W1_3 = 2 s + 2s + 1 Weight

W2

(6.26)

is sele ted to be a high pass lter and must be proper to meet

the rank onsiderations of the augmented plant as required in Se tion 6.2. The gain and bandwidth of the weight are hosen to allow low frequen y ontrol eort and to minimise high frequen y ontrol eort.

20s+4 0 W2 = s+80 20s+4 0 s+80 



A suboptimal ontroller is now synthesised realising an

1:1 opt = 0:5818.

(6.27)

H1

norm of

=

Figure 6.7 shows the frequen y response of the singular values of error between the a tual and the ideal responses. At low and high frequen ies, the error is small. Small error at low frequen ies will give good mat hing to the model resulting in small steady state errors. Around the bandwidth, the error is largest, although it is less than unity. Redu ing the error around the bandwidth will improve the overshoot and rise times of the losed loop system. Above the operating bandwidth the desired response is for low gain, whi h is a hieved as the error is small. The output sensitivity

So response is also shown in Figure 6.7, where there is

small gain at low frequen y for the ontrolled hannels, whilst the rate feedba k has unity gain over these frequen ies. At high frequen ies the gain is unity and around the bandwidth there is peak in the response. The smaller this peak, the more robust the design (as dis ussed in 6.5. Fun tion similarly analysed.

K2 So

and

Si K1

are

For the losed loop system Figure 6.8 shows the response of the rst hannel, its rate and the ideal response of the mat hing model and the response of the se ond hannel and the ideal response of the mat hing model. Iterations of the design y le an now be performed to meet robustness and performan e spe i ations as required.

61

5

0

0 −20

−5 −10 Gain − [dB]

Gain − [dB]

−40

−60

−15 −20 −25

−80

−30 −100

−35 −120 −2 10

−1

0

10

1

2

10 10 Frequency − [rad/s]

−40 −2 10

3

10

10

−1

0

10

1

2

10 10 Frequency − [rad/s]

Figure 6.7: (a) Frequen y Responses of the Dieren e Between

3

10

M

and

(b) Output Sensitivity Frequen y Response

10

S0 GK1

1

1.2

0.9 Ideal Output 1

1

0.8 0.8

0.7 0.6 Units

Units

0.6 Output 1 0.4

Ideal Output 2

0.5 0.4 0.3

Output 3

0.2

Output 2

0.2 0

0.1 −0.2 0

1

2

3

4

5 Time − [s]

6

7

8

9

10

0 0

1

2

3

4

5 Time − [s]

6

7

8

9

10

Figure 6.8: (a) Step Response to the First Input of the Closed Loop System (b) Step Response to the Se ond Input of the Closed

6.9 Con lusions H1 ontrol theory and how it an H1 mixed sensitivity one and two degree-of-freedom design proThe H1 mixed sensitivity method lends itself well to systems whi h

This hapter has given a brief tutorial on be applied in

edures.

are required to meet stability and performan e requirements in the presen e of modelling errors, un ertainty and perturbations arising from disturban es or noise. Un ertainty and disturban e an be expli itly in orporated into the design and stability is guaranteed subje t to bounded perturbations, although robust performan e is not. The frequen y domain pro edures are demonstrated using simple examples.

Weight sele tion an be made to a

ount for model

un ertainty. If model un ertainty is unspe ied, then the weight sele tion is broadly dened by robustness and performan e requirements.

Additionally

in the two degree-of-freedom design, a model is in orporated that is dire tly translated from the performan e requirements that the losed loop system is

62

required to meet. Generally, ontrollers are produ ed by iterative design pro edures. The weights are sele ted and the robustness and performan e analysed. Large order ontrollers an sometimes be generated, but in pra ti e it is usually possible to a hieve signi ant order redu tion.

63

H1 Loop-Shaping

7.

George Papageorgiou and Keith Glover , Alex Smerlas and Ian Postlethwaite 1

1

2

2

7.1 Preliminaries

k2 m A 2 IR nm an be dened as kkAx xk2 where x 2 IR is an input ve tor and k:k2 denotes the Eu lidean 2-norm. It an easily be dedu ed, after a few al ulations, that the gain of A will depend on the dire tion of the input ve tor x. To see this we dene the singular value de omposition (SVD) of 22 a matrix (see pp. 32-35 in [266℄). For example the SVD of a matrix A 2 IR The gain of a matrix

is



A = u1 u2 where

u1; u2 ; v1 ; v2

2 IR 2

and







(A) 0 0  (A)

u1 u2



and





 v1  ; v2 

v1 v2



are unitary matri es

 (A) denotes the maximum singular value of A and  (A) the minimum singular value of A. Therefore, as      v1  v v  = 1 0 1 2 v2  0 1

(see p. 19 in [266℄).

if

x = v1

then the gain of

Similarly, for

x = v2

A

 (A) with Ax in the dire tion of u1 . (A) with Ax in the dire tion of u2 .

will be

the gain will be

Dene the ondition number of

A as

(A) =

(A) : (A)

Hen e an ill- onditioned matrix, i.e. a matrix with a high ondition number, has signi antly dierent gains in dierent dire tions. A round matrix is a matrix with a ondition number near unity. Assume that we are given a linear time invariant (LTI) stable system with

m

inputs and

p

outputs, i.e.

G(j!)

2 Cpm . I

All inputs

system are assumed to have nite energy, i.e. a nite

2-norm of a signal u(t) as

kuk2 = (

Z

+1 1

2-norm.

u(t)

We dene the

1 uudt) 2 :

1

Cambridge University Engineering Department, Cambridge CB2 1PZ, England

2

Lei ester University, Lei ester LE1 7RH, England

64

G

into the

It is easy to prove (see pp. 18-25 in [61℄) that if output

y(t) will be bounded by

kuk2  1 the energy of the

supfkyk2 : kuk2  1g = kGk1

where the

1-norm of G is dened as kGk1 = sup (j!)℄: !

Therefore

kGk1 denotes the maximum gain of G over all frequen ies and all

input dire tions.

Consequently, if for example we wanted good disturban e

1-norm of 1-norm an be a onservative measure if we are

reje tion at the output of a plant we would try to minimise the the output sensitivity. The

not interested in all input dire tions. This motivates the use of the stru tured singular value If

P

 (see pp.

271-300 in [266℄).

is a given plant model, then

~ fa torisation of P where M

~ and N

P = M~ 1 N~

is a normalised left oprime

are stable rational transfer matri es satis-

fying the normalisation onstraint

N~ N~  + M~ M~  = I: The motivation for using

H1

te hniques to design robust ontrollers is

provided by the small gain theorem (see pp. 217-221 in [266℄). From the small

1-norm of a stable 1-norm sense, of the unstru tured perturbation to whi h the system remains stable. Hen e a typi al H1 ontrol problem would be to minimise the 1-norm of a transfer matrix, alled the

gain theorem it an be dedu ed that by minimising the transfer matrix we maximise the size, in an

generalised plant, over all stabilising ontrollers.

This optimisation problem

has an exa t solution [63℄. The transfer matrix we hoose to look at depends on the type of un ertainty present in the plant and the performan e spe i ations. The rst step of a typi al

H1 design pro edure would be to de ide on the type

of un ertainty to be used (see table on p. 227 in [266℄). This is di ult and requires good knowledge of the plant. Normalised oprime fa tor un ertainty is the most general type of unstru tured un ertainty (see pp. 418-419 in [96℄). The se ond step would be to hoose frequen y dependent weights a

ording to performan e spe i ations and solve the optimisation problem (see pp. 213245 in [266℄).

Some well studied

H1

H1 -synthesis (see

ontroller design te hniques are

loop-shaping, the S/KS design pro edure (see Chapter 6) and Chapter 8).

7.2 Overview of the Design Pro edure The

H1 loop-shaping design pro edure [164℄ is des ribed below:

1. Shape

G

W1 and W2 a Gs = W2 GW1 , is

open loop with frequen y dependent weights

ording to losed loop obje tives. The weighted plant, depi ted in Figure 7.1.

65

w1

?

W2 

Figure 7.1: The

w2

to outputs





w1 w2



W1 

? 6

- K1

z2

2. Minimise the



G

w2 z1

H1 loop-shaping standard blo k diagram

1-norm of the transfer matrix from disturban es w1 and z1

and

 ! zz1

2

Che k the a hieved

z2





over all stabilising ontrollers.





=

K1 (I I 1

.

 Gs K1 ) 1 I Gs



1

= : (7.1)

This gives a measure of how robust the desired

loop shape is. 3. Choose the position of

K1 in the loop.

Model redu e the ontroller and

design the ommand pre-lters.

4. Che k the time simulations and frequen y responses of the resulting

losed loop system to verify robust performan e.

Reiterations may be

required.

7.3 Justi ation of the Set-Up H1 loop-shaping developed by M Farlane and Glover [164℄ is a very intuitive

method for designing robust ontrollers as the notions of lassi al loop-shaping readily arry through. The designer an spe ify losed loop requirements su h as disturban e and noise reje tion by simply shaping the loop gains. An important feature of

H1 loop-shaping is that it enables the designer to push for the

best a hievable losed loop performan e subje t to a required level of robustness. This is be ause the designer has ontrol over the ross-over frequen ies of the loop gain singular values. When setting up a robust ontrol problem a de ision has to be made about the type of un ertainty to be used. This an be di ult as it requires good knowledge of the plant. The value of

in Equation (7.1) provides a level of

robust stability to oprime fa tor un ertainty whi h requires no assumptions to be made about the open loop stability of the perturbed plant.

Coprime

fa tor un ertainty is a general type of un ertainty similar to the single-input single-output (SISO) gain and phase margins.

When there is little detailed

knowledge about the un ertainty present in a plant method for designing robust ontrollers.

66

H1 loop-shaping is a good

G with W1 and W2 , Figure 7.1. In [164℄ it is proved that if is not too large (say < 4) the ontroller K1 does not alter the desired loop shape Gs very mu h. Hen e shaping G orresponds to shaping the loop gains Gs K1 and K1 Gs . All losed loop transfer fun tions an be bounded in terms of Gs , W1 , W2 and (see pp. 493-494 in [266℄). In [210℄ it is proved that there are no left half plane pole/zero an ellations between K1 and Gs . This is be ause K1

an be written as an exa t plant observer plus state feedba k. Hen e H1 loopPerforman e is spe ied by shaping the singular values of the plant

weights

shaping ontrollers an be gain s heduled. This was done in [120℄. Left half

plane pole/zero an ellations are undesirable as they an limit the a hievable robust performan e. The ost fun tion minimised provides robust stability.

.

A measure of the

-iteration [164℄. Appli ation of the method to real plants has shown that a value of  = 1= between 0:2 0:3 is satisfa tory in the same way o that a gain margin of 6dB and a phase margin of 45 are for a SISO design. A good value of  should guarantee that the required gain and phase margins

robustness of the desired loop shape is given by

Cal ulating the best

requires no

are a hieved. In [247℄ the phase and gain margins of a stable SISO losed loop system,

PM

and

GM

respe tively, are related to the obtained

GM

 (1 + )=(1 ); P M  2 ar sin():

A lot of resear h has gone into

.

H1 ontrol design te hniques. As a result a

great deal of powerful analysis and model redu tion tools are available to help with the erti ation of su h ontrollers, e.g. approximation, the

 -gap and -analysis.

the gap metri , Hankel norm

7.4 Classi al Loop-Shaping Before being able to design ontrollers using the

H1 loop-shaping design pro-

edure (LSDP) the designer must be ome familiar with the notions of lassi al loop-shaping (see pp. 130-137 in [266℄). Loop-shaping allows the designer to spe ify losed loop obje tives by shaping the loop gains. Closed loop obje tives su h as disturban e reje tion at both input and output of the plant, noise reje tion, output de oupling and tra king an easily be spe ied when loop-shaping.

di

r

-

6

-

K

u-

?up-

d

G

- ? -y ? 6n

Figure 7.2: A typi al losed loop system

67

Simple algebrai manipulations on the losed loop in Figure 7.2 give:

y r y u up

To (r n) + So Gdi + So d (7.2) So (r d) + Ton So Gdi (7.3) KSo (r n) KSod Ti di (7.4) KSo (r n) KSod + Si di (7.5) 1 and T = I S . It is simple to show where Lo = GK , Li = KG, S = (I + L) that So P = P Si and Si K = KSo . Disturban e reje tion at the plant output y , Equation (7.2), an be a hieved 1 by making  (So ) = (I +GK ) small to reje t d and  (So G) =  (GSi ) small to reje t di . Similarly, disturban e reje tion at the plant input up , Equation 1 (7.5), an be a hieved by making  (Si ) =  (I +KG) small to reje t di and (Si K ) = (KSo ) small to reje t d. Sin e (pp. 31-33 in [96℄) = = = =

(KG) 1   (I + KG)  (KG) + 1 (GK ) 1   (I + GK )  (GK ) + 1; we an dedu e that if

(KG) > 1 and (GK ) > 1 1 1  (KG)+1   (Si )   (KG) 1 1  (So )  1 :  (GK )+1  (GK ) 1

Therefore for disturban e reje tion of

d at y and di

at

up

 (So )  1 , (GK )  1  (Si )  1 , (KG)  1: Also, if

 (GK )  1 or (KG)  1 and assuming that G and K

are invertible

(this assumption is made for the purpose of illustration) then

  (So G)  (1K ) . Therefore for disturban e reje tion of di at y the singular values of the ontroller should be high at low frequen ies.

  (KSo)  (1G) . This gives the input required to an el the inuen e of d on up . This will be small if  (G)  1 but an not be set by the designer and onstitutes a physi al limitation of the plant. As designers we are not only interested in disturban e reje tion. For noise reje tion

(To ) must be made small at high frequen y.

Typi ally noise is only

important at high frequen y. Note that noise reje tion at low frequen y oni ts with disturban e reje tion as bandwidth of

G

T + S = I.

dedu ed by examining Equation (7.4). equivalent to reje ting

S to zero for es T

Large loop gains outside the

an make ontrol a tivity quite una

eptable.

This an be

Output de oupling and tra king are

d at the plant output be ause T + S = I .

Hen e for ing

to the identity. Figure 7.3 illustrates the desired loop shapes.

68

 (L)

!l

 (L)

log !

!h

Figure 7.3: Loop gain boundaries

7.5 Choi e of Weights In [164℄ it is proved that

K1

does not modify the desired loop shape signi-

antly, i.e. the loop-shaping ontroller is well- onditioned. Hen e shaping a tually shapes both

Gs K1

and

K1 Gs .

onstitutes the theoreti al justi ation of

Gs

This is a very important result that

H1 loop-shaping. The fa t that K1

is well- onditioned is intuitive by examining the ost fun tion minimised and noting that this transfer matrix an be written in two ways (see p. 485 in [266℄). All transfer fun tions an be bounded in terms of mentioned in Se tion 7.3.

Gs , W1 , W2

and

as

For example the input and output sensitivity are

bounded by

(I + GK ) 1  minf (M~ s )(W2 ); 1 + (Ns )(W2 )g (I + KG) 1  minf1 + (N~s )(W1 );  (Ms )(W1 )g; where

K = W1 K1W2 .

ea h frequen y, large low frequen y and

di at up .

 (M~ s ) = (Ms ) = ( 1+(W12 GW1 ) ) 21 at gain of Gs results in reje tion of d at y

Therefore, as

Choosing ill- onditioned weights ould result in poor disturban e

reje tion. Hen e the bounds give the designer a feel for how the weights affe t the losed loop performan e. It be omes obvious, by examination of the bounds, that the notions of lassi al loop-shaping readily arry through. The designer an not usually augment one of the singular values of the open loop plant (as a fun tion of frequen y) with a diagonal pre- ompensator while leaving the other singular values un hanged.

To over ome this problem the

open loop plant an be augmented with a full blo k pre- ompensator. The singular value de omposition of the plant as a fun tion of frequen y

an be written as

G(j!) = U (j!)(j!)V (j!) .

If ea h element of

V (j!) is V^ then,

approximated with a stable minimum phase transfer fun tion to give

G(j!)V^ (j!) ' U (j!)(j!).

Hen e a diagonal weight an now be designed that

augments ea h singular value of

(j!) dire tly.

Note that the same an be done

when designing the post- ompensator. This method provides great exibility

69

to the designer in terms of understanding how the hoi e of weights ae ts the a hievable performan e. Sele ting the weights in su h a way, does not ae t the robustness of the design, as the plant is not inverted. The resulting ontroller is given by

V^ W1 K1 W2 .

As an be seen disturban e and noise reje tion, output de oupling and tra king an easily be in orporated in the loop-shaping methodology. What has not been dis ussed is translating the time response requirements into frequen y response requirements. Time response requirements are spe ied in terms of overshoot

Mp , settling time ts and rise time tr

with respe t to applying a step

to the referen e of the losed loop. These requirements are set by shaping the loop gain near ross-over and

hoosing the bandwidth of the losed loop.

What we must rst understand

is what kind of information we an extra t from the frequen y response of a stable system.

For example when looking at a singular value plot of the

output sensitivity of a system (stable transfer matrix) one an easily see, at a parti ular frequen y, what the maximum gain is. So requirements of the type that

0 (j!)℄

 0:1 for ! < 0:1 rad/s an easily be in luded in the design  (Lo)  11 at frequen ies smaller than 0:1 rad/s.

pro edure by for ing

The Fourier series of a square wave with period

u(t) =

2 !0

is given by:

N 1 4X sin(2n 1)!0 t:  n=1 2n 1

For a reasonably a

urate representation of a square wave it is su ient to take

N = 6.

!0 = 1 rad/s an be onsidered to 1 to 11 rad/s. In reality we an not a hieve a

Hen e a square wave of frequen y

ontain frequen ies ranging from

perfe t square wave as the high frequen y omponents will be ltered out. If we insert

y(t) =

u(t) at the input of the losed loop then the output y(t) will be

N 1 4X jT [j (2n 1)!0℄j sinf(2n 1)!0 t + 6 To[j (2n 1)!0 ℄g:  n=1 2n 1 o

To follow a square wave of frequen y

!0 (this frequen y is related to ts ) we must !0 ! 11!0, hen e

make the output sensitivity su iently small over frequen ies

ontrol the gain of

To and get it as lose to unity as possible.

This an be done

by in reasing the loop gain in this frequen y range. To a hieve this it might be ne essary to in rease the bandwidth of the system (the bandwidth is related to

tr ).

We must also make sure that the system is su iently well damped,

and therefore ontrol the phase of phase of

To).

Lo (the phase of Lo is related to Mp and the

This may mean de reasing the bandwidth due to a tuator and

sensor limitations. It ould also mean de reasing the phase lags introdu ed by the weights at ross-over. Even though it is not straightforward to translate time response requirements into the frequen y domain there are general trends that an be followed. The rise time and overshoot are related to the damping of the system. The less damped the system the smaller the rise time and the greater the overshoot.

70

For a desired damping ratio

< 1, whi h is usually the ase for air raft, rise time

depends very little on damping. A well damped losed loop is a hieved by making sure that the roll-o of the augmented plant singular values at rossover is dB . Bode's phase-gain relationship and its genertypi ally smaller than de alisation to the multivariable ase [61, 266℄ illustrates how the roll-o of

40

(L)

is related to the phase of the loop gain and hen e the overshoot. Rise time is

ontrolled by setting the bandwidth of the system. A fast system orresponds to a system with a small rise time and in most ases a small settling time as well (see pp. 126-131 in [82℄).

7.6 Design Cy le G is as follows:

The typi al design y le given a plant 1. S ale

G.

The open loop plant must be s aled a

ording to the desired

output de oupling and a tuator usage.

This is be ause the open loop

singular values an not be asso iated with any one input or output (see Chapter 1 in [215℄ and p. 42 in [120℄). A badly s aled plant is equivalent to a badly formulated problem. 2. Choose the weights

W1

and

W2 .

Integrators (or near integrators in the

ase of rate following) are pla ed in

W1

to boost the low frequen y gain.

This ensures a zero steady state error if we are tra king an attitude, disturban e reje tion and output de oupling/tra king.

To in rease ro-

bust stability, hen e de rease phase lag at ross-over (i.e. slope of augmented plant singular values), a proportional gain is added to the pre ompensator. The value of the gain (position of the resulting zero) is a trade-o between speed of response (moves the integrator open loop pole away from the origin in losed loop) and robustness. The bandwidth is made as high as possible within the a tuator and sensor apabilities, i.e. robust stability onsiderations. If the open loop is unstable are must be taken not to make the losed loop too fast so that disturban e reje tion leads to input saturation hen e loss of ontrollability. noise reje tion, hen e it ontains low pass lters.

W2

is hosen for

The over-all design strategy is to make the loop as fast as possible within the limitations of the plant to use the a tuators to their limits for disturban e reje tion.

Open loop pre-lters are then designed to satisfy

handling quality requirements. This is based on the fa t that the disturban e reje tion problem is entirely de oupled from the nominal tra king problem (see [247℄ and referen es therein). 3. Choose the position of the ontroller. The ontroller an be implemented in three ways. Pla ed in the forward path gives a faster response at the expense of overshoots be ause all the ontroller dynami s are ex ited dire tly by the referen es.

Also any right half plane (RHP) zeros of the

ontroller are also RHP zeros of

To .

71

An example of implementing the

ontroller in the forward path is given in Figure 7.11. Pla ing the ontroller in the feedba k path leads to a slower more damped response but any RHP poles of the ontroller lead to RHP zeros of

To.

The ontroller

ould also be implemented in the observer form as proposed in [247℄ (see pp. 72-78). This onstitutes the optimal way of introdu ing the ontroller into the loop. 4. Design the ommand pre-lters.

The pre-lters are designed to meet

the handling quality requirements. Performan e is limited by losed loop non-minimum phase zeros (RHP zeros). 5. Perform time simulations and analysis to prove robust performan e.

H1

loop-shaping readily provides robust stability. We an a hieve nominal performan e but must test for robust performan e.

7.7 Two Degrees-of-Freedom Design Pro edure The two degree-of-freedom (DOF) design pro edure as introdu ed in [117, 154℄ guarantees robust performan e with respe t to an ideal step response model. Figure 7.4 illustrates the blo k diagram of the two DOF setup.

The losed

loop response from the referen e signals to the plant outputs follows that of a spe ied model

K1

Tr .

The ontroller

is the pre-lter and

ontroller lter

K1

K2

K2

K

is partitioned as

is the feedba k ontroller.

K = [K1 K2 ℄ where

The inner feedba k

is used to meet the robust stability requirements while the pre-

optimises the overall system to the ommand input. The use of the

step response model (SRM) is to ensure that

(I where

Gs K2 ) 1 Gs K1 Tr 1   2;

(7.6)

 is the model-mat hing parameter. From Equation (7.6) it is obvious  in reases (I Gs K2 ) 1 Gs K1 ! To. By setting  equal to zero the

that as

two DOF setup redu es to the one DOF problem des ribed earlier in Se tion 7.2.

r

w2

- I - K1 -+ ?- W1 +6 z1

K2

-

w1

G



- ?- - I -z 6 y

Tr

Figure 7.4: Two degrees-of-freedom onguration

The design y le, given a plant

G with no dire t feed-through, is as follows: 72

1. Sele t a pre- ompensator

W1 a

ording to the guidelines given in Se tions W2 is usually a onstant

7.5 and 7.6. Note that in the two DOF setup matrix.

2. Sele t a desired losed-loop transfer fun tion

Tr

between the ommands

and ontrolled outputs. 3. Set the s alar parameter

 to a small value greater than 1; something in

the range 1 to 3 will usually su e.

P . In Figure 7.5 the signals, with respe t to u the ontrol variables (the input to the shaped plant GW1 ), v the measured variables (r; y ), w the exogenous signals (r; w1 ; w2 ) and z the error signals (u; y; z ).

4. Form the generalised plant those in Figure 7.4, are:

-

w

-z

P

u

v



K

Figure 7.5: General ontrol onguration The state spa e representation of

2

As 6 0 6 6 6 0 P =6 6 Cs 6 6 I 6 4 0 C In Equation (7.7)

H = ZCs 

P

is given by :

0 Ar 0 0 2 Cr 0 0

0 H Br 0 0 0 0 I 0 I I 0 0 I

where

Z

Bs 0 I 0 0 0 0

3 7 7 7 7 7 7 7 7 7 5

(7.7)

is the solution to the Gener-

alised Filtering Algebrai Ri

ati Equation. The reader an refer to [164, 215, 246℄ for more information on the Algebrai Ri

ati Equations in loop shaping synthesis.

(As ; Bs ; Cs ) and (Ar ; Br ; Cr ) are the state-spa e GW1 and referen e model respe tively.

realisations of the shaped plant 5. Solve the standard the ontroller

K.

H1

optimisation problem for the plant

P

to get

The ontroller may be written in an observer form

as in [250℄. 6. Partition the ontroller in to a pre-lter

K1 and a feedba k ontroller K2 Sf = K1 (0) 1 K2(0).

and al ulate the s aling fa tor of the pre-lter as

The nal two DOF ontroller is illustrated in Figure 7.6.

73

r

-

Sf

-

-+ +6

K1

K2

Controller

-

W1

G

-y



Figure 7.6: Two degrees-of-freedom loop-shaping ontroller

Note that the s aling fa tor

Sf

is lo ated in the ommand path.

This

has been found to improve the nominal tra king properties of the losed loop.

7.8 Design Example We are going to present a design example to illustrate all the above points and how the designer ould produ e a good or bad design.

7.8.1 Presentation of Model Used We are going to design a longitudinal ontroller for the RCAM [145℄.

G

The

ÆT ; ÆT H , the outputs z;_ V and the states q; ; uB ; wB . The linearised plant model G and the denitions of these variables inputs into the linear model

are

an be found in [145℄. All angles are in rad and velo ities in m/s.

G has poles at 0:011j 0:126, 0:830j 1:107 and zeros at 4:338, 4:390.

We an easily distinguish between the phugoid and short period modes. The phugoid is slower and lightly damped. The plant is open loop stable with a nonminimum phase zero. The RHP zero exists be ause the verti al a

elerometer is lo ated behind the entre of rotation of the air raft. The physi al meaning of a non-minimum phase zero is that the plant goes initially in the opposite dire tion to that desired, so when the air raft pit hes up

z_ is initially going to

be positive. Non-minimum phase zeros within the bandwidth of the plant limit the a hievable performan e (see pp. 90-104 in [61℄). The a tuator dynami s and loop delays are given below. The loop delay transfer fun tion is based on a rst order Padé approximation (MATLAB tools ommands are used).

>> tailplane = nd2sys(1,[0.15 1℄); >> engine = nd2sys(1,[1.5 1℄); >> delay = nd2sys([-0.06 1℄,[0.06 1℄);

7.8.2 Design Spe i ations The tra king requirements are as follows [145℄:

74

-

Vair

f tr < 12 se tr < 5 se ts < 45 se ts < 20 se Mp < 5% for h > 305m Mp < 30% for h < 305m Table 7.1: Time response requirements

z_ = V sin f , hen e for a small ight path angle f the limb rate z_  V f . For good disturban e reje tion a 13 m/s wind step should not indu e a deviation in airspeed greater than 2:6 m/s for more than 15 se . There are no ross- oupling requirements dened between V and z_ . Note that

be omes

7.8.3 S aling The open loop plant is s aled a

ording to output de oupling and a tuator

usage. For onvenien e the units of the a tuators are onverted to degrees (d2r: degrees to radians). It was thought that

0:5

of a degree of thrust.

1 degree of tailplane is analogous to

This is ompatible with the physi al limits of the

a tuators [145℄. In reasing for example

0:5 to 1 will in rease the usage of the

throttle by in reasing the bandwidth of the system.

Bs = diag([d2r 0.5*d2r℄); % input s aling Cs = diag([3 1℄); % output s aling Similarly, in reasing the se ond entry in

Cs from 1 to 2, will in rease the speed

z_ .

of response of the airspeed and the de oupling with open loop s aled RCAM

Cs *G*Bs .

3

Figure 7.7 shows the

4

10

10

2

10

2

10

1

10

0

10

singular values

singular values

0

10

−1

10

−2

10

−2

10

−4

10

−6

10

−3

10

−8

10

−4

10

−5

10

−10

−2

10

−1

10

0

10 frequency (radians/sec)

Figure 7.7:

1

10

10

2

10

−2

10

Cs *G*Bs

−1

10

Figure 7.8:

0

10 frequency (radians/sec)

Gs , Gs K1

Changing the dire tionality of the plant signi antly (i.e.

1

10

and

2

10

K1 Gs

in reasing the

loop gain too mu h in a dire tion of low plant gain) results in redu tion of the a hieved robustness. Hen e weighting the throttle with a big number would

75

result in in reasing the losed loop bandwidth even though the throttle a tuator is slow resulting in poor stability.

7.8.4 Choi e of Weights To ensure that a diagonal weight

W1

augments ea h of the singular values of

the s aled plant independently a weight

V~

7.5. The total pre- ompensator be omes dynami s. In the MATLAB ode

V~

was designed as des ribed in Se tion

V~ W1 . V~

is the variable

used in this design has no

preW_V.

preW_V = 9.8313e-01 -1.0565e-01 -1.0565e-01 -9.8313e-01

W1 to boost the low frequen y gain. Having just an 90Æ of phase is added at ross-over. Hen e a proportional matrix gain is added to W1 . The position of the zeroes is a trade-o between speed of response and robustness. The post- ompensator W2 Integrators are added in

integrator redu es the robustness as

is designed for noise reje tion. Again the loser to ross-over that the singular values are rolled-o at, the bigger the redu tion of the a hieved robustness. The weights hosen are:

w1 = nd2sys(0.25*[3 1℄,[3 0℄); w2 = nd2sys([3 1℄,[3 0℄); preW = daug(w1,w2); w = nd2sys(1,[0.2 1℄); postW = daug(w,w); The bandwidth in ea h loop is pushed up as high as possible subje t to a

Gs is shown in Figure 7.8 (solid K1 does not alter the desired loop shape too mu h. Hen e shaping G open loop is equivalent to shaping both Gs K1 (dotted) and K1 Gs (dash-dot). Note that the singular values of Gs K1 and K1 Gs are virtually the same. The augmented plant has 12 states. The ontroller was synthesised using desired level of robustness. The weighted plant line). Figure 7.8 also illustrates how

>> [sysK,emax℄ = n fsyn(sys,1.1); >> emax emax = 3.3307e-01 >> emargin(sys,sysK) ans = 3.0560e-01 Hen e the a hieved states.

 = 0:31.

The ontroller

Therefore the overall ontroller

the poles in

K1

sysK

V~ W1 K1 W2

was model redu ed to

are lo ated around the bandwidth.

the model redu ed ontroller

sysK2

has

11

states.

7

Most of

The singular values of

(dashed) and the original ontroller

sysK

(solid) ontroller are shown in Figure 7.9. The dieren e an hardly be seen.

76

sysK2

gives an

 = 0:30.

The equations in Se tion 7.3 give a feel for the

magnitude of the a hieved gain and phase margins. The ontroller was implemented in the forward path. The pre-lters were

hosen to be rst order lags.

The singular values of

ted) are shown in Figure 7.10.

-analysis

A

So

(solid) and

To

(dot-

ould be arried out, as in p.

3-36 [18℄, to he k robust performan e. Note that the large positive area under the

So urve is due to the RHP zero (see the waterbed ee t pp.

1

97-103 in [61℄).

1

10

10

0

10

−1

10 0

singular values

singular values

10

−2

10

−3

10

−1

10

−4

10

−5

10

−6

−2

10

−2

−1

10

10

0

1

10 frequency (radians/sec)

Figure 7.9:

10

2

10

−2

−1

10

10

0

10

1

10

10

frequency (radians/sec)

sysK and sysK2

Figure 7.10:

To and So

7.8.5 Time Responses The SIMULINK blo k diagram of the linear model is shown in Figure 7.11.

z_dot

1 3.5s+1

V

1 7.5s+1

u Csc

− +

sysK2

W1

Bsc

W2

delays

actuators

RCAM

y

Csc

Figure 7.11: SIMULINK blo k diagram of the linear model Figure 7.12 shows the response to a ommand on

z_ (solid) at t = 1 s.

Note

the initial undershoot due to the non-minimum phase zero. Figure 7.13 illustrates the response to an airspeed ommand (dash-dot). Figure 7.14 illustrates the reje tion of a wind-shear of

13 m/s.

7.8.6 Two Degrees-of-Freedom Design Having designed a weighting fun tion

W1 that provides good disturban e reje -

tion, the design spe i ations in Table 7.1 an be in luded dire tly in the design pro edure using a two DOF approa h. The user-dened step response model,

77

Step on z_dot

Actuator usage

−0.2

−0.4

−0.6

−0.8

−1

20

40

60

0.4

0.35

1

0.3

0.2

0.1

0

−0.1

0.8

0.6

0.4

0.2

0

−0.2

−0.3 0

20

time (s)

40

−0.2 0

60

20

time (s)

Figure 7.12: Step on

Actuator usage

1.2

tailplane (solid) and thrust (dash−dot) in degrees

tailplane (solid) and thrust (dash−dot) in degrees

z_dot (solid) and V_air (dash−dot) in m/s

0

−1.2 0

Step on V_air

0.5

z_dot (solid) and V_air (dash−dot) in m/s

0.2

40

0.3

0.25

0.2

0.15

0.1

0.05

0

−0.05 0

60

20

time (s)

z_ ommand

40

60

time (s)

Figure 7.13: Step on

Vair ommand

Tr in Figure 7.4, is usually diagonal, emphasising maximum output de oupling and exhibiting ideal handling qualities.

>> z_dot_model = nd2sys(0.5^2,[1 2*0.7*0.5 0.5^2℄); >> V_air_model = nd2sys(0.3^2,[1 2*0.7*0.3 0.3^2℄); >> T_r = daug(z_dot_model,V_air_model); As des ribed in the design y le

 > 1.

The nal value was

 = 1:5.

A few

iterations were required (bearing in mind robust performan e) before arriving to this hoi e. The generalised plant was formed from Equation (7.7) and a slightly suboptimal ontroller was obtained using standard routines [18℄.

H1

The degradation of the stability margin ( ) as

optimisation



in reases is

shown in Table 7.2. It is evident that the better the model-mat hing the less robust is the design we an a hieve. Balan ed residualisation (see pp. 449-454



1.1

1.2

1.3

1.4

1.5

1.6

1.7

4.15

4.30

4.44

4.59

4.73

4.91

5.06

Table 7.2: Stability margin as a fun tion of



in [215℄) was used to redu e the ontroller to 8 states. The ontroller was implemented as in Figure 7.6. Figure 7.15 shows the output response to a unit step input on demand and a

z_ . Figures 7.16 and 7.17 illustrate the responses to an airspeed 13 m/s wind shear respe tively.

By omparing the output oupling in Figures 7.12, 7.13 and 7.15, 7.16 it is evident that the two DOF s heme gives good performan e without signi ant deterioration of the losed loop robustness properties. It an be dedu ed that all the requirements are met. The interested reader is en ouraged to go through the example and hange the s aling and weighting fun tions to obtain a feel of how the dierent parameters inuen e the design.

78

Disturbance on z_dot

Actuator usage

10

5

0

−5

−10 0

20

40

5

0

−5

−10

−15 0

60

20

time (s)

40

0

tailplane (solid) and thrust (dash−dot) in degrees

10

0.4

z_dot (solid), V_air (dash−dot), SRM (dot) in m/s

tailplane (solid) and thrust (dash−dot) in degrees

z_dot (solid) and V_air (dash−dot) in m/s

Step on z_dot

Actuator usage

15

−0.2

−0.4

−0.6

−0.8

−1

0

60

20

40

0.3

0.2

0.1

0

−0.1

−0.2

−0.3 0

60

20

time (s)

time (s)

Figure 7.14: 1DOF wind-shear

40

60

time (s)

Figure 7.15: 2DOF

z_ demand

7.9 Limitations of the Method and Ideal Plant Some plants have features that restri t the a hievable performan e (see pp. 143-153 in [266℄ and Chapters 5, 6 in [215℄). Su h limitations are for example RHP poles outside the bandwidth, RHP zeros within the bandwidth and ill onditioning. These limitations are design method independent.

Step on V_air

Actuator usage

Disturbance on z_dot

0.6

0.4

0.2

0

−0.2 0

20

40

0.35

0.3

0.25

0.2

0.15

0.1

0.05

60

0 0

time (s)

Figure 7.16: 2DOF

20

40

10

5

0

−5

−10 0

60

time (s)

Vair

10

tailplane (solid) and thrust (dash−dot) in degrees

0.8

Actuator usage

15

z_dot (solid), V_air (dash−dot), SRM (dot) in m/s

1

tailplane (solid) and thrust (dash−dot) in degrees

z_dot (solid), V_air (dash−dot), SRM (dot) in m/s

0.4

20

40 time (s)

demand

60

5

0

−5

−10

−15 0

20

40

60

time (s)

Figure 7.17: 2DOF wind shear

H1 ontroller design te hniques are frequen y domain based methods. This

is be ause robustness issues are more easily addressed in the frequen y domain. The potential weakness of

H1 loop-shaping is that there is sometimes

di ulty in translating time response requirements to the desired loop shape. This di ulty an be over ome by pushing for the highest possible losed loop bandwidth. Hen e the designer aims for a bandwidth higher than that required to satisfy the handling quality requirements, subje t to obtaining reasonable robustness. The design problem be omes harder when the plant is open loop unstable in whi h ase a high bandwidth ould lead to input saturation and loss of ontrollability during disturban e reje tion. In su h ir umstan es the bandwidth must be lowered. De oupling be omes a more di ult task, parti ularly if the spe i ations for ea h loop vary signi antly.

79

The ideal plant for ontrolling with an

H1 loop-shaping design would be

a plant that has similar properties in all loops. By similar properties we mean equally fast and powerful a tuators and sensors and not too ill- onditioned. Hen e the ross-over frequen y for all singular values an be made the same. When using lassi al ontrol the designer designs the ontroller dire tly. This is not the ase when using

H1

design te hniques as the ontroller is

the produ t of an optimisation and hen e the designer has to set-up the ost fun tion to be minimised. There is an evident transfer of tasks. As

H1 loop-

shaping provides robustness to a very general lass of perturbed plants the

designer has only got to worry about translating the performan e spe i ations to the desired loop shape. Other examples, tutorials, of designing loop-shaping ontrollers an be found in [18, 120, 215, 247℄.

80

-Synthesis

8.

Samir Bennani1 , Gertjan Looye and Carsten S herer

1

2

8.1 Introdu tion This hapter gives some ba kground theory on the Stru tured Singular Value,

, and provides a ight ontrol design example motivating and demonstrating -synthesis design. The issue addressed in the example illustrates the inherent ontrol paradigm that -synthesis partially the ne essary steps to arry out a

solves. Fundamentally,

 addresses the problem of retaining a desired performan e

level in the fa e of un ertainties, whi h is alled the robust performan e problem. For SISO systems, this is automati ally a hieved when the system has guaranteed robust stability and nominal performan e. This does not hold in the MIMO ase and in this respe t, the

- on ept

is a tool to address the

multivariable robust performan e problem. An important by-produ t of the method is that it rises modeling issues in the most general sense, i.e. that we mean system modeling, spe i ation modeling, un ertainty modeling, open loop or losed loop modeling and their validations are all issues whi h appear on e a designer is fa ed with

.

An attempt in

ta kling and predi ting the real world an be done only by formal tools, and this is where

 is intended to be used.

The singular value loop shape paradigm

as presented in [64℄ was a great leap forward in formalizing robust multivariable

ontrol theory.

This resulted in progress towards

H1

optimal ontrol, for

whi h omputable ee tive solutions are presented in [62, 62, 63℄. The lassi al multivariable feedba k problem is illustrated in gure 8.1. Usually, the plant is an element of a set of plants given by in the set

G~

G~ .

G

We shall onsider that ea h system

is linear, nite dimensional and a time invariant system whi h

an be represented by a transfer fun tion

onsists of three subproblems. nd a ompensator

K

G(s) .

The overall design problem

The Robust Stabilization problem (RS) is to

whi h makes the feedba k loop in gure 8.1 internally

stable for all possible plants

G~ .

The se ond problem, the Robust Performan e

problem (RP), whi h is mu h harder to a hieve, requires the ompensator

K

to make the losed loop system respond well under various external signals. 1

Fa ulty of Aerospa e Engineering, Stability and Control Group, Delft University of Te h-

nology, Kluyverweg 1, 2629 HS Delft, The Netherlands E-mail: s.bennanilr.tudelft.nl 2

Me hani al Engineering Systems and Control Group, Delft University of Te hnology,

Mekelweg 2, 2628 CD Delft, The Netherlands.

81

n r

e -

d K

u

~ G

y

Figure 8.1: Classi al feedba k onguration

This means that for all plants in external ommands noise (

n(t) ).

G~ , the plant outputs y(t)

a

urately tra k

r(t) , even in presen e of disturban es ( d(t) ) and sensor

A third problem alled the Control Eort minimization problem

is a onstraint imposed on the ompensator su h that the ontrol signals

u(t)

and/or other ontrol dependent signals remain within appli able limits. It has been remarked in [64, 65, 58℄ that the singular value on ept leads to

onservative robust performan e predi tions. Therefore, the stru tured singular value

 has been proposed as a more rened robust performan e indi ator.

In Beyond Singular Values and Loop Shapes [227℄ by Stein and Doyle the singular value loop shaping as a paradigm for multivariable feedba k system design in the arrangement as shown in gure 8.1 has been revisited. The main

on lusion drawn was that singular values within the lassi al design framework are ee tive in addressing the performan e robustness problem whenever the problem's design spe i ations are spatially round, but that it an be arbitrarily onservative otherwise. The origin of the problem lies in that onditions for robust performan e based on singular values are not tight (su ient, but not ne essary) and an severely overstate a tual requirements. The onservatism of the singular value loop shape paradigm in the lassi al framework ame from a too narrow denition and representation for a system. Furthermore, a general tight performan e spe i ation pro edure is la king. Finally, the stability analysis and synthesis tools were not addressing the fa t that perturbations arising in the system are stru tured. The onservatism introdu ed when using singular values an be surmounted by using the Stru tured Singular Value (SSV)



as

a tighter multivariable generalization of the stability margin. It will be shown that

 naturally arises from the stability analysis of a general lass of systems

alled Linear Fra tional Transformations (LFT's). Naturally, in the sense that the existen e of LFT's automati ally leads to the formulation of the robust performan e problem. General, in the sense that LFT's are both suitable for the analysis and the synthesis problem. Using LFT's to model sets of systems and the ontrol obje tives in mind, the robust MIMO design problem is formalized by spe ifying, the plant set

G~

over whi h the obje tives must be a hieved and the pre ise mathemati al

statements for the performan e and ontrol eort obje tives. This will be illustrated on a simplied ight ontrol problem that we des ribe rst. The design plant is a linear model of the longitudinal short period dynami s of a Cessna Citation 500 in landing onguration. The model states

82

q and the angle of atta k .

are the pit h rate

The state spa e representation

of the model dynami s is given as:



q_ _



=



Mq M 1 Z

The input is the elevator dee tion



ÆE .

q 



+



MÆE ZÆE



ÆE

(8.1)

The ontrol obje tive is to design a

pit h rate ontroller, su h that the losed loop response mat hes the handling quality model

1:5 . Hid (s) = qq ((ss)) = s+1 :5

From robustness onsiderations we

have to ensure that the system works well in the fa e of un ertain state spa e entries, alled the stability and ontrol derivatives, for trim speed variations up to

 10

m/s. During a full pit h ommand manoeuvre the angle of atta k

jj < 20 deg and the (jÆE j; jÆ_E j) < (10; 30) [deg, deg/s℄. is limited to

elevator dee tion and dee tion rate to

The mathemati al formulation of the performan e spe i ations in the ontrol problem and the model set over whi h these spe i ations have to hold an be done by using linear fra tional transformations and norm bounds. The advantage of the LFT formulation is that it gives a ommon base for un ertainty modelling, stability and performan e analysis of perturbed systems ( alled the analysis problem) and nally for ontroller synthesis (our synthesis problem). Ea h of these three steps will be su

essively illustrated by an appli ation on the air raft example. To illustrate the pra ti allity of

 as mature design tool

we on lude the example with a trade-o study, where the performan e and the robustness in the problem are gradually hanged.

8.2 Linear Fra tional Transformations (LFT's) Denote

M

as a

2  2 blo k-stru tured matrix:

v1 = M11 r1 + M12 r2 v2 = M21 r1 + M22 r2 together with matrix

hannels of by


:

M

with







relating

h

v1 = M11 + M12
(I here

l

v2

to

r2

as

v2 =
r2 .

M

i M22
) 1 M21 r1 = Fl (M;
)r1

indi ates that the lower hannels of

M

have been losed with

the same way we an lose the upper hannels of dimensioned matrix

Closing the lower

gives a Lower Linear Fra tional Transformation of

M


.

In

with some appropriately

that relates r1 to v1 in the following manner: r1 = v1 .

The upper LFT is given by:

h

v2 = M22 + M21 (I

i M11 ) 1 M12 r2 = Fu (M; )r2

Many ontrol problems t within this representation. A well known example is the input-output mapping of a linear system,

83

y = G(s)u.

It an be expressed

in terms of state spa e data as an LFT system.

A) 1 + D an be rewritten as

C (sI

G(s) = Fu





A B ;I C D s

It easy to see that

G(s) =



As we shall see this framework is parti ulary suitable to arry out parametri un ertainty modelling. This is illustrated on the air raft problem where due to hanging operating

onditions the state spa e entries of the nominal model (equation 8.1) vary substantially.

In table 8.1 the nominal values of the elements and the max-

imum relative variations an be found. parameter

M MÆE Mq Z ZÆE

Drawing the system dynami s in a

value

mult. pert.

-1.4796

0.20

-6.7679

0.20

-1.5773

0.20

-0.7441

0.20

-0.0900

0.20

Table 8.1: State spa e elements and perturbations for the design example

. α

+

Zα

+

Zδ

w1

w2

δ1

. q

+ +

Mq

δ2

Mα

z2

w5

∆Z δ δE

z3

w3

w4

δ1

∆Z α α

δ5 Mδ

z1

δ4

z4

∆M δ

z5

δ5

z

∆M q

w

α q

q δ3

.. .

∆Mα

A/C

δE

Figure 8.2: Blo k diagram of exam-

Figure 8.3:

ple system with perturbations

ample system in LFT-form

Representation of ex-

blo k diagram, we obtain insight in how the perturbations ae t the model, see gure 8.2. The perturbations in the table are the maximum absolute hanges

Æi by introdu ing s alings. Æ's arbitrarily within the given bounds. The model

the parameters an undergo whi h are normalized to We may hange any of these parametrized in the

Æ's ree ts a set of models.

To derive an LFT representa-

tion, the invariant part of the model and the un ertain elements (the delta's) are separated. This pro ess is known as pulling out the deltas.

All un er-

Æ1 ... Æ5 are diagonally augmented in the perturbation matrix
= diag(Æ1 ... Æ5 ). In gure 8.2 the signals in and from the delta's have been

ut; the signals (z1 ...z5 ) be ome the outputs from the onstant part and inputs of
, while the opposite holds for the signals (w1 ...w5 ). We an now read o tain elements

84

all signal relations given by the mapping and build the following matrix:

 q w1 w2 w3 w4 w5 ÆE _ Z 1 Z ZÆE 0 0 0 ZÆE q_ M Mq 0 0 M Mq MÆE MÆE z1
Z 0 0 0 0 0 0 0 0 0 0 0 0 0 ÆZ
E z2 0 z3
M 0 0 0 0 0 0 0 0 0 z4 0 ÆMq 0 0 0 0 0 0 0 0 0 0 ÆM
E z5 0  1 0 0 0 0 0 0 0 q 0 1 0 0 0 0 0 0 The so obtained matrix losed with the blo k diagonal matrix




as shown

in gure 8.3 provides the required LFT formulation of the un ertain air raft dynami s overing all possible parameter variations.

Noti e that the para-

metri un ertainty modeling pro ess reveals that un ertainty that is unstru tured at parameter level ( omponent level) be omes stru tured at system level

A/C).

(

Another possible way to apture a set of models is given for the

a tuator. The elevator position is hanged via an a tuator having rst order dynami s to

20%

15 . Ga t (s) = s+15

The devi e is assumed to give position errors up

in a frequen y range up to

may be even more.

1

rad/s, while at higher frequen ies this

The variation in the position error along the frequen y

1+1 . The Wpert (s) = 0:20 s=s=40+1 ~ a t = model set overed by the un ertain a tuator dynami s is given by G fGa t (1 + Æ6 (s)Wpert ) : Æ6 (s) stable kÆ6 (s) k1  1g. The weighting fun tion is used to normalize the unknown perturbation Æ6 and at any frequen y ! the magnitude of Wpert represents the relative un ertainty level in the a tuator is represented by a rst order transfer fun tion

model.

This type of modeling is alled multipli ative un ertainty modeling.

It is unstru tured at omponent level (the a tuator).

On e again, we shall

see how unstru tured un ertainty at omponent level be omes stru tured at system level, when inter onne ting these omponents.

8.3 The Extended Design Framework The performan e robustness problem is addressed in a general framework for system design, whi h onsists of a general problem des ription in terms of LFT systems, some key analysis results, a suitable measure of the magnitude for matrix transfer fun tions, the stru tured singular value

, and some ontroller

synthesis results. An important remark is that all elements onsidered in this design framework have a pra ti al software implementation in order to be useful for the engineering world. This issue is provided by the ex ellent and reliable software of [18℄. Furthermore, the pra ti ing engineer needs good tutorials to keep tra k of the theoreti al advan es in the eld, these are nowadays ri hly provided in referen es as [18, 189, 266, 61℄ and many others. elements of that framework are des ribed next.

85

The various

8.3.1 General problem des ription Using the LFT representation the lassi al multivariable ontrol problem as shown in gure 8.1 an be transformed into a more versatile form. In this way any performan e obje tive from the a tual inter onne tion and its ee t within other system loops is derivable.

w d u

∆ P

z e y

K Figure 8.4: The general problem des ription We re ognize for any un ertain losed loop system the three basi omponents:

P (our problem data) the ontroller K (possibly still to be designed) the un ertain elements
(belonging to a pre-spe ied set).

1. general system 2. 3.

All un ertain elements have been pulled out of the system and pla ed in the


-blo k. For synthesis and analysis the only thing we have to know is that the
-blo k is stable and norm bounded: jj
jj1  1. The always returning subdivision for the general system P onsists of re ognizing three pairs of input-output variables. The rst one (u(t); y (t)) onsist of the ontrol and measurement variables. Then we have (d(t); e(t)), the disturban e and error signals whi h onstitute the generalized performan e variables and nally the third pair

(w(t); z (t))

for the perturbation signals whi h are onne ted ba k

into the system through a norm bounded perturbation The design problem is to nd a ompensator general system

e

P

K


.

internally stabilizing the

while keeping the matrix transfer fun tion between

appropriately small for the whole set of allowable perturbations


.

d

and

In the

transformation pro ess, from the lassi al setup into the more general setup, any un ertainty arising at system omponent (a tuator, plant sensors et .) level be omes automati ally stru tured at the level of the generalized system

P.

Furthermore, the so obtained generalized problem des ription as given in

gure 8.4 is suitable for the synthesis as well as for the analysis problem, and has potential for expansion due to its general stru ture.

8.3.2 Analysis results (Doyle 1984) From the general system representation as shown in gure 8.4, a non onservative, ne essary and su ient ondition for robust performan e an be derived.

86

∆

∆ w

0

M ( P, K )

d

Figure 8.5: Closed loop system

z

w

e

d

M

0

∆p z

M ( P, K )

e

Figure 8.6: Stability onguration for Robust Performan e

Suppose the stabilizing ontroller

K

an a

ept the feedba k loop as shown

M (P; K ) = Fl (P; K ), shown in g2  2 blo k-stru tured transfer M (P; K )(s) whi h together with the
operator in a feedba k

in gure 8.4 to get the losed loop system

ure 8.5. The generalized losed loop system is a fun tion matrix

arrangement, forms the basi obje ts on whi h the system analysis problem is based. Under the ondition the system

M (P; K ) is nominally stabilized by K

the

following results apply:

Theorem 8.1

General Analysis Theorem Doyle (1984)

1. Nominal performan e is satised if and only if

 (M22 (j!)) < 1

8!

(8.2)

8!

(8.3)

2. Stability is robust if and only if

 (M11 (j!)) < 1 3. Performan e is robust if and only if

(M (j!)) < 1

8!

(8.4)

The third result represents the MIMO extension of the robust performan e problem, providing with ne essary and su ient onditions. It is established by starting with the denition that performan e is robust if and only if the transfer fun tion matrix from

d

!e

given by

Fu (M;
), remains in an

norm sense bounded by unity - that is, if and only if

 M22 + M21(I



M11 ) 1
M12 < 1

8 !; k
k1  1

H1

(8.5)

This norm bound is also a ne essary and su ient ondition for the system

M

in gure 8.6 to remain stable if we onne t a se ond norm-bounded per-

turbation, say


p (s)

a ross the

e

! d 87

terminals

In this respe t, robust

performan e is equivalent to robust stability in the fa e of two perturbations,


and
p , onne ted around the system M

in the blo k-diagonal stru tured

arrangement shown in gure 8.6. The system

if the fun tion

det(I

(
;
p )M (j!))

diag

M

is robustly stable if and only

!.

remains non-zero for all

8.3.3 The Stru tured Singular Value The observation to view performan e robustness as a stability test brought the fun tion

(:)

 for omplex matri es M

dened in equation 8.4. The fun tion

is the following:

Denition 8.2

The Stru tured Singular Value

2


4min (M ) = In words,

I


M



8 <  :

det(I

93 = 5 m ;


M ) = 0

for some with




= diag[
1 ;
2 ; : : : ;
℄

 (
i )  1 8 i

is the re ipro al of the smallest value of s alar

singular for some






1 :

(8.6)

whi h makes the matrix

in a blo k-diagonal set. If no su h



exists,



is

taken to be zero. This denition redu es to the onventional maximal singular value in absen e of stru ture ( i.e. when the number of blo ks, reason,



m , in

has been alled the stru tured singular value.

The denition 8.6 is not limited to

2 2




is one ). For this

blo k stru tures.

It an be used

to test stability with respe t to any number of diagonal blo ks: in that ase

robust stability is satised if and only if


pert (M11 (j!)) < 1

8!

for a given blo k-stru tured un ertainty from the set


pert

(8.7) . In this way it is

possible to establish robust stability with respe t to plants ae ted by several stru tured perturbations while tting in the robust performan e paradigm. Denition 8.6 also extends to real-valued perturbations redu ing many parametri system analysis problems to

- al ulations.

More generally still, the

stru tured singular value on ept (not value) extends to time varying systems. The al ulations required for these extended ases expressed in Linear Matrix Inequalities, ontinue to impose substantial hallenges even with the tremendous evolution in the eld of onvex optimization.

8.3.4 Numeri s for the stru tured singular value In general exa t omputation of the stru tured singular value is not possible. Therefore, we work with approximations via the upper and lower bounds of For a omplete tutorial on the stru tured singular value numeri s we refer to [188, 18℄.



.

and the involved

The al ulation of the stru tured singular

value

()

dius.

Based on this generalization omputable bounds an be given and re-

relies always on the parti ular hoi e of the un ertainty stru ture


and generalizes matrix measures as the singular value and the spe tral raned. Dene the following perturbation stru tures

88


= fdiag[Æ1Im ; Æ2;
3 ℄ :

Æ1 ; Æ3 2 C;
3 2 Cnn ; g. The set B
is the sub-set of
for whi h holds B
= f
2
: (
)  1g. We an also dene Q
, Q
= f
:
2
;

= I g   It an be shown that Q
2
,
Q 2
,  (Q
) =  (
) =  (Q
) Asso iI

I




ated with the set

, dene the set of s aling matri es

D = diag[D1; d2 ; d3Ik ℄ :

D

given by:

D1 2 Cmm ; d1 ; d2 2 IR + ; D1 = D1 > 0; I




is full D is a s alar, and vi e versa. It D 2 D and
2
holds, D
=
D. In nn it is easy to see that  (M ) =  (M ). Sin e the perturbation the ase
2 C
is bounded we have 
(M )    (M ). However, this bound is not of pra ti al use sin e the gap between  and the   an be arbitrarily large. On the other hand when
= Æ1 Im , with Æ1 2 C then (M ) = (M ), the spe tral radius of M . Using the transformations D and Q on M the bounds an be rened to: max (QM )  (M )  inf  (DMD 1 ) (8.8) Q2Q
D 2D

Note that where the diagonal blo k of

an be seen dire tly that for ea h I

I

In fa t the left inequality is an equality, but not useful as su h, sin e the optimization over

Q is not onvex; it shows lots of lo al minima and maxima.

More useful is the right inequality (whi h in a limited number of ases is also an equality, but in nearly all ases very tight), sin e the optimization over

D

is onvex. Furthermore, it is an upper bound and therefore safe. The perturbations onsidered, were omplex matri es or s alars. However, in the ase of the parametri un ertainties in the air raft, the perturbations

2 IR ).

are real (

In a ase like this, we would like to know, given the system

with un ertain parameters, the smallest possible ombination of perturbations, that auses the system to be ome unstable. It is obvious that the stru tured singular value for this even more onstrained set of perturbations (un ertainties are only allowed to vary along the real axis) is more di ult to determine. For



al ulations with a mixed omplex/real perturbation set, there exist reasonably tight upper bounds by nding optimal s aling matri es (D-G s ales), for more details the reader is referred to [266, 18℄.

8.3.5 Setting up the design problem For the analysis of the ight ontrol example, we rst have to spe ify the overall

ontrol ar hite ture then translate the design spe i ations into mathemati al obje tives by weighting the signals of interest. To demonstrate the exibility of the proposed framework we shall address the simultaneous design of a feedba k and ommand shaping ompensator whi h is often referred to as two degrees of

freedom ontrol. Upon the hosen ontrol ar hite ture we pla e on the physi al lo ation in the system, namely at signal level our requirements. These requirements are made frequen y dependent and be ome our weighting fun tions. In doing so we end up with the situation depi ted in gure 8.7 whi h is what we

all the inter onne tion stru ture. It onsists of the air raft model parametri un ertainties, the un ertain a tuator

89

Ga t (s),

G(s) with its K (s)

the ontroller

to be designed, an ideal model

Hid (s)

whi h we want to mat h and the per-

forman e weighting lters that pla e emphasis on the frequen y ontent and amplitude on the signals of interest. The inter onne tion stru ture in gure 8.7 is the pi torial equivalent of the mathemati al statement of the plant set together with the ontrol spe i ations (depending on the norm we hoose). It ~.

~

δE

z ~

α

1

αmax

δ1.

α

..

w

δ5

G

δE 1

1

.

δ max

δ max 1 S

δE

.

δE

δ5 15

Wp

qc

u

K

-

q

Win

q nom

Gact

n ~

Wpert

w + 6

q

qe

z6

Wn

-

H id

Figure 8.7: Inter onne tion stru ture of the example system

is often advisable to s ale the systems units appropriately. The nominal pit h rate ommand signal is therefore normalized with respe t to the maximum expe ted ommands with the lter

Win .

The pit h rate ommand input

q

goes

through the ideal model. The dieren e between the ideal model response and the a tual pit h rate measurement

q

is the tra king error. To emphasize how

large and up to what frequen y the error redu tion should o

ur, a lter

Wp (s),

ree ting the tra king obje tive, is pla ed on the error signal. However, tra king should not be a hieved at the ost of ex essive ontrol a tivity. Therefore both the elevator dee tion and rate are penalized. The dee tion and rate are

WÆ_E = 1=Æ_Emax .

W = 1=Æ

Emax and ÆE When one of these weighted signals is larger than one, then

weighted by the inverse of their maximum allowed values the obje tive is violated.

To prevent stall during a full pit h rate ommand

we provide an angle of atta k limiting fun tion by introdu ing a performan e

, using the inverse of the maximum allowable value we get the W = 1=max. Finally, a noise lter Wn is depi ted in gure 8.7. This lter s ales the normalized measurement noise n as a fun tion of the frequen y.

spe i ation on weight

Dis onne ting in gure 8.7 the ontroller and the un ertainties we end up with the open loop inter onne tion stru ture

P

P

as shown gure 8.8. The re-

:~:-sign indi ates z w), the ~ ~ _ performan e hannel given by e = [~ qe ; ~; ÆE ; ÆE ℄, d = [qnom ; n℄ and the measurement/ ontrol hannel with y = [q ; q + Wn n℄, and u = [ÆE ℄. It ontains maining system

has three pairs of inputs and outputs (the

a weighted output) These orrespond to the un ertainty hannel (

all required problem data for design. But sin e the weighting fun tions are in most of the ases our design parameters it is worth to start with the analysis on basis of the hypothesis that we are in possession of a stabilizing ontroller

K (s).

90

3

Wp2 2

10

Wp1 1

Z6 (complex)

10

Hid

0

10

P

gain

~q e ~ α ~ δE ~

. δ

10

w 1 .... w5 (real)

Z 1.... Z 5 (real) Z6 (complex)

q nom

−1

10

Werr −2

10

n

E

qc

Wn

−3

10

δE

(noisy) q

−4

10

−2

−1

10

10

0

1

2

10 10 frequency (rad/s)

10

3

10

Figure 8.8: Open loop inter onne -

Figure 8.9: Weighting fun tions for

tion stru ture

design example (not

Win )

s aled with

8.3.6 Weighting fun tion sele tion onsiderations Up to now, we have dened a set of models (nominal air raft model with perturbations), de ided on the ontroller ar hite ture (two degrees of freedom, measurements et .), and whi h performan e quantities we wish to take into a

ount (tra king

q,

maximum ontrol dee tions et .).

The question now

arising is how do the weights have to look like if we want our losed loop system to a hieve robust performan e?

Mu h an be learned about the system by writing down the transfer fun tions in the inter onne tion stru ture. For simpli ity, we will forget about the parametri un ertainties in the air raft model and onsider only the un ertainty at the a tuator. Re all that the rst step is to obtain the transfer fun tions of the open loop inter onne tion stru ture

P

(gure 8.8). These fun tions an be dire tly read

from the inter onne tion stru ture in gure 8.7. Denote the transfer fun tion from

ÆE to q is Gq and from ÆE to  as G . P is given as:

The

3  3 open loop inter onne tion

stru ture

2

z 6 q~e 6 6  6 ~ 6 Æ~ 6 E 6~ 6 Æ_ E 6 4 q q

3

2

7 7 7 7 7 7 7 7 7 5

6 6 6 6 6 6 6 6 6 4

=

0 0 Wp Gq Ga t Wp Hid Win W G Ga t 0 0 0 0 0 0 Win Gq Ga t 0

0 Wpert 0 Wp Gq Ga t 0 W G Ga t 0 WÆE Ga t 0 WÆ_E sGa t 0 0 Wn Gq Ga t

3 72 3 7 w 7 76 7 6 qnom 7 7 74 5 n 7 7 7 Æ E 5

All expressions required for analysis of the losed loop system

M = Fl (P; K )

an be obtained via the short ut:

Mij (s) = Pij + Pi3 (s)[I

K (s)P33 (s)℄ 1 K (s)P3j (s) i; j = 1; 2 91

(8.9)

K (s) partitioned into [K Kf ℄, here K (s) represents the ommand part, q to u, while Kf (s) for the feedba k task stands for the transfer fun tion from q to u. The input sensitivity fun tion Si is given by: [I K (s)P33 (s)℄ 1 = [I [K Kf ℄ [0 Gq Ga t ℄℄ 1 = [I Kf Gq Ga t ℄℄ 1 = Si

with

given by the transfer fun tion from

so that with the omplementary sensitivity The omplete analysis system

2

Wpert 6 Wp 6 diag 6 W 6 4 WÆE WÆ_E

3T 2 7 7 7 7 5

6 6 6 6 4

M

Ti

we get

Si + Ti = I .

is:

Ti Si K Si Kf Gq Ga t Si Hid Gq Ga t Si K Gq Ga t Si Kf G Ga t Si K G Ga t Si Kf G Ga t Si Ga t Ti Ga t Si K Ga t Si Kf sGa t Si K sGa t Si Kf sGa t Ti

3 2 7 7 7 diag 4 7 5

I Win Wn

3T 5

 approximated by the peak value of the s aled  (DMD 1 ), a -upperbound), preferably to a value lower than

Sin e we try to minimize singular value (

1, the diagonal blo ks (not ae ted by the D-s ales) must have a norm smaller than 1. This leads to the robust stability and nominal performan e onditions:

jjWpert Ti jj1 < 1 and:

2

Wp 0 0 0 32 Hid Gq Ga tSi K

0 76 G Ga tSi K

6 0 W 0

4 0 0 WÆE 0 54 Ga tSi K

0 0 0 WÆ_E sGa t Si K From the rst expressions

(M11 )

3

Gq Ga t Si Kf  G Ga t Si Kf 7 Ga t Si Kf 5 sGa t Si Kf

0 0

it is lear that the weight

loop gain to roll o: at low frequen ies



Win 0 0 Wn

< 1

Wpert

1

for es the

jTi j  1 and Wpert will be taken 20%. At

jTi j  Kf G, while Wpert will in rease (more un ertainty to jWpert Kf Gj < 1 (SISO in this ase), we need at least: jKf Gj < 1=jWpert j. In this example, it will have little ee t,

higher frequen ies

a

ount for unmodeled dynami s et .). For

sin e the plant with its a tuator model have already su ient roll-o. However, the ross-over frequen y of bandwidth of

Ti .

Wpert is used as a design parameter to limit the

This is important to prevent ex itation of, for example, stru -

tural modes by ontrol signals ontaining high frequen ies. A more interesting

ase is the performan e blo k

(M22 ).

To a hieve the tra king obje tive, we need:

jWp (Hid Gq Ga t Si K )Win j < 1. At low frequen ies jHid j = 1 and we assume that K  Kf : jWp (I + Gq Ga t Si Kf )Win j = jWp (I Gq Ga t Kf ) 1 Win j < 1 1 Win j < 1, If the loop gain is high, we have approximately: jWp (Gq Ga t Kf ) so that: jGq Ga t Kf j > jWp Win j. We will hoose Win onstant. Gq Ga t have

onstant gain at low frequen ies, so that we mainly inuen e the low frequen y shape of the ontroller

K

via

Wp .

Observe that the tra king performan e

may be destroyed by the noise input via

Wn :

Wp Gq Ga t Si Kf Wn .

jGq Ga t Si Kf j  1 at low frequen ies, we need at least: jWp Wn j < 1. 92

Sin e But, if

Wp

we give

high gain at low frequen ies to in rease the gain of the ontroller,

this requirement is easily violated. There are two simple solutions: in the rst

Wp is high, so that Wn gets Wp ; in the se ond pla e we an feed the noise to the performan e lter, so that Hid q (q + Wn n) is weighted. This is very obvious: the steady state value (at ! = 0) of Wn a

ounts for example pla e we an make the noise input low where

approximately the inverse shape of

for a sensor bias. This bias may violate the performan e index, be ause this is based on the error between the referen e and the exa t output. By applying the se ond solution, the error is related to the same biased measurement the

ontroller re eives. (In a standard feedba k onguration the transfer fun tion of the noise to the output is hanged from a omplementary sensitivity fun tion to a sensitivity fun tion, whi h has low gain at low frequen ies). We an hoose here to lower the gain of

Wn

if ne essary; this has a desirable ee t when we

design a ontroller for the plant without un ertainties, as will be shown later.

W _ sG S K W

W _ sGa t Si Kf also play an important role, jSi j  1, sGa t = 15j!=j! + 15  15 jK j < 1=jWÆ_E 15Win j and jKf j < 1=jWÆ_E 15Wn j

ÆE a t i in and ÆE mainly at higher frequen ies. In that ase

The terms as

s ! 1,

so that at least:

respe tively. These weightings impose an upper-bound on the high frequen y

gain of the ontroller. In many ases the ontroller rolls o at higher frequen ies, so that the weights do not have a great ee t.

However, in the ase of

plant perturbations or severe disturban es it is very important to penalize the rates to prevent the ontroller from produ ing ontrol signals with rates beyond the physi al limits of the plant, ausing rate saturation. We will design

q ) up to Win = 10=57:3

for ommands (

rad/s. Next, two performan e weights

are hosen, to illustrate their ee t on the ontroller shape:

s=20+1 W = 1000 s=20+1 Wp1 = jW20in j s= 0:5+1 p2 jWin j s=0:01+1

(8.10)

Note, that the ross-over frequen ies are equal. This is an important onsideration. shape of

In the low frequen y range there are two major parameters for the

Wp :

the steady state gain and the ross-over frequen y. We must be

areful to hange one at a time.

If the steady state error appears to be too

large (in a simulation for example) simply in reasing the gain means that also the ross-over frequen y in reases, leading to unintended other ee ts.

Usu-

ally, if a good ross over point is found, one an try to extend the slope into the low frequen y range.

This an be seen for

Note that the weight attens at sient behaviour.

! = 10

Wp1

and

W p2

in gure 8.9.

rad/s. This is useful to limit tran-

The weights on the elevator dee tion and rate are hosen

= 5710:3 rad 1 Æ_E 1 = 5730:3 (rad/s) 1 To limit the angle of atmax 57:3 1 1 ta k we hoose max = 2jWin j = 20 . Finally, we dene the noise lter: s= 0 : 01+1 s= 0 0:01 :01+1 . The DC gain is hosen low taking Wn = 0:0005jWinj s=2+1 = 57 :3 s=2+1 into a

ount that we also have to satisfy the performan e index Wp2 . as:

1

ÆEmax

93

8.4 -Synthesis 8.4.1 Formulation of the synthesis problem The next step is the ontroller synthesis problem. The obje tive is to nd a

K a hieving the desired performan e requirements for the P~ .

stabilizing ontroller whole set of plants

P~ = fFu (P;
pert ) :
pert 2
pert ; k
pert k1  1g

The denition of the

(8.11)

-synthesis obje tive:

Denition 8.3 -synthesis :

K (s) the worst ase performan e, 
(M ) = 
(Fl (P; K )).

Minimize over all stabilizing ontrollers the peak value of

min K (s)

k[Fl (P; K )℄k
< 1

i.e.

(8.12)

stabilizing with the shorthand notation of the to the

-norm

of the operator

1-norm we have kGk
= max! 
(G(j!)).

G

and similarly

The stru tured singular

value does not satisfy the denition of a norm. This notation is adopted only to ree t the fa t that we want to measure the size of the worst ase performan e.


(
) by its upper bound  (D(
)D 1 ). Dene Dpert , the s aling set for the perturbation stru ture
pert . For Dpert 2 Dpert and
pert 2
pert it follows from the denition of the invarian e of  under s aling, Dpert
pert =
pert Dpert , that the s aling set D for the augmented In order to perform al ulations we repla e

perturbation set is dened as:

D=



Dpert 0 0 I



: where Dpert 2 Dpert

 (8.13)

D-s ale orresponding to the performan e blo k
p -blo k is set D 2 D 
an be obtained from the upper bound relation (applied to some onstant matrix M ): 
(M )  min (DMD 1 ) (8.14) D 2D When
pert onsists of F full blo ks, the set D looks like Note that the

to one. With respe t to the s aling stru ture

D=f

diag

[d1 I; : : : ; dF I; I ℄ dj > 0g

(8.15)

D an have any phase without ae ting the value of  (DMD 1 ). Therefore the optimization along the frequen y over D an be repla ed by an optimization over stable minimum-phase D (s). Considering real-rational, stable and minimum-phase s alings D (s) to the a tual optimization formulation The elements of

is given as:

min

K (s) stabilizing

min 2D

D(s) stable, min-phase

kD(j!)Fl (P; K )(j!)D 1 (j!)k1 94

(8.16)

In this way the optimization problem of minimizing the worst ase performan e has been t into the

H1 -synthesis framework. Optimizing over D and K si-

multaneously is in general not onvex. Therefore an indire t s heme is used in

D KK (s) while holding D(s) xed and then optimizes over stable minimum-phase D(s) while holding K (s) xed. More details on erning the pra ti al implementation of the syn-

the hope of nding a ontroller minimizing

.

The pro edure is alled

iteration sin e it iteratively optimizes over the stabilizing

thesis problem an be found in [18, 189℄. In most engineering situations the proposed s heme has been proven to be su

essful.

8.4.2 Controller synthesis and analysis To illustrate the ee ts of the weighting fun tion sele tion, un ertainty model sele tion, i.e the trade-o between performan e and robustness we shall study four design ases in our example: 1. Nominal Performan e Design: All un ertainties are set to zero we designate

Knom the resulting ontroller.

2. Complex Un ertainty Design:

assume plant with only the omplex per-

Æ6 2
at the a tuators. The resulting ontroller will be denoted as K2 . The augmented perturbation
related to the robust performan e index is given by
2 = diag(Æ6 ;
perf ). turbation

3. Real and Complex Un ertainty Design:

Taking all un ertainties into a -

ÆR = diag(Æ1 ; : : : ; Æ5 ) 2
related to robust performan e is denoted as
3 = diag(Æ1 ; : : : ; Æ5 ; Æ6 ;
perf ).

ount leads to ontroller


R

K3 .

The real un ertainty is

. The augmented perturbation

 (M ) (M ) (M ) order

Knom K2 (Wp1 ) K2 (Wp2 ) K3 (Wp1 ) K3(Wp2 ) 0.89

6

28.66

28.66

28.66

28.66

1.71

1.86

1.88

2.11

1.51

1.60

1.78

1.89

9

9

15

15

Table 8.2: A hieved robust performan e levels

In table 8.4.2 the a hieved robust performan e levels expressed in

 values

for all ongurations are summarized. Row # 1 shows the results of the pure

H1

optimization.

Rows # 2 and # 3 reveal the robust performan e levels

a hieved after the rst and se ond

D K iteration.

In the last row the order of

the resulting ontrollers is given. The table ree ts a well known fa t that the robust performan e level de reases as the un ertainty and performan e levels in rease.

95

Controller Shape Analysis In gure 8.10 the frequen y responses of the ontrollers are depi ted. As already noted in se tion 8.3.6, the ontroller will not ne essarily roll o at higher frequen ies, sin e the ombination of the a tuator and the plant model already shows this behaviour. We an see that ontroller shapes atten out at higher frequen ies and lower gains.

In the se tion 8.3.6 we have seen that

K at higher frequen ies has to satisfy: jK j < 1=jWÆ_E 15Win j = 1=(57:3=30  15  10=57:3) = 0:2. This is onrmed by gure 8.10. For Kf we have: jKf j < 1=jWÆ_ 15Wn j = 1=(57:3=30  15  0:1  10=57:3) = 2. This is also E the gain of

satised.

3

10

0

2

Wpert^−1

10

10

Ti

K3 (Wp2)

−1

K2 (Wp2)

10

1

10

K3 (Wp1)

mag

Controller gain

T_i and Wpert

1

10

−2

10

K2 (Wp1)

0

10

K_f −3

10

Knom c −1

10

K_c

−4

10

−2

10

−5

−2

−1

10

10

Figure 8.10:

0

10 frequency (rad/s)

1

10

10

2

10

−2

−1

10

Controller frequen y

10

0

1

10 10 frequency (rad/s)

2

10

Figure 8.11: Input ompl. sensitiv-

responses

ity fun tion with

1 Wpert

Due to the a tuator un ertainty the ontroller will limit its bandwidth at frequen ies where un ertainty starts to be ome important. This is illustrated by the fa t that the bandwidth of the input omplementary sensitivity fun tion

1 are Wpert . The frequen y response plots of Ti and Wpert Indeed Ti rolls o near the ross-over frequen y of Wpert .

is limited by the lter given in gure 8.11.

In this way we prevent unmodelled higher order dynami s from ex itation by keeping the ontrol a tions within the lower frequen y range. for the nominal plant in gure 8.10, and no feedba k

Kf = 0.

Knom

The ontroller

K K approximately inverts the

has only a feedforward a tion

The feedforward a tion

plant and the ideal model is built in as a feedforward lter. We know a priori that in absen e of un ertainty no feedba k is required. It is interesting to see that this out ome is a hieved automati ally by the method.

Wp . By Wp1 ! Wp2 ) the ontroller

Another interesting ee t is the inuen e of the performan e weight in reasing the slope into the low frequen y range ( does exa tly the same.

We an use this to for e the optimization algorithm

to build integration or even double integration.

As in lassi al Bode loop

shape te hniques, the internal model prin iple holds and as we know in reasing tra king error requirements requires in reasing low frequen y ontrol gain.

96

3

10

Nominal Performan e Beside the shape of the ontroller we are interested in the trade-os it makes. In gure 8.12 we have for all ontrollers plotted the nominal performan e level

 (M22 ).

The overall shape is, a high value of

 (M22 ) at low frequen y orre-

sponding to an ee tive tra king requirement at these frequen ies. At higher

 (M22 ) rolls o.  (M22 )  0:9 is a hieved

frequen ies there is no performan e requirement so that plot Given all the ontrollers the best nominal performan e

by the system without un ertainty. For the other ontrollers the level is higher

(worse), be ause there is a trade-o against robustness to the perturbations. We know that the omplex un ertainty is about 20 % in the low frequen y range. This is about of the performan e degradation level of the se ond system with respe t to the nominal system. It is surprising that the third ontroller for the most un ertain plant a hieves a better nominal performan e level than the se ond ontroller. The on lusion is that the real un ertainty at low frequen ies, sets o the ee t of omplex un ertainty with respe t to the nominal performan e and that this ee t is reversed at high frequen ies.

K2 designed with Wp2 is modest in the performan e level of K2 .

For omparison we add the nominal performan e plot for

Wp2 .

Note that the inuen e of

4 1.2 K2 (Wp2) 1

3.5

K2 (Wp1)

Knom 3

K3

2.5

mu(M)

sig1(M22)

0.8

Knom

0.6

K2

2

1.5

K3

0.4

1 0.2

0.5

0 −2 10

−1

0

10

1

2

10 10 frequency (rad/s)

0 −2 10

3

10

10

−1

0

10

1

2

10 10 frequency (rad/s)

Figure 8.12: Nominal performan e

Figure 8.13:

levels ontrollers

levels ontrollers

10

3

10

Robust performan e

Robust Performan e To ompare the robust performan e levels a a hieved ontrollers.

-test

for


3

is applied on all

The result an be found in gure 8.13.

None of the

ontroller a hieves robust performan e. One of the purposes of this omparison is to reveal the ee ts of o-design spe i ations for the ontrollers

Knom.

K2

and

The question we have in mind is how robust is a robustly designed

ontroller? The nominal ontroller

Knom performs worst with a 400 % degradation at

low frequen ies. At higher frequen ies it has a better robust performan e level than

K2

and

K3 .

The result is expe table, sin e it is purely an open loop

ontroller. The shown

-plots

for

K2

and

97

K3

ree t the design result in the

Wp1 . Using Wp2 would show a too Knom ae ting the s ale of the plots. The K2

medium performan e ase, i.e. with weight dramati performan e ollapse of

ontroller has about a 70 % performan e degradation due to real perturbations. Note, that by taking into a

ount the real perturbations in the design, ase

K3 ,

the total robust performan e level improves onsiderably in the low frequen y range at the ost of the level at higher frequen ies. There is an overall better balan e between the performan e and robustness obje tive, whi h improves the better we model the un ertainty in the plant.

Robust Stability

(M11 )
3 , have been

The robust stability properties are shown in gure 8.14, the values of are plotted along the frequen y axis. Again all perturbations, i.e. taken into a

ount. The ontroller

Knom a hieves the best robust stability level, Kf = 0), and

whi h is not surprising anymore sin e there is no loop losure ( apparently there is no perturbation with norm

 1 to destabilize the nominal

K2 (M11 ) < 1). Note that for K2 and K3 two bounds are visible at lower frequen ies; they arise from real  approximations by optimizing an upper bound and a lower bound: the exa t value of  lies in between these bounds. If for K2 only the omplex perturbation is taken into a

ount (not shown) plant (a system with no feedba k has no robust stability problems). For

robust stability is a hieved (

the plot moves approximately 0.1 downwards, whi h means that there is a 10 % stability robustness degradation to unmodelled spe i ations in the design. It is remarkable that the robust stability level for one of

K2 :

K3

is higher (worse) than the

in a small frequen y range it is even possible to nd a ombina-

 (
i )

tion of perturbations (

 1) that destabilizes the plant ((M11 ) > 1).

We must realize that we are optimizing the peak value of

(M (j!)).

Taking

the parametri un ertainties into a

ount in the design improves this value

ompared to the

-test

for

K2.

In this sense we su

eeded in the third de-

sign. However, the balan e between performan e and stability robustness has moved in the wrong dire tion: the rst improved, the se ond got worse, while overall robust performan e improved. The designer has to be areful and has always to nd a right balan e. Espe ially, sin e in pra ti e

(M ) < 1

8 !)

(

is seldomly a hieved. However, for a given situation robust stability should be

(M11 (j!)) < 1 8 !).

preferably the rst to be guaranteed, i.e. (

Then, slowly

and arefully, the designer an buy performan e from the robust stability until the overall requirements are in balan e. We would like to remark that in the multivariable ase this philosophy still holds but things be ome more omplex be ause of dire tionality issues.

Time Simulations, Performan e Sensitivity We shall next analyze the systems performan e via pit h rate step ommand simulations shown in gure 8.17. Three model ases are onsidered:



nom: simulation with the nominal model;

98

1.2

1

0.8

K3

mu(M11)

K2 0.6

0.4

Knom

0.2

0 −2 10

−1

0

10

1

2

10 10 frequency (rad/s)

3

10

10

Figure 8.14: Robust stability ontrollers



pert1: simulation with a perturbed model: all parameters in table 8.1 are perturbed with



Æ = 20%; only MÆE

pert2: identi al perturbation, now

with

Æ = +20%;


MÆE = 20% the others are 20%.

The rst plot shows the rst ontroller: the nominal response oin ides well with the ideal model response. The perturbations have a dramati ee t on the tra king performan e, sin e we are in fa t looking at an open loop simulation. In the responses for the other ontrollers the ee ts are less dramati . Note that for the se ond ontroller

K2

designed with

Wp2

the steady state error

indeed has be ome nearly zero, even under the inuen e of the perturbations. Finally, we an see that

K3

K2 (with Wp1 ) -analysis.

performs better than

perturbations, as ould be expe ted from the

under the

Time Simulations, Robust Stability Aspe t Finally, we are looking at the perturbations that ould destabilize the losed loop systems in the ase of

K2 and K3 .

!peak = 6:5

rad/s and

 plots peak = 0:918 at

We an see in gure 8.14 that the

(lower bounds) of the losed loop systems show peak values of

peak = 1:024 at !peak = 6:9

rad/s respe tively. This

means that we an nd the smallest destabilizing perturbation with appropriate


-stru ture:


= fdiag[Æ1;


; Æ6℄ : with

Æ1 ;




; Æ5 2 IR ; Æ6 2 Cg I

 (
) = 1=0:918 = 1:089 and  (
) = 1=1:024 = 0:977 respe tively. For
in mind,  (
)  1
2
. system with K2 is robustly stable, sin e peak < 1 and the norm of the

the robust stability test we have a norm bounded The

perturbation therefore needs to be larger than 1 to destabilize the system. This

99

is not the ase for

K3 .

Using available software we have found perturbations

that will just destabilize the systems. For

K2

we have (for example):


= diag[Æ1;


; Æ6 ℄ = diag[ 1:0892; 0:8389; 0:7893; 1:0892; 1:0892; 1:0573 0:2618i℄ with norm   (
) = 1:098. For K3 :
= diag[ 0:9768; 0:9768; 0:0073; 0:9768; 0:9768; 0:9624 0:1667i℄ In gure 8.15,8.16 we simulate the losed loop system, without and with perturbations. To see how sensitive results an be, we also implement the pertur-

98 %

bation s aled to

1:02 % of its riti al value. We an see that both K2 and K3 indeed are destabilized while in reasing

and

systems with ontrollers

the perturbation levels over their riti al values, whi h on ludes the example.

controller K2

controller K3

0.35

0.3

nominal

0.3

0.25

0.98*pert 1.02*pert 0.25

1*pert

q (deg/s)

q (deg/s)

0.2

0.2

0.15

0.15 0.1

0.1

nominal 0.98*pert

0.05

1.02*pert

0.05

0 0

1*pert 0 0

1

2

Figure 8.15:

3

4

5 time (t)

6

7

8

9

1

2

3

10

Destabilizing pertur-

Figure 8.16:

bations

4

5 time (t)

6

7

8

9

10

Destabilizing pertur-

bations

Although this example is very simple, it is lear that

-synthesis is a pow-

erful tool where many fa tors an be taken into a

ount: design requirements, un ertainties, disturban e models et . In absen e of un ertainty, the two degrees of freedom ontroller inverts the plant and pla es in the feedforward path almost no feedba k.

This is a desirable strategy only in the absen e of un-

ertainties. We saw that slight un ertainties aused huge performan e degradation.

The di ulty in designing a good ontroller is the to nd the right

trade-os between the many usually oni ting requirements. We believe that the approa h as shown here provides us with tools to make sensible (balan ed) design de isions to a hieve robust performan e. We on lude by saying that the method for es the designer to understand his model and the intimately related spe i ations on it. The method links the design work with the pra ti al world.

8.5 Con lusion We have reviewed a general framework for ontrol system analysis and synthesis. The stru tured singular value



100

arose from the stability analysis of a

more general type of systems, namely linear fra tional transformations. This permits us to ta kle formally the robust performan e paradigm. To over ome the often reported di ulties in the la k of guidan e in the weighting fun tion sele tion during the design we have provided a simple and illustrative example whi h ontains all ingredients and steps that should be arried out in analyzing su h a design problem. Hopefully, we have su

eeded in larifying that a good design is a matter of balan ing requirements.

We might say that



is

the tool to guide us in nding the required trade-os between performan e and robustness. It pla es the hallenge on the side of the pra ti ing engineer. To be su

essful in improving the behaviour of omplex systems he will have to quantify his spe i ations and he will have to rely ontinuously on a better and deeper system knowledge.

The paradigm is no longer ontroller design,

but spe i ation design.

Knom 14

K2 , with Wp1

12

12

pert1

pert1

10 nom

ideal 10

pert2

q (deg/s)

q (deg/s)

8

nom

ideal 8 pert2 6

6

4 4

2

2

0 0

5

10

0 0

15

5

10

time (s)

K2 , with Wp2

12

15

time (s)

12

K3 , with Wp1 nom, pert1

10

10 pert

1 pert2

8

8 q (deg/s)

q (deg/s)

pert2

6

6

4

4

2

2

0 0

5

10

15

time (s)

0 0

5

10 time (s)

Figure 8.17: Step responses for designed ontrollers

101

15

9.

Nonlinear Dynami Inversion Control

Binh Dang Vu

1

9.1 Introdu tion Among the spe i methodologies for the ontrol of systems des ribed by nonlinear mathemati al models, dynami inversion is ertainly the most widely investigated by ontrol resear hers in the last two de ades. A omplete theory is now available for the design of feedba k ontrol laws whi h render ertain outputs independent of ertain inputs (disturban e de oupling and nonintera ting ontrol) or whi h transform a nonlinear system into an equivalent linear system (feedba k linearization or dynami inversion). The theory of nonintera ting ontrol was initiated by the pioneering work on linear systems by Falb and Wolovi h [75℄. The extension to nonlinear systems is due to the work of Singh and Rugh [212℄, Freund [83℄, following an idea due originally to Porter [196℄.

Feedba k linearization is based on some

early work of Krener [139℄ and Bro kett [34℄ demonstrating that a large lass of nonlinear systems an be exa tly linearized by a ombination of a nonlinear transformation of state oordinates and a nonlinear state feedba k ontrol law. A major breakthrough o

urred at the beginning of the eighties with the appli ation of mathemati al on epts derived from the eld of dierential geometry, Isidori et al. [125℄, Byrnes and Isidori [42℄. A good survey of the theory an be found in re ent books : Isidori [124℄, Nijmeijer and Van Der S haft [184℄, Slotine and Li [216℄. The basi feature of feedba k linearization is the transformation of the original nonlinear ontrol system into a linear and ontrollable system via a nonlinear state spa e hange of oordinates and a nonlinear stati state feedba k

ontrol law. The solution of this problem relies on the nonsingularity of the so- alled de oupling matrix. When this ondition is not satised, a dynami state feedba k ontrol law an be investigated.

Su ient onditions for dy-

nami feedba k linearization have been given by Fliess [80℄ who introdu ed the dierential rank of a system. The dierential rank measures the degree of independen e of the system outputs and an be onsidered as the nonlinear equivalent of the rank of the transfer matrix for a linear system. When the ondition of nonsingularity is satised by the given system (stati feedba k) or by a suitable extension of the given system (dynami feedba k), the feedba k ontrol law an be omputed by solving a set of state independent 1

O e National d'Etudes et de Re her hes Aerospatiales (ONERA), BA701, 13661 Salon

de Proven e Air

102

algebrai linear equations.

This is a result of the stru ture of the dynami s

whi h is assumed to be ane in the ontrols. As the input-output behaviour of the resulting state-feedba k system resembles that of a linear time-invariant system, any linear ontrol design te hnique

an be applied to a hieve the design performan e. However, in order to guarantee the internal stability of the system, it is not su ient to look at input-output stability, sin e all internal unobservable modes of the system must be stable as well. The rst step in analysing the internal stability of the system is to look at the zero dynami s. The zero dynami s of a nonlinear system are the internal dynami s of the system subje t to the onstraint that the output, and therefore all the derivatives of the output, are set to zero for all time. There have been many appli ations of nonintera ting ontrol and feedba k linearization to air raft ight ontrol problems : Asseo [15℄, Singh and S hy [213℄, Meyer and Ci olani [170℄, Dang Vu and Mer ier [51℄, Menon et al. [168℄, Lane and Stengel [149℄, Bugajski et al.

[41℄, Adams et al.

[6℄....

The main

advantage of the feedba k linearization te hnique is that it does not require gain s heduling to ensure ight ontrol system stability over the entire operational envelope of the air raft.

Traditional air raft ontrol designs have to rely on

linearized models obtained throughout the ight envelope of the vehi le, with linear ontrollers synthesized for the set of resulting linearized models.

9.2 Plant Model Requirements and Controller Stru ture 9.2.1 SISO ase The essentials of the approa h are most easily understood in terms of the simple single-input single-output problem. The method of synthesis onsiders a lass of nonlinear systems ane in

ontrol

x_ = f (x) + g(x)u y = h(x) where

n IR

(9.1) (9.2)

f , g are smooth ve tor elds on IR n and h is a smooth fun tion mapping

! IR .

Su h a system is feedba k linearizable of relative degree

r if there exist state

and input transformations

z = (x) u = (x) + (x)v where

z 2 IR r v 2 IR

(9.3) (9.4)

(x) 6= 0 and  is a dieomorphism whi h transforms (9.1) into a on-

trollable linear system

z_ = Az + Bv 103

(9.5)

Indeed, we time-dierentiate (9.2) to obtain

h (f (x) + g(x)u) x

y_ = If the oe ient of

u

(9.6)

is zero, we dierentiate (9.6) and ontinue in this way

until a nonzero oe ient appears. This pro ess an be su

in tly des ribed by introdu ing some onventional notation of dierential geometry.

h with respe t to the ve tor eld f

derivative of the s alar fun tion as

Lf h(x) =

The Lie is dened

h f (x) x

(9.7)

Higher order derivatives may be su

essively dened

Lkf h(x) = Lf (Lfk 1 h(x))

(9.8)

With this notation, (9.6) an be written

y_ = Lf h(x) + Lg h(x)u If

Lg h(x) = 0, then dierentiate (9.9) to obtain y = L2 h(x) + Lg Lf h(x)u f

If

(9.9)

Lg Lfk 1 h(x) = 0 for k = 1; :::; r

ends with

The number

(9.10)

1, but Lg Lrf 1h(x) 6= 0, then the pro ess

y(r) = Lrf h(x) + Lg Lrf 1h(x)u

(9.11)

r is alled the relative degree of (9.1). z 2 IR r zk = k (x) = Lk 1 h(x) k = 1; :::; r

Now if we dene the oordinates

then we get the linear

f

(9.12)

r-dimensional

ompletely ontrollable and observable,

ompanion form system

0

0 B0 B z_ = B B :  : 0 where

1 0 : : 0

0 1 0 : :

: 0 1 : :

1

0

1

0 0 B0C :C C B C B C 0C C z + B : C v = Az + Bv A 0A 1 1 0

(9.13)

v = Lrf h(x) + Lg Lrf 1 h(x)u

(9.14)

Su h a system is alled a Brunovsky anoni al form. Exa t linearization is a hieved when the relative degree is equal to the system order

(r = n).

The ontrol law is obtained by transforming the above linear system state variables and ontrol into the original oordinates, with ontrol

u = (x) + (x)v 104

(9.15)

where

(x) =

Lrf h(x) Lg Lrf 1h(x)

(x) =

1 Lg Lrf 1 h(x)

The ontrol law v is hosen depending on the ontrol task. y is required to be stabilized around zero, we hoose v as r 1 X v=

k zk+1 k=0

(9.16) For instan e, if

(9.17)

in order to a hieve the design performan e for the output dynami whi h is given by

y(r) + r 1 y(r 1) + ::: + 1 y(1) + 0 y = 0

(9.18)

Stabilization of (9.18) annot guarantee stabilization of (9.1). A omplete

hara terization of the stability properties of (9.1) requires a view of the entire state spa e. The key result of Isidori [124℄ is that there exists a transformation of oordinates whi h provides a so- alled normal form for (9.1), from whi h a

omplete stability pi ture an be obtained

x

! (z; ) z 2 IR r  2 IR n

r

z_ = Az + Bv _ = q(z; )

(9.19)

(9.20) (9.21)

The zero dynami s of the system (9.1) are dened by the equation

_ = q(0; )

(9.22)

whi h orresponds to the internal behaviour of the system when the ontrol is

hosen to onstraint the output to be identi ally null. For tra king ontrol problems, for instan e if

hoose

v as

v = yd(r)

r 1 X k=0

y

is required to tra k

k (zk+1 yd(k) )

yd, we (9.23)

in order to a hieve the design performan e for the tra king error

e = y yd

(9.24)

e(r) + r 1 e(r 1) + ::: + 1 e(1) + 0 e = 0

(9.25)

whose dynami is given by

Again the internal behaviour must be bounded. It an be shown that for any

 > 0, there exists Æ so that je(k) (t0 )j < Æ k = 0; :::; r 1 =) je(k) (t)j < 

8t > t0 > 0

k(t0 ) R (t0 )k < Æ =) k(t) R (t)k <  8t > t0 > 0

where

_R = q(zR ; R ) and zR = (yd ; yd(1) ; :::; yd(r 1))T 105

(9.26) (9.27)

9.2.2 MIMO ase The multi-input multi-output ase is qualitatively similar to the single-input single-output ase. Consider a nonlinear dynami al system in the form

x_ = f (x) + g(x)u y = h(x)

(9.28) (9.29)

x 2 IR n , u 2 IR m , y 2 IR m , and f , g and h are smooth fun tions of x. The problem onsists of nding m transformations of oordinates and a ontrol where

law

z i 2 IR ri v 2 IR m

z i = i (x) u = (x) + (x)v where

ri

i = 1; :::; m

(9.30) (9.31)

is the relative degree asso iated to the output

yi ,

whi h transform

(9.28) into an equivalent ontrollable linear system

z_i = Ai z i + Bi v

i = 1; :::; m

(9.32)

from whi h the auxiliary ontrol synthesis is performed. Under the ondition of nonsingularity of the matrix


ij (x) = Lgj Lfri 1 hi (x)

i = 1; :::; m

j = 1; :::; m

(9.33)

the linearizing oordinates are given by

zki = Lfri 1 hi (x)

i = 1; :::; m k = 1; :::; ri

u is obtained from (x) =
1 b

and the ontrol law

with

bi = Lrfi hi (x)

The ontrol law

(x) =
1 i = 1; :::; m

v is hosen depending on the ontrol task. rX i 1 ( r i) vi = ydi

ik (zki +1 yd(ki ) ) k=0

(9.34)

(9.35)

(9.36) For instan e, if

(9.37)

then we obtain a nonintera ting ontrol system whi h performs a de oupled tra king of

yd

by

y,

omponent by omponent. In this ase, the matrix




is

alled the de oupling matrix. The input-output behaviour is dened by the diagonal transfer matrix

H (s) = diag(

1 ) di (s) 106

i = 1; :::; m

(9.38)

with

(9.39) di (s) = i0 + i1 s + ::: + iri 1 sri 1 + sri The stru ture of a simple ontrol system (ri = 1 i = 1; :::; m) is depi ted in Figure 9.1. As the output y is required to tra k the ommanded value yd , we hoose v as v = !(y yd) (9.40) where

! = diag( i0 )

i = 1; :::; m

(9.41)

The ontrol law is then given by

u= (

h h h g(x)) 1 !(y yd ) ( g(x)) 1 f (x) x x x h

(

-1

g)

x yd +

ω

v

(

h x

−

-1

g)

− +

u

h

(9.42)

f

x x

.

x=f(x)+g(x)u

y

h(x) ω=diag(c0i) i=1,...,m Figure 9.1: Controller stru ture

9.3 Possible Design Obje tives and Design Cy le Des ription A straightforward appli ation of the linearization te hniques might result in a system :



with unstable unobservable modes resulting in undesirable losed-loop system behaviour;



with large ontrol eort leading to ontrol saturation.

Preliminary physi al onsiderations are ne essary to obtain a good design. By negle ting ertain physi ally small variables, the approximate linearization might result in better performan e. Singular perturbation theory an also be

107

used to reformulate the original dynami model as two or more lower-order systems whi h are better onditioned for linearization; a ommon example is the time-s ale separation between the translation dynami s of an air raft and its rotational dynami s. Good zero-order dynami behaviour and redu ed ontrol a tivity rely on a good hoi e of the ontrolled variables and their dynami s (ve tor

v in the design).

The overall approa h for the ontrol design is as follows :



Step 1. Reformulate if ne essary the original dynami system to obtain an approximate nonlinear model for whi h a state-dependent nonlinear inverse an be easily onstru ted.



Step 2. Constru t the nonlinear inverse. The losed-loop system formed by the nonlinear inverse and the approximate nonlinear model redu es to a linear time invariant system.



Step 3. Use any suitable linear ontrol design te hnique to synthesize a

ontroller for the above linear system (e.g. eigenstru ture assignment).



Step 4.

Transform the linear system state variables and linear ontrol

into the original oordinates and ontrol.



Step 5. Iterate on linear dynami hara teristi s to obtain required performan e with redu ed ontrol a tivity.



Step 6. Eventually omplete the design by synthesizing a robust ontroller using adequate linear te hniques.

9.4 A Simple Design Example The following example on erns the ontrol problem of a simplied V/STOL air raft model and is taken from Meyer et al. [171℄. A simple air raft is used that has a minimum number of states and inputs, but retains many of the features that are onsidered when designing ontrol laws for a real air raft su h as the Harrier. Figure 9.2 shows the PVTOL (planar verti al takeo and landing) air raft, whi h is the natural restri tion of V/STOL air raft to jetborne operation (e.g. hover) in a verti al plane. The air raft state is simply the

y, z , of the air raft entre of mass, the angle  of the air raft relative y-axis, and the orresponding velo ities, y_ , z_ , _ . The ontrol inputs, u1 ,

position to the

u2 , are the thrust (positive downward) and the rolling moment.

The equations of motion for the PVTOL air raft are given by

where -1 is the

y = u1 sin  + u2 os  z = u1 os  + u2 sin  1  = u2 gravitational a

eleration and  is a

(9.43) (9.44) (9.45) small oe ient giving

the oupling between the rolling moment and the lateral a

eleration of the air raft.

108

Φ

z

y Figure 9.2: The planar verti al takeo and landing air raft

Choosing

y and z as the outputs to be ontrolled, we seek a state feedba k

law of the form

u = (x) + (x)v

(9.46)

r = (r1 ; r2 )T ,

su h that, for some

y(r1) = v1 z (r2) = v2 Here,

(9.47) (9.48)

v is our new input and x is used to denote the entire state of the system.

Pro eeding in the usual way, we dierentiate ea h output until at least one of the inputs appears. This o

urs after dierentiating twi e and is given by



y z



=



0 1



+



sin   os 

os   sin 



u1 u2

 (9.49)

Sin e the de oupling matrix is nonsingular (although almost singular as

), we an linearize the system by hoosing the stati state

its determinant is feedba k law



u1 u2



=



sin  os 

os sin  



109

 

0 1



+



v1 v2

 (9.50)

The resulting system is

y = v1 z = v2 1  = (sin  + v1 os  + v2 sin ) 

(9.51) (9.52) (9.53)

This feedba k law makes the input-output map linear, but has the unfortunate

 unobservable. Constraining the outputs v1 = v2 = 0, the zero dynami s are found to

side-ee t of making the dynami s of and derivatives to zero by setting be

1  = sin  

(9.54)

Equation (9.54) is simply the equation of an undamped pendulum. Nonlinear systems, su h as (9.51)-(9.53), with zero dynami s that are not asymptoti ally stable are alled non-minimum phase. From the above analysis, it is lear that exa t input-output linearization of a system an lead to undesirable results. The sour e of the problem lies in



trying to ontrol modes of the system using inputs that are weakly ( ) oupled rather than ontrolling the system in the way it was designed to be ontrolled



and a

epting a performan e penalty for the parasiti ( ) ee ts. For the simple PVTOL air raft, the linear a

eleration should be ontrolled by ve toring the thrust ve tor (using moments to ontrol this ve toring) and adjusting its magnitude using the throttle. The PVTOL air raft is now modelled as

ym = u1 sin  zm = u1 os  1  = u2

(9.55) (9.56) (9.57)

so that there is no oupling between rolling moments and lateral a

eleration. Dierentiating the model system outputs,



ym zm



=



0 1



+



ym and zm , we get   u1 sin  0

os  0 u2

(9.58)

Now, however, the de oupling matrix is singular whi h implies there is no

u2 enters the system  , we must dierentiate (9.55)-(9.56) at least two more times        sin  u1 os  u1 ym (4) = u1_ 2 sin  2u_ 1_ os  +

os  u1 sin  zm (4) u2 u1 _ 2 os  2u_ 1_ sin 

stati state feedba k that will linearize (9.55)-(9.57). Sin e through

The de oupling matrix is invertible as long as the thrust,

u1

(9.59)

is nonzero.

Physi ally, this result in agreement with the fa t that no amount of rolling will ae t the motion of the air raft if there is no thrust to ee t an a

eleration. Linearizing the above system using the dynami state feedba k law



u1 u2



=

u1 _ 2 2u_ 1 _ u1

!

+



110

sin  os 

os u1

sin  u1



v1 v2

 (9.60)

results in

ym(4) = v1 zm(4) = v2

(9.61) (9.62)

Unlike the previous ase, the linearized model does not ontain any unobservable zero dynami s. Thus, using a stable tra king law for

v, we an tra k

an arbitrary traje tory and guarantee that the model will be stable. Of ourse, the natural question that omes to mind is : will a ontrol law based on the model work well when applied to the true system? If

 is small

enough, then the system will have reasonable properties, su h as stability and bounded tra king. This example shows that preliminary physi al onsiderations are ne essary to obtain a good design. By negle ting ertain variables whi h are physi ally small, the approximate linearization results in better performan e.

9.5 Con lusion Feedba k linearization or dynami inversion has drawn onsiderable attention over the last two de ades and oers a potentially powerful alternative ontrol design methodology. Dynami inversion is an attra tive te hnique as it avoids gain s hedules.

Instead, it uses dynami models and full-state feedba k to

globally linearize dynami s of sele ted ontrolled variables. Simple ontrollers

an then be designed to regulate these variables with desirable losed loop dynami s. Theory of feedba k linearization is still gradually developing. There are limitations and open problems. The main drawba k might be that modes be ome unobservable under the linearization or de oupling onstraints, whi h an be unsurmountable in ase they are unstable. The dimension of the unobservable manifold and the omplexity of the ontrol law an vary drasti ally a

ording to the assumptions made on the model used (e.g. small oupling terms negle ted or not).

Preliminary physi al onsiderations are then ne essary to obtain a

good design. The design method requires, more or less, a

urate knowledge of the state of the system, while no satisfa tory theory for the design of the nonlinear observers is available. A suitable nonlinear analogue of the separation prin iple still needs to be developed. One area of resear h, already initiated, is that of ombining the design te hnique developed so far, with appropriate robust te hniques whi h ould take into a

ount unknown parameters and unmodelled dynami s : LQ,

-synthesis,

ontrol, dierential games.

111

QFT, Lyapunov synthesis, adaptive

10. Robust Inverse Dynami s Estimation

Ewan Muir

1

10.1 Introdu tion Robust Inverse Dynami s Estimation (RIDE) [48℄, [33℄,[176℄ has developed from two other methods: the Salford Singular Perturbation Method [115℄ and Pseudo-Derivative Feedba k [194℄, [14℄. Both of these methods use the same type of multivariable Proportional plus Integral (PI) ontroller stru ture but use a high gain to provide the desired de oupling and losed-loop dynami s. RIDE is a development of both these methods whi h repla es the high gain with an estimate of the inverse dynami s of the air raft with respe t to the

ontrolled outputs. This inverse input gives RIDE strong similarities to Nonlinear Dynami Inversion [218℄ and is similar to the equivalent ontrol found in Variable Stru ture Control [244℄.

10.2 General Stru ture The RIDE ontroller onsists of 3 omponents: a model inverse input, a PI

ontroller and a feedforward, as shown in Fig. 10.1 below.

v _ T-1 +

yc

. v

Kudi

KV

^udi +

+ _

KI

+

. r

+

r

+

_

+

x u

Aircraft

y

KP

Figure 10.1: Stru ture of RIDE ontrol law 1

Defen e Resear h Agen y, Flight Dynami s and Simulation Department, Bedford, MK41

6AE, UK

112

- The model inverse provided by the dynami inverse input,

u^di

, a ts to

de ouple the outputs from ea h other and from the other air raft states by using moment an ellation. The inverse is uniquely for the outputs to be ontrolled and is therefore for a subset of the omplete air raft model only. - Having inverted the air raft model with respe t to the outputs and de oupled these, the PI ontroller then assigns to the outputs, the dynami s desired by the ontrol law designer. The integral a tion, with gain

KI , provides robust-

ness against errors in the estimate of the model inverse. The proportional gain matrix

KP

provides stability and is positioned su h that it provides pseudo-

derivative feedba k. - The feedforward omponent, onsisting of a washout lter on the demands and des ribed by equation 10.1, is used to tune the step response hara teristi s to give an appropriate onset of response. feedforward input = where

T

[(T s + I ) 1 KV s℄y

is a diagonal matrix of washout time onstants and

(10.1)

KV

is a matrix of

gains on the feedforward inputs. The stru ture provides de oupling between the outputs and assigns them a se ond order response whi h is spe ied by the designer. The transient response shape an be adjusted using the feedforward. The design method provides simple me hanisms for handling motivator position and rate limiting and it is anti ipated that motivator redundan y an be handled separately from the ontroller design.

10.3 Closed-loop System Chara teristi s and Gain Cal ulation (Output Feedba k Case) The poles of the losed-loop system with a RIDE ontrol law are determined by the following: - the open loop system transmission zeros, - the feedforward washout time onstants given in matrix

2I

- the eigenvalues of the matrix [s

+

CBKP s + CBKI ℄.

T;

As the rst set of poles oin ides with the transmission zeros of the open loop system, it is ne essary to ensure that the feedba k measurements sele ted give stable transmission zero lo ations. The se ond set is spe ied by the designer who sele ts the feedforward time onstants. The third set of eigenvalues an be assigned to the poles of a standard se ond order transfer fun tion of the form given in equation 10.2, through appropriate gain sele tion.

y = (s2 I + 2Zd n s + 2n ) 1 2n y

The proportional, integral and feedforward gain matri es,

(10.2)

KP , KI

and

KV

respe tively, are al ulated from the inverse of the motivator ee tiveness ma-

CB ) 1 , the matrix of desired losed system damping, Zd, natural frequen y, n , and feedforward gain, M , where Zd , n and M are diagonal matrix, (

tri es spe ied by the ontrol law designer.

113

KP = (CB ) 1 2Zd n

(10.3)

KI = (CB ) 1 2n

(10.4)

KV = (CB ) 1 M

(10.5)

For the output feedba k ase and using the gains al ulated in equations 10.3 to 10.5 above, the output equation for the losed-loop system is des ribed by equation 10.6.

y = (s2 I +2Zd n s + 2n) 1 [ 2n y + sM (T s + I ) 1y + s(CB )(^udi udi)℄ As

Zd, n , T

and

(10.6)

M are all diagonal matri es, ea h demand y will ae t y. Therefore, for the output feedba k ase, RIDE will

only one of the outputs

provide tra king of the demands with unity steady-state gain, the dynami s of the response an be spe ied and will be se ond order.

omponent, whose dynami s are spe ied by the matri es

T

The feedforward and

M , will shape

the initial response to any inputs. Any errors in the estimate of the dynami inverse input will be orre ted by the integral loop. happens will be dependent on the integral gain The role of the dynami inverse input, estimate,

u^di , is al ulated from

udi ,

KI .

The rate at whi h this

is to keep

y_ = 0

and thus an

u^di = (CB ) 1 CAx (10.7) 1 where the matrix ( (CB ) CA) is represented by the gain matrix Kudi in Fig. 1. Note that the state ve tor x need only ontain the rigid body states whi h dire tly ae t the outputs y .

10.4 Design Limitations RIDE does not take into a

ount expli itly any motivator or sensor dynami s during the design phase.

In many ases, the dynami s of the a tuators and

sensors will be su iently fast to maintain the desired performan e. Should this not be the ase, a areful hoi e of the design parameters will re tify the problem.

For example, the spe ied bandwidth of the losed-loop response

ould be redu ed and the feedforward used to maintain an adequate speed of response.

10.5 Controller Synthesis Aspe ts The simpli ity of RIDE naturally means that it does not provide the omprehensive solution promised by other more omplex methods.

114

RIDE does not

provide expli it guarantees in terms of either stability or performan e robustness. It is also limited in terms of the amount of spe i ation data whi h an be in orporated dire tly in the design stage.

Therefore separate analysis is

required on e the initial design has been done, to see if the ontroller meets the spe i ation. However, in pra ti e, RIDE has been found to produ e ontrollers with a

eptable time responses, even when performing highly dynami manoeuvres with non-linear air raft models [176℄, and it is possible for designers to a hieve satisfa tory gain and phase margins.

Also the integral a tion

provides robustness to errors in the dynami inverse input estimate. The simpli ity of RIDE, both in terms of the underlying mathemati s and the design pro ess, means that the learning urve is short and undemanding. Also, no spe ialist skills, design software or omputer hardware are required and the resulting ontroller is simple with a lear stru ture. A fuller understanding of the ontroller synthesis pro edure and of the design method an be obtained from the RIDE-HIRM ontrol law in hapter 33.

115

11.

A Model Following Control

Approa h

Holger Duda1 , Gerhard Bouwer1 , J.-Mi hael Baus hat1 and Klaus-Uwe Hahn 1

11.1 Introdu tion Design and development aspe ts for state of the art ontrol systems are based today on improved system models and omputer supported tools. One of the design aims for a ontrol system is a low feedba k authority.

High feedba k

gains, espe ially in multi input/multi output systems with un ertainties, may lead to stability problems, whi h are often di ult to predi t. A well-known re ipe to avoid this is:

Put all available information about the pro ess to be ontrolled into the feedforward bran h of your ontrol system. In view of ight ontrol system design it an be assumed that a detailed database of the air raft to be ontrolled is available, be ause it is usual to improve aerodynami databases of new air raft during ight testing using modern system identi ation te hniques [109℄. Therefore, it is highly re ommended to use this knowledge in the ight ontrol system design pro ess [108℄. One of the most promising approa hes, whi h takes the above mentioned aspe ts into a

ount, is the Model Following Control (MFC) te hnique. Even in the early stages of ight ontrol system resear h MFC on epts seemed to be promising [243℄. An improved theoreti al understanding of the identi ation of system dynami s promotes the appli ation of MFC systems [107℄. The design problem for the MFC on ept an be separated into three independent subtasks: First a ommand blo k has to be dened, whi h in ludes the desired dynami behaviour taking into a

ount the plant's performan e limits. Then a ontroller omplex onsisting of a feedforward and a feedba k ontroller has to be designed, whi h is independent of any ommand blo k hara teristi s, su h as manual ight ontrol laws or autopilot fun tions. The expression  ommand blo k has been introdu ed instead of model in order to avoid misunderstandings on erning the plant model. It predominantly denes the dynami behaviour of the overall MFC system.

The feedforward

ontroller ontains an inverse model of the plant to be ontrolled. Assuming no external disturban es, a perfe t knowledge and an ideal inversion of the plant, 1

DLR German Aerospa e Resear h Establishment, Institute of Flight Me hani s, D-38108

Brauns hweig

116

the omplete ontrol ould be performed by the feedforward ontroller without any feedba k a tivity.

In pra ti e, a feedba k ontroller is required, whi h

only has to manage the remaining ontrol part not overed by the feedforward bran h, whi h will always perform the majority of ontrol a tivity. Sin e the aim of the RCAM Design Challenge is to evaluate ontrol theories

on erning robust ight ontrol system design, one has to answer the rising question:

How an the MFC on ept ontaining high authority feedforward

ontrol enhan e robustness, whi h is mainly ae ted by feedba k

ontrol? The feedforward part of the MFC represents a kind of partly inherent robustness ompared with a pure feedba k system.

By an exa t denition of the

desired performan e and the limitations of the pro ess one omes to oni tfree ontrol a tions and, therefore, to minimum feedba k ontrol a tivity for manoeuvres. This leaves maximum authority to the feedba k ontroller to ope with un ertainties and disturban es reje tion. Besides these robustness aspe ts the MFC on ept provides several additional benets regarding pra ti al appli ations:



The ommand blo k denes predominantly the input ommand behaviour of the overall MFC system.

Therefore, tailored Flying Qualities or au-

topilot fun tions an be easily realised.



The feedforward and feedba k ontrollers are independent from the layout of the ommand blo k.

This separation allows a lear sharing of

responsibilities for the design tasks with well dened interfa es, whi h

an be performed by dierent teams. Therefore, design problems whi h may be observed during simulator or ight testing an be easily lo ated and solved.



The overall ontroller stru ture allows the denition of ertain ommand blo k modules for spe ial tasks, su h as manual ight ontrol laws or autopilot fun tions for an air raft family. A re-design for other (similar) air raft does not have to go through all the individual steps, but only the feedforward and feedba k ontrollers have to be adapted. The attainable

ommonality of ying hara teristi s for an entire ategory of air raft type is a protable element onsidering pilot training and erti ation aspe ts.

11.2 Typi al Appli ations MFC on epts have been utilised in ight systems sin e the sixties [243℄. The main appli ation of the MFC approa h is in-ight simulation.

The aim of

in-ight simulation is to impose the hara teristi s of a ight vehi le to be simulated on airborne simulators, su h as Calspan's TIFS (Total In-Flight Simu-

lator) [175℄, DLR's ATTAS (Advan ed Te hnologies Testing Air raft System)

117

and ATTHeS (Advan ed Te hnologies Testing Heli opter System) [40℄.

Fur-

thermore, MFC on epts have been realised in several experimental heli opter programs in the United States and were even hosen for the new operational Fly-by-Wire heli opter Coman he [93℄. The appli ation potential of MFC systems is demonstrated below by re ent resear h programs arried out at DLR Institute of Flight Me hani s.

They

have been hosen be ause all have been ight-demonstrated in a real-time and real-world environment.

11.2.1 In-ight simulation Various in-ight simulations have been arried out in the xed-wing and heli opter area, su h as the Airbus A3XX air raft, the Indonesian N250 air raft, the HERMES Spa eplane, and the Lynx heli opter. One of the most re ent appli ations of ATTAS has been the airborne simulation of an Airbus A3XX-type transport air raft.

The airframe model is

based on preliminary data of the unaugmented air raft without elasti modes. A typi al ight-test result is illustrated in gure 11.1.

Manually own turn

reversals learly show the ex itation of the low damped dut h roll of the implemented model. The time histories demonstrate a good mat hing between the A3XX model states (solid lines) and the measured ATTAS states (dashed lines) validating the model following me hanism.

Roll Command of the Pilot (deg) 10 0 -10 Bank Angle (deg) 20 0 -30 Angle of Sideslip (deg) 5 0

A3XX ATTAS

-6 0

100

200

300

Time (sec)

400

Figure 11.1: A3XX In-Flight Simulation (Flight-Test Results) Espe ially in the heli opter area the MFC on ept has been proven to be a very valuable tool due to the highly ompli ated ouplings of basi heli opter dynami s [104℄. The in-ight simulation of the Lynx heli opter shall serve as an example [31℄. This heli opter has ouplings opposite to the orresponding

ouplings of ATTHeS in its basi BO 105 mode. Flight tests have been arried out demonstrating, that all ATTHeS states mat h well the ommanded Lynx

118

model states. In general, the in-ight simulation was deemed by the pilots to be representative for the Lynx heli opter.

11.2.2 Flight ontrol system resear h Flight ontrol system resear h proje ts based on MFC on epts have been performed, whi h are summarised below:



LADICO (Lateral/Dire tional Control of an Air raft): This proje t was arried out within the framework of the GARTEUR A tion Group FM (AG 06) Low-Speed Lateral/Dire tional Handling Quali-

ties Design Guidelines. A lateral/dire tional ontrol system for a transport air raft was developed, featuring an MFC on ept [38℄.

Piloted

evaluations of the system in two ground based simulators demonstrated its performan e; all evaluation pilots rated the system as Level 1.



ARCORE (Arti ial Redundan y Con ept for Re onguration): Flight ontrol system re onguration on epts have been developed and ight tested [21℄. The investigated failure was an elevator stu k in the trim position, whi h has been ompensated by the re onguration ontroller using the stabiliser with its poor dynami s for altitude ontrol instead of the stu k elevator.



SCARLET (Saturated Command and Rate Limited Elevator time delay): Air raft-Pilot Coupling (A-PC) problems due to rate saturation have been investigated and ight tested.

An alternative ontrol s heme has

been developed, whi h ompensates for the additional time delay due to rate saturation. The ight test results were very promising [39℄.



ADS-33D riteria (Aeronauti al Design Standard): Flying Qualities Databases for modern Fly-by-Wire heli opters have been developed on ATTHeS [31℄.

11.3 Plant Model Requirements The MFC approa h is well tuned for the usually available pro ess knowledge in the ight ontrol area. Ideally, there should exist a nonlinear pro ess model within the whole ight envelope in luding a tuator systems, sensor systems, engine, time delays, elasti ity, et .:

x_ (t) = f (x(t); u(t)):

(11.1)

But the method also works, if there is only a linear state model of the rigid air raft for one referen e point

x_ (t) = A x(t) + B u(t); as it has been demonstrated in ight tests (se tion 11.2).

119

(11.2)

11.4 Controller Stru ture Ea h MFC system ontains the main three elements ommand blo k, feedfor-

ward and feedba k ontrollers, gure 11.2. The ommand blo k ontains the equations to ompute a sele ted state ve tor

x

and its time derivative

x_

depending on the input signals.

The feedforward ontroller omputes the ontrol inputs whi h are required for model following. It in ludes an inverse model of the plant. The use of the state derivative

x_

together with

x

generated by the ommand blo k (whi h

ontains dynami s) allows the use of pure stati gain matri es in the feedforward

ontroller [35℄. In pra ti e, no perfe t inversion of the plant to be ontrolled an be provided, therefore, the feedba k ontroller has to ope with these un ertainties and disturban e reje tion.

Disturbances . xC Command Block

Feedforward Controller

uFF

Plant to be controlled

+ uFB

xC

Feedback Controller

x

Figure 11.2: General layout of a MFC system

11.5 Possible Design Obje tives The main design obje tive for the omplete MFC system is to full the design requirements.

As already stated, the main three elements ommand blo k,

feedforward and feedba k ontrollers an be designed separately. The design obje tives for these subtasks an be summarised as follows: The ommand blo k must ontain the desired dynami behaviour of the overall MFC system regarding ontrol inputs. Nonlinear ee ts like a tuator rate and dee tion limitations have to be taken into a

ount within the ommand blo k. The feedforward ontroller has to provide an optimum inversion of the plant to be ontrolled. The feedba k ontroller has to ensure rapid and smoothly de aying error dynami s in the presen e of unknown external disturban es and model un ertainties in order to maintain the quality of model following. Nonlinear ee ts (a tuator rate and dee tion limitations) have to be onsidered for its design.

120

11.6 Design Cy le Des ription The design y le for the MFC approa h is separated into the subtasks for the

ommand blo k, the feedforward, and the feedba k ontrollers.

11.6.1 Command blo k A pra ti al and simple way to dene the ommand blo k is to use models of proven systems, su h as an air raft model with Level 1 manual ight ontrol laws (Fly-by-Wire) or optimum autopilot fun tions.

It is obvious that any

ommand hara teristi s implemented in the ommand blo k are limited by the dynami s of the plant to be ontrolled, mainly be ause of the nonlinear

onstraints of the a tuators. The main onstraint to be onsidered is that the dynami s of the air raft model implemented in the ommand blo k are not faster than those of the plant.

11.6.2 Feedforward ontroller For the design of the feedforward ontroller an inverse model of the plant is required. Assuming that the plant model an be represented by a linear state spa e system (equation (11.2)) the following feedforward ontrol law an be applied [35℄:

uF F (t) = B 1 [x_ C (t) A xC (t)℄ :

(11.3)

This equation indi ates that the inversion does not in lude dynami elements (whi h means zero order) if the state derivative

x_ together with x generated by

the ommand blo k are available. For this pro edure the ontrol input matrix of the plant

B

has to be inverted.

This leads to the fundamental problem,

that dierential equations des ribing typi al dynami systems to be ontrolled (air raft, heli opters, industrial robots, et .) often annot be inverted. In most of these ases the number of ontrol inputs is smaller than the number of states, therefore,

B is a non-square matrix.

One approa h to handle this problem is the appli ation of the Pseudo-

Inverse





By = BT Q B BT Q ; whi h strongly depends on the weighting matrix Q.

(11.4) Therefore, an alternative

method is used at DLR Institute of Flight Me hani s:

x is of the order n and the input ve tor u m (with n > m) equation (11.2) an be written as:

Assuming that the state ve tor is of the order









be sele ted properly providing that In this ase

B1







x1 + B1 u (11.5) B2 x2 has the order m. Its elements should the subsystem of x1 is fully ontrollable.

x_ 1 = A11 A12 A21 A22 x_ 2 The state ve tor to be ontrolled x1

is a square matrix and invertible.

121

In order to de ouple

x1 from x2 , the feedforward ontrol law equation (11.3) x2  x2C :

is extended to a de oupling term, with

uF F = B1 1 (x_ 1C |

{z

A11 x1C ) B1 1 A12 x2C : } | {z }

Inversion

(11.6)

De oupling

Dening the ontrol matrixes

M1 = B1 1 ; M2 = M1 A11 ; M3 = M1 A12 ;

(11.7)

equation (11.6) an be simplied to:

uF F = M1 x_ 1C + M2 x1C + M3 x2C :

(11.8)

This pure linear approa h an be extended to nonlinear elements, if required [22℄.

11.6.3 Feedba k Controller The feedba k ontroller has to ompensate for model un ertainties and disturban e reje tion, while the feedforward ontroller performs the majority of the ontrol a tivity. The main requirement for the feedba k ontroller an be dened by:

e(t) = xC (t) x(t)

!

Min.

:

(11.9)

Dierent methods an be utilised to design the feedba k ontroller, su h as ve tor performan e optimisation [113℄ or robust ontrol system design methods. It has been shown that for air raft appli ations a feedba k ontroller using all signi ant states with proportional and integral terms is su ient. Its stru ture is illustrated in gure 11.3.

uFB + 1/s

Command Block states xC

KP

KI

Plant states x

e -

Figure 11.3: Stru ture of the feedba k ontroller The feedba k ontroller is dened by:

uF B (t) = KP e(t) + KI 122

Z

e(t)dt:

(11.10)

The gains of the feedba k ontroller an be optimised independently from the layout of the feedforward ontroller and the ommand blo k. A proven pro edure used at DLR is based on a numeri al optimisation algorithm [126℄. A ve tor ost fun tion allows the formulation of ea h design obje tive separately and its ombination with individual weighting fa tors forman e and

l

for the ontroller a tivity.

k for the ontroller per-

The formal stru ture of the ost

fun tion is given by:

n X

m

Z te

Z t

e X u2l (t) dt + ::: e2k (t) dt + l J = k 0 0 l=1 k=1

(11.11)

This ost fun tion has to be tailored to the a tual design problem. For air raft appli ations mostly a number of about ten gains to be optimised is su ient. However, for highly elasti air raft this number may in rease.

11.7 A Simple Design Example In order to demonstrate the design pro ess a very simple example is dened, gure 11.4: The plant represents a simplied air raft model ontaining only the short period mode, whi h is represented by the following linear model with

  0:2):

a very poor damping ratio (



u





q_ = 0:24 w_ 80:6 . qc Command qc block

0:016 0:67



M1 M2



q + w

+ δt +





2:4 Æ : 6:5 t

(11.12)

q Plant

w

M3 Figure 11.4: Blo k diagram of the design example The ommand blo k in luding the model to be followed is dened as a rst order system, whi h provides a ommand for the pit h rate

q :

q_ = q u: (11.13) For the MFC design the matri es M1 to M3 have to be determined a

ording to equation (11.7). In this ase x1 = q and x2 = w is sele ted. In order to verify this design, the Bode plot from the ommand blo k output

q_C to the plant output q is al ulated, gure 11.5a.

The pure integral behaviour

demonstrates that the inversion works orre tly. The step responses larify the poor damping of the unaugmented plant and the realised rst order behaviour of the MFC system, gure 11.5b.

123

In this linear example, any desired dynami behaviour an be implemented in the ommand blo k, su h as an air raft model with Level 1 Flying Qualities. Under real onditions the nonlinear ee ts of the plant, su h as rate and dee tion limits should additionally be onsidered in the ommand blo k.

. a) Bode Plot of frequency response q/qc Amplitude (dB)

20 0 -20

Phase (deg)

-90

1

0.1

Frequency (rad/sec)

10

b) Step Responses 0 MFC system response q/u

-1 -2 pure aircraft system response q/δ t

-3 0

5

10

15 Time (sec) 20

Figure 11.5: Results in the frequen y and time domain of the design example

124

12.

Predi tive Control

Jan M. Ma iejowski1 and Mihai Huzmezan

1

12.1 Introdu tion Predi tive Control is very dierent from other ontrol te hniques, in that in its most powerful form it requires the on-line solution of a onstrained optimization problem. This makes it an unlikely andidate for ight ontrol. On the other hand, it oers some unique benets, sin e it expli itly allows for hard onstraints, and it an anti ipate pilot ommands if the ight traje tory is known in advan e. This makes it interesting for ight ontrol, parti ularly if higher-level ontrol fun tionality is onsidered. In this tutorial hapter we present the models used by predi tive ontrol, the ontrol problem formulation, dis uss solutions te hniques and ontroller properties, and omment on the problem of tuning the predi tive ontrol problem formulation so as to meet design spe i ations.

12.2 General Chara teristi s Predi tive Control, also known by several other names, su h as Model-Based Predi tive Control (MBPC), Re eding Horizon Control (RHC), Generalised Predi tive Control (GPC), Dynami Matrix Control (DMC), Sequential OpenLoop Optimizing ontrol (SOLO) et , is distinguished from other ontrol methodologies by the following three key ideas:



An expli it `internal model' is used to obtain predi tions of system behaviour over some future time interval, assuming some traje tory of ontrol variables.



The ontrol variable traje tory is hosen by optimizing some aspe t of system behaviour over this interval.



Only an initial segment of the optimized ontrol traje tory is implemented; the whole y le of predi tion and optimization is repeated, typi ally over an interval of the same length. The ne essary omputations are performed on-line.

The optimization problem solved an in lude onstraints, whi h an be used to represent equipment limits su h as slew rates and limited authority

ontrol surfa es, and operating/safety limits su h as limits on roll angle, des ent 1

Cambridge University Engineering Dept, Cambridge CB2 1PZ, England

125

rate, et . Predi tive ontrol has hitherto been applied mostly in the pro ess industries, where the expli it spe i ation of onstraints allows operation loser to onstraints than standard ontrollers would permit, and hen e operation at more protable onditions. The drawba k of this approa h for ight ontrol is of ourse the on-line omputational requirement. But this is a temporary problem, whi h will disappear within a few years as omputing te hnology advan es. If the internal model is linear, the onstraints are linear inequalities, and the performan e riterion being optimized is quadrati , then the optimization problem to be solved online is a onvex quadrati program, whi h is a relatively good situation. (See below for more details.) Most a ademi publi ations on predi tive ontrol deal with un onstrained problems. See [174, 27, 222℄ for some good examples. The usual formulations then be ome losely related to, or even variants of, the LQ te hnique treated in

hapter 5. In this ase ontrollers an often be pre omputed o-line, but mu h of the advantage of the predi tive ontrol formulation is lost. In this hapter we assume that onstraints are an essential part of the problem. The problem with onstraints is treated in some detail in [197℄.

12.3 System Models All ontrol methodologies assume some model of the system being ontrolled. An unusual feature of predi tive ontrol is that an expli it internal model is required as part of the ontroller; this internal model may be of the same kind as the assumed behaviour of the real system, but need not be. For the purposes of analysing behaviour of the omplete ontrolled system it is usual to assume the same kind of model, though not ne essarily with the same parameters. (So the situation is similar to that in Robust Control theory, for instan e  see

hapters 7,8.) The internal model is required in order to generate predi tions of future system behaviour, if a parti ular set of future ontrol a tions is assumed. For this purpose a nonlinear model ould be used, and there have been some studies of using neural network and other nonlinear models with predi tive ontrol. Su h models lead to non- onvex optimization problems, however, and to ontrol s hemes for whi h no analysis is possible. They have not yet been proven to be useful or ne essary in pra ti e [198℄. We will therefore assume that the internal model is linear. Mu h of the predi tive ontrol literature assumes that a linear time-invariant model is available in the form of a (multivariable) step or impulse response, and that predi tions are generated by onvolution: suppose that the multivariable

fgi : i = 0; 1; : : :g, that the ( ontrol) input u(k) and that the (to be ontrolled) output ve tor at time k is y(k). Also let
u(k) = u(k) u(k 1) be the hange in the input at time

step response sequen e is given by ve tor at time

k

is

126

k.

Then the output is given by

k X

y(k) =

i=

1

gk i
u(i) + d(k)

(12.1)

where it has to be assumed that the open-loop system is asymptoti ally stable for this to be valid, and

d(k)

is assumed to be a disturban e a ting on the

output. In this ase predi tions of the output are omputed by

y^(k + j ) = where

N

k+j X i=k+j N

gk+j i
u(i) + d^(k + j )

is a relatively large integer, and

d^(k)

is some estimate of

(12.2)

d(k + j ).

Frequently the disturban e is estimated as

d^(k) = y(k) y^(k)

(12.3)

and it is assumed that future disturban es are the same as the urrent one:

d^(k + j ) = d^(k):

(12.4)

The onvolution model is an ine ient one, sin e the same model an be represented mu h more ompa tly in either transfer fun tion or state-spa e form. Furthermore, representing the system by a model of this kind removes the restri tion to stable models.

The Generalised Predi tive Control (GPC)

form of predi tive ontrol uses a multivariable transfer fun tion form of model:

A(z 1 )y(k) = B (z 1 )u(k) + n(k): z 1

(12.5)

zA(z 1 ), B (z 1 ) are matri es of polynomials in this 1 )℄ 1 B (z 1 ) is the transfer fun tion matrix from the operator  so that [A(z input ve tor u to the output ve tor y . Although it is not ne essary to asso iate in whi h

is the one-step time delay operator (or the inverse of the

transform variable), and

ea h kind of system model with a spe i disturban e model, it is ommonly assumed [47℄ that the disturban e

n(k) in

this model is generated by passing

white noise through a lter whi h in ludes an integrator:

n(k) =

C (z 1 ) e(k) z 1

(12.6)

Inserting an integrator here leads to integral a tion in the ontroller, whi h is also obtained with the use of onvolution models by the assumption of onstant future disturban es. Generating a set of predi tions now involves solving a set of matrix Diophantine equations, but reasonable approximations an be obtained by using simpler pro edures [47℄.

127

The linear model an also be represented in state-spa e form:

x(k + 1) = Ax(k) + Bu(k) + w(k) y(k) = Cx(x) + Du(k) + v(k) where

x(k) is the state ve tor and w(k), v(k) are disturban es.

(12.7) (12.8) For ight on-

trol this model is usually the most appropriate, sin e linearised air raft models are available in this form, with the state variables representing physi ally meaningful quantities. If the disturban es are assumed to be sto hasti then predi tions of the states and outputs an be obtained by using a Kalman lter [151℄. If other assumptions are made then some other observer needs to be used to generate predi tions. To represent sto hasti disturban es with parti ular spe tra, the state of the model has to be augmented by arti ial states in order to use a Kalman lter, in just the same way as is done for LQG design [159℄. Integral a tion in the ontroller an be obtained by in luding integrators in the augmented model.

12.4 Problem Formulations Predi tive ontrol works by hoosing ontrol a tions to minimise some ost fun tion, su h as

J (k) =

N2 X i=N1

jjM x^(k + ijk) r(k + i)jj2Q

+

Nu X i=1

jj
u(k + i)jj2R

(12.9)

subje t to onstraints su h as

j
uj (k + i)j  Vj juj (k + i)j  Uj j(M x^)j (k + ijk)j  Xj where

x^(k + ijk)

is the predi tion of

matrix (for example,

M =C

x(k + i)

(12.11) (12.12)

k, M is some J (k)), and r(k) integers N1 , N2 and

made at time

if only outputs are to appear in

is some referen e (desired) traje tory for

Nu ,

(12.10)

as well as the weighting matri es

Q

Mx(k). and R,

The

are in prin iple hosen to

represent some real performan e obje tives (su h as prot maximisation in a pro ess appli ation [197℄), but in pra ti e they are tuning parameters for the

ontroller. It is assumed that the ontrol signals are onstant after the end of the optimisation horizon, namely that


u(k + i) = 0 for i > Nu .

The inequalities an be used to represent a tuator rate limits (12.10), a tuator authority limits (12.11), and operating/safety limits (12.12). In these inequalities

Uj , Xj

uj (k) denotes the j 'th omponent of the ve tor u(k), et , and Vj ,

are problem-dependent positive values.

The referen e traje tory

r(k) an either be the real pilot ommand ve tor

(set-point), or an be generated by passing the pilot ommand through some lter. In the latter ase the lter design is another tuning parameter. One of

128

the strengths of predi tive ontrol is that if future ommands are known  for example before the start of a turn or other manoeuvre  then these an be anti ipated by the ontroller, leading to smoother manoeuvres, fuel savings, et .


u(k) in the ontrol signals, u(k) themselves, sin e the required steady-state values of u(k ) are not known in advan e. Penalising non-zero u(k ) would `drag' The ost fun tion penalises non-zero hanges

rather than the ontrol signals

the ontrol signals away from the required steady-state values, thus preventing integral a tion, for instan e. The situation is summarised graphi ally in gure 12.1.

PAST

SET POINT

FUTURE

REFERENCE

PREDICTED OUTPUT

r(k+l)

y(k)=r(k) u(k+l) MANIPULATED INPUT

k-n

k-2 k-1 k k+1

k+l

CONSTANT INPUT

Nu

N1

N2

CONTROL HORIZON - Nu MINIMUM OUTPUT HORIZON - N1 MAXIMUM OUTPUT HORIZON - N2

Figure 12.1: Re eding Horizon Strategy

As was said earlier, ombining a quadrati ost su h as

J (k)

with linear

inequalities and a linear model leads to a Quadrati Programming (QP) prob-

 Let the solution of this problem be fu (k + i) : i = 1; : : : ; Nu g. Then  u (k + 1) is applied to the system being ontrolled, and the problem is solved again at time k + 1. (In general one an apply a longer initial segment, and lem.

re-solve the problem at longer intervals.) Other non-quadrati osts are also possible. For instan e, min-max osts are sometimes advo ated in order to obtain robust ontrol, while using absolute values or peak values instead of quadrati fun tions allows the use of Linear Programming, whi h redu es the on-line solution time [197, 8℄.

129

12.5 Solution Te hniques The basi solution te hnique for the onstrained predi tive ontrol problem is to use a standard QP solver (or LP solver if the ost fun tion is appropriate). It is important to appre iate that a solution of a QP problem is required online, and that this problem has to be solved in real time. In pro ess ontrol, where update rates are very low, this is not a big limitation with urrent omputing te hnology. (For example, large multivariable problems with a few tens of inputs, outputs and onstraints, take a few se onds to solve on 486-type

omputers.) But it learly is a problem for ight ontrol, for whi h a speed-up of something like

103 is required.

There are several possible alternatives to the use of standard QP solvers, whi h do not seem to have been investigated thoroughly for predi tive ontrol. The rst is obtained by noting that if there are no onstraints, or if none of the onstraints is a tive, then the solution an be obtained analyti ally, as the solution of a linear equation. (For details, see any of the referen es mentioned previously.)

The problem is that one does not know, before omputing the

solution, whether any onstraints are going to be a tive or not. Se ondly, it is also true that, if one knew exa tly whi h onstraints were a tive, then one

ould again obtain the solution analyti ally. So if one knows that the set of a tive onstraints at time

k is the same as that at time k 1, then one an nd

the solution very qui kly. Furthermore, it will often be the ase that the set of a tive onstraints an only hange in very limited ways from one step to the next; it is then feasible to obtain a small set of analyti solutions qui kly, and

he k whi h one is the a tual solution.

These approa hes exploit knowledge

and understanding of the parti ular optimisation problem being solved  ie ight ontrol of a parti ular air raft  whi h a general QP solver annot do. Another possibility, again not open to a general solver, is to guess that the solution at time

k

will be very similar to that at time

k

1,

and hen e

use that as an initial trial solution. This strategy has been employed in [119℄, for example.

Of ourse su h a strategy will o

asionally go wrong, when a

onstraint is approa hed losely, and a ba k-up pro edure is required for su h o

asions. One problem with standard QP solvers is that they give up if the optimisation problem being solved is infeasible, a situation whi h should not o

ur with proper spe i ation, but nevertheless might.

(Typi ally infeasibilities o

ur

`inside' the algorithm only, and are due to apparently unavoidable onstraint violations some time in the future; the feedba k a tion of the ontroller usually restores feasibility before the problem is en ountered by the system.) In [137℄ the use of Lawson's weighted least-squares algorithm is advo ated, in whi h the weight is iteratively adjusted to emphasise the most-violated onstraint. This algorithm solves the QP problem if it an, and gives a `reasonable' solution if the problem is infeasible.

130

12.6 Controller Properties When a linear model and quadrati ost is used, the resulting ontroller is linear time-invariant providing that either no onstraints are a tive, or that a xed set of onstraints is a tive. (For ea h su h set, a dierent linear ontrol law results.) Thus the ontrol law an be a linear law for long periods of time. However, when hard onstraints are approa hed the ontroller an behave in a very nonlinear way. In parti ular, it may rea t mildly to a disturban e whi h drives the system away from onstraints, but very sharply to a disturban e of similar magnitude but in the opposite dire tion, whi h drives the system towards onstraints. The ontroller stru ture is very dierent from more onventional ontrollers. It onsists of a predi tor, whi h an be ompared with onventional ontrollers, for example by omparing omplexity as measured by the number of state variables, and an optimiser, whi h annot be ompared in that way. Figure 12.2 shows the stru ture of a predi tive ontroller. Clearly a predi tive ontroller is more omplex, in terms of behaviour, in terms of algorithm stru ture, and in terms of omputation y le time, than a onventional ontroller. Veri ation and erti ation is a mu h more formidable task than for a onventional

ontroller.

REFERENCE

OPTIMISER

COMMAND

PLANT

OUTPUT

& PREDICTOR OBSERVER

using

INTERNAL MODEL

STATE ESTIMATE

Figure 12.2: Stru ture of a Predi tive Controller

It is easy to formulate the predi tive ontrol problem in su h a way that the

ontroller displays (multivariable) integral a tion, and reje ts onstant output disturban es.

This is a hieved by the ombination of a suitable disturban e

model and penalisation of non-zero non-zero

u(k).


u(k) in the ost fun tion rather than of

It is not lear, however, how to obtain double-integral (`type 2')

a tion if it is required. An appropriate disturban e model would be required, but it would also seem ne essary to penalise instead of


u(k) in the ost fun tion.


2 u(k) =
u(k)


u(k

1)

This means that steadily-in reasing on-

trol a tions ould result, whi h would not be a

eptable in most appli ations. Reje tion of persistent but bounded-amplitude output disturban es, su h as sinusoids, is easily a hieved by in luding a model of the disturban e (in a

ordan e with the `Internal Model Prin iple') and penalising

131


u(k).

12.7 Design Spe i ations The problem of translating ontrol system design spe i ations into spe i

N1 , N2 , Nu ), weighting maQ, R), predi tor, and possibly a referen e-generating lter, is a di ult and is still a subje t of urrent resear h. Choosing Q, R, and the pre-

hoi es of predi tion and optimisation horizons ( tri es ( one

di tor is losely related to the hoi e of weighting and ovarian e matri es in LQG ontrol; there again the relationship between these parameters and the design spe i ation is very indire t, but experien e gained over many years has led to some rules of thumb, at least. The problem is made onsiderably more

ompli ated by also having to hoose horizons [222, 173, 119℄. If it is assumed that tight ommand-following is attained by the ontroller, then the hoi e of referen e-generating lter approximately denes the behaviour in response to ommands  a kind of model-referen e approa h  at least for the ase of ina tive onstraints. However, the assumption of tight model-following may not be realisti . Time-domain ommand-following spe i ations are, in prin iple, easily a hieved by formulating appropriate inequality onstraints. For example, restri tions on overshoot or rise-time during step responses may be formulated as inequality

onstraints. In pra ti e, however, there are problems if too many onstraints are added, sin e the solution time in reases.

One should, however, be wary

of taking responses to parti ular ommands su h as steps to be representative of behaviour in response to other ommands, sin e the predi tive ontroller is nonlinear (if onstraints be ome a tive). Frequen y domain spe i ations an be he ked under the assumption that no onstraints are a tive, or that a parti ular set of onstraints is a tive. Frequen y response hara teristi s of the ontroller an be omputed (and some software is available to do this [118℄) under either assumption, sin e the ontroller is then linear and time-invariant (assuming a quadrati ost fun tion). No omplete systemati method is urrently known of modifying the optimisation problem parameters in su h a way as to a hieve parti ular frequen ydomain hara teristi s, but signi ant progress towards this is reported in [151℄. This is parti ularly relevant for a hieving stability and performan e robustness. Stability of the losed loop is always part of the design spe i ation, even if only impli itly.

In the absen e of a tive onstraints, it is known how to

enfor e stability. Essentially, either the predi tion horizon

N2

must be made

large enough, or `terminal' equality onstraints, whi h bind at time must be added to the problem formulation.

k + N2 ,

It has been shown that, from

the point of view of stability enfor ement, terminal equality onstraints an be ex hanged for an innite predi tion horizon [199℄.

Furthermore, several

methods are known of ensuring stability even in the presen e of onstraints, but under the assumption that the problem posed always remains feasible. This is a very strong and almost unveriable assumption, and some urrent resear h is aimed at removing it. Most stability proofs are based on proving the monotoni ity of the ost fun tion

J (k) with k, and hen e using the ost fun tion as a Lyapunov fun tion. 132

There have also been some attempts at exploiting the pie ewise-linear nature of the ontroller to prove stability. Whereas obtaining stability is not di ult in pra ti e for predi tive ontrol s hemes, there are not yet standard pro edures for obtaining spe ied stability margins. (This is essentially the same problem as the problem of obtaining spe ied frequen y response hara teristi s, whi h was dis ussed above.) Although tuning of predi tive ontrollers remains di ult, mu h progress is being made, and systemati pro edures, whi h tune only some of the free parameters, are be oming in reasingly lear [173, 151℄.

12.8 Appli ations Predi tive ontrol has mostly been applied in the pro ess industries, and parti ularly in the petro hemi al industries. In these appli ations there is strong motivation to exploit the apability of respe ting onstraints, sin e mu h money is to be made by operating as lose as possible to onstraints. Also time onstants are very big, so there is plenty of time to perform the ne essary omputations. It is important to stress that in these industries predi tive ontrol is a mature te hnology, whi h is used routinely. A few papers report the use of predi tive ontrol with high-bandwidth ele tro-me hani al systems su h as servos and automotive systems [23℄. Typi ally these either restri t the stru ture of the predi tive ontrol law a priori in order to obtain an easier optimisation problem [5℄, or pose a problem without

onstraints [68℄. Several studies of using predi tive ontrol in aerospa e appli ations have been reported, though only a minority of these have really addressed the online omputation problem [105, 223, 24, 214, 252℄.

12.9 Con lusions Constrained Predi tive Control is radi ally dierent from other ontrol approa hes whi h are onventionally used, or might be used, for ight ontrol. Not only is the design method rather dierent, but the implemented algorithm is quite dierent, be ause it works by repeatedly solving an optimisation problem on-line. As a straight repla ement for those approa hes, it is not urrently ompetitive, primarily be ause of the omputational load. Even when further advan es in omputing hardware bring the solution time down to a

eptable levels, whi h they will surely do, the predi tive ontrol approa h will give greater problems of erti ation than onventional ontrollers, be ause of the di ulty of analysing the ontroller behaviour. On the other hand, predi tive ontrol oers some unique benets: its very nonlinear behaviour when onstraints are approa hed, and its ability to anti ipate pilot ommands, instead of merely rea ting to errors propagating round the feedba k loop.

133

We believe that predi tive ontrol is worth investigating further for use in ight ontrol, if its unique benets are exploited to obtain higher-level fun tionality, in addition to routine stability augmentation. This is dis ussed further in Chapter 25.

A knowledgement We would like to thank Dr Angel Perez de Madrid, of UNED, for useful omments during the preparation of this hapter.

134

13.

Fuzzy Logi Control

Gerard S hram , Uzay Kaymak1 and Henk B. Verbruggen1 1

13.1 Introdu tion Designing ontrollers for everyday tasks su h as grasping a fragile obje t, driving a ar, or more ompli ated tasks su h as ying an air raft, ontinue to be a real hallenge, while human beings perform them relatively easily.

Unlike

most onventional ontrol systems, however, humans do not use mathemati al models nor do they use exa t traje tories for ontrolling su h pro esses. Moreover, many pro esses ontrolled by human operators in industry annot be automated using onventional, linear ontrol te hniques, sin e the performan e of these ontrollers is often inferior to that of the operators. One of the reasons is that linear ontrollers, whi h are ommonly used in onventional ontrol, may not be appropriate for nonlinear plants. Another reason is that humans use various kinds of information and a ombination of ontrol strategies, that

annot be easily integrated into an analyti ontrol law.

However, a lot of

experien e and knowledge is available from the experts (e.g. the pilot), whi h

an be made expli it and programmed as a ontrol strategy in a omputer. Knowledge-based (expert) ontrol tries to formalize the domain-spe i knowledge, and uses reasoning me hanisms for determining the ontrol a tion from the knowledge stored in the system and the available measurements. Knowledge-based ontrol systems try to enhan e the performan e, reliability and robustness of urrent ontrol systems by in orporating knowledge that

annot be a

ommodated in analyti models upon whi h onventional ontrol algorithms are based.

Knowledge-based systems an be used to realize the

losed-loop ontrol a tions dire tly, i.e. repla e onventional losed-loop ontrollers, or they an omplement and extend onventional ontrol algorithms via supervision, tuning or s heduling of lo al ontrollers. A ommon type of knowledge-based ontrol is rule-based ontrol, where the ontrol a tions orresponding to parti ular onditions of the system are des ribed in terms of

ifthen rules.

Fuzzy Logi Controllers (FLCs) are rule-

based systems, where fuzzy sets are used for spe ifying qualitative values of the

ontroller inputs and outputs.

Mu h of the expert's knowledge ontains lin-

guisti terms su h as small, large, et ., whi h an be appropriately represented by fuzzy sets. Using fuzzy logi , experts' (linguisti ) knowledge of the pro ess

ontrol an be implemented. The rst appli ation of fuzzy logi ontrol was in 1

Department of Ele tri al Engineering, Delft University of Te hnology.

2600 GA Delft, The Netherlands.

P.O.Box 5031,

{g.s hram}{u.kaymak}{h.b.verbruggen}et.tudelft.nl 135

ement kiln ontrol [114℄. The rules representing the ontroller a tions were derived from the ement kiln operator's handbook. Sin e then, fuzzy logi ontrol has been applied to various systems in the hemi al pro ess industry, onsumer ele troni s, automati train operation, and many other elds [66, 136, 236℄. In se tion 13.2, the basi prin iples of fuzzy sets and fuzzy logi are introdu ed. Next, the ontrollers are onsidered in detail, followed by a dis ussion on ontroller tuning in se tion 13.4. tools are des ribed.

In se tion 13.5, software and hardware

In se tion 13.6, the possibilities of fuzzy logi for ight

ontrol are dis ussed. The hapter ends with on lusions.

13.2 Basi Prin iples The basi idea of a fuzzy logi ontroller is to formalize the ontrol proto ol of a human operator, whi h an be represented as a olle tion of

ifthen rules, in

a way tra table for omputers and for mathemati al analysis. As an example,

onsider the ontrol of the F/A-18 during arrier landing [229℄.

Following a

three dimensional ight path, the task involves the ontrol of speed, sink rate, and angular attitudes in order to allow a safe ship-board landing. The ontrol strategy of the pilot onsists of several subtasks, e.g. roll angle ontrol. If the desired roll angle is positive large (roll angle error positive large), then the pilot imposes a positive lateral displa ement on the sti k:

If roll angle error is positive large Then lateral sti k position is positive

large

The rule des ribes a proportional relation between roll angle error and lateral sti k position. Usually, the rules are a ombination of proportional as well as derivative a tion in order to redu e rates. A typi al rule from the sink rate rule base is:

If sink rate error is near zero AND sink a

eleration is positive large Then longitudinal sti k position is negative medium The rst part of the rules, alled the ante edent, spe ies the onditions under whi h the rule holds, while the se ond part, alled the onsequent, pres ribes the orresponding ontrol a tion.

Both the ante edent and the on-

sequent ontain linguisti terms (large, small, near zero et .) that ree t the pilot's knowledge of the pro ess. The ante edent ondition is dened as a ombination of several individual onditions, using a onne tive, su h as the logi al AND operation.

It is possible that other rules may ombine the ante edent

onditions using dierent onne tives su h as the logi al OR or the omplement NOT. When the rules of the above mentioned type are to be represented in a form tra table for omputers, one needs to dene the linguisti terms and the onne tives that operate on the linguisti terms. In fuzzy ontrol, the linguisti terms are represented by fuzzy sets. Suppose that the pilot has a general idea of what a small or large value is, without a

136

sharp distin tion.

Su h a term an be des ribed by a fuzzy set, represented

by a so- alled membership fun tion [264℄, whi h is dened on the universe of dis ourse

X

as a fun tion:

 : X ! [0; 1℄: The position and shape (e.g.

triangular, trapezoidal or bell-shaped) of

the membership fun tion depend on the parti ular appli ation.

However, in

many ases triangular shapes are preferred be ause they are related to linear a tions. Consider for example the roll angle error. In Figure 13.1, a number of triangular-shaped membership fun tions are shown. Note that in this example the membership fun tions are pairwise overlapping and that their sum is always 1. A roll angle error of 15 degrees belongs for 50% to the set of a positive small error (PS) and for 50% to the set of a positive medium error (PM). In other words, the membership degrees

µ 1

P S (15) and P M (15) are both 0.5.

NL

NM NS ZE PS PM

-30

0

15

PL

30

Figure 13.1: Membership fun tions for roll angle error; negative (N), positive (P), large (L), medium (M), small (S), zero (ZE).

Fuzzy set operations are performed by logi al onne tives su h as AND ( onjun tion), and OR (disjun tion). For this purpose, the logi al onne tives from onvential Boolean logi have been extended to their fuzzy equivalents. The generalization of onjun tion to fuzzy sets is done by fun tions alled tnorms. Disjun tions are generalized by t- onorms. The most ommonly used

onjun tion operators are the minimum and produ t operators. Usually, the maximum or the probabilisti sum operator is used for the disjun tion.

In

Figure 13.2, the onjun tion and disjun tion operations on two fuzzy sets are shown when the minimum and maximum operators are used, respe tively.

conjunction (AND)

disjunction (OR)

Figure 13.2: Conjun tion and disjun tion of two fuzzy sets by minimum and maximum operator, respe tively.

137

13.3 Fuzzy Logi Control Using fuzzy sets and fuzzy set operations, it is possible to design a fuzzy reasoning system whi h an a t as a ontroller [162℄. In Figure 13.3, the stru ture of a typi al fuzzy logi ontroller (FLC) is shown. The ontrol strategy is stored

knowledge base scaling factors

membership functions

rule base

membership functions

fuzzification

reasoning mechanism

defuzzification

scaling factors

control actions

errors dynamic filter & scaling

dynamic filter & scaling

Figure 13.3: Blo k-s hemati representation of fuzzy logi ontroller. in the form of

ifthen rules

in the rule base. They represent an approximate

stati mapping from inputs (e.g. errors) to outputs ( ontrol a tions). The dynami lters are used to introdu e dynami s, e.g.

error and derivative of

error, and to introdu e an integration on the output. Moreover, s aling is performed to keep the signals between the input and output limits for whi h the fuzzy rules are dened. The membership fun tions provide a smooth interfa e from the linguisti knowledge to the numeri al pro ess variables. The fuzzi ation module determines the membership degree of the inputs to the ante edent fuzzy sets. The reasoning me hanism ombines this information with the rule base and determines the fuzzy output of the rule-based system.

In order to

obtain a risp signal, the fuzzy output is defuzzied and s aled. The omputational me hanism of the FLC an be explained on an example of a fuzzy variant of a PD (proportionalderivative) ontroller. Simple PD-like fuzzy ontrol rules an be dened as relations between the ontrol error error derivative


e and the ontrol a tion u.

e, the

As an example, assume that the

following two rules are a part of a fuzzy ontroller's rule base:

If e is small and
e is medium Then u is small If e is medium and
e is big Then u is medium Triangular membership fun tions are dened for the terms small, medium and big in the respe tive domains, see Figure 13.4. The omputational me hanism of the FLC pro eeds in ve steps: 1. Fuzzi ation: The membership degrees of the ante edent variables are

small (e), medium (e), medium (
e), big (
e)).

omputed (

2. Degree of fullment: The degree of fullment for the ante edent of ea h rule is omputed using fuzzy logi operators.

i

determines to whi h degree the

138

The degree of fullment

ith rule is valid.

In the example, the

produ t operator is used:

1 = small (e)
medium (
e) 2 = medium (e)
big (
e): 3. Impli ation: The degree of fullment is used to modify the onsequent of the orresponding rule a

ordingly. This operation represents the

then

if

impli ation dened as a t-norm, i.e. a onjun tion operator (e.g.

produ t). Hen e, the fuzzy outputs of the rules be ome:

0

1 (u) = 1
small (u) 0 2 (u) = 2
medium (u): 4. Aggregation:

the (s aled) onsequents of all rules are ombined into a

single fuzzy set.

The aggregation operator depends on the impli ation

fun tion used; for onjun tions, it is a disjun tion operator (e.g. max): FLC output

0

0

(u) = max(1 (u); 2 (u))

8 u 2 U:

5. Defuzzi ation: the resulting fuzzy set is defuzzied to yield a risp value. Defuzzi ation an be onsidered as an operator that repla es a fuzzy set by a representative value.

There exists a number of defuzzi ation

methods, su h as the entre of area method. In Figure 13.4, a small arrow marks the defuzzied value. The above type of FLC is alled a linguisti ontroller or a Mamdani type of ontroller.

A on eptually dierent type of FLC has been introdu ed by

Takagi and Sugeno [234℄.

In this type of ontroller, the onsequent part is

repla ed by a risp (non)linear fun tion.

The ontroller output is omputed

by taking a weighted average of the individual rule outputs.

Sin e the rule

outputs are risp, there is no need for defuzzi ation. The ontroller an be

ompared to a gain s heduling ontroller whi h has linear, lo al modules whi h are smoothly onne ted. In this way, the lo al linear models an be used for tuning and analysis (e.g. stability) of the FLC, while the global model aptures the nonlinearity of the system. However, the transparen y of the ontroller is de reased. In the RCAM design, the Mamdani type is used be ause this type of ontroller an implement the pilot knowledge most easily (Chapter 26).

Dire t and supervisory ontrol The motivation for many FLC appli ations is to mimi the ontrol behaviour of a human operator in a dire t ontrol onguration or in a supervisory ontrol environment. Many fuzzy logi ontrollers are implemented as dire t ontrollers in a feedba k loop. As the rule base represents a stati mapping between the ante edent and the onsequent variables, external dynami lters are used to introdu e the desired dynami behaviour of the ontroller (Figure 13.3).

139

2 µ

small

1

1 µ small (e)

0

µ

small

medium

µ medium(

1

1

e)

β1 0

µ

medium

1

µmedium(e)

µ 1

1

medium

β2

e)

0 e

u

µ

big µ big (

0

3 0

3 0

u

e

product µ

4

1

max

0 u

5 Figure 13.4: Computational me hanism of a FLC.

A fuzzy inferen e system an also be applied at a higher, supervisory level. A supervisory ontroller is a se ondary ontroller whi h augments the existing

ontroller for various onditions. Supervisory systems are hara terized by the additional exibility they bring to the ontrol system. A supervisory ontroller

an adjust the parameters of a low-level ontroller a

ording to the pro ess information, so that dynami behaviour whi h ould not be obtained by the low-level ontroller due to pro ess nonlinearities or hanges in the operating or environmental onditions an be a hieved. An advantage of a supervisory stru ture is that it an be added to already existing ontrol systems. Hen e, the original ontrollers an always be used as initial ontrollers for whi h the supervisory ontroller an be used for tuning the performan e. A supervisory stru ture an be used for implementing dierent ontrol strategies in one single ontroller (heterogeneous ontrol). These

on epts will be shown in Chapter 26, where separate ontrol strategies for low airspeed and engine failure are in luded.

13.4 Fuzzy Logi Control Design Two dierent methods an be used for designing fuzzy logi ontrollers: 1. Design the ontroller dire tly from the knowledge available from the domain experts.

140

2. Develop a fuzzy model of the plant from measurements, rst prin iples and expert knowledge, and use this model to design a ontroller or in orporate this model in a model-based ontrol s heme. The se ond, indire t method is des ribed in e.g. [17, 36, 127℄. In the rest of this se tion we will only on entrate on the dire t approa h, whi h will serve as a guideline for the design in Chapter 26. The design is hara terized by the following steps: 1. Determine the ontroller inputs and outputs.

For this step, one needs

basi knowledge about the hara ter of the plant dynami s (stable, unstable, stationary, time-varying, low order, high order, et .), the plant nonlinearities, the ontrol obje tives and the onstraints. The simplied plant dynami s together with the basi ontrol obje tives determine the dynami s of the ontroller, e.g. PI, PD or PID type fuzzy ontroller. In order to ompensate for the plant nonlinearities, non-stationarity or other undesired phenomena, variables other than error and its derivative or its integral may be used as the ontroller inputs. It is, however, important to realize that with an in reasing number of inputs, the omplexity of the fuzzy ontroller (i.e. the number of linguisti terms and the total number of rules) in reases onsiderably. In that ase, rule base simpli ation and redu tion te hniques need to be used for keeping the number of rules small [16℄. 2. Determine the rule base. The onstru tion of the rule base is a ru ial aspe t of the design, sin e the rule base en odes the ontrol proto ol of the fuzzy ontroller. Several methods of designing the rule base an be distinguished. One is based entirely on the expert's intuitive knowledge and experien e over all operating onditions. Sin e in pra ti e it may be di ult to extra t all knowledge from the operators, this method is often

ombined with a good understanding of the system's dynami s. Another method is based on using a fuzzy model of the pro ess from whi h the fuzzy ontrol rules are derived. 3. Dene the membership fun tions and the s aling fa tors.

The designer

must de ide, how many linguisti terms per input variable will be used. The number of rules needed for dening a omplete rule base in reases exponentially with the number of linguisti terms per input variable. On one hand, the number of terms per variable should be low in order to keep the rule base maintainable.

On the other hand, with few terms,

the exibility in the rule base is restri ted with respe t to the a hievable nonlinearity in the ontrol mapping. The membership fun tions may be a part of the expert's knowledge, for example the expert knows approximately what a large roll angle error means. If su h knowledge is not available, membership fun tions of the same shape, uniformly distributed over the domain, an be used as an initial setting and an be tuned later. For omputational reasons, triangular and trapezoidal membership fun tions are usually preferred to bell-shaped fun tions. Moreover, the latter

141

fun tions introdu e a nonlinear hara ter whi h may not be desirable in all ases. Generally, the input and output variables are dened on losed intervals. For simpli ation of the ontroller design, implementation and tuning, it is more onvenient to work with normalized domains, su h as the interval

[ 1; 1℄.

S aling fa tors are used to transform the values from the operat-

ing ranges to these normalized domains. However, one should be aware that su h s aling fa tors also s ale the nonlinearity in the ontroller whi h may not always be desirable. 4. Inferen e options. The hoi e of the inferen e operators also inuen es the shape of the mapping between inputs and outputs. The most used inferen e method is the max-min method, where the minimum operator is used for determining the degree of fullment and the impli ation, and the maximum operator for rule aggregation. Another method is the sumprodu t inferen e. The latter ombination is useful for an initial, linear setting of the FLC. This will be explained below. 5. Fine-tuning the ontroller.

The implementation of human heuristi s is

formalized by fuzzy logi in a systemati way. Altough ne-tuning the performan e of the ontroller is essentially a matter of trial-and-error, an understanding of the inuen e of various parameters an guide the pro ess. The s aling fa tors, whi h determine the overall gain of the fuzzy

ontroller and also the relative gains of the individual ontroller inputs, have mainly a global ee t. The ee t of a modi ation of membership fun tions and rules is more lo alized, for example hanging the onsequent of an individual rule. The ee t of the hange of the rule onsequent is the most lo alized and inuen es only that region where the rule's ante edent holds. 6. Stability analysis. The analysis of the ontroller is mainly based on time responses. A stability analysis of the nonlinear FLC is in general di ult. However, results an be obtained by using te hniques from nonlinear systems theory if a model of the pro ess under ontrol is available [66, 235, 251℄. The stability is only proven for the parti ular, simplied model. Re ently, the stability results have also been extended to more general

lasses of systems [43℄. The resulting ontrollers are usually onservative be ause of the onservative nature of the stability riteria. In order to simplify the design, it is possible to initialize the FLC as a linear fun tion between the input and output bounds. This limits the hoi e of membership fun tions and operators, and the ontroller be omes easier to analyse. One way of a hieving linear initialisation is using pairwise overlapping, triangular membership fun tions where the sum of the memberhsip fun tions equals 1. The defuzzied onsequents must be dened su h that the total mapping of the FLC is a linear fun tion. The defuzzied onsequents are the numeri al values after defuzzi ation of ea h individual rule onsequent. Se ondly,

142

produ t operators must be used for determining the degree of fullment and impli ation. The aggregation and defuzzi ation phase are then ombined in one step by the so- alled fuzzy-mean method, whereby the FLC output

y

is

determined as a weighted sum of defuzzied onsequents:

y= with

i

and

i

Nr X i=1

i i

are the degree of fullment and the defuzzied onsequent of

the ith rule respe tively, and

Nr

the number of rules. Note that defuzzi a-

tion is performed for ea h individual rule before aggregation takes pla e.

In

Chapter 26, the FLCs are initialized in this way.

13.5 Available Hardware and Software Tools Sin e the development of fuzzy ontrollers relies on intensive intera tion with the designer, spe ial software tools have been introdu ed by various software (SW) and hardware (HW) suppliers su h as Omron, Aptronix, Inform, Siemens, National Semi ondu tors, et . Most of the programs run on a PC under MSWindows, although some of them are also available for UNIX systems. The general stru ture of most software tools is depi ted in Figure 13.5. The

project editor

Figure 13.5: Generi stru ture of a software tool for fuzzy ontroller design. heart of the user interfa e is a graphi al proje t editor that allows the user to build a fuzzy ontrol system from basi blo ks. Input and output variables

an be dened and onne ted to the fuzzy reasoning unit.

If ne essary, one

an also use pre-pro essing or post-pro essing elements su h as dynami lters, integrators, dierentiators, et . The fun tions of these blo ks are dened by the user. The rule base and the related fuzzy sets are dened using the rule base

143

and membership fun tion editors. The rule base editor is a spreadsheet in whi h the rules an be entered or modied. The membership fun tions editor is used for dening the shape and position of the membership fun tions graphi ally. After the rules and membership fun tions are designed, the fun tion of the fuzzy ontroller an be tested using system analysis and simulation software (e.g. MATLAB/SIMULINK). On e the fuzzy ontroller is tested using various analysis tools, it an be used for ontrolling the plant either dire tly by the environment (via omputer ports or analog inputs/outputs), or through generating a run-time ode. Most of the programs generate a standard C- ode and also a ma hine ode for spe i hardware, su h as mi ro ontrollers or programmable logi ontrollers (PLCs). An alternative implementation is a multi-dimensional look-up table with a simple interpolation routine. This ould simplify validation and erti ation in

ase of ight riti al ontrol systems.

13.6 Fuzzy Logi for Flight Control Re ently, mu h attention has been paid to the appli ation of knowledge-based

ontrol te hniques for ight ontrol [228, 230℄. It shows that te hniques like neural networks and fuzzy systems an provide appropriate tools for nonlinear identi ation [156, 204℄, ontrol of high performan e air raft [183, 229℄ (inner loop as well as outer loop ontrol), heli opters [195, 233℄, spa e raft [26, 106℄, ight ontrol re onguration [142, 182, 263℄, and advisory systems [111, 232℄. In these appli ations, neural networks generally serve as nonlinear, sometimes adaptive, fun tion approximators, while fuzzy systems are used as supervisory, expert systems. An example of a fuzzy logi ontrol appli ation for ight ontrol is [229℄. The ne essary knowledge is extra ted from experien ed pilots. In Chapter 26, pilot heuristi s of ying an air raft are implemented in the design of a FLC as well. The FLC design onsists of longitudinal and lateral outer loop tra king

ontrollers ombined with lassi al inner loop attitude ontrollers. Additional, supervisory rules for low airspeed and engine failure are in luded whi h show how (gain) s heduling and ex eption handling an be readily in orporated.

13.7 Con lusions A fuzzy logi ontroller an be onsidered from the AI point of view as a real-time expert system implementing a part of a human operator's or pro ess engineer's expertise. From the ontrol engineering perspe tive, a FLC is a nonlinear ontroller. Re ently, a lot of resear h eort has been put into fuzzy logi

ontrol. The appli ations in industry are also in reasing. Major produ ers of

onsumer goods use fuzzy logi ontrollers in their designs for onsumer ele troni s, dishwashers, washing ma hines, automati ar transmission systems et . FLC appli ations are beginning to appear in the pro ess industry as well.

144

One of the main reasons put forward for using fuzzy logi is that an expli it mathemati al model des ription is not required for the design of a FLC. Instead the a tions of a human operator, who already has an internal representation of the plant, are modelled.

This an result in a more e ient ontroller

design, saving time and money. This is only true if expli it operator knowledge is available in a suitable form.

Also, for testing and ne-tuning the FLC, a

reasonable simulation model or the pro ess itself should be available. However, if little experien e or knowledge about the pro ess is present, and it is not possible to make eld tests for tuning the ontroller, fuzzy logi ontrol may not be suitable. One has to onsider espe ially the knowledge a quisition bottlene k if the experts' knowledge is not available expli itly.

An alternative is rst

building a fuzzy model of the nonlinear system from measurement data about the system, and then applying model-based ontrol te hniques. Many fuzzy logi ontrollers are implemented as dire t ontrollers in a feedba k loop. In situations where an existing ontroller needs to be extended for several operating onditions or when a more exible ontrol stru ture is required, supervisory fuzzy ontrol an provide an answer.

It is more di ult

to formulate an analyti ontrol law at this level, while a lot of linguisti information may be available, whi h an be used for designing the FLC. At this level, the ontrol problem starts to resemble more and more a de ision making problem, whi h an be solved by te hniques from fuzzy-de ision making. The implementation of human heuristi s is formalized by fuzzy logi in a systemati way. This fa t is also re ognized by the industry, and re ently efforts have in reased to dene a European industry standard for the development methodology of fuzzy logi systems, based on ISO-9000 general system development guidelines [248℄. However, ne-tuning the performan e of the ontroller is a matter of trial-and-error like in lassi al ontrol, but using the provided guidelines and an understanding of the inuen e of ontroller parameters, a satisfa tory ontroller an be obtained.

145

146

Part II

RCAM part

147

14.

The RCAM Design Challenge

Problem Des ription

Paul Lambre hts , Samir Bennani , Gertjan Looye and Dieter Moormann 1

2

2

3 4

Abstra t.

The RCAM design hallenge problem is dened in this

hapter using two main se tions.

The rst se tion dis usses the

basi ight dynami s model, the available inputs, outputs, parameters, et . and the modelling of a tuators, disturban es, et . After that the ontrol design spe i ations are given and the evaluation pro edure to be performed by all design teams is presented.

14.1 Introdu tion This hapter provides the RCAM design problem formulation. It is abstra ted from the GARTEUR FM(AG08) report: Robust Flight Control Design Challenge Problem Formulation and Manual: the Resear h Civil Air raft Model (RCAM) [145℄. This report formed the basis for the RCAM design hallenge, the results of whi h are given in the hapters 15 through 26.

Therefore its

ontents are given here with little modi ation, so that a lear pi ture of the information that was available to the design hallenge teams is given. However, sin e the design and evaluation software that was available to the teams is not supplied with this book, the des ription of this software has been ex luded. It is remarked that both the software itself and the des ription appeared helpful in

larifying the problem formulation, but was not intended to provide additional information. In se tion 14.2, a des ription of the model is given, in whi h analyti al expressions for all the parameters of interest, states, inputs and outputs of the system, are derived.

A detailed, oje t-oriented des ription of the model

omponents is also in luded (air raft, sensors, a tuators, engines, wind, et .). 1

Hoogovens Corporate Servi es B.V., HR&D-RSP-SDC 3G.16, P.O.box 10000, 1970 CA

IJmuiden, The Netherlands. 2

(Formerly: NLR, Amsterdam.)

Delft University of Te hnology (TUD), Fa ulty of Aerospa e Engineering, Kluyverweg 1,

2629 HS Delft, The Netherlands. 3

German Aerospa e Resear h Establishment (DLR), Institute for Roboti s and System

Dynami s Control Design Engineering, Oberpfaenhofen, D-82230 Wessling, Germany. 4

The following authors ontributed to the original RCAM design denition: Pierre Fabre,

Joseph Irvoas, Philippe Ménard (Aerospatiale), Anders Helmersson (LiTH), Jean-François Magni (CERT), Tony Lambre gts (DUT), Alberto Martínez (INTA), Stefano S ala (CIRA), Phillip Sheen (AVRO), Jan Terlouw (NLR) Hans van der Vaart (TU Delft).

149

In se tion 14.3 the design problem is formulated, and the riteria and pro edure adopted for evaluation of the proposed design are des ribed.

14.2 Des ription of the RCAM Model The purpose of this hapter is to dis uss the RCAM model in a general setting, su h that used nomen lature and terminology an be introdu ed, and some of the philosophy behind the stru ture and numeri al al ulations in the software

an be highlighted. The hapter is set up to have some tutorial value, but is by no means omplete in that sense. It is re ommended to onsult a standard referen e su h as [74℄ or [35℄ for more information on the derivation of equations of motion, et .

14.2.1 Blo k diagram of the system A six degrees of freedom nonlinear air raft model in luding nonlinear a tuators (position and rate limited) and a model of disturban es has been proposed by Aérospatiale.

A blo k diagram of this model is given in gure 14.1.

Ea h

box in this blo k diagram will be overed in more detail in following text. In subse tion 14.2.3, an analyti al des ription of the air raft dynami s is given. In subse tions 14.2.4 and 14.2.5, the sensor and a tuator dynami s are detailed. In subse tion 14.2.6, the analyti al models of wind disturban es are presented.

14.2.2 Nomen lature: inputs, states, outputs, parameters As far as appli able, nomen lature is used as dened in the Communi ation Handbook [237℄.

The following tables summarise this nomen lature, as it is

used both for the formulation of the algorithms and the naming of variables in the software. Additional information an be found in Appendix A of this do ument. The inputs to the model are given in table 14.1. In this table,

FE

denotes the

earth-xed referen e frame, whi h is dened as follows. The origin threshold.

OE XE

is lo ated on the runway longitudinal axis at the is positive pointing towards the north, and we as-

sume that the runway is also dire ted towards the north (runway 00), hen e

XE is positive along the runway in the landing dire tion. ZE is positive downward, and YE is in the appropriate

Furthermore,

dire tion for a right handed axis system (positive east).

FB

stands for the body-xed referen e frame, whi h is dened as follows.

OB ZB is

The origin

is at the vehi le entre of gravity.

forward,

positive downward and

(starboard side).

150

YB

XB

is positive

is positive to the right

ACTUATOR MODELS (including nonlinearities)

trim settings controls

uext

feedback path

sim Specific outputs for system analysis

U=[ DA DT DR THROTTLE1 THROTTLE2]

uc

lon

ACTUATORS

Measurements for longitudinal control laws wext WIND=[ WXE WYE WZE WXB WYB WZB ] WIND MODELS (constant wind, turbulence, windshear)

lat Measurements for lateral control laws

WIND

AIRCRAFT

RCAM MODEL ( 6 degrees of freedom, non linear, landing configuration)

Figure 14.1: Blo k diagram of the system

The three earth-xed wind inputs, u(6)u(8), are intended to be used for onstant wind velo ity omponents eg. headwinds, whereas the body-xed wind inputs, u(9)u(11), are intended to be used for gusts. The states used internally by the software are expressed in SI units and are dened in table 14.2. In this table, CoG denotes Centre of Gravity. The outputs from the model are given in SI units and are shown in table 14.3. In this table,

FV

denotes the vehi le- arried verti al frame, whi h is dened as

follows. The vehi le- arried verti al frame is parallel to the earth-xed referen e frame but moves with the vehi le. The origin at the vehi le's entre of gravity. the north,

ZV

XV

OV

is lo ated

is positive pointing towards

is positive downward, and

YV

is positive towards the

east. Only the model outputs labeled as measured an be assumed to be available as inputs to the ontroller that is to be designed.

The simulation outputs

are only intended to be used for evaluation and should not be used in the nal

ontroller. Note that there is some redundan y in the measured signals, e.g.

 an be determined from uV

and

vV :

depending on the ontrol strategy the

most onvenient signals may be used.

151

Symbol

ÆA ÆT ÆR

ÆT H1 ÆT H2 W xE W yE W zE W xB W yB W zB

Alphanumeri DA u(1) DT u(2) DR u(3) THROTTLE1 u(4) THROTTLE2 u(5) WXE u(6) WYE u(7) WZE u(8) WXB u(9) WYB u(10) WZB u(11)

= = = = = = = = = = =

Name aileron dee tion tailplane dee tion rudder dee tion throttle position of engine 1 throttle position of engine 2 Wind velo ity in the x-axis of FE Wind velo ity in the y-axis of FE Wind velo ity in the z-axis of FE Wind velo ity in the x-axis of FB Wind velo ity in the y-axis of FB Wind velo ity in the z-axis of FB

Unit rad rad rad rad rad m/s m/s m/s m/s m/s m/s

Table 14.1: Input denitions

Symbol

p q r  

uB vB wB x y z

Alphanumeri P x(1) Q x(2) R x(3) PHI x(4) THETA x(5) PSI x(6) UB x(7) VB x(8) WB x(9) X x(10) Y x(11) Z x(12)

= = = = = = = = = = = =

Name roll rate (in FB ) pit h rate (in FB ) yaw rate (in FB ) roll angle (Euler angle) pit h angle (Euler angle) heading angle (Euler angle) x omponent of inertial velo ity in FB y omponent of inertial velo ity in FB z omponent of inertial velo ity in FB x position of air raft CoG in FE y position of air raft CoG in FE z position of air raft CoG in FE

Unit rad/s rad/s rad/s rad rad rad m/s m/s m/s m m m

Table 14.2: State denitions

Usually, it is possible to dene geometri air raft parameters within the bodyxed referen e frame. However, in the ase of RCAM this is not allowed, as the CoG is not a geometri ally xed point. referen e frame

FM

For this reason, a measurement

is dened.

The measurement referen e frame is geometri ally xed to the air raft.

The origin

OM

is lo ated at the leading edge of the mean

aerodynami hord, whi h is denoted as ba kwards,

ZM

YM

. XM

is positive pointing

is positive pointing to the right (starboard), and

is positive pointing up.

It is assumed that the aerodynami entre of the wing-body onguration (ACwb ) is also geometri ally xed: its o-ordinates in

FM

0:12 ; 0 ; 0).

are (

With these denitions, it is now possible to spe ify the parameters used in RCAM: they are given in table 14.4. Finally, RCAM provides the possibility to study the ee t of the parameter

hanges dened in table 14.5.

152

Symbol Measured

Alphanumeri

Name

Unit

q nx nz wV z VA V p r  uV vV y 

Q NX NZ WV Z VA V BETA P R PHI UV VV Y CHI

y(1) y(2) y(3) y(4) y(5) y(6) y(7) y(8) y(9) y(10) y(11) y(12) y(13) y(14) y(15)

= = = = = = = = = = = = = = =

pit h rate (in FB ) = x(2) Fx  horizontal load fa tor (in FB ) = mg Fz verti al load fa tor (in FB ) = mg 1 z omponent of inertial velo ity in FV z position of air raft CoG in FE = x(12) air speed total inertial velo ity angle of sideslip roll rate (in FB ) = x(1) yaw rate (in FB ) = x(3) roll angle (Euler angle) = x(4) x omponent of inertial velo ity in FV y omponent of inertial velo ity in FV y position of air raft CoG in FE = x(11) inertial tra k angle

rad/s m/s m m/s m/s rad rad/s rad/s rad m/s m/s m rad

 

x ny

PSI THETA ALPHA GAMMA X NY

y(16) y(17) y(18) y(19) y(20) y(21)

= = = = = =

heading angle (Euler angle) = x(6) pit h angle (Euler angle) = x(5) angle of atta k inertial ight path angle x position of air raft CoG in FE= x(10) lateral load fa tor (in FB )= Fy  see equations 14.1 and 14.5 mg

rad rad rad rad m -

Simulation

Table 14.3: Output denitions

14.2.3 Air raft dynami s model This subse tion des ribes the RCAM dynami s model orresponding to the

AIRCRAFT blo k in gure 14.1.

The dynami obje ts are depi ted in gure 14.2.

These obje ts are:



body des ribes the body dierential equations of motion (see subse tion 14.2.3);



two transformation obje ts des ribe the o-ordinate transformation between the body-xed o-ordinates of the body obje t and the geodeti oordinates of the gravity obje t, and between the body-xed o-ordinates of body and the geodeti o-ordinates of wind, respe tively (see subse tion 14.2.3);



al airspeed des ribes the relationship between the inertial movement, the wind, and the movement relative to the air (see subse tion 14.2.3);



engine_1 and engine_2 des ribe the relevant engine behaviour (see subse tion 14.2.3);

 

atmosphere des ribes the atmosphere model (see subse tion 14.2.3); aerodynami des ribes the aerodynami for es and moments (see subse tion 14.2.3);

153

Symbol

Alphanumeri MASS Aerodynami Parameters

 CBAR lt LTAIL

m

S St
x
y
z

Name air raft total mass

= = =

S STAIL DELX DELY DELZ Engine Parameters XAP T 1 XAPT1

= = = = =

YAP T 1

YAPT1

=

ZAP T 1

ZAPT1

=

XAP T 2

XAPT2

=

YAP T 2

YAPT2

=

ZAP T 2

ZAPT2

=

Default 120 000

Unit kg

6.6 24.8

m m

260.0 64.0 0.23  0 0

m2 m2 m m m

0.0

m

7:94 1:9

m m

0.0

m

7:94 1:9

m

mean aerodynami hord distan e between AC of the wing-body (ACwb ), and AC of the tail (ACt ) wing planform area tail planform area x position of the CoG in FM y position of the CoG in FM z position of the CoG in FM

x position of appli ation point of thrust of engine 1 in FM y position of appli ation point of thrust of engine 1 in FM z position of appli ation point of thrust of engine 1 in FM x position of appli ation point of thrust of engine 2 in FM y position of appli ation point of thrust of engine 2 in FM z position of appli ation point of thrust of engine 2 in FM

=

m

Table 14.4: Parameter denitions

Parameter MASS DELX DELY DELZ

m
x
y
z

: : : :

100 000 kg 0.15  0.03  0.0 

< < < <

Bounds

m


x
y
z

< < < <

150 000 kg 0.31  0.03  0.21 

Table 14.5: Possible parameter hoi es in RCAM



gravity des ribes the gravitational inuen e (see subse tion 14.2.3).

Body equations of motion The following two subse tions give the equations of motion for a rigid body with 6 degrees of freedom and other air raft motion relevant equations whi h as used within RCAM. For a more detailed derivation and explanation of these equations a referen e su h as [74℄ or [35℄ should be onsulted.

Translational motion. The equations for the translational movement in bodyxed o-ordinates are derived from the for e equation,

F = m ( aB + ! F

 VB )

(14.1)

is the sum of for es due to the engines, the aerodynami s and gravity,

is the mass of the air raft,

VB

is the inertial velo ity and

!

m

is the rotational

velo ity expressed in body-xed o-ordinates. The a

eleration (in body-xed

154

RCAM aero

u

sim RCAM

body6DOF equations

COG

engine1

airspeed kinetic

RCAM

air

long

wind

engine2

const. gravity

bodyfixed

bodyfixed

Trafo

Trafo

veh.carried

veh.carried

const.

lat Earth

atmosphere

wind

gust

wind gust

Figure 14.2: Dynami obje ts of RCAM air raft model inside the

AIRCRAFT

blo k of gure 14.1. Conne tion arrows between obje ts hara terise physi al intera tions

system) is the time derivative of velo ity:

2

dV d uB aB = B = 4 vB dt dt w

3 5

(14.2)

B

and the velo ity is the time derivative of the position ve tor expressed in the vehi le- arried verti al frame:

3

2

d x d XV = 4y5 VV = dt dt z

(14.3)

Additionally, some air raft spe i quantities are dened as follows:

h, whi h is the negative z - o-ordinate in the vehi le arried system h= z (14.4) The load fa tor n is dened as the relation of the external for es F (equation 14.1) to the gravity for e mg , with all quantities given in the body-xed The height

oordinate system. In order to have a zero load fa tor for horizontal ight, the z- omponent of

n is redu ed by one. 2 3 nx F n = 4 ny 5 = mg nz 155

2

3

0 405 1

(14.5)

The inertial ight path angle,

, is given as a fun tion of the speed omponents

in the vehi le- arried verti al referen e frame

tan = The tra k angle,

p 2wV 2 uV + vV

(14.6)

, is also dened as a fun tion of the the speed omponents

in the vehi le- arried verti al referen e frame

v tan  = V uV

(14.7)

Rotational motion. The equations of motion for the rotational movement of a rigid body in the body-xed axis system are derived from the moment equation,

M = I !_ + ! M

 I!

(14.8)

is the sum of moments about the entre of gravity due to the engines

and the aerodynami s,

!

is the inertial rotational velo ity, and

!_

is the iner-

tial rotational a

eleration in the body-xed axis system. Using the standard notation [74℄ we get:

2

2

3

3

p_ d 4p5 5 4 !_ = q_ = q dt r r_

(14.9)

Again using standard notation [74℄, the relation between the rotational velo ities and the Euler angles is;

2

d 4 = dt

_ _ _

3

32

2

1

For a normal air raft , the inertia tensor is;

3

p 1 sin  tan  os  tan  5=4 0

os  sin  5 4 q 5 r 0 sin = os  os = os 

2

Ix I = 4 0 Ixz

0 Ixz Iy 0 0 Iz

3

2

5

= m4

I

(14.10)

dened in the body-axis frame

40:07 0 0 64 2:0923 0

3

2:0923 0 5 99:92

(14.11)

2

where all numbers are expressed in square metres, m .

Co-ordinate transformation (Body-Fixed

, Vehi le-Carried)

The rotations between the body-xed and the vehi le- arried o-ordinate system are depi ted in gure 14.3.

To des ribe the angular orientation of the

air raft, a transformation using the three Euler angles

,  ,

and

is ne es-

sary. This transformation is a hieved by initially rotating the vehi le- arried

ZV -axis by the heading k2 -axis by the pit h angle ,

verti al system about the

angle

is rotated about the

and nally the body-xed

1

. Then, the result

Normal air raft are assumed symmetri about the OXZ body axis plane.

156

xB xV

θ

ψ

k1 yV

ψ φ

xV, yV

k2

xB , z v

yB

φ yB , z B zB

θ

k3

zV

Figure 14.3: Co-ordinate transformation body-xed

, vehi le- arried

referen e frame is obtained by rotating the result of that by the roll angle about the

XB -axis.



This results in the following transformation matrix from

the vehi le- arried verti al axis system to the body-xed axis system:

RBV = 32 32 2

os

os  0 sin  1 0 0 4 0 os  sin  5 4 0 1 0 5 4 sin 0 sin  0 os  0 sin  os  Note that

sin

os 0

3

0 05 1

(14.12)

RBV = RVT B .

For example, the transformation of velo ities from the vehi le- arried verti al frame

FV

to the body-xed referen e frame

FB

is given by:

VB = RBV VV with

2

uB VB = 4 vB wB

3 5

(14.13)

2

and

uV VV = 4 vV wV

3 5

(14.14)

Similarly, the a

elerations, rotational velo ities, positions, for es and moments

an be transformed between the o-ordinate systems.

157

Cal ulation of airspeed

Va is the dieren e between the inertial velo ity VB , and the wind velo ities, WB and WE (see table 14.1).

The ve tor airspeed, air raft,

of the

Expressed in the body-xed o-ordinate system this is al ulated as:

Va = VB Hen e, with

the airspeed

WB 2

ua Va = 4 va wa VA

RBV WE

(14.15)

3 5

(14.16)

is given as:

p

VA = (ua 2 + va 2 + wa 2 )

(14.17)

, and the angle of sideslip, , are dened as: w (14.18) tan  = a ua v sin = a (14.19) VA The derivatives of  and with respe t to time are: aax wa a u _ = az a2 (14.20) ua + wa 2 a ( u 2 + wa 2 ) va ( aax ua + aaz wa ) p _ = ay a (14.21) VA 2 ua 2 + wa 2 where aax , aay , and aaz are the x, y , and z -time derivatives of the airspeed in dua ). body-xed o-ordinates. (e.g. aax = dt Next, the angle of atta k,

Aerodynami equations The equations dening aerodynami for es and moments are determined by means of aerodynami oe ients.

Depending on the method of modelling

these oe ients may be dened in dierent referen e frames; e.g.

FB .

FW , FS , or

The referen e frame for aerodynami for es and moments that is used in

RCAM is the stability axis frame

FS .

Aerodynami for es. The aerodynami for es are determined by means of aero-

CD , CY , CL ), whi h are given , and the ontrol

dynami oe ients for drag, sidefor e and lift ( as fun tions of the angle of atta k,

,

the sideslip angle,

surfa e dee tions. The aerodynami lift oe ient,

CL , is dened as (see gure 14.4);

CL = CLwb + CLt 158

(14.22)

C

Lwb is the lift oe ient of the wing and body. It a ts on the aerodynami

entre of the wing and body and is parallel with the S axis. It is only a fun tion of angle of atta k

Z , and for  < 19 degree it is given by the following

equations:

CLwb = Here,

0



 rad 5:5 ( 0 )   14:5 180  rad 3 2 768:5 + 609:2 155:2 + 15:2  > 14:5 180

(14.23)

is the angle of atta k at whi h the wing/body lift is zero:

0 = 11:5

 180

(14.24)

The maximum lift oe ient is obtained at an angle of atta k of

 = 18 degree.

Negle ting the ee t of the tailplane, this is al ulated from equation 14.23 as:

CLmax = CLwb ( = 18

 ) = 2:75 180

(14.25)

CLt a ts on the aerodynami entre of ZS axis. It is given as: S (14.26) CLt = t 3:1 t S

The lift oe ient of the tailplane,

the tailplane and is also parallel with the

where

t denotes the angle of atta k of the tailplane and is al ulated from the

following equations:

t =  " + ÆT + 1:3

q lt VA

d " = ( 0 ) (14.27) d d = 0:25 d Here " is the downwash angle, ÆT is the tailplane dee tion, q is the air raft pit h rate, and lt is the longitudinal distan e between the aerodynami entre of the tailplane and the aerodynami entre of the wing and body. gure 14.4).

;

The aerodynami drag oe ient,

(See

CD , is a fun tion of the angle of atta k CD a ts on the

drag of the tailplane is negle ted and it is assumed that

aerodynami entre of wing and body:

CD = 0:13 + 0:07
(5:5 + 0:654)2 The aerodynami sidefor e oe ient,

(14.28)

CY , is also assumed to a t on the aero-

dynami entre of wing and body and is given by the following equation:

CY = 1:6 + 0:24ÆR where



is the angle of sideslip and

ÆR

(14.29)

is the rudder dee tion.

These non-dimensional oe ients an now be onverted to dimensional for es using the following relationships:

159

CL

wb

xB Va

CL

CoG

CD

α

ε

AC

δT

αt q=0

t

AC t

lt

Figure 14.4: Illustration of aerodynami for es



Aerodynami for e along

XS

X = D = CD



Aerodynami for e along

Aerodynami for e along

(14.30)

YS Y = CY



1 V 2S 2 A

1 V 2S 2 A

(14.31)

ZS

Z = L = CL

1  VA 2 S 2

(14.32)

To al ulate the translational motion of the air raft using equation 14.1, these for es need to be resolved into body axis for e omponents. The resolution from stability axes for es,

(D; Y; L), into the body-axes for es, (FxA , FyA , FzA ),

is given by the following expressions:

FxA = L sin  D os  FyA = Y FzA = L os  D sin  Aerodynami moments.

(14.33)

The moments due to the air raft aerodynami s are

Cl ; Cm ; Cn ),

determined by means of the moment oe ients, (

whi h are as-

sumed to a t about the aerodynami entre of the wing and body and are given

160

by the following equation:

2

Cl 4 Cm Cn

3 5

2

=

4 2

+

6 4 2

+

4

3

1:4 0:59 3:1 SSt lt ( ) 5 180 ) (1  15  3 2 3 11 0 5 p 2 7 0 4:03 SSt l2t 0 5 V A 4 q 5 1:7 0 11:53 2 3r 0:6 0 0:22 ÆA 0 3:1 SSt lt 0 5 4 ÆT 5 ÆR 0 0 0:63

(14.34)

where

p, q,and r are the rotational rates in body axes, ÆA is the aileron dee tion, ÆT is the tailplane dee tion, ÆR is the rudder dee tion. The moment oe ients about the entre of gravity are al ulated from these aerodynami entre based oe ients using the following equation:

2

with

3

2

3

1

0

0

2

31

Cl
x 0:12 CD 5 = 4 Cm 5 + 
y A  RBS
4 CY 5A C3n 02
z CL 31 2 32 Cl
x 0:12 0 CZ CY = 4 Cm 5 + 4 CZ 0
y 5A CX 5 4
z Cn CY CX 0

ClCG 4 CmCG CnCG

2

3

os  0 sin  RBS = 4 0 1 0 5 sin  0 os 

CX = CD os  + CL sin  CZ = CL os  CD sin 

(14.35)

(14.36)

The following expressions are used to onvert these non-dimensional moments oe ients into dimensional moments:



Rolling moment in body axes

LA = ClCG



(14.37)

Pit hing moment in body axes

MA = CmCG



1  V 2 S  2 A 1  VA 2 S  2

(14.38)

1  V 2 S  2 A

(14.39)

Yawing moment in body axes

NA = CnCG 161

These moments, in ombination with the moments due to thrust are then used to al ulate the rotational motion of the air raft from equation 14.8.

RCAM engine thrust al ulation The RCAM is a twin engined air raft model, and the thrust provided by ea h of the two engines is assumed to be aligned with the

x-body axis.

The thrust produ ed by a single engine is given by

Fi = ÆT Hi mg;

i = 1; 2 with m at the nominal mass of 120,000 kg and ÆT H1 the setting of the throttle handles. In equation 14.40,

(14.40) and

ÆT Hi

ÆT H2

determined by

should be expressed

in radians: this has no physi al meaning but appears to be onvenient in the

al ulations. The allowed value of

ÆT Hi

lies between

 0:5 180

and

 10 180

radians.

Note that the maximum thrust to weight ratio is about 0.35 (for both engines together). Hen e, the engine thrust ve tor at the enter of gravity is given in

FB

as:

2

F1 + F2 Fp = 4 0 0

3 5

(14.41)

Due to the geometri lo ation of the engines, see gure 14.5, the engine thrusts also ontribute to the moments a ting on the air raft. These moments

zM

6

6

front

F1

F2

6

6 O M - yM  P2 ACwb

ACwb



F1 ; F2

?xM P1

- xM

front

P1

P1 ; P 2

Figure 14.5: Appli ation points of thrusts. and

P2

are the points where the thrust is applied.

an be al ulated about the entre of gravity as follows:

2

3

2

3


x XAP T i Fi TEi = 4 YAP T i
y 5  4 0 5 (i = 1; 2) (14.42)
z ZAP T i 0 where
x,
y ,
z , XAP T i , YAP T i and ZAP T i are dened in table 14.4. 162

Atmosphere The atmosphere is onsidered to be onstant, irrespe tive of height and position, and onsequently we an dene the following:

kg

 = 1:225 3 m N

P = 101325:0 2 m T = 288:15 K where

 is the density of air, P

is the stati air pressure, and

(14.43)

T

is the absolute

temperature.

Gravity model Due to the restri ted altitude range to be used with this model, gravity is not

onsidered to be a fun tion of altitude. Hen e, gravity is assumed to have a

onstant value of:

g = 9:81 m/s2

(14.44)

14.2.4 Sensor models Models are not provided for the hara teristi s of the sensors:

they are all

assumed to be perfe t.

14.2.5 A tuator models and engine dynami s Both a tuators and engines are assumed to have rst order system dynami s with rate limits and saturations. The time onstants of the rst order system dynami s are:

  

engine models: 1.5 s, ailerons and tailplane a tuators: 0.15 s, and rudder a tuator: 0.3 s.

Numeri al values for rate limits and saturations are given as follows.



Rate limits for throttle movement are: rising slew rate



 = 1:6 180

 rad/s, = 1:6 180   rad  Æ 0:5 180 T Hi  10 180 rad.

rad/s, falling slew rate

throttle limits (saturations) are:

In ase of engine failure we an assume that the throttle setting for the

 rad ÆT Hi = 0:5 180 given by the transfer fun tion 1=(1 + 3:3s).

failed engine redu es to

163

with rst order system dynami s

  

  Æ_  25  rad/s; 25 180 A 180 rad,  saturations of aileron dee tion are: 25 180  ÆA  25 180   Æ_  15  rad/s; rate limits for tailplane dee tion are: 15 180 T 180  rad,  saturations of tailplane dee tion are: 25 180  ÆT  10 180   Æ_  25  rad/s; rate limits for rudder dee tion are: 25 180 R 180  rad.  saturations of rudder dee tion are: 30 180  ÆR  30 180 Rate limits for aileron dee tion are:

14.2.6 Wind turbulen e model Turbulen e is a sto hasti pro ess that an be dened by velo ity spe tra. Commonly used velo ity spe tra for turbulen e modelling are the Dryden spe tra.

V through a frozen turbulen e eld with a

rad/m, the ir ular frequen y of the turbulen e an be

For an air raft ying at a speed spatial frequen y of

al ulated as:

! =V


rad/s

(14.45)

With this, the spe tra an be des ribed as follows:

2Lug 1 V (1 + (Lug V! )2 ) L 1 + 3(Lvg V! )2 vg (!) = v2g vg V (1 + (Lvg V! )2 )2 L 1 + 3(Lwg V! )2 wg (!) = w2 g wg V (1 + (Lwg V! )2 )2 ug (!) = u2g

L

ug , vg , wg

L

L

ug , vg , wg and turbulen e standard deare dependent on altitude and atmospheri onditions.

The turbulen e s ale lengths viations

(14.46)

As an indi ator for the atmospheri onditions it is possible to take the wind

W20 ).

speed at 20 ft above the ground (

For moderat onditions,

m/s (30 kts) is sele ted. The turbulen e standard deviation

wg

W20 = 15:4

is then given

as follows:

wg = 0:1W20 ug and vg are assumed to be fun tions of wg and the altitude h. For h < 305 m (1000 ft): wg ug = vg = (0:177 + 0:0027h)0:4 and for

(14.47)

(14.48)

h > 305 m (1000 ft): ug = vg = wg

The turbulen e s ale lengths of altitude: for

3 < h < 305 m:

Lug , Lvg

Lug = Lvg =

and

Lwg

(14.49) are al ulated as a fun tion

h (0:177 + 0:0027h)1:2 164

(14.50)

and for

Lwg = h

(14.51)

Lug = Lvg = Lwg = 305 m

(14.52)

h > 305 m we take:

With this pro edure, the gust velo ities

ug , vg and wg are dened in the stability W xB , W yB

referen e frame. However, as an approximation the RCAM inputs and

W zB

are used.

To simulate turbulen e, white noise is ltered through forming lters. These lters an be derived from the Dryden spe tra given in equation 14.46. As an example, the transfer fun tion of the lter for simulating the gust velo ity will be onsidered. Given white noise

w, the spe trum of wg

an be obtained as:

wg = jHwg w (!)j2 w Where

wg

(14.53)

w = 1, and Hwg w (!) is the frequen y response fun tion of the forming

lter. Therefore,

L 1 + 3(Lwg V! )2 w2 g wg = jHwg w (!)j2 = Hwg w (!)Hwg w ( !) V (1 + (Lwg V! )2 )2

(14.54)

To obtain a stable and minimum phase lter, the following frequen y response fun tion is sele ted:

r

Hwg w (!) = wg Repla ing the variable

p

Lwg 1 + 3 LVwg j! V (1 + LVwg j!)2

(14.55)

j! by s, the following transfer fun tion is obtained: r

Hwg w (s) = wg The transfer fun tion for generating

p

Lwg 1 + 3 LVwg s V (1 + LVwg s)2

vg

is equivalent.

The transfer fun tion for generating

r

Hug w (s) = ug

(14.56)

ug

an be found as:

2Lug 1 V 1 + Lug s

(14.57)

V

It is important to note that for orre t appli ation of these lters the white noise inputs need to be independent. For a more detailed dis ussion on turbulen e modelling, the reader is referred to for example [35℄.

165

14.3 Design Problem Formulation and Evaluation Criteria 14.3.1 Motivation design and evaluation riteria Within the aerospa e industry there is a large amount of experien e in the ight ontrol system design area. For this reason, the main obje tive of the

ontrol problem stated here is not so mu h to obtain a satisfa tory ontroller, but more spe i ally to exhibit approa hes whi h might redu e the omplexity of ontrol laws and the overall ontrol system design y le. Some of the main features addressed by modern ontrol design te hniques provide the possibility to take into a

ount:

    

the multivariable nature of the ontrol problem the non linear behaviour of the plant the time-varying nature of the plant robustness to parameter hanges and un ertainties simultaneous performan e and robustness spe i ations.

From the onsideration of these features it is expe ted that improvements ould be made in areas su h as:

   

ontrol system ar hite ture development

ontrol law design y le

ontrol design solution

ontrol system implementation

The RCAM design hallenge onsists of the synthesis of a ontrol law apable of fullling an approa h to landing under various external onditions eg. turbulen e and windshear, while being robust to parameter hanges. Furthermore, the air raft guidan e must not degrade under engine failure.

Details on the

design obje tives are given in subse tion 14.3.2. For the uniform omparison of all design entries from the design hallenge parti ipants, a set of evaluation riteria is formulated in subse tion 14.3.3. To evaluate proper ontrol system logi and to make the hallenge more realisti , an evaluation traje tory has been designed to ree t typi al phases during approa h to landing. The evaluation riteria given in this subse tion are based on sets of signals from whi h ertain hara teristi s will be al ulated.

All

designs should be able to tra k the given traje tory within the spe ied bounds. Note that the hoi e of a traje tory as an evaluation riterion is independent of the ontrol law and ontrol design methodology. An important subje t onsidered in this hapter is the translation of design obje tives into evaluation riteria: the evaluation riteria should be su iently

166

representative for the onsidered design obje tives, but will not be able to

over all aspe ts. It is asked that the ben hmark problem parti ipants onsider the design obje tives given in subse tion 14.3.2 and for them to use their own methods to illustrate to what extent these are met by their ontroller design. For instan e, we give robustness spe i ations in terms of real parameter variations, although they are often also onsidered in the frequen y domain or in terms of gain and phase margins. The evaluation pro edure is only aimed at obtaining an obje tive measure for omparison with other designs.

14.3.2 Design riteria Introdu tion The ontroller design problem for the RCAM model is hara terised by a number of fundamental trade-os between oni ting design spe i ations.

For

typi al air raft autopilot systems we re ognise ve lasses of riteria:



performan e riteria: these ree t tra king error and disturban e reje tion hara teristi s of ertain signals;



robustness riteria:

these ree t the stability bounds with respe t to

parameter variations;



ride quality riteria: these ree t the desire to obtain su ient passenger and pilot omfort in the form of bounds on ertain maximum allowable a

elerations and minimum damping levels;

 

safety riteria: these ree t envelope safeguards;

ontrol a tivity riteria: these are a measure of the power onsumed by the ontrols and also give an indi ation of fatigue ee ts.

Performan e riteria The performan e of the ontrolled system an be spe ied in terms of ommand response hara teristi s to normalised referen e signals, tra king error and disturban e reje tion features (see [132℄). The ommand response hara teristi s are dened in terms of rise time

tr ,

settling time

ts

Mp . Rise y(t) takes from y = 0:10

and overshoot

time is dened here as the time the unit step response

y = 0:90, i.e., tr = t(y90% ) t(y10%). Settling time is here dened as the time y(t) to a hieve 99 per ent of its nal value. Finally, overshoot is dened as (ypeak y(1))  100% (see [82℄). the relative peak of y (t), i.e., Mp = y(1) P1- Lateral deviation. The ontrolled air raft's lateral deviation, eyb (t), dened to

for

as the dieren e between the a tual and ommanded lateral air raft position,

y(t) y (t), should be redu ed to 10 per ent within 30 s.

There should be very little overshoot in the response to a unit step in

Mp < 5%. 30% in order to obtain higher tra king

lateral ommand signals at altitudes above 305 m (1000 ft), i.e., At lower altitudes

Mp

may in rease to

167

performan e.

There should be no steady state error due to onstant lateral

wind disturban es. In the nal phase of ight (landing approa h glide path) the lateral deviation from the desired ight path should not ex eed that given in gure 14.6.

6Maximum deviation from lo alizer path

6Maximum verti al deviation

20 m

6m

5m

1.5 m

100 ft

400 ft

-

Altitude

100 ft

-

Altitude

400 ft

Figure 14.6: Maximum lateral de-

Figure 14.7: Maximum verti al de-

viation

viation

P2- Altitude

ommands,

response. The ontrolled system should be able to tra k altitude

h ,

with rise time

tr < 12

s and settling time

ts < 45

s.

There

should be very little overshoot in the response to unit steps in altitude om-

mands at altitudes above 305 m (1000 ft), i.e., Mp < 5%. At lower altitudes Mp may in rease to 30% in order to obtain higher tra king performan e. In the nal phase of ight (landing approa h glide path) the verti al deviation from the desired ight path should not ex eed that given in gure 14.7

P3-

Heading angle response. The ommanded heading angle,

tra ked by the a tual heading angle, settling time

ts < 30 s.

, with a rise time

,

tr < 10

should be s and and

There should be very little overshoot in the response

to unit steps in heading ommands at altitudes above 305 m (1000 ft), i.e.,

Mp < 5%.

At lower altitudes

Mp

may in rease to

30%

in order to obtain

higher tra king performan e. For unit RMS intensity lateral Dryden gust, the RMS of the heading angle error in losed loop should be less than that in open loop.

P4-

, should tr < 5 s and

Flight path angle response. The ommanded ight path angle,

be tra ked by the a tual ight path angle, settling time

ts < 20 s.

,

with a rise time

There should be very little overshoot in the response

to unit steps in ight path angle ommands at altitudes above 305 m (1000 ft), i.e.,

Mp < 5%.

At lower altitudes

Mp

may in rease to

30% in order to obtain

higher tra king performan e.

P5-

,

Roll angle response. In ase of engine failure in still air, the roll angle,

should not ex eed 10 deg; its maximum steady state deviation should not

ex eed 5 deg. During engine failure, sideslip angle



should be minimised; the

steady state roll angle that is needed to a hieve this, should be redu ed to zero with an overshoot of less than 50 % when the failed engine is restarted (the failed engine's throttle setting steps ba k to that of the a tive engine). Under moderate turbulen e onditions (see subse tion 14.2.6)

168

 should remain smaller

than 5 deg.

P6-

Airspeed response. The ontrolled system's airspeed,

to tra k speed ommands,

ts < 45 s.

VA ,

with a rise time

VA ,

should be able

s and settling time

There should be very little overshoot in the step response to speed

ommands at altitudes above 305 m (1000 ft), i.e.,

Mp

tr < 12

may in rease to

30%

Mp < 5%.

At lower altitudes

in order to obtain higher tra king performan e. In

the presen e of a wind step with an amplitude of 13 m/s (25 kts) there should be no deviation in the airspeed larger than 2.6 m/s (5 kts) for more than 15 s. There should be no steady state error due to onstant wind disturban es.

P7-

Heading rate.

In ase of engine failure, the maximum heading rate,

_,

should be less than 3 deg/se .

P8-

Cross oupling between airspeed

manded altitude

h

VA

and altitude

h.

For a step in om-

of 30 m, the peak value of the transient of the absolute

VA and ommanded airspeed VA should be smaller than 0.5 m/s VA of 13 m/s (25 kts), the peak value of the transient of the absolute error between h and h should

error between

(1 kt). Conversely, for a step in ommanded airspeed

be smaller than 10 m.

Robustness riteria R1-

Centre of gravity variation. Stability and su ient performan e should

be maintained for horizontal entre of gravity variations between 15 and 31 % and verti al entre of gravity variations between 0 and 21 % of the mean aerodynami hord (see table 14.5; we will not onsider variations in lateral dire tion).

R2-

Mass variations.

Stability and su ient performan e should be main-

tained for air raft mass variations between 100000 to 150000 kg.

R3- Time delay.

Stability and su ient performan e should be maintained for

transport delays from 50 to 100 ms.

Ride quality riteria Ride quality riteria should ensure su ient passenger and pilot omfort. The following spe i ations are designed to obtain an a

eptable level.

Q1- Maximum verti al a

eleration.

Under normal onditions during manoeu-

vres (no turbulen e) the verti al a

eleration at the entre of gravity should be minimised; it should be less than

Q2-

 0.05 g1.

Maximum lateral a

eleration. Under normal onditions during manoeu-

vres (no turbulen e) the lateral a

eleration at the entre of gravity should be minimised; it should be less than

Q3-

 0.02 g.

Damping. Unless stated dierently, there should be no overshoot in any

step response of any ontrolled variable at altitudes above 305 ft (1000 ft). 1

This value is used in industry during the design phase, in fa t the verti al and lateral

a

eleration limits depend on frequen y. They are even lower at 2 Hz.

169

Below that altitude overshoot may in rease to 30 % in order to obtain higher tra king performan e.

Safety riteria S1-

Airspeed. The airspeed must always be larger than

Vstall denotes the stall speed, i.e. to maintain ight. relation:

1:05  V

, where stall the speed below whi h the air raft is unable

This speed an be found from the following equilibrium

1 2 CLmax mg = SVstall 2

(14.58)

Substituting the relevant values from hapter 14.2, and assuming a mass of

120000 kg, we obtain Vstall = 51:8 m/s. S2-

Angle of atta k. In subse tion 14.2.3 it was given that the maximum lift

oe ient is obtained at an angle of atta k of 18 degree.

Hen e, it an be

on luded from the previous requirement that the stall speed orresponds to:

stall = 18 deg.

A value of 12 deg is onsidered a

eptable.

S3- Roll angle. The maximum roll angle  should be limited to 30 deg. S4- Sideslip angle response. At all times, sideslip angle should be minimised. For unit RMS intensity lateral Dryden gust the RMS of the sideslip angle in

losed loop should be less than that in open loop.

Control a tivity riteria C1- A tuator eort minimisation.

Under moderate turbulen e onditions (see

subse tion 14.2.6), mean a tuator rates for aileron, tailplane and rudder should be less than 33 % of the maximum rates (see subse tion 14.2.5).

C2-

Engine eort minimisation. Under moderate turbulen e onditions (see

subse tion 14.2.6), mean throttle rate should be less than 15 % of the maximum rate (see subse tion 14.2.5).

14.3.3 Evaluation pro edure: RCAM mission and s enario To be able to evaluate all kinds of dierent ontrol design pro edures and resulting ontrollers it is ne essary to nd a uniform evaluation pro edure, independent of the design method.

An established pro edure to do this is to

dene a mission and a typi al landing approa h s enario (see [258, 37, 262℄). This mission onsists of manoeuvres that an be evaluated by means of nonlinear simulations. The performan e of the ontrol law depends on the mission phase, within whi h hard riteria or bounds on ertain signals should be met and/or error signals must be minimised. The mission and s enario to be own by the RCAM model onsists of a landing approa h divided into the following segments (see gure 14.8)

170

Trajectory for RCAM evaluation

altitude (−ZE) [m]

1500 2 e

d

f

1

1000 c

b

3 g

h4

a

500

Runway

0 Wind

0 0 −5 −10 −15 y−position (−YE) [km]

−20

−25

−10

−15

−20

−5

0

x−position (XE) [km]

Figure 14.8: the landing approa h for RCAM



Segment I (0 to 1). Starting at an altitude of 1000 m and with a tra k angle of

 = 90 deg, 1

level ight is to be maintained with a onstant airspeed of 80 m/s . During this segment, the lateral features of the autopilot will be investigated by simulating failure of the left engine (engine 1). This is indi ated in gure 14.8: the failure o

urs at point at point

b.

a, after whi h the engine is restarted

The transient and steady state behaviour of the system will

be analysed.



Segment II (1 to 2). This segment onsists of a ommanded o-ordinated turn from points

to

d

_ =3

with a heading rate of

deg/se .

The obje tives are to

maintain a onstant speed of 80 m/s, to keep the lateral a

eleration

lose to zero, and to restri t the bank angle to

 = 30 deg with onsistent

rudder/aileron dee tions.



Segment III (2 to 3). The des ent phase will be started a

ording to the so- alled Frankfurt des ent pro edure (see [35℄), whi h has been proposed for reasons of environmental noise redu tion. This des ent pro edure is engaged later and is steeper than the lassi al des ent, whi h has a onstant glide slope angle of

= 3

deg. The starting altitude is

h = 1000 m. After a short

= 6 deg at point

period of level ight, the ight path angle is set to

e, and to = 3 deg at point f.

1

The desired airspeed is 80 m/s.

The nominal airspeed during the landing phase depends on the air raft mass, it is taken

equal to 1.3 times m/s.

Vstall :

with a maximum landing weight of 150000 kg this results in

171

 80



Segment IV (3 to 4). The glide slope of

= 3 deg is to be maintained during a wind shear

between points

and

g

h.

The air raft has to maintain safe ight and

should not deviate too far from the desired glide path. The wind shear model used in the evaluation pro edure is a two dimensional model derived from [201℄ (also see subse tion 14.3.4 for more information). The desired airspeed is 80 m/s. Throughout the evaluation pro edure a Dryden turbulen e eld, of s ale length

L = 305 m and

amplitude

 = 0:08 m/s,

is assumed to be a tive. Note that

the amplitude is only 5% of the amplitude for moderate onditions as dened in Chapter 14.2: this is done to prevent that the ee t of turbulen e on lateral and longitudinal a

elerations overrules other ee ts that we are interested in. Superimposed on top of this turbulen e is a 10 m/s onstant wind with a xed heading. This onstant wind is a tive in full respe t during Segments I and II

d, and is slowly redu ed to zero between points d of Segment II g of Segment IV (at the start of the wind shear model). The wind has no

until point and

verti al omponent and is dire ted along the negative earth-xed

x-axis,

i.e.,

it is a ross-wind during Segment I and a headwind during Segment III. To he k robustness properties the entire approa h will be own with a most forward, a nominal and a most aft horizontal entre of gravity lo ation. Furthermore, one ight will be exe uted with a nominal entre of gravity lo ation and a time delay of 100 ms.

14.3.4 Translation of design riteria into evaluation

riteria It should be noted that it is not possible to he k all desired autopilot features by ying a single landing approa h traje tory.

Furthermore, the evaluation

pro edure should be relatively simple and straightforward: we want to be able to apply it to a great variety of ontrollers.

Hen e, the evaluation riteria

should be independent of the type of ontroller used: they should onsist of

al ulable indi ators that enable us to obtain an obje tive omparison between

ompletely dierent ontrollers. For these evaluation riteria we will use the same lassi ation as was given in the denition of the design riteria.

    

performan e robustness ride quality safety

ontrol a tivity

For ea h of these items and for ea h of the four traje tory segments a single number will be al ulated. This number should not be onsidered to be the

172

nal word on overall autopilot performan e: it is merely an indi ator for one or two important aspe ts. In most ases it is hosen su h that a value of smaller than one is a

eptable. To further evaluate the dynami behaviour of the autopilot, we will onsider several plots of key variables during ea h of the segments.

We will ompare

the shape of the a tual traje tory with the demanded traje tory and provide bounds that should be respe ted for good performan e. Similarly, we will plot the most important deviations from the desired traje tory.

Segment I For segment I we will plot a plan view of the referen e traje tory and the four traje tories dened in subse tion 14.3.3, and then superimpose the bounds

a

given in gure 14.9. The points

and

b orrespond to the

beginning and end

First segment: top view

x−deviation [m]

100

50

0

0

a

b

1

−50

−100 −20

−18

−16

−14

−12 −10 −8 y−position (−YE) [km]

−6

−4

−2

Figure 14.9: Segment I: the ee t of engine failure with bounds of the engine failure segment.

Performan e. The bound of 20 m is given to a

ount for the ee t of turbulen e. During engine failure, we allow a maximum lateral deviation of 100 m that should be qui kly redu ed to less than 20 m at the end of the segment (when the air raft should be stabilised again). With

eyb denoting the lateral deviation

in body o-ordinates for the traje tory with nominal entre of gravity and time delay we will use

 j eyb (t)j jeyb (t1 )j =2 + max



t

100

20

(14.59)

as a measure that should be smaller than one for su ient performan e. Note that during the entire segment the maximum deviation of 100 m should not be

t1 orresponds with point 1) the

ex eeded, and that at the end of the segment (

maximum deviation of 20 m is taken into a

ount.

Performan e deviation. The maximum dieren es between the lateral deviation of the traje tories with nominal and perturbed entre of gravity and nominal and maximal time delays are onsidered:


eyb (t) := max (jeybmax(t) eyb (t)j; jeybmin (t) eyb (t)j) 173

(14.60)

We will allow dieren es of 10% of the maximal allowable lateral deviations:






eyb (t)
eyb (t1 ) max =2 + t 10 2

(14.61)

should be smaller than one.

Ride quality. The maximum lateral a

eleration will be limited by:

jny j < 1

(14.62)

0:2

i.e.

jny j should be smaller than 0.2g: under normal ight onditions this value

should be mu h lower (0.02g, see subse tion 14.3.2), but engine failure is an emergen y situation su h that an unusually large lateral a

eleration is a

eptable.

Safety. During the segment, the maximum angle of atta k

max t



 will be limited:

j(t)j 3 < 1 12

(14.63)

 = 12 deg; the power is taken to stress the fa t that  > 12 deg qui kly be omes una

eptable (stall situation).

This implies we a

ept

Control a tivity. The rudder a tuator eort will be onsidered that is needed to stabilise the air raft after engine failure is lifted: this is al ulated as:

Z t1

tb with

ÆR2 dt

(14.64)

tb denoting the end of engine failure ( orresponds to point b in gure 14.9).

This value is not normalised to one as it is not lear what bounds an be obtained: it will a t as a value for relative omparison of ontrollers.

Segment II For segment II we will plot a plan view of the referen e traje tory and the four traje tories dened in subse tion 14.3.3, and then superimpose the bounds given in gure 14.10. Furthermore, to obtain a better insight in the results, we will plot lateral deviations with bounds as given in gure 14.11.

Performan e. The maximum lateral deviation (due to the turn) and the lateral deviation at the end of the segment (when the air raft should be stabilised again) are onsidered:



 j eyb (t)j jeyb (t2 )j max =2 < 1 +

t

200

20

174

(14.65)

Second segment: top view

Second segment: lateral deviations

1

300

0.5 d

2

200 lateral deviation [m]

y−position (−YE) [km]

0 −0.5 −1 −1.5

c

−2 −2.5 −3

1

100

0 1

c

d

2

−100

−200

−3.5 −4 −24

−23

−22

−21 −20 −19 x−position (XE) [km]

−18

−17

−300 0

−16

1

2

3 4 5 6 along track distance from point 1 [km]

Figure 14.10: Segment II: plan view

Figure 14.11:

of the 90 degree turn with bounds

deviations

7

8

Segment II: lateral

during

the

90

degree

turn with bounds

Note that during the entire segment a maximum deviation of 200 m should not

t2 orresponds with point 2)

be ex eeded, and that at the end of the segment (

a maximum deviation of 20 m is taken into a

ount.

Performan e deviation.

As in segment I, the maximum dieren es between

the lateral deviation of the traje tories with nominal and perturbed entre of gravity lo ations and nominal and maximal time delays are onsidered. Again, we will allow dieren es of 10% of the maximal allowable lateral deviations:



Ride quality.




eyb (t)
eyb (t2 ) max =2 < 1 + t 20 2

As in segment I, the maximum lateral a

eleration

onsidered:

i.e.

(14.66)

ny

jny j < 1 0:02

will be

(14.67)

jny j should be smaller than 0.02 g.

Safety. As in segment I, the maximum angle of atta k during the segment is limited:

max t



j(t)j 3 < 1 12

(14.68)

Control a tivity. The rudder and aileron a tuator eort is al ulated as:

Z t2

t1


ÆR2 + ÆA2 dt

(14.69)

This value is not normalised to one as it is not lear what bounds an be obtained: it will a t as a value for relative omparison of ontrollers.

175

Segment III For segment III we will plot a side view of the four traje tories dened in subse tion 14.3.3. Figure 14.12 shows the referen e traje tory, the start and

2

end points of the segment (points

ommand a tions are labelled with

and

e

3) and the onsidered bounds; the f. We will also plot the verti al

and

deviation of the traje tories and overlay the bounds shown in gure 14.13. Third segment: side view

Third segment: altitude deviations

1100

1000

30 2

20 altitude deviation [m]

altitude (−ZE) [m]

e 900

800 f

700

600

500 −17

−15

Figure 14.12:

−14 −13 x−position (XE) [km]

−12

Segment

−11

III:

f

3

0 e

−10

−20

3

−16

10 2

−30

−10

side

−16

−15

Figure 14.13:

−14 −13 x−position (XE) [km]

−12

−11

Segment III: verti-

view of the -6 and -3 degree glides-

al deviations during the -6 and -

lope aptures with bounds

3 degree glideslope aptures with bounds

Performan e.

The maximum verti al deviation during the apture of the -6

degree glideslope and the verti al deviation at the end of the segment (when the air raft should be stabilised again) are onsidered.

Furthermore, speed

variations should be kept small in spite of the hange in required angle of atta k. With

ezb

denoting the verti al deviation in body o-ordinates for the

traje tory with nominal entre of gravity and time delay, we will demand



 j ezb (t)j jezb (t3 )j jVA VA j =3 < 1 max + +

20

t

for su ient performan e.

6

4

(14.70)

Note that during the entire segment a maximum

deviation of 20 m should not be ex eeded, and that at the end of the segment

t3

(

orresponds with point

3)

a maximum deviation of 6 m is taken into a -

ount. Speed variations should not ex eed 4 m/s, i.e. 5% of

VA = 80 m/s).

Performan e deviation. The maximum dieren es between the verti al deviation of the traje tories for the nominal and perturbed entre of gravity lo ations and nominal and maximal time delays are onsidered:


ezb (t) := max (jezbmax(t) ezb (t)j; jezbmin (t) ezb (t)j)

(14.71)

We will allow dieren es of 10% of the maximal allowable verti al deviations:






ezb (t)
ezb (t3 ) max =2 < 1 + t 2 0:6 176

(14.72)

Ride quality. The maximum verti al a

eleration

nz

will be limited:

jnz j < 1

(14.73)

0:1

i.e.

jnz j should be smaller than 0.1 g.

Safety. Again, the maximum angle of atta k during the segment is limited:

max t



j(t)j 3 < 1

(14.74)

12

Control a tivity. The tailplane a tuator eort is al ulated as:

Z t3

t2

ÆT2 dt

(14.75)

This value is not normalised to one as it is not lear what bounds an be obtained: it will a t as a value for relative omparison of ontrollers.

Segment IV For segment IV we will plot a side view of the four traje tories dened in subse tion 14.3.3.

The wind shear model, the desired traje tory through it,

and the bounds are given in gure 14.14. As mentioned before, the wind shear Fourth segment: side view with windshear 1600 1400

altitude (−ZE) [m]

1200 1000 800 600

3 g

400 200

h

4

0 −11000−10000−9000 −8000 −7000 −6000 −5000 −4000 −3000 −2000 −1000 x−position (XE) [m]

Figure 14.14: Segment IV: side view of the nal approa h with wind shear and bounds model is a two dimensional model derived from [201℄. Along the traje tory, the

W xE = 7 m/s, then W xE = 7 m/s, ombined with a

air raft will be fa ed with a headwind going up to about windspeed will hange to a tailwind of about

177

downdraught of about

W zE = 8 m/s (see gure 14.15).

The result of this will

be a drasti de rease in air raft energy: the air raft will not be able to stay on the desired traje tory. The size of the longitudinal deviation and the time until re overy will be measures for evaluation of the ontroller. For this reason we will also plot the longitudinal deviations with bounds as given in gure 14.16. Fourth segment: wind velocities during wind shear

Fourth segment: altitude deviations 30 WXE WZE

8

20

6 altitude deviation [m]

wind velocities WXE and WZE [m/s]

10

4 2 3 g

h

4

0 −2 −4 −6

10

3 g

4

h

0

−10

−20

−8 −10 −11000−10000−9000 −8000 −7000 −6000 −5000 −4000 −3000 −2000 −1000 x−position (XE) [m]

Figure 14.15:

−30 −11

Segment IV: wind

−10

−9

−8

Figure 14.16:

speeds along the traje tory

−7 −6 −5 −4 x−position (XE) [km]

−3

−2

−1

Segment IV: verti-

al deviations during the nal approa h with bounds

Performan e. The maximum longitudinal deviation (due to the wind shear) and the longitudinal deviation at the end of the segment (when the air raft should be within the de ision window) are onsidered for the traje tory with nominal entre of gravity and time delay:



 j ezb (t)j jezb (t4 )j + =2 < 1 max

t

(14.76)

1:5

20

Note that during the entire segment a maximum deviation of 20 m should not

t4 orresponds with point 4)

be ex eeded, and that at the end of the segment (

a maximum deviation of 1.5 m is taken into a

ount.

Performan e deviation. As in segment III, the maximum dieren es between the verti al deviation of the traje tories for the nominal and perturbed entre of gravity lo ations and nominal and maximal time delays are onsidered. Again, we will allow dieren es of 10% of the maximal allowable verti al deviations:






ezb (t)
ezb (t4 ) =2 < 1 max + t 2 0:15

Ride quality. As in segment III, the maximum verti al a

eleration limited:

jnz j < 1 0:2

i.e.

(14.77)

nz

will be

(14.78)

jnz j should be smaller than 0.2g (in segment III this value is lower, but

wind shear is an emergen y situation).

178

Safety. We will onsider whether the air raft is within the de ision window at the end of the segment. Lateral, verti al and speed variations are limited to 5 m, 1.5 m and 3 m/s respe tively as follows:

s 



1 eyb 2 ezb 2 (VA VA ) 2 ( ) +( ) +( ) <1 3 5 1:5 3

Control a tivity. follows:

(14.79)

The tailplane and throttle a tuators eort is al ulated as

Z t4

t3


ÆT2 + (ÆT H1 + ÆT H2 )2 dt

(14.80)

This value is not normalised to one as it is not lear what bounds an be obtained: it will a t as a value for relative omparison of ontrollers.

179

15.

The Classi al Control Approa h

Jim Gautrey

1

Abstra t.

This do ument des ribes the design and analysis of a

lassi al ontroller developed for the Resear h Civil Air raft Model (RCAM) design hallenge. The hallenge onsists of designing an autopilot for an air raft whi h will enable it to meet pre-determined requirements and to follow a pre-dened traje tory. Classi al ontrol methods have been used for this parti ular design hallenge entry, and were found to give satisfa tory performan e and robustness.

15.1 Introdu tion The design approa h adopted here for the RCAM design hallenge [145℄ is based on lassi al ontrol methods. This method uses feedba k of measurable variables plus ontrollers based on proportional, integral and derivative gains to obtain the required performan e. The method is not inherently robust, and therefore many dierent operating ases must be simulated with the ontroller in order to verify its robustness. The design pro ess used for the RCAM ontroller produ es an autopilot whi h is similar to an a tual air raft autopilot, sin e the nature of the air raft leaves little s ope for variation in the design, and lassi al ontrol methods are the most ommon design methods used for autopilot development. The ontrollers were designed using the Control Systems toolbox within Matlab. No other toolboxes were required during the design pro ess. Classi al ontrol methods are urrently used for the majority of ontrol systems designed for air raft as they were the only methods available for many years, and therefore a large amount of knowledge and experien e has been a

umulated. Their use within the air raft industry in ludes appli ations to the design of major systems, su h as piloted ight ontrol, autopilot, and autostabilisation su h as pit h and yaw dampers. These methods are also widely used outside the aerospa e industry. The main advantages of this design methodology are that it is well proven and a large reservoir of knowledge exists about a tual implemented systems. However, a prin ipal disadvantage is that a reasonably large amount of prior knowledge is required on erning the operational and behavioural hara teristi s of the plant sin e the design pro ess is arried out intera tively by the 1

Air Vehi le Te hnology Group, Craneld College of Aeronauti s, Craneld University,

UK.

180

ontrol system designer. It is the designer who has to spe ify the hanges in the

ontroller during the design pro ess in order to generate the required performan e and robustness, as opposed to methods where the ontroller is generated and optimised automati ally from a series of onstraints determined by the designer.

15.1.1 Controller Stru ture Classi al ontrol methods usually result in an autopilot ontroller of the form seen in gure 15.1.

It onsists of inner loops, whi h generally augment the

stability of the air raft and outer loops, whi h generally ontrol the air raft ight path. The system model may, or may not in lude gain s heduling or other non-linearities su h as a tuator rate and position limiting.

Gain s heduling

was not required in the ontroller developed here. However, rate and position limits were required, and the development of these elements is des ribed. Gain s heduling would be required in an autopilot intended for operation over a large ight envelope due to the variation in the aerodynami hara teristi s of the air raft. FEEDBACK LOOPS

INNER LOOPS

AIRCRAFT OUTPUTS

OUTER LOOPS COMMANDS

Figure 15.1: Classi al autopilot ontroller stru ture

S heduling an be organised in two prin ipal ways for an autopilot. The rst is gain s heduling when the autopilot is in one parti ular mode of operation, su h as height hold. In this ase, individual ontroller parameters will be varied to take a

ount of the non-linear hara teristi s within the air raft, su h as variations in its hara teristi s with speed or onguration hange. The se ond form of s heduling is mode swit hing. For example, if the autopilot hanges from height hold to glideslope hold, a dierent set of ontrol laws may be used. In pra ti e, this is a

omplished through the sele tion of an alternate outer loop.

The design presented in this do ument has limited gain s heduling to

take a

ount of autopilot mode hanges, but it does not take a

ount of gross speed or air raft re onguration hanges. These forms of gain s heduling are not restri ted solely to air raft.

15.1.2 Complex ontroller onsiderations The ontroller designed here are simple ontrollers. However, there are many real world onsiderations whi h need to be made. This se tion highlights the areas where the design of a simple ontroller su h as the one illustrated above needs to be modied to ope with real life situations.

181

Gain s heduling Classi al ontrollers, at their simplest level, are designed at a single operating point. The RCAM lassi al ontroller is no ex eption to this. In order to a

ount for hanges in operating onditions, gain s heduling is used to modify the ontroller gains as the operating point of the plant hanges. This is used sin e it is usually a straightforward pro ess as there are relatively few gains in a lassi al ontroller, and the gain s hedules an be very straightforward. For example, most air raft autopilots require a gain s hedule whi h a

ounts for variations in the air raft airspeed, and this is often enough to ensure good performan e over a wide range of operating onditions. However, there is sometimes a requirement to s hedule gains with respe t to other parameters, and this an sometimes ompli ate the ontroller design.

Pra ti al onsiderations A tual ontrol systems suer from problems whi h perfe t systems do not. These in lude non-perfe t dynami s of the plant under onsideration whi h

an result in the ontroller having to ontrol a plant whi h is dierent to the one whi h it was originally designed for. Also, outputs from sensors are rarely perfe t, and therefore this needs to be taken a

ount of.

Filtering may be

required to ompensate for noise in the signals. In addition, the lassi al ontroller onsidered for the RCAM problem does not take a

ount of airframe stru tural modes, whi h it may ex ite.

Again,

ltering is used to take a

ount of this by removing outputs to the air raft

ontrol surfa es at the frequen ies whi h would ex ite the stru tural modes.

Outer loop variations For ontrollers su h as an autopilot, it is desirable to have several dierent modes, or to design the ontroller so it may perform several tasks.

This is

usually a

omplished through the use of dierent outer loops. The same inner loop is used for the dierent modes, but dierent outer loops are engaged depending on the task requirements. This is a ommon pra ti e amongst lassi al ontrol designers as it enables the ontroller stru ture to be kept simple while enabling spe ied tasks to be performed well.

It has been used in the lassi al ontroller for the RCAM

design.

15.2 The Sele tion of the Controller Ar hite ture for the RCAM Problem The autopilot has been split into a longitudinal and a lateral ontroller, with no intera tion between the two.

Ea h is des ribed individually.

longitudinal and lateral ontrollers have inner and outer loops.

182

Both the

Available measurement signals have been used, see referen e [145℄. For an a tual implementation, the design would use the most suitable of the available signals as some are better than others, i.e. measure.

have less noise or are easier to

For the purposes of the design hallenge, all output signals have

been assumed to be perfe t. The only de ien y in the signals available is the la k of a lateral a

eleration and pit h attitude signal. These are ommonly available to autopilots sin e they are relatively easily measured, and are usually

onsidered to be essential.

The pit h attitude signal has been synthesised

for the purposes of the design hallenge, but attempts to synthesise lateral a

eleration were unsu

essful. Inner and outer loops have been used in the ontroller design pro ess. The inner loops are used to provide the ne essary stability augmentation, while the outer loops regulate the augmented air raft's ight path performan e. Fun tional blo k diagrams for the longitudinal and lateral ontrollers an be seen in gures 15.2 and 15.3.

Height error

K

Height rate

+ -

Controller for step height demands

K S

K

+ Attitude Limiter

+

+ Pitch Attitude

Height error

P+I+D Controller Controller for ramp height demands

Mode Switch

Pitch Rate

-

K K

+

Tailplane Demand

+

Rate Limiter Airspeed demand

+ +

Airspeed

Throttle Demand P+I+D Controller

Figure 15.2: Longitudinal ontroller blo k diagram

15.3 The Translation of RCAM Design Criteria into Method Dependent Obje tives The RCAM design riteria are set up in method independent terms in subse tion 3.2 of the RCAM manual [145℄. This subse tion des ribes how these method independent riteria may be translated into method dependent requirements.

183

Track error

K Track rate

+ -

K S

Roll Attitude Limiter

+

+ +

K

Roll Attitude Roll Rate

-

K K

+

Aileron Demand

+

P+I Controller

K

Sideslip Angle

+

Washout Filter Roll Angle

K

+

K

+

Rudder Demand

Washout Filter Yaw Rate

Figure 15.3: Lateral ontroller blo k diagram

Classi al ontrol methods do not rely on the denition of weighting matri es or other optimisation routines sin e the optimisation pro ess is arried out by hand, and therefore there is no dire t translation of design riteria into the design pro ess in this respe t.

However, as the ontroller design progresses,

the various riteria are veried, and ontroller design iteration is based on its performan e with respe t to those riteria. This is addressed in the following subse tions where ea h of the dierent riteria are dis ussed. Most riteria rely on the examination of step responses.

Performan e Criteria The ontroller stru ture is partly determined by the performan e requirements. For example, transient and steady state requirements often di tate whether an integrator should be used in a parti ular ontrol loop. For a step height demand, an integrator is not stri tly required. However, in order to a hieve zero steady state error following a ramp height demand (su h as a glideslope), at least one free integrator is required in the ontroller. Therefore this an generate oni ting ontroller design requirements for dierent autopilot fun tions, whi h require a solution su h as outer loop swit hing or gain s heduling. After the ontroller stru ture has been determined, requirements su h as overshoot and response limits an be ontrolled dire tly by the gain sele tion pro ess. This is one of the easiest riteria to design for sin e responses, su h as the air raft height response to a parti ular demand an easily be evaluated

184

from the earliest stages of lassi al design using the te hniques presented here, and therefore the ee t of gain hanges an easily be assessed. For example, if a parti ular response is ex essively sluggish, it may indi ate that a parti ular value for a damping gain within a ontroller is too large.

Robustness Criteria Classi al ontrol te hniques are not inherently robust. Therefore it is ne essary to test the designed ontroller at many dierent ight ases to ensure that the design is suitable for ea h ase.

For this parti ular design exer ise, a mid-

envelope ight ase is used for ontroller development, and then the result is

he ked at a series of ight ases throughout the ight envelope (see table 15.1),

orresponding to a wide variation in air raft mass, entre of gravity position and omputational time delay. The stability of the ontroller an be he ked at ea h stage by he king the pole lo ations on the s-plane of the linear air raft model plus autopilot. The bandwidth of ea h omponent of the autopilot plus air raft an also be veried. In addition, gain and phase margins may be used.

Passenger Comfort Criteria The passenger omfort riteria are primarily governed by the design of the inner loops sin e these are the loops whi h regulate the air raft's manoeuvre performan e and its transient response to atmospheri disturban es. Therefore it is ne essary to he k for passenger omfort as the design progresses as with the performan e riteria, and to iterate until the design meets the requirements. The outer loops do ae t the passenger omfort riteria, but with the orre t inner loops, the outer loops should not have a large ee t on it. Limiters may also be required in order to limit demand su h as pit h rate pla ed on the

ontroller in order to meet the omfort requirements.

Safety Criteria The safety riteria primarily govern the air raft attitude and speed and are primarily ae ted by the design of the inner loops. For example, it is normal to limit the air raft attitude demands whi h the outer loops an make on the inner loops. This does not permit the outer loop to demand an ex essive pit h attitude when the air raft may not have su ient thrust to hold the required airspeed. For the lateral tra k loops, this onsists of limiting the maximum roll demand that the ontroller an make. Therefore by areful design of the inner loops, and sele tion of appropriate limiters between the outer and inner loops, the air raft safety requirements an be attained. The air raft airspeed an be ontrolled pre isely by the autothrottle design.

By designing the autothrottle loop well, the airspeed an be pre isely

ontrolled for normal ight ases. For abnormal onditions, su h as windshear or mi roburst, simulation will demonstrate whether the air raft meets the requirements.

A orre t inner loop design will ensure that the air raft should

185

meet the requirements for normal ight. For abnormal ight, su h as an engine failure, it is ne essary to simulate and therefore demonstrate that the air raft meets the requirements. However, due to the intuitive nature of lassi al ontrol te hniques, analysis of the air raft response an be used to highlight any problems, and often a solution an be found from this.

Power Consumption Criteria The power onsumption riteria are not signi antly ae ted by the ontroller design. If there is a problem with the design, su h as a large os illatory motions or transients, then the higher power onsumption may be the result of ex essive

ontrol a tivity on one parti ular ontrol surfa e.

For example, this would

indi ate a problem within the inner ontrol loops, su h as ex essive gain, and may therefore be modied by adjusting the inner loop gains a

ordingly.

15.4 The Design Cy le Des ription This subse tion des ribes the a tual design y le. Therefore the design y le for ea h ontroller omponent is onsidered, along its appropriate parameter adjustment strategy. For ea h air raft model omponent, the inner loops are designed rst using linear analysis, primarily using robustness, omfort and some performan e riteria. The design is then tested with the full non-linear air raft simulation to ensure that the design works in this domain. Safety onsiderations are onsidered in order to design any limits whi h need to be pla ed on the inner loop demands. The outer loops are then designed in the linear domain, primarily with performan e requirements in mind.

Finally, the resulting ontroller is

transferred to the simulation environment and formally evaluated with respe t to all of the design riteria. For ea h of the longitudinal and lateral ontroller designs, the following steps are performed. 1. Assemble the appropriate linear model from the non-linear air raft model. This gives a model in state spa e form whi h an be used to represent the air raft. 2. Augment the air raft model with the required sensor and a tuator dynami s, and a linearised time delay if appropriate. 3. Design the inner stabilising loops using the te hniques des ribed below. These are then assessed against the appropriate riteria. 4. Design the outer autopilot loops against the inner stabilising loops, and re he k that the model meets the requirements. 5. Che k the ontroller for robustness by simulating at a number of ight

ases over the ight envelope and with dierent air raft ongurations.

186

6. Perform the nal evaluation. Iteration is required during this pro ess, both within and between steps. A suitable air raft ight ase must also be used for the baseline design pro ess. The ase used has an air raft mass of 120000kg, a g position at [0.23,0,0℄ (see [145℄),

VAIR = 80

m/s and

h = 1000

m.

This is in the middle of the ight

envelope, and seems to provide a reasonable design point.

15.4.1 Longitudinal Controller Design Before des ribing the detailed design of the longitudinal ontroller, it is ne essary to des ribe the prin iples behind its overall operation. It onsists of three main omponents. The rst omponent is the pit h inner loop, whi h augments the longitudinal air raft stability, and whi h has the same ee t as a pit h attitude demand ontroller. The se ond omponent is the autothrottle, whi h uses airspeed feedba k to throttles to regulate the airspeed. The third omponent is the pit h outer loop, whi h regulates the air raft height by translating height error into a pit h attitude demand. In designing the longitudinal ontroller, it is ne essary to design the inner loops rst. These an then be used to augment the air raft, and the outer loops therefore designed around this augmented air raft. Experien e has shown that it was not ne essary to onsider the time delay during the initial design, but slight gain modi ations were made during the simulation of the inner loops with dierent ight ases to allow for time delay variations.

Pit h Attitude Inner loop The pit h attitude inner loop omprises two feedba k paths. The rst is pit h rate feedba k, whi h is used to augment the damping of the short period mode, and the se ond is the a tual pit h attitude demand loop. A tuator or time delay dynami s were not required sin e it was found that their ex lusion from the linearised model used for ontroller design did not have a deleterious ee t on the ontroller performan e.

A short period damping of lose to unity is

desirable sin e this removes any residual pit h os illations from the

 response,

and therefore the gain required an be found from the interse tion of the short period root lo us with the real axis of the plot.

In addition, high phugoid

damping is required. Testing the inner loop ontroller in the simulation environment showed that improved pit h attitude tra king ould be obtained by modifying the gains previously derived. In reasing the gains gave in reased



tra king performan e.

This modi ation pro ess was done iteratively with the autothrottle operational. The autothrottle design is des ribed in the next subse tion.

Speed Inner Loop The speed ontrol omponent is the se ond omponent of the longitudinal autopilot.

Autothrottles have traditionally in luded feedba ks to the throttles

187

from airspeed, and sometimes the pit h attitude error and height error. However, this autothrottle onsists of feedba k of the speed error to the throttles through a Proportional + Integral + Derivative ontroller. This ensures good speed tra king sin e the integrator will ensure that the speed error is redu ed to zero while the dierentiator will smooth the transient response to a speed

hange demand, i.e. ensuring that the longitudinal a

eleration is regulated. Modi ations were made to the baseline autothrottle design on the results from non-linear simulation to step speed and height responses, and also from simulating a range of dierent ight ases. For the design onsidered here, with a maximum permitted longitudinal a

eleration of

 0.02g, a rate limiter pla ed in the demanded speed signal path 0:1962 ms 2 satisfa torily limits the longitudinal a

eleration

having limits of

to a

eptable limits.

Height Outer Loop The third omponent of the longitudinal ontrol system is the height ontrol outer loop. For the outer loop design, two ases were onsidered initially. The rst is the response to a step demand in height.

In order to a

ommodate

air raft trim hanges, it is ne essary to in lude an integrator with the system, otherwise, the air raft may have a onstant steady state error in height if a speed hange is made. This is di ult to do without getting an overshoot in height.

However, the ontroller stru ture used in gure 15.2 does give zero

overshoot with zero steady state error for trim hanges in a height hold situation. The se ond ase was the air raft response to a ramp height demand, or zero steady state error to a non-zero ight path angle demand. This required the addition of an integrator in the height error loop whi h integrates the height error signal to ensure zero steady state error. In order to limit the demanded attitude, an attitude limiter was pla ed between the height outer loop and inner pit h attitude loop. This was limited to approximately +30 degrees (nose up), -12 degrees (nose down). The limits were derived from the autopilot authority, i.e.

with the limits in pla e the

air raft ould hold the sele ted pit h attitude without signi antly a

elerating or de elerating. In pra ti e, lower limits ould be used sin e su h large attitudes are rarely required when under autopilot ontrol. However, analysis showed that the normal a

eleration aused by the ontroller is ex essive, and therefore a rate limiter is pla ed between the inner pit h loop and the outer height loops. This limits the peak normal a

eleration by limiting the pit h rate.

It must be remembered that the ontroller outer

loop design must not make ex essive demands on the inner loop sin e when the limiter is a tive, the air raft is, in open-loop ight, and in a sense, un ontrolled. However the inner loops, retain full authority ontrol of the air raft at all times.

188

15.4.2 Lateral Controller Design As with the longitudinal ontroller, the lateral ontroller has three omponents, shown in the on eptual design, gure 15.3. The rst omponent is the yaw damper. This in ludes the use of the rudder for turn o-ordination, sideslip and lateral a

eleration ontrol. The se ond omponent is the roll attitude inner loop, whi h is used to regulate the roll attitude through the use of aileron. The nal omponent is the lateral outer loop whi h is used to regulate the lateral tra k, and feeds into the roll attitude inner loop.

Yaw Damper and Turn Co-ordination Controller A yaw damper has one main purpose, to in rease the Dut h roll damping in order to enable the autopilot and pilot to y the air raft. A se ond fun tion is to support turn o-ordination, whi h is often alled yaw autostabilisation. For the purposes of this report, the term yaw autostabilisation will refer to both fun tions.

There are several dierent methods whi h may be used to design

the yaw autostabiliser. The nal ontroller used sideslip angle feedba k to the rudder to a hieve this sin e it serves as a useful way to minimise the sideslip angle, whi h then a hieves turn o-ordination. The use of this feedba k has the ee t of destabilising the aperiodi spiral motion, and in reasing the frequen y of the Dut h roll mode, whi h in this ase gives an in rease in dire tional stability. The design an be arried out in the following steps. 1. Sideslip angle to rudder feedba k gain sele tion. The beta to rudder root lo us plot was used to sele t the required Dut h roll natural frequen y. Sele ting a value for the Dut h roll natural frequen y of about 2 rad/s gives a value of 13 rad/rad for the



to rudder

gain. 2. Yaw rate to rudder gain sele tion. The next stage is to sele t the values for the yaw rate to rudder gain. This has the ee t of in reasing the Dut h roll damping ratio. A desirable value of Dut h roll damping is approximately 0.7. It is also ne essary to in orporate a high pass lter sin e the



gain previously sele ted will be

the only gain whi h is not ltered. Filtering is required sin e the other signals have a onstant value in a steady turn whi h must not be passed through to the rudder. Therefore a washout lter with a time onstant of 0.2 s is used. The hoi e of time onstant was arbitrary, and this is typi al of a value used in pra ti e.

The yaw rate to rudder root lo us

plot was used to do this. 3.

 feedba k to the rudder. The addition of washed out



feedba k is the next stage. This has the

ee t of modifying the lateral a

eleration hara teristi s of the ontroller. 4. Addition of



integrator.

The next stage is to add an integrator into the beta feedba k to rudder

189

loop. This is in luded to meet the engine failure requirements sin e this

ase requires a large amount of rudder to be held in order to redu e the ee ts of the failure. An I/P ratio of 0.2 was found to be suitable.

5. Veri ation with simulation. The a tuator ould then be added in, and the ontroller operation veried in the simulation environment. Analysis showed that it had little ee t on the time responses. However the



to rudder gain was in reased to

1.3 rad/rad in an attempt to optimise the lateral a

eleration. It has been stated that the gains used here in rease the RMS

response

ompared to the unaugmented air raft, whi h is not desirable. However, this was ne essary due to the la k of lateral a

eleration signal.

Roll Attitude Inner Loop In the same way as the pit h attitude is ontrolled with pit h attitude and pit h rate feedba k to the tailplane, the air raft is ontrolled in roll by using roll attitude and roll rate feedba k to the aileron. Ordinarily, feeding roll attitude error to aileron (with a suitable gain) results in an air raft whi h is stable in roll attitude sin e roll rate is essentially proportional to aileron input (for a

onventional air raft up to moderate roll angles). However, the RCAM air raft is not onventional laterally.

The roll angle response to a small aileron step

input is a step hange in roll angle, due to the spiral mode whi h is rapid and stable. Therefore it is ne essary to augment the hara teristi s of the air raft before the roll attitude autopilot an be designed. The stages in designing the inner roll loop are as follows.

It has been

assumed that the yaw autostabilisation has been arried out.

1. Roll rate feedba k to aileron. This provides a more step-like roll rate response to a step aileron demand.

2. Roll attitude feedba k to ailerons. Then a

 demand loop is losed around the augmented ailerons so that

the input an be used to demand a given roll angle.

3. A tuator Ee ts. The a tuator was then introdu ed to the design. It had little ee t on the step response. This on ludes the design of the roll inner loops in the linear domain.

Simulation of a roll angle demand showed that the simulation was very

lose to the linear response, and the design was suitable. However, iteration was performed in order to qui ken the response as with the pit h attitude inner loop. This on luded the design of the inner roll loop.

190

Lateral Tra k Hold Autopilot The outer lateral loop ontrols the air raft's lateral tra k and turn performan e. It is a lassi al tra k ontrol loop, where lateral tra k error, with lead ompensation, is fed into the roll attitude demand loop. This gives a zero steady state tra k error when the air raft is tra king a straight line, sin e any tra k error

auses the air raft to a

elerate towards the demanded tra k, and the damping prevents the air raft from overshooting the demanded tra k ex essively. This type of lateral tra k holding has been ommonly used in the past. The lateral tra k error is onverted into a roll angle demand by multiplying it by a predetermined gain. However, in order to be able to ope with a nonsymmetri al ight ase, su h as engine failure, it is ne essary to in lude an integrator, as with the height step ontroller.

A damped omponent of the

lateral tra k error was also in luded in the tra k error to

 demand sin e trials

showed that the response tra k error was very under-damped without it.

Heading Demand Outer Loop A separate heading demand outer loop was designed for the ontroller. This then generates a bank angle demand signal whi h is then fed into the inner roll loop to ontrol the air raft heading. The s hemati for the heading outer loop is similar to the height step demand outer loop.

15.5 Linear Analysis For the purposes of linear analysis, the ight ases from table 15.1

were on-

sidered. All are at an airspeed of 80 m/s.

Flight Case

Mass (Kg)

CGx

CGy

CGz

Time Delay (ms)

e1

120000

0.23

0

0

0.05

e2

120000

0.23

0

0

0.1

e3

120000

0.31

0

0

0.05

e4

120000

0.15

0

0

0.05

t1

100000

0.23

0

0

0.05

t2

150000

0.23

0

0

0.05

t3

100000

0.15

0.03

0.21

0.1

t4

100000

0.15

0

0

0.05

t5

150000

0.15

0.03

0.21

0.1

t6

100000

0.31

0

0.21

0.1

t7

100000

0.31

0

0.21

0.05

t8

150000

0.31

0.03

0.21

0.1

t9

150000

0.31

0.006

0.21

0.1

Table 15.1: Flight ases onsidered for analysis

191

A series of evaluation runs were performed. The longitudinal analysis onsidered height steps, airspeed steps and glideslope inter epts. The design requirements listed in [145℄ were all met. The lateral analysis onsidered heading angle and lateral tra k steps, and again all of the requirements listed in the design manual were met.

15.6 Analysis of the Resulting Controller in terms of the Applied Methodology During the design phase, the prin ipal analysis te hniques are omprised of the following methods. The inner loop designs were, on the whole, arried out in the frequen y domain using root lo us te hniques to modify the air raft eigenvalue hara teristi s. Then the design emphasis swit hed to the time domain, sin e this is the domain in whi h many of the performan e riteria are dened, and the ee t of hanging ontroller gains an be more readily understood. However, the eigenvalues were observed from time to time during this pro ess to ensure that they remained stable.

Problems with ontroller performan e,

su h as overshoot in a parti ular response an generally be ontrolled by intuitive gain modi ation.

The problem hara teristi and knowledge of the

air raft provide the information to assess whi h gain needs adjustment. Simulation was used extensively during the design phase to fa ilitate the observation of the ee ts of gain variation. Linear simulations were used initially with step and ramp input fun tions. Non-linear simulations were used subsequently in order to ne tune the gains and to provide a nal he k that the ontroller performan e in the non-linear domain was a

eptable. Ride quality was assessed using the simulation with various inputs, su h as a step heading demand and a step height demand. A

eptable ride quality an be a hieved by sensible gain sele tion and by the orre t hoi e of feedba k variables.

Again, simulation was used to show whether a parti ular hange

made to the ontroller had an ee t on ride quality. Analyti al methods are presented in [166℄, but they were found to require modifying on the basis of simulation results. Control a tivity an be analysed by monitoring during a parti ular manoeuvre. In the ase when there is high ontrol a tivity on one parti ular surfa e, that a tivity an be redu ed by better gain sele tion, or in extreme ases, by the use of a lter su h as a low pass lter. Resort to su h xes was not required during the design phase for the RCAM ontroller, sin e, in ases where there was ex essive ontrol a tivity, it ould generally be redu ed by a better gain sele tion in the troublesome feedba k path.

15.6.1 Satisfa tion of Design Obje tives Performan e The primary emphasis is on performan e, spe i ally path tra king, and all of the requirements were met in the nal evaluation.

192

However, there were

ompromises with some of the other riteria, spe i ally ride omfort, and to a lesser extent, robustness.

The main robustness variation was in transient

performan e and the steady state performan e was generally onstant. All of the lateral deviation riteria spe ied in the design manual were met.

Altitude Response. All of the altitude riteria spe ied in the design manual were met.

Heading angle. The rise time and settling time riteria were met for all the ight ases onsidered. The open-loop RMS heading angle with unit RMS lateral turbulen e intensity (W20

= 8 m/s) was found to be 0.50 degrees.

engaged, it was found to be 0.89 degrees.

With the heading angle hold

Therefore the ontroller does not

meet the turbulen e riterion. This was found to be due primarily to the integrator in the sideslip to rudder path. If this integrator was removed, the RMS error redu es to 0.54 degrees, whi h just ex eeds the non-augmented ondition.

Flight Path Angle Response. The overshoot riterion was met for all ight ases. However the rise time riterion was not met. This riterion was relaxed in order to allow for the ride quality riterion to be met for the glideslope a quire, as it is required to limit normal a

eleration to less than

 0.1g. Therefore, the rise

time was 8 se onds instead of the stipulated 5 se onds.

Roll angle with engine failure. The maximum roll angle divergen e and steady state value was met with the ontroller following engine failure. However the maximum overshoot of 50% is not met with the lateral tra k loop sele ted in the autopilot sin e the air raft has to roll ba k to attain zero tra k error, ausing an overshoot in roll attitude. If a onstant heading is demanded then the roll attitude overshoot riterion is met (overshoot in roll is 40%) sin e there is no

orre tion required to attain zero lateral tra k error.

Airspeed response. The rise time riterion was met for all CG ases. However, the settling time riterion was not met for the non-zero lateral CG positions. This was unexpe ted sin e the integrator in the speed ontrol loop should for e the error to zero.

Again, the trim routine was suspe ted.

The wind step

deviation riterion was met. From the evaluation routine, it was observed that the largest airspeed deviation during the nal evaluation, in luding the wind shear, was 0.8 m/s.

Heading rate response. The heading rate riterion under engine failure was met sin e the maximum heading rate with an engine failure was 0.5 deg/s. This is observed during the evaluation phase.

Cross oupling from airspeed to altitude.

The requirements to minimise the

airspeed error for a given altitude step input and to minimise the attitude error for a given airspeed step were met.

193

Safety The safety requirements were not designed for spe i ally.

However, during

the nal evaluation, it was veried that the requirements were met.

Airspeed Regulation. The airspeed regulation riterion was always met as the airspeed was always within

 0.8 m/s of the demanded airspeed.

Angle of atta k. This riterion was met in the nal evaluation. The angle of atta k never ex eeded 3.5 degrees or be ame less than 0.5 degrees.

Roll angle. The requirement to limit the roll angle to less than 30 degrees was met.

Sideslip angle. The sideslip angle is ontinually minimised by the a tion of the proportional gain and integrator following engine failure. The RMS value of the lateral gust velo ity was al ulated and a W20 value [145℄ of 8 m/s gave a value of 1.02m/s for the RMS lateral gust intensity, and therefore this value was used to al ulate the response. The RMS value of



was found to be 0.58

degrees for the open-loop air raft, and 0.89 degrees for the augmented air raft. Therefore the requirement was not met. The trade-o here is made in the yaw autostabiliser design between lateral path tra king and the need to redu e the engine failure transient, see subse tion 15.4.2.

Robustness The prin ipal design emphasis was pla ed on satisfying the robustness requirements during the design pro ess. After the performan e requirements had been satised, only minor adjustments were made to the ontroller design in order to meet the robustness requirements.

There was some ompromise between

robustness, omfort and performan e, and a ontroller whi h seemed to give the best overall balan e was sele ted.

Time Delay. The requirement to be able to ope with a time delay of up to 0.1 se onds were met. This aused some problems with the height step response in the nal evaluation, and onsequently the ontroller was modied by in reasing the gains in the height outer loop, whi h gave a more onsistent response for dierent air raft masses and CG positions.

Mass Variations. The ability to ope with variations in air raft mass are met. Centre of Gravity Variation. The ability to ope with variations in the entre of gravity positions were met for CGx and CGz variations. For variations in the lateral entre of gravity position, (CGy), problems were en ountered whi h were thought to be due to the trim routine.

Ride Quality Criteria As the design progressed, the omfort requirements were he ked. However, it was found that the lateral a

eleration requirement ould not be met due to the unavailability of lateral a

eleration signal for feedba k purposes. This would

194

be available in a onventional autopilot, and experimentation showed that the lateral a

eleration ould be redu ed if this signal were available. Some of the peak values of normal a

eleration ex eeded the limit, but not ex essively so.

Longitudinal A

eleration. A maximum longitudinal a

eleration of 0.02g was a hieved, ex ept during the turn, where there was a (-0.08g peak), and during the windshear, where there was a +0.1g / -0.04g peak.

Verti al a

eleration.

The riterion was met, ex ept during the turn when

the verti al a

eleration ex eeded the permitted limits.

The required load

fa tor to maintain the turn rate is approximately 0.1g, and during this phase, the maximum verti al a

eleration is less than 0.2g. The normal a

eleration during the third segment was also slightly greater than the permitted value.

Lateral A

eleration. This was less than 0.025g for the majority of the nal evaluation. It peaked at approximately 0.055g during turn entry and exit, but was redu ed to about 0.025g as the turn be ame established. In order to redu e it further it would be ne essary to in orporate lateral a

eleration feedba k into the lateral ontroller. The lateral a

eleration requirement during engine failure was met.

Damping. This riterion was met for the majority of the ight ases. See the performan e riteria subse tion for more detail.

Control A tivity During the Final Evaluation Control a tivity was within the permitted levels at all times, and it was mostly well under the maximum permitted values.

Control A tivity in Turbulen e Problems arose during ontroller performan e evaluation in turbulent ondi-

to rudder gain resulted in higher rudder displa ements gust, whi h oered no improvement to the RMS gust response.

tions.Having a high for a given

The integrator is in luded in the ontroller in order to omply with the engine failure requirement. The high gain is in the ontroller to redu e



in order to

redu e the lateral a

eleration and also to ope with the engine failure ase. The rudder and aileron rates were measured in the simulated moderate turbulen e, and were less than the maximum values.

15.7 Results of the Automated Evaluation Pro edure This se tion presents the methodology independent results of the ontroller designed in the previous se tions. It is mostly based on the evaluation mission and s enario dened in [145℄: both overall tra king performan e and innerloop behaviour of the ontrolled system will be evaluated by means of bounds on key variables.

195

Segment I: the ee t of engine failure. As the RCAM air raft model is twin-engined, a single engine failure will mainly result in lateral deviation. Hen e gure 15.4 provides a top view of the rst traje tory segment. Second segment: lateral deviations

First segment: top view

300 100

200

0

0

a

b

lateral deviation [m]

x−deviation [m]

50

1

−50

100

0 1

c

d

2

−100

−200 −100 −20

−18

−16

−14

−12 −10 −8 y−position (−YE) [km]

−6

−4

−300 0

−2

1

2

3 4 5 6 along track distance from point 1 [km]

7

8

Figure 15.4: Left: segment I - the ee t of engine failure. Right: segment II lateral deviations during the

3o /s turn

Fourth segment: altitude deviations 30

20

20

10 2

f

3

0

−10

e

altitude deviation [m]

altitude deviation [m]

Third segment: altitude deviations 30

3 g

4

h

0

−10

−20

−20

−30

10

−16

−15

−14 −13 x−position (XE) [km]

−12

−30 −11

−11

−10

−9

−8

−7 −6 −5 −4 x−position (XE) [km]

−3

−2

−1

Figure 15.5: Left: segment III - verti al deviations from the desired glideslope. Right: segment IV - verti al deviations from the desired glideslope.

Segment II: the 3 deg/s turn. Figure 15.4 provides a loser look at the a tual lateral deviations during the se ond segment of the ight, whi h is a 90 deg turn at 3 deg/s.

Segment III: the apture of the -6 and -3 degrees glideslope. The verti al deviations from the desired glideslope are plotted in gure 15.5, whi h is the third ight segment.

Segment IV: the nal approa h with windshear. While on nal approa h with a glideslope of -3 deg the ee t of a windshear model is onsidered.

The verti al deviations from the desired glideslope are

196

plotted in gure 15.5.

Numeri al results Here a table of numeri al results based on the dis ussed simulation results is given 15.2. For the motivation and al ulation prin iple of the various results see [145℄. Segm. I

Segm. II

Segm. III

Segm. IV

Total

Performan e

0.0907

0.1847

0.1523

0.1228

0.1376

Perf. Dev.

0.0727

0.0381

0.1608

0.0933

0.0912

Comfort

0.3732

1.2615

1.1771

0.5771

0.8472

Safety

0.0041

0.0301

0.0063

0.0419

0.0206

Power

0.0027

0.0061

0.0148

0.0298

0.0133

Table 15.2: Numeri al results of the evaluation pro edure

15.8 Con lusions Classi al Control methods are the most ommon methods used for autopilot design. They are not inherently robust, but the ontrollers that are produ ed

an be made robust with orre t ontroller stru ture, time and eort. In addition, they are not initially easy to use, sin e mu h knowledge is required about both the methods and the air raft in question. However, one this knowledge has been obtained, for one parti ular air raft, the methods may readily be extended to other air raft or ight ases without too mu h redesign due to the simple nature of the ontrollers produ ed, and the high visibility of the design. These methods are also rigorously analysable, and therefore they an be readily ertied, and sin e they ontain relatively few omponents, the ee ts of failure of some of those omponents an be assessed relatively easily. There is a great deal of experien e on erning their use and implementation available within most vendors and airframe manufa turers. The ontroller designed here was at a single ight ase.

The pro ess to

extend this design to a number of ight ases is essentially simple due to the relatively small number of gains required in the ontroller, and in addition, there are simple ways of s heduling the gains for variations with airspeed. Their prin ipal disadvantage is the time taken to perform the design pro ess, although if an outline design is available this time an be redu ed. It is

ommon in industry for an existing autopilot design to be modied to suit a new air raft, as opposed to a ompletely new design being performed, and this redu es the design time. A signi ant amount of knowledge on erning air raft and their hara teristi s is also required to support the design pro edure sin e the optimisation of the ontroller depends on the knowledge and intuition of the designer and not a omputer algorithm.

197

The ontroller designed here meets the majority of the requirements spe ied for the RCAM hallenge, and any deviation has been noted. The majority of the dis repan ies would not pose insurmountable problems in pra ti e sin e the way in whi h the ontroller would be operated is dierent. For example, a glideslope inter ept would generally be arried out from below the glideslope, and the autopilot would know in advan e what is being demanded of it. Also, the autopilot would not have to ope with an engine failure in the manner required here sin e a tual engine parameters would be available to it. Sin e an a tual autopilot would not have to ope with these requirements in quite the same way, the design ould be tailored more towards the other requirements su h as ride omfort. Problems were experien ed with ertain ride

omfort levels, and these would be alleviated if other signals had been made available to the ontroller. Other improvements to the autopilot presented here

ould also be made through improved gain sele tion, for example. Finally, this ontroller demonstrates that a simple, robust ontroller an be produ ed with lassi al ontrol design te hniques, although there is a trade-o between some design and performan e requirements.

198

16. Multi-Obje tive Parameter Synthesis (MOPS)

Hans-Dieter Joos Abstra t.

1

Multi-obje tive parameter synthesis (MOPS) provides

a systemati way for omputational ontrol law tuning by dire tly spe ifying and iterating bounds and demands on stability quality,

ontrol performan e, and physi al ontrol realization onstraints. The design an be based on linear or non-linear models and the ontroller stru ture an be hosen in a onvenient way taking advantage of the design engineer's knowledge and prior expertise. Given design spe i ations an be transformed dire tly into the mathemati al design riteria needed for performing the method. Variations in stru tured parameter sets and operating onditions are taken are of by a multi-model formulation. The multi-obje tive/multi-model parameter tuning is done in a goal-oriented way by applying parameter optimisation. In this ontribution it is shown that the MOPS systemati an be used to develop robust ight ontrol systems with good performan e and passenger omfort.

16.1 Introdu tion The RCAM ben hmark problem addresses the design of an autopilot for the nal approa h of a transport-type air raft. The designed ontroller is required to be robust with respe t to variations in speed, weight, entre of gravity position (both verti al and horizontal), time delays, non-linearities, and engine failure. The method whi h is applied is an optimisation-based multi-obje tive/multimodel/multi-parameter design methodology alled Multi-Obje tive Parameter Synthesis (MOPS). Be ause of the inherent freedom in spe ifying ontroller stru tures, this approa h an be used in ombination with any ontroller synthesis method, and be ause of the possibilities in spe ifying the design goals in a onvenient multi- riteria manner this yields a systemati goal-oriented way to solve the ontrol design tuning problem. Two dierent approa hes are adopted for synthesising ontroller parameters. For the lateral ontroller, dire t optimization of the parameters of an appropriate ontroller stru ture was applied. For the longitudinal ontroller an indire t 1

DLR German Aerospa e Resear h Establishment, Institute for Roboti s and System

Dynami s, Control Design Engineering (Prof. G. Grübel), D-82234 Wessling E-mail: dieter.joosdlr.de

199

approa h was hosen: An analyti synthesis method (we used LQR-synthesis) denes the ontroller stru ture and its parameters.

The free parameters of

the synthesis method i.e. the entries of the weighting matri es Q and R serve as tuners that are optimized a

ording to a set of design riteria. Instead of the LQR synthesis method, other methods like parameterized eigenstru ture synthesis ould have been used as well. The ontrollers have been designed by taking into a

ount both performan e and robustness expli itly. The performan e spe i ations given in the RCAM Design Challenge Manual are dire tly used as multiple design obje tives to be treated simultaneously. Robustness is treated by using a multi-model approa h. That is, a ommon ontroller is designed simultaneously for a hosen set of evaluation models.

These models hara terize the variations in weight and

entre of gravity. In multi-obje tive design ea h one of the design-obje tives has to be mathemati ally des ribed as a single riterion.

For the present design task, the

spe i ations for air raft performan e, safety, and ontrol eort as given in Chapter 14 an be easily transformed to suited mathemati al riteria fun tions whi h serve to ompute an individual gure of merit for ea h design obje tive. Expli itly taking are of ea h design obje tive by a orresponding gure of merit makes the ne essary design- ompromise de isions very transparent.

16.2 Applied Control Design Methodology 16.2.1 Controller Stru ture and Parameterisation Both linear and nonlinear ontrollers (e.g. fuzzy-logi ontrollers [129℄) an be used. If a spe i analyti al ontrol law synthesis te hnique is applied within this framework, the ontroller stru ture is naturally bound to this synthesis te hnique. Here ontrol laws are parameterised in two dierent ways.

In a lassi al

P-I-D ontrol stru ture with additional shaping lters the P-I-D gains and the lter parameters are tuned to satisfy losed-loop performan e and robustness requirements. This is for lateral ontrol. An LQR output feedba k ontrol law for longitudinal ontrol is parametrized by the state- and ontrol eort weights Q and R. The free design parameters (i.e. the P-I-D gains and the LQR-weights) are

omputed by a min-max parameter optimisation set up.

16.2.2 System Model Des ription and Robustness Both linear and nonlinear design models are used. There is no methodi restri tion on the representation of system disturban es. Robustness against stru tured parameter variations or, e.g., sensor failures is a hieved by applying a ommon ontroller to a set of worst- ase xedparameter models. Here, this set represents a parameter-nominal model and

200

two dynami worst- ase models, i.e.

the slowest and the fastest one, deter-

mined by a nonlinear a priori parameter study. For ea h model an appropriate set of riteria an be spe ied individually. Thereby the multi-model problem is transformed to a multi-obje tive problem. In general, there exists no theory that guarantees stability or performan e robustness over the range of operation, if only a nite number of operating points is onsidered simultaneously. It depends on the physi al properties of the system to be designed whether runaways an exist.

If they exist, they

have to be added to the set of operating points treated by the multi-model approa h. In this design study we dete ted no runaways. Robustness of the ontroller around an operating point an be enfor ed in the multi-obje tive approa h by applying suitable robustness riteria su h as gain/phase margins in addition to the hosen performan e riteria.

16.2.3 Design Spe i ation The design spe i ations an be requirements in state spa e (e.g. eigenstru ture), time-domain (e.g.

step response, ontrol rate), and frequen y-domain

(e.g. bandwidth, stability margins et ). Ea h design obje tive is to be mathemati ally des ribed by a positive denite riterion

i to be minimized or to be

onstrained by an upper bound, .f. Se tion 16.4.

16.2.4 Design Cy le Des ription The designer formulates a MOPS setup by dire tly spe ifying the design goals as a set of positive-dened omputational riteria.

By this multi-obje tive

formulation all the various oni ting design goals are taken are of individually. To ea h riterion

i a demand value di is asso iated by the designer.

Then the

tuning parameters T of the ontrol law are omputed by solving the min-max parameter optimization problem

min max f i (T )=di g T

i

subje t to performan e and tuning restraints:

gj (T )  dj ; Tkmin  Tk  Tkmax: The solution provides a best-possible pareto-optimal ontrol parameter tuning. It provides dire t quantitative information about the design oni ts and onstraints for a hosen ontroller stru ture. Iterating the demand values

di

in an evolutionary manner for es the trade-o solution into a desired goal

dire tion. For a more detailed introdu tion, see Chapter 2. A typi al design loop is shown in Figure 16.1 illustrating that the MOPS methodology omplements hosen modelling-, ontrol synthesis-, and analysis methods by losing the loop via a goal oriented ontrol parameter tuning.

201

synthesis model

d

MOPS

c

T

synthesis

plant model

P

performance/cost criteria

controller model

closed-loop model

M

simulation/ analysis

I

Figure 16.1: MOPS loses the design loop by ontroller parameter tuning.

16.2.5 Pra ti al Software Aspe ts Appli ation of MOPS requires the set-up of a omputation loop a

ording to Figure 16.1 and availability of a suitable min-max parameter optimization software. The overall omputing time mainly depends on the riteria evaluations. Hen e fast algorithms and software implementations are to be used for the

orresponding analysis omputations. In our ase, synthesis, analysis, simulation, and optimization, are performed

2

in ANDECS

[99℄ whi h provides appropriate design-data management fa ili-

3

ties. In this environment, omputations an also be performed by MATLAB or by MATRIXX

4.

The non-linear air raft model, the a tuators, and the

5

parametrized ontrol stru ture are modelled in Dymola . From Dymola models DSblo k- ode [186℄ is generated automati ally. This simulation-model ode is dynami ally linked to ANDECS or MATLAB/SIMULINK. The latter is used to perform the automated evaluation pro edure des ribed in Se tion 16.7.

16.3 Controller Ar hite ture The ontroller is split into a longitudinal and a lateral ontroller. For onvenien e and visibility in design and maintenan e, these ontrollers are treated separately from ea h other. Sin e MOPS does not pres ribe a spe i ontroller stru ture, we de ided to apply two dierent approa hes. This also shows the exibility of the design method regarding ontroller ar hite tures.

16.3.1 Lateral ontroller For the lateral ontroller we hoose onventional stru tures with P-I elements, wash-out lters, or just onstant gains as they are do umented in textbooks. See for example [165℄ and [35℄.

The parameters of these elements are used

2

ANDECS is a registered trademark of DLR.

3

MATLAB, SIMULINK are trademarks of The MathWorks, In .

4

MATRIX

5

Dymola is a produ t of Dynasim AB, Lund, SE

X

is a trademark of Integrated Systems, In .

202

as tuning parameters during optimisation. The ontroller is omposed of an inner loop ontroller for stabilising and handling-qualities augmentation, and an outer loop ontroller for guiding the air raft along a given traje tory. The stru ture of the overall lateral ontroller is shown in Figure 16.2. The gain of

psidotc

ylat

2

2

da

2

2

dr

80 /v

phic inner loop

outer loop

80 /v

ylatc

p phi r beta

Figure 16.2: Stru ture of the lateral ontroller of dynami order 7.

802=VA2 to over hanges in the ontrol surfa e ee tivity due to speed variations. VA denotes the measured the ontrol ommands is s heduled by the fun tion

airspeed, and 80 m/s is the airspeed the ontroller is designed for. The overall lateral ontrolleris 7th order in luding two integrators and a prelter for lateral tra king ommands. The inner loop, see Figure 16.4, onsists of a bank angle integral ontrol system with onstant gain feedba k roll rate damper and a sideslip suppression system with yaw damper and wash-out network to allow hanges in yaw rate when hanging the air raft's heading.

A ross oupling gain between bank

angle and rudder is introdu ed to improve the de oupling of both motions. The outer loop ontrols the lateral deviation from the ommanded path. For this a P-I-D ontroller and a 2nd order lter is introdu ed. The ne essity of the lter for stabilisation has been found by root lo i onsiderations. Only the lateral deviation is used as feedba k variable, see Figure 16.4. The ontroller distinguishes lateral ommands (no hange of heading) and turn ommands. For lateral ommands, a rst order prelter is used. For turns also the demanded heading rate is ommanded.

16.3.2 Longitudinal ontroller The stru ture of the ontroller results from the synthesis method hosen: LQR state feedba k with a hosen synthesis model, see Figure 16.5. The synthesis model is the linearized 5th order longitudinal model in luding the altitude

z

of the air raft. The parameters of the synthesis model are

hosen right in the middle of the given parameter ranges: m = 125000 kg,


x = 0:23;
z = 0:105.

A design airspeed of 80 m/s is assumed. This 5th

order synthesis model is augmented by linear rst order lag lters to model the a tuators with time onstants given in Chapter 14. It is further augmented by integrators for airspeed and altitude ontrol, respe tively. The resulting on-

203

p

phic

+

2

80 /V

+

kr

da

2

1/s

kr1 -

phi +

kr2

kyr

r

ky s s + ay

2

+

80 /V

2

dr

beta ky1

Figure 16.3: Stru ture of the inner lateral ontroller psidotc

atan(v/g * psidotc)

turn off

1 ylatc

+

fd

Tvf s+1

phic

s 2+fz1s+fz0

+

2

s +fn1s+fn0

ylat

Figure 16.4: Stru ture of the outer lateral ontroller

commands zc

. x = Ax+Bu+Ec y = Cx+Du+Fc

-

vac

2

dT dTH1

actuators

dTH2

-

-

-

-

z0 va0

2

80 /v

q0 nx0

q nx measurements nz wv

nz0 wv0

trim conditions

Figure 16.5: Stru ture of the longitudinal ontroller of dynami order 4.

204

troller is therefore of 4th order, i.e. 2 integrators and an internal model of the a tuators. There is no need for a seperate design of inner and outer loops. The LQR-synthesis method yields a state feedba k ontrol law. However, employing the provided 6 measurements

q; nx ; ny ; wv ; z; and VA ; it is possible

to substitute the state by the available measurements using a (pseudo) inverse of the measurement matrix.

The overall longitudinal ontroller is therefore

given by the following equations

1

2

x_ =

6 6 4 2

+ 

6 6 4



0

TÆT

0 0 1 0

0 0 0 0

3

2

0 6 07 7x+6 5 4 0 0

1 0 TT H 0 0 0 0 3 0  07 7 VAC VA z z 05 1

1

TÆT

0 0 0

0

1

3 7 5

TT H 7

0 0



ÆT ÆT H





ÆT trim ) + K x = Ky1
(ylon ylon y2 ÆT H + Ky = [Ky1 ; Ky2℄ = Kx
Clon

+ denotes Kx is given by the Ri

ati solution of the LQR problem, and Clon pseudo inverse of the linearized longitudinal measurement matrix Clon of

where the

the synthesis model. The tuning parameters for optimization are now the entries of the weighting

Q and R. We de ided to use an output weighting T Q Clon ℄ together with diagonal Q and R matri es. [Clon

matri es

of the form

The longitudinal ontroller distinguishes altitude ontrol and glide slope

ontrol. For altitude ontrol

z

is the onstant altitude for whi h the air raft

is trimmed. For glide slope ontrol,

z is the urrent altitude of the demanded

glide slope. Gain s heduling for airspeed adaptation, similar to the lateral ase, is used for tailplane ontrol.

16.4 Translation of Design Obje tives to Mathemati al Synthesis Criteria MOPS design is well suited to deal with various design obje tives simultaneously. Ea h design obje tive is to be mathemati ally des ribed by a well-dened

riterion

i with a value whi h is the smaller, the better the obje tive is a

om-

plished. For the RCAM ben hmark problem it is straightforward to translate the spe i ations given in Chapter 14, Se tion 14.3.2, into mathemati al obje tive fun tions that an be used for optimisation. For the longitudinal ontroller 18 types of riteria have been used simultaneously, .f. Table 16.1. In this table, the se ond olumn des ribes the spe i ations as they are given in the RCAM Manual. The third olumn gives the

205

mathemati al formulation of the orresponding riteria. The 4th olumn shows the used demand values given as bounds in the Manual; if no spe i value is given min indi ates that the riterion value should be minimized. For min

riteria an appropriate demand value has to be hoosen in an evolutionary manner during design. The last olumn shows the identi ation names given to ea h riterion.

The eigenvalue riteria No.

17, 18, are auxiliary riteria.

They are used to prevent unstable solutions and to enfor e good damping. Overall, there are 18 riteria types to judge the longitudinal behaviour of the ontrolled air raft. However, all obje tive fun tions involved are standard

riteria in the multi-obje tive design environment of ANDECS_MOPS [99℄ and thus an be used o-the-shelf  without extra formulation eort. For a multi-model approa h this set of riteria is applied to ea h one of the models involved. Hen e in our design set up with 3 models we end up with 54

riteria to be taken are of simultanously. For the lateral ontroller we used similar riteria types to rate the step responses of bank angle

 and lateral deviation dy, the disturban e reje tion to

wind, stability and damping of eigenvalues, ontroller eort, or omfort. The

omfort riterion is expressed by means of the lateral load fa tor as

=

Z tend

0

n2y (t)dt :

It should be mentioned that also frequen y-domain riteria or any nonlinear spe i ation ould be dealt with in the multi-obje tive design. However, for the RCAM ben hmark problem only time domain spe i ations are given.

16.5 Des ription of the Design Cy le 16.5.1 Pro edure A priori, the RCAM ontroller ar hite ture was split into a longitudinal and a lateral ontroller to be designed independently.

For both ontrollers the

following design de isions have to be made: (i) Sele tion of an appropriate linear or nonlinear ontroller stru ture.

If

an analyti synthesis method is used, an appropriate synthesis model for synthesising the ontroller parameters has to be hosen also. (ii) Sele tion of a set of dynami worst- ase evaluation models; via these evaluation models the riteria fun tions are evaluated. They should be su ient to ree t the properties of the original (nonlinear model) and yet simple to enable fast omputation during optimisation.

The set of

evaluation models should represent the possible parameter un ertainties to a su ient amount. (iii) Sele tion of the design riteria. The mathemati al design riteria have to be hosen su h that the given design spe i ations are met orre tly.

206

No.

Spe i ations

1

altitude unit step:

Mathemati al Criteria

zero steady state error, settling

< 45

time 2

altitude unit step: rise time

3

s

ross

< 12

s

oupling

speed:

for

a

altitude step

in

air om-

manded altitude of 30 m, the peak

value

of

the

transient

Demand ID

R tend

= t (h(t) 1)2 dt 1 t0 = 10s, tend = 30s

min

HH1

= t2 t1 h(t1 ) = 0:1, h(t2 ) = 0:9

12

RHH1

= max jVA (t)j t

0.5/30

HV1

min

VV1

12

RVV1

= max jh(t)j t

10/13

VH1

= max h(t) t

1.05

OHH1

= max VA (t) t

1.05

OVV1

= max VA (t) t>15

2.6

VW1

R tend h2 (t)dt

= 0 tend = 30s

min

HW1

min

DEZ1

min

THZ1

min

DEV1

min

THV1

min

THW1

min

DTHW1

0.6

DAMP1

0.95

BOUND1

of the absolute error between

VA

and ommanded airspeed

should be smaller than 0.5 m/s

4

airspeed unit step: zero steady state error, settling

< 45

time 5

airspeed unit step: rise time

6

s

ross

< 12

s

oupling

tude:

for

a

airspeed step

in

alti om-

manded airspeed of 13 m/s, the peak value of the transient of the absolute error between and ommanded

h

R tend

= t (VA (t) 1)2 dt 0 t0 = 10s, tend = 30s

= t2 t1 , VA (t1 ) = 0:1, VA (t2 ) = 0:9

h

should be

smaller than 10 m 7

altitude unit step:

8

airspeed unit step:

overshoot

overshoot 9

<

<

5%

5%

airspeed wind disturban e: for a wind step with amplitude of 13 m/s there should be no deviation in the airspeed larger than 2.6 m/s for more than 15 s

10

altitude wind disturban e: no expli it spe i ation given

ontrol a tivity riteria: eort minimization for

11

tailplane, altitude ommand

12

throttle, altitude ommand

13

tailplane, airspeed ommand

14

throttle, airspeed ommand

15

throttle, wind step

16

throttle rate, wind step

17

relative stability of eigenvalues

evi :

no expli it spe i ation 18

absolute stability of eigenvalues

evi :

no expli it spe i ation

=

=

R tend

0

R tend

=1

0

u2 (t)dt

u_ 2 (t)dt

min i



Re(evi ) jevi j



= exp(max(Re(evi ))) i

Table 16.1: Design riteria used for longitudinal ontroller

207

(iv) Sele tion of the demand values. For ea h design riterion a demand value has to be asso iated. By iterating the demand values, the resulting ompromise trade-o solution an be for ed into a desired dire tion.

The

demand value an be - an upper bound to a design riterion (method of inequalities), - a weighting value that expresses the designer's preferen es related to the other riteria to be minimised.

16.5.2 Performing the design y le for longitudinal ontrol Sele tion of the evaluation models The evaluation models are used to al ulate the losed loop design-obje tives during optimisation. The set of models should be representative for the parameter variations assumed. In order to determine a set of representative linear longitudinal models we assessed, for an airspeed of 80 m/s, the un ontrolled air raft pit h rate step responses and eigenvalue lo ations for a su ient number of parameter variations in mass and entre of gravity. This proved that the family of time responses and eigenvalues are overed by only two parameter onstellations. Taking those together with the hosen nominal parameter

onstellation yields a 3-model problem with parameters


x


z

nominal 1

125 000

0.23

0.105

variation 2

100 000

0.31

0

variation 3

150 000

0.15

0.21

mass

nd

The air raft models are augmented by linear a tuator models and a 2 order Padé-approximation to des ribe a time delay

 = 100 ms.

In multi-obje tive design, evaluation models need not ne essarily be linear. However, numeri al simulation of linear systems an be done more e iently than non-linear simulation and is therefore to be prefered in optimisation. Moreover, for linear models one an ompute eigenvalues whi h serve as additional stability indi ators.

Sele tion of design riteria The design riteria des ribed in Se tion 16.4 are sele ted. These basi riteria, amongst others, are available in the design environment ANDECS_MOPS. They ree t the design spe i ations in a su iently dire t manner.

Sele tion of the demand values and design iterations The demand values are now asso iated to the riteria values.

If the spe i-

ations in Chapter 14 provide quantitative values, those are used as demand values. For example, in the presen e of a wind step with an amplitude of 13 m/s

208

there should be no deviation in the airspeed larger than 2.6 m/s for more than 15 s. This demand results immediately in the riterion

= max jV (t) VAC j  2:6 t>15 A with demand value 2.6. Other riteria su h as riteria for the ontroller eort that are not spe ied by a spe i value are weighted relative to ea h other. Su h demand values are iterated in an evolutionary manner during the following design steps in order to for e the design into a desired dire tion. Several onse utive design iterations have been done. This means several optimisation runs with dierent demand values were arried out, ea h one resulting in a trade-o solution a

ording to the hosen demand values. The general design pro edure is illustrated by an example design step: The design goal is to improve the rise time of airspeed from about 12 s to less than 8 s. To a hieve this, only the demand values orresponding to the rise-time riteria ( riterion RVV1 of table 16.1) have to be hanged from 12 to 8. Starting an optimisation run yields - after about 3 min of omputing time the result indi ated by the thi k lines in Figure 16.6. The airspeed response is thc wind step

thc wind step

thc wind step

0.00

E

thc

E

thc

thc

E

0.00

0.00

-0.30

-0.30

-0.30

-0.60

-0.60

-0.60

0.50

1.25

2.00

E1

0.50

1.25

2.00

t

0.50

0.25

-0.25 2.00

E1

0.75 0.25

-0.25 0.50

1.25

2.00

t

E1

0.50

E

E

E

0.8

0.8

0.4

0.4

0.0

0.0 2.00

E1

0.0 0.50

1.25

2.00

t/s

E1

0.50

E

E

E

0.8

0.8

0.4

0.4

0.0

0.0 2.00

E1

E1

height step

0.4

1.25

2.00 t/s

height step

0.8

0.50

1.25

t/s

height step

E1

velocity step

0.4

1.25

2.00 t

velocity step

0.8

0.50

1.25

t

velocity step

E1

E1

0.75

v,z

v,z

0.25

-0.25 1.25

2.00

wind step

E1

0.75

0.50

1.25

t

wind step

E1

v,z

E1 t

wind step

0.0 0.50

1.25

2.00

E1

0.50

1.25

t/s

2.00

E1 t/s

IDEXP HH1 RHH1 HV1 VV1 RVV1 VH1 OHH1 OVV1 VW1 HW1 DEZ1 THZ1 DEV1 THV1 THW1 DTHW1 DAMP1 BOUND1 HH2 RHH2 HV2 VV2 RVV2 VH2 OHH2 OVV2 VW2 HW2 DEZ2 THZ2 DEV2 THV2 THW2 DTHW2 DAMP2 BOUND2 HH3 RHH3 HV3 VV3 RVV3 VH3 OHH3 OVV3 VW3 HW3 DEZ3 THZ3 DEV3 THV3 THW3 DTHW3 DAMP3 BOUND3

t/s

Figure 16.6: Comparing dierent design out omes by time response indi ators. The diagrams in one olumn belong to the same evaluation model. mu h faster, rise time is less than 8 s. Altitude response and reje tion of wind disturban e are almost un hanged. However, the ontrol eort is in reased for

209

both tailplane and throttle a tivity between 10% and 25%. This is indi ated by the normalized riteria values represented in the parallel oordinate display also in luded in Figure 16.6.

16.5.3 Performing the design y le for lateral ontrol Sele tion of evaluation models Similar onsiderations as for longitudinal ontrol lead to a 3-model problem with parameters


y

mass nominal 1

125 000

0

variation 2

150 000

-0.03

variation 3

100 000

0.03

For evaluation, the linearized air raft models are augmented by rst order lag lters for modelling the a tuators for aileron and rudder. Contrary to the evaluation models for the longitudinal ontroller no additional time delay was taken into a

ount during design.

Sele tion of the design riteria For the inner roll and yaw rate damper no expli it spe i ations are given in Chapter 14. The riteria we sele ted are eigenvalue riteria to improve damping and absolute stability margin. For bank angle ontrol, step-response riteria are sele ted as introdu ed in Se tion 16.4. The same holds for sideslip suppression and lateral deviation. Control eort was taken into a

ount by integral-square

riteria.

Sele tion of the demand values and design iterations The demand values are asso iated with the orresponding riteria value by applying the same methodology as in the ase of longitudinal ontrol. A

ording to the ontroller stru ture sele ted, the design of the lateral ontroller was divided into three major design steps. In a rst step, the roll and yaw rate damper is designed su h that a relative damping of 0.6 and a minimum absolute stability margin of 0.25 was a hieved for all evaluation models simultaneously. In a se ond step, yaw suppression and bank angle ontrol is designed. Using the standard riteria des ribed above, it is straightforward to a hieve the a

ording spe i ations. The third step deals with the design of the lateral tra king performan e. The parameters of the inner loop, designed before, remain xed during these design iterations.

210

16.5.4 Design of a turn ompensation using the nonlinear air raft model Nonlinear assessment of the ontrollers designed so far for a trimmed 3 deg/s turn showed poor damping for lateral or altitude step responses. To improve damping the ontroller stru ture is augmented by a turn ompensator feeding bank angle

 on tailplane dee tion ÆT

as shown in Figure 16.7.

trim condition phi0

phi

dT

kuko1s+kuko

2

2

phi0* 80 /v

s+kuko2

Figure 16.7: Stu ture of turn ompensator.

Sele tion of evaluation models To tune the parameters of the ompensator we use the nonlinear model of the

ontrolled air raft within the MOPS design framework. For this the Dymola model was augmented by the ompensator. The air raft was trimmed for a 3 deg/s turn for nominal parameter values and nominal speed. Using the nonlinear air raft model was the simplest way of treating the oupled lateral and longitudinal motions during a turn. Furthermore it illustrates the exibility of the MOPS method on erning appli ation of models and ontroller stru tures.

Sele tion of the design riteria Design riteria steady-state errors of lateral and altitude step responses are

omputed as integral riteria similar to riterion No. 1 of Table 16.1.

Sele tion of the demand values and design iterations All demand values are set to the urrent riteria values.

This means that

all riteria are normalized and treated with the same preferen e.

This is a

ommonly used pro edure in MOPS when no spe i demand values are invoked by the spe i ations. After a few design iterations, an optimized parameter set was found that improves damping onsiderably. This is shown by the lateral step response in Figure 16.8.

211

Lateral deviation step response

Deviation from altitude during lateral step E1

dz

DY

E1 1.00

0.75

0.75

0.50

0.50

0.25

0.25

0.00

0.00

-0.25

-0.25

-0.50

-0.50

-0.75

-0.75

-1.00 0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

3.2

3.6

4.0 E 1

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

TIME

5.0

5.5 E 1 TIME

Figure 16.8: Deviation of y and z for a lateral step response during a trimmed 3 deg/s turn with and without turn ompensation.

16.6 Analysis of the Controlled Air raft in Terms of the Applied Methodology 16.6.1 On-line Analysis In multi-obje tive ontroller design the obje tive fun tions are omputed from indi ators obtained by system analysis and simulation.

The resulting anal-

ysis data are therefore available, even during optimization.

This is utilized

in ANDECS_MOPS for monitoring the optimization pro ess on-line. Figure 16.6 shows su h an on-line visualization with two optimization results. Whi h analysis data should be plotted and how they should be arranged, is under

ompletely free ontrol by the user. To ea h indi ator orresponds a set of design riteria. By dire tly omparing riteria values with the indi ators plotted in the diagrams, one an easily

he k whether the obje tive fun tions are well dened. By means of su h a visualization of the underlying dynami s indi ators one an also gain knowledge about the proper size of the demand values whi h are not expli itely given in the spe i ation.

If, for example, steady-state behaviour is good enough by

inspe tion, the demand value for the steady-state riterion an be xed to the resulting riterion value in the subsequent design iterations.

16.6.2 Parallel Coordinates To visualize the whole set of riteria values over a omplete optimization run, we use parallel oordinates [122℄, [79℄. The oordinate axis of the n-dimensional

riteria spa e are arranged in parallel. A point in the riteria spa e, i.e. the

riteria values for a given set of tuning parameters, is represented in parallel

oordinates by a line whi h inter onne ts all individual riteria values. Parallel

oordinates are well suited to elu idate

 

how well riteria are fullled: the deeper the line the better the result. where oni ts between riteria exist: the lines ross.

212



where on urrent riteria exist: the lines are (almost) parallel.

Figure 16.6 shows the parallel oordinate representation of all riteria involved in the longitudinal design.

The thi k oordinate line orresponds to

thi k time responses in the diagrams above.

16.6.3 Nonlinear Analysis In order to dete t nonlinear ee ts, an a

ompanying assessment based on the non-linear RCAM model was performed for some important requirements. Figure 16.9 shows 32 automated omputation experiments for minimum and maximum parameter values of mass,


x,
z ,
y,

as well as minimum and

maximum time delay for xed design airspeed of 80 m/s. The various diagrams 1.1 Altitude step [30m]

1.2 Altitude for VA step [13m/s]

1.3 Deviation from 3deg/s turn

1.020

1.002

1.012

0.996

1.004

E2

1.0

2.0

3.0

4.0

5.0

E1

0.25

-0.25

0.990 0

deviation [m]

Z

E3

Z

E3

-0.75

0

1.0

2.0

3.0

4.0

5.0

TIME

E1

0

2.1 Speed step [13m/s]

7.75

7.980

7.25

0.82

7.950 4.0

5.0

E1

1.0

2.0

3.0

4.0

5.0

E1

0

3.1 Y step [10m]

3.2 Psi step [0.1 rad]

0.45

0.15

0.15 4.0

5.0

E1

4.1 Engine failure: phi, beta

1.0

2.0

3.0

4.0

5.0

E1

0

4.2 Engine failure: Y deviation

0.0

-2.0

4.0

5.0

-0.4 E2

E1

0.30 0.00

-0.30 -0.60 0.0

0.4

0.8

1.2

1.6

TIME phi

3.0

4.3 Engine failure: heading rate

psidot

Y

0.4

-1.2

1.6

2.0

E -1

0.8

-0.4

1.2

1.0

TIME

E2

0.8

0.75

TIME

0.4

E1

0.15 0

E -1

0.4

5.0

0.45

TIME

0.0

4.0

3.3 Gamma step [0.1 rad]

0.75

0.45

3.0

3.0

E -1

GAMMA

PSI

Y 0.75

2.0

2.0

TIME

E -1

1.0

1.0

TIME

E1

E2

6.75 0

TIME

0

1.0

V

V

V

8.010

0.86

3.0

0.8

E1

0.90

2.0

0.6

2.3 VA for wind step uw [13m/s]

E1

1.0

0.4

t [s]

2.2 VA for altitude step [30m]

E2

0

0.2

TIME

E2 TIME

0.0

0.4

0.8

1.2

1.6

E2 Time

beta

Figure 16.9: Nonlinear assessment for 32 parameter variations and dierent ight manoeuvres, in luding engine failure. show the following analysis results with dotted lines indi ating the allowed toleran es: 1.1 altitude response for a step ommand of 30 m 1.2 altitude response for a airspeed ommand of 13 m/s 1.3 lateral deviation in ase of a 3 deg/s turn 2.2 airspeed response for an altitude step of 30 m 2.3 airspeed response for a headwind step of 13 m/s 3.1 lateral response for a step ommand of 10 m

213

3.2

response for a step ommand of 0.1 rad

response for a step ommand of 0.1 rad 4.1  and in ase of an engine failure

3.3

4.2 lateral deviation in ase of an engine failure 4.3 heading rate

_

in ase of an engine failure

16.7 Results of the Automated Evaluation Pro edure This hapter presents the methodologyindependent evaluation results for the

ontroller design des ribed in the previous hapters. The results relate to the evaluation s enario and mission dened in Chapter 14. They are obtained by using the RCAM automati evaluation software. Both overall tra king performan e and inner-loop behaviour of the ontrolled air raft are evaluated by means of bounds of key variables like tra king error (as performan e measure), normal a

elerations (as omfort riterion), variation in tra k angle (as robustness measure), and angle of atta k (as safety riterion).

Segment I: the ee t of engine failure This segment of the traje tory is a straight line of about 16 km length at

onstant altitude. A onstant wind with 10 m/s is assumed, dire ted along the negative earth-xed axes. First segment: top view

Second segment: lateral deviations 300

100

0

0

a

b

lateral deviation [m]

x−deviation [m]

200 50

1

−50

100

0 1

c

d

2

−100

−200 −100 −20

−18

−16

−14

−12 −10 −8 y−position (−YE) [km]

−6

−4

−300 0

−2

1

2

3 4 5 6 along track distance from point 1 [km]

7

8

Figure 16.10: Segment I: the ee t

Figure 16.11:

of engine failure.

view of the 3 deg/s turn and lateral

Segment II: plane

deviations.

A single engine failure is simulated in this segment.

Sin e RCAM is a

twin-engine air raft, a single engine failure will result in a signi ant lateral deviation. Figure 16.10 provides a plane view of the rst traje tory segment. The given bounds provide an a

eptable level of performan e. The stati lateral deviation should be below an a

eptable value of 20 m.

214

Segment II: the 3 deg/s turn This segment begins with a 90-degree turn followed by a straight line segment, all at onstant altitude. The onstant wind is still blowing and hen e be omes a head wind in the progress of Segment II. The perfe t following of the required traje tory would require a sudden

hange in the air raft's bank angle, whi h is physi ally impossible and obviously not desirable for omfort reasons. Hen e deviations from the desired traje tory are unavoidable. This is shown in Figure 16.11 as a plane view of the lateral deviations from the required traje tory.

Segment III: the apture of the -6 and -3 degree glideslope The performan e of the air raft during this segment is measured by the deviation from a -6 degree glideslope, whi h hanges to a -3 degree glideslope afterwards. A maximum deviation of 20 m should not been ex eeded. Furthermore, speed variations should be kept small during the ne essary variations of the angle of atta k and while the head wind redu es to zero. In Figure 16.12 the verti al deviations of the air raft following the glideslope are plotted together with bounds of a

eptable behaviour.

Segment IV: the nal approa h with windshear While on a nal approa h with a glideslope of -3 degree, the ee t of a windshear is onsidered. Figure 16.13 shows the bounds that dene an a

eptable longitudinal response to the windshear together with the verti al deviations from the desired glideslope.

Fourth segment: altitude deviations 30

20

20

10 2

f

altitude deviation [m]

altitude deviation [m]

Third segment: altitude deviations 30

3

0

−10

e

−20

−30

10

3 g

4

h

0

−10

−20

−16

−15

−14 −13 x−position (XE) [km]

−12

−30 −11

−11

−10

−9

−8

−7 −6 −5 −4 x−position (XE) [km]

−3

−2

−1

Figure 16.12: Segment III: the ap-

Figure 16.13:

ture of the -6 and -3 degree glides-

tudinal deviations for the nal ap-

Segment IV: longi-

lope and verti al deviations.

proa h with windshear and verti al deviations from the desired glideslope.

215

Numeri al Results Using the riteria given in Chapter 14, omparison indi ators for ea h segment of the evaluation ight prole have been al ulated using the RCAM automati evaluation software. These values are given in Table 16.2 together with an overall average for ea h of the ve riteria. For performan e, perfoman e deviation,  omfort, and safety, values smaller than one indi ate a

eptable behaviour a

ording to the requirements stated in Chapter 14. Power measures the ontrol a tivities and is not normalized to one. Its value serves for a relative omparison of ontrollers. Segm. I

Segm. II

Segm. III

Segm. IV

Total

Performan e

0.1270

0.1436

0.2587

0.1781

0.1769

Perf. Dev.

0.0438

0.0184

0.1121

0.0972

0.0679

Comfort

0.5500

1.6295

1.3224

0.5175

1.0049

Safety

0.0051

0.1281

0.0081

0.0736

0.0537

Power

0.0026

0.0104

0.0150

0.0308

0.0147

Table 16.2: Numeri al results of the evaluation pro edure The table shows that besides omfort during Segment II and III all values are less than 1 and hen e satisfa tory. It is up to the designer to judge whether this is an a

eptable trade-o or not. The above diagrams and the numeri al values of the table show that the designed ontrollers are quite robust with respe t to the investigated parameter variations. Moreover, the analysis of Se tion 16.6 shows that the ontrollers remain stable with su iently good performan e even for mass variations up to 50 000 kg. This shows again that the multi-model approa h is well suited for designing robust ontrollers.

16.8 Con lusions Flight Control is not a new and unsolved ontrol appli ation area. Hen e there is a lot of proven knowledge about using suitable ontroller stru tures and setting up meaningful spe i ations. But a best possible appli ation-spe i parameter tuning of ontrol laws is still a time- onsuming trial and error task. The multi- riteria/multi-model/multi-parameter optimisation approa h as demonstrated here aims at relieving that tuning task in a transparent and systemati way. As shown, this approa h results in low-order ontrollers with good performan e. In parti ular,a quite robust solution resulted, as measured by the respe tive ben hmark evaluation gure of merit.

This indi ates the

usefulness of the multi-model approa h. Note, that for longitudinal ontrol we intentionally applied an LQR-state feedba k approa h whi h is realized as omplete output feedba k.

This was

done in order to demonstrate the feasibility of using an analyti al synthesis method within the parameter tuning optimization loop as well. In an industrial

216

appli ation, more likely, another ontroller stru ture would have been hosen. It is a strength of the MOPS method that it is not bound to a spe i ontroller stru ture or spe i ontrol law synthesis method. The multi- riteria/multi-model design approa h is easy to omprehend be ause problem formulation and design- y le setup is straight forward in terms of the design spe i ations. The omputer, not the engineer, takes on the work of numeri ally tuning the appropriate ontrol design parameters. The eort related to setting up the design y le, i.e.

setting up design

models and ontroller stru tures, is essentially the same as in any other general

ontroller synthesis method. More emphasis than in other methods is put on setting up the design riteria and this is an engineering appli ation strength of the MOPS method. For a spe i appli ation area like ight ontrol, generi design-obje tives as well as problem-spe i obje tives like handling qualites

riteria an be implemented for re-use in an appropriate design environment. The design y le is exe uted intera tively. Synthesis of new ontroller parameters is done automati ally by the omputer, whereas the de ision making for appropriate models, riteria, or demand values, requires an intera tion by the design engineer. Computation time for optimization may be in the order of minutes, depending on the problem to be solved. The a tual eort ne essary for performing a redesign after a major design de ision hange, depends on what has been hanged: models, ontroller stru tures, or performan e spe i ations.

The essential feature is that the multi-

riteria/multi-model design approa h is most exible and systemati to deal with su h hanges.

217

17.

An Eigenstru ture Assignment

Approa h (1)

Lester Faleiro1 and Roger Pratt Abstra t.

1

This hapter is a des ription of the use of one parti ular

methodology of eigenstru ture assignment on the Robust Control Air raft Model (RCAM) hallenge as set out in [145℄.

Emphasis

is put on the general methodology in order to on entrate on the similarities between traditional ontrol system design methods and eigenstru ture assignment. The results show that although the nal

ontroller in this ase is not the ideal one, the design pro ess is dened learly enough for eigenstru ture assignment to be used as a design tool in a wider design pro ess.

17.1 Introdu tion Various methodologies of eigenstru ture assignment are detailed in hapter 3, and some of them are ustilised in this hapter to produ e one possible solution to the design problem.

The design pro ess that is presented here is not the

ideal one by any means, but the intention is to demonstrate the attributes that eigenstru ture assignment has as a design tool, and the potential results it an deliver. The se tions of the hapter are set out in the order in whi h the design pro ess an be arried out. However, it must be remembered that the pro ess is an interative one, as will be ome lear.

17.2 The Sele tion of the Controller Ar hite ture for the RCAM Problem For most xed-wing air raft, there is very little ross- oupling between the longitudinal and lateral dynami s. Consequently, it was de ided that two ontrollers would be used; one for the longitudinal dynami s, and one for the lateral dynami s.

218

SPPO

Eigenvalue

 !n q  u w z

Phugoid

0:8303  1:1069i

0:0114  0:1264i

Displa ement 0

0.6

0.0898

1.3837

0.1269

-

0.0136

0.0002

0

0.0098

0.0016

0

0.0144

0.1216

0

0.9430

0.0175

0

0.3320

0.9924

1

Table 17.1: Eigenve tors of the longitudinal open-loop system

17.2.1 Longitudinal ontroller The open-loop longitudinal dynami s for RCAM at a nominal trim ondition (mass at 120 tonnes, horizontal entre of gravity at 23% of mean aerodynami

hord (MAC) and verti al entre of gravity at its lowest point) an be determined using GARTEUR software. The eigenstru ture of this system is shown in Table 17.1. For the sake of simpli ity, a tuator dynami s will be omitted. The dynami s ontain the traditional Short Period Pit hing Os illation (SPPO) and phugoid modes, and an additional verti al displa ement mode, oupled only to the verti al displa ement state.

VA z

command +

VA z

error

∫

Llon +

δ T δ TH

+

Trim conditions +

Klon

Aircraft and actuation

q nZ VA wV z VA z

Figure 17.1: Longitudinal losed-loop system stru ture The hosen design task was to regulate pit h rate, verti al a

eleration, airspeed, verti al velo ity and verti al displa ement(

[q nz VA wv z ℄).

Based on

the design requirements to tra k hanges in verti al displa ement and airspeed, these two variables were hosen as tra ked outputs. The full linear losed-loop system an now be depi ted as shown in Figure 17.1.

There are two main

omponents to the stru ture: 1. The ve feedba k signals are used to regulate the air raft. This is done by multiplying the error between the output signals and the trim onditions 1

Department of Aeronauti al and Automotive Engineering and Transport Studies, Lough-

borough University, Loughborough, Lei estershire LE11 3TU, United Kingdom.

219

by the stati gains in the matrix

Klon,

whi h produ es taileron and throttle

signals to return the air raft to the trim ondition. 2.

The errors between the referen e signals and their respe tive outputs

are integrated and fed through a gain matrix,

Llon , whi h will ensure that the

error between the referen e signal and the output signal is always zero.

17.2.2 Lateral ontroller stru ture As with the longitudinal air raft model, the lateral linear dynami s for the given trim ondition an be determined using GARTEUR software. The eigenstru ture of this system is shown in Table 17.2. Again, the modes of the open-loop system are onventional.

Eigenvalue

 !n p r 

vB ylat

Dut h Roll

Heading

ylat

0.00

0

Roll

Spiral

-1.3017

-0.1837

-

-

0.3684

-

-

-

0.6405

-

0.2177

0.0008

0.0170

0.0008

0

0.0145

0.0005

0.0049

0.0005

0

0.1669

0.0043

0.0264

0.0043

0

0.0111

0.0028

0.0076

0.0028

0

0.0138

0.0472

0.8901

0.0472

0

0.9614

0.9989

0.4546

0.9989

1

0:2360  0:5954i

Table 17.2: Eigenve tors of the lateral open-loop system

There are eight measured outputs available in the lateral dynami s of RCAM. Only ve of these are ne essary to implement su ient ontrol over the four modes des ribed in the open-loop dynami s.

It was hosen to regulate the

hanges in sideslip angle, roll rate, yaw rate, roll angle, tra k angle and lateral

[ p r   ylat℄).

deviation (

Based on the design requirements to tra k hanges

in heading rate and lateral displa ement, the tra ked outputs were hosen as roll angle (whi h is dire tly related to heading rate) and lateral displa ement. All other lateral demands an be translated into a ombination of these. The linear losed-loop system an be depi ted s hemati ally as shown in

ylat is used both as a regulation feedba k signal and as  is given by dieren e between the initial  and  output. However, when a turn rate is demanded, the desired steady-state  has to be altered to prevent the regulator from Figure 17.2. Note that

an output signal. For normal operation, the error from

attempting to keep it at its initial ondition. This is done by integrating a demand to get a desired initial

.

at any time step and simply adding that value to the

Some aspe ts of the stru ture des ribed here were added on during

the design pro edure itself, and the pro ess is inevitably iterative.

220

˙ ψ

 ψ˙ Va  Va c φ c = atan  -------------   g 

c

ylat

c

-

φ ylat

∫

error

+ Trim conditions

χ

+ βprφ

ψ

c

+

0.2

+ +

χ

+ -

trim error

ylat -

+ Llat +

Klat φ

χ

∫

Aircraft and actuation

βp rφχ

error error

δA δR

additional

Figure 17.2: Lateral losed-loop system stru ture

17.3 The Translation of RCAM Design Criteria into Method Dependent Obje tives The RCAM hallenge design spe i ations are detailed in [145℄.

The simple

eigenstru ture assignment design synthesis performed for the RCAM design only requires two sets of inputs. The rst are the system matri es

C.

A, B

and

The se ond onsists of a set of desired losed-loop eigenvalues and a set of

d

desired losed-loop eigenve tors (

and

Vd ).

This se tion des ribes how the

various design spe i ations an be transformed into desired eigenvalues and eigenve tors.

17.3.1 Performan e riteria Performan e requirements are the most signi ant requirements in RCAM. The

riteria are omposed of rise time and settling time spe i ations, overshoot limits and ross- oupling limitations. The following se tions relate to the requirements des ribed in [145℄, and only spe i ations pertinent to the use of eigenstru ture assignment have been in luded. Both the longitudinal and the lateral systems ontain se ond order modes, and all the tra king requirements are given for step input onditions.

The

response of a se ond order system to a step input is well do umented.

The

dierent spe i ations for rise time and overshoot for a parti ular system an

221

thus be transformed into eigenvalues, and any states that should not exhibit a mode an then be de oupled from that mode in the eigenve tors (see hapter 3 for a fuller explanation).

Lateral deviation The tra king requirement for lateral deviation provides a minimum limit for the natural frequen y and damping ratio of the se ond order modes onne ted with

ylat.

This allows a ertain exibility in the eigenstru ture assignment

pro edure, as an improved de oupling solution may be obtained by allowing the eigenvalues to roam within this limit.

Altitude response This is a similar spe i ation to the lateral deviation requirement. Again, the overshoot limit means that verti al displa ement should not be oupled into modes that have a damping ratio of less than 0.7. The verti al deviation limits for low level ight is more di ult to in orporate into the eigenstru ture assignment design.

Essentially, this an be ta kled by simply improving ro-

bustness as mu h as possible, so that the hange in altitude will not result in a large deviation of the performan e of the air raft from the nominal losed-loop system.

Roll angle response This spe i aiton relates to engine failure. The only thing that an be done in eigenstru ture assignment is to have a relatively qui k response to

ylat devia-

tions, so that in a regulation situation, roll angle will not have the opportunity to be ome large enough to violate the given limits. Additionally, de oupling roll angle from modes that might be ome ex ited during an engine failure, su h as the Dut h Roll, should prevent large roll angles from o

urring during regulation. The requirement to de ouple sideslip from roll motion an be in orporated into the eigenstru ture assignment pro edure easily. Lateral velo ity hanges (and hen e hanges in sideslip for a onstant forward velo ity) are de oupled from the modes relating to rolling motion by inserting a zero into the relevant eigenve tor element. This will produ e a o-ordinated turn.

Airspeed response The requirements for airspeed tra king an again be transformed into bounded eigenvalues for the modes oupled to airspeed. The limit on airspeed deviation in the presen e of a step disturban e in wind velo ity an only be addressed by having su ient damping on the mode involving velo ity to insure that it is redu ed to allowable levels in the time required. The requirement to have no steady state velo ity error an be easily dealt with by using the velo ity error integrator des ribed in the longitudinal ontroller stru ture.

222

Cross- oupling between airspeed and altitude This requirement is a de oupling spe i ation, and an be in orporated into the desired eigenve tors of the longitudinal ontroller. As shown in Table 17.3,

u has been de oupled from z tra k and w has been de oupled from VA

tra k.

17.3.2 Robustness riteria Unfortunately, the method do umented here annot yet be used to design for exa t parameter variations. Instead, the obje t is to make the system as insensitive to parameter variation as possible, thus attempting to preserve nominal stability and performan e whilst arbitrarily improving robustness.

17.3.3 Spe i ation of eigenstru ture A set of initial estimates, based on the desired performan e, robustness and stability of the losed-loop air raft an now be proposed. Note that this set is not unique, and it is very likely that the desired eigenstru ture will have to be updated before a suitable design an be found.

Longitudinal mode spe i ation In keeping with onventional air raft behaviour, the losed-loop longitudinal system an be assigned three modes: the SPPO, the phugoid and a verti al displa ement mode. In addition to these, the eigenstru ture assignment pro edure augments the open-loop system matri es with the two integrated error states of the outer-loop tra king. The eigenstru ture that is hosen for the rst design is shown in Table 17.3. SPPO

Eigenvalue

q  u w z R V R A z

0:8  0:8i

Phugoid

0:15  0:15i

z

VA

tra k

z tra k

-0.3

-0.4

-0.5

x

x

x

x

x

x

x

x

x

x

0

x

0

x

0

x

0

x

0

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

Table 17.3: Desired eigenstru ture of the longitudinal losed-loop system

Lateral mode spe i ation As with the longitudinal system, there are traditional modes of behaviour that

an be dened for the losed-loop system. These in lude a Roll mode, a Spiral mode and a Dut h roll. Additional modes that we an now dene are a heading

223

mode and the two integral tra king modes. An initial desired eigenstru ture is shown in Table 17.4. Roll Eigenvalue

p r 

vB ylat R R  ylat

Spiral

Dut h Roll

Heading

0:182  0:157i

ylat



ylat

tra k

tra k -0.5

-4.4

-0.2

-0.13

-0.55

-1.5

x

x

0

x

x

x

x

x

x

x

x

x

x

x

x

x

0

x

x

x

x

x

x

x

x

x

x

0

0

0

x

0

0

0

x

x

x

x

x

x

x

x

x

x

0

x

x

x

x

x

x

0

x

x

x

x

Table 17.4: Desired eigenstru ture of the lateral losed-loop system

17.4 The Des ription of the Design Cy le Only a few tools are required in order to produ e and analyse an eigenstru ture assignment design that works.

However, the pro ess is still a di ult one.

This is not to do with the tools themselves.

Rather, it has to do with the

interpretation of the results of the analysis and the onsequent de isions that the designer must make in spe ifying an updated eigenstru ture.

17.4.1 Initial synthesis The rst design synthesis is a simple one.

All that is required is the robust

eigenstru ture assignment algorithm. There are two basi sets of inputs to this

A, B , C and Ca. The last of C whi h des ribe the outputs that are The se ond is the desired eigenstru ture d and Vd , The values

program. The rst is a set of system matri es, these is a matrix that ontains the rows of to be tra ked.

of the initial estimates for these are given as method dependent obje tives in Table 17.3 and Table 17.4. These are simply entered into a program to produ e the stati gain matri es

K

and

L.

17.4.2 Intermediate analysis The intermediate analysis of the ontroller takes up most of the time in this design pro ess.

This is done using a ombination of linear simulation and

eigenstru ture analysis to ensure that performan e goals are satised, and some sensitivity analysis to determine how robustness goals are being met.

The longitudinal ontroller Eigenstru ture analysis an be used to examine the system. The losed-loop eigenstru ture of the initial design is given in Table 17.5.

224

The shaded ele-

ments show that the de oupling that was spe ied in the desired losed-loop eigenve tors has been attained exa tly. Klon =

  !n q  u w z Æt Æth



1:4559 0:3091

SPPO 0 :8 0 :8 i 0.71 1.13 0.0086 -0.0076 0 0.8279 -0.4198 -0.0026 -0.0012 (V ) 30400



0:1129 0:0300

0:0025 0:0123

0:0241 0:0006

0:0075 0:0020

Phugoid z (Taileron) 0:150:15i -0.3 -5.0285 0.71 0.21 0.0001 -0.0002 0.0654 0.0005 0.0005 -0.0130 -0.0457 0 0.0155 0 -0.0428 -0.9910 0.2024 0.2871 -0.0095 0.0002 0.0004 0.1150 0.0005 0.0004 -0.0004 Gain margin -4.3 dB, 9 dB



Llon =



0:0008 0:0017

0:0009 0:0002



(Throttle) VA tra k z tra k -0.8875 -0.4 -0.5 -0.0022 -0.0006 -0.0003 0.0025 0.0015 0.0005 -0.0051 -0.2483 0 0.6469 0 -0.1782 -0.5061 0.2763 0.4400 -0.0045 0.0015 0.0014 -0.0012 0.0053 0.0009 Phase margin 38 degrees



Table 17.5: Eigenstru ture of the longitudinal losed-loop system The intera tion of the tra king ommands with the outputs an be followed qualitatively by using the losed-loop input oupling ve tors.

The oupling

between a ommanded input and the modes of the system is given by:

Wlon Blon Llon

(17.1)

These ve tors are shown in Table 17.6. These oupling ve tors show us that when there is a demand in

VA ,

the Phugoid, SPPO,

z, z

tra k and

VA

tra k

modes will be strongly ex ited. The other modes will be involved, but not to as great an extent. Looking ba k at Table 17.5, when these modes are ex ited, it will ause an ex ursion in both forward velo ity and in verti al displa ement.

VA

demand

z demand

SPPO

0.3901

0.4595

Phugoid

0.9686

0.0430

z

4.0196

1.4312

(Taileron)

0.0680

0.0792

(Throttle)

0.7187

2.0467

0.3574

0.0486

1.7478

3.1458

VA tra k z tra k

Table 17.6: Input oupling of the longitudinal losed-loop system When a hange in

z

is demanded, the SPPO, the verti al mode and the

z

tra k mode are the most involved. These modes do alter the z state, but are not very dominant in the forward velo ity state. Thus, our ursory examination of

VA is de oupled z demand, but z does not appear to be de oupled from a VA demand.

the eigenstru ture of the system indi ates that it is likely that from a

This evaluation of the eigenstru ture an be tested by using linear simulation of step demands on the tra ked variables. Figure 17.3 shows the time response of this system to a step demand in rease of 13 m/s in

VA .

that, as predi ted by eigenstru ture analysis, there is a large ex ursion in

225

Note

z.

A

step ommand of 30m in rease in Again, as predi ted,

VA

z produ es the results shown in Figure 17.3. z

has been de oupled well enough from a demand in

to satisfy the design spe i ation. 15

35

VA

10

30

m, m/s

m, m/s

25 5

wv

0

−5

z

−10

20

10 5 0

−15 0

10

20

30

40

50

z

15

−5 0

wv VA 10

Time (seconds)

20

30

40

50

Time (seconds)

Response to 30m step in z

Response to 13 m/s step in VA

Figure 17.3: Step responses of initial longitudinal system It is now up to the designer to alter the design parameters on the basis of this analysis. Although this initial system has satised most of the design spe i ations, the verti al a

eleration involved is outside the allowable limits, espe ially with a

z

demand. A glan e at the time responses shows that this is

likely to be the result of an under-damped SPPO mode. The damping on the desired SPPO mode an now be altered by the designer to obtain the desired ee t.

The lateral ontroller The lateral system an be analysed in exa tly the same way as the longitudinal, as ontroller stru tures are similar.

Again, it is important to take designer

experien e into a

ount during the design pro ess. An initial ontroller for this system was designed and analysed, and was found to be useful for linear simulations whi h satisfy the step demand requirements detailed in [145℄.

17.4.3 Robustness and goal attainment One of the most evident problems with eigenstru ture assignment is that any small alteration of the eigenvalues alters the ve tor spa e from whi h the a hievable eigenve tors an be sele ted. This means that although it is easy for the designer to hoose arbitrary eigenvalues to produ e desired responses, it is not possible to examine the ee t of this hoi e on the nal eigenve tors of the

losed loop system. Ea h hoi e of eigenvalues will ae t the many parameters by whi h the nal design an be measured. One su h parameter is stability robustness. A set of design goals an be used to spe ify desired stability robustness, whi h an be des ribed by the minimum singular value of the return dieren e matrix (RDM) at the input and output of the system. These values

an be translated into onservative gain and phase margins. The exa t relations are do umented in [152℄. Goal attainment oers a method of using any freedom inherent in the system to sear h the allowable set of solutions iteratively for a design that satises as many of these robustness goals as possible.

226

A simple diagrammati al approa h to the goal attainment pro edure, as well as a fuller explanation of the mathemati al ba kground involved an be found in [95℄. To preserve the desirable qualities of a system designed by eigenstru ture assignment, goal attainment was implemented with an internal eigenstru ture assignment algorithm.

The pro edure begins with an arbitrary set of eigen-

values, and produ es a feedba k gain matrix as a solution to a robust eigenstru ture assignment problem spe ied by the designer. matrix,

K,

This feedba k gain

is then used to evaluate the nearness of the stability robustness

measures to the desired robustness goals. The desired eigenvalues and/or desired eigenve tors are then altered a

ordingly, and the eigenstru ture assignment pro edure is repeated until a solution has been obtained. This pro edure is depi ted s hemati ally in Figure 17.4 and the resulting ontrollers are shown in Table 17.7.

Algorithm to Have NO goals compute been YES updated parameters attained? Λ V d d

K and L (final)

A,B,C Initial Λ V d d Goals for performance and stability robustness

Eigenstructure Assignment Procedure

Designer interaction

K and L Obtain performance and stability measures

Goal attainment algorithm

Figure 17.4: S hemati of goal attainment pro edure

These ontrollers have been designed using only eigenvalue perturbations. Comparing them with that given in Table 17.5, it an be seen from both the

ondition numbers of the eigenve tor matri es and the multivariable stability margins that goal attainment an be used with eigenstru ture assignment to produ e a system with improved robustness.

17.5 Analysis of the Resulting Controller The ontroller was analysed with respe t to the a hievement of the design spe i ations. Where step responses have been demanded, these are performed on the non-linear simulation. Unless otherwise spe ied, the results are for the

227

Klon = 

Klat = Llon =



First design

(V )



3:088 0:269 0:008 0:048 1:441 0:131 0:012 0:018

2:435 5:165 4:599 19:571 0:005 2:232 0:001 3:755 0:0008 0:0019 0:0013 0:0008

(V )



Llat =



0:014 0:07



8:077 0:050 0:437 0:005



0:3006 0:0028 0:1888 0:0003

Longitudinal

Lateral

30400

34730

8439

30640

-4.8 dB, 11.3 dB

Phase margin

43 degrees

-3.7 dB, 6.7 dB

No. of Iterations

200

100

Gain margin



31 degrees

Table 17.7: Chara teristi s of the nal ontroller designed with goal attainment

losed-loop air raft at nominal ondition (120 tonnes, horizontal 23% mean aerodynami hord and verti al 0% mean aerodynami hord entre of gravity; time delay at 0.05s). Any design spe i ations that are satised have not been do umented, but all the required tests are shown in graphi al form in Figure 17.6.

Deviation (metres)

17.5.1 Performan e riteria 1

Lateral demand Lateral disturbance

0.5

0

−0.5 0

20

40

60

80

100

Time (seconds) Figure 17.5: Lateral deviation performan e

Lateral deviation The lateral deviation of the air raft from an initial ondition, 1m o the desired traje tory, is shown in Figure 17.5. A redu tion of lateral deviation to 10% of its original value o

urs in less than 8 se onds, but also results in an overshoot of 25%. This is not a

eptable performan e. Earlier designs did not have this

228

bad performan e, but it was required in order to ensure that the air raft did not stray outside the allowable bounds in the event of an engine failure.

Flight path angle response This response has an overshoot of only 1.6%, but does not respond as qui kly as desired to ommands in ight path angle. This situation annot be easily remedied using the eigenstru ture assignment ontroller, as it has been designed to solve independent a on urrent

VA

and

z demand and VA demand situations. wV demand.

A

demand implies

Roll angle response The design spe i ations require that a small roll angle be maintained after engine failure to keep sideslip minimised. However, the eigenstru ture assignment design is su h that roll angle is minimised instead.

As an be seen in

Figure 17.5, this results in a steady sideslip of about 3 degrees. It also invalidates the remaining design riteria of not overshooting 50% of the steady roll angle on engine start-up.

Cross oupling between airspeed and altitude Although ross oupling was insigni ant during the early designs, later alterations have aused a 1.4 m/s deviation (desired at less than 0.5 m/s) in

z

VA for

a demand of 30 metres in height ( ).

17.5.2 Robustness riteria It was found that for a time delay of 50 ms, the system had robustness over all other parameter variations spe ied ( hanges in entre of gravity and mass). With the maximum time delay of 100 ms, in a small part of the parameter envelope, namely at aft and high entre of gravity with a mass of greater than 145 tonnes, the system qui kly began to lose robustness, and eventually be ame unstable. However, it must be mentioned that this o

urred during the 3 degree/s turn, whi h has already been shown to be a bad segment for this parti ular ontroller.

17.5.3 Ride quality riteria These riteria relate to passenger omfort under normal manoeuvres. As su h, the results from the evaluation simulation have been taken as representative.

Maximum verti al a

eleration This is desired to be less than violated twi e.

0:05g.

Figure 17.7 shows that this value is

The less harsh violation is during the wind shear, where we

would expe t an un omfortable ride. However, the worst violation, at

229

0:4g,

PSI (degrees)

Z (metres)

30

25

20

15

10

10 9 8 7 6 5 4 3 2

5 1 5

10

15

20

25

30

35

GAMMA (degrees)

Response to 30m step in Z 3

2

1.5

1

0.5

5

10

15

20

25

30

35

40

10

15

20

25

30

10 8 6

φ

4 2 0 −2 −4

β

−6 −8 −10 0

50

100

150

200

Roll and sideslip during engine failure and re-start

Flight path angle step response

2

3

PSIDOT (degrees)

PHI (degrees)

5

Response to 10 deg step in heading angle

2.5

0 0

0 0

40

PHI / BETA (degrees)

0 0

2 1 0 −1 −2 −3 −4 0

5

10

15

20

25

30

35

40

Roll angle under moderate turbulence

1.5 1

failure

0.5 0 −0.5

restart

−1 −1.5 0

50

100

150

Heading rate during an engine failure 6

1.2

4 2

δVA (m/s)

VA (m/s)

1

0.8

0.6

0.4

0 −2 −4 −6 −8 −10

0.2

−12 0 0

10

20

30

40

50

60

70

−14 0

80

Step response of airspeed VA, Z (m/s, m)

VA, Z (m/s, m)

14 12 10

VA

8 6 4

z

2 0 −2 0

10

20

30

40

50

60

70

80

Airspeed deviation response to a wind step in the direction of UB 30 25 20

z

15 10

VA

5 0

10

20

30

40

50

60

70

80

0

10

20

30

40

50

Response to a 30m Z demand

Response to a 13 m/s VA demand

Figure 17.6: Time responses of nal system

230

60

omes during the steady turn. Of ourse, this is to be expe ted, and has already been explained.

0.1

0.2

0.05

0.15 0

ny(g)

nz (g)

−0.05 −0.1

−0.15 −0.2

−0.25

0.1 0.05 0

−0.05

−0.3

−0.1

−0.35 −0.4 0

100

200

300

400

500

0

100

Time (seconds) Vertical acceleration

200

300

400

500

Time (seconds) Lateral accleration

Figure 17.7: A

elerations during evaluation

Maximum lateral a

eleration This should be less than

0:02g.

Figure 17.7 shows that this requirement is

violated twi e. The rst time is during the engine failure, whi h is onsidered allowable for this emergen y situation. The se ond time is during the steady turn, where large a

elerations of

0:08g o

ur.

17.5.4 Safety riteria Only two of the safety riteria need to be examined with respe t to the way in whi h the design has ae ted them. The roll angle over the evaluation simulation for the nominal system with a 21% mean aerodynami hord verti al entre of gravity are shown in the rst plot in Figure 17.8. The required roll angle limit is 30 degrees. This is ex eeded by the air raft with a high entre of gravity. However, it an be seen that this is always aused by the bad performan e of the system during the steady turn. Without a lateral deviation ommand, the response of the system would be as shown in the se ond plot in Figure 17.8, but the lateral deviation regulation of the system is introdu ing another 7 to 15 degrees of roll angle. 25

35 20

φ (degrees)

φ (degrees)

30 25 20 15 10 5 0 −5

15

10

5

0

−10 −15 0

50

100

150

200

250

300

350

−5 0

10

20

30

40

50

Time (seconds) Roll angle during independent turn

Time (seconds) Roll angle during evaluation

Figure 17.8: Roll angle hara teristi s

231

60

RMS rate (deg/s)

Maximum in design spe s (deg/s)

Aileron

15.5

Tailplane

4.3

8.3 5

Rudder

0.6

8.3

Throttle

0.32

0.24

Table 17.8: Satisfa tion of the maximum rate requirements

17.5.5 Control a tivity riteria The requirements are that mean a tuator rates (taken as RMS a tuator rates here) should be less than 33% of the maximum rates. Figure 17.9 shows the a tuator movements under `moderate' turbulen e onditions.

Note that the

aileron exhibits ex essive movement. This is again due to the fa t that it is trying to regulate lateral deviation too qui kly. This an be seen more vividly in Figure 17.9. The dependent axes on these plots are s aled to the minimum and maximum rates of the a tuator on erned.

The aileron onstantly rate

saturates. The remainder of the a tuators perform well.

17.6 Results of the Automated Evaluation Pro edure This se tion is based on the evaluation mission and s enario dened in the RCAM manual [145℄. Figure 17.10 shows the time plots of the air raft with the eigenstru ture assignment ontroller. The dotted lines indi ate performan e bounds that should not be ex eeded. Four runs are displayed on ea h plot to determine a measure of the robustness of the system under parameter variation. The only segment that indi ated possible problems with the ontroller was the se ond segment, the 3 deg/s turn. These problems and the reasons for their o

urren e have already been analysed during linear simulation. Segm. I

Segm. II

Segm. III

Segm. IV

Total

Performan e

0.0707

0.1454

0.3427

0.1707

0.1824

Perf. Dev.

0.0324

0.0121

0.0828

0.2120

0.0848

Comfort

0.8864

3.3991

1.2201

0.4523

1.4895

Safety

0.0159

0.0749

0.0122

0.1013

0.0511

Power

0.0079

0.0098

0.0148

0.0329

0.0163

Table 17.9: Numeri al results of the evaluation pro edure Quantitative measures of the ontroller are shown in Table 17.9. All the values in the table have been normalised to maximal allowable bounds ex ept for power. Thus, ex eeding a value of unity indi ates that the relevant bounds

232

Actuation (degrees)

δA δR

20 15

δTH

10 5 0 −5 −10

δT

−15 −20 100

110

120

130

140

150

20

δR rate (deg/s)

25

20 15 10 5 0 −5

−10 −15

10 5 0 −5

−15

−20

−20

−25 100

110

120

130

140

−25 100

150

δTH rate (deg/s)

15

10

5

0

110

120

130

140

150

140

150

1.5

1

0.5

0

−0.5

−5

−10

−15 100

110

120

130

140

−1

−1.5 100

150

110

120

130

Actuator rates during moderate turbulence Unless specified, the independent axis describes time in seconds Figure 17.9: A tuator hara teristi s

Second segment: lateral deviations

First segment: top view 300 100 200

0

0

a

b

lateral deviation [m]

x−deviation [m]

50

1

−50

100

0 1

c

d

2

−100

−200 −100 −20

−18

−16

−14

−12 −10 −8 y−position (−YE) [km]

−6

−4

−300 0

−2

1

2

30

20

20

2

f

3

0

−10

e

8

10

3 g

4

h

0

−10

−20

−20

−30

7

Fourth segment: altitude deviations

Third segment: altitude deviations 30

10

3 4 5 6 along track distance from point 1 [km]

Segment II: lateral deviations during the 3 deg/s turn

Segment I: the effect of engine failure

altitude deviation [m]

δT rate (deg/s)

15

−10

altitude deviation [m]

δA rate (deg/s)

Actuator movements during evaluation under moderate turbulence 25

−16

−15

−14 −13 x−position (XE) [km]

−12

−11

Segment III: vertical deviations from the desired glideslope

−30 −11

−10

−9

−8

−7 −6 −5 x−position (XE)

−4 [km]

−3

−2

−1

Segment IV: vertical deviations from the desired glideslope

Figure 17.10: Evaluation pro edure results

233

has been violated. For the eigenstru ture assignment ontroller, almost all the values in this table are adequate. The large values for omfort are due to the importan e pla ed on ensuring good performan e through a

urate tra king of the desired traje tory. Thus, in its `eagerness' to keep to the desired traje tory, the air raft uses short, sharp movements that ause a

elerations to ex eed the re ommended design bounds.

17.7 Con lusions 17.7.1 Requirements for the designer This hapter has attempted to provide a omprehensive appli ation of simple eigenstru ture assignment to the RCAM design hallenge problem.

The

progression of a ontroller design for RCAM involves the need for two things: 1. The designer needs to have a good knowledge of air raft's dynami s. 2. The designer needs to understand the way in whi h the air raft's eigenstru ture relates to these dynami s. This knowledge is required so that, as with

lassi al design, elements in the desired eigenstru ture an be altered with a knowledge of the onsequen es of that alteration. It has been demonstrated in this do ument that it is fairly easy to understand, at least qualitatively, the

onne tion between eigenstru ture and system behaviour.

17.7.2 The design pro ess On e the knowledge des ribed above had been gained for RCAM, design spe i ations were used to produ e a simple ontroller stru ture. A desired eigenstru ture was then produ ed and used to nd an initial ontroller. This ontroller was then analysed and altered until a nal ontroller was produ ed. Along the way, goal attainment was used to improve ea h su

essive design.

The entire design pro ess is shown in Figure 17.11.

The on ept of

eigenstru ture was used throughout the pro ess in some form or other (analysis and assignment). Nevertheless, as the diagram shows, the designer intera ts with the pro ess throughout. The solid boxes represent pro esses. These are either omputer programs, or sub-pro esses in the design that are performed using eigenstru tural tools, as des ribed in the relevant se tions. The dotted boxes represent the initial design spe i ation as produ ed by the designer, based on her/his knowledge and experien e. The variables in these boxes an be altered as the design pro ess progresses. For RCAM, it was only ne essary to alter the desired eigenstru ture, and no hanges to the ontroller stru ture were required during the design. The worst aspe t of the nal ontroller was found to be the behaviour of the air raft when subje ted to both a roll demand and a lateral deviation at the same time. The aileron response to these on urrent ommands is una

eptable, and should be altered. The reason for this response was a re-design to a

ount for large variations in lateral deviation during an engine failure. It is

234

RCAM information Open-loop system analysis

Design Process Controller structure Desired

Design specification

Λ V d d

Eigenstructure Assignment Procedure

A, B, C, Ca

K and L Intermediate analysis Design specification NO met? YES Goal attainment Goals for performance and stability robustness

Detailed final analysis K and L (final) Non-linear simulation

Figure 17.11: S hemati of the design pro ess

235

thus likely that the best way to ounter the problem is to design a separate

ontroller for an engine failure ase.

17.7.3 Advantages of eigenstru ture assignment When ompared with lassi al methods, one advantage of using eigenstru ture assignment is in the time taken to produ e a robust design. Classi al methods

an produ e a system with good performan e, but robustness is a matter of tedious iteration. With eigenstru ture assignment, the designer is able to spe ify performan e and some robustness limitations during the design pro ess. As with other modern methods, another advantage is in the use of all the freedom that is not otherwise utilised in lassi al design to produ e a more

e of eigenstru ture assignment, though, is in being able

robust system. The fort

to use some of this freedom to de ouple modes from states. Thus, the method is primarily a performan e improvement method, and within that, robustness

an also be improved. The ontroller stru ture is a simple gain matrix. Again, this is omparable to lassi al methods, where the relation between the outputs of the system and the a tuation demanded is visible to the designer, who is used to examining stati gains as part of a design pro edure.

This also eases maintenan e and

implementation of the ontroller.

17.7.4 Disadvantages of eigenstru ture assignment Although the method is able to take a

ount of any available freedom in the

losed-loop system, it is unable to a

ount for the unassignable modes. These in lude the min

(m; p)

(n max(m; p)) unassignable modes and the limitation of tra king

outputs (see hapter 3). This means that it might be ne essary to

insert the algorithm into a goal attainment routine to simply assure that the system will be stable. These optimisation routines may take a lot of time to

onverge.

Additionally, the result of the optimisation depends on the initial

ondition.

Thus, altering the initial ondition will result in a dierent on-

troller. Un ertainties in this pro ess mean that rather than using eigenstru ture assignment on its own as a omplete design pro ess, the method is best used as part of an array of tools in analysis and synthesis of a system.

17.7.5 Improvements to eigenstru ture assignment The method des ribed here has been extended to the addition of dynami

ompensation. This was unne essary for the RCAM ase, and should only be used if the system annot be designed with a simple stati gain. As eigenstru ture assignment is a very exible methodology, it should be possible to extend it to improve the ability to tra k more outputs and to de ouple inputs from outputs, rather than modes from outputs. The idea behind these improvements is that the method should eventually be extremely visible

236

and exible enough for lassi al ontrol designers to be able to understand how it relates to lassi al ideas.

17.7.6 Summary Overall, the pro ess was su

essful for use with the RCAM design hallenge. It was possible to use the majority of the design spe i ations given in the RCAM manual to produ e an a

eptable ontroller in linear simulation. It was shown that with this pro ess, it was possible to onstru t and implement a simple system that had good performan e and robustness hara teristi s. In addition, the designer was able to ontrol the trade-o between the two by using their intuition to alter the desired eigenstru ture. The only disappointment in this ase was that the same ontroller was not able to guarantee good performan e for both roll ommand following and lateral displa ement orre tion, although this was an anomaly for this parti ular ase where the two requirements were in strong oni t. It has been shown that eigenstru ture assignment has a simple ontroller ar hite ture, a design y le that involves the designer to a large extent, a solution that satises most of the design spe i ations, and a simple implementation due to the visibility of the method by lassi al designers. Although the this parti ular utilisation of eigenstru ture assignment as a full design method does not provide all the ne essary answers that designers using lassi al ontrol are looking for, the pro edure is ertainly an useful tool in the analysis and synthesis of air raft ontrol design systems.

237

18.

An Eigenstru ture Assignment

Approa h (2)

Jesús M. de la Cruz , Pablo 2Ruipérez and Joaquín Aranda 1

Abstra t.

2

In this hapter the Eigenstru ture Assignment Method

is applied to the Resear h Civil Air raft Model (RCAM) ben hmark problem. The design is done by making use of the lassi al approa h whi h onsists of splitting up the ontroller into two parts, a longitudinal and a lateral ontroller, and in using the standard inner-outer loop ontrol stru ture.

This method allows us to dire tly satisfy

spe i ations in terms of transient response and modes de oupling. However, it does not ope dire tly with system un ertainties.

In

order to ope with robustness, measurements of multiloop gain and phase margins are used in the hoi e of a robust eigenstru ture. Both hannels make use of a onstant gain matrix in the inner loop and a s alar gain in the outer loop. In spite of the ontroller simpli ity, good performan e and robustness results are obtained, although robustness may be in reased by means of gain s heduling with respe t to airspeed.

18.1 Introdu tion The eigenstru ture te hnique has been widely applied to the design of ight

ontrol systems and it is a well-known fa t that its use requires an in-depth knowledge of the system to be ontrolled [11℄.

Although this method is not

intended to deal with robustness, many robusti ation pro edures have been proposed [155℄. Here the eigenstru ture is hosen so that good multiloop gain and phase stability margins are obtained. The hapter is organised in the following way. Se tion 18.2 des ribes the

ontroller stru ture used.

The ontroller has been de oupled into the longi-

tudinal and the lateral hannels.

Both have been designed a

ording to an

inner/outer loop ontrol stru ture.

The inner loop ontrollers are designed

following the eigenstru ture method. A onstant gain matrix is used in both

hannels. The outer loop only uses a s alar gain that is al ulated by means of the root lo us method. 1

Dpt. Informáti a y Automáti a. Fa ultad de Cien ias Físi as. Universidad Complutense.

28040-Madrid. Spain. (Funded by proje t CICYT TAP94-0832-C02-01). 2

Dpt. Informáti a y Automáti a. Fa ultad de Cien ias. U.N.E.D. 28040-Madrid. Spain.

238

Se tion 18.3 deals with the way the RCAM design riteria are translated into the desired eigenstru ture. Se tion 18.4 des ribes the design y le.

The linear model of the plant is

analysed and the most appropriate eigenstru ture is hosen. Next, the feedba k ontroller is obtained and the performan e of the losed-loop system for the linear models is analysed. Multiloop gain and phase margins are used to measure robustness and to guide eigenstru ture hoi e. In se tion 18.5 the veri ation of all the design spe i ations with the nonlinear system is presented. Simulations for the worst possible ombinations of delay, mass and entre of gravity are given. In se tion 18.6 the results of the automated evaluation pro edure are given. The theory has been presented in hapter 3. More details, in luding program ode for the design written in Matlab an be found in [52℄

18.2 Sele tion of the ontroller ar hite ture We make use of the lassi al approa h onsisting of splitting up the ontroller into two parts: a longitudinal and a lateral ontroller. Both of them have been designed in two stages: the inner loop and the outer loop.

The fun tion of

the inner loop is to make the air raft easy and pleasant to y, and it is often

alled a stability augmentation system. The outer loop fun tion is to repla e the pilot for ertain ight manoeuvres su h as maintaining height and speed, turning onto a spe ied heading, limbing at a spe ied rate, et . Figure 18.1 shows the stru ture of the longitudinal ontroller, and gure 18.2 shows the stru ture of the lateral ontroller.

zc

+ z

LonKo wVc VAc

-

∫

+

+

+

q nx nz wV VA

∫

δT LonKin

δTH

Aircraft + Actuators

( z , q , nx , nz , wV , VA )

Figure 18.1: Longitudinal ontroller

18.2.1 Longitudinal ontroller The hoi e of the signals to be used in a design is based on the analysis of the system, the spe i ations and the design method.

Measurement signals Table 18.1 shows the measurements as used by the longitudinal ontroller.

239

vc uc yc

-

atan(vc/uc)

χc

Lateral deviation

+ LatKo

+

-

∫

+

y β=0

β p r φ χ

δΑ

∫

LatKin

δR

Aircraft + Actuators

(y,β,p,r,φ,χ)

Figure 18.2: Lateral ontroller

Inner Loop ontroller

q nx nz wV VA

Pit h rate Horizontal load fa tor Verti al load fa tor

z omponent of inertial velo ity

Air speed

Outer Loop ontroller

z

z position of air raft CoG

Table 18.1: Longitudinal measurements used

Although in the inner loop only four measurements are needed to assign four eigenvalues, two for the short period and two for the phugoid, we make use of ve in order to prevent the slower mode of the a tuators (that of the throttle) from be oming unstable, see [52℄. Finally, the outer loop provides altitude tra king by adding a feedba k of the altitude

z.

A tuator signals These signals are the elevator dee tion or tailplane dee tion position

ÆT H .

ÆT , and throttle

Referen e signals The sele tion of signals hosen as referen es has been guided by the spe i ations given as design riteria, see Ÿ18.3. The sele ted ones are the referen e velo ity position

wV and the referen e airspeed VA for the inner loop, and the referen e zC for the outer loop. 240

Controller stru ture The inner loop ontroller has a stati gain matrix a ting on the ve hosen measured signals and on the integral of the errors of the ommanded variables

wV

and

VA , in the order here spe ied.

The two integrators result in two addi-

tional states that must be in orporated into the linear model for the ontroller design. The outer loop has a proportional a tion a ting on the altitude error. No integral a tion is needed here to avoid steady state errors related to altitude step ommands or disturban es, sin e the altitude dynami s in lude a pole at the origin. The output of the outer loop a ts as a referen e for

wV .

18.2.2 Lateral Controller The lateral ontroller has a stru ture analogous to the longitudinal ontroller.

Measurement signals Table 18.2 shows the measurements as used by the lateral ontroller. Inner Loop ontroller

p r  

Angle of sideslip Roll rate Yaw rate Roll angle Inertial tra k angle

Outer Loop ontroller


y

Lateral deviation

Table 18.2: Lateral measurements used

A tuator signals These signals are the aileron dee tion

ÆA , and rudder dee tion ÆR .

Referen e signals Again, the sele tion of referen es has been guided by the spe i ations given as design riteria, see Ÿ18.3. Those sele ted are the lateral deviation for the outer

C and sideslip angle C for the inner loop. C is not given as a referen e signal but it is obtained from the referen e velo ity

omponents uC and vC as atan(vC =uC ). C has a null onstant referen e value in order to keep always lose to zero.

loop, and the inertial tra k angle

Controller stru ture The inner loop ontroller has a stati gain a ting on the ve hosen signals measured, and on the integral of the errors of the ommanded variables

241

and

,

in the order spe ied here. The introdu tion of two integrators results in

two additional states that must be in orporated into the linear model for the

ontroller design. The outer loop has a proportional a tion a ting on the lateral error. No integral a tion is required here to avoid steady state errors relating to lateral step ommands or disturban es, sin e the lateral dynami s in lude a pole at the origin. The output of the outer loop a ts as a referen e for the inertial tra k angle.

18.3 Translation of design riteria into method dependent obje tives Out of the ve lasses of design riteria given in hapter 14, for the RCAM design, the performan e riteria are the most signi ant. These riteria are given in terms of transient response hara teristi s to ommand signals and ross

oupling onstraints. The main hara teristi of the eigenstru ture assignment method is that it allows the designer to satisfy spe i ations dire tly in terms of transient response and mode de oupling. Then, these are the most suitable

riteria to be used as a guide in the design phase. Our design is based mainly on these riteria, but we made use of linear and non-linear losed-loop time responses and robust analysis tools as a guide in the sele tion of the eigenstru ture.

18.3.1 Performan e Criteria The performan e riteria an be lassied into two groups: longitudinal and lateral. We dis uss separately for ea h group the way in whi h the spe i ations have been in orporated into the design.

Longitudinal spe i ations There are ommand response spe i ations in terms of overshoot, rise time and settling time for three ommanded signals: airspeed, ight path angle and altitude. These spe i ations provide a lower limit for the damping ratio and natural frequen y of se ond order modes and for the time onstant of rst order modes oupled with the signals. The ight path angle

is neither available as an output nor as a referen e

signal. To ope with this situation we use the relationship where

V

sin( ) = wV =V ,

is the total inertial velo ity. Therefore, for onstant inertial velo ity

the ight path angle may be ontrolled by means of

wV .

This leads us to an

interpretation of the spe i ations in terms of ommands in

wV .

We add a

verti al velo ity error integrator state to get good low-frequen y tra king. We add an outer loop for tra king the referen e altitude. A signal proportional to the altitude error is used as a referen e signal for

242

wV .

The proportional

gain is al ulated using the root lo us method in su h a way that it fulls the transient riteria. The spe i ation of de oupling between airspeed and altitude may be obtained by de reasing ross- oupling between

VA

and

wV .

Lateral spe i ations There are now two ommand signal spe i ations (heading angle and lateral deviation) in terms of transient response hara teristi s that may be transformed into bounded eigenvalues. The remaining spe i ations are given in terms of behaviour in ase of engine failure and under disturban es. The heading angle is a lateral motion state but it is neither available as an output nor as a referen e. Instead, we make use of the inertial tra k angle that is an output signal and is related to the heading angle by means of the equation

=

+ V .

Although there is no referen e signal for

be obtained from the referen e velo ity omponents equation

C = tan 1 (vC =uC ).



,

su h a signal may

and

vC

by means of

An integrator is introdu ed to eliminate sideslip

errors, where the ommand signal for the safety riteria of keeping

uC

is hosen as 0.

Doing so, we ope with

minimised at all times. An inertial tra k angle

error integrator is also introdu ed in order to avoid heading angle steady state errors. The lateral deviation is ontrolled in the outer loop. A signal proportional to the lateral error is used as a referen e signal for the inertial tra k angle. The proportional gain is al ulated using the root lo us method in su h a way that it fulls the transient riteria. The engine failure requirements an not be easily interpreted in terms of eigenvalue and eigenve tor spe i ations and shall not be tested before the phase of analysis of simulation results. However, the requirement of keeping sideslip angle to a minimum in ase of engine failure an be translated into a spe i ation of de oupling between the eigenvalues related with the roll motion and those related with the lateral velo ity.

18.3.2 Robustness riteria The eigenstru ture method is not a robust ontrol method and, although many dierent algorithms have been proposed to enhan e the robustness, no method will be used here. However, we make use of the stability margins given in hapter 3 to measure the robustness of the feedba k system. For a hosen eigenstru ture the stability margins are measured and the eigenstru ture hanged to get better stability margins. After a few steps a suitable eigenstru ture is sele ted.

18.3.3 Other riteria The ride quality riteria, safety riteria and ontrol a tivity riteria an not be in orporated in an ad ho manner into the eigenstru ture method.

243

As done

with robustness, they will be analysed in the ourse of the analysis of results phase and physi al relations between their behaviour and the eigenstru ture a hieved will be established in order to ope with them.

In the sele tion of

the eigenvalues we should have in mind to assign the mode values lose to the open-loop air raft modes to minimise the ontrol a tivity.

18.4 Design y le and ontroller derivation The design y le is summarised in the following s heme 1. Analysis of the linear and non-linear model of the plant. 2. Sele tion of the eigenstru ture and gain al ulation. 3. Analysis of the open and losed-loop system (linear and non linear model). 4. Robusti ation pro edure (iterate). 5. Iterate 1 - 4. As we have seen in the previous se tion, among all the design riteria only those of the performan e riteria related to the transient response of the system

an be interpreted almost dire tly in terms of eigenstru ture. The rest of them must be analysed after an eigenstru ture has been hosen and the ontroller found.

From this analysis another eigenstru ture will be hosen and so on.

That has been the most onsuming time task. A robusti ation pro edure or any other form of "optimal" solution may help to break the iterative pro edure. We will now explain the method we have used to sele t an eigenstru ture.



We analysed the design riteria and the oupling of the modes of the plant with the states, the inputs and the outputs. This analysis was used as a guide in hoosing the outputs for feedba k and the eigenstru ture. After sele ting the outputs to be used for feedba k the integrators were added to the loop.



We hose a set of eigenvalues that fullled the transient response limits in a

ordan e with the requirements and were lose to the natural air raft modes. After that, their asso iated eigenve tors were hosen to get the desired de oupling between the modes.



With the eigenstru ture hosen, we omputed the feedba k gain and analysed: - the stability (sin e one of the non-assigned eigenvalues might be ome unstable or badly damped) and the stability margins - the time response of the system - the de oupling of the obtained modes

244

Now the y le begins. New outputs and/or eigenstru ture must be hosen to improve the results obtained. We have rst sele ted an eigenstru ture that provides a

eptable design riteria. We have tested dierent eigenvalues with xed eigenve tors. On e the eigenvalues that give better stability margins have been hosen, the eigenve tors have been hanged to try to improve robustness and, when ne essary, de oupling. After a few steps we onvin ed ourselves that the hosen eigenve tors ould not be improved.

18.4.1 Longitudinal ontroller Longitudinal model The non-linear model is used to generate linear models for ontrol law design and to generate non-linear time histories for evaluating ontrol designs. On e a trim ondition is established for the non-linear air raft model within the simulation environment, a linear model is generated to apture the perturbational dynami s around the equilibrium point. The model has been linearised around the following operating ondition: kg,

gx = 0:23 and gz = 0:1.

V = 80 m/s, h = 1000 m, mass = 120000

The aerodynami model is augmented with rst

order a tuator models. Two integrated error states are added to the linearised model, one for

wV

and another for

VA .

The number of outputs is now in reased

by two and the eigenvalues for the modes of the integrators may be spe ied. The maximum allowed transport time delay of 0.10 s is added to the model with a rst order Padé approximation.

Inner loop ontroller design As mentioned in hapter 3 we an spe ify as many losed-loop eigenvalues as outputs for feedba k used. Therefore, only four measurements are needed to spe ify the phugoid and short period modes. However, we used ve measurements so that the slower mode of the a tuators (that of the throttle) is spe ied, in order to avoid it be oming unstable.

Mode Eigenvalues

q  uB wB XT XT H delay ÆT delay R ÆT H R wV VA

Phugoid

Short Period

Throttle

xx x1 1x 00 xx xx xx xx 00 xx

1x xx 00 x1 xx xx xx xx xx 00

x x x x x 1 x x x x

0:4376 0:0624{

0:9059 0:4388{

0:5

R

wV 2:0000 x x 0 x x x x x 1 0

R

VA

1:9000 x x x 0 x x x x 0 1

Table 18.3: Desired eigenstru ture of the longitudinal losed-loop system

245

Table 18.3 shows the eigenstru ture hosen for the system. The state om-

XT

ponents are given in the rst olumn, where the rst order tailplane model and

XT H

is the state orresponding to

the state orresponding to the rst

order engine model. The rst row shows the desired eigenvalues, and the desired eigenve tors are shown underneath, where the symbol "x" represents the

unspe ied elements in the eigenve tors. The resulting gain is:



LonKin = 00::4755 0455



0:0532 0:0838 0:0169 0:0055 0:0033 0:0014 1:3063 0:3047 0:0152 0:1221 0:0004 0:0227

Loop stability margins Figures 18.3 and 18.4 show the singular value plots of the sensitivity fun tion the omplementary sensitivity fun tion

S+T

T

S,

and the balan ed sensitivity fun tion

at the a tuator inputs and at the sensor outputs. Tables 18.4 and 18.5

show the gain and phase margins obtained from the sensitivity fun tions. S, T and S+T (HG)

1

S, T and S+T (GH)

1

10

10

S+T

S+T 0

0

magnitude

10

magnitude

10

S

−1

10

T

−2

10

T

S

−1

10

−2

−2

−1

10

0

10

10

1

10

10

−2

−1

10

Figure

18.3:

0

10

frequency (rad/s)

1

10

10

frequency (rad/s)

Longitudinal

inner

Figure

18.4:

Longitudinal

inner

loop singular values of the input

loop singular values of the output

sensitivity fun tions

sensitivity fun tions

Fun tion

S T

S+T



1.42 1.03 1.96

!

0.57 0.09 0.48

Km = 1= 0.70 0.98 0.51

Gain margin (dB) [-4.6,10.6℄ [-29.7,5.9℄ [-9.8,9.8℄

Phase margin (deg) 41.2 57.8 54.1

Table 18.4: Longitudinal inner loop stability margins at the inputs

Fun tion

S T

S+T



1.47 1.00 1.96

!

0.54 0.02 0.46

Km = 1= 0.68 1.00 0.51

Gain margin(dB) [-4.5,10.0℄ [-54.4,6.0℄ [-9.8,9.8℄

Phase margin (deg) 39.9 59.9 54.1

Table 18.5: Longitudinal inner loop stability margins at the outputs Good stability margins are obtained, but we must remember that these margins are onservative, and even better stability margins should be expe ted.

246

Outer loop ontroller design Choosing

LonKo = 0:1027,

the slower roots are

0:14  0:14{,

having a rise

time of 11 s, whi h is less than the spe ied rise time for an altitude ommand (15 s), and a settling time of 35.4 s whi h is well below the required 45 s. The gain margin for the outer loop is 13 dB at is 63 deg at

w = 0:1 rad/s.

w = 0:35 rad/s and the phase margin

18.4.2 Lateral ontroller Lateral model The pro edure followed in dealing with the lateral model is analogous to the one used with the longitudinal model. A linear model is generated from the same trimmed ondition used to obtain the linearised longitudinal model. The a tuator dynami s have been added to the linear model by augmenting it with orresponding states. Also, we added the two integrated error states and a 0.10 s delay with a rst order Padé approximation.

Inner loop ontroller design Table 18.6 shows the eigenstru ture hosen. The state omponents are given in

XA is the state orresponding to the rst order aileron XR the state orresponding to the rst order rudder

the rst olumn, where dee tion model and dee tion model.

Mode Eigenvalues p r



vB XA XR

Delay Æa Delay R Ær R



Dut h roll

0:8  0:6{ 00 1x 00 xx x1 xx xx xx xx xx xx

Spiral -0.4 x x 1 x 0 x x x x x x

Roll Sub. -1 1 0 x x 0 x x x x x x

Heading -0.8 x x x 1 0 x x x x x x

R



-1.5 0 x 0 x x x x x x 1 x

R



-1.4 x x x x 0 x x x x x 1

Table 18.6: Desired eigenstru ture of the lateral losed-loop system

The resulting gain is:

LatKin =



3:6246 1:7016 2:9057 3:0480 13:1933 0:6869 2:2288 1:5216 0:0782 2:4251 0:2268 1:0320 0:7237 0:1820 247



Loop stability margins The results obtained at the a tuator inputs and at the sensor outputs are summarised in Table 18.7 and Table 18.8, respe tively. Figures 18.5 and 18.6 show the singular values plots of the sensitivity fun tions.



Fun tion

S T

S+T

Km = 1=

!

1.43 1.18 2.19

0.65 0.21 0.46

0.70 0.85 0.46

Gain margin (dB) [-4.6,10.4℄ [-16.2,5.3℄ [-8.6,8.6℄

Phase margin (deg) 40.8 50.0 49.2

Table 18.7: Lateral inner loop stability margins at the inputs

Fun tion

S T

S+T



1.71 1.01 2.19

Km = 1=

!

0.50 0.04 0.46

0.59 0.99 0.46

Gain margin(dB) [-4.0,7.7℄ [-43.3,6.0℄ [-8.6,8.6℄

Phase margin (deg) 34.1 59.5 49.2

Table 18.8: Lateral inner loop stability margins at the outputs

S, T and S+T (HG)

1

S, T and S+T (GH)

1

10

10

S+T

S+T 0

0

magnitude

10

magnitude

10

S

−1

10

T

−2

10

T

S

−1

10

−2

−2

10

−1

0

10

10

1

10

10

−2

10

−1

0

10

frequency (rad/s)

10

1

10

frequency (rad/s)

Figure 18.5: Lateral inner loop sin-

Figure 18.6: Lateral inner loop sin-

gular values of the input sensitivity

gular values of the output sensitiv-

fun tions

ity fun tions

Outer loop ontroller design Choosing

w = 0:29

LatKo = 0:001,

the gain margin for the outer loop is 13.9 dB at

rad/s and the phase margin is 63.9 deg at

slower roots are

0:11  0:10{

w = 0:08

rad/s.

The

that have a rise time of 15 s and a setting

time of 44 s. Therefore, the transient spe i ations for a lateral deviation step

ommand are met.

248

18.5 Analysis of the resulting ontroller The ontroller was designed by an iterative method onsidering riteria of performan e and robustness in the linear model, without taking into a

ount non linearities. In this se tion we present the veri ation of all the design spe i ations with the non-linear system. The des ription of the design riteria is given in hapter 14. All the simulations are run using the following onguration (see Ÿ18.4):

kg,
x = 0:23 ,
y = 0 and
z = 0:1 ; and the initial VA = 80 m/s, altitude = 1000 m, = 90 deg, = 0 deg and

mass = 120.000

onditions are:

 = 1:65 deg.

18.5.1 Non-linear simulation observing performan e riteria Figure 18.7 shows the response to lateral deviation, altitude, heading angle, ight path angle and airspeed ommands. It an be seen that all performan e

riteria on erning altitude, heading angle, inertial ight path angle, airspeed and ross oupling from airspeed to altitude are fullled. In the lateral deviation step ommand a step hange of 100 m has an overshoot of

13 %, however in the 5 %, but the lateral

linearised model the overshoot is smaller than the spe ied

50 40

10

20

30

40

−88 10

20

30

80 79.5 0

40

20 30 time (sec)

VA (m/s)

150

200

50

100

150

200

10

20

30

40

0 −5 0

50

100

150

200

50

100

150

200

50

100

150

200

100 150 Time (sec)

200

10 0 −10 0

2 0 0

10

20

30

40

81

0.2

80 79 0

50

100

150

0 −0.2 0

200

Step of 13 m/s in VA

40

Altitude (m)

10

100

2 0 −2 0

Step of 3 deg in Gamma

Step of 13 m/s in VA 90 85 80 0

50

5

Phi (deg)

80.5

Gamma (deg)

Step of 3 deg in Psi

−90 0

−95 0

Step of 30 m in altitude

1020 1000 0

−90

ny (g)

30

1010 1000 990 0

10

20 30 time (sec)

40

2 1.5 1 0

50

100 150 Time (sec)

200

Altitude (m)

20

Beta (deg)

10

−85

V_A (m/s)

0 0

VA (m/s)

Altitude (m)

Psi_dot (deg/sec)

Psi (deg)

Step of 100 m 100

Step of 30 m in altitude

Psi (deg)

10 % in less than 30 s, a

ording to spe i ations.

Alpha (deg)

Lat. desv. (m)

deviation is redu ed to

1005 1000 995 0

50

Figure 18.7: Response of the non-

Figure 18.8: Response of the non-

linear model to ommand signals

linear model to an engine failure

18.5.2 Non-linear simulation of an engine failure The results are given in Figure 18.8. We an see that:  is less than the spe ied 10 deg, its steady state deviation does not ex eed 5 deg and, when the engine is restarted,  is redu ed to zero with an overshoot that sligtly ex eeds the spe i ation of 50 %; is qui kly minimised; and _ is well below the spe ied 3 deg/s. The omfort and safety riteria are also veried: higher than

79

the airspeed is always

m/s, whi h is mu h bigger than the spe ied

249

62:2

m/s

=

1:2Vstall ;

the angle of atta k

and the lateral a

eleration spe ied

0:2 g.

 ny

 1:7 deg is less than the spe ied 12 deg; in reases up to

0:1

Finally, the variations in altitude are less than

g, that is less than the

2 m.

18.5.3 Non-linear simulation observing ride quality and safety riteria The spe i ations for ride quality riteria are a maximum verti al a

eleration and a maximum lateral a

eleration. Figure 18.9 shows the lateral a

eleration

90 deg turn and the verti al a

eleration in a hange from = 0 deg to

= 6 deg and from -6 deg to -3 deg. These values orrespond to segments

in a

II and III of the Automated Evaluation Pro edure (Ÿ18.6), respe tively. It an be seen that lateral a

eleration a

eleration

nz

ny is always within the limits and the verti al

slightly surpasses the maximum level.

90 degrees turn

gamma rotation to −6 and −3 degree 0.05 nz (g)

ny (g)

0.02 0 −0.02 0

20

40 time (sec)

60

0 −0.05 0

80

50 time (sec)

100

Figure 18.9: Ride quality riteria observation Figure 18.10 depi ts the results of the non-linear simulation showing the safety riteria.

The simulations orrespond to the landing approa h of the

evaluation pro edure as used in Ÿ18.6, whi h is a good representation of all

80 70 0

Phi (deg)

Alpha (deg)

90

200

20 0 −20 0

200

10 0 0 10

400 Beta (deg)

V_A (m/s)

possible ight onditions.

400

200 time (sec)

400

0 −10 0

400

200

Figure 18.10: Safety riteria observation The safety riteria are fullled sin e the airspeed is always well above

1:2Vstall = 62:2 m/s; the maximum angle of atta k  remains within the limits,

its maximum value is observed during the turn, but is well below the limit of

12 deg; the roll angle  remains also within the limits, but in the turn is near to the limit of 30 deg; and the sideslip angle is qui kly minimised at all times. 250

Detailed response to an engine failure for airspeed and angle of atta k has been given in the previous se tion.

18.5.4 Non-linear simulation under moderate turbulen e

onditions We will give an analysis of the non-linear systems under moderate turbulen e

onditions. The spe tra of the turbulen e we used in the simulations are des ribed in hapter 14 (Ÿ14.2.6). Figure 18.11 shows a tuator behaviour. All the requirements on the mean of a tuators are fullled. Moreover, all RMS values are less than the limits (the spe ied values are given in parentheses):

< 8:25 deg/s) with a RMS of 7.30

- The mean aileron rate is 0.42 deg/s ( deg/s

<5

- The mean tailplane rate is 0.12 deg/s (

deg/s) with a RMS of 1.03

deg/s

< 8:25 deg/s) with a RMS of 1.97

- The mean rudder rate is 0.08 deg/s ( deg/s

< 0:24 deg/s) with a RMS of 1.50

- The mean throttle rate is 0.09 deg/s (

50

20 0 −20 0

50

10 5 0 0

50 time (sec)

100

50

100

20 0 −20 0

50

100

5 0 −5 0

50 time (sec)

100

Phi (deg)

Phi (deg)

20 0 −20 0

−5 0

50

100

5

Beta (deg)

0

100

0

0 −5 0

50

100

−85 −90 −95 0

50

−90 −95 0

50 time (sec)

100

−5 0

50

100

50

100

50

100

50 time (sec)

100

5 0 −5 0

−85

−90 −95 0

−90 −95 0

Figure 18.11: Control a tivity un-

Figure

der moderate turbulen e onditions

moderate turbulen e onditions of angles

18.12:

0

−85

100

−85

5

Psi (deg)

20 −20 0

50

Close loop

5

Chi (deg)

100

Beta (deg)

50

Psi (deg)

dr_dor (deg/sec)

−20 0

Open loop 20 0 −20 0

Chi (deg)

dt_dot (deg/sec) da_dot (deg/sec)

100

0

dth_dot (deg/sec)

100

20

dth1 (deg)

dr (deg)

dt (deg)

da (deg)

deg/s

, ,

Responses and

under



, the sideslip  under these moderate

Figure 18.12 shows in open and losed-loop, the roll angle angle

, the heading angle

turbulen e onditions.

and the inertial tra k angle

In losed-loop,



always remains within the spe ied

limit of 5 deg. Table 18.9 shows the RMS values of the errors in open and losedloop. The losed-loop values of

,

and

open-loop values, however the RMS of loop.

251



are less than their orresponding

is higher in losed-loop than in open-





Open loop RMS (deg)

2.04

1.06

0.89

0.97

Closed loop RMS (deg)

0.65

0.96

1.45

0.28

Table 18.9: RMS of the errors in angles

, ,

and

  in open and losed-loop

18.5.5 Non-linear simulations observing robustness riteria Figures 18.13-18.15 show the simulation results obtained at the nominal, minimum and maximum values of time delay, mass and entre of gravity, all ases at the design airspeed of 80 m/s, giving a set of 81 simulations. In the plots, the angles are given in radians, the displa ements in meters and the velo ities in m/s. Figure 18.13 shows the lateral step response and the altitude step response. We an see that there is very little ontrol a tivity, there are almost no overshoots and the settling times are well below the spe ied values. Figure 18.14 shows the airspeed-altitude ross- oupling. We an see that the deviations are always within the limits. Figure 18.15 shows the roll and heading angles at engine failure. In all ases the performan e riteria are fullled: the roll angle never ex eeds 10 deg and its steady state deviation does not ex eed 5 deg; the heading rate is always well below the spe ied 3 deg/s, and the sideslip angle is qui kly minimised. In [52℄, the simulation results an been found for the inner loop response to heading and ight path angle steps ommands, and for air speed step and wind step responses. With respe t to the inner loop response to heading and ight path angle steps, in all ases the overshoot is lower than the spe ied

5 % and

the settling time is less than the spe i ations. Regarding the airspeed step and wind step response, all the performan e riteria are fullled in the airspeed

ommand. The response to a wind step has three dierent hara teristi s whi h

orrespond to the three dierent values of the mass whi h have been onsidered. In ase of minimum and maximum mass values, the deviation in airspeed is larger than 2.6 m/s after 15 s of the step, and in ase of nominal mass value the spe i ation is fullled. In [52℄, the simulation results obtained for all possible worst onditions of time delay, mass and entre of gravity, at a speed of 63.7 m/s and at maximal ap speed (90 m/s) an also be found. The worst results are obtained in the engine failure at 63.7 m/s, with minimum mass and with


x = 0:31 .

Consequently, a new set of ontroller gains

was omputed using the linearised model at 63.7 m/s velo ity, but without

hanging the rest of the trimming parameters, and with the same eigenstru ture as before.

The resulting ontroller and simulation results, obtained by

making use of this ontroller when simulating at 63.7 m/s and for the worst

ondition of time delay, mass and entre of gravity, are given in [52℄. Similar performan e and robustness results to those obtained with the ontroller designed at the original speed of 80 m/s are obtained, thus gain s hedul-

252

Lateral step response y − y_c −.

Altitude step response z − z_c −.

1.5

1.5

1

1

0.5

0.5

0

0

−0.5 0

10

20 30 time [s] vv − vv_c −.

−0.5 0

40

1

1

0.5

0.5

0

10

20 30 time [s] wv − wv_c −.

40

0

−0.5 0

10

20 time [s] da

30

−0.5 0

40

10

20 time [s] dt

30

40

20 30 time [s] throttle L− R−−

40

0.1 0.02 0.05

0.01

0

0 −0.01

−0.05

−0.02 −0.1 0

10

20 time [s] dr

30

40

0

0.05

10

0.01 0.005

0

0 −0.005

−0.05 0

10

20 time [s]

30

40

−0.01 0

10

20 time [s]

30

40

ident: td012:m012:x012:z012:ex7

Figure 18.13: Lateral and altitude step response at the design airspeed

Airspeed − Altitude cross coupling z − z_c −.

z (va_c) 20

0

10

−20

0 −10

−40 0

10

20 time [s] va (z_c)

30

40

−20 0

10

20 30 time [s] va − va_c −.

40

2 15

1

10 0 5 −1

0

−2 0

10

20 time [s] dt (z_c)

30

40

−5 0

0.2

0.2

0.1

0.1

0

0

−0.1

−0.1

−0.2 0

10

20 30 time [s] throttle L− R−−

40

−0.2 0

0.2

0.2

0.1

0.1

0

0

−0.1

−0.1

−0.2 0

10

20 time [s]

30

40

−0.2 0

10

10

10

20 time [s] dt (va_c)

30

40

20 30 time [s] throttle L− R−−

40

20 time [s]

30

40

ident: td012:m012:x012:z012:ex7

Figure 18.14: Airspeed-altitude ross oupling at the design airspeed

253

Roll angle and heading response to right engine failure phi − R e_f −. psi − R e_f −. 1 0.2

0.5

0.1 0

0

−0.1

−0.5

−0.2 0

10

20 30 time [s] beta − R e_f −.

40

−1 0

10

20 30 time [s] psid − R e_f −.

40

0.1 0.2

0.05

0.1 0

0

−0.1

−0.05

−0.2 0

10

20 30 time [s] da − R e_f −.

40

−0.1 0

10

20 30 time [s] dt − R e_f −.

40

10

20 30 time [s] throttle L− R−−

40

0.1

0.5

0.05 0

0 −0.05

−0.5 0

10

20 30 time [s] dr − R e_f −.

40

−0.1 0

0.2

0.5

0.1 0

0 −0.1

−0.5 0

10

20 time [s]

30

40

−0.2 0

10

20 time [s]

30

40

ident: td012:m012:x012:z012:ex7

Figure 18.15: Roll angle and heading response to right engine failure at the design airspeed

ing with respe t to velo ity should be used.

18.6 Results of the automated evaluation pro edure This se tion presents the results of the evaluation pro edure, as proposed in Ÿ14.3.3 onsisting of a landing approa h.

Segment I Figure 18.16 shows the performan e of the ontroller in this segment, and it

an be seen that the lateral deviation is always less than 20 m. Therefore, the

ontroller omplies with the orresponding spe i ation.

Moreover, the four

plots are almost the same, whi h means small sensitivity to time delay and to horizontal entre of gravity variations.

Segment II Figure 18.17 gives the behaviour of the model in this manoeuvre. It an be seen that the traje tory of the model surpasses the bounds marked in the plots but the lateral deviation never ex eeds the maximum value of

200 m and at the end

the lateral deviation is lose to zero. The lateral a

eleration never surpasses

254

the maximum allowable value (see Figure 18.9).

Moreover, the model has a

very smooth turn, fulls all the performan e design riteria and our attempts to have a traje tory within the bounds diminished the stability margins, so we a

epted it as is.

Segment III Figure 18.18 represents the behaviour of the model in the des ent phase.

It

an be seen that the traje tories of the model surpass the bounds marked in the plots although the verti al deviation never ex eeds the maximum value of 20 m and at the end of the segment the deviation is lose to zero. In Figure 18.10 we an see that the speed variation is well below the allowed 4 m/s. Moreover, the model has a very smooth transition during the entire segment, although the verti al a

eleration is a little bit high at some points. We an see in Figure 18.9 that the verti al a

eleration slightly surpasses the maximum allowed value. This is ree ted in the omfort index in Table 18.10. Sin e the rest of the design riteria are fullled and our attempt to diminish this value produ es worse results, we a

epted it.

Segment IV Figure 18.19 shows the behaviour of the model in this segment. It an be seen how the traje tories fall inside the bounds during the entire segment. The rest of the spe i ations are fullled by the ontroller.

Numeri al results Table 18.10 summarises the results as obtained by the ontroller along the landing approa h. For full details see hapter 14.

In general the results are

good, ex ept for the omfort riterion in Segment III. The problem with the

omfort has already been explained. It is basi ally due to a small high level of the verti al a

eleration. First segment: top view

Second segment: lateral deviations 300

100

0

0

a

b

lateral deviation [m]

x−deviation [m]

200 50

1

−50

100

0 1

c

d

2

−100

−200 −100 −20

−18

−16

−14

−12 −10 −8 y−position (−YE) [km]

−6

−4

−300 0

−2

1

2

3 4 5 6 along track distance from point 1 [km]

7

8

Figure 18.16: Segment I: The ee t

Figure 18.17:

of engine failure with bounds

deviations during the 90 degrees

Segment II: Lateral

turn with bounds

255

Fourth segment: altitude deviations 30

20

20

10 2

f

altitude deviation [m]

altitude deviation [m]

Third segment: altitude deviations 30

3

0

−10

e

−20

−30

10

3 g

4

h

0

−10

−20

−16

−15

−14 −13 x−position (XE) [km]

−12

−30 −11

−11

−10

−9

−8

−7 −6 −5 −4 x−position (XE) [km]

−3

−2

−1

Figure 18.18: Segment III: Verti al

Figure 18.19:

deviations during the -6 and 3 de-

al deviations during the nal ap-

Segment IV: Verti-

grees glidslope with bounds

proa h with bounds

Segm. I

Segm. II

Segm. III

Segm. IV

Total

Performan e

0.0764

0.4964

0.3285

0.1905

0.2730

Perf. Dev.

0.0309

0.0161

0.4926

0.1873

0.1817

Comfort

0.5432

0.7340

1.1808

0.4674

0.7314

Safety

0.0038

0.0382

0.0070

0.0345

0.0209

Power

0.0037

0.0027

0.0150

0.0309

0.0131

Table 18.10: Numeri al results of the evaluation pro edure

18.7 Con lusions In this hapter the eigenstru ture method has been applied to the RCAM ben hmark problem. The design was ompleted by making use of the lassi al approa h. This

onsists of splitting up the ontroller into two parts, a longitudinal and a lateral

ontroller, and in using a standard inner-outer loop ontrol stru ture. In every inner loop the feedba k outputs, ommand signals and integrated outputs have been hosen guided by the design spe i ations. For both inner loops a onstant gain feedba k matrix has been al ulated using the eigenstru ture te hnique. This method allows the designer to satisfy dire tly performan e riteria given in terms of damping, settling time and de oupling, but not to ope dire tly with system un ertainties. However, the eigenstru ture was hosen in an iterative way, so that good stability margins were obtained.

In both outer loops a

onstant s alar gain has been used. When analysing the ontroller with the non-linear model (Ÿ18.5) all the design riteria are fullled, but the maximum verti al a

eleration is surpassed in ertain onditions and the RMS of the heading angle error in losed-loop is greater than in open-loop. Good robustness results are obtained with respe t to variations of the mass,

entre of gravity and transport time delay. However, gain s heduling should be used with respe t to velo ity sin e bad results are obtained with speed

256

variations, mainly for the engine failure ase, as explained in Ÿ18.5. The results obtained in the automati evaluation pro edure demonstrate good good fulllment of all of the design riteria ex ept the omfort riteria. In our design this is due to the fa t that the verti al a

eleration annot be diminished without violating other performan e riteria. The sele tion of a good eigenstru ture follows an iterative pro ess that an be time onsuming. The pro ess ould be shortened if some optimisation method were used.

Without any optimisation pro ess, doubts about how good the

ontroller is will always remain.

257

19.

A Modal Multi-Model Approa h

Carsten Döll , Jean-François1 Magni1 and Yann Le Gorre 1

Abstra t.

This paper presents a modal multi-model approa h ap-

plied to the RCAM design hallenge.

Inner feedba k loops (sta-

bilization) are designed by eigenstru ture assignment.

Equations

orresponding to de oupling properties for a representative set of linearized models are olle ted. In the lateral ase, the number of equations is small enough so that these equations an be solved using a proportional feedba k. In the longitudinal ase, the use of a dynami ontroller turns out to be ne essary. Outer loops (tra king) are designed separately using lassi al ontrol.

19.1 Introdu tion The RCAM design problem is aimed at keeping as lose as possible to industrial requirements. Espe ially, proportional or dynami gains with minimum number of states are required.

Under this onstraint we tried to redu e the usual

s heduling as mu h as possible by improving robustness. This task was even more di ult on a

ount of the unusual nature of the RCAM problem, having its own evaluation riteria whi h diered somewhat from usual industrial ones. The main dieren e between a tual industrial problems and RCAM problem omes from the fa t that the autopilot we are looking for is the same for ruising, beam apture and landing.

It would have been possible to de-

sign three stru tures and to onsider ommutations, but this kind of problem is well understood in industry, therefore no improvement ould be expe ted from our ontribution. So, we preferred to design a single stru ture and onsequently, the autopilot design we propose has faster than a typi al autopilot (indu ing some overshoot and omfort deterioration). Nevertheless we believe that the methodologies proposed to redu e s heduling are still valid (maybe more e ient) in a more realisti industrial setting. Se tion 19.2 details the ontroller stru ture used and the hosen referen e inputs and measurements. Then, the spe i ations are translated into method dependent obje tives in Se tion 19.3. In Se tion 19.4 the proposed ontroller is derived. An analysis of robustness and performan es is proposed in Se tion 19.5. 1

CERT ONERA, Département d'études et Re her hes en Automatique, BP 4025, F31055

Toulouse Cedex, Fran e.

258

Outer loop design. A simple preliminary inner loop is used for this purpose. It appeared that it is ne essary to use a sophisti ated outer loop with feedforward. Indeed, the use of feeforward permits a designer to meet the design spe i ations with a mu h slower inner loop; in turn, most of the inner loop degrees of freedom an be devoted to robustness and performan e. The proposed outer loop turned out to be so robust that it was not ne essary to update it when several kinds of inner loops were being tested.

Inner loop design. First, standard eigenstru ture assignment was used. Performan e is easily met by this approa h but it is more di ult to ensure robustness.

For the lateral hannel it is shown that the initial eigenstru ture

assignment an be improved, without gain s heduling, so that robustness requirements ould be met. For the longitudinal hannel, we did not nd a satisfa tory proportional gain. A non-s heduled low dimensional dynami feedba k is proposed. The analysis of robustness is made by onsidering a set of linearized models (see page 265). In order to assess damping ratio and settling time, the poles are plotted; for ross- oupling, the step responses of all the linearized models are plotted. More details on erning the results an be found in [55℄.

The theory is

briey presented in Chapter 3 and more details an be found in [150℄.

19.2 Sele tion of the Controller Ar hite ture We shall onsider inner and outer loops. Generally speaking,



the inner loop stabilizes and augment the handling qualities as well as providing robustness



the outer loops guide the air raft along a given traje tory.

A traditional approa h to the ontroller ar hite ture is to split the ontroller into two omponents: longitudinal and lateral ontrollers. These two parts are treated separately from one another, ex ept for the turn ompensation (denoted

q oord).



2

longitudinal ontroller : tra king of the total inertial speed altitude

z.

integral ee t, relative to the verti al speed



yB .

and of the

and the inertial speed



V.

and of the lateral de-

The inner loop onsists of a non-intera tive ontroller, with

integral ee t, relative to the sideslip angle 2

wV

lateral ontroller: tra king of the sideslip angle viation

V

The inner loop onsists of a non-intera tive ontroller, with



and roll angle

.

We dis overed too late that we implemented tra king of the inertial speed instead of

tra king of the airspeed. Results on erning tra king of the airspeed an be found in [55℄. Robustness issues are similar in both ases. The main dieren e is that, on a

ount of the ee t of turbulen es on ontrol surfa e a tivity, the bandwidth of the longitudinal ontroller proposed in this hapter needs to be redu ed.

259

Longitudinal measurements (ordering is onsistent with the feedba k gains given

z dire tion (wV ), total velo ity (V ), pit h rate nz ), integrator over wV wV; , integrator over V V loop: position of CoG in earth-xed frame (z ). See also

later). Velo ity in vehi le-axis

q

( ), verti al load fa tor ( and for the outer Figure 19.1.

p r ), integrator over , integrator over   and for the outer loop: ight path heading angle (), lateral deviation (eyb ), integrator over eyb . See

Lateral measurements. Angle of sideslip ( ), roll rate ( ), yaw rate ( ), roll angle (

also Figure 19.2.

The used referen e signals are given in Figures 19.1 and 19.2. The turn ompensation is

q oord = The generation of

V = 0):

g

and

g

g sin2 g V os g

omes from the well known relations (in whi h

 = + V ;  = atan





V _ g

The longitudinal ontroller is shown in Figure 19.1 and the lateral one in Figure 19.2.

Note that the subs ript

g

indi ates signals oming from the

traje tory generator.

wV;g

- - Ko ?wV; 6 Vg

zg

- - RR 6 -q oord 6 6 6

---- Ki ÆT-ÆT H --

Figure 19.1: Longitudinal autopilot stru ture.

Air raft delays a tuators

-wV-R Vqnz-

z

Ki is a non-s heduled 4th order

transfer matrix.

19.3 Translation of Design Criteria into Method Dependent Obje tives 19.3.1 Settling time and overshoot riteria These riteria are easily taken into a

ount by appropriate hoi e of eigenvalues (real part to ensure settling time, damping ratio to redu e overshoot).

260

The

g g

-= 0- R -K ?- -K ? -- RR -- Æ6 6  0 --6 --- K Æ6 6

yB;g

.

o2



o1

A

.g

_ g (0)

( V

-

i

atan

delays

R

a tuat.



R



R y

p r

g

( Vg
)

- - ..... 6 6  0) R

Air raft

g

g

Figure 19.2: Lateral autopilot stru ture with a referen e signal generator based on

_ . Ki

is a non-s heduled proportional feedba k.

hoi e of eigenvalues in the longitudinal hannel ase is made using the Naslin rule (see [30℄). Note that the use of feedforward terms interferes quite a lot with the hoi e of eigenvalues. The onstraints on eigenvalues that we use were derived by trial and error (see Figure 19.1 and Ÿ19.2) after feedforward terms were introdu ed into the outer loops.

Related design riteria. Altitude response, ight path angle response, inertial speed response, heading angle response and overshoot limitations (passenger

omfort riteria).

19.3.2 De oupling riteria These riteria are taken into a

ount by appropriate hoi e of eigenve tors (see

V wV .

Chapter 3). De oupling between

ross- oupling between

V

and

and

h

is taken into a

ount by redu ing

Related design riteria. Cross oupling between tween



and

.

V

and

h,

ross- oupling be-

19.3.3 Tra king riteria The stru ture (integrators) of the ontrol law permits the designer to take these

riteria into a

ount. In the inner loop, an integrator is put in series with the

261

V , wB , 

dieren e between a measurement (

and

)

and the orresponding

input referen e (see Figures 19.1 and 19.2). In the outer loop, a simple gain su es to ensure zero steady-state tra king error, provided that there is at least one open-loop integrator.

In Figures 19.1 and 19.2, these integrators

are shown expli itly (integrators between

Related design riteria.

wV

and

z and between  and y).

Speed response, lateral deviation, heading angle re-

sponse and roll angle response in the ase of engine failure.

19.3.4 Robustness riteria Robustness is treated by using a multi-model approa h as presented in Chapter 3. This approa h permits us to treat performan e robustness together with stability robustness. The hanges in stability and performan e due to parameter variations should be a

eptable, for position of the enter of gravity, mass and delays.

19.3.5 Other riteria The design riteria whi h remain here on ern maximum values of several signals (asso iated with safety riteria) and RMS values. To take them into a

ount we just  rossed the ngers. In fa t, these riteria give limitations to the fastness of our ontrol laws, while other riteria are more easily met when dynami s are a

elerated. They were he ked during simulation runs.

Related design riteria. Maximum verti al a

eleration, maximum lateral a

eleration, angle of atta k, airspeed, roll angle.

Under moderate turbulen e

onditions mean a tuator rates of aileron, tailplane and rudder and mean throttle.

19.4 Design Cy le and Controller Derivation Briey, the design y le is divided into two steps. The rst step (Ÿ19.4.1,19.4.2)

on erns the outer loop design, for whi h a preliminary inner feedba k relative to the nominal model is designed. Then, the se ond step (Ÿ19.4.4,19.4.5) onsists of nding a robust ontroller whi h has similar properties all over the

ight domain. The rst step is an iterative pro edure involving:

  

tuning of the eigenvalues assigned by the preliminary inner loop, outer loop gain omputation,

hoi e of feedforward signals in the stru ture (Figures 19.1, 19.2).

The iterations ended when all ex ept robustness riteria were met at the onsidered nominal operating point. The details an be found in Chapter 5 of [55℄. Here we prefer to present more pre isely the se ond step: robust inner loop derivation, as it is the most innovative part of our ontribution.

262

19.4.1 Outer longitudinal loop design First, a preliminary inner ontrol law is omputed. For that we use a model

3

(nominal) in the middle of the ight domain . Four eigenve tors orresponding to eigenvalues

f 0:99j 0:90; 0:63j 0:57g are assigned su h that C1 vi = 0 i.e. wV (= C1 x). Two eigenvalues f 0:14  j 0:15g

these modes are de oupled from are assigned su h that

C2 vi = 0 i.e.

this mode is de oupled from

V (= C2 x).

These eigenvalues are the roots of a Naslin polynomial. The resulting gain is:

Ki;lon =



0:0217 0:0036

0:0026 0:451 0:0113 0:342

0:116 0:072

0:0063 0:0009

0:0007 0:0013

 (19.1)

In Ÿ19.4.4 a robust ontrol will be derived. It must orrespond to a behaviour similar to the one obtained using

Ki;lon over all the ight envelope, so that the

outer loop omputed now will not need to be updated. With the above initial inner loop, the outer loop is designed as follows. Considering settling time, the outer loop must be designed in su h a way that all poles have a real part about

0:35.

Using the root lo us approa h, three

eigenvalues are assigned on the same verti al line gain is:

0:325.

The orresponding

Ko;lon = 0:124

(19.2)

19.4.2 Outer lateral loop design As for the longitudinal hannel, a preliminary inner loop is needed. The eigenvalues assigned here were obtained by trial and error within a design loop in luding outer loop feasibility tests and feedforward signal onsiderations. Three

f 0:7j 0:7; 0:62g are assigned su h C4 vi = 0 i.e. these modes are de oupled from (= C4 x). Three eigenvalues f 1:7; 1:48; 1:30g are assigned su h that C1 vi = 0 i.e. these modes are de oupled from (= C1 x). The resulting gain is:

eigenve tors orresponding to eigenvalues that

Ki;lat =



2:68 1:64

1:98 1:86 0:049 2:38

3:58 0:16

0:30 1:51 0:93 0:14

 (19.3)

With the above initial inner loop, the outer loop is designed in two steps.

First outer loop (see Fig. 19.2). Considering settling time onstraints, must be su h that no pole has real part larger than

0:3.

Ko1;lat

Using the root lo us

approa h, it was possible to pla e the slowest eigenvalue at

7 = 0:3 + 0:2 j .

This hoi e also satises damping ratio requirements. The orresponding gain is:

Ko1;lat = 1:41

(19.4)

Se ond outer loop (see Fig. 19.2). Outer loop 2 onsists of a single-input multioutput system. 3

Mass

m

Measurements are the lateral deviation

VA = 80 m/s, horizontal

 and a time delay 0.075 s.

= 120000 kg , airspeed

verti al CoG lo ation

gz

= 0.0

263

eyb

and its integral

CoG lo ation

gx



= 0.23 ,

value; the input is the aileron ontrol signal. Two poles are assigned using the te hnique of Chapter 3. Finally:

Ko2;lat = [ 0:0012 0:000028 ℄

(19.5)

19.4.3 General omments on inner loop design Considering the initial inner loop design given in Ÿ19.4.1 and 19.4.2, in view of the results shown Figures 19.3 (left), 19.4 (top) and 19.5 (top-left) the following

omments hold:

 

Robustness is not satisfa tory. All performan es are good.

Therefore, it remains to design ontrol laws whi h lead to the losed-loop behaviour of the initial step, but all over the ight domain. Robustness will be assessed by onsidering a bank of models overing, more or less, the entire ight envelope. This bank of models is dened in Table 19.1. We shall onsider pole maps and step responses showing ross oupling for all

V of 13 m/s, wV must be smaller than 0.7 m/s, V must be smaller than 1 m/s. For a demand o o o o of  of 20 , must remain smaller than 1 and  < 1 while = 1 . these models. For a demand of

for a demand of

wv

of 4.2 m/s,

19.4.4 Inner longitudinal loop design Closed loop poles: longitudinal

Closed loop poles: longitudinal with dynamic gain

4.5

4.5

4

4

3.5

3.5

3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0 −0.5 −5

0

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

−0.5 −5

0

Figure 19.3: Longitudinal losed-loop poles.

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

Left: initial design, right: pro-

posed 4th order dynami gain

In order to have an e ient multi-model based approa h, ompatibility between design obje tives from one model to one other must be ensured. For instan e, we annot expe t to assign exa tly the poles of two models whi h are very lose. It is suggested:

264

Model des ription Number

Mass

hor. CoG

vert. CoG

0 1

120

0.23

0

80

0.075

150

0.15

0

80

0.075

2

150

0.15

0.21

80

0.075

3

150

0.31

0

80

0.075

4

150

0.31

0.21

80

0.075

5

100

0.15

0

80

0.075

6

100

0.15

0.21

80

0.075

7

100

0.31

0

80

0.075

8

100

0.31

0.21

80

0.075

m [t℄

gx [ ℄

gz [ ℄

Airspeed

V [m=s℄

Delay

Delay [s℄

9

150

0.15

0

70

0.075

10

150

0.15

0.21

70

0.075

11

150

0.31

0

70

0.075

12

150

0.31

0.21

70

0.075

13

150

0.15

0

90

0.075

14

150

0.15

0.21

90

0.075

15

150

0.31

0

90

0.075

16

150

0.31

0.21

90

0.075

17

120

0.15

0

80

0.05

18

120

0.31

0

80

0.05

19

120

0.23

0

80

0.05

20

120

0.23

0

80

0.1

21

100

0.15

0

60

0.075

22

100

0.15

0.21

60

0.075

23

100

0.31

0

60

0.075

24

100

0.31

0.21

60

0.075

25

100

0.15

0

90

0.075

26

100

0.15

0.21

90

0.075

27

100

0.31

0

90

0.075

28

100

0.31

0.21

90

0.075

29

120

0.23

0

60

0.075

30

120

0.23

0

65

0.075

31

120

0.23

0

70

0.075

32

120

0.23

0

75

0.075

33

120

0.23

0

85

0.075

34

120

0.23

0

90

0.075

35

150

0.31

0.21

75

0.075

36

150

0.31

0.21

85

0.075

Table 19.1: Considered linearized models

265

W_V versus demand of V: 13 m/s

W_V versus demand of W_V: 4.2 m/s 6

1

4 W_V : m/s

W_V : m/s

0.5 0 −0.5 −1 0

2 0

20

40

−2 0

60

V versus demand of V: 13 m/s

20

40

60

V versus demand of W_V: 4.2 m/s

15

1

V : m/s

V : m/s

0.5 10

0

5 −0.5 0 0

20

40

−1 0

60

20

40

60

t:s

t:s

W_V versus demand of V: 13 m/s

W_V versus demand of W_V: 4.2 m/s 5

0.2

4 W_V : m/s

W_V : m/s

0 −0.2

3 2 1

−0.4 0 −0.6 0

20

40

−1 0

60

V versus demand of V: 13 m/s

20

40

60

V versus demand of W_V: 4.2 m/s

15

0.4

V : m/s

V : m/s

0.2 10

5

0 −0.2 −0.4

0 0

20

40

60

−0.6 0

t:s

20

40

60

t:s

Figure 19.4: Longitudinal step responses. posed 4th order dynami gain

266

Top: initial design, bottom: pro-



to onsider models as far as possible from ea h other (for example, models 12 and 25 in Table 19.1) so that ompatibility problems be ome less stringent.



to assign eigenve tors by means of proje tions, to ensure ontinuity of requirements.

The design y le redu ed to rstly ompute a new initial gain relative to model 12. Then, analysis of all models is done onsidering this initial feedba k. Poorly damped high frequen y poles arise. model number 25 (low mass, high speed).

They belong to models around For robustness improvements the

poles of model 25 are shifted towards the left, while some poles of the low frequen y domain belonging to model 12 and hara terizing well assigned time behaviour are either slightly shifted to the left or frozen at their a tual lo ation. Several eigenstru ture assignment equations orresponding to both models being olle ted; it remains to nd the orresponding dynami gain (see Ÿ3.4.4, Equation (3.31)) that will be in parallel with the initial one. The order of the dynami gain is hosen as being as low as possible. During a trial and error approa h it is found that a se ond-order ontroller would already provide good performan e and de oupling, however, the matrix elements would be too high. Therefore, a fourth-order ontroller is onsidered. The state-spa e representation of the gain is given in [55℄. Bode plots of ea h individual gain are also given in this report. Some gains appear to be high pass. The resulting pole maps and the step responses an be found in Figures 19.3 (right) and 19.4 (bottom).

19.4.5 Inner lateral loop design Let us now analyse the limitations that are related to the use of proportional feedba k.

In [55℄, two te hniques are used (multi-model as presented here

and pole migration as in [160℄).

In both ases, our on lusions are similar:

proportional gains and damping ratio larger than 0.7 over all the ight domain are in ontradi tion. The proposed proportional ontroller indu es a damping ratio of about 0.4 (Figure 19.5, top-right) whilst time responses are perfe t (Figure 19.6 top). When damping is augmented some poles be ome slower along the real axis (Figure 19.5, bottom-left) and in turns, settling time be omes too slow. A dynami feedba k, exa tly as proposed in Ÿ19.4.4 leads to Figure 19.5, (bottom-right) for the poles and Figure 19.6 (bottom) for the time response. These results are perfe t. For the derivation of the proportional law the te hnique used is quite straightforward; it onsists of assigning three slow poles (the same as for the initial law, see Ÿ19.4.2) de oupled from

 for the riti al model 12 (low speed,

high mass) and three fast poles (the same as for the initial law) de oupled from

, based on the nominal model.

Six eigenvalues are xed and six measurements

are available, therefore (3.22) an be solved without dynami s. The resulting

267

Closed loop poles: lateral

Closed loop poles: lateral with multimodel

4

4

3.5

3.5

3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0 −0.5 −4

0

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

−0.5 −4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

Closed loop poles: lateral

Closed loop poles: lateral 4

4.5 3.5

4 3

3.5 2.5

3 2.5

2

2

1.5

1.5 1

1 0.5

0.5 0 −0.5 −4

0

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

−0.5 −5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

Figure 19.5: Lateral losed-loop poles. Top-left: initial design; top-right: proposed proportional gain; bottom-left: well damped proportional gain but too slow ; bottom-right: 4th order dynami gain

gain is:

Ki;lat;mult =



2:45 3:96

1:96 2:32 0:11 4:81

3:52 0:45

0:27 1:51 1:34 0:14

 (19.6)

19.5 Analysis of the Resulting Controllers In this hapter further, mainly nonlinear veri ations, due to gust and ight man÷uvres, will be presented.

The simulations are made for the nominal

onguration.

19.5.1 Nonlinear simulation of an engine failure Fig. 19.7: in ase of engine failure, the steady state deviation of the roll angle is

 = 4:3o = 0:075 rad, thus smaller than the expe ted 0:090 rad = 5o and is

redu ed to zero with an overshoot of 60 % after the restart of the failed engine, ex eeding slightly the spe i ation of 50 %. Furthermore, the maximal value

268

Beta versus demand of phi: 20^o 1.5

2

1 Beta : deg

Beta : deg

Beta versus demand of beta: 2^o 2.5

1.5 1 0.5

0.5 0 −0.5

0 0

20

40

−1 0

60

Phi versus demand of beta: 2^o

20

40

60

Phi versus demand of phi: 20^o

0.1

25

Phi : deg

Phi : deg

20 0

−0.1

15 10 5

−0.2 0

20

40

0 0

60

20

t:s

40

60

t:s

Beta versus demand of Beta:2^o

Beta versus demand of Phi:20^o 1

2 0.5

1.5

0

1

−0.5

0.5 0 0

20

40

−1 0

60

Phi versus demand of Beta:2^o

20

40

60

Phi versus demand of Phi:20^o

0.3 20 0.2 15

0.1 0

10

−0.1 5

−0.2 0

20

40

0 0

60

20

40

60

Figure 19.6: Lateral step responses. Top: proposed proportional multi model design (damping design (damping

> 0.4 all over the ight domain), bottom: > 0.7 all over the ight domain)

269

dynami multimodel

max = 6:3o = 0:110 rad whilst up to 0:175 rad = 10o would be allowed (see Fig. 19.7). As an be seen in the same gure, slideslip angle is minimized. The resulting lateral a

eleration ny in reases up to 0:08 g. This passes the normal level of 0:02 g but is still well within the limit of 0:2 g spe ied for o an engine failure. The heading rate has to remain less than 3 =s = 0.052 rad. _ max = 0:013 rad/s = 0:7o=s. Besides the This stipulation is a tually met by is

guarantee of performan e and omfort, the safety riteria are also respe ted. The airspeed is

VA  77 m/s  62:2 m/s = 1:2 VStall (120 t) o and thereby, the angle of atta k meets with   1:7 = 0:03 rad the demand o of max  0:2 rad = 12 . Finally, the altitude is almost kept at 1000 m. Measurements Psi_dot [rad/s]

Psi [rad]

−1.5 −1.6

50

100

150

200

0

−0.05 0

50

100

150

200

75 0 0.2

100

150

0 −0.2 0

50

100 t [s]

150

150

200

50

100

150

200

50

100

150

200

50

100 t [s]

150

200

0

0

−0.2 0 1005

200 H [m]

50

100

−0.1 0 0.2

80

50

0.1

ny [−]

V_A [m/s]

−0.05 0 85

alpha [rad]

0

Phi [rad]

Beta [rad]

−1.7 0 0.05

0.05

1000 995 0

200

Figure 19.7: The observation of the design riteria during engine failure

19.5.2 Nonlinear simulation observing safety riteria Fig. 19.8 depi ts the results of the nonlinear simulations referring to safety

riteria on airspeed

VA ,

angle of atta k



and roll angle

.

The man÷uvre

own orresponds to those during the landing approa h of the evaluation pro edure (see Chapter 19.6): engine failure, standard turn, des ent with a ight path angle

of

6o

and

3o

and wind shear. It is onsidered to be a good

representation of all arising ight onditions.

270

Safety criteria V_A [m/s]

100 90 80 70 0

50

100

150

200

250

300

350

400

450

500

50

100

150

200

250

300

350

400

450

500

50

100

150

200

250 t [s]

300

350

400

450

500

alpha [rad]

0.2

0

−0.2 0

Phi [rad]

1 0.5 0 −0.5 0

Figure 19.8: Safety riteria observation (the four segments are onsidered)

Generally speaking, the safety riteria are fullled:

VA . The minimal airspeed is rea hed during the fourth segment with 75 m/s. Hen e, it is still 20 % larger than 1:2 VStall . For engine failure

Airspeed about

see also se tion 19.5.2.

. The maximal angle of atta k is dete ted during the turn but  remains less than half of the limit of 0:2 rad = 12o. Results orresponding

Angle of atta k

the engine phase failure are given in Fig. 19.7.

Roll angle

.

The roll angle

 remains well within all borders, ex ept during 0:5

the beginning of the standard turn. There, it ex eeds slightly, the limit of rad

= 30o.

It is onsidered to be a

eptable.

19.5.3 Nonlinear simulations under moderate turbulen e

onditions In the following, the nonlinear simulations of the ontrolled air raft under mod-

ug = vg = wg = 1:54 Lug = Lvg = Lwg = 305 m, with altitude h = 1000 m wind speed at 20 ft above the ground W20 = 15:4 m/s for

erate turbulen e onditions are dis ussed. We have m/s =

0:1 W20

> 305 m

and

and the

moderate onditions.

With Fig. 19.9, a tuator eort minimisation is proved. Throttle rate

271

Æ_T H

−0.2

dthr1 [rad]

0 −0.2

da_dot [rad/s]

100

100

50

100

50

100

50 t [s]

100

0.05 100

0.15 0.1 0.05 0 0

0.2

50

0.1

50

100

0

0.15

0 0

50

0.4 0.2 0 −0.2 −0.4 0

dr_dot [rad/s] 50

0.4 0.2 0 −0.2 −0.4 0

100

0 −0.5 0

dthr2 [rad]

50

dthr1_dot [rad/s]

dr [rad]

−0.4 0 0.5

Phi [rad]

dt_dot [rad/s]

0

50

100

50 t [s]

100

0.02 0 −0.02 0

dthr2_dot [rad/s]

da [rad] dt [rad]

Actuator outputs and measurements 0.4 0.2 0 −0.2 −0.4 0 50 100

0.02 0 −0.02 0

0.05 0 −0.05 0

Figure 19.9: A tuator dee tions and rates under moderate turbulen e onditions

272

0:0042 rad/s = 0:15
0:028 rad/s = 0:15 Æ_T H;max. The same 0:144 rad/s = 0:33
0:436 rad/s = 0:33 Æ_A;R;max. Only tailplane eort is slightly elevated. Tailplane rate Æ_T ex eeds - but not signi antly - the limit of 0:086 rad/s = 0:33
0:262 rad/s. However, a tuator deviations due to turbulen e are very

remains well within

applies to aileron and rudder rates. They are normally less than

small. That means that there is still a lot of ontrol power left whi h ensures

The maximum roll angle max in Fig. 19.9 is 1:5o = 0:025 rad whilst the spe i ation maximum is xed to 0:087 rad

man÷uveribility and agility. about

o

= 5 .

Open loop

Closed loop

0.04

0.04 RMS = 0.0097

RMS = 0.0120 0.02 beta [rad]

beta [rad]

0.02 0 −0.02

−0.02

−0.04 0

50

−0.04 0

100

50

100

0.03 RMS = 0.0094 Psi_c − Psi [rad]

Psi_c − Psi [rad]

0.03 0.02

0

0.01 0 −0.01 −0.02 0

50 t [s]

Figure 19.10: Comparison of

100



and

0.02

RMS = 0.0144

0.01 0 −0.01 −0.02 0

50 t [s]

100

responses, open and losed-loop, due to

unit RMS gust

In Fig. 19.10, additional disturban e redu tion hara teristi s of the proposed ontroller are shown. Obviously, for the unit RMS lateral gust input the

= 0:0097 = 0:0120
= 0:0144 rad is higher than RMS
ol = 0:0094 rad, i.e. disturban es are slightly redu ed for the inner loop quantities roll angle  and sideslip angle , but outer loop heading angle response is slightly deteriorated and gener-

losed-loop RMS sideslip angle RMS l rad is smaller than the openrad). On the other hand, the RMS heading angle error

loop (RMS ol RMS

l

ally speaking, natural behaviour is not hanged signi antly by the ontroller. In ase of a failure in the ontrol system, this would be desirable. Nevertheless, it would be better if the disturban e redu tion was in reased by the ontroller.

273

19.6 Results of the Automated Evaluation Pro edure The landing s enario orresponding to four segments is des ribed in Ÿ14.3.3.

Segment I See Figure 19.11 left side. Obviously, the ontroller fulls all these spe i ations.

Segment II See Figure 19.11 right side.

The tra k ex eeds the bounds, sin e su

essful

attempts to redu e the deviations have in reased the lateral a

eleration

ny

in an una

eptable way and we are penalised for ex essive lateral a

eleration (explanation see se tion 19.2).

Segment III See Figure 19.12 left side.

The time responses remain within the maximal

bounds, unfortunately, they ex eed the limits in terms of agility. Basi ally, the

ontroller is too slow for the given bounds. But it was not possible to in rease the gains without violating other onstraints.

Segment IV See Figure 19.12 right side.

Obviously the ee t of wind shear is very well

redu ed, almost evened out.

It is possible to redu e the gains and to keep

the responses still within the bounds. This would also lead to smaller verti al a

eleration

nz

and hen e to better omfort. However, when the gains of the

perturbation redu tion loop are redu ed, the tra k following system be omes inevitably slower whi h would result in even bigger deviations than we already have (see Segments II/III).

Numeri al results Table 19.2: All the numeri al values are a

eptable ex ept for omfort indi ators (see [55℄ for more details).

19.7 Con lusions RCAM autopilot design is performed in two steps. The outer loop designs are based on lassi al ontrol. For the inner loops, a new multi-model eigenstru ture assignment te hnique is proposed, it is briey des ribed in Ÿ3.4 and 3.4.4. In the lateral ase the proposed multi-model ontroller is proportional, it is

omputed solving an equation similar to (3.29). However, the use of a dynami gain is shown to lead to better results.

274

In the longitudinal ase, a dynami

First segment: top view

Second segment: lateral deviations 300

100

0

0

a

b

lateral deviation [m]

x−deviation [m]

200 50

1

−50

100

0 1

c

d

2

−100

−200 −100 −20

−18

−16

−14

−12 −10 −8 y−position (−YE) [km]

−6

−4

−300 0

−2

1

2

3 4 5 6 along track distance from point 1 [km]

7

8

Figure 19.11: Left: segment I - the ee t of engine failure. Right: segment II - lateral deviations during the

3o/s turn

Fourth segment: altitude deviations 30

20

20

10 2

f

3

0

−10

e

altitude deviation [m]

altitude deviation [m]

Third segment: altitude deviations 30

3 g

4

h

0

−10

−20

−20

−30

10

−16

−15

−14 −13 x−position (XE) [km]

−12

−30 −11

−11

−10

−9

−8

−7 −6 −5 −4 x−position (XE) [km]

−3

−2

−1

Figure 19.12: Left: segment III - verti al deviations from the desired glideslope. Right: segment IV - verti al deviations from the desired glideslope.

feedba k is proposed. It is omputed solving an equation similar to (3.31). In pra ti e, gain s heduling is used for designing su h an autopilot. The proposed dynami feedba k has a low order and does not need to be s heduled.

The

degrees of freedom introdu ed by the dynami feedba k repla e with benets those introdu ed by s heduling.

In fa t, this is not a surprise, as the most

important parameter for robustness is the mass. It is also known that mass variation indu es hanges in the dynami s, therefore the proposed autopilot an be viewed as inherntly s heduled with respe t to mass variations via frequen y variations.

Design y le. Any ontrol design te hnique is dened in order to deal, with some priority, with a given subset of usual design riteria.

For modal te h-

niques, the riteria that are naturally tra table are: settling time, overshoot, damping ratio (see Ÿ19.3.1) and de oupling (see Ÿ19.3.2). Tra king (see Ÿ19.3.3) was taken into a

ount by using integrators. The hoi e of signals that are integrated was not analysed, we just reprodu ed the stru ture that is in use in industry. Other riteria su h as omfort, fatigue... (see Ÿ19.3.5) were onsidered mainly during the outer loop design by redu ing their dependen y on inner loop design. More pre isely, adding several feedforward signals made possible the

275

Segm. I

Segm. II

Segm. III

Segm. IV

Total

Perform.

0.3310

0.2516

0.7132

0.0464

0.3355

Perf. Dev.

0.0580

0.1585

0.0740

0.1230

0.1034

Comfort

0.4058

1.0015

1.3254

0.4756

0.8021

Safety

0.0080

0.0683

0.0062

0.0214

0.0260

Power

0.0030

0.0049

0.0120

0.0265

0.0116

Table 19.2: Numeri al results of the evaluation pro edure

use of slower inner loops; in turn, riteria of Ÿ19.3.5 ould almost be ignored. Briey, the rst step of the design y le onsisted mainly of the hoi e of the outer loop stru ture and of orrelated onstraints on erning the settling time of the inner loops at the nominal point. Therefore, to deal with robustness, the se ond step onsists of satisfying the above time domain onstraints but all over the ight domain.

User friendliness of the method. Ÿ3.4.4, Equ.

(3.29)-(3.31).

The te hnique used here is illustrated in

The required ontrol theory ba kground redu es

to the knowledge of the notion of modes (see Ÿ3.1).

When dynami gains

are onsidered (longitudinal ase Ÿ19.4.4), numerous degrees of freedom are introdu ed, most of them do not orrespond to eigenstru ture assignment onstraints. Therefore they need to be optimized. The riteria we used are dened in [150℄. In this ase, e ient numeri al tools are ne essary. Con erning the multi-model feature of the proposed te hnique, a good ba kground in ight me hani s (good sense also) is required. Indeed, when several obje tives are dened for a given model, they are also more or less dened for adja ent models, hen e we must be very areful when additional obje tives on erning other models are onsidered. It is very tempting to expe t oni ting results. To prevent oni ting requirements we onsidered the most distant models (high mass - low speed and low mass - high speed). Moreover, the hoi e of losed-loop eigenve tors was based on orthogonal proje tions in order to ensure oheren y. In the lateral ase, the proposed te hnique is mu h simpler be ause oni ting obje tives are avoided (see Ÿ19.4.5).

E ien y for robustness. The proposed te hnique is very e ient, provided that problems inherent in multi-model te hniques are well understood. After having designed a good rst design, no more than one hour was required to treat a new similar problem. To treat a problem for whi h no previous experien e is available, a mu h longer analysis is ne essary to assess multimodel oni ting obje tives.

Complexity of design. In the lateral ase, the ontroller redu es to a simple proportional gain. In the longitudinal ase, the advantage of the proposed design te hnique is that dynami s are added only in order to treat robustness; usually, dynami s are added to make solvable, a problem whi h would be otherwise NP-hard. As a onsequen e, the number of states of the ontroller is

276

minimized (2 or 4 states).

Blo k diagrams. The used stru tures are given

 

in Figure 19.2 with

Ki in Equ.

in Figure 19.1 with

Ki

(19.6).

being a 4-dimensional dynami gain. Numeri al

values are given in [55℄.

Further work. Clearly, even if a proportional gain is proposed for the lateral

hannel, our on lusion is that the use of dynami feedba k is more e ient. It permits us to repla e s heduled gains by simpler gains. However there are two main drawba ks on erning the approa h used in this paper.

The rst

problem is that it is di ult to tune our laws without oming ba k to the use of design tools. The se ond problem is that we have not analysed the ee t of our gains on the exible modes. But we believe that by using stru tured gains (i.e.

dynami s only between relevant inputs and outputs, with some

gains set to zero) both above problems are easily handled. Methods for solving the stru tured gain multi-model eigenstru ture assignment problem are under development and do not present any major theoreti al problem.

A knowledgement The work presented in this hapter and the parti ipation of CERT-ONERA in the denition of the RCAM design hallenge (Ÿ14) was supported by the Servi e Te hnique des Programmes Aéronautiques (STPA).

277

20.

The Lyapunov Approa h

Jamal Daafouz1;2 , Denis Arzelier1, Germain Gar ia1;2 and Ja ques Bernussou1 Abstra t.

This hapter presents the dierent aspe ts of the design

methodology that has been used to ope with the RCAM design problem. A quadrati approa h involving robust pole pla ement in a disk and the solution of parameter-dependent Ri

ati equations is proposed. Robustness as well as performan e is a

ounted for by robust disk pole lo ation that ensures robust stability and provides a good transient behaviour together with a spe i sele tion of the weighting matri es involved in the Ri

ati equations.

20.1 Introdu tion Robust ontrol design has now rea hed a fairly high level of development and industrial pa kages are available for su h purpose. Among the various methods in robust ontrol,

H1 optimization using state spa e models seems the most

popular. The reason being that, in its early stages, it appeared as a way to extend the margin on epts used in the frequential single variable domain to

the multivariable one. Another method omes from the extension of Lyapunov method to the study of un ertain systems, and it is termed quadrati approa h in the literature. This hapter is quadrati approa h oriented.

This an be justied by the

fa t that the design obje tives are dened in the time domain. Moreover, when dealing with multiobje tive problems, the

H1 optimization approa h often on-

sists of dening a stru tured multi-blo ks un ertainty formulation for whi h the

stru tured eigenvalue tool is needed with the asso iated D-K iterations. Su h algorithms still la k a strong onvergen e result so that their e ien y often relies on the intuition of their users.

The situation is a bit dierent for the

quadrati approa h whi h, at rst sight, presents a serious drawba k ompared to the

H1 one.

The multiobje tive problems approa hed by single obje tive

with multiple onstraints problems are in fa t generally solved using only su ient onditions in the quadrati approa h. This is due to the fa t that a single Lyapunov fun tion is involved in the onditions. However, this inferiority is, in some sense, greatly redu ed by the fa t that the onditions possess, in most

ases, the onvexity property so that onstru tive numeri al methods an be 1

LAAS-CNRS, 7 avenue du olonel Ro he, 31077 Toulouse Cedex 4

2

Also with INSA, Complexe S ientique de Rangueil, 31077 Toulouse Cedex

278

proposed for their solution.

For instan e, the LMI, (Linear Matrix Inequal-

ities), tools an be implemented when the onditions are expressed as ane relations with respe t to these variables.

20.2 Sele tion of Controller Ar hite ture From the Design Challenge spe i ations, the ontroller ar hite ture an be

hosen arbitrarily. It is however lassi al in air raft ontrol law design to arry out a two-stage synthesis : one for the longitudinal motion and one for the lateral motion leading to two separate ontrollers : a longitudinal ontroller and a lateral ontroller. Ea h ontroller is designed separately. Furthermore, ea h ontroller is then omposed of two ontrollers, (inner and outer ontrollers), forming two loops : an inner loop and an outer loop. This is also natural in air raft ontrol law design sin e the inner loop an be used for robust stabilization purposes while the outer is used for robust tra king purposes. Figure 20.1 shows the ontroller ar hite ture used.

K_lt_in and K_lg_in

are respe tively the lateral and longitudinal inner ontrollers. K_lt_out and

01

02

0 1 and K0 g2

K_lg_out are the orresponding outer ontrollers. K l , K l , K g are pre ompensators ne essary to have a unit stati gain.

20.2.1 Measurement signals Longitudinal ontroller As shown in gure 20.1, the longitudinal ontroller uses three measurement signals in its inner loop : pit h rate

wV

and airspeed

VA .

q, z omponent of inertial velo ity in FV ,

This hoi e is onsistent with our hoi e of the ommands

in the inner longitudinal loop. In the outer longitudinal loop, the

z position of

the air raft is used as measurement signal.

Lateral ontroller The inner lateral loop uses two measurement signals : sideslip angle angle

.

,

roll

20.2.2 A tuator signals ÆT , ÆT H1 ; ÆT H2 , whi h are aggregated in one signal, ÆT H , are the longitudinal ontroller outputs. Aileron dee tion ÆA and rudder dee tion ÆR , are the lateral ontroller outputs. Here, all the available a tuator signals are used, namely tailplane dee tion

throttle positions of engine 1 and 2,

20.2.3 Filters and models First of all, no lters of any kind are used or onsidered in our design pro edure. Dierent kind of models have been used in our study : linearized models and non linear model.

The non linear model, the RCAM model, is dened

279

in the RCAM Design Challenge Manual, [145℄, and is a 6 degrees of freedom, non-linear model of the air raft in landing onguration. This model is mainly used in non-linear simulations in the analysis of the resulting ontroller in terms of the applied methodology, (see se tion 20.5), and, of ourse, in the automated evaluation pro edure, (see se tion 20.6). The longitudinal and lateral

ontrollers are synthesized using un ertain linear models (see se tion 20.4).

Lateral Controller K_lt_out

K_lt_in

. ψc

.

atan(80/g* ψc)

+

+

yc K 0l2

+

+ +

φc

K 0g 2

K 0l1

β

+

+

δA

φ

+

+

Vv

δR

Aircraft + Actuators q

zc

βc

+

+

+

δT

c

wc Vc

K 0g 1

q

+ +

wv

δTH

V

+

K_lg_in

K_lg_out Longitudinal Controller

Figure 20.1: Controller ar hite ture

280

20.2.4 Referen e signals From the nine referen e signals available, the longitudinal ontroller uses:

3

2

z the altitude referen e Lgref = 4 wV inertial verti al velo ity referen e 5 V total airspeed referen e and the lateral ontroller uses:




y the lateral deviation referen e Ltref = _

the rate of hange of tra k angle referen e



20.2.5 Resulting ontroller omplexity and implementation issues Stru tural de oupling As previously mentioned, a stru tural de oupling between longitudinal and lateral design has been arried out. For ea h loop, a two-stage design from the inner loop to the outer loop has been ondu ted.

Dynami order of the ontroller The omputation of the dynami order of the ontroller takes the presented design pro edure further into a

ount. Indeed, for ea h motion, the design is

omposed of the following steps: Linearized lateral model

5

Balan ed realization

4

A tuators

2

Integrators Augmented plant

2 8

Extended plant

16

Balan ed realization

10

Global lateral ontroller

20

Inner ontroller

Outer ontroller

8 10

Table 20.1: Dynami order: lateral ontroller

- Extra tion of a linearized un ertain model. - Constru tion of a linear augmented model, with in addition, the a tuators and the integrators. - Computation of an inner ontroller having the same dynami order as the augmented plant.

281

- Constru tion of an extended model omposed with the augmented one and the inner ontroller.

- Computation of a balan ed realization of the previous extended model for numeri al purposes.

- Computation of an outer ontroller having the same dynami order as the redu ed extended plant.

This pro edure is the same for the longitudinal and the lateral motion ex ept that it is ne essary to perform an additional redu tion for numeri al purposes on the lateral linearized undertain model. This pro edure is detailed in se tion 20.4.

Here is the summary of the dierent dynami orders involved in this

design.

Linearized longitudinal model

4

A tuators

2

2

Integrators Augmented plant

8

Extended plant

16

Balan ed realization

20

10

Global longitudinal ontroller

Inner ontroller

Outer ontroller

8 10

Table 20.2: Dynami order: longitudinal ontroller

So, we get a nal global ontroller of dynami order 40 whi h has four integrators. It is to be noti ed that no model redu tion method has been used on the nal ontrollers whi h ould be easily performed and would provide a great redu tion of this dynami order. In addition, it is lear from the design pro edure that no s heduling has been used and that nonlinear ee ts are not present in our ase.

Stability of the ontroller As mentioned in the tutorial hapter, the methodology ensures that the resulting ontroller will be stable. In fa t, the ontroller has the stru ture of an observer. It an be shown, by using results from

H1 theory that the ontroller

dynami matrix veries a Lyapunov equation whi h proves its stability.

282

20.3 Translation of Design Criteria into Method Dependent Obje tives 20.3.1 Linear un ertain model In this paragraph, we derive a linear un ertain model whi h takes into a

ount un ertainty due to mass and entre of gravity variations, that is:

m x y z

   

M x y z

   

m  x y z

mass

entre of gravity

The range of variations are des ribed in details in the RCAM manual [145℄. Considering the extreme values of mass and entre of gravity,

24 extreme models

an be determined. Examining these models, an un ertain model is des ribed where only un ertainty on matrix

A

has been onsidered. The global model

will be de omposed later into a lateral and a longitudinal un ertain model. These models will be des ribed in the next se tion.

20.3.2 Steady state error In order to satisfy the steady state error requirements, integrators will be added. This is also presented with details in the next se tion.

20.3.3 Rise time and settling time Inner loops The requirements expressed in terms of time response, (rise time and settling time), are dire tly taken into a

ount by lo ating the poles appropriately in the omplex plane. For the inner loops of the lateral and the longitudinal on-

4 + j 0 with radius 3.6. 0:4 leads for the inner loops to a settling time about

trollers, the poles are lo ated in a disk entreed at An absolute stability of

ts ' 04::45 = 11s. On the other hand, the lo ation of some open loop poles (near 7 + j 0) is satisfa tory so the ontrol does not have to move them. This justies the hoi e of the disk entre at 4 + j 0 with radius 3:6. Lateral outer loop

For the lateral outer loop, the ir le has to be hosen in a way whi h guarantees a good tra king of referen e signal but also whi h preserves the dynami al

hara teristi s of the inner loop. After some trials, the disk entred at with radius

10 gave a good result.

10+ j 0

It is to be noted that, the disk orresponding

to the inner loop lies within the outer loop disk.

283

Longitudinal outer loop The remarks for the longitudinal outer loop also apply here. A disk leading to good results is entred at

6 + j0

with radius

5:84.

The inner loop disk lies

also within this disk.

20.3.4 Other spe i ations The other spe i ations are not handled expli itly. To take them into a

ount in the ontroller design, we use the degrees of freedom of the proposed approa h for robust pole lo ation, that is the weighting matri es

R1 ; R2 ; Q.

A tuators

movement and engine power an be minimized by an appropriate sele tion of the weighting matri es

R1 ; R2 ; Q.

In order to ontrol robustness margins

and wind ee ts, we have onstrained the sensitivity fun tions to be about 6

1

dB. This ensures a satisfa tory modulus margin .

We have onsidered the

sensitivity fun tions at the input of the a tuators and at the output of the sensors.

20.4 Controller Derivation - Design Cy le The design of the

r am

ontroller is de omposed into two separate subprob-

lems : a lateral hannel ontrol and a longitudinal hannel ontrol. For ea h subproblem, an inner loop to ensure stabilization and an outer loop for tra king are designed. The riterion onsidered in the design of the inner loop on erns the magnitude of the sensitivity fun tions. The goal is to minimize perturbation ee ts due to the wind.

Limiting the sensitivity fun tion magnitude at

the input of the a tuators is a way to satisfy this requirement. In the outer loop ase, the fundamental riterion is a good time response indu ing the desired requirements on the settling time. This does not in lude overshoot. The prin ipal steps in designing ea h hannel ontrol are : i- Build the appropriate norm-bounded un ertain linear model.

In lude

dynami a tuators and integrators and improve numeri al onditioning if ne essary using balan ed realizations. ii- For ea h loop, sele t a ir le in the left half plane and three weighting matri es

R1 ; R2 ; Q.

iii- Che k the magnitude of the sensitivity fun tions and time responses using the linearized model. iv- Che k robustness and performan e riteria on the losed loop non-linear model. If the result is unsatisfa tory, the design an be restarted from the step (ii-). 1

The modulus margin is dened as the radius of the ir le entred in

to the Nyquist plot of the open-loop transfer fun tion.

284

[ 1; j 0℄ and tangent

20.4.1 Numeri al tools for ontroller synthesis The synthesis and analysis are performed in a

Matlab

and

Simulink Matlab

envi-

ronment. To design the ontroller, a spe ial fun tion running under

environment is used to assign the system poles in a disk. This fun tion omputes a d-stabilizing ontroller in a

ordan e with the algorithm proposed in

Matlab

se tion 5.3 of the tutorial hapter 5. The main toolbox used is Control toolbox with

. Some fun tions from the

-Analysis Toolbox are used

to

build losed-loop systems and in some ases to ompute balan ed realizations of ill- onditioned systems.

20.4.2 Intermediate analysis Lateral design model In order to apply the d-stabilizing approa h, an un ertain linear model of the form (20.1) is required.

x_ = (At + D1t F E 1t )x + Bt u y = C t x + Dt u

F

2F

(20.1)

From the omplete linearized model we build the appropriate lateral design model by pi king out the states, inputs and outputs appli able to the lateral

hannel :

3

2

p roll rate 7 6 r yaw rate 7 6 7 States : xlt = 6  roll angle 7 6 5 4 heading angle vB y omponent of inertial velo ity 

Æ aileron defle tion Inputs : ult = A ÆR rudder defle tion



3

2

angle of sideslip 5 Outputs : ylt = 4  roll angle vV y omponent of inertial velo ity The resulting

2

Alt = 6 4

ABCD matri es are given by:

1:2667 0:5498 0 0 0:0242 3 0:0522 0:5207 0 0 0:0045 7 1 0:0284 0 0 05 0 1:0004 0 0 0 2:2678 79:7679 9:7879 0 0:1699  0 0 0 0 0:0125  1 0 0 Clt = 0 0 0 0 2:2640 79:8667 1 285

2

0:8402 0:2904 3 0 :0176 0:3325 7 0 05 Blt = 6 4 0 0 0 2:0384  0 0 Dlt = 0 0 0 0

From the un ertainty matri es of the omplete linearized model, we pi k out the lines and olumns orresponding to the lateral omponents to get the lateral

hannel un ertainty domain:

2

0:2600 0 0:0420 3 0 0 0:0103 7 D1lt = 6 0 0:1281 05 4 0 0 0 7:4300 0 0

E 1lt =

The design is started by using sideslip angle signals.





1 0 0 0 0 0 1 0 0 0 0 0 0 0 1

and roll angle



as feedba k

In order to improve numeri al onditioning we use a balan ed real-

ization of this un ertain model.

A way to obtain this onsists of omputing

two balan ed realizations, one for the nominal model and the other one for the

F = 1).

un ertain model where the un ertainty is taken to be maximum (

D1

resulting un ertainty matri es ( these balan ed realizations.

and

E1 )

The

represent the dieren e between

To obtain the lateral norm bounded un ertain

model, we add a tuator dynami s and integrators to the nominal balan ed realization. We assume that there is no un ertainty on the a tuator dynami s or on the integrators.

Longitudinal design model Similarly to the lateral design, an un ertain linear model of the form (20.2) is needed for applying the d-stabilizing approa h to the longitudinal hannel.

x_ = (Ag + D1g F E 1g )x + Bg u y = Cg x + Dg u

F

2F

(20.2)

From the omplete linearized model, the appropriate longitudinal design model is built by pi king out the states, inputs and outputs appli able to the longitudinal hannel :

3

2

q pit h rate 7 6  pit h angle 7 States : xlg = 6 4 uB velo ity in body axis x dire tion 5 wB velo ity in body axis z dire tion 

Æ tailplane defle tion Inputs : ulg = T ÆT H throttle position



3

2

q pit h rate 4 Outputs : ylg = wV z omponent of inertial velo ity 5 VA airspeed The resulting

2

Alg = 4

ABCD matri es are given by: 0:9825 0 0:0006 0:0161 3 1 0 0 0 2:1937 9:7758 0:0325 0:0743 5 77:3570 0:7675 0:2264 0:6684 286

2

Blg = 4

2:4379 0:2912 3 0 0 0:1837 9:8100 5 6:4785 0

Clg =



1 0 0 0 0 79:8667 0:0283 0:9996 0 0 0:9996 0:0290

Dlg =



0 0 0 0 0 0

From the un ertainty matri es of the omplete linearized model we pi k out the lines and olumns orresponding to the longitudinal omponents to get the longitudinal hannel un ertainty domain:

2

0:0804 03 0 05 D1lg = 4 0 0 0 1:2851

E 1lg =

h

0 0 0 1i 0 1 0 0 q; wv ; VA )

The design is started by using the three outputs (

as feedba k

signals. As in the lateral design ase, we add the dynami a tuators and the integrators to the previous model. Contrary to the lateral model, there is no need to ompute balan ed realizations of the longitudinal model.

20.4.3 Design parameter adjustment strategy A ir le in the left half-plane where the losed-loop poles should lie and three

R1 ; R2 ; Q)

weighting matri es (

have to be hosen. The weighting fun tions

sele tion riterion is based on minimizing the magnitude of the sensitivity fun tions. The sensitivity fun tions onsidered here are the sensitivity fun tions at the input of the a tuators and the sensitivity fun tions at the output of the sensors. A way to a hieve this is to start the design by omputing a ontroller with

R1 = 1 1; R2 = 2 1; Q = 3 1 where 1 = 2 = 3 = 1. Then, iteratively 1 ; 2 ; 3 are su

essively adjusted and at ea h iteration a on-

the parameters

troller is omputed. We stop when an a

eptable magnitude of the sensitivity fun tions is obtained. Here the adjustment of only one parameter namely is su ient to a hieve sensitivity fun tions magnitudes of about

6 dB.

1

Lateral Channel - Inner loop As explained in se tion 20.3, the lateral inner loop ontroller is su h that the

4 + j 0 and radius r = 3:6. The R1 = 1000  122 ; R2 = 122 ; Q = 188 . Taking 0 = 1, the solution is obtained for  = 3:8147:10 6 and the losed-loop poles

losed-loop poles lie in the disk with entre weighting matri es are :

are shown in gure 20.2.

Figures 20.3-20.4 show the sensitivity fun tions and the omplementary sensitivity fun tions of whi h the maximum is about

7:7 dB.

Lateral Channel - outer loop The inner loop provides robust stabilization using ommanded sideslip

R  as inputs. For
y = vV dt as the input.

roll angle



and

lateral tra king we use the lateral tra king error We start by omputing the inner losed-loop for

the nominal lateral model. An integrator is added to the third output order to re onstru t the lateral tra king error

287


y .

vV

in

The riterion in designing

Poles location 4 3 2

Imag

1 0 −1 −2 −3 −4 −8

−7

−6

−5

−4 Real

−3

−2

−1

0

Figure 20.2: Closed-loop poles of the lateral inner loop Sensitivity (Su) and compl. sensitivity (Tu) at the input

Sensitivity (Sy) and compl. sensitivity (Ty) at the output

10

10 0 Singular values (dB)

Singular values (dB)

0

−10 Su Tu −20

−30

−40

Sy Ty −20 −30 −40

−50 −2 10

−1

10

Figure 20.3:

0

1

10 Frequency (rad/s)

10

−50 −2 10

2

10

Sensitivity fun tions

of

the

a tuators

0

10 Frequency (rad/s)

1

10

2

10

Sensitivity fun tions

of the lateral hannel at the out-

Tmax =

of the sensors (Tmax 7:25dB Smax = 7:72dB)

(

4:91dB Smax = 5:78dB)

−1

10

Figure 20.4:

of the lateral hannel at the input

−10

put

=

the outer loop ontroller is to a hieve good time response : no overshoot and settling time less than 45s. This is done by hoosing a ir le with entre -10+j0

R1 = 106  122 ; R2 = 1; Q = 11010 . Taking 0 = 1, the pro edure su

eeds with  = 3:7253 10 9 .

and radius 10 and the weighting matri es are xed as :

Longitudinal Channel - Inner loop The longitudinal inner loop ontroller is su h that the losed-loop poles lie in

4 + j 0 and radius r = 3:6. The weighting matri es R1 = 104  122 ; R2 = 133 ; Q = 188 . Taking 0 = 1, the solution is  4 and the losed-loop poles are shown in gure 20.5. obtained for  = 2:44 10 the disque with entre

are :

Figures 20.6-20.7 show the sensitivity fun tions and the omplementary sensitivity fun tions of whi h the maximum is about

7 dB.

Longitudinal Channel - outer loop To a hieve altitude tra king, the design of an outer loop ontroller is onsidered. An integrator is added to the inner losed-loop to obtain the altitude output

288

z

Closed loop poles 4 3 2

Imag

1 0 −1 −2 −3 −4 −8

−7

−6

−5

−4 Real

−3

−2

−1

0

Figure 20.5: Closed-loop poles of the longitudinal inner loop Sensitivity (Su) and compl. sensitivity (Tu) at the input

Sensitivity (Sy) and compl. sensitivity (Ty) at the output 10

0

−10

Singular values (dB)

Singular values (dB)

0

Su Tu −20

−10 Sy Ty −20

−30

−30

−40

−40

−50 −2 10

−1

10

Figure 20.6:

0

10 Frequency (rad/s)

1

10

−50 −2 10

2

10

Sensitivity fun tions

−1

0

10

1

10 Frequency (rad/s)

Figure 20.7:

10

Sensitivity fun tions

of the longitudinal hannel at the

of the longitudinal hannel at the

input of the a tuators (

output

4:64dB Smax = 4:89dB)

Tmax =

of

the

sensors

wv .

Tmax =

(

4:21dB Smax = 7:11dB)

from the z- omponent of the inertial velo ity

2

10

The design of the outer loop

is now a single input-single output problem sin e we only use the ight path

ommand to a hieve the altitude tra king. The riteria in designing the outer loop ontroller is to ensure good tra king result in terms of time response : no

6 + j 0 with r = 5:84 and the weighting matri es set to: R1 = 40; R2 = 1; Q = 11010 give an a

eptable losed-loop altitude response. Taking 0 = 1, the 5. solution is obtained for  = 6:1035 10 overshoot and settling time less than 45 s. A ir le entreed at

radius

The ontroller design is now omplete. The analysis of the resulting ontroller in terms of the applied methodology is the purpose of the next se tion.

20.5 Analysis of the Resulting Controller The analysis of the resulting ontroller is twofold. First, we have performed an analysis on the nominal linearized models presented in the se tion 20.4, separating longitudinal and lateral motions.

Se ondly, a non linear analysis

has been performed using the design environment des ribed in [145℄.

289

20.5.1 Linear analysis The performan e of the ontroller is he ked by analysing the ommand response hara teristi s to step referen e signals. The ommand response hara teristi s are dened in terms of rise time

Mp .

tr , settling time, ts

and overshoot

These hara teristi s are dened in [145℄.

Lateral deviation The response to a step of

1

m in the lateral position is given in gure 20.8.

tr = 11:33 Mp = 0 %) are met.

ts = 28:56

The requirements for rise time, (

s), settling time, (

s),

and overshoot, (

Lateral response

Step response to an altitude command

1

1

0.8

Altitude (m)

Laterl position (m)

0.8

0.6

0.6

0.4

0.4 0.2

0.2

0 0

0

5

10

15

20 25 Time (s)

30

35

40

−0.2 0

45

5

10

15 Time (s)

20

25

30

Figure 20.8: Closed loop lateral re-

Figure 20.9: Altitude response to a

sponse to a unit lateral ommand

step in altitude of

1m

Altitude response The tra king of altitude ommands has been tested by plotting the altitude response to a step in altitude of

tr = 6:92 s),

time, (

1

m (gure 20.9). The requirements for rise

ts = 16:07 s),

settling time, (

Mp = 0 %)

and overshoot, (

are met.

Flight path angle response The system is subje ted to a step of

3 deg in ommanded ght path angle (gure tr = 5:26 s), settling time, (ts = 11:87

20.10). The requirements on rise time, (

Mp = 0 %) are met.

s), and overshoot, (

The ross oupling between ight path

angle and velo ity is weak (maximum value of the absolute error of V of 0.04 m/s). The elevator and engine ommands are also given and appear to be not too large. Similarly, a response to a step of 13 m/s in ommanded airspeed leads to :

tr = 6:65 s, ts = 13:72 s and Mp = 0 %. 290

Response to roll angle command

4 25

path angle (deg)

phi

3 20

2

15

Angle (deg)

1 V (m/s) 0

10

5

de (deg) beta

−1

0

thr (deg)

−2

−5

dr da

−3 0

2

4

6

8

10 Time (s)

12

14

16

18

−10 0

20

2

4

6

8

10 Time (s)

12

14

16

Figure 20.10: Response to a step in

Figure 20.11: Response to a

ight path angle of

step in roll angle ommand

3 deg

18

20

20 deg

Roll angle response The gure 20.11 presents the response of the system in terms of roll angle and

20 deg step in roll angle ommand. The ross oupling from 0:17 deg. The hara teristi s of the response are tr = 4:18 s, ts = 9:26 s and Mp = 0 %. Jointly to the response to a step in roll angle, the response to a step of 2 deg in sideslip ommand is

hara terized by : tr = 4:55 s, ts = 9:21 s and Mp = 0 %.

sideslip angle to a

roll angle to sideslip angle is less than

20.5.2 Robustness riteria Centre of gravity variations and mass variations Robustness spe i ations are dire tly taken into a

ount by the norm-bounded un ertainty modelling, (see the un ertain lateral and longitudinal models). If the algorithm su

eeds, the dedu ed ontroller, using the quadrati d stabilizability on ept, ensures robust stability and performan e against entre of gravity variations and mass variations in the range des ribed by the un ertain modelling. This range overs those des ribed by the robustness riteria. Here, minimum performan e is a hieved by the robust pole pla ement in a disk. So, this property is he ked by s anning the losed-loop poles of the linearized models when the un ertainty term

F

varies in

F (gures 20.2 and 20.5).

Velo ity variations Robustness is also assessed by onsidering a set of linearized models dened in table 20.3. This set ontains the nominal model

M0 and other models obtained

by the trimr am pro edure for dierent values of mass, entre of gravity and velo ity. Models

M1 , M2

and

M3

are the ones spe ied in the addendum to

the RCAM Design Challenge [143℄. Responses of the system in terms of sideslip angle for the lateral hannel and velo ity for the longitudinal hannel are shown in gures 20.12 and 20.13. For

291

Model M0 M1 M2 M3 M4 M5 M6 M7 M8 M9 Mass 120 100 125 150 100 100 100 100 100 110 Cxg 0.23 0.31 0.31 0.31 0.31 0.31 0.15 0.31 0.15 0.31 Czg 0 0 0 0 0.21 0 0 0 0.21 0 Va (m/s) 80 58 90 71 58 71 80 90 90 58 Model M10 M11 M12 M13 M14 M15 M16 M17 M18 M19 Mass 110 120 120 120 130 130 140 140 150 150 Cxg 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.15 Czg 0 0 0 0.21 0 0 0 0 0.21 0.21 Va (m/s) 80 58 71 90 71 90 71 90 71 90 Table 20.3: A set of models to he k robustness

1

1 V (m/s)

Angle (deg)

Beta 0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0 Path angle (deg)

Phi −0.2

−0.4 0

−0.2

5

10

15

20

25 30 Time (sec)

35

40

45

−0.4 0

50

5

10

15

20 Time (sec)

25

30

35

40

Figure 20.12: Response to a 1 deg

Figure 20.13: Response to a step in

step in sideslip angle

Velo ity of 1 m/s

the onsidered set of linearized models, the system remains stable and performan es are well preserved.

Time delay To he k robustness with respe t to time delay variations, all the previous simulations have been performed with a delay equal to delay equal to

100 ms.

50 ms and with a time

As an example, the augmentation of the delay does not

ae t the hara teristi s of the response of the sideslip angle, the roll angle or the ross oupling between these two variables.

20.5.3 Non linear analysis The non linear analysis has been ondu ted in a similar way to the linear one. Here, of ourse, the simulation environment we have onsidered is the one given in [145℄. The general trim onditions for the non linear simulations are given in the table below : It is to be noted that onstant wind speed and turbulen e are not onsidered

292

Mass

120000 kg

Centre of gravity x-pos.

0.23

Centre of gravity y-pos.

0

Airspeed

80 m/s

Inertial ight path angle

0 deg.

Inertial tra k angle

-90 deg.

Initial position in FE

[0 0 -1000℄ m

Computational time delay

0 s

S aling st. deviation gust

15.4 m/s

Constant wind speed

[0 0 0℄ m/s

Table 20.4: Trim onditions for the non linear analysis

here ex ept where expli itly mentioned.

Performan e riteria The simulation parameters and the results of the time simulations, rise time, settling time and overshoot are given here. The later hara teristi s are indi ated on the plots by dashed lines.

Altitude response The tra king of altitude ommands has been tested by plotting the altitude

50 m (gure 20.14). The hara teristi s of the tr = 4:32 s, ts = 15:9 s and Mp = 2:13 %.

response to a step in altitude of

−940

−76

−950

−78

−960

−80

Heading angle (deg)

Altitude (m)

response are

−970

−980

−82

−84

−990

−86

−1000

−88

−1010 0

10

20

30

40

50 Time (s)

60

70

80

90

−90 0

100

10

20

30

40

50 Time (s)

60

70

80

90

100

Figure 20.14: Altitude response to

Figure 20.15:

a step in ommanded altitude of

sponse to a step in ommanded

50 m

Heading angle re-

heading angle of

293

13 deg

Heading angle response The tra king of heading angle is illustrated in the gure 20.15 where the re-

13 deg is represented. tr = 4:4 s, ts = 10:7 s and Mp = 0 %.

sponse to a step in ommanded heading angle of

hara teristi s of the response are

The

Roll angle response An engine failure is simulated at t = 10 s and the failed engine is restarted at t = 100 s (gures 20.16 and 20.17) .The maximum value of the absolute error of the roll angle  is of 3:82 deg. There is no steady state error due to the engine failure. After the restart of the failed engine, the maximum value of the absolute error of the roll angle



is of

5:56 deg.

After the engine failure, the

maximum value of the absolute error of the sideslip angle

is of 2:7 deg.

After

the restart of the failed engine, the maximum value of the absolute error of the sideslip angle



is of

4:27 deg.

6

10 8

4

6

Sideslip angle (deg)

Roll angle (deg)

4 2 0 −2

2

0

−2

−4 −6

−4

−8 −10 0

20

40

60

80

100 Time (s)

120

140

160

180

−6 0

200

Figure 20.16: Roll angle response

20

40

60

80

100 Time (s)

Figure 20.17:

120

Sideslip

140

160

180

200

angle re-

sponse

Both throttle responses are shown here (gure 20.18). In gure 20.19, roll angle response is represented for moderate turbulen e, that is, we have sele ted wind at

20 ft above the ground, W20 = 15:4 m/s.

Heading rate Here, engine failure is observed at

t = 50 s (gure 20.20).

t = 10

s and engine is starting again at

The peak maximum heading rate,

_

is equal to

0:0349

deg/s.

Cross oupling between airspeed VA and altitude h: The ross oupling between step in velo ity ommand of

h and VA is illustrated in the gure 20.21. For a 13 m/s, we get a maximum value for the absolute 294

0.18

6 Throttle 2

0.16

4 0.14 2

Roll angle (deg)

Throttle 1 − 2 (rad)

0.12 0.1 0.08 0.06

0

−2

0.04 −4 Throttle 1

0.02 0 0

20

40

60

80

100 Time (s)

120

140

160

180

−6 0

200

Figure 20.18: Throttle responses

10

20

30

40

Figure 20.19:

50 Time (s)

60

70

80

90

100

Roll angle response

for moderate turbulen e 0.04

−986

−988

0.03

−990

−992

Altitude (m)

Psi_dot (deg/s)

0.02

0.01

0

−994

−996 −0.01 −998 −0.02

−0.03 0

−1000

10

Figure

20

30

40

20.20:

50 Time (s)

60

70

Heading

80

rate

90

−1002 0

100

re-

20

30

40

50 Time (s)

60

70

80

90

100

Figure 20.21: Altitude response for

sponse for an engine failure

error of the altitude

10

a step in airspeed of

h equal to 12:93 m.

13 m/s

For a step in altitude of

a maximum value for the absolute error of

V

of

0:23 m/s.

30 m, we get

20.6 Results of the Automated Evaluation Pro edure This se tion presents the methodology independent results of the ontroller designed in the previous se tions. It is mostly based on the evaluation mission and s enario dened in [145℄ : both overall tra king performan e and innerloop behaviour of the ontrolled system will be evaluated by means of bounds on key variables.

The ee t of engine failure As the RCAM air raft model is twin-engined, a single engine failure will mainly result in lateral deviation. Hen e gure 20.22 provides a top view of the rst traje tory segment. The stati lateral deviations satisfy all the required onstraints with very small overshoots orresponding to engine failure and engine

295

restart. First segment: top view

Second segment: lateral deviations 300

100

0

0

a

b

lateral deviation [m]

x−deviation [m]

200 50

1

−50

100

0 1

c

d

2

−100

−200 −100 −20

−18

−16

−14

−12 −10 −8 y−position (−YE) [km]

−6

−4

−300 0

−2

1

2

3 4 5 6 along track distance from point 1 [km]

Figure 20.22: The ee t of engine

Figure 20.23:

failure

during the 3 deg/s turn

7

8

Lateral deviations

The 3 deg/s turn At the beginning of the turn, the perfe t following of the required traje tory and the desire to perform a oordinated turn would imply a sudden hange in the air raft's bank angle, whi h is only possible with an innitely high roll rate. Obviously this is undesirable but deviations from the desired traje tory at the start, (and the nish), of the turn are unavoidable. As it an be seen on the gure 20.23, whi h is a loser look at the a tual lateral deviations, the traje tories are onned within the xed bounds.

The apture of the -6 and -3 degrees glideslope We start with a glideslope of -6 deg; again it is unavoidable that the air raft leaves the desired traje tory.

It returns to the traje tory without overshoot

and well within a period of 30 s. After that, we go to a glideslope of -3 deg su h that we get an inverse behaviour with respe t to the desired traje tory, that is about half the size of the rst response. The verti al deviations from the desired glideslope are plotted in gure 20.24.

The nal approa h with windshear While on nal approa h with a glideslope of -3 deg the ee t of a windshear model is onsidered.

The verti al deviations from the desired glideslope are

plotted in gure 20.25. The behaviour of the ontrolled system is satisfa tory.

Numeri al results Table 20.5 summarizes the numeri al results obtained with the designed ontroller in terms of performan e indi ators. For the motivation and al ulation prin iple of the various results see [145℄.

296

Fourth segment: altitude deviations 30

20

20

10 2

f

altitude deviation [m]

altitude deviation [m]

Third segment: altitude deviations 30

3

0

−10

e

−20

−30

10

3 g

4

h

0

−10

−20

−16

−15

−14 −13 x−position (XE) [km]

Figure 20.24:

−12

−30 −11

−11

Verti al deviations

−10

−9

−8

−7 −6 −5 −4 x−position (XE) [km]

Figure 20.25:

from the desired glideslope

−3

−2

−1

Verti al deviations

from the desired glideslope

Segm. I

Segm. II

Segmm. III

Segm. IV

Total

Performan e

0.1753

0.1058

0.5940

0.1698

0.2612

Perf. Dev.

0.0888

0.3484

0.4737

0.2678

0.5011

16.7323

0.1603

Comfort

1.3541

0.6029

4.7976

Safety

0.0056

0.1544

0.0054

0.0365

0.0505

Power

0.0055

0.0252

0.0139

0.0278

0.0181

Table 20.5: Numeri al results of the evaluation pro edure

All the indi ators ex ept omfort are less than

1

(maximal bound spe ied

in [145℄) indi ating a relatively good behaviour of our ontroller. In fa t, we get non a

eptable value of the lateral a

eleration be ause in segment II, (whi h

orresponds to a turn of

90Æ), the ontrol is not smooth enough.

Lo ating the

inner loop poles in a ir le larger than the previous one provides a smooth

ontrol in the lateral hannel.

A redesign of the lateral ontroller has been

onsidered. A lateral inner ontroller has been omputed to lo ate the poles in a ir le entered at

8 + j 0 with radius r = 7:9 and with the same weightings

matri es as previously. It is not ne essary to hange design parameters of the outer loop ontroller. An outer loop ontroller is obtained with the same hoi e of ir le and weightings matri es as in the previous design.

The evaluation

pro edure has been run with the new ontroller. Lateral deviations are shown in gures 20.26 and 20.27 and the new numeri al results are given in table 20.6. The omfort indi ator has been signi antly improved by hanging only one design parameter, namely the ir le. We believe that a renement in the lateral inner ontroller design to get omfort indi ator less than 1 is possible by hanging other design parameters, namely weightings matri es.

20.7 Con lusions and Lessons Learned After a phase of analysis and evaluation, the results produ ed by the use of the designed ontroller appear to be quite good. The main strong features of the

297

First segment: top view

Second segment: lateral deviations 300

100

0

0

a

b

lateral deviation [m]

x−deviation [m]

200 50

1

−50

100

0 1

c

d

2

−100

−200 −100 −20

−18

−16

−14

−12 −10 −8 y−position (−YE) [km]

−6

−4

−300 0

−2

1

2

3 4 5 6 along track distance from point 1 [km]

Figure 20.26: The ee t of engine

Figure 20.27:

failure

during the 3 deg/s turn

7

8

Lateral deviations

Segm. I

Segm. II

Segm. III

Segm. IV

Total

Performan e

0.2053

0.1380

0.2115

0.2864

0.2103

Perf. Dev.

0.0748

0.0937

0.2484

0.1417

Comfort

0.5185

3.8774

0.1500 1.3632

0.4857

1.5612

Safety

0.0068

0.2426

0.0088

0.0586

0.0792

Power

0.0046

0.0080

0.0150

0.0309

0.0146

Table 20.6: Numeri al results of the evaluation pro edure

methodology an be established in the following way. - It does not require any pre ise knowledge or prior experien e about ight

ontrol systems design. This point an be illustrated by the fa t that it is the rst appli ation of this methodology in the aeronauti al eld by the team of the LAAS-CNRS. - This methodology is well suited to the two-stage design, whi h appears to be inherent to the ight ontrol systems design. - The use of the methodology does not need a very large number of synthesis parameters whi h are learly and easily identiable. A tions on these parameters an be translated in lear frequential interpretations. - The pro edure of synthesis an be easily automated. - The numeri al tools whi h have been used are reliable and widespread in the s ienti world. - The stru ture of the ontroller is simple and natural for su h problems. - The time spent on designing the ontroller was not too important. Most of the spent time was used to understand the problem and to dene the ar hite ture of the ontroller. On the other hand, some drawba ks need some improvements:

298

- The dimension of the resulting ontroller may be large when applying a multistage design whi h is the ase here with the inner and outer loops. Indeed, for ea h design a ontroller is obtained with a dimension equal to the one of the model used. Getting a redu ed order ontroller may be a hieved by using standard expli it model redu tion method but with the known di ulties linked with the ne essity of ontrolling the performan e degradation. - It is ne essary to have a pre ise model of the un ertainty in order to as ertain the robust stability and pole lo ation. - If un ertainty had to be onsidered in other terms of the model,

(B; C; D),

the parameter-dependent Ri

ati equations ould not be still available. The Ri

ati type approa h in quadrati design is orrelated to spe i un ertainty formulation su h as the norm-bounded one and does no longer hold when interval-type or polytopi un ertainty is onsidered. An

LMI

formulation should be onsidered. Possible extensions and improvements are possible, mainly the size of the ontroller an be redu ed by using expli it model redu tion methods. Moreover, other ar hite tures, (use of other measurement signals...), for the ontroller should be explored.

299

21.

An

H1 Approa h

Mark R. Tu ker and Daniel J. Walker 1

Abstra t.

This hapter des ribes an

2

H1 approa h to the design

of a ontroller for the Resear h Civil Air raft Model (RCAM). The

ontroller produ ed onsists of an inner and outer loop.

The in-

ner loop ontrols the manual ying of the plane and is designed using a multivariable

H1 two degree-of-freedom mixed sensitivity

approa h. The design in orporates performan e and robustness requirements to produ e a losed loop de oupled system to meet the ying spe i ations. The outer loop deals with autopilot tra king and is designed using

H1

normalised oprime fa tor one degree

of freedom loop shaping te hniques.

Three outer loop ontrollers

are synthesised to tra k height, heading and to reje t lateral deviations. Analysis tests and simulation results look promising. The

ontrollers produ ed are of a high order, although in prin ipal it should be possible to a hieve signi ant order redu tion.

For a

predened mission s enario, the performan e, robustness, omfort, safety and power results satisfy the spe i ation.

21.1 Introdu tion In this hapter, the design methodology used for the RCAM ben hmark problem is based on

H1 ontrol theory.

Sin e the 1980's,

H1 ontrol theory has

been establishing itself as as powerful te hnique for the design of multivariable

ontrollers that are robustly stabilising as well as meeting performan e requirements. It is a frequen y domain te hnique whi h an a

ommodate robustness issues in luding disturban e reje tion and model un ertainty.

Frequen y de-

pendent weights are sele ted to a

ommodate robustness and performan e requirements. The losed loop system an also be mat hed to an ideal system by in luding mat hing models into the design. Owing to variations in speed, weight, entre of gravity, time delays, engine failure and disturban es from wind shear and gusts, and the need to meet the performan e spe i ations dened in [145℄, the RCAM problem is very suitable to the appli ation of 1

H1 te hniques.

Engineering Department, University of Lei ester, University Road, Lei ester LE1 7RH,

United Kingdom. E-mail: mrtsun.engg.le.a .uk Tel: +44 116 252 2567/2874 Fax: +44 116 252 2619 2

Engineering Department, University of Lei ester, University Road, Lei ester LE1 7RH,

United Kingdom. E-mail: wjdlei ester.a .uk Tel: +44 116 252 2529 Fax: +44 116 252 2619

300

The subsequent se tions of this hapter detail the hosen ar hite ture and how the design requirements are in orporated into the design y le. Two distin t

H1 te hniques are used.

Analysis of the resulting ontrollers and evalu-

ation results are then given.

21.2 Controller Ar hite ture The ar hite ture used is based on an inner and an outer loop stru ture (Figure 21.1). The two loops are employed to separate the manual ying requirements, with the pilot in the loop, from the tra king of the desired traje tory when in autopilot mode. A number of sele tors exist, so that the ontroller an swit h between dierent modes; eg full authority autopilot to manual.

Outer Loop vx_d

χ _d

Demanded Track Angle

vy_d

Inner Loop Outer Heading

φ _d

Controller χ ylat

Lateral Deviation Controller

V_d Inner Loop

Gain Scheduling

RCAM Model

Controller

For Speed

and Actuators

z_d

Outer Height Controller

q, vz, Vair, beta, p, r, phi

wv _d

z wv _d

ylat ,χ, z

Figure 21.1: RCAM Controller

21.2.1 Inner Loop The inner loop is simply one multivariable ontroller designed to meet the manual ying performan e and robustness spe i ations. Rather than designing separate longitudinal and lateral ontrollers one ombined ontroller has been designed. The

H1 optimisation allows a single ontroller to be designed

expli itly so that the ross- oupling error between the ontrolled hannels is redu ed.

Referen e Signals

wV _d),

The referen e signals to the inner loop are the verti al velo ity demand (

VA_d )

airspeed demand (



and roll angle demand ( _d ).

These signals are

adequate for manual ying and an be used by the outer loop to tra k the height and heading as well as reje ting lateral deviations.

301

Measurements Signals The measurements signals used by the inner loop are the measured verti al

wV ), r

velo ity (

VA ) p

airspeed (



q

and roll angle ( ), as well as the pit h rate ( ),



yaw rate ( ) and roll rate ( ) and sideslip ( ). The three rates are fed ba k to



enhan e the ontrol and stabilisation of the inner loop. The sideslip ( ) is fed ba k to be ontrolled to zero.

A tuator Demands

ÆA ),

The aileron (

ÆT ),

tailplane (

ÆR )

rudder (

ÆT H )

and engines (

are all on-

trolled by the inner loop, with the two engines ontrolled as one. For robustness to airspeed variations, the ontrol surfa e a tuator signals are gain s heduled to a

ount for the fa t that the ontrol surfa e ee tiveness is a fun tion of the square of the airspeed.

21.2.2 Outer Loop The outer loop omprises three separate ontrollers, used spe i ally for tra king a traje tory when in autopilot mode.

Height Tra king The height tra king outer loop ontroller has three inputs; height demand, height rate demand and the feedba k of the a tual height.

The ontroller

generates a verti al speed demand for the inner loop. Note that the referen e height rate demand input is swit hed in for tra king ramped inputs, and is swit hed out if stepped demands are required.

Inertial Tra k Angle Tra king The inertial tra k angle ontroller has three inputs; inertial tra k angle demand, inertial tra k angle rate demand and the feedba k of the a tual inertial tra k angle.

The ontroller generates a roll angle demand for the inner loop to

o-ordinate a turn. Note that the referen e inertial tra k angle rate demand input is swit hed in for tra king ramped inputs, and is swit hed out if stepped demands are required.

Lateral Deviation Reje tion This ontroller takes the lateral deviation error as the input signal and orre ts for this deviation by generating an inertial tra k angle demand. The inertial tra k angle demand is dire tly fed into the outer loop heading referen e ommand whi h generates a roll angle demand to oordinate a turn to orre t for the deviation.

302

21.3 Translation of RCAM Design Criteria The two degree of freedom mixed sensitivity design for the inner loop allows for the in lusion of some design riteria dire tly into the design methodology. A number of weights are used in the design and many of these are derived dire tly from the design riteria. The outer loop design is mu h simpler in omparison, and design goals are simply met by tuning the shaping parameters.

21.3.1 Performan e Spe i ation From the spe i ation, a mat hing model for the referen e signals an be onstru ted for in lusion in the design.

Verti al Velo ity

wV ),

The verti al velo ity (

V ) are related by

inertial ight path angle ( ) and inertial velo ity

(

sin( ) =

wV V

As the ight path angle ( ) is typi ally small,

V)

inertial velo ity (

wV

(21.1)

sin( )

 ,

so for onstant

/

(21.2)

Thus the ight path angle spe i ation an be used as the verti al velo ity

tr ), overshoot (ts ) and settling time (Mp )

spe i ation. To meet the rise time (

spe i ations, the following se ond order model is hosen

!n2 1 M1 = 2 s + 21 !n1 s + !n2 1 with

!n1 = 0:85 rad/s and 1 = 1.

(21.3)

This model a hieves the following results,

with the spe i ation requirements shown in bra kets for omparison.

tr = 4:0s (< 5s) ts = 7:8s (< 20s) Mp = 0:0% (< 5%)

(21.4)

Airspeed The rise time, overshoot and settling time spe i ation for airspeed is also given. Using the mat hing model

!n2 2 M2 = 2 s + 22 !n2 s + !n2 2 with

(21.5)

!n2 = 0:3 rad/s and 2 = 1 yields the following results, with the spe i-

ation requirements on e again shown in bra kets for omparison.

tr = 11:2s (< 12s) ts = 22:1s (< 45s) Mp = 0:0% (< 5%) 303

(21.6)

Roll Angle

 V ) and heading rate (_ ) by   V _  = atan g

For a oordinated turn, the roll angle ( ) an be approximated in terms of the inertial velo ity (

Hen e for small angles

(21.7)

 / _

(21.8)

The spe i ation gives the ideal heading response, so to ontrol the roll angle (whi h is proportional to heading rate), the heading rate bandwidth should be 2-5 higher than for the heading. response rise time is

In terms of the time domain, the heading

tr < 10 se s, so the heading rate rise time should be 2 5

se s. The ideal se ond order model hosen was

!n2 3 M3 = 2 s + 23 !n3 s + !n2 3 with !n3 = 1 rad/s and 3 = 1, leading to tr = 3:4s ts = 6:6s Mp = 0:0%

(21.9)

(21.10)

Cross-Coupling Using the se ond order systems 21.3, 21.5 and 21.9 the ideal system as the transfer fun tion matrix

 0:09 1 M = diag s2 +10:7:7225 s+0:7225 ; s2 +0:6s+0:09 ; s2 +2s+1

M is formed



(21.11)

All o-diagonal elements are zero, orresponding to zero ross- oupling between the hannels.

21.3.2 Robustness Robustness to the plant un ertainties of entre of gravity and mass variations are not expli itly in orporated into the design. It is noted that su h parameter variations ould be modelled as additive perturbations and in orporated into the design weights. Here however, the parameter variations are only onsidered for sele ting a nominal ight ondition for the RCAM linearisation. The linear model was produ ed for air raft mass of

0:23m and entre of gravity in z of 0:0m.

120000kg,

entre of gravity in

x

of

The following robustness issues are expli itly in luded in the design.

Time Delays For the time delays, a nominal delay of

0:05s was hosen for ea h of the ontrol

signals and was represented in the model as rst order Padé approximations: i.e.

40 s 40 + s 304

(21.12)

Wind and Output Perturbations The ontroller is designed to be robust to output perturbations and wind disturban es.

The design takes advantage of the RCAM model's wind inputs.

The inputs are used in the ontroller synthesis to minimise the ee t of wind disturban e.

Airspeed As ontrol surfa e ee tiveness is a fun tion of the airspeed squared, the RCAM is made robust to airspeed variations by gain s heduling the ontrol surfa e

ÆA ),

demands to aileron (

ÆT )

tailplane (

ÆR )

and rudder (

as a fun tion of the

airspeed squared.

21.4 Design Cy le The design y le onsists of rst the inner, then the outer loop design. The inner loop design is based on a linearization taken from the non-linear RCAM model, whi h is inserted into a spe ially sele ted system design stru ture. Weights are then hosen. Having synthesised an

H1 sub-optimal ontroller, ontroller re-

du tion, then testing are performed. The outer loop design pro eeds in a simi-

lar manner, ex ept the linear models used are derived from the ideal mat hing models used for the inner loop design. Further testing is performed on the omplete system, and iterations of the design y le are performed. This next se tion des ribes the inner and outer loop detail.

H1 designs3 in more

21.4.1 Inner Loop Design Linearization The model used for the ontroller synthesis omprises three subsystems: the linearization of the RCAM rigid -body dynami s, an a tuator linearization and a time delay Padé approximation. The RCAM itself is linearised around the

120000 kg), entre of gravity in x (0:23m), entre y (0:0m) and airspeed (80 m/s). The RCAM linearised model has 8 states of roll rate (p), pit h rate (q ), yaw rate (r ), roll angle (), pit h angle ( ) and the inertial body axis velo ities in x (uB ), y (vB ) and z (wB ). There are 4 ontrol inputs. The a tuator signals are aileron dee tion (ÆA ), tailplane dee tion (ÆT ), rudder dee tion (ÆR ) and throttle position (ÆT h ). operating onditions of mass (

of gravity in

Additionally, there are 6 wind perturbation inputs of wind velo ity in earth

axis o-ordinates (Wxe , Wye , Wze ) and in body axis o-ordinates (Wxb , Wyb , Wzb ). The 7 output measurements used are pit h rate (q), verti al velo ity (wv ), airspeed (VA ), sideslip ( ), roll rate (p), yaw rate (r ), roll angle (). 3

Program ode for the designs, written using Matlab and Simulink in luding the ontrol

and robust ontrol toolboxes is given in [242℄.

305

The a tuator model linearization is ee tively the dynami models of the a tuators without saturation and rate limits. Port and starboard engines are ganged together. The linear a tuator transfer fun tion matrix used is diag



6:7 6:7 3:35 0:67 s+6:7 ; s+6:76 ; s+3:35 ; s+0:67

(21.13)

The nal linearised model for the design has 16 states (8 RCAM, 4 a tuators, 4 delays), 4 ontrol inputs, 6 wind disturban e inputs and 7 sele ted outputs.

System Design Stru ture The system design stru ture of Figure 21.2 is used. In this system referen e inputs,

and

r3

are perturbation inputs,

z1

r1

are the

are the performan e

z2 are the weighted ontrol outputs, e1 are the ontroller referen e e2 are the fed ba k measurements and u are the a tuator ontrol signals.

outputs, inputs,

r2

W2

z2 r2

r1

+

e1

u

e2

K

r3

+ G

z1

+

W1 -

M

Figure 21.2: Two Degree of Freedom RCAM Design System

Minimisation Fun tion The variables in Figure 21.2 are related by

3

2

2

32

3

r1 W1 M W1 W1 G2 W1 G1 z1 6 7 6 r2 7 6 z2 7 W 0 0 0 2 7 6 7 76 6 4 e1 5 = 4 I 0 0 0 5 4 r3 5 0 I G2 e2 G1 u Eliminating u, e1 and e2 , the transfer fun tion from r to z is given by 

W1 (So GK1 M ) W1 So W1 So G2 Tzr = W2 Si K1 W2 K2 So W2 K2 So G2

(21.14)



(21.15)

where the input and output sensitivities are dened as

So = (I + GK2 ) 1 Si = (I + K2G) 1 306

(21.16)

Errors and ontrol eort due to the wind disturban es are minimised via

W1 SoG2

and

W2 K2So G2 respe tively. Tzr the state spa e stru ture

To minimise

of this plant in orporated into

the design software of [242℄ is used.

Weight Sele tion Weight

W1

has been sele ted as

W1 = where

diag



 = 10 6 .

0:7225s 2:5 1 10 s2 +1:7s+0:7225 ; s+ ; s+ ; s+ ; s 10 0:1225s s2 +0:7s+0:1225 ; s2 +2s+1 ; s+

(21.17)

The four integral terms are hosen to give zero steady state errors on these

hannels, and will provide good tra king to referen e demands . The integral gains are a design parameter. In reasing gain gives the losed loop system better mat hing to the ideal model and in reases the bandwidth of the disturban e reje tion.

wv ) and airspeed (VA ) have

The longitudinal hannels of verti al velo ity (

unity gain giving good losed loop performan e and disturban e reje tion. The





lateral hannels of sideslip ( ) and roll angle ( ) have gains of 10. These values are hosen to enable better disturban e reje tion of both the sideslip and roll angle as well as better tra king the roll angle demands. This is so that errors on sideslip and roll angle, su h as might o

ur during an engine failure, are qui kly redu ed. The three rate terms are fed ba k to the ontroller to enhan e ontrol and stabilisation.

The weights on these rates are sele ted to be bandpass lters.

Se ond order lters were sele ted to reje t disturban es and ross- oupling

q

ee ts at the sele ted frequen ies. For the pit h rate ( ), the frequen y sele ted was taken as the natural frequen y of the verti al velo ity response. For the

p

roll rate ( ), the frequen y sele ted was set to the natural frequen y of the roll

r

angle response, and for the yaw rate ( ) the sele ted frequen y was taken to be the bandwidth of the heading response (

r

), as yaw rate ( ) is approximately

the same as heading rate. The ontrol weight

W2

low frequen y tra king.

needs to limit high frequen y a tivity and to allow

Hen e

W2

needs to be a high-pass lter to bound

these requirements. First order weights are sele ted with design parameters of

rossover frequen y and low and high frequen y gains. To redu e the tuning parameters, the high and low frequen y gains were hosen to be the inverse of one another. There is no physi al reason why this needs to apply, so more tuning with an additional parameter ould be applied at a later stage. These parameters may be more a

urately sele ted to represent the un ertainty arising in the model as this weight an be used to produ e robust designs to plant additive un ertainty [242℄. So representing the RCAM entre of gravity and mass variations as additive un ertainties an help the sele tion riteria for this weight and lead to robust designs against these variations. The basi weighting

307

fun tion is

Parameter

w

ks + w s + kw

(21.18)

is hosen so that high frequen y ontrol is minimised above a

ertain threshold frequen y, and

k

is hosen to allow low frequen y ontrol

eort and at the same time minimise the high frequen y ontrol eort. Weights for the a tuators ae ting longitudinal motion, (the tailplane and engines), were hosen with

k = 10 and w = 200 rad/s.

k

Keeping the gain ( )

low is desirable so that a tuators operate within a known linear region, in order to avoid a tuators rate limiting or saturating. Similarly, for the lateral a tuators (the aileron and rudder),

hosen as

40 and 200 rad/s.

k

and

w

were

More freedom of ontrol was allowed in the lateral

k

hannels than the longitudinal hannels so the gain ( ) is higher, enabling more a tuator eort to be used, whi h was parti ularly useful meeting spe i ations relating to engine failure. The transfer fun tion of the ontrol eort weight is

 s+200 10s+200 40s+200 10s+200 ; s+2000 ; s+8000 ; s+2000 W2 = diag 40s+8000



(21.19)

Controller Synthesis The augmented plant is onstru ted by ombining the weights and the linearised model. realised with a

H1 optimisation is performed and a suboptimal ontroller was

= 1:1 opt = 5:58. i.e. k Tzr k1 = 5:58, where Tzr is given

in 21.15. This value gives an indi ation of robustness. A more robust solution

would yield a lower value whi h might be obtained at the expense of performan e for example by redu ing the hannel bandwidths in the weight through more tuning of the parameters of ontrol weight

W2

W1

or

for example by

in orporating plant additive perturbations into the sele tion riteria to model the entre of gravity and mass variations.

H1 optimisation produ es a ontroller whose order will be equal to that

order of the augmented plant. This order is hen e the sum of the number of states of the plant (8 RCAM + 4 a tuators + 4 delays), weight

W2

(4) and the mat hing weight

M

W1 (10), weight

(6): a total of 36 states. This high order

is redu ed by 10 states using balan ed trun ation. Some iteration was required to arrive at a reasonable inner loop ontroller.

Having done so, the outer loops were designed.

21.4.2 Outer Loop Design The inner loop design resulted in a losed loop system that losely approximated the mat hing model

M

of 21.11. Therefore, it was de ided to utilise

M

as the basi design model for the outer loops design.

Height Tra king Figure 21.3 shows the loop shaping arrangement for the outer loop height tra king. The ontroller implementation is shown in Figure 21.4. This arrangement

308

wv 0.7225 s2+1.7s+0.7225

1 _ s

Kz

1 _ s

z Wz

Shaped Plant

Figure 21.3: Height Tra king Loop Shaping System

r1 r2

+ -

Wz

+

wv 0.7725 s2+1.7s+0.7725

1 _ s

Kz

1 _ s

z

s Controller Figure 21.4: Height Tra king Controller System

was hosen as it an be used to provide good responses to step height demands (using referen e

r1 )

or ramped height demands (using referen es

su h as when following a glide slope.

r1

and

r2 ),

The shaped plant takes the approximate model for the verti al speed from

wv ) is integrated to give height

the mat hing model 21.11. The verti al speed (

z

( ) as an output. An integrator weight is pla ed at the input to the plant. This ensures that the system is type 2 and so ensures that there is zero steady state error to ramped height demands or disturban es. The two outputs of the plant are shaped using a stati diagonal weight



Wz

of the form:

Wz = kz1 1 0 0 kz 2

 (21.20)

Using this loop shaping system, a stabilising sub-optimal ontroller is synthesised using the normalised oprime fa tor method des ribed in [164℄. Finally, the ontroller is augmented with the shaping weight to give the stabilising feedba k ontroller.

Wz being tuned. Wz k1 and then tuning k2 to meet the required step

Design y le iterations are performed, with the weight is tuned by sele ting a nominal response spe i ations.

k1

with minor adjustments to

an then be tuned to improve the ramp response,

k2

if ne essary. The nal weight hosen was



Wz = 0:55 1 0 0 3:64 This led to a

= 3:31

 (21.21)

whi h is su iently low to indi ate good robustness.

The ontroller generated has 5 states.

309

Kh

φ 1 s2+2s+1

1 _ s

g

1 _ s

V

χ Wh

Shaped Plant

Figure 21.5: Inertial Tra k Angle Loop Shaping System

r1 r2

+ -

+

Wh

Kh

φ 1 s2+2s+1

1_ s

g

χ

sV

s Controller

Figure 21.6: Inertial Tra k Angle Controller System

Inertial Tra k Angle Tra king Figure 21.5 shows the loop shaping arrangement for the outer loop inertial tra k angle. The ontroller implementation is shown in Figure 21.6. This arrangement is essentially the same as for the height tra king, with two referen e inputs to allow for step and ramp demands, and an integrator at the input to ensure zero steady state error to ramped disturban es and referen es. The shaped plant is based on the approximate model for the roll angle response from the mat hing model 21.11.

The roll angle is integrated and

s aled to give inertial tra k angle as an output. Lapla e transforms yields:

where

g = 9:81 m/s2 and V



Linearising 21.7 and taking

g  sV

(21.22)

is the nominal design inertial velo ity. The inertial

tra k angle rate is also made available as an output. Design y le iterations are then performed as for the height tra king loop. A stati diagonal weight



Wh = 1 0 0 4:15 was hosen. The design realised a



= 2:85 indi ating good

(21.23)

robustness. The

ontroller generated has 5 states, redu ed to 3 using balan ed trun ation.

Lateral Deviation Reje tion Figure 21.7 shows the loop shaping arrangement for the outer loop lateral deviation tra king. The shaped plant takes the losed loop inertial tra k angle dynami s, resulting from the ontroller

Kh designed in the previous subse tion.

310

∆_ d

+

K∆

W∆

Kh

-

φ 1 s 2+ 2s + 1

g sV

χ

V s

∆

Controller Inertial Track Angle Loop

Shaped Plant

Figure 21.7: Lateral Deviation Loop Shaping System

In earth oordinates, it is assumed that

Y=

Z

V sin dt

(21.24)


)

Assuming the tra k to be along a bearing of 0 deg the lateral deviation (

Y. small, so   sin . will be equal to

The heading deviations when on tra k are assumed to be Hen e


=

V  s

(21.25)

V is the nominal design inertial velo ity. W
is sele ted to be a simple gain and a stabilising sub-optimal ontroller is

where

synthesised using the normalised oprime fa tor method. The ontroller order is then redu ed using balan ed trun ation and the ontroller is augmented with the shaping weight. Design y le iterations are then performed, with value hosen was

whi h realised

W
being tuned.

W
= 0:014

= 1:85 suggesting good robustness.

The nal (21.26)

The ontroller generated

has 7 states due to the high order of the plant model used, but this is redu ed to 2.

21.4.3 Design Cy le - Dis ussion Weight Sele tion Weight sele tion in

H1 designs an be a time onsuming pro ess with many

design iterations needed. The RCAM problem was ta kled in su h a way that

weight sele tion was performed e iently. The outer loop design in parti ular provided three simple manner.

H1 ontrollers that ould easily be tuned in a lassi al

The inner loop has in prin iple more weights to sele t than the outer loops. However, the mat hing model is dened by the spe i ations, and the weight

W1

is all but dened, ex ept for the integrator gain values.

W2 ,

the ontrol

eort weight, is dened broadly by robustness requirements to be a high pass

lter; only the gains and ut-o frequen ies need to be sele ted, although more thorough design sele tion might onsider the parameter variations of the RCAM as additive perturbations and in orporate them a

ordingly.

311

Dening the ontroller ar hite ture an be more time onsuming than the tuning. Hen e it is important to sele t a good ar hite ture so that the weights are easily dened.

Controller Order The design y le presented inherently generates ontrollers with a large number of states.

For the inner loop design, the ontroller will have as many states

as the augmented plant, and for the outer loop, the ontrollers will have more states than the shaped plants. The redu tion s heme used is a relatively simple one, introdu ed partly to speed up the performan e tests and simulations. Further ontroller order redu tion has not been investigated, as it was felt that the priority was to get su

essful designs working. Lower order ontrollers would be more desirable in an a tual implementation. It is probably possible to redu e the order signi antly, and so more rigourous ontroller order redu tion would be re ommended for future work.

Design Time Overall, the method adopted in this hapter has been found to be very appli able to the RCAM problem. Mu h time was needed to nd a suitable system design ar hite ture, but on e obtained, the produ tion of suitable results was fairly straightforward with weight sele tion methods as previously dis ussed. Some time was spent rening and tuning the design.

21.5 Controller Analysis 20

20 0

0

−20

Singular Values − [dB]

Singular Values − [dB]

−20

−40

−60

−80

−100

−60 −80 −100 −120

−120

−140 −8 10

−40

−140

−6

10

−4

10

−2

10 Frequency − [rad/s]

0

10

2

10

4

10

−160 −2 10

−1

10

0

10

1

10 Frequency − [rad/s]

2

10

3

10

4

10

Figure 21.8: Output Sensitivity and the Sensitivity of the Control Eort to Input Demands

Frequen y Analysis - Inner Loop The design seeks to minimise a number of weighted sensitivities as seen in 21.15.

312

For the output sensitivity of Figure 21.8, low gain is a hieved over the oper-

wv ), airspeed VA ) and roll angle () and the ontrolled sideslip ( ). Pit h (q), roll (p) and yaw (r ) rates have unity gain at these low frequen ies. At high frequen ies, ating bandwidth for the three mat hed signals of verti al velo ity (

(

the output sensitivity gain is unity and so high frequen y disturban es are not

ontrolled. Between

1

rad/s and

10 rad/s,

there are high peaks. Minimising

these peaks in reases the robustness. By the small gain theorem, the higher the maximum singular value of

So , the smaller the unstru tured output inverse

multipli ative perturbation that will de-stabilise the system. The fun tion

Si K1, representing the sensitivity of the ontrol eort to input

demands is also shown in Figure 21.8. At high frequen ies, the gain is small thus limiting high frequen y ontrol eort as spe ied. Figure 21.9 shows the frequen y response of the dieren e between the

losed loop system and the mat hing model.

At high and low frequen ies

the dieren e is very low. The dieren e is maximised around the operating bandwidth, whi h may degrade the overshoot or settling time to step demands.

100

0

50

−50 Singular Values − [dB]

Singular Values − [dB]

Also shown is the ontroller frequen y response (Figure 21.9).

0

−50

−100

−150

−100 −4 10

−2

10

0

10 Frequency − [rad/s]

2

10

4

10

−200 −4 10

−2

10

0

10 Frequency − [rad/s]

2

10

4

10

Figure 21.9: Controller Frequen y Response and Dieren e between the Closed Loop System and the Mat hing Model

Frequen y Analysis - Outer Loop Figure 21.10 shows the frequen y responses for height tra king.

The open

Gz ), shaped model (Wz Gz ) and the ontrolled model (Kz Wz Gz )

loop model (

responses are shown as well as the frequen y response of the losed loop and the

ontroller. Similar analysis is performed for the other outer loop ontrollers.

Controller Eigenvalue Analysis Figure 21.11 shows the lo ations of the poles of eigenvalues for the stable inner and ombined outer loop ontrollers.

313

150

50

100

50

K

0

G

−50

Gain − [dB]

Gain − [dB]

0 WG

−50

−100

−100

−150

GK/(I+GK)

KWG −200

−150 −250

−200

−250 −3 10

Figure 21.10:

−2

−1

10

10

0

10 Frequency − [rad/s]

1

10

2

10

−300 −3 10

3

10

−2

10

−1

10

0

10 Frequency − [rad/s]

1

10

2

10

3

10

Frequen y Response of the Height Model, Shaped Plant and

Controlled Plant and for the Height Closed Loop System and the Controller 15

2

1.5

10 1

5 Imag Axis

Imag Axis

0.5

0

0

−0.5

−5 −1

−10 −1.5

−15 −25

−20

−15

−10

−5

0

Real Axis

−2 −5

−4

−3

−2 Real Axis

−1

0

1

Figure 21.11: Eigenvalue Lo ations for the Inner and Outer Loop Controllers

Time Domain Analysis The subsequent analysis shows time domain results of the ontrolled RCAM. Tables give numeri al results of tests for the nominal system used in the design. Additionally plots show the linear and non-linear responses for the tests under dierent ight onditions. The ight onditions used orrespond to those used in the evaluation s enario. The ontrolled RCAM has been shown to have good performan e and robustness at these ight onditions but linear analysis predi ts robustness issues at some extreme ight onditions.

The eigenvalues of the losed inner loop,

onsidered at 81 dierent ight onditions a ross the whole ight envelope, show inner loop stability. However, when the outer loop is losed, the eigenvalues indi ate instability for the ight ondition of extreme aft entre of gravity and high mass. This maybe a onsequen e of using an experimental design that used low order `ideal' models, and where the ontrollers were subsequently redu ed to simpler low order realisations. Further work on the outer loop design, in parti ular for extreme ight onditions is needed. Using more a

urate design models and in orporation of perturbation and parameter variations into the design might be onsidered. Further iterations of the design y le would be needed in pra ti e.

314

The ontrol signals are of parti ular interest to monitor to see if ex essive

ontrol eort is being used or if the a tuators are rea hing rate or saturation limits. For example, the results obtained for the ight path angle are good with respe t to the spe i ations. However, from the linear simulations, it an be seen that large and fast tailplane dee tions are alled for. This results in the non-linear simulation rate limiting, whi h would need to be improved upon. Rise Time

tr

Settling Time

ts

(se s)

Linear

3.9

Non-linear

Mp

(%)

20.0

4.8

1.5

16.4

< 5.0

Spe i ation

Overshoot

(se s)

-0.3

< 20.0

< 5.0

Table 21.1: Flight Path Angle Response

6

6 THRUST

4

THRUST

0

GAMMA

2 Angles − [deg] / Velocities − [m/s]

2 Angles − [deg] / Velocities − [m/s]

4

GAMMA

VCAS

−2 −4 −6 −8

0 VCAS −2 −4 −6 −8 DT

−10 −12 0

DT

−10

2

4

6

8

10 12 Time − [s]

14

16

18

20

−12 0

2

4

6

8

10 12 Time − [s]

14

16

18

20

Figure 21.12: Linear and Non-Linear Response to a Flight Path Angle Demand

Rise Time

tr

Linear Non-linear Spe i ation

(se s)

Settling Time

ts

(se s)

11.2 9.4

< 12.0

22.0 37.2

< 45.0

Overshoot

Mp

(%) -0.0

Table 21.2: Airspeed Response

315

0.0

< 5.0

Height Couple (m) 0.0 1.6

< 10.0

15

10

Angles − [deg] / Velocities − [m/s] / Height − [m]

Angles − [deg] / Velocities − [m/s] / Height − [m]

15

VCAS

5 THRUST

DT

Z

0

GAMMA

−5

VCAS 10

THRUST 5 Z 0 GAMMA

−5

DT −10 0

10

20

30 Time − [s]

40

50

−10 0

60

10

20

30 Time − [s]

40

50

60

Figure 21.13: Linear and Non-Linear Response to an Airspeed Demand

Rise Time

tr

Settling Time

ts

(se s)

Overshoot

(m/s)

Linear

11.3

23.1

0.0

Non-linear

11.0

23.2

0.0

< 12.0

Spe i ation

Airspeed Couple

Mp (%)

(se s)

< 45.0

0.1 0.1

< 5.0

< 0.5

Table 21.3: Height Response

30

30 Z

25 Angles − [deg] / Velocities − [m/s] / Height − [m]

Angles − [deg] / Velocities − [m/s] / Height − [m]

25 20 15 10 5

DT GAMMA VCAS

0 THRUST −5 −10 −15 0

20 15

THRUST 5 GAMMA 0 VCAS −5 −10

10

20

30 Time − [s]

40

50

Z

10

DT

−15 0

60

10

20

30 Time − [s]

40

50

60

Figure 21.14: Linear and Non-Linear Response to a Height Demand

Rise Time

tr

Settling Time

ts

(se s)

(se s)

Linear

9.4

21.3

Non-linear

9.4

21.4

Spe i ation

< 10.0

< 30.0

Table 21.4: Heading Response

316

Overshoot

Mp

(%) -0.0 -0.0

< 5.0

15

15

CHI 10

5

5

Angles − [deg]

Angles − [deg]

PHI 10

BETA

0

CHI

PHI

BETA

0

DR

DE −5

−5

DA

DA −10 0

10

20

30 Time − [s]

40

50

−10 0

60

10

20

30 Time − [s]

40

50

60

Figure 21.15: Linear and Non-Linear Response to a Heading Demand

Heading Rate

_ max

Roll Angle

Roll Angle

(deg)

(deg)

max (%)

172.0

max

(deg/s)

ss

Linear

1.3

9.9

3.8

Non-linear

0.7

8.3

3.7

< 3.0

Spe i ation

< 10.0

Overshoot

149.0

< 5.0

< 50.0

Table 21.5: Engine Failure Response

15

15

10

10

5

PHI CHI

0

Angles − [deg]

Angles − [deg]

5

BETA −5

DA

−10

BETA −5

DA

−10

−15

−15

DE

−20

−25 0

PHI CHI

0

DR

−20

20

40

60

80

100 120 Time − [s]

140

160

180

−25 0

200

20

40

60

80

100 120 Time − [s]

140

160

180

200

Figure 21.16: Linear and Non-Linear Responses to Engine Failure

Rise Time

tr

(se s)

Linear

28.0

Non-linear

27.8

Spe i ation

< 30.0

Overshoot

Mp (%)

-0.0 -4.2

< 5.0

Table 21.6: Lateral Deviation Response

317

10

10 LAT

LAT 8 Angles − [deg] / Deviation − [m]

Angles − [deg] / Deviation − [m]

8

6

4

2 DA

6

4

2

PHI

DE

DA

0

PHI

DE

0 CHI

CHI −2 0

5

10

15

20 Time − [s]

25

30

35

−2 0

40

5

10

15

20 Time − [s]

25

30

35

40

Figure 21.17: Lateral Deviation Linear and Non-Linear Responses

21.6 Results of the Automated Evaluation Methodology-independent results of the ontroller using the evaluation mission and s enario dened in [145℄ are now given. Ea h plot shows responses at 4 ight onditions.

Segment I: the ee t of engine failure As the RCAM represents a twin-engined air raft, a single engine failure will result in lateral deviation. Figure 21.18 shows the air raft's deviation from the tra k when an engine fails and is then restarted. The dotted bounds spe ify the a

eptable level of performan e. First segment: top view

Second segment: lateral deviations 300

100

50

0

0

a

b

lateral deviation [m]

x−deviation [m]

200

1

−50

100

0 1

c

d

2

−100

−200 −100 −20

−18

−16

−14

−12 −10 −8 y−position (−YE) [km]

−6

−4

−300 0

−2

1

2

3 4 5 6 along track distance from point 1 [km]

7

8

Figure 21.18: Segment I: the ee t

Figure 21.19:

of engine failure

deviations during the 3 deg/s turn

Segment II: lateral

Segment II: the 3 deg/s turn Figure 21.19 shows the a tual lateral deviations and the a

eptable bounds whilst o-ordinating a

90 deg turn along the tra k. 318

Segment III: the apture of the -6 and -3 degrees glide slope Initially, a glide slope of -6 deg is demanded, followed by -3 deg. The verti al deviations from the desired glide slope and the a

eptable bounds are plotted in gure 21.20. Fourth segment: altitude deviations 30

20

20

10 2

f

altitude deviation [m]

altitude deviation [m]

Third segment: altitude deviations 30

3

0 e

−10

−20

10

3 g

4

h

0

−10

−20

−30

−16

−15

−14 −13 x−position (XE) [km]

Figure 21.20:

−12

−30 −11

−11

Segment III: verti-

−10

−9

−8

−7 −6 −5 −4 x−position (XE) [km]

Figure 21.21:

−3

−2

−1

Segment IV: verti-

al deviations from the desired glide

al deviations from the desired glide

slope

slope

Segment IV: the nal approa h with wind shear While on nal approa h with a glide slope of -3 deg the ee t of a wind shear is evaluated. The verti al deviations from the desired glide slope and the a

eptable bounds are plotted in gure 21.21.

Numeri al results Table 21.7 gives the numeri al results for the automated evaluation. Segm. I

Segm. II

Segm. III

Segm. IV

Total

Performan e

0.1237

0.1483

0.0545

0.0721

0.0997

Perf. Dev.

0.0249

0.0180

0.0479

0.0833

0.0435

Comfort

0.3674

0.5878

0.5827

0.6235

0.5404

Safety

0.0056

0.0433

0.0077

0.0478

0.0261

Power

0.0030

0.0052

0.0151

0.0303

0.0134

Table 21.7: Numeri al results of the evaluation pro edure

21.7 Con lusions This hapter applies

H1 optimisation to the ontrol of the RCAM. 319

The ontroller designed onsisted of an inner loop to meet performan e and robustness requirements during manual piloted ight and an outer loop to perform the autopilot tra king. The inner loop was designed using an

H1 two degree-of-freedom mixed sen-

sitivity formulation. Performan e spe i ations were translated into method-

dependent obje tives for the verti al velo ity, airspeed and roll angle responses. Robustness riteria relating to time delays and to wind and output disturban es as well as to ontrol eort minimisation were all dire tly in orporated into the design environment. Perturbation and disturban e inputs as well as frequen y dependent weights were used. Tuning of the weights to meet design requirements was needed, but tuning was eliminated where the exa t spe i ations were in orporated. The dire t in lusion of the design riteria meant that results produ ed both for ying performan e and robustness analysis were good at the expense of high order ontrollers. The outer loop design was simpler in on ept.

The inner loop design is

air raft dependent, and mat hes the RCAM response to an ideal air raft response. The outer loop design is ee tively air raft independent, making use of the ideal inner loop response to produ e the autopilot tra king system.

H1

loop shaping te hniques generated three outer loop ontrollers to tra k the height, heading and to reje t lateral tra k deviations. To meet the autopilot requirements, a number of design iterations to tune the responses were performed, produ ing good tra king results. At extreme ight onditions further outer loop design iteration is required to improve robustness, possibly at a ost of some performan e. The majority of design eort went into reating a suitable system design ar hite ture. This was very important, as the ar hite ture used allowed many design spe i ations to be dire tly in orporated. This enabled the time spent tuning design weights to be redu ed.

320

22.

A

-Synthesis Approa h

(1)

Samir Bennani1 and Gertjan Looye

1

Abstra t.

An appli ation of

-synthesis to the RCAM design prob-

lem is presented. Our ontrol obje tive is to a hieve robust performan e. This results in two subproblems that have to be addressed rst, the stability robustness and the nominal performan e problem. Under these onditions the stru tured singular value on ept provides us a tool to a hieve a desired performan e level and keep this in the presen e of un ertainties. To demonstrate the exibility of the method separate ontroller ar hite tures have been adopted for the longitudinal and lateral air raft motions. Further, it is shown how the spe i RCAM problem requirements are naturally translated into the design framework. The resulting ontroller performs satisfa torily and inherits the imposed robustness and performan e spe i ations.

22.1 Introdu tion The theoreti al ba kground of

-Synthesis

-analysis and synthesis is dis ussed in hapter 8.

is a multivariable losed-loop design methodology addressing the

robust performan e issue. We believe that to su

essfully apply the method to the RCAM problem, both a ight me hani s and a lassi al Bode shaping ba kground are required. The ight me hani s insight leads to a sensible problem formulation that is in rst instan e translated into the hoi e of the ontroller ar hite ture. It is also needed to understand the problem spe i ations, whi h have to be translated mathemati ally into so alled weighting fun tions. These weighting fun tions usually ree t (the inverse) frequen y responses of the desired losedloop transfer fun tions. With a little Bode shaping ba kground these weighting fun tions an be onstru ted. Controller synthesis is automated, so that the major a

ent lies on the weighting fun tion sele tion. This means that the design a tivity has moved from produ ing a ontroller towards produ ing spe i ations.

This in turn

requires the ability to physi ally interpret the requirements. For the longitudinal motion an energy study has been arried out to analyse the intera tion between airspeed and ight path. We would like to ontrol these 1

Delft University of Te hnology, fa ulty of Aerospa e Engineering,

Kluyverweg 1, 2629HS Delft, The Netherlands. E-mail: s.bennanilr.tudelft.nl

321

simultaneously through appropriate throttle and elevator ommands. Understanding these dynami s provided us guidelines on how to a hieve de oupled altitude and speed responses with minimum a tuator eort and a pilot like

o-ordination of the ontrols. For the lateral motion we have studied several trim options for the engine failure ase [157℄.

This study lead us to opt for the aerodynami ally best

solution: the zero-sideslip option. In the following se tion we dis uss the hoi e of the ar hite ture. In se tion 3 we show how to ree t the design requirements in a mathemati al format, suitable for design and analysis: the so alled inter onne tion stru ture.

In

se tion 4 we pro eed with the a tual ontroller design where we optimize the stru tured singular value

.

In se tions 5 and 6 simulations are presented. We

on lude the work in se tion 7.

22.2 Sele tion of Controller Ar hite ture The general breakdown of the ontroller ar hite ture is lassi al in the sense that separate ontrollers for the longitudinal and lateral motions are designed. The intention is to a hieve the spe i ations as stated in se tion 14.3 without mode swit hing or other logi . We design for a speed of 80 m/s. The two main tasks for both ontrollers are stability augmentation (innerloop ontrol) and guidan e along a ight path by tra king altitude, lateral position and airspeed referen es:

href , yref

and

VAref

(outer-loop ontrol).

22.2.1 Longitudinal ontroller The basi ar hite ture of the longitudinal ontroller is depi ted in gure 22.1.

ÆT ) and the throttle positions (ÆT H 1;2 ) Klon , on-

This system uses the tailplane dee tion (

to ontrol longitudinal air raft dynami s. The input to the ontroller sists of attitude angle

, pit h rate q

(providing for inner-loop bandwidth and

short period damping respe tively) and errors in airspeed and altitude and the rates thereof. Although



is not available as a measurement, we an estimate

it from other measured signals.

V_A

is approximated through omplementary

ltering of inertial and air data measurements [148℄. The two degrees of freedom lie in the appli ation of a feedba k ontroller and of feedforward lters, whi h generate smooth altitude and speed ommands. These lters are dened by the designer, while the feedba k ontroller is obtained from optimization via DK-iteration, as des ribed in se tion 8.4. The losed-loop system has to tra k the ltered altitude and speed referen es as tightly as possible. In fa t, the input lters represent ideal models of the losed-loop system. This ar hite ture allows the input lters to be repla ed by any other generator of smooth signals, like the traje tory generator in the evaluation pro edure, des ribed in se tion 14.3.3. Note that the feedba k ontroller ombines both inner-loop as well as guidan e fun tions.

322

.

h ref

feedforward, h

VA ref

feedforward, V

δT

K lon

δ TH1,2

hc hc

.

VAc VAc

. .

h h

-

RCAM VA VA (lon) Actuator, Engines

-

θq

Figure 22.1: Longitudinal ontroller ar hite ture

22.2.2 Lateral ontroller To ontrol the lateral motions in a o-ordinated way, a dierent ar hite ture is adopted, see gure 22.2.

φ ffw

yref

y

−

χ

+

K out

φc

βc

y

δA

K in (lat)

δR

RCAM

χ

(lat) Actuators

βprφ

Figure 22.2: Lateral ontroller ar hite ture

Separate inner-loop and outer-loop systems are designed. The inner-loop

ontroller provides stability augmentation and has to tra k roll and sideslip angle ommands. The inputs to the ontroller are roll and sideslip ommands

 , , , ) and roll and yaw rates (p, r), while the outputs ÆA , ÆR ). In most ight ases the sideslip angle will be regulated to zero ( = 0). Roll angle ommands are provided by and measurements (

are aileron and rudder ommands (

the outer-loop ontroller, whi h has the task of lateral guidan e. A o-ordinated turn is hara terised by a steady state roll angle if there is no wind. This ar hite ture allows a roll angle to be ommanded in a feed-forward path, improving tra king performan e in turns. The inner-loop ontroller also has a two-degrees of freedom ar hite ture, sin e full ommands and their orresponding feedba k signals enter the ontroller instead of error signals. This gives an extra degree of freedom in the

ontroller synthesis (se tion 8.4). The outer-loop ontroller minimizes lateral ight path errors via roll ommands to the inner-loop ontroller.

Feedba k of the tra k angle

damping of the lateral outer-loop dynami s.

323



improves

22.3 Translation of Design Criteria into Method Dependent Obje tives -framework,

This se tion des ribes the translation of design riteria into the

presented in se tion 8.3. We will on entrate on the the longitudinal ontroller design.

22.3.1 Design riteria in the general framework .

u u

(h VA y) ref

W pert

z

∆

w +

K

Act effort

n y nz

Act.

u

Comfort

α

Nominal Plant

Safety

Go

Set of Plants

(h VA y)

other feedback signals -

Tracking (h VA y) e

noise

Figure 22.3: Implementation of design requirements In the design spe i ations in se tion 14.3, performan e, safety, a tuator eort and omfort requirements are distinguished. As we will see, these an be put in a single performan e ve tor. To ree t the robustness problem the

nominal plant an be extended to a set of plants we design for, using the

z w, with w =
z ).

perturbation hannels ( ,

In gure 22.3 we see that all performan e variables an be observed by looking at the signals of interest at their physi al lo ations in the loop.

For

example, to wat h a tuator eort, we an pull out the ontrol signals at the servos. For safety, we an look at the angle of atta k. As a basis for

-analysis

as well as synthesis, this representation enables us to take are of nearly all obje tives simultaneously. The RCAM problem formulation implemented in the

-design

framework

an be divided into the following subproblems:

Robust Stability (RS) The losed-loop system should be stabilized for the spe ied set of plant models. For example: (input multipli ative un ertainty)

G~ (s) = fG0 (1 + Wpert (s)
) j k
k1  1 g where

G0 represents the linearized air raft and a tuator dynami s at the nom
is a norm bounded (diagonal) matrix and Wpert (s) is

inal design onditions,

324

a diagonal transfer fun tion matrix shaping the magnitude of the un ertainty at ea h a tuator input as a fun tion of the frequen y. The framework allows for implementation of many kinds of un ertainties, like parametri un ertainties arising in the state spa e matri es, see the design example in se tion 8.2.

Nominal Performan e (NP) The losed-loop system should satisfy the performan e spe i ations for the nominal plant:

G0 (s).

Robust Performan e (RP) This problem goes one step further: the losed-loop system should a hieve the required performan e level for the whole set of plants we wish to take into a

ount:

G~ (s).

Next, we transform the representation in gure 22.3 into the framework depi ted in g. 22.4. We pull the un ertain element




out of the system, as

well as the sub-system to be designed, the ontroller. To this end we dene two new sets of inputs and outputs to the nominal plant

P

in ludes

G0

P.

and all losed-loop transfer fun tions indu ed by all input

output signals on the nominal system

G0 .

∆ robustness tracking act. effort safety comfort

perturbations Nominal Plant

P

references disturbances

Κ Figure 22.4: Design requirements in a general framework Thus far, we have indi ated the losed-loop quantities of interest in a qualitative way. The next step is quanti ation of requirements on them by augmenting weighting fun tions. First, all input signals have to be normalized with respe t to their maximum value. In other words, we assume that the input signals have magnitude

1 and

need to be s aled to a suitable magnitude (relative to other inputs) before entering the system. In this way physi al system onditioning is provided. At the outputs of the performan e hannel we add weighting fun tions spe ifying the relative importan e of keeping the signal of interest small as a fun tion of the frequen y. In a same fashion we add lters to the inputs and/or outputs of


to shape the magnitudes of the perturbations. 325

The weighted losed-loop system now looks like:

M = Wout Fl (P; K )Win where

Win

and

Wout

ontain input and output weighting fun tions on the

signals, in luding s alings, performan e weights, perturbation lters, noiseshaping lters, referen e lters et . This is shown in gure 22.5. Finally we an apply the general theorems for analysis (se tion 8.3): 1. Robust Stability

(M11 (j!)) < 1 8 !

2. Nominal Performan e 3. Robust Performan e

(M22 (j!)) < 1 8 !

(M (j!)) < 1 8 !

In gure 22.5 the groups of signals involved in ea h of the theorems have been indi ated (RS, NP, RP).

RS RP NP

RS Nominal Plant

Wout

Win

RP

NP

P

K M = F l (P,K) Figure 22.5: Augmentation of weightings

22.3.2 Appli ation to the longitudinal ontroller In this se tion we will illustrate the prin iples des ribed above with the appli ation to the longitudinal ontroller design. The main performan e obje tive we address here is de oupled tra king of

r = [href ; VAref ℄T ), see gure 22.6.

speed and altitude referen es (

we

Wsout Wp e -

w

∆

z

Wid Ws in

r

Wpert

Go

u

K

+ y=e -

Figure 22.6: Simplied inter onne tion stru ture The ommands that enter the ontroller are smoothed by a (se ond order, diagonal) input lter matrix,

Wid .

We want the losed-loop system to tra k the

326

ltered referen es as lose as possible. This is ree ted by a diagonal weighting lter matrix

Wp , pla ed on the error signals between ideal model response and e = [he ; VAe ℄T ).

the a tual altitude and speed responses (

In order to normalize the referen e and output signals and to shape the intera tion level between the altitude and speed hannels, we use input and output s aling matri es

WSin

and

WSout .

Usually, we take

WSout = WSin1 .

Un ertainties in the model are ree ted by an input multipli ative un ertainty model:

(1 +
Wpert ).

This way of modeling un ertainties is somewhat

onservative. In a rened design we ould alleviate this by redu ing the magnitude of

Wpert

and onsidering parametri un ertainties instead, for example

for un ertainty in the mass and the entre of gravity lo ation. In order to apply the analysis theorems in se tion 22.3.1 we will rst derive the open-loop inter onne tion stru ture ne tion stru ture

M

P

and obtain the losed-loop inter on-

for a given ontroller

K.

Open-loop inter onne tion stru ture P First, we redraw g. 22.6 into g. 22.7 a

ording to the repartitioning depi ted in g. 22.5 so that the general analysis theorems and synthesis pro edures hold (se tion 8.3).

P are we (we = [whe wVe ℄T ).

The performan e hannels of weighted errors

given by the input signals

r

and the

We dis onne t the ontroller from the losed-loop stru ture and we write down the equations for the partitioned open-loop system system

P from g. 22.7.

∆

z we

w Wsout Wp

e

Wid Ws in

-

r

Wpert + -

Go y

u

P K

Figure 22.7: Transformation to the general system representation

2

z 4 we y

3 5

2

3

w P11 P12 4 5 r = P21 P22 u 2 0 0 = 4 WSout Wp G0 WSout Wp Wid WSin G0 Wid WSin 



327

(22.1)

Wpert WSout Wp G0 G0

32

3

w 54 r 5 u

Closed-loop inter onne tion stru ture To obtain the losed-loop stru ture

M = Fl (P; K )

we apply the denition of

the lower linear fra tional transformation:

M = Fl (P; K ) = P11 + P12 K (I P22 K ) 1 P21 1 , Ti = KG0 (I + KG0 ) 1 [159℄ and the relation: With: So = (I + G0 K ) K (I + G0 K ) 1 G0 = (I + KG0 ) 1 KG0 , we substitute eq. 22.1 and nd:   W T W KS W W pert i pert o id Sin M= WSout Wp So G0 WSout Wp So Wid WSin Robust Stability As we an see from

M,

robust stability is hara terized by a weighted input

omplementary sensitivity

(Wpert Ti ) < 1.

M11 = Wpert Ti .

For robust stability we require

Nominal Performan e The performan e hannel is hara terized by a weighted output sensitivity

WSout Wp So Wid WSin .

Now let us on entrate on the spe i ation of de ou-

pled tra king of altitude and speed ommands. The performan e loop in more detail is:



weh weV



"

weh weV



"

Wouth Wph hhe Widh Winh Wouth Wph VhA e WidV WinV = WoutV WpV VhAe Widh Winh WoutV WpV VVAe WidV WinV A 1: Usually we take WSout = W 

Sin

Wouth Wph VhAee WidV WinV Wph hhe Widh = V Ae WoutV WpV h Widh Winh WpV VVAe WidV Ae

#

#

href VAref



href VAref



We an see that with this hoi e of input and output s alings only o-diagonal

(Winh ; WinV ) = diag(1=10; 10), and diag(Wouth ; WoutV ) = diag(10; 1=10), the diagonal terms remain un hanged while one oterms are ae ted. Taking diag

diagonal term is amplied and the other is redu ed, both with a fa tor 100. In this way we an inuen e intera tion between the hannels in ontroller synthesis. We an now formulate the nominal de oupled tra king performan e ondition as:

kWSout Wp So Wid WSin k1 < 1.

Robust Performan e Finally, to obtain the robust performan e ondition for this problem, we require

(M ) < 1.

From se tion 8.3.4 we know that this is equivalent to:

(M ) = inf  D



Wpert Ti DWpert KSoWid WSin WSout Wp So G0 D 1 WSout Wp So Wid WSin 328



An upperbound for

(M )  If

q

(M ) is:

[266℄

(WSin1 Wid 1 G0 )(kWpert Ti k + kWSout Wp SoWid WSin k)

WSin Wid = I , the a hievable robust performan e level is proportional to the (G0 )). By properly hoosing

square root of the ondition number of the plant (

WSin ,

we an improve the robust performan e level and provide for physi al

system onditioning.

22.4 Controller Design Cy le 22.4.1 Introdu tion In this se tion the design of the ight ontroller for RCAM is presented. In se tion 22.2 the ar hite ture of the ontroller is des ribed. Three subsystems are designed using



-synthesis:

a longitudinal ontroller, with integrated inner-loop and outer-loop fun tions;

 

a lateral inner-loop ontroller, for tra king roll and sideslip ommands; a lateral outer-loop ontroller, for lateral guidan e.

The design work for these ontrollers onsists of the following steps:



hoose air raft onguration and ight onditions for the design point and obtain a linear model for that point;



P ), onsisting

dene a lay-out for the general inter onne tion stru ture (

of the plant model, ontroller, weighting fun tions, modeled un ertainties;



iteratively synthesize and analyse ontrollers and adjust weighting fun tions. Analysis is based on

-plots, ontroller properties (e.g.

frequen y

responses, order) and losed-loop time responses;



preparation of the ontroller for implementation: removal of fast poles, balan ing, order redu tion;



implementation with the nonlinear air raft model to perform time simulations.

Of ourse these steps together are also arried out in an iterative way. The longitudinal ontroller will be des ribed in detail. For the lateral designs the reader is referred to [25℄. lab/Simulink with the

The design work is arried out in Mat-

-Analysis and Synthesis Toolbox (-Tools) [18℄. 329

22.4.2 Sele tion of the design model Some basi knowledge of the longitudinal air raft hara teristi s an be obtained from trim urves. These depi t stati (longitudinal) ontrol dee tions, required for horizontal equilibrium ight, as a fun tion of the airspeed. The most interesting urves an be found in gure 22.8 (For a more detailed analysis of the RCAM hara teristi s and performan e, see [157℄). a) Trim curves tailplane, varying Xcg

b) Trim curves throttles, varying mass

20

15 0.15c

−DT (deg)

DTH1+DTH2 (deg)

m = 120000 kg

15 0.23c 10 0.31c 5

150000 kg

10

120000 kg

Xcg = 0.23c 100000 kg

0 50

60

70

80 90 VA (m/s)

100

5 40

110

50

60

70

80 VA (m/s)

90

100

110

Figure 22.8: RCAM trim urves From gure 22.8(a) we an see that for a forward entre of gravity lo ation the required tailplane dee tion be omes quite large at lower airspeeds. Stati longitudinal stability is guaranteed, sin e always

dÆT =dVA > 0 [35℄.

More inter-

esting are the throttle trim urves: these show a minimum, whi h o

urs at the

minimum drag speed,

VMD (whi h depends on air raft onguration and mass). VMD , it is ying on the ba kside of the

If the air raft is ying at speed below

power urve: a lower trim speed requires a higher throttle setting. For more details, see [35℄ and [157℄. For an air raft equipped with separate autopilot and autothrottle systems, this results in so alled speed instability, when only one of these systems is engaged [146℄. This is a strong motivation for a multivariable approa h, as is

-synthesis, to design an integrated autopilot/autothrottle

system. We design for a speed of 80 m/s. The nominal design model is obtained by trimming and linearizing the RCAM in a horizontal ight at a speed of 80 m/s, a mass of 120 000 kg, a verti al CoG position of 0



 and a horizontal CoG

lo ation of 0.23 . An impression of the model hara teristi s of the model an be obtained by looking at damping, eigenfrequen ies and time onstants: mode

frequen y (rad/s)

damping

time onstant

phugoid

0.1269

0.0898

-

short period

1.3837

0.6000

-

Dut h roll

0.6404

0.3684

-

Spiral

-

-

5.4449

aper. roll

-

-

0.7682

All modes are stable. The Dut h roll and the short period modes show reasonable damping while naturally, the phugoid is badly damped.

330

22.4.3 The longitudinal ontroller As a rst step in

-synthesis design, an inter onne tion stru ture is set up.

The

stru ture for the longitudinal ontroller is shown in gure 22.9. We will have a

WS out

Wp

h h VA VA Wp2

+ -

h C h C VA CVAC Wact

1 0

RCAM 1 (lon) 0

uu

w

act. eng.

∆

z

Wid

Wpert

δ T δ TH1,2

+ 11 00 11 00

K

noise

qθ

nz q

h nom VAnom

WS in

+ +

Wn

n

h h VA VA Figure 22.9: Inter onne tion stru ture for longitudinal ontroller

loser look at ea h of the blo ks in the diagram.

RCAM (lon): This blo k ree ts the linearized longitudinal air raft dynami s. The general stru ture has been dis ussed in se tion 22.2.1.

a t./eng. : Tailplane a tuator and engine dynami s are modeled as rst order lters:

HÆT (s) = 0:151s+1

and

HÆT H1;2 (s) = 1:51s+1

K: This blo k ontains the ontroller to be designed.

Wpert :

This diagonal weighting fun tion matrix is a rude way to a

ount

for un ertainties in the air raft model, without addressing a spe i un ertain parameter. With the loop losed via the omplex diagonal matrix


, k
k1  1

a set of models has been dened for whi h we want to guarantee stability and to a hieve our performan e spe i ations.

For


=0

we have our nominal

model. We assume that the set of models is large enough to a

ount for several un ertainties in the model, like un ertain a tuator responses, un ertain aerodynami parameters, time delays, entre of gravity shifts and mass variations. The

-synthesis methodology enables us to address these perturbations as para-

metri un ertainties. This an be done in a rened design. The weighting fun tions on the diagonal of

Wpert have a low value in the lower

frequen y range, while in the mid-frequen y range the magnitude in reases. Another ee t of this shape is that the ontroller is for ed to roll o at higher frequen ies (refer to se tion 8.3.6).

Wpert

is a diagonal matrix:

Wpert = diag(WpertT ; WpertT H1;2 ), with:

5s+1 s+1 WpertT H1;2 (s) = 0:25 s=0:200+1 WpertT (s) = 0:25 s=200+1 In gure 22.10 the frequen y responses of are determined after a few iterations.

331

Wpert

are given. These weightings

5

10

Wp (h,VA) mag

throttles 0

tailplane

10

Wpert

−2

10

−1

10

0

1

10 10 frequency (rad/s)

Figure 22.10: Frequen y responses of

Wid : Wid

2

3

10

Wp

10

and

Wpert

generates speed and altitude ommands and the rates thereof (se -

tion 22.2.1). The speed and altitude ommands are related to resp.

href (s) as follows:

Widh (s) =

VAref

and

V (s) 0:152 h (s) = A = 2 href (s) VAref (s) s + 2  1  0:15s + 0:152

The lters represent the desired dynami s of the losed-loop system with bandwidth and damping as ommonly adopted in autopilot design [132℄. Choosing the bandwidthes equal results in de oupled speed and altitude

ontrol with lowest throttle a tivity. This an be explained from point-mass energy onsiderations. We will give more details in se tion 22.5.1.

Wp :

Up to frequen ies beyond the bandwidth of the ideal model lters, the

dieren es between the altitude and speed responses and respe tively

VA

h

and

should be small. This requirement is ree ted by the diagonal weighting

matrix

Wp .

At lower frequen ies

Wp

is small, while it rolls o at higher fre-

quen ies. For both the altitude and speed hannels, the weighting fun tions are taken as

 1=50s + 1 2 : Wph (s) = WpV (s) = 15000 1=0:015s + 1 

The weights of the rates are set to zero so that the resulting weighting for the tra king error is given by state errors the loop gain

Wp =

(Wph ; 0; WpV ; 0).

diag

For small steady

jGK (j!)j of the system has to be large in the lower

frequen y range. This an be a hieved by in reasing the low frequen y gain of the these weighting fun tions. By moving the pole to the origin and in reasing the gain, the ross-over frequen y is held onstant. This is dis ussed in more detail in [25℄. In gure 22.10 the frequen y responses of

Wp

are given. Note

that weighting ontains approximate double integration. From se tion 8.3.6 we know that the ontroller will at lower frequen ies have approximately the same shape. This enables the ontrolled system to tra k ramp ommands with very small steady state error.

332

WSin

and

WSout :

We motivated in se tion 22.3 that we have to normalize the

h max = 5 m and the VA max = 1 m/se , WSin = diag(h max; VA max). Wp2 : Wp2 = diag(Wpnz ; Wpq ). This weighting is applied in order to keep referen e inputs.

The altitude ommand is s aled to

speed ommand to

ontrol over the pit h rate and over the normal a

eleration. Good tra king of the feedforward lter outputs should not be at the ost of extreme pit h rates.

W = 57:3=0:5

; the The weight is onstant over all frequen ies and is set to pq pit h rate should never ex eed 0.5 deg/s for the ommand levels adopted in the inter onne tion stru ture.

1=0:02. Wa t : This

Normal a

eleration

nz

is weighted with

Wpnz =

weighting fun tion is diagonal and puts onstant weights on the

Wa t is set to:  1 ; 1 ; 1 1 Wa t = diag ÆT (max ; _ _ ) ÆT (max)
ÆT H1;2 (max) ÆT H1;2 (max) = 57:3 diag 101 ; 151 ; 21 ; 11 ; rad 1

ontrols and ontrol rates.



The maximum dee tions and rates are within the limits spe ied in se tion 14.2.5.

Wn :

Of ourse, measurements will always be orrupted with some noise.

Wn

is again diagonal, onsisting of the following weightings:

:01s+1 Wnh = 5  10 4 11==0500 Wnh_ = 0:02 s+1 :01s+1 W Wnv = 2:5  10 4 11==0500 nv_ = 0:02 s+1 Wn = 0:05=57:3 Wnq = 0:025=57:3

22.4.4 Controller synthesis Based on the inter onne tion stru ture dened in se tion 22.4.3, we optimize

 via DK-iteration (see se tion 8.4). -Tools [18℄.

the performan e robustness index purpose routines are available in The

-plots are the most important indi ators in the design pro ess:

give information about whi h weightings need to be adjusted next. longitudinal ontroller the of



For this

-plot an be found in gure 22.11.

they

For the

The peak value

(robust performan e) along the frequen y axis has been minimized via

DK-iteration.

Robust performan e is not a hieved:

(M ) > 1. Stability is (M11 ) < 1).

guaranteed for the whole adopted set of plants (robust stability:

Nominal performan e is not a hieved in this ase, but onsidering only tra king, we have met our spe i ations. The lateral inner-loop and outer-loop ontrollers are designed following the same pro edure.

The resulting ontrollers have high order so that order re-

du tion is ne essary. For this purpose we use balan ed Hankel-singular value trun ation. Prior to nding a balan ed realization, we have to make sure that all ontroller poles are stable. Very fast poles that sometimes o

ur, have to be removed before running simulations. The ontrollers are always he ked for su ient roll o to prevent the ontrol system from ommanding high-frequen y signals to the a tuators.

333

Stability/Performance Indicators

1.6 1.4

Robust Performance

1.2

mag

1 Nominal Performance

0.8 0.6 0.4 0.2

Nominal Tracking Performance

Robust Stability

0 10−2

10−1

Figure 22.11:

100

101 freq rad/sec

102

103

-plots for the longitudinal ontroller

The ontroller order after redu tion is still high. For the longitudinal system 24, for the lateral inner-loop ontroller 24 and for the lateral outer-loop system 8. Mu h lower orders an be a hieved with more advan ed redu tion methods [187℄.

22.5 Analysis of the Resulting Controllers 22.5.1 Longitudinal ontroller For the longitudinal system we on entrate on tra king performan e and de oupling of the altitude and speed hannels and on minimum ontrol a tivity. An interesting way to evaluate speed ight path de oupling and throttle a tivity is to look at energy prin iples. Considering the air raft as a point-mass, we an write the total energy state (sum of potential and kineti energy) as follows:

1 Etot = Epot + Ekin = mgh + mV 2 2 (we assume there is no wind) Taking the time derivative, dividing by and with

h_ = V sin , we get the spe i energy rate: V_ E_ s = V sin + g

W = mg

!

(22.2)

From equilibrium along the ight path we an derive: [35℄

T Where

T

and

D

W

D

=

V_ E_ + sin = s g V

are resp. thrust and drag.

(T

D)=W

(22.3)

is alled the spe i

ex ess power. Sin e air raft are mostly own near their minimum drag speed (in that ase

dD=dV

 0), thrust gives dire t ontrol over the spe i energy 334

rate: an in rease in thrust is fully available for in reasing the ight path angle and/or the a

eleration. The distribution between the two an be ontrolled by the (energy onservative) elevator. An interesting way to see whether the ontroller responds a

ording to these prin iples, we an dene an energy ex hange manoeuvre. We ommand altitude and speed steps so that potential and kineti energy respe tively de rease and in rease with the same amount (or vi e versa). Sin e we have hosen identi al feedforward lters for both hannels (see se tion 22.4.3), the manoeuvre. This means that

E_ s

V_ =g  sin during

 0 and therefore in the ideal ase the

throttles should not respond. Sin e the air raft is not a point-mass, we expe t some response, but this still should be low. The energy ex hange is ontrolled with the (energy onservative) tailplane. The prin iple of using throttles for energy input and elevator for energy distribution is the basis for the Total Energy Control System (TECS), as des ribed in [146, 148℄.

altitude response

speed response

0

2.5 2 V (m/s)

h (m)

−5 −10 −15 −20 0

1.5 1 0.5

20

40

0 0

60

20

time (s) throttle input 0.02 contr. ad−hoc 6

0.01 Edot_s (−)

DTH1,2 (deg)

60

Spec. energy rates

8

4 2 0 0

40 time (s)

0 −0.01

20

40

−0.02 0

60

time (s)

vdot/g Edot_s

sin(gamma) 20

40

60

time (s)

Figure 22.12: Energy manoeuvre responses We now ommand a in rease speed (so that

20 m altitude hange and a orresponding 2:42 m/s
Ekin =
Epot , starting at VA = 80 m/s). In

gure 22.12 the nonlinear simulation results are plotted. For omparison also the throttle response is given of an "ad-ho " (in this ontext) design. In the third sub-gure an be seen that the throttle response is indeed negligible ompared to the ad-ho  ontroller. In the fourth gure the spe i energy rate and its ontributions are depi ted. The energy rate remains very small, while indeed

V_ =g  sin .

335

22.5.2 Lateral ontroller For the lateral design we show responses of the linear model to a lateral position step ommand, see gure 22.13a. The outer-loop ontroller produ es a roll angle ommand, whi h has to be tra ked by the inner-loop ontroller. From gure 22.13b we an see that this o

urs in a very smooth way. The sideslip response is small, indi ating su ient de oupling of the roll and sideslip hannels.

12

0.3

10 PHI, BETA (deg)

0.2

Y (m)

8 6 4

0.1 BETA

0 PHI_C

PHI

−0.1

2 0 0

20

40 time (s)

60

80

−0.2 0

a) Lateral step response

20

40 time (s)

60

80

b) Inner-loop responses

Figure 22.13: Lateral ommand responses

For a more detailed analysis of the ontrol system, we refer to the design report, [25℄.

22.6 Results of the Automated Evaluation Pro edure In this se tion we present simulation results and the a hieved performan e of the omplete ontrol system when implemented in the automated evaluation mission and s enario, as dened in se tion 14.3.3. Both overall tra king performan e and inner-loop behaviour of the ontrolled system will be evaluated by means of bounds on key variables. We will fo us the dis ussion on the four separate ight segments dened in se tion 14.3.3. A summary of the numeri al results of the evaluation will be given at the end of this se tion.

Segment I: the ee t of engine failure As the RCAM air raft model is twin-engined, a single engine failure initially results in a lateral deviation.

Figure 22.14 provides a top view of the rst

traje tory segment. The given bounds indi ate an a

eptable level of performan e. The ontrol system behaves very well in this situation. We did an extra simulation without wind disturban es, to obtain insight in how the ontroller responds to an engine failure, see g. 22.15. The ontrol dee tions are smooth and the air raft is trimmed to the new situation.

336

First segment: top view

15 THROTTLE2

10

100

control inputs (deg)

x−deviation [m]

5 50

0

0

a

b

1

−50

THROTTLE1 0 DA

−5

DT

−10 −15

DR

−20

−100 −20

−18

−16

−14

−12 −10 −8 y−position (−YE) [km]

−6

−4

−25 0

−2

20

40

60 time (s)

80

100

120

Figure 22.14: Segment I: the ee t

Figure 22.15: Segment I: ontrol in-

of engine failure

puts during engine failure

Segment II: the 3 deg/s turn At the beginning of the turn there is a deviation from the referen e traje tory; the air raft needs some time to a hieve the bank angle required for a

o-ordinated turn [35℄. At the end of the turn the opposite o

urs: some time is needed to return to the initial attitude. Figure 22.16 provides a loser look at the lateral deviations. During the turn, the lateral error is de reased slowly.

Third segment: altitude deviations 30

200

20 altitude deviation [m]

lateral deviation [m]

Second segment: lateral deviations 300

100

0 1

c

d

2

−100

−200

−300 0

10 2

f

3

0

−10

e

−20

1

2

3 4 5 6 along track distance from point 1 [km]

Figure 22.16:

7

−30

8

Segment II: lateral

−16

−15

−14 −13 x−position (XE) [km]

−12

−11

Figure 22.17: Segment III: verti al

deviations during the 3 deg/s turn

deviations from the desired glideslope

Segment III: the apture of the -6 and -3 degrees glideslope The nal approa h starts with a glideslope of -6 deg and after some time this is in reased to -3 deg. The longitudinal ontroller is implemented without the feedforward lter for altitude (g. 22.1), sin e the traje tory generator in the

h

software produ es smooth referen es for altitude ( ). Sin e we required tight tra king of the signals from this feedforward lter (se tion 22.4.3), the verti al deviations from the referen e ight path depi ted in gure 22.17 are very small.

337

Segment IV: the nal approa h with windshear Along the glideslope of -3 deg a windshear with downdraft is en ountered. Equation 22.3 gives the spe i ex ess power for zero-wind onditions.

In a

windeld, we would like to onsider the the air raft kineti energy relative to the airmass. With this in mind, we an a

ount for wind disturban es by

orre ting eq. 22.3 with the so- alled

T where

W

D

F -fa tor [180, 179℄:

V_ = A + sin + F g

horizontal F = W_ Xg E os a W_ gZE sin a + WVZAE : WXE and WZE are resp. 

and verti al wind omponents (in verti al omponent of

Va

in

FV ).

FE )

and

A positive

F

waV

, (

is the

has a performan e de reasing

(T

ee t, sin e a part of the spe i ex ess power this term.

waV VA

a = ar sin

D)=W

is onsumed by

Sin e the airspeed and ight path referen es are onstant, thrust

F (we assume D  onstant).
T=W =
(ÆT H 1 + ÆT H 2 )  F .

must hange only to an el gure 22.18 where indeed

This is depi ted in

The windshear en ountered along the ight path learly is not beyond the performan e limits of RCAM [157℄. In g. 22.18 we see that thrust is initially de reased.

In severe windshear this is very undesirable, sin e in this phase

the air raft should in rease its energy level in order to y through the ore of the windeld without loosing too mu h airspeed and altitude. The verti al deviations from the desired glideslope are plotted in gure 22.19: the deviations are small sin e the wind disturban e is ompensated by the thrust. Fourth segment: altitude deviations

20

F−factor

0.1

altitude deviation [m]

F−fact. (−), THROTTLE 1+2 (rad)

30

throttle 0

10

3 g

4

h

0

−10

−20

−0.1 0

10

Figure

20

30

40

50 60 time (s)

22.18:

windshearF -fa tor

70

80

90

Segment and

−30 −11

100

−10

−9

−8

−7 −6 −5 −4 x−position (XE) [km]

−3

−2

−1

IV:

Figure 22.19: Segment IV: verti al

throt-

deviations from the desired glide-

tle response

slope

Numeri al results Table 22.1 gives numeri al results based on the dis ussed simulation results. For the motivation and al ulation prin iple of the various numbers see se tion 14.3.3.

338

Segm. I

Segm. II

Segm. III

Segm. IV

Total

Performan e

0.1305

0.1626

0.0511

0.0903

0.1086

Perf. Dev.

0.0293

0.0215

0.0371

0.0837

0.0429

Comfort

0.4204

1.1386

1.6027

0.6393

0.9502

Safety

0.0048

0.1038

0.0109

0.0448

0.0411

Power

0.0042

0.0058

0.0152

0.0302

0.0138

Table 22.1: Numeri al results of the evaluation pro edure

22.7 Con lusions In the previous se tions the appli ation of

-synthesis for a ight ontrol system

has been dis ussed. Blo k diagrams depi ting the ontroller ar hite ture an be found in gure 22.1 and 22.2. Most eort was put in a hieving good performan e. We designed with general (input-multipli ative) un ertainty models, whi h appeared to be su ient to over variations in mass, enter of gravity lo ation and un ertainty in time delay. We designed for a speed of 80 m/s. In order to simplify the system we avoided mode swit hing. Performan e of the designed ontroller is satisfa tory in terms of tra king and de oupled responses. A great feature of the methodology is that we an a

ount simultaneously for un ertainties in multiple spe i parameters (e.g. mass,

X g ).

The

 analysis

gives us bounds on the magnitudes of the un ertainties whi h an be tolerated until instability o

urs. The term robustness should in this view be used with

are. The a hieved robustness is in fa e of the addressed un ertainty level. The plant, ontroller, modeled un ertainties, performan e weights, disturban e models et .

all t in a single framework: the general inter onne tion

stru ture (e.g. gure 22.9) and a suitable matrix measure namely the stru tured singular value. This framework easily opes with:

  

the multivariable nature of the plant; dierent kinds of modeled un ertainties; simultaneous performan e spe i ations.

As an analysis tool, it enables the designer to assess the performan e and robustness level of a ontroller in fa e of the modeled un ertainties. On the other hand, a ontroller an be synthesized from the very same framework.

As a

synthesis tool, the framework allows the designer to trade-o between performan e and robustness obje tives and between dierent performan e obje tives. Design work onsists of tuning weighting fun tions, while ontroller synthesis is automated. A short oming of the method it that the resulting ontrollers are of the order of the s aled inter onne tion stru ture. Although ommer ially available model redu tion te hniques give some relief these do not provide satisfa tory redu tion levels to be a

eptable for industrial appli ations as this one. However, mu h

339

eort has been made and en ouraging results have been provided by Pa kard etal [187℄. The redu tion problem has been approa hed from a and in orporated in the original

-synthesis

problem.

 perspe tive

The redu tion levels

reported are doubled without loosing the a hieved performan e levels. For our problem this would mean that it is possible to easily obtain ontroller orders of six to eight with the proposed te hnique. As an indi ator for stability,



is generally appli able to assess stability

robustness of a ontrolled plant in fa e of modeled un ertainties. Any (linear)

ontroller designed for this plant an be onne ted in the loop and analysed [158, 245℄.

 is a robust performan e indi ator as well.

This requires quanti ation be-

tween requirements (e.g. in terms of nonlinear simulations) and weights. From the design experien e with RCAM we feel that a more fundamental approa h to this quanti ation needs to be developed. Beyond

,

mu h progress has been made in the robust ontrol area. The

potentials of using Linear Matrix Inequalities (LMI's) for ontroller analysis and synthesis have been widely re ognized. Based on LMI's algorithms have been developed that an nd robust gain-s heduled ontrollers, taking not only un ertainties into a

ount, but also nonlinearities and the time varying nature of the plant [260, 13, 112, 257, 185℄.

340

23.

A

-Synthesis Approa h

(2)

Jan S huring - and Rob M.P. Goverde 1 2

Abstra t.

3

Robust ontrol theory deals expli itly with the un er-

tainties in the hara teristi s of the air raft and in the environment in whi h it has to operate. A modern approa h to robust ontrol is

H1 optimal ontrol.

Generally, un ertainty in a model is present

at various omponents and is in this way highly stru tured. The in-

uen e of the stru tured un ertainty on system performan e an be analysed by applying of

-analysis.

A ombination of the te hniques

H1 optimal ontrol and -analysis is known as -synthesis. This

method has been applied to the design of an Autopilot for RCAM,

-synthesis inner-loop ontrollers. The formal riterion of the -synthesis method ould not be met by these ontrollers. However, using

the resulting design performs rather well.

23.1 Introdu tion Stability and ontrol is one of the major te hni al hallenges in the design of an air raft. Flight ontrol systems must fun tion properly in all ight onditions and air raft states in order to ensure safety of ight. The air raft model is a very omplex system with lots of un ertainties due to unmodelled dynami s, parameter un ertainty, sensor noise, a tuator errors, et . Also the intera tion between the air raft and its environment has to be dealt with, in luding disturban es su h as wind shears, gusts, and turbulen e.

Furthermore, spe ial

situations su h as engine failure may be taken into a

ount. Besides providing stability and ontrol to air raft, ight ontrol systems play an in reasingly important role in meeting ost and performan e obje tives for modern ivil and military air raft. Robust ontrol theory oers a systemati approa h to design and evaluate ight ontrol systems that fun tion properly for all relevant ight onditions and air raft states. It deals expli itly with the un ertainties in the hara teristi s of the air raft and in the environment in whi h it has to operate.

1

National Aerospa e Laboratory NLR, Anthony Fokkerweg 2, 1059 CM Amsterdam, The

Netherlands. E-mail s huringnlr.nl, fax +31 20 511 3210. 2

The NLR ontribution to GARTEUR A tion Group FM(AG08) was funded by the

Netherlands Agen y for Aerospa e Programs. 3

Delft University of Te hnology, Fa ulteit der Civiele Te hniek, Se tie Verkeerskunde Stev-

inweg 1, Delft, The Netherlands.

341

A modern approa h to robust ontrol is

H1 optimal ontrol theory.

This

method an be interpreted as the minimization of the transfer matrix of the system in a worst ase s enario. The resulting ontrol law guarantees stability of the losed-loop system and a hievement of the obje tives for the whole lass of systems indu ed by the un ertainty (i.e. robust performan e), if a ertain inequality is satised. Throughout the system, un ertainties may be present. Instead of modelling these un ertainties as unstru tured as in a worst ase s enario, the un ertainties at the various omponents an be taken into a

ount one by one expli itly, and rearranged to form a stru tured un ertainty at the system level. A te hnique to measure robustness, taking expli itly into a

ount the stru tured un ertainty,



is stru tured singular value ( ) analysis. A ombination of the te hniques of

H1

optimal ontrol and

-analysis

provides an approa h to design and analyse robust ontrol systems. This design approa h is alled

-synthesis.

It is des ribed extensively in Chapter 8.

Variations in the behaviour of a system, aused by ertain varying parameters, an be modelled as stru tured un ertainty in the system using parametri

un ertainty modelling, making extensive use of the theory of linear fra tional transformations. In the se tions to follow a design proje t is des ribed, illustrating a pro edure to deal with variations in many parameters. The design will be distributed over a

ontroller, a

-synthesis

-synthesis

longitudinal inner-loop

lateral/dire tional inner-loop ontroller and a hand-

tuned outer loop ontroller. A more detailed des ription of the proje t an be found in [208℄.

23.2 Control Con ept 23.2.1 Introdu tion A six degrees-of-freedom non-linear Resear h Civil Air raft Model (RCAM) in luding a tuators and a model of disturban es have been des ribed in Chapter 14. The air raft has two engines, to be operated in unison. An Autopilot shall be designed, a

ording to the spe i ations in Chapter 14. The Autopilot has to be robust to a number of parameter variations, a

ording to the spe ied operating envelope, listed in Se tion 23.3.2. Owing to the de oupling property of the air raft dynami s into symmetri and asymmetri dynami s, the ontrol system design an be divided into longitudinal and lateral/dire tional ontroller designs. Moreover, a division will be made into inner and outer loop ontrollers. The inner loop ontrollers, designed by

-synthesis, will provide de oupled

robust ontrol of as many ontrolled variables as there are ontrol ee tors. The outer loop ontrollers have to provide the link between the spe ied Autopilot fun tions and the ontrolled air raft.

342

Owing to the de oupled inner-loop ontrol, the outer-loop ontroller only has to a t as a number of single-input, single-output ontrollers, redu ing omplexity onsiderably. Therefore, its parameters an be tuned by hand to satisfy the spe i ations.

23.2.2 Longitudinal inner loop ontroller The following ontrolled variables (equal in number to the number of available longitudinal ontrol ee tors) have been hosen:

VA wV

airspeed,

z omponent of air raft velo ity in FV .

The ontrolled variables shall have responses resembling those of an ideal model dened in Se tion 23.3. Additionally, the ontroller will use the pit h rate for damping purposes. This will lessen the burden of the inner loop ontroller to estimate this quantity.

23.2.3 Lateral/dire tional inner loop ontroller The following ontrolled variables have been hosen:

vV

angle of sideslip,

y omponent of air raft velo ity in FV . The variable vV represents the lateral velo ity w.r.t.

the ommanded heading.

For an ommanded heading dierent from zero, the a tual lateral velo ity w.r.t. the a tual ommanded heading has to be al ulated from

C , vV

and

uV .

The ontrolled variables shall have responses resembling those of an ideal model dened in Se tion 23.3. Additionally, the ontroller will use the roll rate, the yaw rate and the roll angle, for damping purposes only.

23.2.4 Longitudinal outer loop ontroller In the outer loop ontroller the ommands for the inner loop are reated. The

ommanded airspeed redu ed by the trim speed, is put through. The ommanded verti al speed is onstru ted from altitude error, referen e verti al speed and a tual verti al speed, all provided with appropriate gains.

23.2.5 Lateral/dire tional outer loop ontroller In the outer loop ontroller the ommands for the inner loop are reated. As no other requirement for the angle of sideslip has been given than to minimize it, a zero ommand has been applied. For de rabbing purposes during landing, it may be used to ontrol air raft heading, to line the air raft up with the runway dire tion. The ommanded lateral speed is onstru ted from lateral deviation, referen e lateral speed, a tual lateral speed (w.r.t.

ommanded heading) and

referen e heading rate, all provided with appropriate gains.

343

parametri un ertainty modelling

-

?

sele tion weights

?



reation standard plant

?

H1 synthesis

-

D -K



iteration

tuning weights

-analysis

?

ontroller order redu tion

?

losed-loop analysis

?

non-linear simulation

?

end

Figure 23.1: The ontrol system design y le.

23.3 RCAM Controller Design 23.3.1 Control system design y le Figure 23.1 illustrates the generi ontrol system design y le. The rst step

ontains the parameter un ertainty modelling using the theory of LFTs (Chapter 8). Ea h linear system model onsists of four system matri es. For ea h element of these system matri es, the minimum and the maximum value o

urring over all operating points onsidered is established. From these values a mean value and a range for ea h element is al ulated.

The elements are

lassied based on mean and range: when both are below a ertain toleran e, the element is a zero; when only the range is below that toleran e the element is a onstant; otherwise it is a variable. Applying the tools from the Parametri Un ertainty Modelling Toolbox [144℄ a Linear Fra tional Transformation is al ulated from the mean, range and las-

344

perturbation

noise

ommands






input

--

-

turbulen e

-

ideal

perturbation output

air raft

-

model

-+ ? -

eort weighting

performan e

model

ontroller

ontrol

weighting



-

weighted

-

eort

weighted errors

measurement

Figure 23.2: Inter onne tion stru ture for ontroller design.

si ation matri es. The LFT represents a omplete set of linear system models. It ontains the mean model and oe ients, oupling the normalized perturbations into the model.

The perturbation matrix is a diagonal matrix with

normalized elements ranging from

1

to

1.

An all zero perturbation matrix

will result in the mean model. Any linear model in the respe tive set an be re onstru ted from the LFT by a parti ular ombination of perturbation element values. In order to apply the

-synthesis method an inter onne tion stru ture has

to be reated in the standard plant format, see Figure 23.2. It in orporates both the air raft LFT and the environment of the air raft. The air raft environment

onsist of disturban es (turbulen e), an ideal model that shall be mimi ked by the ontrolled air raft, and weightings (eort and performan e weighting). The

omplete standard plant again is formulated as an LFT. The tuning of weighting fun tions is a manual pro ess, based on notions of physi al relevan e of the weighting parameters, in view of the way the resulting time responses of the ontrolled air raft satisfy the ontroller spe i ations. This tuning pro ess forms the outer loop of Figure 23.1 (not to be onfused with a ontroller outer loop). It involves a heavy mental eort of the designer. The

-synthesis pro ess is applied to the standard plant. -Synthesis is an

iterative pro ess, hara terized by the D-K iteration. Generally 3 or 4 iterations are su ient. This iteration forms the inner loop of Figure 23.1. It is a more or less automated pro ess.

Only the D-s aling part of it involves manual

345

intervention. The resulting ontroller generally has a high order. Moreover, its eigenvalues may have absolute values that well ex eed pra ti al limits. redu tion and residualisation an be advantageous.

Therefore order

Of ourse, the intended

purpose of the ontroller has to be preserved in these operations, in terms of stability and performan e. Finally, response al ulations of the ontrolled air raft shall reveal if the weightings have been hosen properly. If not, the

-synthesis pro ess has to be

repeated for an updated standard plant.

23.3.2 Operating envelope The spe i ation envelope is spanned over a number of operating points dened by all ombinations of the parameter values given below: Parameter

Nominal Value

Lower Bound

time delay

0.075 s

0.05 s

0.1 s

120 000 kg

100 000 kg

150 000 kg

air raft mass longitudinal CoG verti al CoG

0.23 0.1





0.15 0





Upper Bound

0.31 0.21





The spe i ation envelope is used in ontroller design. An impli ation of the mass sele tion has to be mentioned. Sin e the design airspeed has been dened as 1.23

Vstall , the design speed depends on air raft mass: Mass

Air speed

100 000 kg

58.1 m/s

120 000 kg

63.7 m/s

150 000 kg

71.2 m/s

23.3.3 Trimming, linearisation and submodel sele tion In order to apply parametri un ertainty modelling, the non-linear RCAM is trimmed and linearised in ea h element of an array of 81 operating points, spanning the operating envelope dened in Se tion 23.3.2. The air raft model has been augmented by the a tuator dynami s and the time delay (modelled as a Padé lter to enable linearisation). From ea h omplete linear model two submodels are sele ted for the longitudinal and the lateral/dire tional dynami s respe tively. This results in two sets of submodels with mating trimming onditions. These are used in parametri un ertainty modelling and in linear simulation.

23.3.4 Parametri un ertainty modelling The method presented applies equally to the longitudinal and the lateral/dire tional submodel. The ontents of ea h submodel over the spe i ation envelope depend on time delay, mass, longitudinal CoG lo ation, and verti al

346

CoG lo ation at the same time. Therefore we have to deal with a 4-dimensional parametri un ertainty modelling. A preliminary investigation [94℄ has shown that ea h varying entry in the state spa e matri es varies monotonously with respe t to ea h un ertain parameter, both for separate and for ombined parameter variation. Therefore the set of values for ea h varying entry only has to ontain a lower and an upper bound. From a set of submodels two sets of system matri es are obtained ontaining the minima and maxima of the entries over all system matri es. In our approa h the next step is to dene independent variables ea h orresponding to a varying entry in the system matri es with given lower and upper bounds.

Then, the varying entries are repla ed by the orresponding

independent variables whi h an vary between the lower and upper bound of the parti ular entries. In this way the system matri es ontain the whole set of linear models indu ed by the un ertainty. For the longitudinal submodel 39 entries are lassied as varying entries, whi h yields 39 independent variables to des ribe the un ertainty.

Of ourse, the introdu tion of 39 independent variables implies

some onservatism w.r.t. the situation with the 4 original varying parameters. The varying entries an be seen as a nominal value with a perturbation, whi h an be regarded as an LFT (Chapter 8). The system an be seen as an inter onne tion of LFTs whi h an be reformulated as one single LFT, using the properties of LFTs. The nal perturbation blo k-stru ture for the longitudinal submodel ontains 39 real perturbation blo ks of unit dimension orresponding to the 39 varying entries. The perturbation blo k-stru ture for the lateral/dire tional submodel ontains 31 real perturbation blo ks of unit dimension orresponding to 31 varying entries. The above pro ess is implemented in MATLAB using routines of the

-

Analysis and Synthesis Toolbox (mutools) [18℄ and the Parametri Un ertainty Modelling (PUM) Toolbox [144, 238℄.

23.3.5 Creation of the inter onne tion stru ture The rst step in the ontroller design y le is to build the inter onne tion stru ture for ontroller design. This involves dening the inputs and outputs, the various models and weighting fun tions, the perturbation blo k-stru ture, and the inter onne tions of the omponents.

Perturbed air raft model The LFT des ription of the air raft has been obtained in the un ertainty modelling step of the former Se tion. The

-synthesis

pro edure is appli able to

omplex perturbations only, be ause of the following reason.

347

In the D-K iteration used in

-synthesis, the frequen y dependent D-s alings

have to be tted with stable minimum-phase transfer fun tions. In the present version of the mutools toolbox this an only be realized for diagonal D-s ale matri es, whi h implies full omplex blo ks in the perturbation blo k-stru ture. This is implemented by repla ing the

n

n

1-dimensional omplex blo ks (where

1-dimensional real s alar blo ks by

n

equals 39 or 31 depending on the

submodel). Obviously, a 1-dimensional blo k is always a full blo k. Due to the repla ement again some onservatism is introdu ed in a subsequent ontroller design. Finally, a ontrol performan e perturbation blo k is dened, and the perturbation blo k-stru ture is augmented with that blo k in order to express the robust performan e problem formulation.

Inter onne tion stru ture Additionally to the perturbed air raft model a number of other blo ks has to be in orporated into the inter onne tion stru ture, see Figure 23.2. As an example the longitudinal inter onne tion stru ture will be elaborated. The turbulen e model has been dened a

ording to [145℄. Turbulen e is a sto hasti pro ess that an be des ribed by velo ity spe tra. It an be simulated by passing white noise through a lter modelled a

ording to a velo ity spe trum. Commonly used velo ity spe tra for turbulen e modelling are the Dryden spe tra. Figure 23.3 shows the frequen y response of the turbulen e lters.

Turbulence model : longitudinal (−), vertical (−−)

1

Ideal model : q (−), wv (−−), va (−.)

0

10

10

0

10

−1

10

−1

10

−2

10

−2

10 −3

10

−4

10

−3

−2

0

10

2

10 10 Frequency (rad/s)

10

4

10

−4

10

−2

10

0

10 Frequency (rad/s)

2

10

Figure 23.3: Frequen y response of

Figure 23.4:

longitudinal and verti al turbulen e

of ideal model for longitudinal re-

models.

sponse.

Frequen y response

An ideal model has been dened pres ribing the way the ontrolled air raft should behave to satisfy the spe i ations. For ea h of the 3 ontrolled output variables an ideal response is dened. The pit h rate should be minimized for

omfort purposes, so its ideal behaviour is dened as a onstant small gain of

0:001 .

The ideal models for the other ontrolled variables, verti al speed

348

4

10

and airspeed are dened by respe tively:

WwV ideal := 0:01

s + 20 s + 0:2

WVA ideal := 0:01

and

s + 15 : s + 0:15

Figure 23.4 shows the frequen y response of the ideal models. Next, performan e weighting fun tions and eort weighting fun tions have to be dened. These are the main inputs to the iterative designing pro ess. The performan e outputs for the ontroller design inter onne tion stru ture are dened as the weighted errors of the air raft output variables w.r.t.

the

ideal behaviour. The performan e weighting fun tion is an important tool to obtain desired performan e of the ontroller to be designed. A frequen y dependent weighting is applied. At high frequen ies the ontroller will not be able to ountera t any errors due to the inertia of engines and air raft.

In the lower frequen ies a

more pre ise ontrol is desired, and in the lowest band, where (approximate)

onstant errors o

ur, a perfe t removal is required. A performan e weighting aimed at su h ontrol hara teristi s behaves like a lag-lead lter:

with

! s + !1 ; K 2 !1 s + !2

!1  !2 .

(23.1)

The ontroller resulting from su h a weighting exhibits proportional and integral a tion.

Moreover, the synthesis tools automati ally add derivative

a tion for damping purposes. The integral a tion, evoked by the emphasis on low frequen y performan e, may ause overshoot.

On the other hand, if no

integral a tion is applied, a steady-state error is introdu ed. In parti ular, the weighting fun tion of the verti al speed error was nally dened as:

WwV := 0:01

s + 20 ; s + 0:02

and the weighting fun tion of the airspeed error was dened as:

WVA := 0:01

s + 10 ; s + 0:01

whereas the weighting of pit h rate has been dened at a onstant

0:001 .

Fig-

ure 23.5 shows the performan e weighting fun tions. The eort weighting fun tion is implemented to weight ontrol eort. Espe ially when saturations and rate limiters are present in the a tuators, are should be taken not to overload the a tuation, whi h is a main purpose of effort weighting. The weights are sele ted su h that a ertain de oupling in the tailplane and throttle ontrols is obtained. The tailplane eort weighting was nally dened as:

WÆT := 0:5

s + 0:2 : s + 10

349

Performance weighting : q (−), wv (−−), va (−.)

1

Actuator effort weighting : tailplane (−), throttle (−−)

0

10

10

0

10

−1

10 −1

10

−2

10 −2

10

−3

10

−3

−4

−2

10

10

0

10 Frequency (rad/s)

2

10

4

10

10

−2

10

−1

10

0

10 Frequency (rad/s)

1

2

10

10

Figure 23.5: Performan e weighting

Figure 23.6: Eort weighting fun -

fun tions for longitudinal ontrol.

tions for longitudinal ontrol.

This eort weighting will a t as a lead-lag lter, whi h is of the form of (23.1) but with

!1  !2 .

It applies the largest weight at the higher frequen ies. The

engine eort weighting was dened as:

WÆT H := 0:5

s + 0:02 : s+2

Also this eort weighting a ts as a lead-lag lter. It applies less weight at the lowest frequen ies than the tailplane weighting but more at medium frequen ies. In this way a ertain preferen e is expressed to use throttle at the lowest frequen ies, and the tailplane at medium frequen ies. At the highest frequen ies the weightings for both a tuators are equally high. Figure 23.6 shows the eort weighting fun tions. The weighting fun tions presented above are the result of an iteration pro ess. The initial sele tions were based on physi al onsiderations. Finally, the omponents are onne ted a

ording to in Figure 23.2.

23.3.6 D-K Iteration The D-K iteration, based on the mutools routine a ontroller

K

using

H1 synthesis.

unity for all frequen ies.

dkit,

starts by al ulating

The initial D-s alings are assumed to be

As an example a few Figures are shown from the

longitudinal ontroller design pro ess. Figure 23.7 shows the singular values of the resulting losed-loop system, in the rst iteration. Next,

 is omputed for

the losed-loop system at various frequen y points. Figure 23.7 also shows the ( omplex)

 plot.

The se ond iteration starts with tting the D-s alings intera tively.

For

ea h full blo k in the perturbation stru ture the D-s aling has to be tted. As an example, Figure 23.8 shows a rst order t for the third D-s aling. The se ond iteration pro eeds with a new system and with omputing

 as before.

350

H1 synthesis for the s aled open-loop

SINGULAR VALUE PLOT: CLOSED−LOOP RESPONSE

CLOSED−LOOP MU: CONTROLLER #1

30

18 16

25 14 12 10 MU

MAGNITUDE

20

15

8 10

6 4

5 2 0 −3 10

−2

10

−1

10

0

1

2

10 10 FREQUENCY (rad/s)

10

0 −3 10

3

10

Figure 23.7: Singular values (left) and



−2

−1

10

10

0

1

10 10 FREQUENCY (rad/s)

2

10

(right) of the losed-loop system in

the rst iteration. Iteration Controller order Total D-s ale order



peak-

1

2

3

17

25

43

0

8

26

29.595

2.927

1.521

16.565

1.886

1.513

Table 23.1: Summary of the D-K iteration for longitudinal ontroller design.

This pro edure is repeated until no more (signi ant) improvement an be made. Table 23.1 gives the iteration summary for the longitudinal ontroller design. The maximum singular value and



plot of the longitudinal losed-loop

system is given in Figure 23.9. The nal value of

 is 1:513, so robust perfor-

man e is not satised for the given weightings and the ( onservative) un ertainty model that was used. Table 23.2 gives the iteration summary for the lateral/dire tional ontroller design. The nal value of

 is 2:329, so again robust performan e is not satised.

Iteration Controller order Total D-s ale order



peakTable 23.2:

1

2

3

4

17

43

45

45

0

26

28

28

43.055

7.449

4.268

2.693

9.140

6.279

3.222

2.329

Summary of the D-K iteration for lateral/dire tional ontroller

design.

351

3

10

FITTING D SCALING #3 of 39, W/ORDER = 1

0

10

−1

10

Max. singular value and complex mu of closed−loop system −2

1.6

−3

1.4

10 10

1.2

−4 −3

10

−2

10

−1

0

1

2

10 10 10 10 1) mag data 2) newfit 3) previous D−K SCALED TRANSFER FUNCTION: OPTIMAL & RATIONAL

3

10

1 Magnitude

10

30

0.8

0.6

20

0.4

10 0.2

0 −3 10

−2

−1

0

1

2

10 10 10 10 10 1) MU UPPER BND 2) UPPER BND WITH RATIONAL FIT

Figure 23.8:

0 −4 10

3

10

Example of a rst-

Figure

order D-s aling t.

−2

10

23.9:

value and



0

10 Frequency (rad/s)

Maximum

2

10

singular

of longitudinal losed-

loop system.

23.3.7 Controller order redu tion The ontrollers obtained by D-K iteration both have an order of more than 40. It is natural that the order of ontrol systems designed by

-synthesis is very

high. However, the order of the ontrol system an be redu ed by model order redu tion te hniques. For this purpose the Hankel state redu tion te hnique is applied using the

mutools routine

hankmr.

A redu ed ontroller is reated su h, that its transfer

fun tions are equivalent to those of the original ontroller. Figures 23.10 and 23.11 show the bode plots of the 43th order original longitudinal ontroller obtained by the D-K iteration and the 16th order redu ed ontroller. An additional he k on the appli ability of a ontroller order redu tion is the

omputation of

 for the losed-loop system with the new (redu ed) ontroller.  does not satisfy the riterion  < 1 this additional he k

In this ase, where

is felt to be less appropriate. The order of the lateral/dire tional ontroller has been redu ed from 45 to 15. Moreover, from this ontroller a few states with very high natural frequen y have been removed by residualisation using the mutools routine

sresid, be ause

high natural frequen ies are not very favourable for the pra ti al appli ation of a ontroller.

23.3.8 Linear analysis inner loop ontrollers The ontrollers have been analysed to he k whether they satisfy the purposes of the inner loops. Of ourse the losed-loop eigenvalues have to ree t a well damped behaviour over the envelope. The losed-loop eigenvalues all have a satisfa tory real part over the spe i ation envelope (Se tion 23.3.2).

352

4

10

4

4

10

2

10

Log Magnitude

Log Magnitude

10

0

10

−2

10

−4

10

0

10

−2

10

−4

−4

10

−2

10

0

2

10 10 Frequency (radians/sec) Reduced controller, order 16, output 1 (original ooo )

10

4

10

−2

0

2

4

10 10 10 Frequency (radians/sec) Reduced controller, order 16, output 2 (original ooo )

10

Phase (degrees)

300

200 0 −200 −400 −4 10

−4

10

400 Phase (degrees)

2

10

−2

10

0

10 Frequency (radians/sec)

2

10

200 100 0 −100 −4 10

4

10

−2

10

0

10 Frequency (radians/sec)

2

4

10

10

Figure 23.10: Original and redu ed

Figure 23.11: Original and redu ed

order longitudinal ontroller bode

order longitudinal ontroller bode

plots, tailplane ontrol.

plots, throttle ontrol.

Another linear analysis is the omputation of losed-loop time responses. The results from su h analyses are essential in tuning the weighting fun tions for performan e improvement. Important design riteria are the response to step inputs and the magnitude of ross- oupling between separate hannels. In Figure 23.12 these results are presented for the longitudinal inner loop

ontrol, for all 81 parameter ombinations of the spe i ation envelope.

23.3.9 Outer loop design The ontroller shall provide the ontrol a tions whi h have been spe ied expli itely in Chapter 14. The longitudinal part of the omplete ontroller will be des ribed in more detail. The longitudinal ontroller subsystem is presented in Figure 23.13.

The

subsystem ontains both an outer loop ontroller and an inner loop ontroller. The inner loop ontroller onsists of an ideal model input shaping lter

lon_idmod) and an a tual ontroller (lon_ tl).

(

The feedba k signals fed to the inner loop ontroller ea h re eive a separate treatment. The pit h rate is ltered by a lag-lead lter in order to attenuate

losed-loop gain at high frequen ies. A marginal instability whi h did appear in non-linear simulation for some parameter ombinations had to be suppressed by this lter. The verti al velo ity is ltered by a high-pass lter, whi h a ts as a kind of omplementary lter for the ideal model. The low-frequen y ontent of the a tual verti al velo ity is fed via the forward hannel, whi h will be dealt with in the following. The airspeed is redu ed by the trim speed. The forward hannel of the inner loop ontroller omprises the outer loop

ontroller, produ ing referen e values for the inner loop ontroller. The pit h rate referen e value is zero, as only two motion variables an be ontrolled independently using the two longitudinal ontrol ee tors.

353

wv <> wv cmd

wv <> va cmd

1

0

0.8 −0.02 0.6 −0.04 0.4 −0.06 0.2 0 0

10

20 time (s)

30

40

−0.08 0

10

va <> wv cmd

20 time (s)

30

40

va <> va cmd 1

0 0.8

−0.02 −0.04

0.6

−0.06 0.4 −0.08 0.2

−0.1 −0.12 0

10

20 time (s)

30

0 0

40

10

20 time (s)

30

40

Figure 23.12: Response of longitudinal losed-loop submodel: verti al velo ity

ontrol, airspeed ontrol and ross- oupling between both.

354

2 WV_C

1 Z_C

kw1

+ −

0 q cmd

+ + −

kz

x’ = Ax+Bu y = Cx+Du lon_idmod

Mux Rate Limiter

kw2 3 V_C

+ −

x’ = Ax+Bu y = Cx+Du lon_ctl

+ −

v0

2 THROTTLE1

wv q

4 Demux longitudinal measurements

wv z va

3 THROTTLE2

s+40 4s+40 Lag−lead s s+0.2 High−pass + −

1 DT

Demux

Mux

Figure 23.13: Longitudinal subsystem of the designed ontroller.

The referen e value for the verti al velo ity is omposed of the altitude error, via a gain, and the verti al velo ity outer loop error, via separate gains for ommanded and a tual verti al velo ity. The total referen e value is fed via a rate limiter to avoid saturation of the rate limiter in the tailplane a tuator. The airspeed referen e value again is redu ed by the trim speed. The outer loop gains have been tuned by hand. Owing to the robust inner loop ontrollers, simple, onstant outer loops without s heduling, are su ient. The gains have been dimensioned su h, that satisfa tory responses are obtained for small outer loop ommands in the rst pla e. Then, when in reasing input magnitudes aused problems, e.g. due to a tuation non-linearities, the ommands fed to the inner loop had to be limited. Finally, the evaluation performed a

ording to Chapter 14.3.3 indi ated the need for onsiderable feedforward.

23.3.10 Resulting ontroller omplexity and implementation issues As dis ussed in the previous Se tions, the resulting ontroller is a stable, invariant ontroller with fairly high order.

No swit hing is applied, the only

non-linearities are a rate limiter and a number of axis transformation fun tions. In an a tual implementation the ontinuous dynami parts have to be translated to dis rete equivalents. A problem still to be solved is the rather high bandwidth the ontrollers assume. environment.

This may pose a problem in a real-time

The number of states as su h should not be problemati for

state-of-the-art omputing hardware. The hara ter of this type of bla k box ontrollers, in whi h the fun tion of individual gains and dynami properties annot be explained to a ertifying

355

authority, might pose a more fundamental problem.

23.4 Non-Linear Assessment of the Designed Controller In order to assess whether the ontroller designed in Se tion 23.3 satises the spe i ations established in Chapter 14, a set of assessment routines has been written, performing non-linear simulations of RCAM with ontroller.

The

assessment produ es standardized results, enabling mutual omparison between dierent ontrollers. The assessment pro edure is des ribed in Version 3 of [145℄. The assessment is able to deal with a number of spe i ations in Se tion 14.3.2 of [145℄. The user an spe ify whi h operating ondition or ombination of operating onditions shall be applied. A huge number of assessment results an be obtained this way. As an illustration the results for lateral deviation ontrol assessment and altitude response assessment are reprodu ed in Figure 23.14, representing all 81 parameter ombinations of the spe i ation envelope.

23.5 Results of the Automated Evaluation Pro edure The landing s enario onsisting of four segments is des ribed in Chapter 14.3.3.



Segment I:

see Figure 23.15, left side.

Engine failure is handled very

well apparently.



Segment II: see Figure 23.15, right side.

The redu tion of lateral error

is too slow to omply with the given bounds.



Segment III: see Figure 23.16, left

side. Verti al error redu tion is not

able to omply in time with the requirements.



Segment IV:

see Figure 23.16, right side. The verti al deviations stay

within the bounds.



Table 23.3:

Most of the numeri al values are below one, whi h means

omplian e with the requirements. The ex eptions are the Comfort req. in Segment II and the Safety req. in Segment IV.

23.6 Con lusions and Lessons Learned -Synthesis is dened as an iterative pro ess applying alternately H1 synthesis -analysis. An LFT des ription was used to model air raft parametri

and

356

Lateral step response

Altitude step response

y − y_c −. [m]

h − h_c −. [m]

1.5

0.5

1

0

0.5

−0.5

0

−1

−0.5 0

10

20 30 time [s] vv − vv_c −. [m/s]

−1.5 0

40

1

1

0.5

0.5

0

0

−0.5 0

10

20 time [s] da [deg]

30

−0.5 0

40

5

10

20 30 time [s] wv − wv_c −. [m/s]

40

10

30

40

20 30 time [s] throttle L− R−− [deg]

40

10

40

20 time [s] dt [deg]

1 0.5

0

0 −0.5

−5 0

10

20 time [s] dr [deg]

30

−1 0

40

10

0.5 2 1 0

0

−1 −2 0

10

20 time [s]

30

−0.5 0

40

20 time [s]

30

Figure 23.14: Assessment of lateral deviation ontrol and altitude response for all parameter ombinations in the spe i ation envelope.

357

First segment: top view

Second segment: lateral deviations 300

100

50

0

0

a

b

lateral deviation [m]

x−deviation

[m]

200

1

−50

100

0 1

c

d

2

−100

−200 −100 −20

−18

−16

−14

−12 −10 −8 y−position (−YE) [km]

−6

−4

−300 0

−2

1

2

3 4 5 6 along track distance from point 1 [km]

7

8

Figure 23.15: Left: segment I - the ee t of engine failure. Right: segment II - lateral deviations during the

3Æ/s turn.

Fourth segment: altitude deviations

20

20

10 2

f

[m]

30

3

altitude deviation

altitude deviation [m]

Third segment: altitude deviations 30

0

−10

e

−20

−30

10

3 g

4

h

0

−10

−20

−16

−15

−14 −13 x−position (XE) [km]

−12

−30 −11

−11

−10

−9

−8

−7 −6 −5 −4 x−position (XE) [km]

−3

−2

−1

Figure 23.16: Left: segment III - verti al deviations from the desired glideslope. Right: segment IV - verti al deviations from the desired glideslope.

un ertainties.

A somewhat onservative method was used to deal with the

multi-dimensional aspe t of the un ertainties.

-synthesis

The appli ation of

to the inner-loop ontroller design of the

RCAM Design Challenge ould formally not su

eed in a robust perform-

 < 1), neither in the longitudinal, nor in the  values in the order of 2 were obtained.

ing ontroller design (

lat-

eral/dire tional ase. Instead,

The outer loop design applied hand-tuning of parameters. Finally, an apparently rather satisfa tory ontroller has resulted. Only its behaviour at higher frequen ies and the relation between small input response and large input response still ould benet from some improvement. It may be argued, that a less onservative un ertainty modelling pro edure might have resulted in lower

 values and maybe in a more satisfa tory 358

Segm. I

Segm. II

Segm. III

Segm. IV

Total

Performan e

0.0552

0.5857

0.5888

0.8875

0.5293

Perf. Dev.

0.0088

0.0363

0.1055

0.1906

0.0853

Comfort

0.4412

2.0553

0.9889

0.5791

1.0161

Safety

0.0032

0.0159

0.0056

4.1336

1.0396

Power

0.0024

0.0033

0.0151

0.0296

0.0126

Table 23.3: Numeri al results of the evaluation pro edure

ontroller. On the other hand, an only slightly more demanding envelope spe i ation ould an el su h an improvement readily.

After all, it annot be

the intention to re ommend a method whi h is only ee tive for very limited robustness spe ied. Therefore one may wonder, if it is desirable from a ontrol performan e standpoint, to apply a xed gain robust ontroller in air raft ight ontrol, instead of a ontroller with gain s heduling. A robust ontroller is not optimized for a single situation, but is the result of a trade-o between performan e and robustness, yielding an optimum for a set of operating onditions.

Gain s heduling, on the ontrary, uses a tual

information on parameter values to optimize ontroller performan e on line for any situation, based on existing air raft dynami s knowledge. A method whi h models parameter variations as un ertainties, as shown in this Chapter, might not make the most ee tive use of the available data on

urrent air raft state and dynami properties. A hara teristi of the

-synthesis method,

and more spe i ally the

H1

synthesis part thereof, whi h annot be appre iated, is that the designer has no dire t inuen e on ertain properties of the ontroller, e.g. its bandwidth. There is even no dire t inuen e on the stability of the ontroller itself, other than reje ting it. (Although an unstable ontroller formally might be stabilized by the plant, su h that the ombination is stable.) Given the plant LFT des ription, the means the designer has at his disposal,

omprising sele tion of the ontroller top level ar hite ture and the dimensioning of weighting fun tions, are of limited power. Perhaps even more than the extent, to whi h an

H1 synthesis ontroller will H1 synthesis

be viewed as a bla k box devi e by the ertifying authorities, the

is therefore experien ed as a bla k box method by the designer, whi h is not quite appre iated.

359

24.

Autopilot Design based on the

Model Following Control Approa h

Holger Duda1 , Gerhard Bouwer1 , J.-Mi hael Baus hat1 and Klaus-Uwe Hahn1 Abstra t.

The appli ation of the Model Following Control (MFC)

te hnique to the RCAM autopilot design problem is presented. The overall ontroller stru ture and the design y le are dis ussed. A detailed ontroller assessment onrms that the design riteria are met in the design point and for several additional o-design points,

onsidering varied entre of gravity lo ation, air raft mass, airspeed and additional time delays. The presented design approa h is insensitive to additional time delays (up to 100 mse ) and entre of gravity variations. Air raft mass and airspeed variations had small adverse ee ts on the ontroller performan e.

The results of the

automated evaluation pro edure demonstrate that the verti al and lateral deviations from the tra k are well within the limits for all investigated ases.

24.1 Introdu tion In hapter 11 a tutorial review of the applied ontrol design methodology is given. In this hapter the Model Following Control (MFC) te hnique is applied to the RCAM autopilot design problem. The already stated feature of the MFC

on ept to separate the main three elements ommand blo k, feedforward and feedba k ontrollers is utilised.

Ea h module an be designed and analysed

separately leading to a straight forward design with a transparent overall

ontroller stru ture. The design y le and the analysis of the ontroller in terms of the applied methodology are summarised.

The robustness of the MFC design is investi-

gated by nonlinear simulations of typi al manoeuvres in time domain for several o-design points.

24.2 Overall Controller Stru ture The overall MFC system stru ture of the RCAM design is presented in gure 24.1. 1

The main three elements of the MFC approa h an be identied: the

DLR German Aerospa e Resear h Establishment, Institute of Flight Me hani s, D-38108

Brauns hweig

360

ommand blo k, the feedforward ontroller and the feedba k ontrollers. The ommand blo k in ludes linear air raft models in the longitudinal and lateral axes without intera tions between the two. The following referen e signals from the traje tory generator are used as inputs to the ommand blo k:

eyb , the desired heading rate _ r , the desired verti al velo z_r , the airspeed error VA r VA , the position error zr z and the measured verti al velo ity wv . The outputs of the ommand blo k are the state ve tor to be ontrolled x1C = [p ; r ; uB ; wB ℄, its time derivative x _ 1C , and the state ve tor for de oupling x2C = [q ;  ; vb ℄. Additionally, the states , p , r ,  and q are utilised as input signals to the feedba k ontroller. the lateral deviation

ity

x1C = [pc rc uBc wBc] . x1C Inverse Aircraft . x1C M1x1 +M2x1

Reference Signals eyb . Ψr . zr VAr zr

-

Gain Schedule + δA + + x2C = [qc Θ c vBc] GS δT Decoupling + + Command δR M3 x 2 + + Block δ TH1 + δ TH2 Feedforward Controller βc pc rc Feedback Controller Φc qc q Φ r p β wV z VA

Turn Correction

Figure 24.1: RCAM overall ontroller stru ture

The feedforward ontroller only ontains the matri es to equation (11.8).

M1 to M3 , a

ording

The feedba k ontroller ontains several proportional and integral terms to redu e the error between the ommand blo k states

, p , r ,  , q

and

the orresponding air raft states. No additional lters or nonlinear elements are onsidered for the feedba k path with the ex eption of a turn orre tion term for the measured pit h rate

q.

This blo k is required in order to keep the

altitude onstant during a turn. Note that no inner loop airspeed ontrol is utilised. Additionally, gain s heduling is used for the ontrol surfa e ommands in

361

order to enhan e the robustness against airspeed variations as has been suggested in [143℄.

No further gain s heduling or adjustments of the ontroller

stru ture are taken into a

ount.

24.3 The Translation of RCAM Design Criteria into Method Dependent Obje tives For the appli ation of the MFC on ept the RCAM design hallenge has to be separated into three independent subtasks: - to design a ommand blo k, whi h meets the design riteria, - to design a feedforward ontroller whi h in ludes an inverse behaviour of the plant, - and to design a feedba k ontroller ope with plant model un ertainties and disturban es reje tion. The rst subtask an be a

omplished without any referen es to the feedforward and feedba k ontroller. A simple (linear or nonlinear) air raft model with autopilot fun tions has to be designed with respe t to the design requirements.

Pursuing the idea that numerous autopilot have been designed and

implemented in air raft ight ontrol systems in the past, a relevant autopilot/air raft ombination an be hosen for the ommand blo k for this design problem. For obvious reasons it was de ided to use an available Classi al Con-

trol approa h to the RCAM problem for the presented on ept [88℄. Therefore, with respe t to the translation of the design riteria, the same statements as summarised in hapter 15 are appli able to the present example. The design goal for the feedforward and feedba k ontrollers is to for e the plant to follow ommand blo k states in an optimal way.

If the ommand

blo k is hanged for new requirements or any other reason the feedforward and feedba k ontrollers remain un hanged. They are designed only on e for a spe i system for all demands generated by the ommand blo k. The design obje tives for these omponents are dis ussed in se tion 11.5.

24.4 The Des ription of the Design Cy le TM

The omplete design pro ess is performed within the Matlab/Simulink

en-

vironment using the fun tions from the Control System Toolbox ex ept the optimisation of the feedba k ontroller. For this subtask an o-line optimisation based on a quadrati sear h algorithm was used [126℄. The design y le steps are presented in gure 24.2: This pro ess in ludes several iterations regarding the denition of the ommand blo k, the design of the feedba k ontroller and the he king of the omplete MFC system against the requirements. It has been shown that mainly the feedba k ontroller had to be adjusted after he king the omplete system, if the ommand blo k has been designed arefully, e.g. in luding nonlinear a tuator models.

362

Definition of a design point for linearisation Linearisation of the nonlinear aircraft model Definition of an appropriate controller structure Determination of the feedforward controller Definition of the command block based on an available RCAM design Analysis of the isolated command block Design of the feedback controller Analysis of the complete MFC system Assessment of the MFC system for off-design points Figure 24.2: RCAM design y le for the MFC approa h

Due to the pure linear approa h in this design example, the determination of the feedforward ontroller did not require any iterations. Note that also the last step ( ontroller assessment, se tion 24.5) did not require any iterations, be ause it has been shown that the ontroller, whi h was developed in the design point, fullled the design riteria also in the investigated o-design points (one-shot

approa h). The single design steps are dis ussed below:

Design point The design point has been dened to be approximately in the middle of the ight envelope (nominal ase of the evaluation pro edure):

m = 120:000 Kg, CG = [0:23; 0; 0℄, VA = 80 m/se , h = 1000 m. Linearisation The linearised air raft model was obtained using the trimr am routine, whi h is supplied with the RCAM design software.

A problem was found in this

linearisation routine: the element of the dynami matrix to zero.

be no oupling between the roll angle velo ity

A(9; 4) was not equal

This oni ts with the physi al ba kground, be ause there should

w_ B in the design point.



and the

z

omponent of the inertial

It is thought, that this problem is aused by the

fa t that the gradient for the linearisation is only al ulated in one dire tion. However, for the determination of the feedforward ontroller this element has been set to zero:

A(9; 4) = 0.

Controller stru ture Sin e the basi stru ture of a MFC system is xed in prin iple (gure 11.2), only the input and output signals to the main three elements ( ommand blo k,

363

feedforward and feedba k ontrollers) have to be dened. The input signals to the feedforward ontroller are summarised a

ording to equation (11.5) in the ve tors

x2C

x 1C

and

x2C .

The elements of the ve tor

x 1C

have to be dened, while

automati ally ontains the remaining elements of the omplete air raft

state ve tor

x.

Assuming that no asymmetri thrust for yaw ontrol is available, the plant has four input signals:

ÆA , ÆT , ÆR

and

ÆT H 1 .

Hen e, the ve tor

x 1C

also

has to ontain four elements; these have been dened based on a ontrollability analysis:

x1C = [p ; r ; uB ; wB ℄.

Note that the sideslip angle

is not in luded,

therefore, the requirement to minimise the sideslip angle has to be implemented in the lateral model of the ommand blo k.

Regarding the sele tion of the

longitudinal states a further promising alternative would have been to use instead of

wB .

However, the use of

wB provided su

essful results.

q

The sele tion of input signals to the feedba k ontroller should be appli able to ivil transport air raft with onventional ontrol surfa es throttle

ÆT H 1 .

ÆA , ÆT , ÆR and the

Feedforward ontroller The pure air raft dynami s without the a tuator models were onsidered to determine the feedforward ontroller. It is dened by the matri es

M1

to

M3

a

ording to equation (11.7). The sele tion of the state ve tor to be ontrolled

x1C = [p ; r ; uB ; wB ℄

x2C , whi h inq ,  ,  , M3 a

ording two

leads to a state ve tor for de oupling

ludes the remaining elements of the omplete state ve tor ( and

vB ).

In this example the feedforward ontrol matrix

equation (11.8) has two zero olumns, therefore, the de oupling state ve tor was redu ed to

x2C = [q ;  ; vB ℄ (gure 24.1).

Command blo k The ommand blo k is separated into a lateral and a longitudinal part without any oupling, gure 24.3.

The stru ture of the autopilot fun tions and the

blo ks of the augmented air raft, whi h ontain linear air raft models in luding

ontrol loops, are summarised in [88℄. The augmented air raft blo ks ontain linear air raft models in both axes, whi h have been obtained from a linearisation of the nonlinear RCAM. In the lateral axis the augmented air raft represents a roll angle ommand system, while the inner loops have been designed in order to obtain minimised sideslip angle during rolling.

In the longitudinal axis the augmented air raft model

represents a pit h angle ommand system. Both augmented air raft models ontain nonlinear a tuator models, whi h has been shown to be very important regarding the MFC performan e. The outer loops around the ommand blo k represent typi al autopilot fun tions su h as lateral tra k hold, glideslope hold, altitude hold and autothrottle. Note that the demanded verti al velo ity

z_r

from the traje tory generator is

only used for mode swit hing between altitude and glideslope hold.

364

The smoothing lter for the demanded heading rate improves the omfort during a turn.

The blo k 

_



to

ontains the following relationship for a

steady state turn [35℄:

 = tan 1 ( _ V0 =g): eyb

Lateral track hold s -0.0054 0.05 s + 1

. Ψr

7.2

+

0.001

1/s

Φcmd

+ + +- 30 deg

. tan-1 (Ψ V0/g) . Ψ to Φ

1 s2 + 1.8s + 1 Smoothing

(24.1)

. pc . rc Augmented βc Lateral Aircraft: pc Roll Angle Command, rc No Sideslip Φc Angle vBc yc

. zr zr - z wV

0.13

1/s

-

Altitude hold 3 P=1 I = 0.056 D=5 PID

VAr - VA

-

. zr = 0

-0.022 -0.018

Glideslope hold

P=1 I = 0.04 D = 2.5 PID

. zr > 0 Mode Select

Engine Model

0.2

Θ cmd Augmented Longitudinal Aircraft Pitch Angle Command

δ thc

. uBc . wBc qc Θc zc VAc . zc

Autothrottle

Autopilot functions

Command state vector generation

Figure 24.3: RCAM ommand blo k stru ture

Analysis of the ommand blo k The isolated ommand blo k an be analysed against the requirements without the nonlinear air raft model, the feedforward and feedba k ontrollers. For this purpose the additional outputs

y , z , VA , z_

(broken arrows in gure 24.3)

were used to simulate the losed loop ommand blo k. In omparison to the original Classi al Control approa h some gains had to be adjusted, for example in the outer lateral tra k hold (gure 24.3). Table 24.1 presents the performan e of the isolated ommand blo k in terms of the design riteria regarding the rise time overshoot

Mp

tr ,

the settling time

ts

and the

for unity step responses in lateral tra k, altitude and airspeed.

Additionally, the airspeed and altitude ross oupling parameter error peak after a 30 m altitude step demand) and

z13

V30

(speed

(altitude error peak

after a 13 m/se airspeed step demand) are presented. All requirements are met for the isolated ommand blo k.

365

Lateral tra k step

Altitude step

Airspeed step

Cross oupling

tr

ts

Mp

tr

ts

Mp

tr

ts

Mp

V30

z13

(se )

(se )

(%)

(se )

(se )

(%)

(se )

(se )

(%)

(m/se )

(m)

6.4

12.2

1.3

9.2

17.6

1.7

5.1

8.3

0.5

0.2

6.1

Table 24.1: Che king the isolated ommand blo k against the design riteria

Feedba k ontroller The feedba k ontroller was obtained from an o-line optimisation. A quadrati sear h algorithm minimises the following ost fun tion by hanging the values of the feedba k gains [126℄:

J=

Z t=20se

t=0

w2 + 602 [15 2 + p2 + q2 + 2 ℄ dt:

(24.2)

For every optimisation step, the linearised air raft with the nonlinear a tuator models and an additional time delay of 150 mse was simulated over 20 se . Between 0.5 se and 1.0 se the system is disturbed with pit h, roll and yaw

2

a

elerations of 2 rad/se . After 10 se simulation time, one engine fails. The weighting fa tor of 60 relates the error of 1 deg or 1 deg/se to 1 m/se . The sideslip angle



is weighted 15 times stronger than the remaining angles or

angular rates. Additionally, up to the engine failure the ontrol surfa e dee tions and

ÆR

ÆA , ÆT

were weighted with a fa tor of 100 in order to suppress disturban es

with low ontrol a tivity.

Initial values for the gains to be optimised were

obtained from the ontrol matrix of the linearised air raft. The stru ture and the nal gains of the feedba k ontroller are presented in gure 24.4.

Proportional

Integral

state error output

δT

q

q -1.33

-0.133

r

Longitudinal

p

δA

0.

0.

-6.0

-4.0

0.

δR

-5.1

4.0

0.

0.

4.16

Lateral

Figure 24.4: RCAM feedba k ontroller: stru ture and gains

The measured pit h rate is orre ted using the following equation, whi h is ne essary to keep the altitude onstant during a turn [35℄:

366

g sin2  : V os 

qT =

(24.3)

Analysis of the omplete MFC system The omplete MFC system in luding the ommand blo k, the nonlinear air raft, the feedforward and feedba k ontrollers was analysed using a spe ial assessment environment. The following items have been examined:

  

Performan e of the ontroller, Quality of the inversion, Ee ts of turbulen e and engine failure.

The performan e of the ontroller at the design point is presented in table 24.2 onsidering an additional time delay of 50 mse . A performan e is a hieved similar to the isolated ommand blo k (table 24.1) indi ating that the ontrol system ensures a more or less exa t following of the ommand blo k. Lateral tra k step

Altitude step

Airspeed step

Cross oupling

tr

ts

Mp

tr

ts

Mp

tr

ts

Mp

V30

z13

(se )

(se )

(%)

(se )

(se )

(%)

(se )

(se )

(%)

(m/se )

(m)

5.6

10.4

3.3

9.2

17.6

1.9

5.2

12.4

0.7

0.27

4.9

Table 24.2: Che king the omplete nonlinear system against the design riteria (design point)

The quality of the inversion was investigated by means of a omparison of the ommand blo k states and the air raft states.

Figure 24.5 shows the

time histories of a lateral tra k step response. The step time was one se ond. The upper two diagrams demonstrate the performan e of the ontroller at the design point and the orresponding ontrol surfa e dee tions. In the lower two diagrams the roll and sideslip angles of the air raft are ompared with those of the ommand blo k. A very good mat h is a hieved. The investigation of turbulen e ee ts proved that the requirement regarding roll angle was met for the design point. The ee ts of engine failure are presented in gure 24.6: After 1 se simulation time the left hand engine fails and after 25 se it is restarted. The maximum lateral deviation after the engine failure is less than 20 m, whi h is well within the requirement. The ontrol surfa e signals show that the engine failure is mainly ompensated by the rudder. The roll angle never ex eeds 10 deg and its steady state value is less than 5 deg. A sideslip angle of about

1 deg built up after the engine failure and restart.

The peak of the lateral a

eleration during this manoeuvre is about 0.064 g.

367

Lateral Step Response (Y (m)) 2 1 0 Surface Deflections (solid: DA (deg), broken: DR (deg)) 5 0 −5 Roll Angle (solid: PHI (deg), broken: PHI_c (deg)) 0.5 0 −0.5 Sideslip Angle (solid: BETA (deg), broken: BETA_c (deg)) 0.02 0 −0.02 0

5

10

15

20 Time (secs)

25

30

35

40

Figure 24.5: Lateral step response of the omplete MFC system (design point)

This failure ase demonstrates the hara teristi of the MFC system: the sideslip angle of the ommand blo k



is about zero during the whole ma-

noeuvre time interval and the dieren e between



and



is redu ed within

se onds. This proves the performan e of the feedba k ontroller.

24.5 Analysis of the Resulting Controller in Terms of the Applied Methodology In order to investigate the ontroller robustness against parameter variations a set of 15 test ases is dened, table 24.3 (page 370). The ases e1 to e4 are taken from the evaluation pro edure, while the 11 additional ases a1 to a11 have been dened with the following strategy: The ases a1 to a5 are dened in order to analyse the robustness against mass, verti al entre of gravity lo ation and airspeed variations. The ase a6 to a11 ontain several variations regarding mass and airspeed, but all with a most aft entre of gravity lo ation and a time delay of 100 mse a

ording to the re ommendations from [143℄. At these test points nonlinear simulations have been performed, whi h are dis ussed below:

Lateral tra k step responses Figure 24.7 presents the time histories of lateral tra k step responses for the

Mp < 5 % is met for all ases. Mp is slightly in reased for the low airspeed

dened test ases. The overshoot requirement Generally, it has been observed that

ases, for example 4.2 % for ase a4 against 3.3 % for ase e1 (design point), but it be omes lear that the ontroller is very robust regarding this requirement.

368

Lateral Deviation (Y (m)) 20 0 −20

Engine Failure

Engine restarted

Surface Deflections (solid: DA (deg), broken: DR (deg)) 20 0 −20 Roll Angle (solid: PHI (deg), broken: PHI_c (deg)) 10 0 −10 Sideslip Angle (solid: BETA (deg), broken: BETA_c (deg)) 2 0 −2 0

5

10

15

20 Time (sec)

25

30

35

40

Figure 24.6: Ee ts of engine failure (design point)

An os illation in the roll angle is observed for ases a6 and a7, whi h are

hara terised by a low airspeed and a low mass. The ombination of low airspeed and low mass forms the worst ase for this type of manoeuvre. 1 m Lateral Step Response

Y (m)

2

1

0 Roll Angle PHI (deg)

0.5

0

−0.5 0

5

10

15 Time (sec)

20

25

30

Figure 24.7: RCAM ontroller assessment: lateral tra k step responses

Altitude step responses Figure 24.8 presents the time histories of 30 m altitude step responses for the dened test ases. Additionally, unity altitude step responses have been investigated in order to he k the performan e riteria. The overshoot requirement

Mp < 5 % is met for all ases ex ept a9 (Mp = 5:4 %).

The airspeed to altitude

ross oupling requirement is met for all ases. The maximum airspeed error

369

Mass Case

(Kg)

CGx

CGy

Air

Time

Trim

Speed

Delay

File

(m/se )

(mse )

Comments

e1

120000

0.23

0

80

50

r x0017

Design point

e2

120000

0.23

0

80

100

r x0017

Evaluation ase

e3

120000

0.31

0

80

50

r x0217

Evaluation ase

e4

120000

0.15

0

80

50

r x0117

Evaluation ase

a1

100000

0.23

0

80

50

r x1017

low mass

a2

150000

0.23

0

80

50

r x2017

high mass

a3

120000

0.23

0.21

80

50

r x0027

upper CG

a4

120000

0.23

0

63.7

50

r x0010

low airspeed

a5

120000

0.23

0

90

50

r x0016

high airspeed

a6

100000

0.31

0

58.1

100

r x1210

see [143℄

a7

100000

0.31

0.21

58.1

100

r x1220

see [143℄

a8

150000

0.31

0

71.2

100

r x2210

see [143℄

a9

150000

0.31

0.21

71.2

100

r x2220

see [143℄

a10

120000

0.31

0

90

100

r x0216

see [143℄

a11

120000

0.31

0.21

90

100

r x0226

see [143℄

Table 24.3: Denition of test ases for the ontroller assessment

peak of

V30 = 0:49 m/se is again obtained for ase a9.

Additionally, a lightly

damped airspeed os illation is present for ases a8 and a9, whi h are the low airspeed ases with high mass. The upper verti al entre of gravity lo ation worsens the situation slightly. The ombination of low airspeed, high mass and upper entre of gravity lo ation forms the worst ase for this type of manoeuvre.

30 m Altitude Step Response

H (m)

1000 980 960

VA − VA0 (m/sec)

Airspeed Error 0.4 0.2 0 −0.2 0

5

10

15 Time (sec)

20

25

30

Figure 24.8: RCAM ontroller assessment: altitude step responses

370

Glideslope aptures Figure 24.9 presents the time histories of 6 deg glideslope aptures for the dened test ases. Additionally, the airspeed errors are presented indi ating that two ases are hara terised by a low performan e regarding the airspeed

ontrol after the glideslope apture.

These are ases a6 and a7, whi h are

hara terised by a low airspeed and a low mass.

Another ase with a poor

airspeed ontrol performan e is the low airspeed ase a4 with nominal mass, whi h has an airspeed error of about 2 m/se at the end of the simulation time interval. The ombination of low airspeed and low mass forms the worst ase for this type of manoeuvre. 8 m/s Glideslope Capture

H (m)

1200 1000 800 600 VA − VA0 (m/sec)

Airspeed Error 10

5

0 0

5

10

15 Time (sec)

20

25

30

Figure 24.9: RCAM ontroller assessment: glideslope aptures

Airspeed step responses Figure 24.10 presents the time histories of 13 m/se airspeed step responses for the dened test ases. In this gure a large overshoot is noti ed for the ases a6 and a7, whi h are hara terised by a low airspeed and a low mass. However, for unity airspeed step responses the overshoot requirement is met for all ases. The airspeed to altitude ross oupling requirement is met for all ases, the maximum altitude error peak is less than 10 m. The ombination of low airspeed and low mass forms the worst ase for this type of manoeuvre.

Engine failures Figure 24.11 presents the time histories of nonlinear simulations with engine failures for the dened test ases.

ases.

The roll angle requirement is met for all

Maximum sideslip angle ex ursions of about 1.1 deg after the engine

failure and about -1.5 deg after the engine is restarted are observed for ases a8 and a9, whi h are the low airspeed ases with high mass.

371

For these two

VA − VA0 (m/sec)

13 m/sec Airspeed Step Response 15 10 5 0

H (m)

Altitude Error 1010 1005 1000 995 0

5

10

15 Time (sec)

20

25

30

Figure 24.10: RCAM ontroller assessment: airspeed step responses

ases an os illation in the sideslip angle is present indi ating a low dut h roll damping. The ombination of low airspeed and high mass forms the worst ase for this type of manoeuvre.

Roll Angle 10 PHI (deg)

Engine Failure

Engine restarted

0

−10

BETA (deg)

Sideslip Angle 1 0 −1 0

5

10

15 Time (sec)

20

25

30

Figure 24.11: RCAM ontroller assessment: ee ts of engine failures

Turbulen e Figure 24.12 presents the time histories of nonlinear simulations in turbulen e for the dened test ases. The requirement regarding the roll angle

 < 5 deg

is met for all ases. A maximum verti al load fa tor of about 0.3 g is rea hed for the high airspeed ases a10 and a11, whi h is only slightly higher than the maximum value for the design point of 0.25 g. The performan e requirements regarding overshoot and airspeed to altitude

ross oupling for the dened test ases are summarised in gure 24.13. The

y

h

overshoots for step responses of the lateral tra k ( ), the altitude ( ) and the

v

airspeed ( ) are presented.

Mp < 5

The overshoot riterion (

372

%) is slightly

Roll Angle PHI (deg)

5

0

−5 Vertical Load Factor

NZ (g)

0.5

0

−0.5 0

5

10

15 Time (sec)

20

25

30

Figure 24.12: RCAM ontroller assessment: ee ts of turbulen e

violated only on e for ase a9 (altitude step). The airspeed to altitude ross

oupling riteria are met for all ases. These investigations demonstrate the performan e and stability robustness of the ontroller. Generally, it has been shown that the ee ts of time delays and

entre of gravity deviations are negligible, while airspeed and mass variations have the most adverse ee ts on the ontroller performan e. However, only one slight violation of the performan e riteria was found, so that the results are

VA30 (m/s)

Mp (%)

highly satisfa tory.

6 5 4 y 3 h 2 v 1 0 e1 e2 .5 .4 .3 .2 .1 0

e3

e4

a1

a2

a3

a4

a5

a6

a7

a8

a9 a10 a11

Airspeed Error after 30 m Altitude Step

e1

e2

e3

e4

a1

a2

a3

a4

a5

a6

a7

a8

a9 a10 a11

Altitude Error after 13 m/sec Air Speed Step

z13 (m)

10 8 6 4 2 0

Overshoot Criteria

e1

e2

e3

e4

a1

a2

a3

a4

a5

a6

a7

a8

a9 a10 a11

Case Figure 24.13: RCAM performan e requirements: altitude ross oupling

373

overshoot and airspeed to

24.6 Results of the Automated Evaluation Pro edure This se tion presents the results of the automated evaluation pro edure dened in [145℄:

both overall tra king performan e and inner-loop behaviour of the

ontrolled system are evaluated by means of bounds on key variables.

Segment I: the ee t of engine failure As the air raft model is twin-engined, a single engine failure will mainly result in yaw a

eleration and, therefore, lateral deviation.. Figure 24.14 provides a top view of the rst traje tory segment. The lateral deviations are well within the limits. Furthermore, the airspeed should not drop below

m = 120000 Kg).

m/se (

1:2  Vstall  61:2

The minimum airspeed during the engine failure is

about 79 m/se in this ase. First segment: top view

Second segment: lateral deviations 300

100

0

0

a

b

lateral deviation [m]

x−deviation [m]

200 50

1

−50

100

0 1

c

d

2

−100

−200 −100 −20

−18

−16

−14

−12 −10 −8 y−position (−YE) [km]

−6

−4

−300 0

−2

1

2

3 4 5 6 along track distance from point 1 [km]

7

8

Figure 24.14: Segment I: the ee t

Figure 24.15:

of engine failure.

deviations during the 3 deg/s turn

Segment II: the 3

deg/s

Segment II: lateral

turn

At the moment when the turn is initiated, the perfe t following of the required traje tory and the desire to perform a oordinated turn would imply a sudden

hange in the air raft's roll angle, whi h is only possible with an innitely high roll rate. Obviously, this is unwanted, su h that deviations from the desired traje tory at the start (and the end) of the turn are unavoidable. Figure 24.15 provides a look at the a tual lateral deviations, whi h are within the limits.

Segment III: the apture of the -6 and -3 degrees glideslope A des ent with a glideslope of -6 deg is initiated; again it is unavoidable that the air raft leaves the desired traje tory.

It should return to the traje tory

without overshoot within a period of 30 s. After that, a glideslope of -3 deg has to be established The verti al deviations from the desired glideslope are plotted in gure 24.16, whi h are well within the limits.

374

Fourth segment: altitude deviations 30

20

20

10 2

f

altitude deviation [m]

altitude deviation [m]

Third segment: altitude deviations 30

3

0

−10

e

−20

−30

10

3 g

4

h

0

−10

−20

−16

−15

−14 −13 x−position (XE) [km]

−12

−30 −11

−11

−10

−9

−8

−7 −6 −5 −4 x−position (XE) [km]

−3

−2

−1

Figure 24.16: Segment III: verti al

Figure 24.17: Segment IV: verti al

deviations from the desired glides-

deviations from the desired glides-

lope

lope

Segment IV: the nal approa h with windshear During the nal approa h with a glideslope of -3 deg the ee t of a windshear model is onsidered.

The verti al deviations from the desired glideslope are

plotted in gure 24.17, whi h are well within the limits.

Numeri al results Table 24.4 represents the numeri al results based on the previously dis ussed simulation out ome. For the motivation and al ulation prin iple of the various results see [145℄. Segm. I

Segm. II

Segm. III

Segm. IV

Total

Performan e

0.1058

0.1618

0.1541

0.0686

0.1226

Perf. Dev.

0.0177

0.0185

0.2252

0.2215

0.1207

Comfort

0.3651

1.2611

1.4841

0.6054

0.9289

Safety

0.0042

0.0423

0.0080

0.1049

0.0399

Power

0.0026

0.0083

0.0148

0.0308

0.0141

Table 24.4: Numeri al results of the evaluation pro edure

24.7 Con lusions The Model Following Control (MFC) te hnique is a very exible tool for ight

ontrol law design. It has been utilised for several years at DLR Institute of Flight Me hani s in both in-ight simulators ATTAS (Advan ed Te hnology Testing Air raft System) and ATTHeS (Advan ed Te hnology Testing Heli opter System), as well as for dierent ight ontrol system resear h proje ts. The basi philosophy of this method is to put all available information about the plant to be ontrolled into the feedforward bran h of the ontrol system.

375

Due to an exa t denition of the desired performan e and the limitations of the pro ess in the feedforward path, one omes to oni t-free ontrol a tions and, therefore, to minimum feedba k ontrol a tivity for manoeuvres. This leaves maximum authority to ope with un ertainties and disturban e reje tion enhan ing the robustness of the design. The a

urately determined separation of the main three elements ommand blo k, feedforward and feedba k ontrollers helps to provide a simple and modular ontrol system ar hite ture (gure 11.2). The expression  ommand blo k, whi h in ludes the model to be followed, has been introdu ed in order to avoid misunderstandings regarding the plant, whi h is also often alled model. It is obvious that any model hara teristi s implemented in the ommand blo k are limited by the dynami s of the plant to be ontrolled. In omparison to Classi al Control approa hes the MFC te hnique seems to require an unne essary additional eort regarding the model denition/design for the ommand blo k, the design of feedforward and feedba k ontrollers. But several benets were obtained:

 

low feedba k ontrol a tivity in omparison to pure feedba k systems, separated design of the main three elements by dierent teams with a

lear sharing of responsibilities,



saving time and money by implementing qualied ontrol laws in the

ommand blo k of new ight ontrol systems,



transparent ontroller stru ture allows an easy lo ation of design problems and re-design for other air raft,



modules for dierent appli ations an be dened, whi h are appli able to a omplete air raft family ( ommonality).

Espe ially, the last item demonstrates the power of the MFC on ept, when looking at a broader spe trum of appli ations.

The ommonality of ying

hara teristi s for an air raft family is a protable element onsidering pilot training and erti ation aspe ts. On e an optimum ommand blo k module is designed it an be used for a omprehensive variety of dierent air raft, while only the feedforward and feedba k ontrollers have to be adapted to the new air raft. On the other hand, on e the feedforward and feedba k ontrollers are designed for a spe i air raft, the ommand blo k an be hanged for any reasons, su h as dierent tasks and requirements, while the ontroller parts remain un hanged. A representative example for this te hnique is the in-ight simulation. But this feature an also be applied to operational ight ontrol systems; the ommand blo k an be hanged for dierent ight ontrol modes, su h as autopilot or Fly-by-Wire modes. The main onstraint to be onsidered in this general approa h is that the dynami s of the air raft model implemented in the ommand blo k are not faster than the dynami s of the air raft to be

ontrolled.

376

For the RCAM design problem an idealised air raft model with typi al autopilot fun tions has been installed in the ommand blo k. The air raft model has been obtained by a linearisation of the nonlinear RCAM at the design point. The autopilot fun tions are based on an available Classi al Control approa h to the RCAM design problem (gure 24.3). Nonlinear a tuator models are implemented in the ommand blo k, be ause otherwise its outputs may be too demanding. However, if required for any reasons it is also possible to implement a omplete nonlinear model in the ommand blo k. Utilising this available solution, the RCAM design problem was essentially redu ed to the sele tion of the ontroller stru ture, the determination of the feedforward ontroller and the optimisation of the feedba k ontroller.

The

ontroller stru ture for the ivil transport air raft lass with onventional ontrol surfa es su h as ailerons, elevator, rudder and throttle levers is xed in prin iple (gure 24.1). The separation of the three independent subtasks allows a straight forward design.

Assuming, that the designer has a basi knowledge of ight

dynami s, the method is very user-friendly.

This statement is supported by

the fa t that the main design work was performed by one person who had no previous experien es of the MFC on ept. He had only limited knowledge on ontroller design, but a profound ba kground on Handling Qualities and

TM

Matlab/Simulink

appli ations. The eort for the design and assessment of

the RCAM problem in luding preliminary do umentation was less than three man-months. The feedforward and feedba k ontrollers are linear with the ex eption of a turn orre tion for the measured pit h rate, whi h keeps the altitude onstant during a turn, and a simple gain s heduling for the ontrol surfa e dee tions. In view of an extension to all ight onditions mainly the ontroller part has to be extended. Additional design points have to be dened in order to al ulate a set of feedforward ontroller matri es and feedba k ontroller gains. A gain s heduling with sele ted blending fun tions an be utilised for the al ulation of the a tual values depending on the ight ondition or air raft onguration (mainly ap extension). If required, the feedforward ontroller an be extended to nonlinear equations, but this an in rease the ontroller omplexity tremendously. If the ommand blo k has been dened arefully, it an remain un hanged within the entire ight envelope. The assessment of the MFC system design for the various parameter hanges demonstrated adequate ontroller performan e and robustness against parameter variations. The design riteria regarding overshoot, airspeed to altitude

ross oupling, rise and settling time were met with only one single ex eption. Generally, the ases with varied airspeed and mass form the worst ases, while the ee ts of additional time delays and CG variations were negligible. The results of the automated evaluation pro edure yield a similar overall ontroller performan e ompared to the Classi al Control approa h, whi h forms the basis for the ommand blo k. It would be interesting to perform a dire t omparison between the two approa hes, whi h is not yet available.

377

The omplexity of the overall MFC system is basi ally determined by the

omplexity of the ommand blo k. The feedforward ontroller is dened only by three matri es. A feedba k ontroller using all signi ant states with proportional and integral terms has been shown to be su ient.

The order of

the omplete ontroller is 24 ( ommand blo k: 22, feedforward ontroller: 0, feedba k ontroller: 2). The stru ture of the MFC on ept is well suited for error dete tion and ontrol system re onguration strategies. These aspe ts will gain more importan e and are of high interest for future developments. It is proposed to sear h for a unied ontroller stru ture for ivil transport air raft types. Optimum ommand blo k modules for a omplete air raft family have to be dened, whi h in lude models of air raft with proven ight ontrol laws for manual ight (Fly-by-Wire) or existing autopilot fun tions.

378

25.

Flight Management Using Predi tive

Control

Mihai Huzmezan1 and Jan M. Ma iejowski Abstra t.

1

This hapter investigates the use of Model Based Pre-

MBP C ),

di tive Control ( trol based on

together with more onventional on-

H1 loop-shaping, to implement an autopilot for the

RCAM whi h performs ight management in addition to stability augmentation and guidan e. We believe that

MBP C

represents a

te hnology whi h should be onsidered in the transition from stability augmentation to ight management systems.

25.1 Introdu tion Model Based Predi tive Control (MBPC) a

ording to many authors has some very appealing attributes:

Simpli ity  the basi idea of

MBP C

is fairly

intuitive, and an be understood without advan ed mathemati s; Ri hness 

MBP C s hemes, su h as models, obje tive fun tions,

the ommon elements of

predi tion horizons, et , an be tailored to spe i problems; and Pra ti ality  the usual ombination of linear dynami s and inequality onstraints allows realisti nonlinearities to be handled [47, 173, 265, 222℄. See Chapter 12 for a tutorial presentation of

MBP C .

Notation in this Chapter is onsistent with

that used in Chapter 12. Constrained

MBP C

an remedy some of the drawba ks asso iated with

xed gain ontrollers:



Disturban es of large amplitude on the plant output might saturate the a tuators. This ould lead to poor output de oupling and potential loss of stabilisability in the ase of an unstable open loop plant.



Respe ting ight envelope onstraints is not straightforward and requires a priori de isions.



Pre-lters are usually hosen to give the fastest possible response without saturating the a tuators during a typi al pilot demand. This implies that small demands are a hieved in the same time as large demands.

We believe that

MBP C

is a te hnology whi h should be onsidered in the

transition from stability augmentation to ight management systems.

This

hapter summarises our experien es and gives some dis ussion of the potential of

MBP C

1

for ight ontrol. It is based on our experien e of:

Cambridge University Engineering Dept, Cambridge CB2 1PZ, England

379

1. Applying

MBP C to the RCAM Design Challenge as a stability augmen-

tation system, see [119℄.

2. Investigating a ombined see [192℄.

MBP C /H1

autopilot for the RCAM model,

25.2 Remarks on MBP C for Stability Augmentation This Se tion is in luded in order to give the reader a avour of the design

y le, ontrol strategy and ar hite ture employed when Model Based Predi tive Control (MBPC) method was used to a hieve stability augmentation. The level of su

ess a hieved, an overall ritique and some re ommendations for designers are given. For further details see [119℄.

The ontroller ar hite ture The ontroller stru ture we have adopted for the RCAM Design Challenge was an

MBP C

ontroller (inner loop) in ombination with a onventional

ontroller (outer loop).

The design onsisted of separate design of the inner

and outer loops, for ea h of the longitudinal and lateral hannels.

We have

adopted this hybrid stru ture for several reasons, partly te hni al and partly strategi . The RCAM Design Challenge was a good opportunity to he k strong and weak points of

MBP C .

Our de ision to use

MBP C in onjun tion with a MBP C

onventional outer-loop ontroller allowed us to investigate the limits of as a Stability Augmentation System (SAS).

The plant internal model The model used within the

MBP C

algorithm to generate the ontrol law is

a linearised representation of the nonlinear RCAM model (des ribed in Chapter 14.2) around an operating point situated in the middle of the ight envelope. The de oupling into longitudinal and lateral hannels for the air raft model followed onventional pra ti e on ight ontrol systems for ivil air raft. The linear internal models for the

MBP C

ontrol were obtained in two stages.

Firstly a linear model was produ ed using the RCAM nonlinear model, then a orresponding redu ed model was employed for ea h hannel, based on the de oupling between hannels and the previous experien e. This Design Challenge imposed restri tions on the available measurements (no a tuator states or outputs were assumed to be measurable), therefore the inner

MBP C

ontroller does not in lude su h a tuator models as part of its

internal model.

380

Measurement and Referen e Signals The hoi e of measurement signals depends on the quality of available measurements. Sensor models are not provided and therefore not in luded be ause they were assumed to be perfe t, whi h makes the usual hoi e of measurement signals di ult. Like other ontrol te hniques measurements are available.

MBP C

works better when state

Therefore our approa h to the RCAM Design

Challenge assumes estimation of the state from the nonlinear plant output measurements using the nonlinear plant equations instead of a onventional estimator. In general, the generation of the referen e traje tory for

MBP C

assumes

some knowledge of the set-point to be followed. For the SAS we assume no knowledge of the future set-point, therefore we generate the referen e traje tory as

r(k + l) = r(k)

for

l = 0 : : : N2

.

For other appli ations of

MBP C

this

issue has to be re onsidered be ause the lter whi h generates the referen e from the set-point an be regarded as another design variable whose ee t is approximately equivalent to an adjustment of weights, but more transparent for the designer in some ases. Sin e the RCAM is a transport air raft, the assumption of a onstant setpoint over the predi tion horizon  20 steps ahead an be maintained be ause of the slow variation of the ommand from one time step to another.

The

ommand is provided either by the pilot or the outer loop ontroller. For the longitudinal hannel the

MBP C q

ontroller is provided with the

Va ), verti al rate (z_ ).

following referen e signals: pit h rate ( ), air speed (

The

lateral ontroller has the following referen es provided: side slip ( ) and roll



angle ( ).

The design y le To summarise, the following steps o

ur during the design of an

MBP C

on-

troller for stability augmentation: 1. Determine the requirements for the losed loop behaviour. 2. Make a hoi e of measurement and referen e signals and produ e the internal model of the plant. 3. Dene the onstraints related to inputs, rates of hange in the inputs, outputs and states; onstru t matri es that represent these. 4. Choose appropriate dimensions for ontrol and predi tion horizons. 5. Have an initial hoi e of the ost fun tion weighting matri es. 6. Tune the performan e of the losed loop system by performing losed loop linear and nonlinear simulations. This involves iteration of steps 3, 4, 5. 7. Perform a stability and robustness analysis using analyti expressions, in the un onstrained ase, or simulations in the onstrained ase.

381

First segment: top view

Second segment: lateral deviations 300

100

0

0

a

b

lateral deviation [m]

x−deviation [m]

200 50

1

−50

100

0 1

c

d

2

−100

−200 −100 −20

−18

−16

−14

−12 −10 −8 y−position (−YE) [km]

−6

−4

−300 0

−2

1

2

30

30

20

20

10 2

f

3

0

−10

e

−20

−30

3 4 5 6 along track distance from point 1 [km]

7

8

Fourth segment: altitude deviations

altitude deviation [m]

altitude deviation [m]

Third segment: altitude deviations

10

3 g

4

h

0

−10

−20

−16

−15

−14 −13 x−position (XE) [km]

−12

−30 −11

−11

−10

−9

−8

−7 −6 −5 −4 x−position (XE) [km]

−3

−2

−1

Figure 25.1: Results of the automated RCAM Evaluation Pro edure in the ase of

MBP C

as a Stability Augmentation System

The ontroller behaviour The ontroller behaviour is assessed from the perspe tive of performan e, robustness, ontrol a tivity, passenger omfort and safety. Passing the ontroller through the Design Challenge Evaluation Pro edure des ribed in Se tion 14.3.3 we are able to laim relatively good operation a

ording to the various design

riteria des ribed in Se tion 14.3.2. Table 25.1 gives numeri al results based on the simulation results. For the motivation and al ulation prin iple of the various results see Se tion 14.3.3. In general, performan e requirements were met but, as seen in [119℄ and Figure 25.1. Segm. I

Segm. II

Segm. III

Segm. IV

Total

Performan e

0.5153

0.2792

0.1943

0.1138

0.2756

Perf. Dev.

0.0797

0.0675

0.0882

0.0601

0.0739

Comfort

0.6027

1.7584

0.6723

1.2348

1.0671

Safety

0.0081

0.0228

0.0100

0.0250

0.0165

Power

0.0055

0.0077

0.0151

0.0295

0.0145

Table 25.1: Numeri al results of the evaluation pro edure We onsider that at a high level the stru ture of the

MBP C

ontroller, as

well as the idea, are simple, but the transparen y of the ontroller is redu ed by the use of an on-line optimiser.

Although the elements of the ontroller,

382

namely the predi tor and the optimiser, orrelate learly with the fun tionality of the ontrol strategy, as des ribed in Chapter 12, the onstrained optimisation makes a pre ise task allo ation harder.

25.3 A Combined MBP C /H1 Autopilot for Flight Management Guidan e and Stabilisation We have onsidered a ight management system as an alternative appli ation for the

MBP C te hnique going beyond the stability augmentation system [192℄.

Here a novel ar hite ture is presented and tested in the same framework as that of the RCAM model. This stru ture provides a robust ontroller, from the point of view of performan e and stability, maintaining relatively low omplexity and over oming the negative aspe ts in the role onsidered so far.

Flight Management System

ATC

MBPC Guidance System Stability Augmentation System Hinf Aircraft Terrain Map

Figure 25.2: The autopilot ar hite ture

25.3.1 The autopilot ar hite ture A ight management system must optimise long term obje tives su h as passenger omfort and fuel onsumption over a priori known way-points provided by air tra ontrol. Moreover, urrent requirements in ight management are in reasingly on erned with 4D navigation  the ight path is time stamped and orre ted with the along-tra k error as dened in Se tion 14.3.3. The use of

MBP C

as a method to design a ight management system seems desir-

able, as all the above goals an be expressed by the standard

MBP C

problem

formulation. Stability augmentation and guidan e systems must be designed by a method that provides disturban e reje tion at both input and output of the plant, noise reje tion, robust stability, exibility in spe ifying the bandwidth of the ontroller, tra king and a ri h variety of analysis tools that will aid erti ation. For these reasons

H1 loop-shaping was used to design the stability augmenH1 loop-shaping ontroller

tation and guidan e system. The stru ture of the

is shown in Figure 25.5. The reader should onsult the Chapter 7 for the design pro edure employed. We do not laim that an

383

H1 loop-shaping ontroller

Delay

Aircraft States Actuators States

Aircraft Measurements

Mux

Mux Mux References H−inf with Measurements Controller

Transformation Matrix Selector States

Demux

Input Scaling

Demux Actuators Commands from H−inf Controller States

Aircraft Actuators

MBPC

Mux States

Figure 25.3: The ombined

MBP C /H1 ontroller

is the only suitable hoi e, but in the aerospa e eld, where it is appropriate to address the worst ase signals, the goals on erning guidan e and stability augmentation are well expressed by the standard

H1 loop shaping formulation.

By monitoring the a tuators and plant outputs and imposing onstraints on their behaviour

MBP C

[173, 137, 199℄

an help deal with the drawba ks

asso iated with the xed gain ontrollers. For example, the a tuators an be used to their limits.

MBP C

will adjust the referen e to the inner losed loop

H1 ontroller and RCAM air raft) so as to avoid violation of the ight enve-

(

lope and ensure good time response hara teristi s. All the above motivate the

following stru ture shown in Figure 25.3, few explanations being ne essary for a better understanding. The internal model of the the redu ed model of the air raft, the sele tor redu ed from

15

to

7

MBP C

ontroller ontains

H1 loop-shaping ontroller, a tuators and states.

The sele tor denes the dynami

system used for the blending of the altitude and verti al speed.

The mean-

ing of the transformation matri es shown in Figure 25.3 is asso iated with the redu tion of the linear inner-loop model. The ight management system must not destabilise or an el the ee t of the inner-loop and must be robust to modelling un ertainty. When employing

MBP C ontroller, see Figure 25.4, has to be robust MBP C ontroller an be less robust than the H1 ontroller as the un ertainty within the bandwidth is mu h smaller be ause it has been redu ed by the H1 ontroller. It is our intention that the MBP C ontroller, shown in Figure 25.4, provide this novel ar hite ture the

to un ertainty within the inner losed loop bandwidth. Hen e the

an optimised referen e to the inner losed loop without interfering with the

MBP C de ision variable,
u(k + l), MBP C ontroller (inputs to the inner-loop) are the blended altitude and verti al speed (z + z_ ) and the airspeed (Va ). The referen es are assumed to be provided from a data stabilisation and guidan e. Note that the

depends on the inner loop design. The outputs of the

base provided by the air tra ontrol (ATC) via Data Link. The estimated or measured states used within the

MBP C

H1 ontroller and the output mixer.

belong to the air raft, a tuators,

The ombined ontroller stru ture, shown in Figure 25.3, operates as a

Ts = 0:01 s) for the H1

multi-rate system having a high sampling rate ( troller and a low one,

Ts = 1 s, for the MBP C 384

ontroller.

on-

+ − 1/z Inner Loop Zero−Order Time States Hold5 Unit Delay2 Sum Mux Correction Mux OPTIMISER Inner Loop Otuputs

MBPC

Zero−Order Hold6

Mux

Memory Block

u + Delta u 1/z + Unit Delay Inner Loop Sum1 Command

Reference Data Base Reference

Mux MBPC Measurements Figure 25.4: The outer MBPC ontroller

Altitude and Vertical rate Mux Reference

− +

Airspeed Reference Mux References

Altitude Measurement

Sum Precompensator Input Scaling

Mux Vertical Rate Measurement

Selector Subsystem

H−inf Controller

Air Speed Measurement Mux Measurements

Figure 25.5: The inner

385

H1 ontroller

Actuators

A single ontroller, shown in Figure 25.5, for longitudinal hannel was designed for both stability augmentation and guidan e even though it is onventional to design an inner loop to provide stability and outer loop for tra king. The advantage of this stru ture is that the designer obtains insight in how robustness is traded o for good performan e in altitude following and lateral deviation minimisation. A tuator states and loop delays to simulate omputational delays were also used. For further details of the the design example in Chapter 7.

H1 ontroller onsult

Moreover, a single ontroller for both SAS and guidan e system (GS) has the added advantage of redu ing the number of states of the This be omes very important when employing an

MBP C

MBP C

H1

ontroller.

outer loop as the

uses an internal model of the losed loop for predi tion. The higher

the omplexity of the losed loop the longer the optimisation problem will take to be solved. Note that the

H1 loop-shaping ontroller K1 is pla ed in the

forward loop with no pre-lters.

The ight management role of the ate the traje tory for the

MBP C

MBP C

ontroller required us to gener-

o-line using the generator provided within

the onventional RCAM environment. This generator is not a part of our management system, be ause we have assumed the traje tory information oming from a data base given via Data Link by the air tra ontrol (ATC). Then the

MBP C

ontroller in a re eding horizon manner will use this information

on-line in order to produ e the optimised referen es for the inner loop.

time Clock

Memory

Mux

Time Base Demux

Aircraft Model Demux Trajectory Generator

Reference time Time Correction

Zero−Order Hold

Data Base Reference

Figure 25.6: The ight manager MBPC traje tory generator

The s heme shown in gure 25.6 was employed to produ e, using the dynami al model of the air raft, the 4D referen e traje tory sampled using the same sampling rate of the

MBP C

ontroller. In pra ti e this will be repla ed

by a system whi h interpolates the ATC way-points using the air raft dynami model.

386

25.3.2 The ight management ontroller design pro edure The design y le 1. Determine the requirements for the ight management, guidan e and stabilisation systems behaviour 2. Design an

H1

loop-shaping ontroller that will ensure disturban e re-

je tion, noise reje tion and robust stability as des ribed in Chapter 7 by translating the pertinent requirements into dynami pre and post ompensators. 3. Choose the implementation and lo ation of the

H1

loop-shaping on-

troller in the inner-loop. Produ e a low order state-spa e model of the inner-loop that will serve as the

MBP C

internal model.

4. Dene the onstraints related to inputs, rates of hange of the inputs, outputs (ight envelope limits) and states (a tuator limits) and onstru t matri es that represent these over the ontrol and predi tion horizons. Choose appropriate values for the

MBP C

tuning parameters: ontrol

and predi tion horizons and the ost fun tion weighting matri es in order to meet safety, omfort and overall ight management system performan e requirements. 5. Tune the losed loop ost fun tion parameters via losed loop linear and nonlinear simulations. This involves iteration of step 4. 6. Perform a stability and robustness analysis using the analyti expressions in the un onstrained ase or time simulations in the onstrained one.

The translation of design riteria into MBP C method dependent obje tives The design riteria for the outer

MBP C ontroller involve safety, omfort, on-

trol a tivity and performan e of the overall system. These have to be translated into the hoi e of several

MBP C

tuning parameters  ontrol and predi tion

horizons, weighting matri es, sampling time and sometimes even onstraint boundaries. The hoi e of these parameters is based on several theoreti al results, but ertain rules of thumb as well, integrated together in a trial-and-error tuning pro edure. The safety riteria, ree ting the envelope safeguards, will provide the onstraint limits upon the variables involved in the on-line optimisation. Table 25.2 gives the output onstraints used and their physi al interpretation. Having a tuator rates and positions available as states of the air raft model augmented with models of the a tuators and the

H1 ontroller, we are able to

impose onstraints on their behaviour as stated in Table 25.3.

387

Flight envelope onstraints

Minimum value

51:8  1:05 30 0

Airspeed (m/s) Verti al speed (m/s) Altitude (m)

Table 25.2: Flight envelope onstraints as Constrained Variable Name

Maximum value

51:8  2:5 +30 15000

MBP C

onstraints

Limits

0:436  ÆT  +0:174 0:261  Æ_T  +0:261 +0:009  ÆT h  +0:174 0:028  ÆT_ h  +0:028

Tail-plane Dee tion Tail Plane Dee tion Rate Engines Throttle Limits Engine Throttle Slew Rates

Table 25.3: A tuator onstraints as

MBP C

Unit rad rad/s rad rad/s

onstraints

The way to translate requirements upon omfort within

MBP C

is by em-

ploying ost fun tion weights that give suitable trade-o between tight following of a given traje tory and large loads on the air raft stru ture.

MBP C /H1 autopilot is obtained by apH1 loop shaping ontroller taking advantage of

The robustness of the ombined propriate design of the inner its main feature.

Now we larify how the performan e riteria of the are translated into the available tuning parameters.

MBP C

ight manager

The main performan e

variables that an be measured by using a step as the referen e traje tory are

tr ),

the rise time (

ts )

the settling time (

Mp ),

and the overshoot (

as dened

in [82, 159℄ for example.

The tuning of the ight management MBP C parameters In the following se tion, where we use the same notation for the tuning parameters as in Chapter 12, we give all the te hni alities of the design y le step involving tuning of the

The ontrol

Nu

MBP C

and predi tion

parameters.

N2

horizons. The inuen e of the ontrol and

predi tion horizons is mainly upon the performan e of the ontrolled system, but they have some inuen e upon robustness as well.

In general a smaller

ontrol horizon makes the ontrolled system more robust to un ertainties su h as parameter variations [222℄. In general the hoi e of these horizons takes into a

ount knowledge of the dynami s of the inner-loop. In ase of the ontrol horizon we perform a step response analysis of the system assuming a predened sampling period.

Our nal hoi e for the ontrol horizon was

in reasing it from the minimum value of

1.

Nu =4

after

The predi tion horizon is derived from the settling time having in general a lenght greater than the system order. A small horizon will redu e the

388

omputational omplexity, but must ontain at least the non-minimum phase behaviour in the ase of su h systems to be ontrolled using

MBP C .

The

robust performan e and stability impose extra boundaries on the horizons. An in rease of the predi tion horizon should be onsidered only if the ontrol system proves to have long settling time in order to avoid de reasing the speed of the ontrol algorithm to an una

eptable extent. The ee t of the predi tion horizon upon step responses (via its dening parameters: rising and settling time) was studied. The inuen e upon settling

1

time is relatively large for a predi tion horizon from smaller for horizons greater than

8.

to

8,

but signi antly

For our ost fun tion trading o ompu-

tational omplexity against robust stability we have in reased the predi tion horizon from

7 (the MBP C

internal model order) to

N2 =10.

The small ontrol and predi tion horizons ensured that the optimisation

0:9 s on a Sun SPARCstation 20).

an be solved in real time (

The sampling period

MBP C

Ts .

The sampling period plays an important role in

ontrollers. A possible hoi e for this parameter is ten times smaller

than the fastest settling time in the losed loop system (the value of

Ts

is ob-

tained using linear time response analysis assuming onstraints are ina tive). There is a tradeo between de reasing the sampling period and in reasing dimensions of matri es involved in the

MBP C

optimisations performed in the time unit.

algorithm or the number of

If the sample period is small in

relation with the servo performan e of the system (e.g.

settling time) this

will result in large ontrol and predi tion horizons, possibly ausing numeri al problems. However, the smaller the sampling period, the better an a referen e traje tory be tra ked or disturban e reje ted. In order not to interfa e with the inner ontroller, the

MBP C ontrol loop

should have a lower bandwidth than the inner loop. This allows a big value for the sampling time

Ts =

1 s, but does not require it. This makes possible

a real time implementation of the ontroller and simulation results with this value proved satisfa tory.

The weighting matri es

R, Q.

The weighting matri es upon the outputs and

ontrol in rements are important design parameters. Both give a measure of the tra king properties required from the losed loop system. Sin e the referen es for the inner-loop have already been s aled, as part of the inner loop design, su h that a unit hange of ea h referen e is equally

R

signi ant, it is possible to set

= diag

(1; 1),

and avoid tuning this weight

alltogether. This leads to onsiderable simpli ation of the tuning pro edure. The tuning is an iterative pro ess typi ally starting with rst step in the

Q

Q = diag(1; 1).

The

ontroller parameter design was to tune it in the absen e

of onstraints. This tuning of

Q

is not a one step pro ess. At rst we tune

using the time simulations involving the linear model of the plant. This step is followed by ne tuning, done by time simulations employing the full nonlinear model of the plant. In order to improve the passenger omfort, whi h means that the ontrol is less tight, we have to redu e

Q.

Be ause the

389

H1

loop-shaping ontroller

redu es the amount of un ertainty in the inner-loop we do not require so mu h robustness from the

MBP C

ontroller. Hen e this allows small values for

The nal value of the output weighting matrix was

Q = diag(0:007; 0:02).

Q.

On e we have de ided the initial tuning parameters we an pro eed to time simulations. For the rst stages of the tuning pro edure it is re ommended to have a short simulation time (six up to ten times the maximum time onstant of the plant) and at the beginning to start in the un onstrained ase and then to move towards the onstraint one.

Software tools for ontroller synthesis and analysis Software whi h we all a Development Spa e was onstru ted using MATLAB SIMULINK in order to enable us to perform analysis and simulations with various

MBP C

ongurations. This environment uses linear state-spa e ontin-

uous or dis rete

MBP C

an be nonlinear.

internal models although the simulation plant model

It handles non square systems, disturban es and an in-

lude state-estimation of the plant model as an external fa ility by adding the

orresponding SIMULINK blo k.

Expli it onstraints on manipulated input

hanges, input variables, outputs and states are handled.

It allows the user

to handle un onstrained problems, hara terised by an analyti solution, and

onstrained problems solved by means of hill limbing algorithms.

Clock

Reference Plot facilities Double click block to plot the variables involved in MBPC control Print graphs MBPC

Double click block to print the graph window Print graphs to file

Aircraft

The Plant Model

Double click block to save as "results.ps" the graph window Save results to file

Double click block to edit the model parameters START UP

Double click block to save the variables in "results.mat"

Double click block to load initial parameters

MBPC Tuning Parameters

Frequency plots

Double click block while the simulation is running to change the MBPC parameters

Double click block to plot the frequency plots in the unconstrained case

Figure 25.7: The SIMULINK MBPC Development Spa e The environment shown in Figure 25.7 provides fun tions and has features that give the user apability to design and simulate simple and omplex multivariable plants.

390

MBP C

ontrollers for

Only state spa e internal models

are used be ause of the reliability of the numeri al algorithms involved, and the possibility of interfa ing them with other state-spa e based tools. as the subspa e method  used in identi ation.

MBP C

the

method are passed immediately to the

su h

The tuning parameters of

MBP C

algorithm without

waiting for a simulation run to nish. This feature an be used to understand the inuen e of the various parameters and the way of tuning them. The main requirement of the Development Spa e is that the state measurements from the plant model are available if estimation of them is not used. When a redu ed model of the plant is employed, as the

MBP C internal model,

a state transformation is needed in order to provide the redu ed model, employed for predi tion, with the ne essary initial measurement. Therefore, in the SIMULINK plant blo k we have in luded the ne essary transformation matrix.

25.3.3 Analysis of the longitudinal hannel autopilot At this stage a trial and error pro ess omplemented by the designer's knowledge have been performed in order to a hieve the required performan e for the

ontrolled plant. The performan e was tested with the full nonlinear model by providing steps in the referen e signal of the ommanded altitude and velo ity. The altitude response for various aircraft configurations

The crosscoupling in airspeed for various aircraft configurations

10

0.6

0.4 The airspeed and the airspeed reference

The altitude and the altitude reference

5

0

−5

−10

−15

−20

0.2

0

−0.2

−0.4

−0.6

−25

−30 0

20

40

60 Time t [s]

80

100

−0.8 0

120

20

40

The engine throttles actuators

60 Time t [s]

80

100

120

The tailplane actuators

0.095

−0.07 −0.08

0.09 −0.09 0.085

−0.1

rad

rad

−0.11 0.08

−0.12 0.075

−0.13 −0.14

0.07 −0.15 0.065 0

20

40

60 Time t [s]

80

100

120

−0.16 0

20

40

60 Time t [s]

80

Figure 25.8: Results of the longitudinal analysis in the ase of Flight Management System (altitude

30m step response)

100

MBP C

In the altitude ase the ontrolled air raft was subje ted to a step of

391

120

as a

30m for

various gravity entre positions, mass variations and time delays (as des ribed in Se tion 14.3.3).

The altitude response for various ombinations of these,

together with the ross ouplings in air speed and orresponding a tuator (the engine throttle and tail-plane) movements are shown in Figure 25.8. The step tra king is within the spe i ations. The biggest variation from the nominal

ase being

4m.

bounds (eg.

The ross oupling between altitude and airspeed is within

smaller than

onstraints imposed for the

1kt). The ontrol MBP C design.

a tivity is limited within the

Conversely, he king the airspeed response subje t to a step of 13 m/s we

an on lude that the response is satisfa tory in the fa e of the same un ertainties as in the previous ase, see Figure 25.9.The traje tory following of the nonlinear air raft was studied using time simulations for various ongurations of the plant.

Channel

Responses

Longitudinal

altitude

Performan e a hieved

tr tr

velo ity

= 30 s, ts = 50 s, Mp = 0% = 4 s, ts = 30 s, Mp = 5%

De ision +,,+ +,+,+

Table 25.4: Analysis results a hieved with the nonlinear longitudinal plant

As shown in Figures 25.8, 25.9 and Table 25.4 results were satisfa tory. The use of the a priori information on the referen e traje tory an be observed when the re eding horizon me hanism brings it into the predi tion horizon of the

MBP C

ontroller.

In Figure 25.10 we depi t time responses of the RCAM longitudinal hannel full non-linear model.

MBP C

was implemented with onstraints pla ed

upon the a tuator dee tions and rates and on the outputs as ight envelope restri tions. We have subje ted the ontrolled air raft to dierent s enarios, relevant for the longitudinal ase (Figure 25.10). At

20 s the RCAM goes into a des ent at

a rate of 10 m/s. At 10 s there is a wind-shear of 10 m/s for a 10 s duration. The predi tion horizon of the

MBP C

is

N2

= 10 s.

The

H1

ontroller by

itself tries to re over from the disturban e as fast as possible and return to the original altitude of 1000 m. The ombined

MBP C /H1 stru ture, though,

takes into a

ount that the air raft is going to start des ending at 20 s and

does not try to rea h the original set-point hen e improving passenger omfort. In ee t the

MBP C

ontroller is modifying the referen e to the inner losed

loop. At the bottom of the des ent there is no overshoot for similar reasons. The transients in the rst 5 s (Figure 25.10) are due to the non-linear model not being perfe tly trimmed. The ombined ontroller minimises the overshoot and follows the referen e well within the design spe s from Se tion 14.3.2. At 75 s there is a head wind of 3 m/s that lasts for 17 s. As the predi tion horizon is 10 s it an be dedu ed that the disturban e reje tion apability of both

ontrollers is the same in these ir umstan es.

392

The airspeed response for various aircraft configurations

The crosscoupling in altitude for various aircraft configurations

14

8

12 The airspeed and the airspeed reference

The altitude and the altitude reference

6 10

8

6

4

2

4

2

0

−2 0

−2 0

20

40

60 Time t [s]

80

100

−4 0

120

20

40

0.18

−0.04

0.16

−0.06

0.14

−0.08

0.12

−0.12

0.08

−0.14

20

40

60 Time t [s]

80

100

120

−0.1

0.1

0.06 0

60 Time t [s]

The tailplane actuators

rad

rad

The engine throttles actuators

80

100

−0.16 0

120

20

40

60 Time t [s]

80

Figure 25.9: Results of the longitudinal analysis in the ase of

100

MBP C

120

as a

Flight Management System (airspeed 13 m/s step response)

Altitude descend response − Comparative results H−inf (dash−dot) and MBPC+H−inf (solid) controllers 1050

Speed response − Comparative results H−inf (dash−dot) and MBPC+H−inf controllers (solid) 85

1000 80 950 900

speed [m/s]

altitude [m]

75 850 800 750

70

65 700 650 60 600 550 0

10

20

30

40 time [s]

50

60

70

55 0

80

50

100

150

time [s]

Figure 25.10: Altitude and airspeed response (subje t to wind-shear) omparative results: dash dot the

H1 ontroller and solid the MBP C /H1 ontroller

393

25.3.4 Results of the automated evaluation pro edure The RCAM design hallenge involves designing a ontrol law that is able to perform an approa h to landing in the presen e of turbulen e and wind-shear, whilst remaining robust to mismodelling 14.3. This Se tion presents the methodology-independent results of the designed

ontroller. It is mostly based on the evaluation mission and s enario dened in Se tion 14.3.3. Both `overall tra king performan e' and `inner-loop behaviour' of the ontrolled system will be evaluated by means of bounds on key variables. A dis ussion of the behaviour of the ontrolled air raft will be based on the relevant ight segments for the longitudinal hannel. A omparison between the guidan e and stabilisation fun tions and the overall autopilot is provided. Finally, a summary of the omparative numeri al results of the evaluation will be presented. To prove the idea of the ombined ontroller stru ture the most relevant segment of the approa h manoeuvre is extra ted. This segment, that represents

6Æ and 3Æ glide-slopes. Æ 6 ; again it is unavoidable that the air raft

the nal des ent to land, requires the apture of We start with a glide-slope of

leaves the desired traje tory. It should return to the traje tory without overshoot and well within a period of 30 s. After that we go to a glide-slope

3Æ of

su h that we get an inverse behaviour with respe t to the desired traje tory, that should be about half the size of the rst response (if the system has a more or less linear behaviour). In Figure 25.11 the longitudinal response of the air raft is plotted for three entre of gravity lo ations and with bounds that spe ify a

eptable behaviour. Both ontrollers,

MBP C /H1

(dotted) and

H1

on its own (dash-dot),

behave in an a

eptable manner. The dieren e arises when onsidering the verti al deviation from the desired glide slope. While the

H1 ontroller tries

to follow the des ent referen e traje tory as losely as possible resulting in overshoots, the

MBP C /H1

ontroller takes advantage of the a priori known

traje tory, optimising and improving omfort and safety. The verti al deviations from the desired glide-slope are plotted in Figure 25.11. While on nal approa h with a glide-slope of

3Æ the ee t of a wind-shear

is onsidered (see segment IV gh in Figure 25.11, se ond graph). The verti al deviations from the desired glide-slope are plotted. Table 25.5 gives omparative numeri al results based on the two simulations with the two distin t ontrollers, the ombined autopilot the

H1 loop shaping ontroller on its own.

MBP C /H1

and

Ea h segment of the evaluation

pro edure is onsidered via the design riteria:

omfort, safety and power.

The performan e and robustness riteria addressed by the inner ontroller are not in luded.

Be ause the evaluation riteria are independent of the type of

ontroller used the table ontains al ulable indi ators that enable us to obtain an obje tive omparison between this one and ompletely dierent ontrollers from other design hapters. For ea h of the traje tory segments a single number was al ulated. The smaller the numbers the better the design.

394

The motivation and al ulation

Fourth segment: altitude deviations 30

20

20

10 2

f

altitude deviation [m]

altitude deviation [m]

Third segment: altitude deviations 30

3

0

−10

e

−20

−30

10

3 g

4

h

0

−10

−20

−16

−15

−14 −13 x−position (XE) [km]

−12

−30 −11

−11

−10

Third and forth segment: The engine throttle actuators

−9

−8

−7 −6 −5 −4 x−position (XE) [km]

−3

−2

−1

Third and forth segment: The tailplane actuator movement −0.08

0.14 0.12 −0.1 rad

rad

0.1 0.08 0.06

−0.12 0.04 0.02 0 300

350

400

450

500

550

−0.14 300

350

400

Time t [s]

450

500

550

Time t [s]

Figure 25.11: Segment III: verti al deviations from the desired glide-slope and Segment IV: verti al deviations from the desired glide-slope and orresponding a tuators movements (The MBPC/H-inf ombined autopilot (dotted) and the H-inf ontroller (dash-dotted)

Criteria

MBP C /H1

H1

Comfort

0.5137

1.3316

Safety

0.0075

0.0084

Power

0.0155

0.0150

Table 25.5: Comparative numeri al results

395

prin iples of these gures an be found in Se tion 14.3.3.

25.4 Con lusions and Lessons Learned The main di ulty in designing

MBP C

is in the tuning of the many param-

eters available in the algorithm. But we found that, with experien e, partly systemati tuning pro edures were developed, also see [151℄. It was possible to obtain a satisfa tory ontroller despite a very simple model used as an internal model by the

MBP C .

In order to move towards

more systemati design, from the robustness point of view, it was ne essary to use an inner robust stabilising

H1 loop-shaping ontroller to ensure robustness

against large perturbations (eg. delays, mass and gravity entre variations).

The absen e of well dened rules for hoosing the tuning parameters requires serious experien e of the designer in order to rea h a

eptable results.

But

assuming a good analysis tool set and adequate rules of thumb, the problems

an be over ome. It may be noted that this problem is not spe i for

MBP C ;

for example the learning urve that will bring a lassi al ontrol engineer to the stage when a omplex MIMO design an be su

esfully designed using

H1 will

take up to several months, as dis ussed in Chapter 7. The dieren e is that for the

H1 loop-shaping ontroller synthesis more systemati pro edures have

been developed.

On e the above mentioned obsta les are over ome, a fairly

systemati redesign pro ess an be developed, if the designer who inherits the redesign has experien e of the te hnique and of the pro edures used at the rst stage of design. The onstrained optimisation, whi h has to be solved on-line, augments the omputational omplexity of implementing the ontroller. The di ulty to be over ome was redu ing the omputation time su iently to allow real-time operation. This was ensured by the low sampling rate employed for the outer

MBP C

ontroller. Moreover, progress an be expe ted both in the e ien y

of solution algorithms for they run. Constrained

MBP C

MBP C

and in the power of the hardware on whi h

may not oer any advantages over more onventional

ontrol algorithms apart from straightforward onstraint handling. As a result we have presented in Se tion 25.3 an automati pilot based on a ombined use of an

H1 loop-shaping ontroller that will provide the stability augmenta-

tion and guidan e fun tions and an manager and overall supervisor.

MBP C

ontroller that will a t as a ight

As expe ted, the

MBP C

proved to be an

ee tive te hnique for ight ontrol, on e this higher-level obje tive of ight management was in luded. Current quali ation and erti ation pro edures are not appropriate for

MBP C

or some other modern ontrol solutions.

As long as onstraints are

not a tive our solution, having an analyti al form, an be ertied just as well as any other linear ontrol law. But that provides only a partial analysis of the

ontroller, when it is operating in its linear mode. Moreover, it ould be argued that if the autopilot goes unstable the

MBP C ontroller ould be swit hed o.

396

A knowledgements The work referred in Se tion 25.3 was developed in ollaboration with our

olleague George Papageorgiou [192℄. We are grateful for the omments given by the evaluators regarding our Report [119℄: Prof. Rudolf Bro khaus (DASA), Dr. M.P.S hifaudo (ALENIA), Dr. J.F.T.Bos (NLR), Dr. G.S hram (DUT), Dr.

R.de Vries (DUT). Mihai Huzmezan is supported by Pembroke College

Cambridge, the Lundgren fund, ORS s holarship, Cambridge Overseas Trust and CT Taylor fund.

397

26.

A Fuzzy Control Approa h

Gerard S hram and Henk B. Verbruggen1 1

Abstra t.

Pilot heuristi s of ying an air raft are implemented in

the design of a fuzzy logi ontroller (FLC). The FLC design exists of longitudinal and lateral outer loop tra king ontrollers ombined with lassi al inner loop attitude ontrollers. The rst step in ludes tuning the inner loop ontrollers using lassi al ontrol te hniques.

Next, the FLCs outer loop ontrollers are initialized

as linear, MIMO ontrollers in whi h the rules are derived from

ontrol strategies based on the experien e of pilots. The nal step in ludes ne-tuning of the FLCs by modifying the rules based on an initial evaluation of time responses with respe t to the spe i ations. Additional rules for low airspeed and engine failure are in luded as well, whi h show that gain s heduling and ex eption handling an be in orporated in a straightforward way. Following this three-step pro edure, a ompromise is found in whi h performan e and robustness properties are met at the ost of a slightly too high verti al and lateral a

eleration. The obtained FLC is of low order, deterministi and transparent.

26.1 Introdu tion Re ently, mu h attention has been paid to the appli ation of knowledge-based

ontrol te hniques for ight ontrol [228, 230℄. It shows that te hniques like neural networks and fuzzy systems an provide appropriate tools for nonlinear identi ation [156, 204℄, ontrol of high performan e air raft [183, 229℄, heli opters [195, 233℄, and spa e raft [26, 106℄, ight ontrol re onguration [142, 182, 263℄, and advisory systems [111, 232℄. In these appli ations, neural networks generally serve as nonlinear, sometimes adaptive, fun tion approximators, while fuzzy systems are used as supervisory, expert systems. The RCAM problem as formulated in Chapter 14 is a unique hallenge to investigate the feasibility of knowledge-based ontrol te hniques for a realisti

ivil ight ontrol problem. Be ause the problem is relatively well-understood, pilot heuristi s are available and linearized air raft models an be derived. Therefore, a hybrid ontroller stru ture is proposed. The inner loop ontrollers

onsist of lassi al attitude ontrollers whi h are tuned using the air raft models (see also Chapter 15). Then, for outer loop (tra king) ontrol, longitudinal and 1

Department of Ele tri al Engineering, Delft University of Te hnology.

2600 GA Delft, The Netherlands. Email:

P.O.Box 5031,

{g.s hram}{h.b.verbruggen}et.tudelft.nl 398

lateral Fuzzy Logi Controllers (FLCs) are developed. For this purpose, the

pilot heuristi s of ying an air raft are used. Experien ed human pilots know how to handle an air raft satisfa tory, i.e.

they know how to ombine om-

peting riteria su h as performan e, robustness, safety, and passenger omfort without mathemati al formulations. Fuzzy logi provides a transparent interfa e between the low-level, attitude ontrol of an air raft and the high-level reasoning of human pilots. The knowledge is aptured through the use of if-

then rules and linguisti terms like small, big, et . for ea h variable. Additional rules for low airspeed and engine failure are in luded as well, whi h shows that gain s heduling and ex eption handling an be in orporated straightforwardly in the same framework. A tutorial on FLC is provided in Chapter 13. This hapter introdu es the

ontroller stru ture in se tion 26.2, followed by the dis ussion in se tion 26.3 on how to translate the requirements whi h are imposed on the RCAM problem into the ontroller design.

Next, the ontroller is tuned in se tion 26.4.

An

extensive evaluation is given in se tion 26.5, and in se tion 26.6 the landing results are shown. Finally, on lusions follow in se tion 26.7. A detailed report on the FLC design an be found in [207℄.

26.2 Sele tion of the Controller Ar hite ture 26.2.1 Classi al inner loop ontrollers The design starts with a pit h attitude ontroller. In the lassi al ontroller, both pit h angle and pit h rate are fed ba k to the ontrol input of the tailplane by the stati gains

Kq

K , respe tively. An estimate of the pit h angle is _(t) = q(t) os((t)) r(t) sin((t)). In Figure 26.1 the

and

obtained by integrating

ontrol stru ture is shown.

- - K - 6 6



+

+

-

ÆT

 air raft

-

Kq



-

q

Figure 26.1: Blo k-s hemati representation of pit h attitude ontrol system.

Then, two lateral inner loops are sele ted: a roll attitude ontrol system, and a yaw damper. In Figure 26.2 the roll attitude ontrol system is shown: proportional feedba k to aileron in luding roll rate damping by the gains and

Kp , respe tively.

K

In Figure 26.3, the yaw damper is shown. The fun tion of

the yaw damper is to in rease the damping of the Dut h roll motion. In the yaw rate feedba k loop, a washout lter is added in order to allow a onstant yaw rate in ase of a o-ordinated turn (while sideslip is redu ed to zero, see outer loop ontroller). An extra (washed out) feedba k, whi h is in luded like in the

lassi al ontrol approa h ( hapter 15), is the gain and rudder dee tion

ÆR .

KR

between roll angle



This feedba k de reases the lateral a

eleration in

399

ase of roll attitude hanges.



- - K - 6 6 +

+

-

ÆA

 air raft

-

Kp



-

p

Figure 26.2: Blo k-s hemati representation of roll attitude ontrol system.

s s+1

ÆR

-? 6 s +

-

ÆR

KR



 r

air raft

-

Kr

s+1



Figure 26.3: Blo k-s hemati representation of yaw damper.

26.2.2 Fuzzy logi outer loop ontrollers The attitude ontrollers serve as inner loops for the longitudinal and lateral tra king ontrollers. Suppose that the longitudinal task of the pilot is to ontrol the altitude and velo ity of the air raft. The ontrol behaviour an be related to the total energy of the air raft [147℄. For example, if the pilot realizes that the air raft ies too low and too slow, he or she will in rease thrust in order to in rease the energy of the air raft. On the other hand, if the air raft ies too low but the velo ity is too high, than the pilot will in rease the pit h angle. Thrust in fa t in reases the total energy, while an ex hange between kineti and potential energy an be a hieved by pit h angle hanges (via the tailplane). Based on these heuristi s, two rule bases are dened for throttle setting and pit h angle ommands, respe tively (Tables 26.1 and 26.2).

The labels

are dened negative (N), positive (P), and extreme (E), very big (VB), big (B), medium (M), small (S), very small (VS), and zero (ZE). The number of rules is hosen quite large in order to make lo al modi ations of the ontroller possible. The terms are related to a series of membership fun tions between

[ 1; 1℄.

The position and shape of the membership fun tions are dened su h

that the initial input-output fun tion of the FLC is linear between the bounds (see Chapter 13). The membership fun tions are shown in Figure 26.4. In Figure 26.5, the longitudinal ontroller is shown.

The air raft blo k

represents the air raft dynami s in luding the inner loop pit h angle ontroller. The fa tors

[ 1; 1℄

S

used for s aling big and small errors are translated into the

domain, and ontrol ommands

[ 1; 1℄

are s aled towards minimum

and maximum ontrol a tions. The ontrol a tions are throttle settings

ÆT H1 = ÆT H2 ) and pit h angle ommands  .

ÆT H (=

It is well-known that human pilots

are able to mimi derivative and integral a tions as well [167℄. The derivative signals of altitude and velo ity are required for extra damping. For this purpose,

400

Table 26.1: Rule base for throttle setting, e.g. if velo ity error is PM (air raft too slow), and altitude error is ZE, then throttle hange is PS (more thrust). velo ity error

(!) (#)

NB

NM

NB

NE

NM

NVB

NS

NB

ZE

NM

altitude error

NS

ZE

PS

PM

NVB

NB

NB

NM

NM

NS

NS

NVS

PB

NM

NS

NVS

ZE

NS

NVS

ZE

PVS

NVS

ZE

PVS

PS

ZE

PVS

PS

PM

PS

NS

NVS

ZE

PVS

PS

PM

PB

PM

NVS

ZE

PVS

PS

PM

PB

PVB

PB

ZE

PVS

PS

PM

PB

PVB

PE

Table 26.2: Rule base for pit h angle ommands, e.g. if velo ity error is PM, and altitude error is ZE, then pit h angle hange is NS (nose little down). velo ity error

(!) (#)

NB

NM

NB

ZE

NM

PVS

NS

PS

ZE

PM

altitude error

NS

ZE

PS

PM

NVS

NS

ZE

NVS

PVS

ZE

PS

PVS

PB

NM

NB

NVB

NE

NS

NM

NB

NVB

NVS

NS

NM

NB

ZE

NVS

NS

NM

PS

PB

PM

PS

PVS

ZE

NVS

NS

PM

PVB

PB

PM

PS

PVS

ZE

NVS

PB

PE

PVB

PB

PM

PS

PVS

ZE

limb rate and horizontal a

eleration are used. The integral signal paths are required for zero steady-state errors.

The integral and derivative gains are

simply set at ommonly used values in lassi al ight ontrollers:

Ki = 0:1.

Kd = 5 and

Extra tuning of these values did not appear to be ne essary. Noti e

that the s aling, and the derivative and integral ompensation are performed in the rst and last blo k of Figure 13.3. Pilot heuristi s are again onsidered for the lateral outer loop ontroller. The sele ted variables to be ontrolled are heading angle

ylat , the sideslip angle

and heading rate

_

, lateral deviation

for o-ordinated turns. A human

pilot will generally handle heading and lateral deviation by roll/aileron ommands, and sideslip by rudder dee tion. This strategy is implemented in two

_ in ase _ ref ) is added separately.

FLCs. For a desired heading rate roll of

sin 1 ( Vg




of a o-ordinated turn, an extra

Lateral deviations must rst be translated into roll angle ommands. Suppose that a lateral deviation

ylat

is present.

In order to redu e the lateral

deviation, we want to initiate a negative lateral velo ity 10 smaller than lateral deviation. about 30s. Be ause

sin

 Vy_

y_

of e.g.

a fa tor

Then the lateral deviation will vanish in

(for small heading angles



), an extra desired


= 101
V1
ylat . In Table 26.3, a rule base is dened with heading error e = ref and the extra desired heading hange
as ante edents, and roll angle ommand heading angle hange an be determined by

401

input: altitude error

input: airspeed error

NB NM NS ZE PS PM PB

NB NM NS ZE PS PM PB

1

1

0.5

0.5

0 −1

−0.5

0

0.5

0 −1

1

−0.5

0

0.5

1

outputs: pitch angle and throttle NE

NVB

NB

NM

NS

NVS

ZE

PVS

PS

PM

PB

PVB

PE

1

0.5

0 −1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1 + Kd s 

Figure 26.4: Membership fun tions for longitudinal FLC.

- ?- SV - - Sh 6

VA;ref + href

-

[-1,1℄

+

- S - 1 + Ks -SÆ - 1 + Ks

ÆT H1 + Kd s 

[-1,1℄

FLC

air raft

-VA -h

+

inner loops

i

TH

-



i

Figure 26.5: Blo k-s hemati representation of longitudinal ontroller.

as onsequent. The rule base for sideslip error

e = ref is shown as well.

Table 26.3: Rule bases for roll angle and rudder ommands.

e (!)
(#)

sideslip

rudder

NB

NS

ZE

PS

PB

NB

NE

NVB

NB

NS

ZE

NB

NM

NS

NVB

NB

NS

ZE

PS

NS

NS

ZE

NB

NS

ZE

PS

PB

ZE

ZE

PS

NS

ZE

PS

PB

PVB

PS

PS

PB

ZE

PS

PB

PVB

PE

PB

PM

error

e

ÆR

The membership fun tions for heading and roll angle are shown in Figure 26.6. With the fuzzy sets whi h are used in the rule base, an initial linear fun tion is obtained in the interval

[ 1; 1℄.

The other indi ated membership

fun tions will be used in a later stage. For the sideslip error and rudder dee tion, the membership fun tions are shown in Figure 26.7. The fuzzy sets (NM,NS,ZE,PS,PM) are on entrated between

[ 0:25; 0:25℄.

In se tion 26.5,

it will be shown that in ase of an engine failure, the membership fun tions (NVB,NB,PB,PVB) must be used to represent the more aggressive ontrol behaviour of the (human) pilot. In Figure 26.8, the lateral ontroller is shown.

The

parameters as part of fuzzi ation and defuzzi ation.

S -gains

are s aling

Noti e that an extra

Ki = 0:2) is added for zero steady-state sideslip suppression.

integrator (

402

input: heading error NB

NS

ZE

PS

input: delta heading PB

NB

1

1

0.5

0.5

0 −1

−0.5

0

0.5

0 −1

1

NS

ZE

PS

−0.5

0

0.5

PB

1

output: roll angle NE

NVB

NB NM NS NVS ZE PVS PS PM PB

PVB

PE

1

0.5

0 −1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Figure 26.6: Membership fun tions for lateral tra king ontroller. output: rudder deflection

input: sideslip NB

NS

ZE

PS

NVB

PB

1

1

0.5

0.5

0 −1

−0.5

0

0.5

1

0 −1

−0.8

NB

−0.6

NM

−0.4

−0.2

NS ZE

0

PS PM

0.2

PB

0.4

PVB

0.6

0.8

1

Figure 26.7: Membership fun tions for sideslip ontroller.

26.3 Translation of Criteria into Design In the following, guidelines are provided on how to translate the design riteria as dened in se tion 14.3 into the ontroller design (see also se tion 13.4).



The desired performan e is addressed in two ways:

the hoi e of the

ontroller stru ture and the tuning pro edure. A natural bandwidth separation is a hieved by fast inner loops and slow outer loops.

This

means that the desired response time of roll and pit h angle ommands are dened a fa tor 4 to 5 smaller than for tra king. Furthermore, (small) integrators are added to ensure that altitude and airspeed steady-state errors are zero. Lateral deviation a ts as an integral signal for heading. Next, the ontroller is tuned based on time responses under nominal onditions. Tuning the s aling fa tors of the inputs of the FLCs determines the overall gain of the losed loops. For example, if the time responses are in general too slow, the s aling fa tors are in reased. Individual rules are adjusted su h that lo al desired hanges are a hieved, e.g. more aggressive ommands around a set-point. Finally, by adding new rules for

ertain onditions, impli it gain s heduling an be a hieved, e.g. in ase of an engine failure.



Robustness with respe t to enter-of-gravity and mass variations appeared to be no problem for the FLCs. Generally, sin e human operators often perform as low gain ontrollers, the obtained FLCs are quite robust.

403

_ ref

- sin 1 ( Vg
) ylat - 1 y_ 1
- S 10 V ref - -S 6 ref - - S 6 [-1,1℄

+

- S - ?

FLC

+

[-1,1℄

+

[-1,1℄

-



+

[-1,1℄

-

FLC

-

- SR - 1 + Ks ÆRi

[-1,1℄

air raft + inner loops

-

Figure 26.8: Blo k-s hemati representation of lateral ontroller.

Furthermore, the pilot is able to adapt his or her ontrol behaviour; a strategy whi h an be implemented in the FLC. In the heading FLC for instan e, it appeared to be ne essary to dene separate rule bases for low airspeed (impli it, smooth gain s heduling).



The ride quality requirements are essentially a

eleration limits and suf ient damping. The verti al and lateral a

eleration are losely related to pit h rate and roll rate, respe tively. In order to a hieve low lateral a

eleration, a ross-link from roll angle to rudder dee tions is added. In order to de rease the verti al a

eleration, a pit h rate limiter ould be added. In Chapter 15, these problems are dis ussed in detail. In order to a hieve damped responses, derivative signals in the form of horizontal a

eleration and limb rate are added in the longitudinal outer loop ontroller. Unfortunately, lateral a

eleration is not assumed to be known, and annot be used for damping of lateral responses like heading.



Con erning the safety requirements, no spe ial attention is paid to stall speed and angle of atta k prote tion.

If ne essary, an angle of atta k

prote tion an be added in the pit h inner loop ontroller, and a stall speed prote tion in the related FLC. Furthermore, the roll angle limit is simply a hieved by s aling the FLC su h that {-1,1} orresponds to the minimum and maximum values. Finally, sideslip angle redu tion is addressed by adding an integrator in the sideslip angle outer loop.



Finally, ontrol a tivity appeared to be a minor problem. As mentioned earlier, the ontroller is mainly of low gain. Additional tuning of ontrol a tivity an be addressed dire tly by the rule bases of throttle a tivity and rudder dee tion, and indire tly by the rule bases for pit h and roll angle, e.g. less aggressive ommands.

It is lear that the translation of human heuristi s into a FLC is a systemati pro edure. This fa t is also re ognized by industry, and re ently eorts have

404

in reased to dene a European industry standard for the development methodology of fuzzy logi systems, based on the ISO-9000 general system development guidelines [248℄. However, ne-tuning the performan e of the ontroller is a matter of trial-and-error like in lassi al ontrol, but using the provided guidelines and an understanding of the inuen e of ontroller parameters (see Chapter 13), a satisfa tory ontroller an be obtained.

26.4 Tuning of the Controller Parameters The hybrid ontroller, is tuned in three steps. First the inner loop ontrollers are tuned. Then the s aling fa tors of the initial FLCs are determined. Finally, the FLCs are improved by ne-tuning the rule onsequents and eventually adding new rules.

Step 1: Tuning the lassi al ontrollers In Chapter 15, it is shown how root lo us te hniques for tuning of lassi al

K = 1 and Kq = 1:5 are found for appropriate damping of phugoid motion and

inner loop gains an be used. For the pit h attitude ontroller, values of a response time of 5s.

Good yaw rate damping is a hieved with gain values

Kr = 2 and a washout lter time onstant  = 2s. A roll angle response time of 5s is a hieved with the gains K = 1:5 and Kp = 1:5. Finally, lateral a

eleration in ase of roll angle ommands is minimized with the gain from roll to rudder

KR = 0:4, and a washout lter time onstant  = 2s.

Step 2: Tuning the initial FLCs Tuning the initial FLCs of se tion 26.2 involves tuning the s aling fa tors. The

;S

S

S ;S

s aling fa tors of the FLC outputs ÆT H  and R  are dened su h that minimum and maximum allowed values orrespond to the {-1,1} limits. For throttle setting: 0.5 to 10 degrees, for pit h angle -10 to 20 degrees, and for roll angle and rudder dee tion -30 to 30 degrees. In the implementation phase, the integrators are pla ed in front of the s aling fa tors. Next, the s aling fa tors of the FLC inputs must be dened. A rst hoi e for the s aling fa tor of altitude

Sh

is

1 100 .

In other words: an altitude error

of 100m is assumed to be very big. An airspeed hange of 12 m/s results in approximately an equal total energy hange

1 Therefore, a fa tor of SV = 12

2

as an altitude hange of 100m.

is dened (nominal airspeed

VA = 80 m/s).

The initial longitudinal FLC is tested with a simultaneous step on altitude

and airspeed. The results show that the response time is too large. Therefore, the s aling fa tors are in reased to

Sh = 251

and

SV = 31 .

Noti e that the

s aling fa tors determine the overall gain of the altitude and airspeed loop. The ee t is plotted in Figure 26.9. 2

mgh + 1=2mV 2 . For onstant g

h = V

V + 1=2(
V )2 .

Total energy is dened as

equation an be derived:

405

total energy, the following

Airspeed response

Altitude response

81

1030

80

−− Sh=1/100,Sv=1/12

79

−. Sh=1/25,Sv=1/3

1020 1010

78 1000

77 76 0

20

40

990 0

60

20

40

Figure 26.9: Ee t of tuning the s aling fa tors

60

Sh and SV .

Similarly, the s aling fa tors for heading and sideslip are determined:

180 1  5

and

1 S = 180  5.

S =

Errors of 5 degrees are here interpreted as big errors.

Step 3: Fine-tuning the FLCs The initial longitudinal ontroller appeared to show good performan e ex ept (1) too mu h overshoot for altitude responses, and (2) too mu h altitude deviation for a large airspeed hange.

In order to improve the responses, the

following lo al modi ations are performed: (1) more throttle a tion around set-point for faster settling, and (2) greater priority for altitude with respe t to airspeed. The desired modi ations are a hieved by only hanging the rule

onsequents. The rule bases for throttle setting and pit h angle are shown in Table 26.4 and 26.5 where the arrows indi ate the modied onsequents. The ee t of the modi ations is shown in Figure 26.10. The ross- oupling on altitude is de reased without deteriorating the airspeed step response. The Airspeed response

Altitude response

85 80

1010 −− initial FLC

1005

−. tuned FLC 75

1000

70

995

65 0

20

40

990 0

60

20

40

60

Figure 26.10: Ee t of ne-tuning the rules. ee t of ne-tuning the rules an be further analyzed by al ulating the output of the FLC as a fun tion of altitude and airspeed errors. In Figure 26.11, the outputs are shown: the rule modi ations introdu e lo al nonlinearities into the initial, linear loop gain. The initial lateral ontroller shows good responses and needs no further onsequent modi ations, ex ept in ase of an engine failure. The lateral deviation of the ight path is too large and the roll angle ommand is larger than the allowed 10 degrees. In general, in ase of an engine failure, the human pilot

406

Table 26.4: Modied rule base for throttle setting: the throttle ommands are more aggressive around the set-point.

Ve (!) he (#)

NB

NM

NS

ZE

NB

NE

NM

NVB

NS

NB

NM

ZE

NM

NS

PS

PM

NVB

NB

NB

NM

NM

NS

NVS

ZE

NS

NVS

ZE

PVS

!NM !NS ZE !NS ZE !PS ZE !PS !PM

PB

PVS

PS

PS

PM

PS

NS

NVS

PM

PB

PM

NVS

ZE

PVS

PS

PM

PB

PVB

PB

ZE

PVS

PS

PM

PB

PVB

PE

Table 26.5: Modied rule base for pit h angle: the onsequents hanges are dened su h that ontroller a tions with respe t to altitude errors are more aggressive, and with respe t to velo ity errors less aggressive.

Ve (!) he (#)

NB

NM

PS

!NVS !ZE !PVS !PS !PM

!NS !NVS !ZE !PVS !PS

PM

PVB

PB

PB

PE

NB NM NS ZE

!PE

NS

ZE

!NM !NS ZE PVS PS

!PB !PVB

PS

PM

!NB !NVB !NE !NM !NB NB NVS NS !NS ZE NVS !NVS PVS ZE !ZE !PM !PS !PVS !PB !PM !PS

PB NE NVB

!NM !NS !NVS !ZE !PVS

will in rease the power of the remaining engine in order to keep altitude and airspeed (energy on ept), and he will add extra rudder dee tion in order to keep the desired heading angle with minimal sideslip. Rudder dee tion in ase of motor failure is a general pilot pro edure. The power in rease is a hieved by the longitudinal FLC based on energy prin iples. The extra rudder dee tion is in prin iple also a hieved with the lateral ontroller. However, the human pilot will rea t more aggressively than for the ase of no engine failure, with respe t to heading and sideslip errors. The pilot behaviour is modelled by two extra rule bases, depi ted in Table 26.6. The onsequents are more aggressive than in the rule bases for no engine failure (Table 26.3).

The ee t of the

modi ations is that the gains of the lateral outer loops are in reased. Table 26.6: Rule bases in ase of engine failure.

e (!)
(#)

NB

NS

ZE

PS

PB

NB

NE

NE

NVB

NM

ZE

NB

NVB

NS

NE

NVB

NM

ZE

PM

NS

NB

ZE

NVB

NM

ZE

PM

PVB

ZE

ZE

PS

NM

ZE

PM

PVB

PE

PS

PB

PB

ZE

PM

PVB

PE

PE

PB

PVB

407

sideslip error

e

rudder

ÆR

throttle command

pitch command

12

30

10

20

8 10

6 0

4

−10

2 0 40

−20 40

20

5

5

20

0

0

0

−20 −40

altitude error (m)

−5

0

−20 −40

altitude error (m)

airspeed error (m/s)

−5

airspeed error (m/s)

Figure 26.11: FLC outputs as fun tion of airspeed and altitude errors.

Smooth swit hing between the rule bases for engine ON and OFF is automati ally a hieved by dening two membership fun tions for the extra input engine failure. The membership fun tions are shown in Figure 26.12. In ase of an engine failure, the rules for engine OFF are dire tly initiated by swit hing from 0 to 1. When the engine is restarted, the pilot swit hes gradually its ontrol behaviour from engine OFF to engine ON sin e he/she has to re over rst. In Figure 26.13, the extra input signal for the ontroller is shown: the engine breaks down at t=0s and is restarted at t=100s. The ee t of the extra rules for engine failure is also shown. The lateral deviation and roll angle responses are de reased by the more aggressive rudder ommands. input: engine failure ON

input: airspeed

OFF

LOW

1

1

0.5

0.5

0 0

0.5

0 60

1

NOMINAL

65

70

75

80

Figure 26.12: Membership fun tions for engine and airspeed inputs.

lateral deviation

engine failure signal 1.5

1

roll angle

20

15

0

10

−20

5

−40

0

0.5

− initial FLC −− with extra rule base

−60 0

0

50

100

150

200

−80 0

50

100

150

200

−5 −10 0

50

100

150

200

Figure 26.13: Ee t of extra rule bases in ase of engine failure. After tuning the ontroller for nominal onditions, robustness is studied.

408

The only real problem appeared to be airspeed variations for the lateral ontroller. Using similar roll ommands for low and high airspeed indu es high

_

and low heading rates, respe tively, sin e



g Va


.

In order to make the

lateral heading ontroller more robust against airspeed variations, extra rule bases with smaller roll ommands are dened for low airspeed (Table 26.7). Table 26.7: Rule bases roll angle for low airspeed.

e (!)
(#)

NB

NS

ZE

PS

NB

NB

NS

NM

ZE

NS

PS PB

NM

NS

NVS

ZE

NS

NVS

ZE

PVS

NVS

ZE

PVS

PS

NVS

ZE

PVS

PS

PM

ZE

PVS

PS

PM

PB

e (!)
(#)

NB

NB NS

PB

NS

ZE

PS

PB

NB

NB

NM

NS

ZE

NB

NM

NS

ZE

PS

ZE

NM

NS

ZE

PS

PM

PS

NS

ZE

PS

PM

PB

PB

ZE

PS

PM

PB

PB

(Engine ON)

(Engine OFF)

Smooth s heduling between the rule bases for low (60m/s,

Vstall

for m =

150.000kg) and nominal airspeed is a hieved by dening the two membership fun tions whi h are depi ted in Figure 26.12. The ee t of the extra rule bases is shown in Figure 26.14.

heading initial FLC

heading with extra rules

8

8

6

6

4

4

2 0 −2 0

2 − nominal airspeed −− low airspeed 20

40

0 −2 0

60

20

40

60

Figure 26.14: Ee t of extra rule base for low airspeed on heading response.

26.5 Analysis of the Resulting Controller The performan e and robustness of the ontroller is tested under variations of the enter of gravity, mass, airspeed, and time delay.

In Table 26.8, nine

onditions under whi h the ontroller is tested are dened. se tions, the longitudinal and lateral results are dis ussed.

409

In the following

Table 26.8: Flight onditions: variation of airspeed, enter of gravity ( .o.g.), mass, and time delay. No.

Mass(kg)

Airspeed(m/s)

CGx ( )

CGz ( )

Time delay(s)

Variation

1

120.000

80

0.23

0

0

nominal

2

120.000

80

0.15

0

0.1

.o.g.

3

120.000

80

0.31

0

0.1

.o.g.

4

120.000

80

0.23

0.21

0.1

.o.g.

5

100.000

80

0.23

0

0.1

mass

6

150.000

80

0.23

0

0.1

7

120.000

67.4

0.23

0

0.1

1.3

8

120.000

90

0.23

0

0.1

max.

9

150.000

80

0.31

0.21

0.1

mass


Vstall Va

worst ase

26.5.1 Analysis of longitudinal ontroller First, in order to show that the ontroller indeed mimi s the pilot's behaviour a

ording to energy prin iples, an airspeed ommand of -3 m/s and an altitude ommand of +25 m are simultaneously given under nominal onditions. In Figure 26.15, the responses are shown. The a tuator responses show that throttle setting remains pra ti ally onstant, and that by a tailplane dee tion the energy is ex hanged from airspeed to altitude. Airspeed response (−−)

Altitude response (−−)

81

tailplane (−) and throttle (−−)

1030

80

0.1

1020

0

79 1010

−0.1

1000

−0.2

78 77 76 0

20

40

60

990 0

20

40

60

−0.3 0

20

40

60

Figure 26.15: Simultaneous airspeed and altitude ommand. The performan e for only altitude hanges is tested by a step ommand of -5m.

The (nearly identi al) step responses are plotted in Figure 26.16. The

requirements are fullled for the 9 ongurations.

Noti e the dieren es in

trimmed throttle and tailplane settings. Altitude response

tailplane (−) and throttle (−−)

1002

0.1

1000 0 998 −0.1 996 994 0

20

40

−0.2 0

60

20

40

60

Figure 26.16: Time responses altitude hange -5m.

410

The ross- oupling ee t on velo ity is tested by a step response of -30m. The maximum allowed velo ity deviation is 0.5m/s. The responses are plotted in Figure 26.17. From the gure it an be on luded that the ross- oupling spe i ation is met. Altitude response

Airspeed response

1010

0.4

1000

0.2

990 0 980 −0.2

970 960 0

20

40

−0.4 0

60

20

40

60

Figure 26.17: Time responses altitude hange -30m. Next, the ontroller is tested for airspeed ommands. In Figure 26.18, the responses are shown for an airspeed hange of -5m/s. Again, the spe i ations

on erning rise time, settling time and overshoot are met. Noti e the throttle setting: the energy is slightly de reased orresponding to the lower airspeed. Airspeed response

tailplane (−) and throttle (−−)

2

0.1

0

0

−2

−0.1

−4

−0.2

−6 0

20

40

60

−0.3 0

20

40

60

Figure 26.18: Time responses airspeed hange -5m/s. The ross- oupling ee t on altitude is tested by a step response of -13 m/s. The maximum allowed altitude deviation is 10m. Moreover, the airspeed deviation must be smaller than 2.6 m/s after 15s. The responses are plotted in Figure 26.19. Both requirements are met under all onditions. Airspeed response

Altitude response

5

1010

0

1005

−5

1000

−10

995

−15 0

20

40

990 0

60

20

40

60

Figure 26.19: Time responses airspeed hange -13m/s. Finally, in Figure 26.20, step responses for

411

= 3 degrees are shown.

Only

the settling time spe i ation is not met (30s instead of 20s). Be ause of the integrator, the verti al deviation from the ight path is zero. Noti e that the ight paths are dierent for low, nominal, and high airspeed. Gamma response

Altitude response

1

1100

0

1000

−1 900 −2 800

−3 −4 0

20

40

700 0

60

20

40

60

Figure 26.20: Time responses ight path hange -3 degrees/s.

The ride quality riteria are dened as maximum a

elerations and minimum damping.

Evaluating the verti al a

eleration during large step om-

mands, indi ates that the maximum value of values of

0:2g.

0:05g is ex eeded with peak

The problem is that verti al a

eleration is mainly aused by

pit h angle hanges. A rate limit on pit h angle ommands is a possible inner loop ontroller solution (see Chapter 15). On the other hand, the damping of the responses is very good. This was a hieved through the ompensation of altitude and airspeed errors by limb rate and horizontal a

eleration, respe tively. Noti e therefore that the horizontal a

eleration is very low as well.

1:05
Vstall bound, 12 degrees as dened as

During all simulations, the airspeed did not ex eed the and the angle of atta k did not ex eed the a

eptable

safety riteria.



The ontrol a tivity is evaluated under moderate turbulen e onditions ( = 1.5 m/s).

The response is shown in Figure 26.21 for nominal onditions.

The mean (absolute) throttle rate is 23% of the maximum rate (0.36 and 1.6 degrees/s), while the mean (absolute) tailplane rate is 21% of its maximum rate (3.2 and 15 degrees/s).

The throttle a tivity ex eeds the allowed 15%.

The problem is that the gain around the set-point was in reased to obtain less overshoot in altitude responses (Figure 26.11). However, the mean value of tailplane rate satises the allowed 33%. tailplane (−) and throttle (−−)

Ailerons(−) and rudder(−−)

10

10

5

5

0

0

−5

−5

−10 0

50

−10 0

100

50

100

Figure 26.21: A tuator responses moderate turbulen e (degrees).

412

26.5.2 Analysis of lateral ontroller The performan e and robustness of the lateral ontroller is also evaluated by a series of time responses under the onditions of Table 26.8.

The rst test

is to follow the lateral displa ement of 50m. The time responses are plotted in Figure 26.22. Both requirements on rise time and overshoot are satised. Noti e that the sideslip angle is redu ed to zero. lateral deviation (−−)

heading(−),sideslip(−−)

60

6

40

4

20

2

0

0

−20 0

20

40

−2 0

60

20

40

60

Figure 26.22: Time responses lateral displa ement 50 m. The heading response is tested by a heading ommand of 5 degrees. The step responses are plotted in Figure 26.23. The rise and settling time requirements are satised. However, the overshoot is about 20% in order to make the lateral deviation from the ight path zero. lateral deviation (−−)

heading(−),sideslip(−−)

10

8 6

0

4 −10 2 −20 −30 0

0 20

40

−2 0

60

20

40

60

Figure 26.23: Time responses heading 5 degrees. In ase of a o-ordinated turn, the roll angle must not be larger than 30 degrees. In Figure 26.24, the responses for a turn of

90 degrees in 30s is shown.

The roll angles are limited to 30 degrees. The lateral displa ement is dierent for low and high airspeed, but onverges to zero in all ases. The responses for engine failure are plotted in Figure 26.25.

Under all

onditions, the air raft is stabilized in about 40s. The requirements are fullled ex ept for the large overshoot of roll angle when the engine is restarted. The slightly os illatory responses belong to the air raft with large mass. Noti e the ee tive, aggressive rudder ommands, and the zero sideslip responses.

Ride quality in ludes lateral a

eleration and damping. The lateral a

eleration initially had large peak values for normal manoeuvres.

The lateral

a

elerations are mainly due to sudden roll angle hanges, like in o-ordinated turns. In order to redu e the lateral a

eleration, a feedba k is added in the inner loop ontroller from roll angle to rudder dee tion.

413

This redu es the

roll angle (−−) 40

lateral deviation (−−)

heading(−),sideslip(−−)

100

150

0

100

−100

50

−200

0

20

0

−20 0

20

40

60

−300 0

20

40

−50 0

60

20

40

60

Figure 26.24: Time responses o-ordinated turn 3 degrees/s for 30s. lateral deviation (−−)

heading(−),sideslip(−−)

10

4

0

2

−10 0 −20 −2

−30 −40 0

50

100

150

−4 0

200

roll angle (−−)

50

100

150

200

Ailerons(−),rudder(−−)

10

20 10

5

0 0 −10 −5 −10 0

−20 50

100

150

−30 0

200

50

100

150

200

Figure 26.25: Time responses in ase of engine failure and restart.

lateral a

elerations, but for a o-ordinated turn with heading rate of 3 degrees/s, the peak value is still 0.04g. During the simulated heading and lateral manoeuvres, the a

eleration is smaller than the allowed 0.02g. During the engine failure, the maximum lateral a

eleration is about 0.08g, whi h is smaller than the allowed 0.2g. The se ond riterion in ludes good damping. Responses to lateral displa ements are ni ely damped with no overshoot. Heading angle responses have too mu h overshoot; priority is given to rise time and settling time requirements at the low altitude. For safety reasons, the roll angle is limited by 30 degrees. This was a hieved by the bounds of the FLC for roll angle (see simulation of o-ordinated turn). The se ond riterion in ludes sideslip angle. Sideslip angle is de reased to zero at all times, even in ase of an engine failure, by using an integrator. Moreover, the RMS of sideslip angle is 0.52 degrees in open-loop, and slightly smaller, 0.49 degrees, in losed loop for unit RMS intensity lateral Dryden gust. The RMS values of heading angle are 0.58 and 0.29 degrees, respe tively. The lateral ontrol a tivity is evaluated under moderate turbulen e on-



ditions (

= 1.5m/s).

The responses are shown in Figure 26.21.

The mean

aileron rate and the mean rudder rate are 8% (2.1 and 25 degrees/s) and 6%

414

of their maximum values (1.5 and.

25 degrees/s), respe tively.

The ontrol

a tivity riterion of 33% is met. It is noti ed that the roll angle response under moderate turbulen e onditions is smaller than the allowed 5 degrees.

26.6 Results of the Automated Evaluation Pro edure Finally, the ontroller is tested by the automated evaluation pro edure under varying enter of gravity and time delay (four test ases, see se tion 14.3). During the rst traje tory segment, an engine failure (a) and restart (b) takes pla e. In Figure 26.26, the (nearly identi al) lateral displa ements are shown. The steady-state value of lateral deviation does not onverge to zero. On the other hand, roll angle and lateral a

eleration are limited and sideslip angle is redu ed to zero (see also Figure 26.25). The se ond segment involves a o-

Second segment: lateral deviations

First segment: top view

300 100

0

0

a

b

lateral deviation [m]

x−deviation [m]

200 50

1

−50

100

0 1

c

d

2

−100

−200 −100 −20

−18

−16

−14

−12 −10 −8 y−position (−YE) [km]

−6

−4

−300 0

−2

1

2

3 4 5 6 along track distance from point 1 [km]

7

8

Figure 26.26: Lateral responses Segment I and II. ordinated turn of 3 degrees/s. In Figure 26.26, the lateral deviation is shown. The roll angle and lateral deviation are within the bounds, but the peak of lateral a

eleration is slightly too high (0.04g) (see also Figure 26.24). During the third segment, a landing approa h of -6 and -3 degrees is simulated. In Figure 26.27, the longitudinal responses of the air raft are plotted; all are within the bounds that spe ify a

eptable behaviour. Verti al a

eleration is during the approa h of -6 degrees slightly too large. Finally, the ee t of a windshear model is onsidered. In Figure 26.27, the verti al deviations from the desired path are shown. The responses are just inside the predened bounds. This response may be improved by in orporating forward-looking (pilot-like) guidan e strategies in ase of windshear. For example, extra throttle setting at the en ounter of the windshear, in order to ompensate the energy loss due to the mi roburst and outow of the windshear. Implementing su h rules into the FLC ould be performed in a similar way as for engine failure. The numeri al results based on the simulations are given in Table 26.9. The values indi ate that all spe i ations are satised (smaller than 1) ex ept the

415

Fourth segment: altitude deviations 30

20

20

10 2

f

altitude deviation [m]

altitude deviation [m]

Third segment: altitude deviations 30

3

0 e

−10

3

10

g

4

h

0

−10

−20

−20

−30

−16

−15

−14 −13 x−position (XE) [km]

−12

−30 −11

−11

−10

−9

−8

−7 −6 −5 −4 x−position (XE) [km]

−3

−2

−1

Figure 26.27: Longitudinal responses Segment III and IV.

omfort riteria in ase of o-ordinated turn and glide-slope apture.

These

problems were already dis ussed in detail in the previous se tions. The best solution is by modi ations of the inner loop ontrollers, sin e the a

elerations are proportional to attitude rates. Table 26.9: Numeri al results of the evaluation pro edure Segm. I

Segm. II

Segm. III

Segm. IV

Total

Performan e

0.1332

0.1582

0.1593

0.5086

0.2398

Perf. Dev.

0.0319

0.1090

0.0671

0.1574

0.0914

Comfort

0.4078

1.9181

1.0779

0.5829

0.9967

Safety

0.0082

0.0205

0.0078

0.2368

0.0683

Power

0.0035

0.0069

0.0152

0.0293

0.0137

26.7 Con lusions Be ause the ight ontrol problem is relatively well-understood in general, pilot heuristi s are available and linearized air raft models an be derived. Therefore, a hybrid ontroller stru ture is proposed. The inner loop ontrollers onsist of

lassi al attitude ontrollers whi h are tuned using the air raft models.

For

outer loop (tra king) ontrol, longitudinal and lateral Fuzzy Logi Controllers (FLCs) are developed using pilot heuristi s of ying an air raft (Figures 26.5 and 26.8). The on lusions of the FLC approa h are, regarding:



The Methodology: Fuzzy logi provides a transparent interfa e between the low-level, attitude ontrol of air raft and high-level reasoning of human pilots. Experien ed human pilots know how to handle an air raft satisfa torily, i.e. they know how to ombine ompeting riteria like performan e, robustness and passenger omfort without mathemati al formulations. The pilot heuristi s are aptured through the use of ifthen

416

rules and linguisti terms like small, big, et .

for ea h variable (Chap-

ter 13). Additional rules for low airspeed and engine failure are in luded as well, whi h shows that gain s heduling for dierent ight onditions and ex eption handling an be readily in orporated.



Tuning and E ien y: The translation of human heuristi s into a FLC is a systemati pro edure [248℄, but ne-tuning (optimizing) a fuzzy logi

ontroller based on time responses is a matter of trial-and-error like in

lassi al ontrol. However, using the provided guidelines of se tion 26.3, a basi understanding of the inuen e of ontroller parameters (Chapter 13), and following the three-step pro edure of se tion 26.4, a satisfa tory ontroller an be obtained.

Our experien e is that the design

involves a low number of iterations, ertainly when knowledge of lassi al

ontrol theory and ight me hani s is present.



User-friendliness: In order to design a FLC, basi knowledge of fuzzy logi theory and lassi al ontrol is ne essary.

Our experien e is that

fuzzy logi ontrol is not very di ult to learn, with a learning urve

omparable to lassi al ontrol.



General Results: The ontroller is evaluated with respe t to performan e, robustness, ride quality, safety and ontrol a tivity riteria (se tions 26.5 and 26.6). The results show that a ompromise is found in whi h performan e and robustness properties are good, at the ost of slightly too high verti al and lateral a

elerations.



Controller omplexity: The obtained FLC is of low order, deterministi , and transparent. The ontroller stru tures are based on human ontrol strategies whi h are very visible (re ognizable). The interpretation is easy be ause of the rule based hara teristi s, with lear linguisti labels like big and small a tions.

Even for non- ontrol-engineers, the a tions

of the FLC are easy to understand.

The nonlinear, stati fun tions of

the FLCs from error signals to ontrol ommands an be implemented by look-up tables in order to simplify al ulation, validation and erti ation (Chapter 13).



Lessons learned: Not being experts in ight ontrol and aeronauti s, the design involved some extra time to a hieve a realisti ontroller. However, our experien e shows that FLC, ontroversial but applied in many te hni al and non-te hni al areas [136℄, has potential for ight ontrol problems.

417

418

Part III

HIRM part

419

27.

The HIRM Design Challenge

Problem Des ription

Ewan Muir -

1 2

Abstra t.

A ommon ontrol law design problem has been spe i-

ed based on a military ai raft onguration as part of GARTEUR A tion Group AG-08 on Robust Flight Control.

The air raft

mathemati al model des ription is a

ompanied by a set of design spe i ations for a wide ight envelope ontrol law.

The design

spe i ations are mirrored by a set of evaluation riteria to fa ilitate omparison of dierent design methods. The design hallenge do ument also presents parti ipants with a list of attributes, overing the design method, design pro ess and resultant ontroller, whi h they should report on to provide a balan ed assessment of the omplete development of their ontrol law in addition to a presentation of results.

27.1 Introdu tion 27.1.1 Ba kground As many of the parti ipants in the GARTEUR a tion group are primarily interested in military air raft, it was felt that a ontrol design task based around a military air raft should be spe ied in addition to the ivil RCAM problem des ribed in hapter 14. The di ulty with spe ifying su h a problem is that modern military air raft mathemati al models are lassied. The military problem for the GARTEUR group was based on the only un lassied model of high enough delity available at the time. The model is based on the High In iden e Resear h Model (HIRM) s aled up to a size representative of modern ombat air raft. HIRM was originally developed under a Defen e Resear h Agen y programme as a vehi le for high angle of atta k aerodynami investigations. The origin of the model explains the un onventional onguration with both anard and tailplane plus an elongated nose. 1

FDS Dept, DRA Bedford, Bedford MK41 6AE, UK

2

The following authors ontributed to the original HIRM design denition: Mr S Bennanni

(DUT), Dr R Hyde (CCL), Mr P Lambre hts (NLR), Mr D Moormann (DLR), Mr S S ala (CIRA), Mr J S huring (NLR), Mr J Terlouw (NLR). Signi ant ontributions to the design

riteria were also made by Dr J Irving (BAe-MA) and Mrs K Ståhl-Gunnarsson (SMA).

421

The main obje tive of the study is the design of a ontrol augmentation system for HIRM. The ight ontrol system is to give good handling qualities a ross the spe ied ight envelope and also provide robustness to unmodelled plant dynami s, modelling un ertainties and variations in operating point within the ight envelope.

A

eptable noise and disturban e reje tion must

also be demonstrated. The problem is written to provide an aerospa e ontrol law ben hmark, whi h is both on ise and bounded, with a set of representative design aims for both robustness and handling qualities. At the same time, it

ontains most of the elements ae ting the implementation of ontrol laws in real systems and whi h designers need to allow for.

27.1.2 Contents of hapter This hapter summarises the full HIRM model and problem spe i ation des ribed in [177℄. Se tion 27.2 of this hapter details the HIRM onguration and the model stru ture and omponents. Se tion 27.3 des ribes the ontrol law design problem and design spe i ations. The riteria used in the evaluation of the ontrol laws are outlined in se tion 27.4.

27.2 Des ription of the HIRM model 27.2.1 Introdu tion The High In iden e Resear h Model (HIRM) is a mathemati al model of a generi ghter air raft whose onguration is shown in Fig. 27.1. This mathemati al model is based on aerodynami data obtained from wind tunnel tests and ight testing of an unpowered, s aled drop model.

The aerodynami s

therefore ontain degrees of non-linearity representative of modern ombat air raft. The wind tunnel and drop models were originally designed to investigate ight at high angles of atta k whi h is why the dataset extends over a wide angle of atta k and sideslip range (

50 to +120Æ, 50Æ ) but does not in lude

ompressibility ee ts resulting from high subsoni speeds. The HIRM mathemati al model used for this design study was derived by s aling up the data obtained from the drop model to reate an air raft of F-18 proportions. Engine, sensor and a tuator models have been added to reate a representative, non-linear simulation of a twin engined, modern ghter air raft. This gives an air raft with the hara teristi s shown in Table 27.2.1. The air raft is basi ally stable both longitudinally and dire tionally. There are however ombinations of angle of atta k and ontrol surfa e dee tion whi h

ause the air raft to be unstable longitudinally and/or dire tionally.

27.2.2 Blo k diagram of the system To provide a model in a ommon format for ea h of the parti ipants in the design hallenge, the software for the six degree of freedom HIRM model was generated automati ally by Dymola, where ea h identiable omponent of the

422

Figure 27.1: HIRM onguration.

model was oded in the form of equations with the onne tions between those obje ts representing their physi al intera tion. From the physi al des ription set up in Dymola, a onsistent symboli al mathemati al model was built automati ally by the Dymola symboli equations handler, and from that, e ient simulation ode for dierent simulation environments was generated. For the Matlab/Simulink simulation environment, ode was generated in the form of a Matlab m-le and of mex-les for Fortran or C. Fortran or C ode ould also be generated in the neutral DSblo k format whi h an be used dire tly within the ANDECS simulation environment. The Simulink blo k diagram of the six degree of freedom nonlinear High In iden e Resear h Model in luding nonlinear a tuator and sensor models is given in Figure 27.2. Ea h box in this blo k diagram will be des ribed in detail. an analyti al des ription of the air raft dynami s is given.

In Ÿ 27.2.3,

In Ÿ 27.2.4, the

a tuator model dynami s are detailed and in Ÿ 27.2.5, the sensor models are des ribed. The sti k hara teristi s assumed for the model are given in Ÿ 27.2.6. In Ÿ 27.2.7, the analyti al models of wind disturban es are presented.

423

Chara teristi

Magnitude

mass

x Iy Iz Ixz

15296.0

I

xy

kg

2 2 2 kg m kg m2 2 kg m m2

24539.0

kg m

163280.0

kg m

183110.0 -3124.0

yz

0.0

Wing area

37.160

I

Unit

and I

Mean aerodynami hord Wing span

3.511

m

11.400

m

Table 27.1: Air raft model inputs denition

Before starting a simulation symmetrical taileron (dtsc)

load ini****.mat of desired flight condition in the workspace

0 U0 trim_inputs to actuator

differential taileron (dtdc) 0

hirmsim outputs for simulation

symmetrical canard (dcsc) 0 differential canard (dcdc)

Mux

0 rudder (drc)

+ + − Sum_inputs1

time Time

Clock

Double click here for more information

hirmexa Actuator model

0

wind

nose suction (suctionc)

wind_input

Mux

hirmex

Demux

Mux

HIRM model (16 states)

Demux

hirmexs

hirmy measured variables

Sensor model

left engine throttle (thrott1c)

right engine throttle (thrott2c)

CONTROLLER

Mux1

Figure 27.2: Simulink blo k diagram of HIRM air raft system

27.2.3 Air raft dynami s model This se tion des ribes the HIRM dynami s model orresponding to the blo k

hirmex in Figure 27.2. The dynami s obje ts are depi ted in Figure 27.3. These obje ts are:

 

body des ribes the body dierential equations of motion. (see Ÿ27.2.3) two transformation obje ts des ribe the oordinate transformation between the body-xed oordinates of the body obje t and the geodeti

oordinates of the gravity obje t, and between the body-xed oordinates of body and the geodeti oordinates of wind, respe tively.

(see

Ÿ27.2.3)



al airspeed des ribes the relationship between the inertial movement, the wind, and the movement relative to the air.

  

aerodynami s des ribes the aerodynami for es and moments. (see Ÿ27.2.3) engine1 and engine2 des ribe the relevant engine behaviour. (see Ÿ27.2.3) atmosphere des ribes the model of the atmosphere. (see Ÿ27.2.3)

424

atmosphereConst ( as a function of height )

control inputs

aerodynamic calcairspeed (calculate airspeed)

engine_1

outputs for system analysis

body Equations of motion engine_2

for control BodyFixed

Transfor-

HIRM dynamics model

mation

BodyFixed

Transfor-

Geodetic

gravitationConst ( g = 9.81 m/s 2 )

mation

Geodetic

wind

wind inputs

Figure 27.3: Dynami s obje ts of HIRM air raft model inside blo k hirmex of gure 2.2



gravity des ribes the gravitational inuen e (see Ÿ27.2.3)

Model inputs, states, outputs and parameters The following inputs are available for ontrolling HIRM:

   

Symmetri al and dierential taileron dee tion Symmetri al and dierential anard dee tion rudder dee tion nose su tion (this was a feature in the original HIRM and was retained in this model for ompleteness but it was not used for the purposes of this design exer ise)



left and right throttle settings (these are used symmetri ally)

Be ause the vehi le started as a large s ale model for wind tunnel and drop testing, HIRM's wings were too slender to a

ommodate aileron a tuators and so dierential tailplane and anard dee tions are used for roll ontrol. In addition to the ontrol inputs, wind omponents in the 3 body axes an be in luded.

425

The air raft states onsist of:



the 3 body axis velo ities



the 3 body axis angular rates



the 3 air raft attitudes



the 3 omponents of air raft position



4 engine states (ea h engine is represented by a 2nd order model)

Control law designers an sele t from the following list of outputs as feedba k signals:



the 3 body axis angular rates



the 3 air raft attitudes



the 3 body axis a

elerations



air speed



Ma h number



altitude



angle of atta k



sideslip

In addition to these measured outputs, other signals are available for monitoring purposes:



ight path angle



ground speed



earth axis position North and East



thrust from ea h engine

Rigid body Equations of motion The equations of motion and transformations between the various axis systems used in the model are standard. As they an be found in any textbook in ight me hani s su h as [74℄, they are not des ribed in detail here.

426

Aerodynami s The aerodynami for e and moment oe ients are given by the summation of several omponents. Most omponents have the form for for e or moment

a

with respe t to parameter

b

C ( ; d)

. The oe ient ab is determined by linearly

interpolating between the values given in a look-up table as a fun tion of the variables

d.

and

Variable

is usually angle of atta k and variable

sideslip or relevant ontrol surfa e dee tion. Be ause of a dis ontinuity in the data at

20Æ ,

d

either

it was suggested that

designers should not evaluate their ontrol laws at this angle of atta k.

Engine model Ea h engine is modelled as shown in gure 27.4.

Throttle demand Throttle non-linearity

Thrust demand +

-

π 1+0.07958s

’Rate’ limiter

1 s

Sea level thrust

Thrust at altitude ρ ρ0

Figure 27.4: Engine model

A throttle demand of 0 sele ts idle whi h is 10 kN of thrust.

A throttle

demand of 1 orresponds to a maximum dry thrust of 47 kN. Full reheat is sele ted when the throttle demand equals 2 at whi h time the thrust is 72 kN. The rate at whi h the thrust hanges is dependent on whether the engine is in dry thrust or reheat. For dry thrust, the maximum rate of hange of thrust is 12 kN/s whereas in reheat it is 25 kN/s. If the rate limit is ignored, the engine dynami s an be represented by a se ond order transfer fun tion with a damping of unity and a natural frequen y of

2 rad/s.

The sea level engine thrust

FE0

is s aled with density as follows:

FE = FE0

 : 0

The engine setting angles are zero and so the thrust a ts parallel to the air raft x-body axis. Therefore:

FEx = FE ;

FEy = 0;

FEz = 0

Atmosphere The atmospheri model is that of a standard atmosphere with the following pressure and temperature variations.

427

T = T0 p = Tgrad = T0 = p0 = In the equations above, and

Tgrad

p

8 <

Tgradh 

 R Tg

grad h < 11000 m p0 TT0 g (h 11000) : p0 exp R T0 h  11000 m  0:0065 K=m h < 11000 m 0:0 K=m h  11000 m  288:15 K h < 11000 m 216:65 K h  11000 m  101325:0 Pa h < 11000 m 22632:0 Pa h  11000 m :

is the stati pressure,

its rate of hange w.r.t.

T

the absolute temperature

height. The subs ript 0 for temperature

and pressure represents the starting point for the interpolation, for heights of either

h = 0 m or h = 11000 m.

Gravity The gravitational onstant is assumed to be invariant with altitude and has a

2

value of g = 9.80665 m/s .

27.2.4 A tuator models Available motivators The motivators available for use in ontrolling the air raft are listed in se tion 27.2.3. The a tuator representations for ea h aerodynami ontrol surfa e are des ribed in se tions 27.2.4 to 27.2.4 below.

Taileron a tuator The taileron onsists out of a starboard taileron and a port taileron, whi h an be ontrolled independently. The taileron a tuator is modelled by the third order transfer fun tion

1 (1 + 0:026s)(1 + 0:007692s + 0:00005917s2) with a

80 deg/s rate limit.

Dee tion limits for starboard and port taileron are

+10 deg to 40 deg.

A positive dee tion is dened as trailing edge down. The inputs of the taileron a tuator model are dierential and symmetri al tailplane demand.

428

Rudder a tuator The rudder a tuator is represented by a se ond order transfer fun tion with a

80 deg /s rate limit:

1 (1 + 0:0191401s + 0:000192367s2) Dee tion limits are

30 deg.

A positive dee tion is dened as trailing

edge to port.

Canard a tuator The anard onsists out of a starboard anard and a port anard, whi h an be ontrolled independently. The following transfer fun tion is valid for both parts of the anard. The anard a tuator is represented by a se ond order transfer fun tion with

a

80Æ=s rate limit:

1 (1 + 0:0157333s + 0:00017778s2) Dee tion limits for starboard and port anard are

+10 deg to 20 deg.

A

positive dee tion is dened as trailing edge down. The inputs of the anard a tuator model are dierential and symmetri al

anard demand.

27.2.5 Sensor models The list of sensor signals available for ontrol feedba k purposes is given in se tion 27.2.3. Two dierent types of sensor dynami s models are used for the air data and attitudes, and the rates and a

elerations, as shown in Fig. 27.5

Sensor dynami s and signal onditioning The transfer fun tions for the individual omponents are as follows (assuming a Flight Control Computer (FCC) operating at



80Hz):

air data sensor dynami s:

1=(1 + 0:02s)



attitude sensor dynami s:



anti-aliasing lter:

1=(1 + 0:0323s + 0:00104s2) 1=(1 + 0:00398s + 0:0000158s2) 429

Air data and attitudes Sensors

Sensor dynamics

Actuators

Anti-aliasing filters

FCC compute delays

D/A conversion

Anti-aliasing filters

FCC compute delays

D/A conversion

Rates and accelerations Sensors

Sensor dynamics & notch filters

Actuators

Figure 27.5: Sensor hardware assumptions



rate and a

eleration sensors ( ombined sensor and not h lter):



averaging on rate and a

eleration data:

(1 0:005346s + 0:0001903s2)=(1 + 0:03082s + 0:0004942s2) (1 0:00208s)=(1 + 0:00417s)



ompute delay:



D/A onverter:

(1 0:0062s)=(1 + 0:0062s) (1 0:00208s)=(1 + 0:00417s)

Simplied sensor dynami s The sensor dynami s from Ÿ 27.2.5 an be approximated by the following transfer fun tions:



Rates and a

elerations:



Air data:



Attitudes:

(1 0:0173s + 0:00019s2)=(1 + 0:0401s + 0:000704s2) (905:92 14:437s + 0:116s2)=(908:77 + 29:573s + s2 ) (7161:8 82:317s + 0:3417s2)=(7162:3 + 190:85s + s2 )

Designers were instru ted that the approximations above ould be used during the design phase, but the full sensor dynami s des riptions from Ÿ 27.2.5 were to be used for ontrol law analysis.

430

Sensor noise The measurement noise is generated by passing pink noise of unit rms power through the following lters and adding this to the feedba k signal.



Noise hara teristi s for angular rates

p, q, r,

air speed, angle of atta k

and sideslip:







Noise hara teristi s for linear a

elerations





0:05 1 + 0:0089s + 0:000041s2

0:05 1 + 0:011s + 0:000063s2



0:053s 1 + 0:053s

0:053s 1 + 0:053s

ax , ay

and



az :

1 + 0:0039s + 0:000078s2 1 + 0:0018s + 0:000078s2



It is assumed that the altitude, Ma h number and attitudes



,  and

signals are relatively noise free. However these signals are not be dierentiated around the ross-over frequen y, as it is assumed that su h a dierentiated signal is more noisy than the orresponding rate measurement.

Measurement errors The measurements listed in se tion 27.2.3 above an be assumed to be a

urate ex ept for the following:

 and :

to within

2 deg.

These errors are assumed onstant during the period of a simulation.

27.2.6 Sti k hara teristi s The following sti k hara teristi s are to be used during the ontrol law evaluation against the performan e riteria of Ÿ 27.3.5. In eptor Longitudinal sti k Lateral sti k

Dee tion amplitude

Sti k for es

-72 to +120 mm

1.2 N/mm

-80 to +80 mm

0.5 N/mm

Table 27.2: Sti k hara teristi s.

27.2.7 Atmospheri turbulen e models The turbulen e model used is the Dryden spe tral model des ribed in

431

[2℄.

27.3 Control problem denition 27.3.1 Introdu tion Modern ghter air raft are designed with either unstable or only marginally stable ongurations whi h ne essitate ontrol augmentation systems.

The

obje tive of the HIRM design hallenge is to design a ontrol augmentation system whi h will tra k the demands listed in se tion 27.3.2 with a response whi h is in keeping with the handling qualities listed in se tion 27.3.5 a ross the ight envelope dened in se tion 27.3.3. They should also demonstrate good disturban e reje tion apabilities and insensitivity to sensor noise.

27.3.2 Control strategy Pilot ommands The pilot ommands should ontrol the following responses.



lateral sti k dee tion should demand velo ity ve tor roll rate. The velo ity ve tor roll is a roll performed at onstant angle of atta k and zero sideslip. The velo ity ve tor roll will therefore vary from a pure body-axis roll rate at

  

0 deg angle of atta k to pure body-axis yaw rate at 90Æ.

longitudinal sti k dee tion should demand pit h rate. rudder pedal dee tion should demand sideslip. the throttle lever should ontrol velo ity ve tor air speed.

Requirements of the ontrol system The automati ontrol system should satisfy the following requirements



The following ross- ouplings should be minimised. Changes in air speed should not ause pit h transients.

Pit h transients due to rolling ma-

noeuvres should be minimised. Velo ity ve tor rolls should ause minimal sideslip ex ursions.



10 deg and +30 deg angle 3g and +7g normal a

eleration. Overshoots of 5 deg and

The pit h rate demand system should limit at of atta k and

0:5g are allowable on angle of atta k and normal a

eleration respe tively.

The overshoots must be washed out and the air raft should return to the limiting values within 2 se onds.



The ontrol system must make use of the motivators in an e ient manner. For example, the anards and tailerons must trim the air raft in a manner whi h minimises drag.

432

27.3.3 Robustness onsiderations The ontrol system should maintain its good ying qualities and robustness a ross the ight envelope dened in se tion 27.3.3.

In addition, it should

demonstrate a toleran e to the model un ertainties listed in se tion 27.3.3, the measurement errors given in se tion 27.3.3 and omplian e with the hardware implementation issues identied in se tion 27.3.3.

Design envelope The design envelope for the HIRM ontrol law is

   

Ma h Angle of atta k Sideslip Altitude

0:15 to 0:5, 10 deg to 30 deg, 10 deg, 100 to 20000 ft.

Modelling errors For linear assessments, the ontrol laws need to be robust to the following errors in the aerodynami moment derivatives

Cmw Clv Cnv Cmq ; Clp ; Clr ; Cnp ; Cnr CmT S ; CmCS ; ClT D ; ClCD ; ClRUDDER ; CnT D ; CnCD ; CnRUDDER

0:001 0:01 0:002 10% 10%

For nonlinear assessments, the ontrol laws need to be robust to the following errors in the total moment oe ients

Cm Cl and Cn

0:03 0:008

The engine, a tuator and sensor models an be assumed to be a

urate and have zero toleran es.

Measurement errors Sensed values of angle of atta k and sideslip may not be a

urate. The ontrol system must be robust to the following measurement errors:

 and

2 deg 433

Hardware implementation onsiderations The ontrol laws must be designed to operate at

80Hz, the iteration rate of the

ight ontrol omputer (FCC). They an be designed by negle ting hardware implementation issues, but must be robust to the dynami s of stru tural lters, D/A onverters and omputational delay, whi h are given in se tion 27.2.5 above.

27.3.4 Robustness requirements In the multi-loop ase, the losed-loop system should be able to withstand the appli ation of simultaneous and independent gain and phase osets at the input of ea h one of the a tuators as shown in Fig. 27.6 without be oming unstable (note that the losed-loop system shown in Fig. 27.6 is only an example and does not represent a mandated ontrol system stru ture). The orresponding perturbation matrix

P

will be of the form

P = diag(K1 e |1 ; : : : ; K6 e |6 ) with

Ki and i

taking values in the regions shown in Fig. 27.7 (no toleran es)

and Fig. 27.8 (toleran es applied).

Add additional gain and phase or break for open loop analysis here Pilot demands

Command path filtering

e +

Outputs

u Actuators

Aircraft

Controller

Sensors

-

u = actuator demands e = error signals

Figure 27.6: Closed-loop system showing point for analysis

27.3.5 Performan e requirements 1. For single loop analysis, the open-loop Ni hols plot of the frequen y response between ea h a tuator demand signal

e,

u

and the orresponding error

obtained by breaking the loop at the point shown in Fig. 27.6

while leaving the other loops losed, should avoid the regions shown in Fig. 27.9 (no toleran es applied) and Fig. 27.10 (toleran es applied). The toleran es are listed in se tion 27.3.3 and should be applied to the linearised air raft models.

Note that when performing the frequen y re-

sponse, a gain of -1 needs to be in luded on the input or output to obtain

434

the orre t phase response. These hara teristi s should be valid for ea h loop. 2. For tests 2 and 3, the results from non-linear frequen y responses should be used. These responses are obtained by supplying a sinusoidal input of in reasing frequen y to the system.

The frequen y should in rease

logarithmi ally from 0.5 to 20 rad/s over a 20 s period. Sti k for es of 3.24 and 9.71 pounds (14.4 N, 12 mm and 43.2 N, 36 mm) should be used in pit h and 0.9 and 2.7 pounds (4 N, 8mm and 12 N, 24 mm) in roll. The time response of the pit h or roll attitude should be analysed using a Fast Fourier Transform to obtain the gain and phase hara teristi s of the response. The gain and phase an then be plotted in gures 27.11, 27.13 and 27.14. To ensure PIO resistan e, the hara teristi s of the frequen y

1

response between sti k for e in pounds (

pound for e =

4:448N)

and

pit h or bank attitude in degrees should meet the following requirements.

 

The pit h and bank attitude absolute amplitude gains at phase should be

< 16dB.

The magnitude of the average phase rate,

j_ averagej = 2f f

deg

_ average , dened as

f

should lie either within the level

180

;

1 or level 1

boundaries shown in

Fig. 27.12. It should be noted that the plot in Fig. 27.11 is only an example.

3. This riterion omplements the handling qualities metri s illustrated in gures 27.11 and 27.12 [3℄. The non-linear frequen y response from longitudinal sti k for e to pit h attitude should lie within the level 1 boundary of Fig. 27.13. The non-linear frequen y response from lateral sti k for e to roll angle should lie within the region labelled as providing a good response in Fig. 27.14. The frequen y responses are non-linear and are between the pilot demand and the air raft pit h attitude or roll angle (i.e. the integral of the velo ity ve tor roll rate). For these riteria, the term open-loop means that there is no pilot losing the loop between air raft response and pilot demand. The responses should be obtained for diering sti k for es: pit h sti k for es of

3:24 and 9:71 pounds (14.4 N, 12 0:9 and 2:7 pounds

mm and 43.2 N, 36 mm) and roll sti k for es of (4 N, 8 mm and 12 N, 24 mm) should be used.

4. The ratio of dropba k db to steady state pit h rate

qss

should be

0 < db =qss < 0:25s for pre ision tra king. Dropba k is dened in Fig. 27.15 [3℄.

435

5. The peak pit h a

eleration in response to a step input should be a hieved in

< 0:15 s.

6. The roll mode time onstant should be

 0:4s.

7. The maximum roll a

eleration in response to a step input should be

< 600 deg/s2.

8. The maximum roll rate should be approximately

70 deg/s.

9. Sideslip response requirements. The oupling in sideslip due to roll should be minimised and not ex eed

 > 15 deg.

0:5

deg for

 < 15

deg and 2 deg for

The step response to sideslip demand should lie within the

boundaries shown in Fig. 27.16. The sideslip response should also have

> 0:5).

an a

eptable level of damping ( 10. Speed ontrol requirements.

< 3%).

overshoot (

< 1%

(

The speed response should have minimal

The time onstant for small amplitude speed demands

throttle travel) should be in the range

0:75

to

1:5

se onds. For

large amplitude speed demands, maximum use should be made of the engine performan e. 11. Avoidan e of stru tural oupling. To avoid airframe/FCS stru tural oupling the following limitations should be observed.



The maximum high frequen y (frequen ies above

4

Hz) gain from

pit h rate (rad/s) to ontrol surfa e dee tion (rad) should be



The maximum high frequen y (frequen ies above

4

< 3:0.

Hz) gain from

normal a

eleration (g) to ontrol surfa e dee tion (rad) should be

< 0:09 rad/g.

12. Disturban e reje tion.

The ontrol system should minimise the ee t

of atmospheri disturban es on the air raft's ight path.

The ee t of

turbulen e on the air raft should be assessed during straight and level ight.

27.3.6 S heduling onsiderations 1. The amount of s heduling should be minimised. 2. Any s heduling must be against measurable air raft states as dened in se tion 27.2.3 above. 3. If angle of atta k is used for s heduling, any linear analysis must take into a

ount the impli it feedba k whi h this generates.

436

27.4 Evaluation riteria The nal ontrol law design should meet the requirements spe ied in se tion 27.3 a ross the ight envelope. For veri ation purposes, a set of spe i evaluation riteria are set out below, against whi h all entries to the design

hallenge should be measured. The riteria have been divided into four sub- lasses whi h are: robustness, performan e, physi al onsiderations and ontrol a tivity. Most of the evaluation riteria are based on two parti ular ight onditions whi h are:



0:4, 10000 feet altitude, 8:67 degrees in iden e

Flight ondition 1: Ma h and zero sideslip.



Flight ondition 2: Ma h

0:24, 20000 feet altitude, 28:9 degrees in iden e

and zero sideslip. The rst ight ondition an be thought of as a nominal ight ondition, and the se ond as an edge of the envelope ondition, likely to ause greater stability and a tuator limiting problems. The robustness assessments in se tion 27.4.1 are based on linear analysis.

The remaining evaluations in se tions

27.4.2, 27.4.3 and 27.4.4 are based on nonlinear responses and analysis.

27.4.1 Robustness riteria Nominal ight Ni hols plot For ight onditions 1 and 2, Ni hols plots are obtained by breaking the losedloop at the 6 plant a tuator demands (symmetri and dierential taileron, symmetri and dierential anard, rudder and symmetri engine), linearising the model and performing a frequen y response. robustness test des ribed in Figure 27.9.

These orrespond to the

The Ni hols plots are he ked to

ensure that they do not infringe the ex lusion zone shown in Figure 27.9.

Robustness to multivariable un ertainty For ight ondition 1, the losed-loop system stability is he ked when the simultaneous gain and phase osets des ribed in Ÿ27.3.4 and shown in gure 27.7 are added. Identi al gain and phase osets are added to ea h of the loops. As a minimum, the gain and phase osets at the

4 orners of the plot on gure 27.7

are tested.

Robustness to parametri un ertainty The test in se tion 27.4.1 is repeated for ight onditions 1 and 2 with the following perturbations added to the linear model aerodynami oe ients as des ribed in se tion 27.3.3

437

Cmw Clv Cnv Cmq ; Clp ; Cnr Clr ; Cnp CmT S ; CmCS ; ClT D ; ClCD ; ClRUDDER ; CnT D ; CnCD ; CnRUDDER

0:001 0:01 0:002 10% +10% 10%

27.4.2 Performan e riteria Disturban e reje tion Using the Dryden turbulen e model with moderate turbulen e levels, the RMS variations of pit h rate, roll rate, yaw rate and normal a

eleration are to be re orded for ight ondition 1.

Gibson riteria For ight onditions 1 and 2, the plots orresponding to Figures 27.13 and 27.14 are onstru ted. Due to rate limiting and other non-linear ee ts, the frequen y responses are, in pra ti e, a fun tion of sti k for e. Hen e the plots are onstru ted using sinusoids of

3:24 and 9:71 pounds for e (14.4 N, 12 0:9 and 2:7 pounds for e (4 N, 8 mm

mm and 43.2 N, 36 mm) in pit h and and 12 N, 24 mm) in roll.

Assessment manoeuvres The nonlinear response of the air raft should be tested by applying the step inputs given in Table 27.3. Responses showing the pit h rate, velo ity ve tor roll rate, sideslip angle, angle of atta k and air speed should be plotted for all of the above responses. Normal a

eleration should be plotted if this rea hes its positive or negative limiting value as dened in se tion 27.3.2. The above series of manoeuvres are repeated with the following un ertainties and measurement errors:

Cm Cl Cn

0:030 0:008 0:008

in ombination with a measurement error of

2 deg on in iden e and +2 deg

on sideslip.

27.4.3 Physi al onsiderations Airframe loading and departure prevention It should be re orded whether the following riteria (see Se tion 27.3.2) are satised:

438

Manoeuvre

1

5Æ/s pit h rate demands

2

70Æ/s roll rate demands

3

10Æ sideslip demands

4

+100

kn (w.r.t.

M = 0:3)

step

in air speed demand 5a

+10Æ/s

pit h rate demand fol-

M M M M M M M M M

360Æ velo ity ve tor +70Æ/s when air raft

roll on

at

30Æ

= 0:2, h = 1000 ft = 0:3, h = 5000 ft = 0:5, h = 15000 ft = 0:3, h = 5000 ft = 0:5, h = 15000 ft = 0:2, h = 1000 ft = 0:3, h = 5000 ft = 0:5, h = 15000 ft = 0:3, h = 5000 ft

M = 0:5, h = 15000 ft

lowed by

5b

Flight ondition

on

30Æ angle of atta k limit

limit. Followed by 5 5d

10Æ/s

pit h rate demand. Fol-

when

360Æ

velo ity ve tor roll

lowed by

omplete

raft has unloaded.

sume straight and level ight ap-

0Æ/s pit h rate demand when air-

when air raft has unloaded to reproximately.

Table 27.3: Step inputs

439

    

Max normal a

eleration

< 7g (with +0:5g maximum overshoot)

Min normal a

eleration

> 3g (with 0:5g maximum overshoot)

Max in iden e

< 30 deg (with +5 deg maximum overshoot)

Min in iden e

> 10 deg (with 5 deg maximum overshoot)

Settling time

< 2s following the manoeuvre

Stru tural oupling To avoid airframe/FCS stru tural oupling, the ontrol system should be analysed to ensure that the limitations stated in se tion 27.3.5 are observed.

27.4.4 Control a tivity riteria Max roll rate and pit h rate For ight ondition 2, the maximum roll and positive pit h rates obtainable should be re orded ( orresponding to full lateral and full longitudinal sti k). Time histories of the response and the ontrol surfa e dee tions should be given.

Control a tivity due to noise and turbulen e For ight ondition 1, the RMS a tuator demand a tivity, relative to the steadystate values of the a tuators when no turbulen e is present, due to the Dryden turbulen e model should be re orded for ea h of the six a tuator demands. The above should be repeated but in response to the sensor noise hara teristi s dened in se tion 27.2.5.

440

-4.5

6

Gain oset (dB)

-2.5

0Æ

30Æ

Phase

-

Æ

oset ( )

+2.5

+4.5

Figure 27.7: Gain and phase osets ex luding toleran es

6

Gain oset (dB)

-3.0

-1.0

0Æ

30Æ

Phase

-

Æ

oset ( )

+1.0

+3.0

Figure 27.8: Gain and phase osets in luding toleran es

441

6

6

Gain (dB)

3

180Æ

145Æ

-

Æ

Phase ( )

-3

-6

Figure 27.9: Gain and phase ex lusion zones on Ni hols plot for single loop analysis frequen y response ex luding toleran es.

6

Gain (dB)

4.5

1.5

180Æ

145Æ

-

Æ

Phase ( )

-1.5

4.5

Figure 27.10: Gain and phase ex lusion zones on Ni hols plot for single loop analysis frequen y response in luding toleran es.

442

Gain: degrees attitude per pound stick force (dB)

10

5

0 f -5

bw

-10

-15

-20 fc -25 2f c -30 -220

-200

-180

-160 Phase (degrees)

-140

-120

-100

Figure 27.11: Pit h/roll sti k for e to pit h attitude/roll angle frequen y response riterion.

150

Average phase rate (Degs/Hz)

Level 2

100

Level 1

50

Level 1*

0 0.2

0.4

0.6 0.8 1 1.2 1.4 Frequency, fc, at -180 degrees phase (Hz)

1.6

Figure 27.12: Phase rate riterion.

443

1.8

20 (−100,18) (−80,15)

Relative open loop amplitude (dB)

15

(−75,10)

10 (−100,6) 5

(−75,4)

L1 (−85,2)

0 (−110,0)

(−150,−3) −5 L1 −10

(−140,−12)

−180

−160

−140

−120 −100 −80 Open loop phase (degrees)

−60

−40

Figure 27.13: Pit h sti k for e to pit h attitude frequen y response riterion.

0.25 db 25

0.5 db

20

1 db

−1 db

15 3 db

10 Magnitude (dB)

−3 db 6 db

5 Sluggish response 0

−6 db

PIO activity

−5 −12 db

−10 Oscillation ratcheting

−15

Quick jerkey response

Good response

−20 db

−20 −25 −350

−300

−250

−200 −150 Phase (degrees)

−100

−50

0

Figure 27.14: Roll sti k for e to roll angle frequen y response riterion.

444

q (rad/s)

pitch attitude

qstat

qmax

dropback = db

db/qstat

time (s)

time (s)

Figure 27.15: Denition of dropba k.

Sideslip command requirement 2

1.8

1.6

Normalised sideslip angle

1.4 Upper boundary 1.2

1

0.8 Lower boundary 0.6

0.4

0.2

0 0

1

2

3

4

5 Time (seconds)

6

7

8

9

Figure 27.16: Sideslip requirements.

445

10

28.

Design via LQ Methods 1, Massimiliano Mattei Fran es o Amato 2 Stefano S ala and Leopoldo Verde

1

2

Abstra t.

In this hapter the appli ation of the LQ based te h-

niques to the HIRM design problem is onsidered. The stru ture of the proposed ontroller is basi ally a PI a tion designed with optimality riteria in order to a hieve the performan e and robustness requirements. Moreover, the ontrol s heme is ompleted by a nonlinear ompensation of the dynami pressure ee t, a feedforward ontrol a tion, some demand shaping lters and swit hing logi based on angle of atta k and Ma h number, as well as on the pilot demand. Most of the design requirements are easily satised; some others need a greater eort for tuning of the ontroller design parameters. A pro edure to tune su h parameters, formulated in terms of an optimization problem, is proposed. Simulation results from the Automated Evaluation Pro edure are also provided.

28.1 Introdu tion The following ontribution to the FM(AG08) Garteur Group is the result of a ooperation between the Flight Me hani s and Control Group of C.I.R.A. and the System and Control Group of the Dipartimento di Informati a e Sis-

temisti a of the University of Naples Federi o II. It onsists of the appli ation of the LQ based te hniques to the HIRM design problem.

These te hniques

(see also Chapter 4), whi h are basi ally linear plant oriented, annot be applied in a straightforward way to take into a

ount all the requirements given in Chapter 27. We propose a design y le based on the introdu tion of some auxiliary fun tions to a hieve an optimal hoi e of the design parameters of the

ontroller, namely the LQ matrix weightings. The main hara teristi s of the proposed ontrol s heme an be summarised as follows: 1. the nature of the ontroller is basi ally a multivariable Proportional plus

Integral (PI) ontrol law with matrix gains omputed by means of the standard LQ te hnique; 2. the ontroller is adaptive on the basis of the pilot demand (velo ity ve tor roll demand requires a s heduling of the PI gains); 1

Dipartimento di Informati a e Sistemisti a, Università degli Studi di Napoli Federi o II

via Claudio 21, 80125 Napoli, Italy, Tel.+39(81)7683172, Fax+39(81)7683686 2

Centro

Italiano

Ri er he

Aerospaziali

Via

Tel.+39(823)623949, Fax+39(823)623335

446

Maiorise,

81043

Capua

(CE),

Italy

3. the ontroller gains are s heduled, on the basis of the Ma h number and of the angle of atta k, so as to over the whole operating envelope of the air raft; 4. the robustness and performan e issues are addressed by utilizing a design

y le whi h optimizes a suitable ost fun tion, over the possible weighting matri es of the LQ (see Chapter 4). The hapter is organized as follows. The ontroller ar hite ture hosen and the translation of the HIRM design riteria into method dependent obje tives are dis ussed in Se tions 28.2 and 28.3 respe tively.

Se tion 28.4 is devoted

to the des ription of the design y le,swit hing while in Se tion 28.5 some numeri al results from the automated evaluation pro edure are presented. Finally Se tion 28.6 deals with some on lusions about the overall design a tivity

arried out by the CIRA+UNAP team. A more detailed des ription of su h a tivities an be found in the Garteur Report TP-088-26 [9℄, whi h is the main referen e of the hapter.

28.2 The Sele tion of the Controller Ar hite ture for the HIRM Problem p w demand

Mach, α

Switching Commands

Feedforward Generator

w Pilot demand

Demand shaping filters

Ref +

1 s

-

ym

+

PI gains with AW

+

+

+

Nonlinear Compensation

xm

Reference Signal Generator

2

ρ Vm

HIRM outputs

HIRM

+ -

+

Velocity vector roll rate computation

-

LQO Error Detector

Figure 28.1: HIRM Control S heme

The ontrol system ar hite ture proposed for the HIRM design problem is s hemati ally shown in Figure 28.1.

From a fun tional point of view, the

ontroller an be divided into the following sub omponents: i) a feedforward ontrol generator;

447

ii) a referen e signal generator; iii) a state feedba k PI a tion designed with the LQ te hnique; iv) swit hing logi based on the pilot demand; v) a nonlinear ompensation of the ee t due to the variation of dynami pressure; vi) Anti-windup a tion due to the presen e of multiple integrators; vii) a nonlinear transformation to ompute the velo ity ve tor roll rate from the measured variables; viii) a set of demand shaping lters; ix) swit hing logi based on the

 and Ma h variables;

x) a Luenberger observer to dete t and eliminate the onstant measurement errors on the

 and

angles.

28.2.1 The feedforward ontrol signal and the referen e signal generator The feedforward ontrol signal generator, trained o-line with the pro ess, provides the ontrol ommands to keep the air raft in straight and level ight at ea h point of the ight envelope. Let us onsider the nonlinear equations of the air raft motion in absen e of un ertainties and disturban es

x_ = f (x; u)

(28.1)

where

x = (V  p q r   Z engF engF 1 )T u = ( ÆT S ÆT D ÆCS ÆCD ÆR ÆT H )T

(28.2a) (28.2b)

are the state and the input of the air raft respe tively. To obtain the straight

Vd and to hd , sin e an analyti al approa h is not possible, we have to

and level ight ommands orresponding to a desired true airspeed a nominal altitude

numeri ally solve the following optimization problem

min z opt

where ments,

s X

i=1;::;7

qi2 fi2 (~x(zopt ); u~(zopt )) ;

(28.3)

fi is the i-th omponent of the ve tor fun tion f , qi are weighting elezopt = (  engF uT )T is the optimization variable, and

x~(zopt ) = ( Vd zopt (1) 0 0 0 0 0 zopt(1) 0 hd zopt (2) 0 )T u~(zopt ) = ( zopt (3) zopt (4) zopt (5) zopt(6) zopt(7) zopt (8) )T : The redundan y of the ontrol surfa es (presen e of anards and tailerons), in balan ing the moments a ting on the air raft, makes the solution of the

448

problem (28.3) indeterminate.

We an use this degree of freedom to sear h,

among the possible solutions, for the one whi h minimizes the drag a ting on the airplane, so as to satisfy the requirement given in Se tion 27.3.2.

This

ondition an be translated into a modied optimization problem

min z opt

qe

s X

i=1;::;7

qi2 fi2 (~x(zopt ); u~(zopt )) + qe2 engF2 ;

(28.5)

being a further weighting element. Hen e the solution of problem (28.5) provides the ontrol ommands, as

well as the angle of atta k value, to a hieve a straight and level ight ondition at a minimum drag. If we perform the omputation of su h ommands for a representative number of points in the ight envelope, we an store these values in the Flight

Control Computer (FCC) and use them as a feedforward ontrol law. Furthermore, sin e the synthesis of the feedba k ontroller is performed using the linearized models of the air raft in the neighbourhood of the states and inputs resulting from the above des ribed pro edure, the linearity of the PI ontrol a tion alls for the variable output variables and

ytr

y

ytr , y

being the a tual value of the

being the value of the output variables orresponding

to the straight and level ight. The task of the referen e signal generator is that of providing the value of

ytr

asso iated with the straight and level ight

ondition of the air raft at the beginning of ea h manoeuvre.

28.2.2 The PI feedba k ontrol a tion The ore of the proposed ontrol s heme is a multivariable PI feedba k a tion. We used the LQ ontrol te hnique to ompute the PI matrix gains following the guidelines drawn in [10℄ (see also Chapter 4). Let us onsider the linearized model of the HIRM air raft in the form

x_ = Ax + Bu u y = Cx

(28.6a) (28.6b)

orresponding to a ertain ight ondition and assume the omplete measurability of the state variables (28.2a).

To satisfy the requirements given in

Se tion 27.3.2 we have to synthesize a ontroller whi h allows for the regulation of four output variables: the velo ity ve tor air speed, the pit h rate, the velo ity ve tor roll rate and the sideslip angle. We introdu e also

 among the

variables to be regulated for reasons to be laried in Se tion 28.2.3; therefore

y = ( V q pw  )T = Cx.

Let us now make referen e to the losed-loop s heme shown in Figure 28.1, where we onsider the linear plant to be the HIRM linearized model.

The

state-spa e realization of the integrator is

x_ i = e yi = xi ; 449

(28.7a) (28.7b)

where

e=r

y

is the tra king error and

r

is the referen e signal. We have

the following losed-loop system state-spa e equation





x^_ = A + BCu Kp 

where

x^ = xx i

 

Bu Ki x^ + 0 r 0 I

(28.8)

 . Equation (28.8) an be rewritten in a ompa t form as





x^_ = A^ + B^ K^ x^ + B^2 r where







A 0 ; B^ = Bu 0 C 0



(28.9)

 

and B^2 = 0I ; ^ = ( Kp Ki ) is a are the state-spa e matri es of the augmented system and K

A^ =

state feedba k gain a ting on su h a system. Obviously, the design of a state feedba k on the augmented system

K^ , portional and integral gain matri es for the original model (A; B; C ).

allows to ompute, via an appropriate partitioning of the matrix

^ B^ ) (A;

the pro-

As always it happens when adopting LQ based te hniques, the key point for the design is the hoi e of the weighting matri es for the quadrati ost fun tion related to the auxiliary system

^ B^ ), say Q^ and R^ . Indeed, our main obje tive (A; e, while maintaining the ontrol

is to keep as low as possible the tra king error

variables within the pres ribed ranges. This means that in the quadrati ost fun tion, dened on system

^ B^ ), the last ve states, whi h are the states of (A; Q^ terms (the

the integrators, should be emphasized by in reasing the relative

 is an ex eption, sin e in general it does not need to R^ matrix, a good trade-o between ^ and performan e and ontrol a tivity must be found; in any ase we assume Q R^ to be diagonal matri es. integrator of the error on

be regulated). To establish the hoi e of the

The PI ontroller stru ture proposed has three appre iable properties: i) the low order of the ompensator (only ve linear states oming from the integrators); ii) the simpli ity in the omputation of the gain matri es (we use the solution of a standard LQ problem); iii) some intrinsi robustness properties guaranteed by the LQ ontrol (see Chapter 4). However, there are several problems in using this te hnique whi h for ed us to add other omponents to the ontrol s heme: i) it is not possible to take into a

ount the nonlinear nature of the plant; ii) it is supposed to have the omplete a

essibility of the state; iii) there is no way to dire tly take into a

ount all the robustness and performan e spe i ations as required by the HIRM design problem;

450

iv) due to the presen e of multiple integrators there is a potential for windingup, espe ially sin e some of the ontrolled variables are related through kinemati relationships. To over ome problem i) we introdu ed two a tions:

a nonlinear s aling

of the ontrol ommands (see Se tion 28.2.5) and a swit hing logi between dierent sets of matrix gains, s heduled with respe t to Ma h number, angle of atta k and pilot demand. Con erning point ii), we have omplete a

essibility of the six DOF air raft model states; the problem is the presen e of the additional states of sensors and a tuators. By means of an extensive ampaign of simulations on the omplete linearized model of the air raft we ould verify that the dynami s of sensors and a tuators do not ae t the performan e of our ontroller. Hen e we propose to negle t them in the synthesis of the ontroller and to re-introdu e them when verifying the performan e of the losed-loop system. A possible way to over ome problem iii) will be dis ussed in the next hapter. Nevertheless, if the designer has a deep knowledge of the plant, it is not di ult to nd the proper weighting matri es to a hieve the performan e requirements.

Robustness requirements, in the form given in Chapter 27, an

only be he ked a posteriori if applying the LQ te hniques in a lassi al way. Finally, the winding-up potential due to the presen e of the multivariable integrators has been eliminated by a te hnique based on resetting integrators. This te hnique, dis ussed in the TP-088-26 Garteur report [9℄, prevents from wind-up phenomena, at the ost of a small performan e de rease, in the presen e of saturated ommands.

28.2.3 Swit hing logi on the pilot demand The velo ity ve tor roll manoeuvre requested in Chapter 27 opens a problem in the ontrol law design: there are oni ting a tions to be performed depending on the pilot demand. Indeed a speed variation whi h brings the air raft from a straight and level ight ondition to another one annot be performed at a onstant angle of atta k; onversely, when demanding velo ity ve tor roll rate, the angle of atta k has to be onstant as spe ied in Se tion 27.3.2. In order to satisfy these oni ting requirements we propose a swit hing ontroller depending on the pilot demand: we introdu e the denomination Manoeuvre

Oriented Swit hing Logi (MOSL) for the logi whi h governs the s heduling. If the pilot demands a velo ity ve tor roll rate greater than 10

deg=se the

gains of the ontroller swit h to a new set. The new gains have to take into

, whi h means that, in hoosing the matrix weight Q^ , the term relative to the  error integrator has to be in reased. Note that, for demands less than 10 deg=se , it is not ne essary to swit h the ontroller sin e there are only small os illations indu ed on . a

ount the regulation of

Assume that we have found two optimal ontrollers: one for the velo ity ve tor roll manoeuvre and one for all the other manoeuvres; we have to solve the problem of swit hing from one to another. Indeed swit hing between two

451

or more ontrollers in a dis ontinuous way ould ause instability of the nonlinear losed-loop system due to sharp transients and also, performan e may deteriorate. Many papers have appeared on the problem but there is still no general result. In order to guarantee the ontinuity of the ontrol ommands, we introdu e the following logi .

The feedba k part of our ontroller is, as illustrated in

Kp1 and Ki1 the proportional Kp2 and Ki2 the pro~ the portional and integral matrix gains of the se ond ontroller. Denoting by t Se tion 28.2.2, a PI ontroller. Let us denote by

and integral matrix gains of the rst ontroller, and by

swit hing time instant between the two ontrollers, we have:

u(t~ ) = Ki1 xi (t~) + Kp1 x(t~) (28.10a) + u(t~ ) = Ki2 xi (t~) + Kp2 x(t~) : (28.10b) If we add a feedforward input to the ontroller, say w , equations (28.10) be ome u(t~ ) = Ki1 xi (t~) + Kp1 x(t~) + w(t~ ) (28.11a) u(t~+ ) = Ki2 xi (t~) + Kp2 x(t~) + w(t~+ ) : (28.11b) By setting

w(t~+ ) = (Ki1 Ki2 )xi (t~) + (Kp1 Kp2 )x(t~) + w(t~ ) the ontrol signal u turns out to be ontinuous. Also the zero steady state error e = r y in the presen e of the additional

onstant input w is preserved by the presen e of the integrators in the loop. Note that, in the digital implementation of the ontroller, unit delays (that means additional states) would be used to store the values of the ontrol ommands during a sampling time.

28.2.4 The omputation of pw Sin e lateral sti k dee tion demands velo ity ve tor roll rate whi h is not a measured variable, we need to nd an e ient way to estimate or to ompute this variable on the basis of the set of measured variables.

A nonlinear relation between the angular rates in the wind axes pw , qw and rw and the angular rates in the body axes p, q and r as a fun tion of the angles  and is dened by:

0

1

0

10 1

pw

os  os sin sin  os p  qw A =  os  sin os sin  sin A  q A : (28.13) rw sin  0

os  r Sin e the  and state variables are measured, we an use transformation (28.13) to obtain the a tual value of pw . Note that, when demanding velo ity ve tor roll rate,  is kept almost onstant and is minimised for the manoeuvre to be satisfa torily performed. This means that the matrix in transformation (28.13) turns out to be almost onstant; hen e the nonlinearity introdu ed by this new sub omponent is expe ted to be su iently mild to not deteriorate the losed-loop system performan e.

452

28.2.5 The nonlinear ompensation of the dynami pressure variation As usual when using a linear ontroller for a nonlinear plant, it doest not guarantee the stability and performan e of the nonlinear losed-loop system; indeed we are in the presen e of strong variations of the linearized models of the plant around dierent operating onditions. A possible way to avoid these problems is to ompensate some of the nonlinearities of the model to make as similar as possible, the linearized model matri es in a large region of the operating envelope. In the ase of the air raft it is possible to take into a

ount the nonlinear ee t of the velo ity and the altitude on the aerodynami for es and try to

ompensate it by means of a nonlinear s aling of some of the gains of the ontroller. Indeed the aerodynami for es and moments generated by the ontrol surfa es are linearly dependent on the dynami pressure

q = 0:5V 2 .

If the LQ ontroller has been designed on the linearized model around one operating point, where the nominal dynami pressure is

q0 = 0:50V02 , when the

ontroller is applied in other operating onditions the ontrol law is amplied or redu ed in relation to an in reasing or a de reasing dynami pressure. In order to normalize the aerodynami surfa e ontrol power we s ale the

ontrol ommands at the output of the PI a tion with a term

q0 =q.

28.2.6 Demand shaping lters As usual in the aeronauti al appli ations, an important rle is played by the DS (Demand Shaping) lters (see for example [28℄ and [3℄). In our ontroller s heme, the lters have been used to a hieve the following tasks: i) to avoid the saturation of the ontrol ommands; ii) to allow the satisfa tion of the Gibson riteria and the spe i ation on dropba k a

ording to the requirements given in Se tion 27.3.5. Indeed, if the pilot inputs a sudden demand on one of the variables to be

ontrolled, it is possible that the ontrol ommand ould exhibit peaks or high rate of variations. When dealing with a nonlinear plant, where saturations and rate limiters are a ting, it may happen that the losed-loop system state is brought out of the stability region. On the other hand, the requirements on the input-output behaviour of the plant expressed by Gibson riteria an be a

omplished by introdu ing linear lters at the input of the plant with an adequate pla ement of their poles and zeros. Two kind of lters have been onsidered in our ontroller: 1. a nonlinear lter whi h avoids the demand to be too high or too fast for some riti al points of the ight envelope; 2. a linear lter onsisting of a stable linear dynami system, whose poles and zeros lo ation allow for the satisfa tion of the Gibson riteria.

453

28.2.7 Swit hing logi on the  and Ma h values The linear nature and the low order of the feedba k PI ontroller proposed in Se tion 28.2.2, as said, has the drawba k that the gains designed around one linearized model of the air raft are not assured to work well in the whole ight envelope: therefore ontroller gain s heduling is needed.

This is mainly due

to the fa t that, in the ase of high angle of atta k, the air raft has strong

oupling between lateral and longitudinal dynami s. In prin iple we tried to s hedule only with Ma h number.

If we refer to

straight and level ight onditions, the ight envelope is redu ed to be only

 is univo ally determined by the solution of  is de reasing with the Ma h number, hen e

dependent on two variables, namely the Ma h number and altitude. Indeed, is for ed to be always zero while problem (28.5). In level ight,

we have that at high Ma h number there are no ouplings between lateral and longitudinal dynami s.

Nevertheless it happens, during transients, that the

air raft is for ed to y at high angle of atta k even if the Ma h number is high. In su h ases we have that the ontroller designed for straight and level ight, due to the ouplings between longitudinal and lateral dynami s, does not work adequately.

Hen e it is ne essary to s hedule the ontroller gains also as a

fun tion of the angle

.

In summary we have that a rst partition of the ight envelope is based on

 values. 2nM n :

the Ma h number. A se ond partition is based on the of PI gains to be stored in the FCC turns out to be

The number

i) 2 for ea h point of the envelope based on the pilot demand (MOSL); ii)

iii)

nM is the number of subregions based on the Ma h number, (in our ase nM = 2 sin e the envelope is divided into two subregions orresponding to Ma h> 0:27 and Ma h0.27); n is the number of subregions based on the  value for ea h Ma h number (in our ase n = 2 for  > 25 deg and   25 deg).

Assume now that we have found the optimal ontrollers to over the whole ight envelope; in order to guarantee the ontinuity of the ontrol signal when swit hing between two dierent sets of gains, on e again we use the auxiliary variable

w introdu ed in Se tion 28.2.3.

28.2.8 The observer As spe ied in Se tion 27.3.3, an un ertain but onstant error on the interval

[ 2; +2℄ deg is present.



and



in

Sin e the availability of the omplete measurement of the state of the six

degrees of freedom (DOF) air raft model guarantees good performan e and robustness properties of the losed-loop system, a reasonable way to operate is that of identifying the measurement error value when ying straight and level and then an elling it during the manoeuvre. To a hieve the dete tion of the error we suggest to use a Linear Quadrati Observer (LQO). The design of the LQO, as usual, results in hoi e of some weighting matri es. Our obje tive is to estimate the a tual values of

454

 and

on the basis of

the available information about the ontrol inputs and the measured outputs of the air raft. Let us onsider the linear model of the air raft in the form 28.6, where the states and the ontrol inputs are dened a

ording to Se tion 28.2.1. The lassi al Luenberger observer is des ribed by the following linear equation (see [140℄):

x~_ = Ax~ + Bu u + L(y C x~) y~ = C x~ where

x~ and y~ = ( ~ ~ )T

(28.14a) (28.14b)

are the estimated state and output ve tors and

L

is the gain matrix of the observer. If we design the gain of the observer with an LQG strategy we have to dene the weighting matri es

Q~

and

R~

in the

following Ri

ati equation

A + AT + Q~ C T R~ 1 C  = 0 :

(28.15)

Sin e this is a deterministi problem, the degree of freedom oming from the

hoi e of

Q~ and R~

an be used to emphasize the estimate of



and



and to

speed up the onvergen e of the estimator. Hen e, when the air raft is in a straight and level ight ondition, the LQO

~ and ~ and the dete tion of the m and ~ m , m and m being the

onverges as soon as possible to the estimates errors is given by the dieren es

~

measured variables. Until the next straight and level ight of the air raft, the error is assumed to be known. This kind of pro edure has three main drawba ks: i) as said, the speed of onvergen e of the estimate has to be as fast as possible; indeed it may happen that the air raft remains in a straight and level ight ondition only for a few se onds; ii) if, when performing the estimation, the measurement

y

is ae ted by

noise there ould be an in orre t estimation of the variables; iii) the observer matri es annot be onsidered onstant over the whole operating envelope be ause they are representative of the nonlinear plant behaviour only in a neighbourhood of the design point. To over ome problem i) we have to adequately hoose the weighting matri es of the ontroller. Indeed if we try to speed up the time response of the observer we ould have a de ay of the estimation performan e due to the fa t that the measurement noise enters into the bandwidth of the ontroller.

Hen e, as

always, it happens in the hoi e of the weighting matri es, a good trade-o has to be found. Con erning problem ii), we have veried via simulations that, if the disturban es are zero mean, the designed estimator works well. Finally to over ome problem iii) we have to s hedule the observer matri es on the basis on the Ma h number and the altitude so as to over the whole operating envelope.

455

28.3 The Translation of HIRM Design Criteria into Method Dependent Obje tives As previously said, one of the drawba ks of the LQ ontrol te hinque is that it is not possible to take dire tly into a

ount spe i ations on performan e and robustness as spe ied in the HIRM problem denition Chapter 27. As explained in Se tion 28.2.2, the designer has to work on the weighting matri es to try to a hieve his obje tives; this trial and error pro edure to synthesize a suitable ontroller ould take a very long time. In order to save part of this time we introdu ed some auxiliary fun tions, dependent on the design parameters, whi h allow us to he k, at ea h trial, if the given requirements are satised. We dened two kinds of auxiliary fun tions:



the fun tions denoted by

gi translate the more stringent spe i ations of

the ontrol problem; these fun tions are positive if the given requirement



is not satised;

fi translate less stringent spe i ations; in parfi is to zero the loser we are to the omplete satisfa tion

the fun tions denoted by ti ular the loser

of the spe i ation. Hen e the designer has to look, at ea h trial, at the

gi

fi

and the

fun -

Q^ = ^ diag(^q1 ; :::; q^n ) and R = diag(^r1 ; :::; r^m ) be the weighting matri es of the LQ problem dened in Se tion 28.2.2, n and m being the number of states and ^ B^ ); the design parameter ve tor is dened inputs of the augmented system (A; T as xopt = ( q ^1 ; :::; q^n ; r^1 ; :::; r^m ) . The following optimization problem has to

tions to understand in whi h dire tion to move for further trials.

Let

be solved

f (xopt ) s:t: min x

(28.16a)

opt

gi (xopt ) < 0 ; i = 1; ::; n ; (28.16b) 2 where f (xopt ) = i=1;::;no qi fi (xopt ) and gi (xopt ) are the fun tions to be minimized and the onstraints respe tively, qi are weighting elements, n is the number of onstraints and no is the number of terms in the ost fun tion whi h P

ome out from the performan e and robustness requirements. The solution of problem 28.16 an also be automated by means of a numeri al optimization pro edure. In the following we will des ribe some of the

fi

and

gi fun tions utilised to

translate the majority of the HIRM requirements. Re all that the Gibson riteria, as well as the spe i ations on the

 drop-

ba k given in Chapter 27, have been satised open-loop, by means of the demand shaping lters. In parti ular, we will detail the te hnique used to hoose the parameters of a rst order linear time-invariant lter for the

q-demand

hannel.

Modelling Errors Considerations.

Sin e the design is performed on the

linearized model of the plant, in the synthesis phase we onsidered the modelling

456

errors as un ertainties entering the system matri es of the air raft.

Let us

onsider the un ertain linearized model of the air raft in luding sensors and a tuators in the form

x_ = A(paer )x + Bu (paer )u y = Cx where

(28.17) (28.18)

paer is the ve tor of the model un ertainties whi h is assumend to belong

to the hyperretangle

Paer a

ording to Se tion 27.3.3.

We an he k the stability of the losed-loop system in presen e of the un-

16 by analysing the losed-loop eigenvalues for the 2 ( i) verti es of Paer , say paer ; sin e for many parameters the range of variation is

ertainties

paer

2 Paer

small, stability on the verti es should guarantee stability for all values of the un ertainties. As the stability of the perturbed system is a stringent spe i ation, it will be onsidered in an auxiliary fun tion whi h takes into a

ount the maximum real part of the eigenvalues of the losed loop system

ACL(paer ):

i) )))) ; i = 1; ::; 216 gi (xopt ) = max(real(eig(ACL (p(aer

(28.19)

Stability Analysis under Perturbation at the Input of the Plant.

The

stability test required in Se tion 27.3.4 deals with the simultaneous and independent gain and phase osets at the input of ea h one of the as shown in Figure 27.6.

6

a tuators

In Chapter 4 it is re alled that the LQ ontroller

guarantees robust stability against purely real or purely imaginary matrix perturbations at the input of the plant while it does not guarantee stability for mixed real- omplex perturbations, as those onsidered in the HIRM problem. Nevertheless, to take into a

ount the robustness requirements, further auxiliary fun tions have been introdu ed.

Let us onsider the region plotted in

Figure 27.7 (Figure 27.8 in luding toleran es), say

F

 IR2 ; our matrix pertur-

bation is

P (pp ) = diag(K1 e j1 ; :::; K6 e j6 ) where

( Ki i ) 2 F

(28.20)

pp = ( K1 ; 1 ::: K6 ; 6 )T ; pp 2 F = F 6

an be seen as a ve tor of un ertain parameters.

Consider now the perturbed system matrix transfer fun tion Gp (s; pp ) = G(s)P (pp ); an analysis of the stability of the perturbed system an be arried 12 verti es of the set F , say pp(i) , i = 1; : : : ; 212 . out by onsidering it for the 2 i h (i) be the Ni hols plot of det Gp (s; p(pi) )K (s) whi h, in a parametriLet Ni zed form, is des ribed by the equations

Mag(i) = Mag(i) (!) P h(i) = P h(i) (!) : 457

(28.21) (28.22)

We introdu e a new fun tion whi h represents the relative distan e in the Ni hols plane between

Ni (i) and the point ( 180; 0) in the interval of interest

! 2 [!1 ; !2 ℄ (see Figure 28.2): 8 <

q

min! qMag(i) 2 + (P h(i) + 180)2 dNi (i) = : + min! Mag(i) 2 + (P h(i) + 180)2

if Ni (i) passes under zero ; if not : (28.23)

If we have a plant with no poles with positive real part a ondition for stability is that the

212

fun tions

g(216 +i) (xopt ) = dNi (i) i = 1; ::; 212

(28.24)

be negative.

Single Loop Analysis: Ni hols Plots.

In Se tion 27.3.5 a performan e

spe i ation is given in terms of the single-input single-output Ni hols plot of the frequen y response between ea h a tuator demand and the orresponding error signal, obtained by breaking the loop at the point shown in Figure 27.6 while leaving the other loops losed. These Ni hols plots should avoid the regions shown in Figure 27.9 (no toleran es applied) and in Figure 27.10 (toleran es applied). Let us onsider the parametrized equation of the Ni hols plot obtained

i-th hannel ! 2 [!1 ; !2 ℄, say Ni 2(i) by breaking the

of the loop in the frequen y interval of interest

Mag2(i) = Mag2(i)(!) P h2(i) = P h2(i) (!) and let

(28.25) (28.26)

(Mag (t); P h (t)) t 2 [0; 1℄ be the parametrized equation of the ontour

of the region to be avoided, denoted by

D. We an establish a relative distan e D (see Figure 28.3):

between the Ni hols plot and the region

p min!;t T(Mag Mag2(i))2 + (P h P h2(i) )2 if Ni 2p (i) D = ; ; ! 2 [!1; !2 ℄ ; t 2 [0; 1℄ ; d2(D; Ni 2(i) ) = (T Mag Mag2(i)(~!))2 + (P h P h2(i))2 (~!) + min > t > : if Ni 2(i) D 6= ; ; t 2 [0; 1℄ ; 8 > > <

(28.27)

!~ = (~!1 !~2 )=2, and !~1 and !~ 2  !~ 1 are the values of ! for whi h Ni 2(i) interse ts the ontour of D. We an now dene 12 new onstraints to be onsidered (6 without toleran es where

and 6 with toleran es) whose negativeness guarantees the satisfa tion of the performan e requirement about the Ni hols plots:

g(216 +212 +i) (xopt ) = d2(D; Ni (i) ) i = 1; ::; 12 : 458

(28.28)

40

10

30

8 6

20 4

d>0

d2>0

10 Mag (db)

Mag (db)

2

[−180,0] 0

0 d2<0 −2

−10

d<0 −4

−20 −6

−30

−8

−40 −300

−250

Figure

dNi (i)

28.2:

−200

−150 phase (deg)

−100

Distan e

−10 −200

−50

fun tion

−190

−180

−170

Figure 28.3:

from instability

−160 −150 phase (deg)

−130

−120

Distan e fun tion

for performan e

Physi al Considerations.

−140

−110

d2

In Se tion 27.3.5 other requirements, whi h take

into a

ount the physi al behaviour of the plant, are given. Due to the nonlinear nature of these spe i ations it is not possible to take a

ount any of them dire tly in the LQ ontroller synthesis. Moreover, they an be onsidered in the formulation of a number of auxiliary fun tions, part of them omputed on the basis of numeri al simulations performed on the losed-loop nonlinear system. As an example, onsider the two spe i ations given on the sideslip time response; the former on erns the oupling between roll and sideslip, while the latter regards the response to a sideslip demand. Let

(t; xo pt) be the sideslip response of the losed-loop nonlinear system to [0; 10℄, obtained for the design parameter

a roll rate demand in the time interval

xopt .

We introdu e the following auxiliary fun tion

f5 (xopt ) = max 2 (t; xopt ) : t2[0;10℄

(28.29)

To satisfy the se ond requirement we make referen e to the region shown in Figure 27.16, say

B.

The ontour of this region is dened by the following

parametri equations

L = L(t) U = U (t) ;

t 2 [0; 10℄

whi h represent the lower and upper bound of

(28.30) (28.31)

B respe tively. Performing a ten

se ond simulation of the nonlinear losed-loop system, we obtain the sideslip time response

(t; xopt ).

Let us dene a time dependent distan e fun tion in

the form:

d3(t; xopt ) =

8 <

0 if L(t)  (t; xopt )  U (t) ; L(t) (t; xopt ) if (t; xopt )  L(t) ; : (t; xopt ) U (t) if (t; xopt )  U (t) 459

(28.32)

We introdu e the following auxiliary fun tion

f6 (xopt ) = max d3(t; xopt ) : t2[0;10℄ The Gibson Criteria.

For the

(28.33)

q demand- hannel, two spe i ations have

to be satised: the ratio of dropba k to steady-state pit h rate should be in the interval

[0; 0:25℄ (see Se tion 27.3.5 and Figure 27.15) and the Gibson plots spe -

ied in Se tion 27.4.2 should pass through the regions shown in Figures 27.13 and 27.14 of the HIRM manual. The rst one is a time domain spe i ation, while the se ond one is a frequen y domain spe i ation. We propose to operate in the time domain (whi h is oherent with our whole design te hnique) and to verify a posteriori the satisfa tion of the Gibson plots requirements. Let us onsider the transfer fun tion of the linearized HIRM system on the

q

demand-

q

hannel, namely

Wq (s).

The relation between this transfer

q demand- W (s) = Wq (s)=s. Hen e, for ing W with a unit step signal is equivalent to for e Wq (s) with a ramp signal of unit slope. From linear system theory we know that if an asymptoti ally stable system W (s) is for ed with a ramp

fun tion and the transfer fun tion orresponding to the hannel is

signal, the asymptoti behaviour of the output is generally (if no zeros in the origin are present) still a ramp, having the following equation



d yr (t) = W (s)js=0 t + W (s) : ds s=0

(28.34)

From the above onsideration and from Figures 27.14 and 27.15 it is lear that the transfer fun tion

Wq

has to satisfy the following requirements

Wq (0) = 1 d 0  Wq (s)  0:25 : ds s=0 In order to a hieve this obje tive by means of a

(28.35a) (28.35b)

q

demand shaping lter

we propose to nd, by means of the numeri al solution of an optimization problem, a rst order, asymptoti ally stable, unit stati gain, transfer fun tion in the form

Wf (s; z; p) =

ps+z zs+p

whi h satises the ondition

Wqf (0; z; p) = 1 d 0  Wqf (s; z; p)  0:25 ds s=0 where

z and p are parameters to be optimized and Wqf (s; z; p) = Wf (s; z; p)Wq (s) : 460

(28.36a) (28.36b)

28.4 The Des ription of the Design Cy le Two points are ru ial in the design pro ess: the s heduled nature of the ontroller whi h requires a denition of the s heduling regions, and the di ulty of onsidering, during the LQ based matrix gains design, the whole set of spe i ations given in Chapter 27. The design y le an be synthesized by means of the following pro edure 1. Choose a re tangular region, say

Ej , in the straight and level ight enve-

lope (Ma h-altitude plane) to design a ontroller with xed (non s heduled) parameters; 2. Design a xed set of matrix ontroller gains via the trial and error pro edure des ribed in Se tion 28.3 to take into a

ount the HIRM requirements; 3. Design, if ne essary, dierent gains of the ontroller depending on the angle of atta k; 4. Test the performan e of the ontroller designed by means of the evaluation fun tions

fi and gi introdu ed in Se tion 28.3.

redu e the region

Ej and go to point 1.;

If they are not satisfa tory,

5. Sele t a new region adja ent to (or in luding) the previous one and repeat points 1. to 4. until the whole igth envelope is overed; 6. Design the demand shaping lters looking at the Gibson plots of the

losed loop system; 7. Design the LQO for the error dete tion for a su iently large number of points within the operating envelope; 8. Test the performan e of the omplete losed-loop nonlinear system by means of numeri al simulation. The numeri al tools whi h were used to support the above design y le are well-established for solving the LQ problems, with some numeri al optimization odes, problem oriented, developed to qui ken ea h step of the design pro edure. Among these, a Simulink based User Interfa e for the Design and

Analysis of the Flight Control System has been built, by whi h the designer

an run MATLAB fun tions developed for the HIRM problem.

28.5 Numeri al Results In this se tion the set of gures, gure 28.4 to gure 28.9, show the results of the Automated Evaluation Pro edure that reveal the performan e and robustness of the proposed ontrol s heme.

461

Parametric uncertainty, Nichols plot, Mach 0.4, 10000 ft

Parametric uncertainty, Nichols plot, Mach 0.24, 20000 ft

20

20

15

− dts

15

− dts

−. dtd

−. dtd 10

o dcs x dcd

5

+ dr * thr

0

−5

Open−Loop Gain (dB)

Open−Loop Gain (dB)

10

+ dr

−15

−15

Figure

−250

−200 −150 Open−Loop Phase (deg)

28.4:

−100

Parametri

−50

−20 −350

0

un er-

* thr

−5

−10

−300

x dcd

0

−10

−20 −350

o dcs

5

−300

Figure

−250

−200 −150 Open−Loop Phase (deg)

28.5:

−100

Parametri

−50

0

un er-

tainty, Ni hols plot, Ma h 0.4, al-

tainty, Ni hols plot, Ma h 0.24, al-

titude 10000 ft

titude 20000 ft

28.6 Con lusions To evaluate the suitability of the proposed ontrol s heme and of the related design y le for problems su h as the HIRM design ontrol problem, we would like to summarize the aspe ts whi h, in our opinion, an be onsidered to be the main advantages and the main drawba ks of the method. Of ourse, we have the following advantages: i) the ontroller has a low order and the s heme is quite simple to be implemented, sin e it is essentially a PI ontroller; this kind of stru ture also simplies the understanding of its physi al a tion on the air raft; ii) it does not require a deep theoreti al ba kground to apply the proposed method; furthermore the onditions that the plant should verify are very minor and an be easily he ked; iii) the used software is well established (all the software developed for the LQ methods) or quite easy to implement; iv) most of the design requirements are easily satised; some other requirements an be satised by tuning the ontroller parameters; v) re-design of ight ontrol laws with this method should be straightforward, provided that the physi al problem is not signi antly hanged. On the other hand, the main drawba ks are: i) the design is partially based on a trial and error pro edure whi h does not guarantee a short design time; however, based on our experien e, the pro edure does not need many iterations; ii) some of the given requirements annot be easily taken into a

ount in the design phase;

462

1) HIRM problem understanding

12%

2) Study of the literature on the LQ methods

5%

3) Translation of HIRM requirements into LQ spe i ations

10%

4) Denition of the ontroller stru ture

12%

5) Tuning of the ontroller parameters

12%

6) Software development

16%

7) Report writing

18%

8) Running the evaluation pro edure

10%

9) Others

5%

Total

100% Table 28.1: Time spent for the HIRM design

iii) a omplete simulation ampaign on the nonlinear losed loop system is needed to validate the ontroller at ea h ight ondition, but it is ertainly true that this last point is also ommon to any ight ontrol system design (as stressed for example in [3℄), espe ially if linear plant oriented. We would nally like to give some information on erning the design time spent during this a tivity by our proje t team. The total amount of work on the design problem was about 45 man/weeks. In Table 28.1 the time spent for ea h developed a tivity is shown. From this table it is evident that the majority part of the design time has been spent to understand the HIRM problem, hoose the ontrol s heme and translate the whole set of requirements into method dependent fun tions. On e the ar hite ture of the ontroller was hoosen, only tuning the design parameters.

463

12% of the time was spent

pw

q

0.05

az

8

−8

6

−10

(m/s^2)

(deg/s)

(deg/s)

4 0

2

−12 −14

0 −16

−2 −0.05

0

5 time (s)

−4

10

0

5 time (s)

−18

alpha 8

20

6

18

4

99.5 99

(deg)

22

101 100.5 100

16

5 time (s)

12

10

5 time (s)

−15

−0.01

10

−10

−15

0

5 time (s)

dcd

−3

10

−5

0

5 time (s)

x 10

5 time (s) dcs

0.01

−20

0

0

0

(deg)

0.02

(deg)

(deg)

−10

2

−2

10

dtd 0.03

10

beta

0

0

dts −5

−25

5 time (s)

x 10

2

14

0

0

−3

101.5

(deg)

(m/s)

Va

10

−20

10

0

5 time (s)

dr

10

thr

0.06

60

0

50 0.04

−4

(%)

40

(deg)

(deg)

−2 0.02

−6

30 20

0 −8 −10

10 0

5 time (s)

10

−0.02

0

5 time (s)

0

10

0

5 time (s)

10

Assessment manoeuvre: pitch rate demand, Mach 0.3, 5000 ft

Figure 28.6: Assessment manoeuvres: pit h rate demand at M=0.3, h=5000 feet

pw

q

80

az

10

5

60

0

(deg/s)

(deg/s)

(m/s^2)

5 40 20

−5

0 −10

0 −20

0

2

4

−5

6

0

2

time (s)

4

−15

6

0

2

time (s)

Va 104

4

6

4

6

4

6

4

6

time (s)

alpha

beta

15

1

103

0

101

(deg)

(deg)

(m/s)

10 102

−1 −2

5 100 99

−3

0

2

4

0

6

0

2

4

time (s)

time (s)

dts

dtd

10

−4

6

0

2 time (s) dcs

5

10

0

5

5

(deg)

(deg)

−5

(deg)

−5 0

−10

0

−15 −5

−10

0

2

4

−25

6

0

2

4

time (s)

time (s)

dcd

dr

6

−10

4

5

3

−5

(%)

10

0

0

−5

0

2

4 time (s)

6

−10

2

thr

10

5

0

time (s)

15

(deg)

(deg)

−15

−20

2

1

0

2

4

6

time (s)

0

0

2 time (s)

Assessment manoeuvre: roll rate demand, Mach 0.3, 5000 ft

Figure 28.7: Assessment manoeuvres: roll rate demand at M=0.3, h=5000 feet

464

pw

q

15

az

0.2

−9

0.1

−9.5

10

0

(m/s^2)

(deg/s)

(deg/s)

5 0

−10

−5 −0.1

−10.5

−10 −15

0

5 time (s)

−0.2

10

0

5 time (s)

Va 14.5

15

14

10

100.4

(deg)

(m/s)

100.6

100.2

13.5

13

100

0

5 time (s)

12.5

10

5 time (s)

5 time (s)

0 −5

−12.4 −12.6

−6.5

−10

−20

10

−7

−7.5

−15

5 time (s)

0

5 time (s)

dcd

−8

10

0

5 time (s)

dr

1

10

dcs −6

(deg)

−12

(deg)

(deg)

0

dtd

−12.2

10

5

−5

10

5

0

5 time (s)

0

0

dts −11.8

−12.8

0

beta

(deg)

100.8

99.8

−11

10

alpha

10

thr

20

12

15

10

10

(%)

(deg)

(deg)

8 0.5

6

5 4 0

0 0

5 time (s)

−5

10

2 0

5 time (s)

0

10

0

5 time (s)

10

Assessment manoeuvre: sideslip demand, Mach 0.3, 5000 ft

Figure 28.8: Assessment manoeuvres: sideslip demand at M=0.3, h=5000 feet

pw

q −8.5

0

−9 0

(deg/s)

(m/s^2)

−0.01

(deg/s)

az

0.01

−0.02

−9.5 −10

−0.03 −0.5

−10.5

−0.04 −0.05

0

10

20 time (s)

30

40

0

10

Va

20 time (s)

30

−11

40

alpha

160

0

10

8

150

20 time (s)

30

40

30

40

30

40

30

40

beta

−3

14

x 10

6

12

130

(deg)

(deg)

(m/s)

140 10

4 2

120 8

0

110 100

0

10

20 time (s)

30

6

40

0

10

−10

0.005

−11

−0.005

−13

−0.01

10

20 time (s)

30

40

0

10

20 time (s)

30

−11

40

10

30

40

20 time (s) thr

80 60

0.02

40 0

20 time (s)

0

100

(%)

(deg)

(deg)

−8

10

−9 −10

0.04

0

−8

dr

−7

20 time (s) dcs

0.06

−7.5

−8.5

10

−7

dcd

−3

x 10

0

−6

0

−12

0

−2

40

(deg)

0.01

−6.5

30

dtd

−9

(deg)

(deg)

dts

20 time (s)

−0.02

20

0

10

20 time (s)

30

40

0

0

10

20 time (s)

Assessment manoeuvre: air speed demand, Mach 0.3, 5000 ft

Figure 28.9: Assessment manoeuvres: velo ity demand at M=0.3, h=5000 feet

465

29.

The

H1 Loop-Shaping Approa h

George Papageorgiou1 , Keith Glover and Ri k A. Hyde Abstra t. The H1 loop-shaping design pro edure (referred to 1

2

hereafter as LSDP) was used to design xed gain ontrollers for the HIRM. It was found that a single xed gain ontroller designed for the linearisation

ini3005, performed adequately over the whole

ight envelope of the HIRM. As s heduling would most probably be required for a real ghter air raft with a mu h wider ight envelope, the ontrol law has been developed with a stru ture suitable for s heduling.

As the longitudinal and lateral dynami s are es-

sentially de oupled, two separate ontrollers were designed.

The

lateral ontroller has an inner loop for stabilisation and an outer loop for tra king. The longitudinal ontroller has one single loop that provides both fun tions.

A se ond xed gain ontroller was

designed for the HIRM using the multi-model approa h des ribed on pp. 67-72 in [247℄. This method enables the designer to nd a xed gain ontroller that makes the normalised oprime fa tor ost fun tion at over the whole ight envelope. Details about how to apply this method an be found in [191℄. The design team found it relatively easy, even with the limited time available, to satisfy the spe i ations set out in Chapter 27.

29.1 Introdu tion The LSDP is an intuitive method for designing xed gain robust ontrollers. A tutorial on

H1 loop-shaping is given in Chapter 7. A ontroller that has been H1 loop-shaping provides robust stability to oprime fa tor

designed using

un ertainty. Coprime fa tor un ertainty is a general type of un ertainty mu h

in the same way that single-input single-output (SISO) gain and phase margins are. Therefore, when there is little detailed knowledge about the un ertainty present in a plant the LSDP is a good method for designing robust ontrollers. The dieren e to gain and phase margins is that oprime fa tor un ertainty

an be used to dire tly address robustness in systems with multiple feedba k loops, i.e. multivariable systems (see pp. 240-244 in [266℄). The LSDP has been used in a variety of appli ations and studies.

Most

relevant to the HIRM design hallenge is the work in [120℄ where a ight on1

Cambridge University Engineering Department, Cambridge CB2 1PZ, England

2

Cambridge Control Limited, Cambridge CB4 4WZ, England

466

trol law was developed for the DRA Bedford Resear h Harrier XW175 and subsequently ight tested. The ontrol law performed well and is the subje t of on-going work. The experien e of developing this ontrol law showed that

H1

loop-shaping has a number of key attributes whi h make it parti ularly

suitable for this type of appli ation.

Perhaps the most important attribute

is that the resulting ontroller an be written as an exa t plant observer plus state feedba k. This stru ture allows gain s heduling, as designs at dierent operating points have the same state-spa e stru ture and hen e the observer and state feedba k gains an be linearly interpolated. The observer also allows the handling of input limitations (e.g.

authority and rate limits) by driving

the observer with a hieved plant inputs rather than demanded ones. A third very important attribute of

H1 loop-shaping ontrollers is that the observer

stru ture adds visibility, in that physi al units and interpretation an be applied to the ontroller oe ients and states. This may be advantageous with respe t to ight learan e. The advantages of the observer stru ture are fully exploited for the HIRM ontrol law design.

H1 loop-shaping has mu h in ommon with the design approa h urrently

used by industry, sometimes referred to as  lassi al ontrol. The loop-shaping part of the design pro edure is arried out in exa tly the same way that lassi al design is arried out:

inputs and outputs are mat hed up, and single

loop shaping is arried out to ensure that low frequen y gain is large enough, roll-o at ross-over is not ex essive and that su ient high frequen y roll-o is provided. On e this is done, then the optimal

H1 loop-shaping ontroller

is synthesised for this so- alled weighted plant, see Se tion 7.2. Hen e, it is

perfe tly possible to take a lassi al design, and to augment it with a orresponding

H1 ontroller whi h will then modify the feedba k stru ture so as to

allow for the multivariable nature of the system.

When designing using lassi al ontrol where the system has inherent ross oupling (as for example with most yaw - roll augmentation systems), orresponding ross-terms are put into the ontroller. Design of these terms is not always straightforward in that their ee t on the feedba k loops is not addressed dire tly, and some iteration may be required. With the LSDP, these

ross-terms an be left to the

H1 synthesis part of the design. In the authors' H1 loop-shaping is the potential for de-

view, one of the prime motivations for

sign time redu tion, parti ularly when dealing with multi-input multi-output systems with strong ross- oupling.

29.2 The Controller Ar hite ture The rst stage of the design pro ess is to sele t the ontrol law ar hite ture. Sele tion of the ar hite ture is an essentially design method independent task, and the reasoning used for the presented design is mu h the same as would be used for a lassi al ontrol law. However,

H1 loop-shaping and related robust

optimal ontrol methods ould have been used to help sele t the ar hite ture by looking at the robustness impli ations of ea h andidate ar hite ture, e.g. ex-

467

amination of the robustness impli ations of dierent anard and taileron blending s hemes would be possible. Limited time available pre luded this type of analysis. If a full ontrol law design for a prototype or produ tion air raft was undertaken, ontrol law stru ture sele tion with referen e to robustness impli ations is denitely re ommended. In sele ting the stru ture here, robustness requirements are taken into a

ount in a more heuristi way from knowledge of the system to be ontrolled. All of the a tuators available for the design hallenge are used with ex eption of the dierential anards. This is be ause omitting any of them will ne essarily ompromise performan e in terms of a hievable for es and moments. The reasons for not using the dierential anards are given in Se tion 29.2.2. There are four primary feedba k loops to design, three rotational ones and airspeed. Multivariable ontrol allows the designer to design all four simultaneously. However, the longitudinal motion  pit h and airspeed  is essentially de oupled from the lateral loops  yaw and roll. What oupling there is between lateral and longitudinal motion is due to kinemati ross- oupling and/or asymmetri aerodynami for es due to, for example, dierent ow regimes over ea h of the wings. The linearisations provided for wings level, steady ight do not apture these ee ts. Benets from designing on the omplete 4-input 4output system are likely to be more prominent in a Linear Parameter Varying (LPV) framework [259℄ within whi h these oupling terms ould be modelled. For example, parametri dependen e on roll rate ould be modelled and hen e designed for. However, given the time onstraints, an LPV solution was not investigated and hen e the de ision was taken to separate the longitudinal and lateral ontrol law designs. Figure 29.1 shows the top level SIMULINK spe i ation of the ontroller. The two gitudinal

H1 loop-shaping ontrollers are ontained within the lateral and lonH-infinity ontroller

blo ks. They are implemented in dis rete

time observer form and hen e have two sets of inputs, the measurements and the a hieved air raft inputs. The pre- ompensator weights, and

lateral W1,

longitudinal W1

ontain all the integrators, phase advan e terms and roll-o

terms designed in the same way as for a lassi al ontrol law. They are implemented in a modied Hanus self- onditioned form (see Chapter 7 in [120℄ and [110℄). This is exa tly the stru ture used for the Harrier ontrol law developed in [120℄. Note the two s aling blo ks in the feedba k paths. These are used to trade-o the relative amounts of oupling whi h are to be tolerated between outputs, e.g. s aling speed in knots and pit h rate in degrees implies that a 1 knot variation in airspeed is as equally undesirable as in pit h rate. The lters on the

1Æ /s oupling

outputs blo k implements rst order high frequen y roll-o

p, q and r measurements.

The ut-o frequen y is 50 rad/s.

29.2.1 Longitudinal Controller The primary feedba k variables used to design the longitudinal ontroller are pit h rate

q

and airspeed

V.

This hoi e is straightforward in that these are

the quantities the pilot wishes to ontrol. Use of pit h attitude

468



to stabilise

normal acceleration limits

incidence limits 5 RH inceptor (stick demand in Newtons)

+ −

K longitudinal H−infinity controller

long scaling

2

Mux

4

Longitudinal RH inceptor (pitch rate demand in deg/s)

+ +

LH inceptor (speed demand in m/s)

longitudinal dc gain compensation

longitudinal W1 7

6 Lateral RH inceptor (roll stick force) 1

command filter

−K−

1

stick force to deg/s demand

Control outputs

+ +

Lateral RH inceptor (roll rate demand in deg/s)

Mux

Measurements

actuator demands

+ + convert to velocity vector roll rate demand

[1/hrv2 1/rho hrv2 rho]

outputs

lateral dc gain compensation

lateral W1

3 pedals (side−slip demand in degrees)

side−slip controller

K lateral H−infinity controller

lat scaling

Figure 29.1: The ontrol law ar hite ture

the air raft would require phase advan e, whi h in ee t dierentiates the measurement over some frequen y range. This would produ e a noisier signal than the measured

q

and a less robust design.

Furthermore, the dynami s of the

pit h attitude sensor are slower than those of the

q

sensor and pit h attitude

an not be used at large roll angles. The HIRM has tailerons and anards available for longitudinal ontrol. There are several dierent strategies whi h ould be used to determine how to apportion a required pit hing moment between the surfa es. Before sele ting a strategy a number of onsiderations must be taken into a

ount:



The anards have mu h faster dynami s, and hen e an be used to higher frequen ies with less phase lag.

 

The tailerons an generate mu h larger pit hing moments. The tailerons generate a non-minimum phase ight path response, whereas the anards produ e a response in the ommanded dire tion immediately.



Produ ing ountering pit hing moments simultaneously from tailerons and anards is ine ient.

 

The ar hite ture may have stru tural loading impli ations. Consideration must be made of what happens when the surfa es rate or authority limit. Rate limiting of the surfa es an lead to pilot-indu ed os illations.

469

Two possible s hemes are:



Frequen y blending of the two a tuators. Using a omplementary lter, the higher frequen y omponent of the demand is fed to the anards, and the lower frequen y omponent to the tailerons. In this way the anards are used to obtain a fast initial response. The anard pit hing moment is then transferred to the tailerons whi h take the low frequen y part of the demand. In this way the trim is taken by the tailerons whi h have the moment generating power. This is energy e ient in that the trim is taken only on one surfa e and hen e the surfa es do not oppose ea h other. Using the anards at higher frequen ies may also allow more bandwidth to be extra ted from the system. Rate limiting an be addressed by rossfeeding the demand not generated by the surfa e that has rate limited to the other surfa e.



Driving both surfa es in tandem. The inputs an be s aled su h that the demand is a per entage of total travel so that both surfa es saturate at the same point. One of the motivations for this approa h is that rate limiting is less likely to o

ur.

If both surfa es ee t the demanded pit hing

moment they both have less far to travel than if one surfa e had been used. A disadvantage is that the extra agility of the anards is not being exploited. Additionally, small high frequen y disturban es drive the mu h heavier tailerons whi h may be less energy e ient in terms of required hydrauli power. A se ond advantage of this s heme is that a failure of one of the surfa es still gives a system whi h generates pit hing moments a ross the required frequen y range, albeit with redu ed authority. Either of the two above strategies ould be employed. The rst s heme was

hosen for this design. The omplementary lter is of the form

F anard (s) =

s ; s + !f

Ftaileron (s) =

!f : s + !f

These transfer fun tions are implemented in dis rete time. The anard demand is also normalised with the gain

N .

This gain is su h that the gain per unit

demand to pit h rate at open-loop ross-over frequen y is the same for both

anard and taileron. Therefore if one surfa e saturates or rate limits, its demand an be fed dire tly a ross to the other surfa e. All of the limiting and

ross-feeding o

urs in the

a tuator demands blo k, Figure 29.1. outputs blo k in Figure

A pit h attitude hold is implemented in the

29.1.

Although not listed as a spe i ation of the ontrol law in Se tion 27.3.2, some kind of hold is required in pra ti e. This enables the pilot to go sti k-free if he so wishes. Figure 29.2 shows the

pit h attitude hold blo k, the output of out_1 be omes q +0:25err ,

whi h is a pit h rate demand. In ee t the variable where

err =  urr prev .

The robustness properties of the losed-loop are not

altered signi antly by feeding ba k

q + 0:25.

This is be ause the

 portion of

the signal does not modify the loop gain at ross-over too mu h. The attitude hold is only engaged when the ag input,

470

hold flag

in Figure 29.2, is set to

1

1/z

hold flag 4

Abs

<

last out *

AND

phi NOT

0.1

+ +

− +

0.

0.1 radians

0.25 *

3

+ +

1 out_1

2

theta

q

Figure 29.2: The

1

and the roll angle



pit h attitude hold blo k

is smaller than 0.1 rad.

A pit h attitude hold is not

desirable at large bank angles. The onstru tion of the total pit h rate demand,

ommand filter blo k in

Figure 29.1, onsists of the following terms:



Longitudinal sti k demand. The sti k ommands normal a

eleration in

g.

This is onverted into an equivalent pit h rate demand. The demand

is also fed through a pre- ompensator blo k whi h is used to meet the pit h drop-ba k requirement, see Figure 27.15.



Normal a

eleration limiting term. If the normal a

eleration limits are ex eeded, the amount by whi h the limit is ex eeded is onverted into the required pit h rate demand to remove the ex ess.



In iden e limiting term. The ex ess in iden e is turned into a pit h rate demand that will restore in iden e to within the spe ied limits.

attitude-hold engaged flag blo k is shown in Figure 29.3. This

ommand filter blo k mentioned above. The output of this blo k engage flag, is the input hold flag of the blo k depi ted The

blo k is ontained within the

in Figure 29.2. The attitude hold is only engaged if the pit h rate is less than

1Æ /s,

and the pit h rate demand is less than

0:1Æ /s

(ee tively zero).

On e

engaged, it stays engaged whatever the pit h rate, until the pilot ommands a non-zero pit h rate.

29.2.2 Lateral Controller The primary feedba k variables used to design the lateral ontroller are roll rate

p and yaw rate r.

Roll angle and heading angle were not used as primary

feedba k variables for the same reason that pit h attitude was not used when designing the longitudinal ontroller (see Se tion 29.2.1). In addition, note that

p and r

are valid for any orientation in spa e, whi h is not the ase for Euler

angles.

A velo ity ve tor roll rate and side-slip demand system is spe ied

in Se tion 27.3.2.

The designer might therefore be tempted to use side-slip

as a primary feedba k variable.

This will almost ertainly produ e poorer

471

1

Abs

<

q_dem

pilot demand zero 0.0017 0.0017

2

Abs

<

q

OR

pitch rate less than 1 degree/s

0.017

AND

q zero or alpha−hold engaged

1/z alpha−hold engaged

1/z

1

delay

engage flag

0.017

Figure 29.3:

1 lateral stick demand

engage flag logi

0.

*

p demand *

2 side−slip controller demand 3

1

degrees to radians

+ +

2 r demand

f(u)

alpha

sin f(u) cos

Figure 29.4: Lateral demands

performan e in that the side-slip measurement is both more noisy and a slower measurement than

r.

These two ee ts will mean that less bandwidth would

be extra ted from the yaw loop. Designing a tight primary feedba k ontroller using

p and r robustly stabilises the air raft and provides an inner losed-loop

system around whi h an outer loop side-slip tra king system an be built. The side-slip ontroller an be seen in the lower left-hand orner of Figure 29.1. Its output is a yaw rate demand whi h enters the inner loop as illustrated in Figure 29.4. A yaw rate demand is also ross-fed from lateral sti k to ee t a velo ity ve tor roll  the required term is

sin 

times the lateral sti k demand as is

illustrated in Figure 29.4. Omitting this ross-term would leave it to the sideslip ontroller to reje t the side-slip indu ed when a roll rate is ommanded. As the side-slip outer loop has a lower bandwidth than the inner loop, ex essive side-slip oupling will o

ur. The ross-term puts in a fast yaw rate demand to a hieve the velo ity ve tor roll. The HIRM has dierential anards and tailerons available for roll ontrol. However, the anards are very inee tive in roll as they generate smaller for es, and are lo ated mu h loser to the entreline of the air raft.

Hen e using

anards requires large surfa e dee tions and gives little benet. Furthermore, it limits their availability for longitudinal ontrol for whi h they have denite

472

benets over the tailerons. By studying the aerodynami s of the HIRM it an be dedu ed that the dierential anards also have a very signi ant inuen e on the ee tiveness of the symmetri al anards and tailerons thus reating a robustness issue. Hen e, only the dierential tailerons are used for roll ontrol. For yaw ontrol, only the rudder is available.

29.3 Controller Design This hapter dis usses how the weighting fun tions and ontroller parameters were sele ted to meet the design requirements.

29.3.1 Lateral Primary Feedba k Controller In designing the feedba k ontroller the main obje tive is to push the ross-over frequen y as high as possible, whilst retaining an a

eptable level of robustness. No referen e is made to the handling qualities required. This is be ause if the feedba k ontroller is as fast as possible given robustness onstraints, design of a pre-lter to meet handling qualities should be straightforward. A fast feedba k

ontroller redu es losed-loop response lag, and gives a robust response i.e. one whi h should not vary signi antly with envelope operating point. If this approa h gives better handling qualities than spe ied, the high bandwidth is still justied in terms of the a hieved disturban e reje tion. The argument against in orporating handling quality requirements when designing the feedba k ontroller is that ne essarily a trade-o between robust stability and handling qualities will be ee ted.

One ould argue that the

primary purpose of the feedba k ontroller is to maximise robustness for the spe ied open-loop ross-over frequen ies, i.e. disturban e reje tion properties. The design pro edure an be divided into the following steps. 1. S ale the dierential taileron and rudder inputs by

1= 21 V 2 .

This nor-

malises the moment generated so as not to vary signi antly with ight

ondition. 2. Sele t the lateral states from the linearisation i.e.

v, p, r and .

Append

this linearisation with the a tuator models, full order sensor models, antialiasing lters and omputational delay. The full order sensor models are used sin e the nal ontroller is model redu ed anyway. 3. S ale the outputs to ree t the oupling requirements. The s aling used



is ( -tools and MATLAB ommands are used)

>> out_s aling = diag([0.3 1℄); A ouple of design iterations were arried out before arriving at this s aling. The s aling ree ts the inherent oupling within the plant i.e. that a unit oupling in roll happens a lot more easily than a unit oupling it yaw. Attempting to s ale the outputs to make the oupling into yaw

473

and roll approximately equal results in poorer robustness.

In general,

trying to hange the dire tionality of the plant is not good pra ti e. The roll rate

p +  .

p

is augmented with the roll angle to give the output variable This boosts the low frequen y gain, and enables a roll angle

hold to be ee ted for zero lateral sti k demand. The during a roll-rate demand.

 term is removed

This an be justied (in terms of robust

 is hosen su h that the open-loop ross-over p part of the onstru ted output variable.

stability) provided that is entirely set by the

4. The desired ross-over frequen y for both loops is 10 rad/s. This is the highest the ross-over frequen y an go before robustness margins are ne essarily redu ed, due primarily to a tuator roll-o.

To verify this,

a few design iterations at slightly higher ross-over frequen ies an be

arried out, and the resulting a hieved robustness margin

, monitored.

Both loops have suitable roll-o rates at ross-over, and so all that is required is to boost low frequen y gain, and add high frequen y roll-o lters. The sele tion of the appropriate transfer fun tions is exa tly as for a lassi al design. Both loops are rolled o with the lter

50

s+50 .

The lters

are dis retised using a bilinear transformation with frequen y warping to mat h the lters at 10 rad/s. Low frequen y gain is boosted with the pre- ompensator

>> W_p = nd2sys([1 1℄,[1 0℄); >> W_r = nd2sys([1 2℄,[1 0℄); >> W_1 = daug(W_p,W_r); 6

10

5

10

4

10

singular values in db

3

10

2

10

1

10

0

10

−1

10

−2

10

−3

10

−2

10

−1

10

0

10 log frequency

1

10

2

10

Figure 29.5: Singular values of the weighted plant

5. Multiply the two plant inputs with gains

kw4

and

kw6,

an input s aling

whi h gives the required ross-over frequen ies. Plot the singular values of the shaped 2-input 2-output plant, Figure 29.5. Che k that the desired loop shapes have been a hieved. 6. Design the

H1 loop-shaping ontroller, K1 , for the shaped plant. Che k

that the resulting robustness margin is su ient (typi ally

474

 > 0:3 indi-

ates a robust design).

Che k the step responses. These are shown in

Figures 29.6 and 29.7. Step on p

Step on r

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

−0.2 0

0.2

0.4

0.6

0.8

1 time (s)

1.2

Figure 29.6: Step on

1.4

1.6

1.8

−0.2 0

2

p demand

0.2

0.4

0.6

0.8

1 time (s)

1.2

Figure 29.7: Step on

1.4

1.6

1.8

r demand

If gain s heduling is to be arried out, then the above pro edure is applied to ea h of the design points. Typi ally, the dynami weighting fun tions will not need to be hanged between design points, and so all that hanges are

kw6

and the ontroller

K1 .

kw4,

The ontroller is then implemented in observer

form, and the state-spa e matri es gain s heduled using linear interpolation. Details on how to do this an be found in [120℄. For the HIRM design hallenge, the ight envelope is not that wide and hen e a single xed gain ontroller an be used. used was the one designed for the operating point

orresponds to envelope.

12Æ

The xed gain ontroller

ini3005.

This linearisation

in iden e, and is somewhere in the middle of the ight

The orresponding values of

kw4

and

were then used for all ight ases. The a hieved

kw6

for this ight ondition

 was 0:32.

Should the level of

robustness have proven insu ient, then the loop-shaping exer ise above ould have been re-run with less stringent performan e requirements.

29.3.2 Side-slip Controller An outer loop side-slip ontroller was designed using loop-shaping and normalised oprime fa tor robust stabilisation.

By examining the Bode plot of

yaw rate demand to sideslip, the following ompensator was determined

W1 =

4(s + 2) s(0:1s + 1)

This has a rst order roll-o at 10 rad/s to attenuate side-slip sensor noise, and additional low frequen y gain below 2.0 rad/s. The Bode plot of the shaped loop is shown in Figure 29.8. The resulting

H1 ontroller was model redu ed to 4 states by arrying out

a least squares mat hing of the gain and phase plots of the full order ontroller. This an be done using the MATLAB fun tion robustness margin is

 = 0:42.

invfreqz.m.

The a hieved

This guarantees a gain margin of at least

475

2:4,

2

5

1.2

10

4

10

1

3

10

0.8 2

singular values

10

0.6

1

10

0

10

0.4

−1

10

0.2 −2

10

0

−3

10

−4

10

−2

−1

10

10

0

10 frequency (radians/second)

1

−0.2 0

2

10

10

Figure 29.8: Shaped loop

and a phase margin of at least

1

2

3 time (s)

Figure 29.9: Step on

4



5

6

response

45Æ (see Se tion 7.3 for the relevant formulae).

Figure 29.9 shows the side-slip step response whi h meets the spe i ation set out in Figure 27.16.

29.3.3 Longitudinal Primary Feedba k Controller The longitudinal ontroller initially had a multivariable stru ture with feedba k variables

q

and speed. However, the dieren e in the required ross-overs for

these two loops (approximately a fa tor of 10) meant that there was little or no advantage to having a multivariable stru ture. Hen e, two SISO ontrollers were designed given that the added omplexity of the ross-terms in the MIMO

ase ould not be justied in terms of robustness or performan e. The

q

feed-

ba k was designed using the LSDP, and the velo ity feedba k using lassi al loop-shaping. The losed-loop time response shows a good de oupled response with no overshoot. Full details an be found in [191℄.

29.3.4 Pilot q - ommand Filter Design The losed-loop step response gives little or no overshoot. In order to a hieve the drop-ba k requirement, a pilot ommand lter of the following form was used:

T s+1 ppilot (s) = 1 T2 s + 1

By putting T1 > T2 pit h rate overshoot is introdu ed. The values of T1 and T2 were determined by performing a sear h over a range of values of T1 and T2 , and then sele ting all pairs whi h give a drop-ba k of approximately 0:125 s. The other handling quality riteria were then examined for these pairs, and a

T1 = 0:336 and T2 = 0:144 0:1 s, an average phase rate of 57Æ/Hz and f = 1:7

parti ular solution sele ted. The values hosen were whi h gave a drop-ba k of Hz.

From Figure 27.12, it an be seen that this gives Level 1 performan e.

Figure 29.10 shows the time response of a pit h rate demand.

476

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 0

0.5

1

1.5

time (s)

Figure 29.10: Step on

q demand

29.4 Results from the Evaluation Pro edure This hapter ontains results from the evaluation pro edure. These results are from the rst appli ation of the pro edure, and indi ate that very little modi ation is required to the ontroller in order to meet all of the requirements. The authors believe that this is in part due to one of the key strengths of the design method in that loop-shaping gives good visibility as to what performan e an be a hieved from the system.

Hen e, the design is likely to exploit the sys-

tem's inherent performan e limits fully. At the same time, robustness must be maintained so as to respe t the Ni hols ex lusion regions. This is equivalent to a hieving a large enough value of

.

It should also be borne in mind that the ontroller designed for

ini3005 is

used at all operating points in the analysis, whereas in pra ti e it would be gain s heduled with airspeed. Note that the ontrol law tested here is non-linear in that it ontains models of a tuator rate and authority limits, and is in dis rete time. Hen e it is very realisti and implementable. All of the Ni hols plot tests were satised, with the ex eption of the one shown in Figure 29.11 where the ex lusion region is just lipped.

A small

in rease in the phase advan e in the dierential taileron weight is required. The results indi ate a suitably robust design a ross the envelope despite not gain s heduling the ontroller. Figure 29.12 shows the Gibson Ni hols plot riterion for small perturbations of the sti k. It an be seen that the response is Level 1. The Ni hols plot for large sti k demands has not been produ ed yet. The drop-ba k and phase rate

riteria are given in Table 29.1. All meet the Level 1 requirements. Table 29.2 gives data on the response to turbulen e. Figures 29.13, 29.14, 29.15 and 29.16 show responses resulting from ommands in pit h, roll, side-slip and speed. Dierent ight ases have been used to demonstrate that performan e is good a ross the ight envelope. In Figure

5Æ /s pit h rate ommand is made. Note the pit h rate overshoot used to give the required drop-ba k. At 4 s the in iden e limit is hit, and the alpha 29.13 a

limiter for es the pit h rate down, so as to maintain alpha on the limit. Note also that the oupling into side-slip at high alpha is well ontrolled. In Figure

477

Flight ondition

1000 ft Ma h 0.24,20000 ft Ma h 0.30,5000 ft Ma h 0.40,10000 ft Ma h 0.50,15000 ft

f

Drop-ba k

Ma h 0.20,

0:17 s 0:23 s 0:10 s 0:07 s 0:12 s

1.32 1.40 1.58 1.40 1.35

_ 56:0Æ/Hz 56:0Æ/Hz 56:6Æ/Hz 57:0Æ/Hz 57:1Æ/Hz

Table 29.1: Drop-ba k and phase rate riterion

RMS yaw rate due to turbulen e

Æ /s Æ 0.2470 /s Æ 2.1431 /s

RMS normal a

eleration due to turbulen e

0.1317 g

RMS roll rate due to turbulen e

1.7272

RMS pit h rate due to turbulen e

Table 29.2: Response to turbulen e

29.14 a maximum roll-rate ommand is made.

Performan e is reasonable in

that this is a severe manoeuvre whi h results in the air raft banking over well past

90Æ.

Coupling to side-slip is small (about

omparison to the demand on In Figure 29.15 a

10Æ

p.

1Æ ), and oupling to q small in

step on side-slip is ommanded. The response time

is well within the requirements. De oupling from other ontrolled outputs is also good. In Figure 29.16 a rapid speed demand hange is made. Note how full use of the engine is made, and that the ontrol law respe ts the engine rate limits. However there is some speed overshoot whi h should be redu ed, possibly by redu ing the ross-over frequen y of the speed loop, or by limiting the dynami s of the speed demand. Overall, reasonable performan e was a hieved from the design. As with any design approa h, a degree of renement will be possible with some iteration. However our design ame lose to meeting all requirements on the rst attempt, and this was with very limited time available to the authors.

29.5 Con lusions A stability augmentation system has been developed for the HIRM using

H1

loop-shaping. In the authors' view, the design problem was relatively simple, parti ularly given the narrow ight envelope of the HIRM. However, the approa h whi h has been taken will extend in an easy manner to a mu h wider ight envelope. Key to this is the observer stru ture used to implement the

H1 ontroller. The ontrol law stru ture has mu h in ommon with a lassi al

ontrol law, the main dieren e being the addition of the lateral and longitudinal

H1 ontrollers to the feedba k path. The observer stru ture is also used

to handle a tuator authority and rate limiting in a systemati fashion.

478

Ap-

Nominal flight, Nichols plot, Mach 0.24, 20000 ft 20

15

− dts −− dtd

Open−Loop Gain (dB)

10

o dcs x dcd

5

+ dr 0

* thr

−5

−10

−15

−20 −350

−300

−250

−200 −150 Open−Loop Phase (deg)

−100

−50

0

Figure 29.11: Ni hols plot of the nominal plant at Ma h 0.24,

20000 ft

Gibson criterion, pitch: linear analysis 20

15

Gain (dB)

10

5

L1

0

−5 L1 −10 −180

−160

−140

−120 −100 Phase (deg)

−80

Figure 29.12: Gibson Level 1 boundaries

479

−60

−40

q

0

5

−0.5 −1

0

5 time (s)

−9

0 −5

10

0

66

0

5 time (s)

15

10

0

0

10

−0.2

(deg)

−10

−30

10

0

0

dcs

5 time (s)

−2

10

10 (deg)

0.5 (deg)

10

0 −0.5

0

5 time (s)

−1

10

0

5

0

5 time (s)

10

−5

0

thr2

(%)

100

50

0

5 time (s)

10

50

0

10

0

thr1 100

5 time (s) dr

15

−20

0

dcd 1

0

10

2

20

−10

5 time (s) dtd

−20 5 time (s)

0

4

0 (deg)

(deg/s)

5 time (s) dts

2

10

0.2

10

4

5 time (s) beta

25

q cmd

0

0

0.4

20

6

(deg)

−12

10

(deg)

30 (deg)

(m/s)

68

64

(%)

5 time (s) alpha

35

0

−10 −11

Va 70

62

az −8

(m/s^2)

10

(deg/s)

(deg/s)

pw 0.5

0

5 time (s)

10

Assessment manoeuvre: pitch rate demand, Mach 0.2, 1000 ft

Figure 29.13:

5Æ/s pit h rate demand

480

5 time (s)

10

pw

q

60

5

0

20 0

−2

0

2

4

−6

6

0

2

10

102

2

4

0

6

0

2

−2

6

4

(deg)

0

2

4

−40

6

0

10 (deg)

0.5 (deg)

6

0

−1

0 −10

0

2

4

6

−20

0

time (s)

thr1

4 (%)

4 (%)

6

2

2

0

2

4

6

0

0

time (s)

2

4

6

time (s)

Assessment manoeuvre: roll rate demand, Mach 0.3, 5000 ft

Figure 29.14:

70Æ/s roll rate demand

481

2 time (s)

thr2

6

6

dr 20

−0.5 4

4

time (s)

1

time (s)

0

2

dcd

−10

2

0 −20

dcs

0

6

20

time (s)

10

4

dtd

−5

−15

6

6

40

time (s)

0

2 time (s)

−10

20 2

0

dts

(deg)

(deg/s)

4

0

40

4

0 −1

p cmd

60

6

beta

time (s)

80

4

1

time (s)

0

2

2

5

0

0

time (s)

(deg)

104

(deg)

−10

6

alpha 15

(deg)

(m/s)

Va

−20

4 time (s)

106

0

−5

−4

time (s)

100

0

(m/s^2)

(deg/s)

(deg/s)

40

−20

az

2

pw

q

1

0 −0.5 0

5 time (s)

−9.4

0

(m/s^2)

(deg/s)

(deg/s)

0.5

−1

−0.05 −0.1

10

0

Va

5 time (s)

10

−9.6

−9.8

(deg)

(deg)

10 7

0

5 time (s)

10

0

betcmd

5 time (s)

−5

10

0

(deg)

(deg)

(deg)

10

dtd

−5

−6.7 0

5 time (s)

10

0

dcs

5 time (s)

10

0

0.5

10 (deg)

15

0 −0.5

5 time (s)

0

−1

10

5

0

5 time (s)

10

−5

0

thr2

12

12 (%)

14

10 8

10 8

0

5 time (s)

10

6

10

0

thr1 14

5 time (s) dr

1

(deg)

0.05

0

−10

dcd

0.1

(deg)

5 time (s)

5 −6.6

(%)

0

dts

5

6

5 0

6.9

10

−0.05

10

15

161

0

5 time (s) beta

7.1

161.2

160.8

0

alpha

161.4

(m/s)

az

0.05

0

5 time (s)

10

Assessment manoeuvre: sideslip demand, Mach 0.5, 15000 ft

Figure 29.15:

10Æ side-slip demand

482

5 time (s)

10

pw

−10

x 10

q

0

0

10

20 time (s)

30

−9 −9.5

0

(m/s^2)

5

−5

−0.5 −1

40

0

10

140 120 0

10

20 time (s)

30

0

10

30

0

40

20 time (s)

30

(deg)

−6

−8

40

0

10

20 time (s)

30

dcd

(deg)

(deg)

0

10

20 time (s)

30

−1

40

0

10

thr1

0

(%)

0

10

20 time (s)

30

40

40

20 time (s)

30

40

30

40

dr

5

20 time (s)

30

40

30

40

−5

0

50

0

10

x 10

100

50

30

dtd

x 10

thr2

100

20 time (s)

0

−0.5 0

10

−10

10

0.5

−1

0

0

−2

40

1

0

40

−1

dcs 1

30

beta

x 10

−10

1

−7 10

20 time (s)

2

dts

(deg)

(m/s)

20 time (s)

−5

0

10

1

−4

120

0

−11

8

4

40

140

(deg)

−11

3

Va cmd

(%)

40

6

160

0

30

(deg)

10 (deg)

(m/s)

160

−2

20 time (s) alpha

12

100

−10 −10.5

Va 180

100

az

0.5

(deg/s)

(deg/s)

10

0

10

20 time (s)

Assessment manoeuvre: air speed demand, Mach 0.3, 5000 ft

Figure 29.16: A 50 m/s speed hange

483

10

20 time (s)

propriate handling of these non-linearities is essential for a hieving the desired handling qualities, and ensuring sensible apportioning of ontrol power.

H1 loop-shaping is a fairly intuitive ontroller design method that an be

pi ked up in a relatively short amount of time, parti ularly from someone with a ight ontrol ba kground. The design strategy employed bypassed the step of trying to express in detail, the time response requirements in the frequen y domain (a drawba k of frequen y domain based methods). The order of the ontrol law is not seen as an issue as regards implementation provided that the omplexity an be justied in terms of desired robustness and performan e. Experimentation with model redu tion showed that the lateral

H1

ontroller ould be redu ed to 10 states and the longitudinal to 8

states, both with minimal hange to the robustness and performan e.

Bal-

an ed trun ation of a oprime fa torisation of the ontroller was used for this. However, some are would be required when model redu ing a gain s heduled design, as the physi al interpretation of the model redu ed states must be the same for all designs to allow gain s heduling. The type of un ertainty ( oprime fa tor un ertainty) adopted, although quite general, did not prove to be too onservative. S heduling was avoided partly be ause the open-loop HIRM was s aled with dynami pressure, a well known te hnique within industry. The paradigm of

H1 loop-shaping is extremely powerful.

There are other

extensions whi h have not been demonstrated here due to la k of time available for the proje t. One ex iting new area is that of self-s heduled design methods whereby a parameter dependent ontroller is synthesised in one step for a parameter dependent plant. For the HIRM, the obje tive would be to nd a

ontroller dependent on airspeed given a speed dependent model of the HIRM. The synthesis of the ontroller relies on solving a set of linear matrix inequalities for whi h there are numerous algorithms available. In the authors' view, self-s heduled methods and linear parameter varying (LPV) plant des riptions will provide very powerful and relevant tools for the aerospa e industry.

In

parti ular, they provide a framework in whi h to address aerodynami nonlinearities and rate dependent ee ts. Further investigation of the best ontrol stru ture is also a possibility. Designing a single ontroller for roll, pit h and yaw would be worth arrying out to see what the ross-terms between lateral and longitudinal feedba k loops are as a fun tion of ight ondition. This might in lude looking at non-steady ight onditions su h as a non-zero roll rate. For new types of ontrollers to be used by industry, there are two major pre-requisites. Firstly benets need to be quantied so as to justify a hange in approa h. Se ondly, on e the benets have been established, the ight learan e aspe ts need to be addressed. The HIRM design hallenge was not set up to address these issues dire tly. Its fo us was to demonstrate the methods to industry, and to highlight what the advantages might be. As regards potential benets of the method, there are two main ones worth mentioning. Firstly, for

omplex multi-input multi-output systems, the

H1 loop-shaping method pro-

vides a way of synthesising a ontroller whi h ree ts the ross- oupling in the

plant. As su h, better performan e and robustness may be obtained sin e se-

484

le ting ontroller diagonal terms with non-multivariable methods an be a trial and error task. Se ondly, the to redu e design time.

H1 loop-shaping approa h has great potential

This is partly a result of its ability to handle om-

plex multivariable systems, and partly that given the nature of the robustness optimisation, it is hard to design a bad ontroller. Flight learan e of this type of ontrol law may be easier than for many other non- lassi al design methods.

In the main, this is be ause the loop-

shaping aspe t has dire t onne tions to lassi al design. The main dieren e to a lassi al design is the addition of the multivariable element.

However, the

H1

H1 feedba k ontroller

feedba k element is implemented in an observer

form for whi h the stru ture is lear, and for whi h interpretation of the physi al units of individual gains are lear. However, this is still (in the UK and probably many other ountries) a new approa h to have state-spa e elements within the ontrol law as opposed to SISO transfer fun tions.

However, this will

probably not be an insurmountable problem, parti ularly given the numeri al and e ien y advantages of using state-spa e implementations. The next step therefore has to be to quantify benets in some meaningful manner.

485

30.

Design of Stability Augmentation

System using

-Synthesis

Karin Ståhl Gunnarsson Abstra t.

It is des ribed how the

1

-synthesis method was used to

design a ontroller for the HIRM ben hmark problem. Controllers for the longitudinal and lateral axes were designed. For the longitudinal axis a xed-gain ontroller was designed, but for the lateral axis designs were arried out for a low and a high angle of atta k ight ase. The main on lusions on erning the use of

-synthesis

are: Handling qualities requirements as formulated in the HIRM problem an easily be in orporated in the design.

Robustness to

modelling errors an be handled in a straightforward way whereas robustness due to variations in ight ase an be more di ult to in lude. It may be a bit di ult to hoose weighting transfer fun tions that ree t the requirements in an appropriate way, and quite some time is spent on tuning the weighting transfer fun tions.

30.1 Introdu tion The following se tions des ribe the design and analysis of a ight ontrol system for the HIRM, see 27. Sin e the requirements on the ontrol system emphasize robustness and de oupling, the use of a multivariable robust design method is motivated.

The approa h hosen in this hapter is the

-synthesis

te h-

nique, see 8, [60℄ and [59℄, whi h gives the designer a dire t way to in orporate robustness and de oupling aspe ts into the design pro ess. In Se tion 30.2 the ontrol stru ture is presented.

Next, in Se tion 30.3

the translation of the HIRM design riteria into an inter onne tion stru ture is dis ussed. Then the design y le is des ribed in Se tion 30.4. Se tion 30.5 treats a

- analysis of how the

requirements are fullled and in Se tion 30.6,

the results from the automated evaluation are given. Finally, some on lusions are given in Se tion 30.7.

30.2 Sele tion of Controller Ar hite ture The approa h taken here is to split the ontroller into two parts: a longitudinal and a lateral/dire tional ontroller. 1

Saab Military Air raft, S-581 88 Linköping, Sweden

486

The longitudinal ontroller onsists of a linear part providing robustness and handling qualities. The linear ontroller of the pit h axis has xed gains, and is designed to operate over the omplete ight envelope.

Longitudinal

sti k dee tion and throttle position are transformed to pit h rate and velo ity ommands respe tively.

az

normal load fa tor dee tion

ÆT S

The ontroller uses velo ity

V,

pit h rate

q

and

for feedba k. The ontroller outputs symmetri taileron

ÆCS demands, as well as right ÆT H 2 . The intention has been

and symmetri anard dee tions

and left throttle position demands

ÆT H 1

and

to use both symmetri taileron and symmetri anard for manoeuvering, with

only symmetri taileron being used for trimming. The engines are onsidered to operate symmetri ally. In addition to the linear ontroller there is a manoeuvering load limit (MLL) fun tion blo k whose purpose is to limit the angle of atta k and load fa tor at

10o and +30o angle of atta k and 3g and +7g normal load fa tor.

There are

also nonlinear ompensations of the symmetri anard dee tions and throttle positions. The lateral ontroller onsists of a linear part and a blo k al ulating the gravity ompensation for yaw rate. The linear ontroller onsists of one ontroller for low angle of atta k onditions and one ontroller for high angle of atta k onditions. In the intermediate angle of atta k region there is a blending of the outputs of the two ontrollers. The lateral sti k dee tion and rudder pedal signals are transformed to velo ity ve tor roll rate and sideslip ommands respe tively. The ontroller uses roll rate

, pit h attitude  and speed V . to reate the signal

p, yaw rate r,

roll angle

 , sideslip

Roll angle, pit h attitude and speed are used

r0 = r

g sin() os( ) V

whi h is used to oordinate the turn. The lateral ontroller outputs dierential taileron dee tion

ÆT D

and rudder dee tion

ÆR

demands. Dierential anard

dee tions are not used sin e it is usually not very ee tive to use for roll manoeuvres. The ontrollers resulting from

-synthesis are generally of high order even

after the use of model redu tion s hemes. In this ase the order of the longitudinal ontroller is 13 and the lateral ontroller is of order 13. The ontrollers are stable. The longitudinal ontroller in ludes two integrators.

30.3 Translation of Design Criteria into Method Dependent Obje tives This se tion des ribes how the dierent robustness and performan e design

riteria of the HIRM problem dened in 27.3 are transformed into an inter onne tion stru ture that will be used in the

-synthesis

design pro edure. The

inter onne tion stru ture is shown in Figure 30.3 and onsists of the air raft model augmented with un ertainty models, ideal response models and weighting fun tions. If the un ertainty models


r ,
and the ontroller K are "pulled

487

MLLfunction + + qc Vc V q az

Σ

dTS A

+

M,dTS

B

Σ

dCS - +

C

Σ -

α Vtrim aztrim

D

0.5

Linear controller

dTH1 dTH2

Figure 30.1: Stru ture of the pit h ontroller

M pc βc β p r

+

Σ -

A

B

dTD

C

D

dR

Linear controller

g/Vsin(φ)cos(θ) Figure 30.2: Stru ture of the lateral ontroller

out of the inter onne tion stru ture, one gets the general problem des ription shown in 8.3.

The three input-output pairs are the ontrols and the mea-

(u; y), the disturban es and the errors (d; e) and the perturbation signals (w; z ). Here d = (d md ; dnoise ; dgust ), e = (eperf ; ea t ), w = (wr ; w ) and z = (zr ; z ). Below the dierent elements of the inter onne tion stru ture surements

are des ribed. The elements of the inter onne tion stru ture that onstitute the air raft, a tuator and sensor models are the blo ks HIRM,

A tuator and Sensor.

The

rigid body air raft model is ontained in the blo k HIRM and onsists of a linear state-spa e model of the longitudinal or lateral dynami s at the design ight ase. In the longitudinal ase, the engine dynami s are in luded as well. The a tuator models are found in the blo k

A tuator and onsist of the linear Sensor onsists of an

a tuator models as given in Se tion 27.2.4. The blo k

approximation of the sensor dynami s. The sensor dynami s are approximated with a rst order Padé model orresponding to a time delay of 60 ms.

488

e act

Wperf

Wact Ideal model

zc

∆c

K

Wcmd

u

Σ

wc wr

Wdel d cmd

e perf

∆r

Wr Σ

Actuator

zr

Wl Hirm

y d noise

d gust

Wnoise

Σ

Sensor

Wgust

Figure 30.3: Inter onne tion stru ture

The un ertainty model, a

ording to the problem denition, overs both the variation of the HIRM model due to dierent operating points in the ight envelope and modelling errors. The variations due to operating points hanges were modelled as an unstru tured omplex multipli ative un ertainty,


,

at

the input of the air raft model, while the modelling errors were modelled by a stru tured real un ertainty,


r , in the air rafts state-spa e model. Wdel , Wr and Wl .

The size of

the un ertainties are s aled by

The elements of the inter onne tion stru ture that dene the performan e model are ontained in the blo ks and

Wa t .

The blo k

Ideal model, W md, Wnoise , Wgust , Wperf

Ideal model ontains the ideal response models.

These are hosen

to ree t the desired handling qualities. In the longitudinal ase, the dierent pit h axis requirements are in luded in the design by the reation of an ideal pit h rate response model. In this ase it was possible to dene a model su h that both the pit h attitude frequen y response requirements as well as the pit h rate time response requirements were fullled.

The transfer fun tion

from sti k dee tion to pit h rate that was found to full all requirements is:

0:83s + 1 q =9 q (0:05s + 1)(s + 4:8s + 9) The motivation for the hoi e of this transfer fun tion is the following: From nz is 

= 4:8 1=rad wsp of about 3rad=s is obtained.

a simulation at Ma h 0.3 and altitude 5000 ft the value of found. From [2℄ a short period frequen y value A damping ratio

sp

 0:8 was onsidered to be reasonable.

In order to full

the frequen y response requirement a term giving some extra phase lag was introdu ed. In this design only the requirements for the speed ontroller for small throttle travel is taken into a

ount. The requirement leads to the following hoi e

489

of ideal speed response model:

v 1 = v 1 + s The ideal response models for pit h rate and speed respe tively are diagonally augmented.

This stru ture of the ideal response model gives de oupling be-

tween pit h rate and speed. The lateral requirements are also taken into a

ount by dening the roll rate and sideslip ideal response models. The roll rate requirement leads to the following hoi e of ideal roll rate response model:

1 p = p 1 + 0:4s For the sideslip demand system the requirement is satised by the following

hoi e of ideal sideslip response model:

1 = 1 + s The ideal response models for roll rate and sideslip respe tively are also diagonally augmented. The signals that onstitute the overall design obje tives are then reated. These error- or performan e variables onsist of tra king errors, a tuator dee tions and rates.

The tra king errors are the dieren e between the ideal

response and the a tual signal and are weighted by

Wperf .

The inverse of this

weight indi ates how large the allowed steady-state tra king error due to ommand inputs is.

The a tuator dee tions and rates are weighted with

Wa t .

The weightings an be thought of as being about the inverse of the maximal allowable value. The weight of the ontrol surfa e rates an be used as a tool to ontrol the bandwidth of the system.

d md , dgust . The magnitude and frequen y ontent of these signals are shaped with W md , Wnoise and Wgust . The disturban e signals ae ting the errors onsist of pilot ommands,

sensor noises,

dnoise

and wind gusts

30.4 Controller Derivation - Design Cy le 30.4.1 Longitudinal ontroller design The approa h taken for the former is to perform the linear design for the ight

ase Ma h 0.3 and altitude 5000 ft and make the design robust enough to fulll the requirements for all ight ases. The rst step in the design y le is to dene numeri al values of the transfer fun tions in the inter onne tion stru ture dened in Figure 30.3 in Se tion 30.3.

u ÆT S , ÆCS , ÆT H , WXE WZE ,) and three Note that ÆT H = ÆT H 1 + ÆT H 2 , i.e. the thrust

To reate the air raft model ontained in the blo k HIRM, ve states, ( ,

w, q ,

engine 1, engine 2) ve inputs, (

V , q, az ) were sele ted.

outputs, (

490

ommand used in the longitudinal model, is the sum of the individual engine

ommands.

WXE

and

WZE

are the axial and verti al wind gust omponents.

Next, inputs and outputs are added in order to model un ertainties in the

oe ients of the linear state spa e model. In order to study ee ts of variations in the aerodynami stability derivatives four extra inputs and outputs,

zr

and

wr ,

Cm ; Cmq ; CmÆT S

and

CmÆCS ,

were added to the nominal system

des ription. The weighting fun tions of the un ertainty model

Wdel , Wr

and

Wl

are

then dened. The un ertainties due to operation point variations around the nominal ight ase at Ma h 0.3 and altitude 5000 ft, were found to be about 60 % for all frequen ies. This gives

Wdel (s) = diag[0:6; 0:6; 0:6℄.

The sizes of

the stru tured un ertainties are 10 % of their nominal value for all un ertainties ex ept for the rst one.

For this oe ient the absolute error of 0.001

is re al ulated into a value in per ent using the nominal value.

Wl =

This gives

[abs(0:001=A(5; 3)); 0:1; 0:1; 0:1℄, where A(5,3) is the nominal value of the oe ient in the state-spa e model. Wr is taken as an identity matrix. diag

In the longitudinal design model, the disturban es signals onsist of wind gusts, sensor noises added to the measured signals and the pilot ommands. In order to ree t the size and frequen y ontent of these signals, weighting fun tions are hosen. The wind gust weighting fun tion is where

Wgust =

[WWXE (s); WWZE (s)℄

diag

s

+1 WWXE (s) = WWZE (s) = 2 s+1

ree ting a frequen y ontent of 1 rad/s and a magnitude of 1 m/s. The weight-

W

=

[W (s); W (s); W

(s)℄

. The ing fun tion of the sensor noise is az q noise diag v dierent sensor noise omponents are supposed to be given all by the same weighting fun tion:

s +1 Wv = Wq (s) = Waz (s) = 0:003 10s 200 + 1 The load fa tor weighting fun tion was then in reased by a fa tor

1000.

pilot ommands are pit h rate,

ontent of

q

V . The frequen y these signals are modelled by W md = diag[Wq (s); WV (s)℄, where s +1 s +1 Wq (s) = 0:2 20s WV (s) = 10 2s 5 +1 0 :5 + 1 and speed,

The

The bandwidth of the pilot ommands in pit h rate and speed, are assumed to be about 5 rad/s and 0.5 rad/s respe tively. The maximal amplitude of the

ommands are assumed to be 0.2 rad/s and 10 m/s respe tively. The ideal models of the pit h rate and the speed ommand responses were dened in Se tion 30.3. The pit h rate and speed tra king errors are weighted by

Wperf = diag[Wqerr (s); Wverr (s)℄ where s s +1 30 + 1 Wqerr (s) = 500 300 s + 1 Wverr (s) = 50 s + 1 0 :3 0:03 491

This means, the pit h rate ommands should be followed with an a

ura y of 0.002 rad/s for low frequen ies, while the requirement is relaxed for higher frequen ies. Emphasising on the tra king requirement results in integral a tion to be in luded in the ontroller. The requirement on the speed response is not as high. The ommand should be followed with an a

ura y of 0.02 m/s at low frequen ies. The dee tions and rates of the a tuator signals should be minimized. This is done by weighting the taileron dee tion and rate and anard dee tion and rate and throttle dee tion with where

Wa t = diag[WÆT S ; WÆ_ T S ; WÆCS ; WÆ_ CS ; WÆT H ℄

s +1 WÆT S (s) = 0:1 0s:5 50 + 1 ( s + 1)( 2s + 1) WÆCS (s) = 10 s0:5 s + 1) ( 0:005 + 1)( 200 ( s + 1)(s + 1) WÆT H (s) = s0:1 s + 1) ( 0:01 + 1)( 100

WÆ_ T S (s) = 1:5 WÆ_ CS (s) = 1:5

The rate weighting fun tions are onstant, while the dee tion weighting fun tions are frequen y dependant.

In order to avoid the use of the anard for

trim, the anard dee tion weighting fun tion as well as the thrust weighting fun tion are large for low frequen ies, small for intermediate frequen ies and large for high frequen ies. Now, the inter onne tion stru ture of the longitudinal design problem is dened.

If the


's

and the

K

are pulled out, the open-loop stru ture

P,

with three pairs of inputs and outputs is a hieved. Here, the inputs and out-

puts that orrespond to the un ertainty hannel are given by z = [zr ; z ℄, w = [wr ; w ℄. The inputs and outputs to the performan e hannel are d = [WXE ; WZE ; V ; q ; az ; q ; V ℄, e = [qerr ; verr ; ÆT S ; Æ_T S ; ÆCS ; Æ_CS ; ÆT H ℄, The measurement/ ontrol hannel are given by y = [q ; V ; V; q; az ℄, u = [ÆT S ; ÆCS ; ÆT H ℄. In this ase, it was not found ne essary to in lude the stru tured un ertainties in the design, sin e the design be omes robust enough anyway. Sin e the inter onne tion stru ture in ludes the un ertainty des ription for analysis purposes, the system is redu ed by taking away the four inputs and outputs that orrespond to the un ertainty des ription, that is the input output pair

(zr ; wr ).

K , that fullls the D K -iteration pro edure, see 8.4. In the rst D K -iteration, an H1 ontroller is a hieved with all D-s alings equal to unity. The optimal value of a hieved in the rst iteration is 18.15. The -value of the

losed-loop system is then al ulated together with the frequen y dependent D The next step in the design pro ess is to nd a ontroller,

design obje tives by using the

s alings. Low order transfer fun tions are tted to these frequen y responses. The inter onne tion stru ture is then augmented with the transfer fun tions

D and D 1 .

In this ase we have four un ertainty blo ks: three orresponding

to the unstru tured un ertainties and one orresponding to the performan e blo k.

This implies that three

D-s aling 492

transfer fun tions should be found

D is kept to I . Se ond order transfer fun tions were D-s alings. Now, a new H1 ontroller an be designed. After three iterations the redu tion of leveled o. The result of ea h iteration is summarized in Table 30.1. The nal a hieved was 2.91, whi h implies that the performan e requirements

sin e the last blo k in

hosen for all

are not a hieved.

Table 30.1: Summary of longitudinal ontroller design

Iteration

Order of system

value a hieved

1

31

18.15

2

43

2.91

3

43

2.88

4

43

2.91

In Figure 30.4 the maximum singular value of the frequen y response of the

 (M ), where M = M (P; K ) dened in 8.3, and the maxi(M ) after the fourth iteration are shown. Note that if (M ) had been less than   (M ) for all frequen ies would have indi ated that additional D -s alings would have been useful for further optimisation of the design. Sin e the -values in Figure 30.4 are larger than

losed-loop system,

mum value of the stru tured singular value,

1, this indi ates that the design obje tives are not a hieved. However, from time simulations it was found that the properties of the ontroller is quite satisfa tory. This indi ates that some of the weighting fun tions do not ree t the a tual requirements to full extent and should be hanged. It ould also indi ate that the design method is onservative or that it an not take all the a tual requirements into a

ount without making the design too onservative. However, due to the limited time available for this study, no hanges of weighting fun tions were arried out. 3

14 sigma mu

mu with total controller mu with truncated controller 12

2.5 10 2

mu

mu

8

6

1.5

4 1 2

0.5 −3 10

−2

10

−1

10

Figure 30.4:

0

10 Frequency [rad/s]

1

10

-analysis

2

10

0 −3 10

3

10

of longitu-

−2

10

Figure 30.5:

dinal design

−1

10

0

10 Frequency [rad/s]

1

10

2

10

3

10

-analysis of total and

trun ated ontroller

The resulting ontroller after this design has order 43. It was redu ed using

493

standard model redu tion te hniques. The order of the redu ed ontroller was 13. In Figure 30.5 the

-values showing the

robust performan e level for the

ontrollers of order 43 and 13 respe tively are shown. As an be seen in the gure, the

-value

in reases at low frequen ies for the 13-th order ontroller.

This ontroller was hosen anyway, sin e time simulations did not show signifi ant performan e deterioration when ompared to the 43-th order ontroller as is shown in Figure 30.6. 14

0.5

12

0

10

V [m/s]

q [deg/s]

−0.5 8 6

−1

−1.5 4 −2

2 0 0

2

4

−2.5 0

6

2

Time [s]

4

6

4

6

Time [s]

0

5

−1 4 −2 3

−4

dCS [deg]

dTS [deg]

−3

−5 −6 −7

2 1 0

−8 −1 −9 −10 0

2

4

−2 0

6

Time [s]

2 Time [s]

Figure 30.6: Time simulations with total (-) and trun ated (- -) ontroller.

30.4.2 Lateral ontroller design The linear design was again performed for the ight ase Ma h 0.3 and altitude 5000 ft. The lateral air raft model in HIRM was reated by pi king out the

v p,

states, inputs and outputs appli able to the lateral hannel: three states ( ,

r), three inputs (ÆT D , ÆR , WY E ) and four outputs (p, r, ay, ).

In order to des ribe un ertainties in the oe ients of the linear state spa e model, extra inputs and outputs were added. The pilot ommands are roll rate,

p and sideslip, . The frequen y ontent W md = diag[Wp (s); W (s)℄

and magnitude of these signals are shaped by where

s

s +1 Wp (s) = 1:25 20s 5 +1

+1 W (s) = 0:2 2s 0 :5 + 1

The roll rate ommand is given in velo ity axis and hen e the roll rate and

494

sideslip tra king errors are dened as:

perr = p os(o ) + rsin(o ) pideal

err =

ideal

Wperf = diag[Wperr (s); W err (s)℄ where s +1 s +1 Wperr (s) = 50 300 W err (s) = 200 30s s +1 0:3 0 :3 + 1

The error signals are weighted by

In order to a hieve satisfa tory turn- oordination a modi ation of the yaw rate

r is performed

r0 = r

g sin() os( ) V

But the approa h taken here is to make the design as if

r is the feedba k signal.

This probably results in an unstable spiral mode whi h an be a

epted if the time to double is long enough. The dee tion and rates of the a tuator signals should be minimised. This is done by weighting the taileron and rudder dee tions with

WÆR (s)℄ where

s +1 WÆT D (s) = 0:15 0:s35 35 + 1

Wa t = diag[WÆT D (s);

s +1 WÆR (s) = 0:05 0:s35 35 + 1

In order to make the design robust for all ight ases, a robustness weighting giving about

6 dB amplitude margin and 35 degrees phase margin has been

used to over unstru tured un ertainties. The weighting fun tion ree ting this requirement is given by:

Wdel (s) =

p10:60:62

!

p10:06:62

The size of the stru tured un ertainties is 10 % of their nominal value for all un ertainties ex ept for the

Cl

and

Cn

derivatives. For these derivatives the

absolute error of 0.01 and 0.002 was re al ulated into values in per ent using the nominal values. However the stru tured un ertainties were not in luded in the design. The ontroller design is arried out in a similar way to the longitudinal

ase.

There are only two unstru tured un ertainties and therefore only two

D-s alings.

The orders are hosen to be two for both transfer fun tions that

are tted to the

D-s alings.

Four iterations were arried out. The results of the iterations are summarized in Table 30.2. In Figure 30.7 the

-value showing the robust performan e of the resulting

ontroller is shown. Also in this ase the properties of the ontroller are quite satisfa tory despite that the

-analysis shows that the performan e obje tives

are not a hieved. The ontroller resulting after this design has order 32.

ontroller of order 13 was al ulated.

A redu ed order

-analysis of the total and the trun ated

ontroller is shown in Figure 30.8.

495

Table 30.2: Summary of lateral ontroller design

Iteration

Order of system

1

24

value a hieved 3.74

2

32

1.83

3

32

1.82

4

32

1.82

Another design was arried out for the ight ase at Ma h 0.24 and altitude 20000 ft.

In order not to violate the ontrol surfa e dee tions limits, the

requirements on roll mode time onstant and sideslip due to roll rate ommands were relaxed. 2 sigma mu

1.8

2

1.6

mu

mu

1.4 1.5

1.2

1

mu with total controller, upper bound 1

mu with total controller, lower bound mu with truncated controller, upper bound

0.8

mu with truncated controller, lower bound

0.6 −2 10

−2

−1

0

10

1

10 10 Frequency [rad/s]

Figure 30.7:

-analysis

2

10

10

3

−1

10

10

of lateral

Figure 30.8:

design.

0

1

10 10 Frequency [rad/s]

2

10

3

10

-analysis of total and

trun ated lateral ontroller.

30.5 -Analysis of the Resulting Controller Analysis of the properties of the resulting ontrollers was arried out for ve ight onditions in the ight envelope of HIRM. The Ma h number and altitude of the ight ases together with the symbol of the line type whi h represent the ight ase throughout this se tion are given in Table 30.3 below.

30.5.1 Analysis of the longitudinal ontroller For ea h of the ve ight ases

-analysis has been arried out.

In Figures 30.9

and 30.10 the results of the robust stability and nominal performan e tests are shown. If the system is robustly stable and fullls the nominal performan e requirements, the

-value

is less than one for all frequen ies. From the gures it is

found that at all ve ight ases the system is robustly stable, while nominal

496

Table 30.3: Ma h number, altitude and line type for the ve ight ases Ma h

Altitude

Line type

[ft℄ 1

0.20

1000

-.

2

0.24

20000

- -

3

0.30

5000

-

4

0.40

10000

x

5

0.50

15000

o

0.35

80

70

0.3

60 0.25 50

Mu

Mu

0.2 40

0.15 30 0.1 20 0.05

0 −3 10

10

−2

10

−1

10

0

10 Frequency [rad/s]

1

10

2

10

0 −3 10

3

10

−2

10

−1

10

0

10 Frequency [rad/s]

1

10

2

10

3

10

Figure 30.9: Robust stability tests

Figure 30.10: Nominal performan e

of longitudinal system at dierent

tests of longitudinal system at dif-

ight ases.

ferent ight ases.

performan e is not a hieved. However for this design, the

-value expressing

nominal performan e was not less than one for the design ight ase, whi h is ight ase 3, sin e the weighting fun tions expressing the nominal performan e were not well tuned. But sin e time simulations for the design ight ase look satisfa tory, it an be expe ted that the same result is a hieved for ight ase 4 and 5 as well, sin e the from ight ase 3.

-values

for these ight ases do not dier a lot

It an however be expe ted that ight ase 1 and 2 will

exhibit poorer performan e, sin e the

-values for these ight ases are larger.

That this is the ase is shown in Figure 30.11 and 30.12 that show the linear responses to a pit h rate and a speed step ommand respe tively.

30.5.2 Analysis of the lateral ontroller The properties of the lateral ontroller were also analysed. In Figure 30.13 and 30.14 the results of the

-analysis arried out for the dierent ight ases are

shown. From Figure 30.13 it is found that all ight ases ex ept at Ma h 0.5 altitude 15000 ft fullls the robust stability requirement.

The results of the

nominal performan e test are shown in Figure 30.14. This plot indi ates that the low frequen y properties at the dierent ight ases diers from the design

497

1.4

12

1.2

10

1

Speed [m/s]

Pitch rate [deg/s]

14

8

6

0.8

0.6

4

0.4

2

0.2

0 0

0.5

1

1.5

2

Figure 30.11:

2.5 Time [s]

3

3.5

4

4.5

0 0

5

Responses to pit h

0.5

1

1.5

2

Figure 30.12:

rate ommands for the ve ight

2.5 Time [s]

3

3.5

4

4.5

Responses to speed

ommand for the ve ight ases.

ases.

ight ase. That this is the ase, was veried by simulations. 80 1.2

70 1

60

50

Gain

Gain

0.8

0.6

40

30 0.4

20 0.2

10

0 −3 10

−2

10

−1

10

0

10 Frequency [rad/s]

1

10

2

10

0 −3 10

3

10

−2

10

−1

10

0

10 Frequency [rad/s]

1

10

2

10

3

10

Figure 30.13: Robust stability test

Figure 30.14: Nominal performan e

of lateral system for dierent ight

test of lateral system for dierent

ases.

ight ases.

30.6 Results of the Automated Evaluation Pro edure In the automated evaluation pro edure dierent tests are arried out in order to verify that the dierent requirements on erning robustness and performan e are fullled. In this ase, the evaluation pro edure shows that the ontroller resulting from this design exhibits quite satisfa tory properties. The robustness properties are exemplied with Ni hols plots for a ight ase at Ma h 0.4 and altitude 10000 ft. Ni hols plots for nominal system and with parametri un ertainties are shown in Figures 30.15 and 30.16 respe tively. From the gures it is found that the ex lusion region is not interse ted ex ept for the symmetri anard loop at low frequen ies.

498

5

Nominal flight, Nichols plot, Mach 0.4, 10000 ft

Parametric uncertainty, Nichols plot, Mach 0.4, 10000 ft

20

20

15

− dts

15

− dts

−− dtd

−− dtd 10

o dcs

Open−Loop Gain (dB)

Open−Loop Gain (dB)

10

x dcd

5

+ dr 0

* thr

−5

+ dr

−10

−15

Figure

30.15:

Ni hols

plot,

−180 Open−Loop Phase (deg)

−90

−20 −360

0

Nominal Ma h

0.4

* thr

−5

−15

−270

x dcd

0

−10

−20 −360

o dcs

5

−270

−180 Open−Loop Phase (deg)

−90

0

system,

Figure 30.16:

altitude

tainty, Ni hols plot, Ma h 0.4 alti-

10000 ft

Parametri un er-

tude 10000 ft

Figures 30.17 and 30.18 show the evaluation of the pit h and roll Gibson

riterion at Ma h 0.4 and altitude 10000 ft. From the gures it is found that these requirements are fullled.

Gibson criterion, pitch, Mach 0.4, 10000 ft, 10%− 30%−−

Gibson criterion, roll, Mach 0.4, 10000 ft, 10%− 30%−−

20

25 20

15 offset added 10.8 dB − 10.4 dB −−

10

15

offset added 11.0 dB − 10.0 dB −−

10

Gain (db)

Gain (dB)

5 5

f (Hz) 0.5 o 1.0 x 2.0 + 3.0 *

L1

0

f (Hz) 0.5 o 1.0 x 2.0 + 3.0 *

0 Sluggish PIO

−5

Oscillation −10

−5

−15 L1

Good −20

−10 −180

−160

−140

−120 −100 Phase (deg)

−80

−60

−25 −350

−40

−300

−250

−200 −150 Phase (deg)

−100

−50

0

Figure 30.17: Ni hols plots of fre-

Figure 30.18: Ni hols plots of fre-

quen y response from sti k dee -

quen y response from sti k dee -

tion to pit h attitude together with

tion to roll attitude together with

requirement.

requirement.

Some examples of non-linear time simulations are shown in Figures 30.19 to 30.21. Figures 30.19 and 30.20 show the responses to a pit h rate ommand of 5 degrees/s at Ma h 0.3 altitude 5000 ft and Ma h 0.5 altitude 15000 ft respe tively. Figure 30.21 shows the responses to a roll rate ommand of 60 degrees/s at Ma h 0.3 and altitude 5000 ft. The evaluation pro edure also showed low ontrol a tivity due to turbulen e and sensor noise.

499

x 10

pw

q

(deg/s)

(deg/s)

4 2 0 −2

0

5 time (s)

−5

5

−10

0 −5

10

0

Va

99

0

5 time (s)

2

20

1

10

10

0

5 time (s)

−1

10

dts

4

−5

0

(deg)

5

−10

0

5 time (s)

−15

10

0

1 (deg)

0.5 0

5 time (s)

−1

10

0

beta

5 time (s)

x 10

0

10

dtd

5 time (s) −3

−0.5 0

−10

2

x 10

10

dr

0 −1

0

thr1

5 time (s)

10

−2

0

5 time (s)

10

thr2 80

60

60 (%)

80

40 20 0

10

1

(deg)

−5 −10

5 time (s) dcd

0

10

−5

dcs 5

x 10

−4

0

(deg)

(deg/s)

q cmd

0

5 time (s)

0

6

2

0

−5

25

15

98

(deg)

−20

10

(deg)

(deg)

(m/s)

100

(%)

5 time (s)

−15

alpha

101

97

az

10

(m/s^2)

−4

6

40 20

0

5 time (s)

10

0

0

5 time (s)

10

Assessment manoeuvre: pitch rate demand, Mach 0.3, 5000 ft

Figure 30.19: Responses to pit h rate ommand at Ma h 0.3 altitude 5000 ft

500

x 10

pw

q

(deg/s)

(deg/s)

0.5 0 −0.5 −1

0

5 time (s)

−5 −10

5 0 −5

10

−15 −20

0

Va

5 time (s)

−25

10

alpha

(deg)

160

4

10

158 156

0

5 time (s)

5

10

0

q cmd

0

5 time (s)

2

−6

1

−8

−12

10

beta

0

5 time (s)

1

0.5

0.5

0 −0.5

0

5 time (s)

10

0

dtd

5 time (s) −3

1

−1

x 10

10

0

−2

10

(deg)

(deg)

−5

5 time (s)

−1

dcd

0

0

−3

−4

dcs

(deg)

−2

10

−10

5

x 10

10

dr

0 −0.5

0

thr1

5 time (s)

10

−1

0

5 time (s)

10

thr2 100

(%)

100

(%)

5 time (s)

(deg)

(deg)

(deg/s)

2

50

0

x 10

10

2

dts

4

−10

5 time (s)

0

6

0

0

−5

15

(deg)

162

(m/s)

az

10

(m/s^2)

−3

1

0

5 time (s)

10

50

0

0

5 time (s)

10

Assessment manoeuvre: pitch rate demand, Mach 0.5, 15000 ft

Figure 30.20: Responses to pit h rate ommand at Ma h 0.5 altitude 15000 ft

501

pw

q

0

−9

5

−9.5

0

2

4

0 −5

6

(m/s^2)

50

az

10

(deg/s)

(deg/s)

100

0

2

4

time (s)

time (s)

Va

alpha

6

−10 −10.5

2

4

6

4

6

4

6

4

6

time (s) beta

103

1 0.5 (deg)

12

102

(deg)

(m/s)

0

101

0 −0.5

11.5 100

0

2

4

6

0

2

time (s) p cmd

dtd

10

−6 −8

20 2

4

−10

6

0 −10

0

2

time (s)

4

−20

6

dcs

dcd

dr

0.5

5

1

(deg)

2 (deg)

10

0

0

0

−0.5

−5

−1

−1

−10

4

6

0

2

time (s) thr1

6

0

2 time (s)

thr2 8

6

6 (%)

(%)

4 time (s)

8

4 2 0

2 time (s)

1

2

0

time (s)

3

0

2

20

(deg)

(deg)

(deg/s)

40

0

0

time (s)

−4

60

(deg)

−1

6

dts

80

0

4 time (s)

4 2

0

2

4 time (s)

6

0

0

2

4

6

time (s)

Assessment manoeuvre: roll rate demand, Mach 0.3, 5000 ft

Figure 30.21: Responses to roll rate ommand at Ma h 0.3 altitude 5000 ft

502

30.7 Con lusions The

-synthesis method was applied to the HIRM ben hmark problem. It was -synthesis an be used to design a ight ontrol system with good

found that

handling and robustness properties. It was found easier to design a longitudinal than a lateral ontroller. This is however usually the ase.

It was no problem to design a ontroller that

satised the requirements on roll rate response and sideslip response, but it was more di ult to keep the sideslip small due to roll rate ommands.

To

over ome this problem the s heduling of the lateral ontroller has to in rease, or the inter onne tions stru ture must be hanged in order to better take this requirement into a

ount. Another problem on erned the in lusion of integrator a tion into the ontroller.

Espe ially in the design of the longitudinal ontroller it is a benet

if the design of integrator a tion is in luded in the design pro ess instead of adding integrators afterwards. However it was found that the way integrator a tion was in luded in this design was not satisfa tory, sin e it seemed as if more integrators than ne essary were in luded in the ontroller. A better way to in lude integrators is to in lude integrators in the inter onne tion stru ture and feed the output from the integrators to the ontroller. However, due to time limitations this solution was not tried. To make the ontrollers robust over the omplete ight envelope might also ause problems.

In the design of the pit h axis ontroller is was found

that it was su ient to des ribe the model variations due to varying ight

ase as a variation in gain at ea h input of the system. This approa h gave a ontroller that works satisfa torily in almost the omplete ight envelope. An alternative would be to des ribe the model variations over ight ases as un ertainties in the dierent elements of the state-spa e model of the air raft. However, onsidering the un ertainties as un orrelated will probably give a too

onservative result. Instead a linear parameter varying (LPV) model ould be

reated, see for instan e [260℄, [112℄.

This results in a model with repeated

s alar un ertainties, but sin e the software tool that has been used, see [18℄, does not support design with this kind of un ertainty model, this solution was not tried.

-analysis

was found to be a useful tool in the design pro edure.

If the

weighting fun tions are properly tuned it an be a useful tool to test if the requirements are fullled or not. In the latter ase,

-analysis

ould also be

used to investigate whi h requirement is driving the problem. To summarize, the advantages of the

-synthesis design method are:

 The same framework an be used both in the design and in parts of the analysis of the ontroller.

 Requirements that an be expressed by linear transfer fun tions an easily

be handled by the method, as well as requirements on tra king error magnitude,

ontrol a tivity level and disturban e reje tion.

 Robustness

onsiderations like modelling errors are straightforward to

in lude in the design.

503

 Within the framework it an be determined whi h requirement drives the problem.

 Commer ially available software tools exist that an be used in the design.  Parts of the design an be automated. There are also disadvantages with the

-synthesis design:

 Non-linear requirements annot be handled.

This implies that require-

ments su h as angle of atta k and load fa tor limits and other non-linear requirements annot be in luded in the design, but must be handled by separate fun tions.

 Un ertainty models where the un ertainties annot be onsidered as in-

dependent annot be handled by the present version of the software tools.

 The resulting ontroller is often of high order.  There is a need to hoose a large number of weighting transfer fun tions

whi h might be time onsuming for a non-experien ed designer.

The on lusion of this study is that the

-synthesis method an be used to

design a ontrol augmentation system with good properties. It might however be a problem to make the design robust over a large ight envelope due to problems in properly modelling un ertainties in the present version of the software tool. In that ase gain-s heduling of

 ontrollers must be onsidered.

Alter-

natively an inner/outer-loop ontroller ould be used, where the inner-loop is designed to equalize the plant over the ight envelope using for instan e dynami inversion and where the outer-loop is designed by the

-synthesis method

to give good handling and robustness properties, see [7℄, [200℄. Together these methods form a powerful tool to ontrol un ertain nonlinear systems and should be onsidered in future resear h.

504

31.

Design of a Robust, S heduled

Controller using

-Synthesis

Johan Markerink Abstra t.

1

H1 -, -synthesis is used to design H1 -, -synthesis dire tly in orporates

In this hapter,

a ontroller for the HIRM.

both performan e and robustness obje tives in the design pro ess. The shape of the losed-loop frequen y response is spe ied by the designer with the use of frequen y dependent weighting fun tions in the design set-up alled the inter onne tion stru ture. Using this

approa h two ontrollers for both the longitudinal and the lateraldire tional air raft motions were designed.

Model redu tion was

used to redu e the order of these ontrollers to a total of 31 states. The ontrollers were analyzed using

H1 -, -analysis te hniques,

verifying omplian e with the design obje tives. Linear time simulation showed good robustness and performan e hara teristi s of the resulting ontrolled air raft.

To extend these hara teristi s

over a larger ight envelope, a ontroller output s heduling s heme based upon dynami pressure was adopted. Several nonlinear elements were added to the ontroller to a

ount for nonlinearities in the dynami s and to implement limiting and departure prevention requirements.

Nonlinear test on the resulting losed-loop system

demonstrated that the ontroller provides the airplane with good handling qualities over the designated ight envelope. Air raft responses show little or no deterioration when model un ertainties and perturbations are used, proving the robustness of the ontroller.

31.1 Introdu tion This hapter ree ts the design of a ontroller for the manual ontrol augmentation problem of the High In iden e Resear h Model (HIRM) as dened in

hapter 27. This work served as the Delft University of Te hnology (DUT) entry for the HIRM design hallenge and was performed as a graduation proje t by the author. The method used for this ben hmark is

H1 -, -synthesis; one

of the robust ontrol methodologies re ently developed.

Using this method,

ontrollers for both longitudinal and lateral-dire tional air raft motions are 1

Delft University of Te hnology, Fa ulty of Aerospa e Engineering, Kluyverweg 1, 2629

HS, Delft, The Netherlands.

E-mail address:

erinnlr.nl

505

markerindutlsb3.lr.tudelft.nl or mark-

designed. During this design, the ontroller ar hite ture and the method dependent obje tives are hosen to omply with the HIRM design riteria as far as possible. The stru ture of this hapter is as follows: In se tion 31.2 the layout of the

hosen ontroller ar hite ture and a des ription of the internal ar hite ture is given. Se tion 31.3 will be dealing with the translation of the HIRM design

riteria into the stru ture of a general

H1 -, -synthesis design.

This results

in the set-up of an inter onne tion stru ture in whi h the desired handling qualities play a major role. Se tion 31.4 ontains a des ription of the design

y le and a motivation for some of the hoi es made during this design y le. In se tion 31.5 the resulting linear ontrollers will be analyzed in both the frequen y and the time domain.

In se tion 31.6 the nonlinear results of the

method independent automated evaluation software will be reviewed. Finally, in se tion 31.7, the on lusions and lessons learned are presented.

31.2 The Sele tion of the Controller Ar hite ture This se tion deals with the layout of the ontroller ar hite ture used in the HIRM design problem. Investigation of the linearized models provided with the HIRM software at dierent points in the ight envelope, revealed that there is little intera tion between the longitudinal and lateral-dire tional air raft motions. Therefore the

- ontroller designs for the two sets of air raft motions are - ontrollers and two separate ontroller

performed separately, resulting in two

blo ks. For s heduling purposes, both blo ks have an additional input. This input is used to s ale the ommanded ontrol surfa e dee tion with the inverse of the dynami pressure. The s heduling parameter is onstru ted using:

s heduling fa tor =

6900 q

This means that the ight ondition with a dynami pressure of 6900 N/m

2

an be seen as nominal, requiring no s aling. For other ight onditions, the

hange in ontrol surfa e ee tiveness due to the hange in dynami pressure is an elled by the s heduling fa tor. The sti k gains are hosen to ommand

12:5Æ/s pit h rate and 90Æ/s roll rate at the maximum sti k for es.

31.2.1 Longitudinal ontroller ar hite ture For the longitudinal air raft motions, the proposed ontrol strategy alls for a Pit h Rate Demand (PRD) system and a velo ity ve tor airspeed demand. The longitudinal ontroller ar hite ture is depi ted in gure 31.1. In this ar hite ture the following blo ks are present:



Dynami pressure s heduling: The ontrol surfa e ommands are multiplied by the dynami pressure s heduling fa tor.

506

dynamic pressure scheduling

az_measured

"_measured

dts_demand

pitch rate cmd

q_measured

* Product q-limiting

x' = Ax+Bu y = Cx+Du µ-controller

throttl1_demand

+

air speed cmd

+

V_measured

dcs_demand

throttl2_demand throttle nonlinearity

V_trim

V-limiting

2_measured f(u)

2_trim

sin(2)-sin(2_tr)

3 Gain

Figure 31.1: Longitudinal ontroller ar hite ture



q-limiting: The q-limiting blo k is a group of fun tions whi h is used to limit the pit h rate ommand a

ording to the manoeuvre limits. a

eleration has to stay within the -3g to +7g region, while ex eed



10Æ and +30Æ.

The normal



may not

The pilot's pit h rate ommand is redu ed if these

boundaries are approa hed or ex eeded. V-limiting blo k: The purpose of this blo k is to limit the high frequen y

ontent and the absolute value of the ommanded hange in airspeed. This is performed by a se ond order lter and a feedba k of the throttle position within this blo k.

 - ontroller: This is the a tual linear ontroller, represented by a state-spa e 

system with 3 inputs and 3 outputs. Throttle nonlinearity: This blo k is used to resolve the nonlinear throttle response explained in se tion 27.2.3.

This nonlinearity results from the

dieren e between the dry and reheat thrust in rease as a fun tion of the



throttle ommand. Additional throttle demand: If the pit h attitude hanges, the gravitational for e omponent from the mass of the airplane will indu e an additional for e in the airplane X -axis. This for e will in turn indu e an unwanted hange in airspeed. The additional throttle demand is designed to ompensate for a

hange in attitude by generating the following additional throttle demand:

additional throttle demand = KT H (sin() sin(trim )) nonlinear testing revealed that in the dynami pit h manoeuvres onsidered,

KT H =3 was a good hoi e.

507

roll rate cmd

dtd_demand

sideslip cmd

1 0.34s+1 Filter

p_measured

dcd_demand

x' = Ax+Bu y = Cx+Du µ-controller

* Product dr_demand

$_measured

dynamic pressure scheduling

Figure 31.2: Lateral-dire tional ontroller ar hite ture

31.2.2 Lateral-dire tional ontroller ar hite ture The lateral-dire tional ontrol strategy is that of a velo ity ve tor roll rate demand, ommanded by lateral sti k and a sideslip angle demand, ommanded by rudder pedal. In body axis, the velo ity ve tor roll rate is a fun tion of the angle of atta k

:

pW = p os() + r sin()

This poses a problem, be ause it is impossible to in orporate this nonlinear

- ontroller. It would be possible to design a velo ity -synthesis, but this would only be valid for one design strategy for the - ontroller is therefore hanged

relationship in a linear

ve tor roll rate system with angle of atta k. The

to ontrol body axis roll rate and sideslip.

The sideslip tra king however,

is required to be very tight, so that a roll rate demand will only result in small sideslip hanges. Nonlinear testing proved that this way, the ontroller ee tively ontrols velo ity ve tor roll rate at any angle of atta k. To keep the sideslip small, a body axis roll rate ommand automati ally also results in a yaw



rate, orresponding to pw sin( ).

The ar hite ture for the lateral-dire tional

ontroller is plotted in gure 31.2 and onsists of the following elements:



Sideslip ommand lter:

This lter is used to limit the high frequen y

ontent of sideslip ommands thereby alleviating any dynami al roll angle

oupling that might be present.

 - ontroller:

This is the a tual linear ontroller, represented by a state-

spa e system with 4 inputs and 3 outputs.

The ontroller designed with

H1 -, -synthesis is a 2 degrees-of-freedom ontroller. This means that the

ommands d and the measured outputs y enter the ontroller separately. The



- ontroller in orporates both stabilization as well as ommand shap-

ing, removing the need for an additional ommand path lter. Dynami pressure s heduling: The ontrol surfa e ommands are multiplied by the dynami pressure s heduling fa tor.

31.2.3 Resulting ontroller omplexity and implementation issues The omplexity of the resulting ontroller is mainly governed by the number of states of the two

- ontrollers.

These are:

508

frequen y [rad/s℄

damping

9.8927e-006

1.0000e+000

5.0908e-003

1.0000e+000

8.6871e-002

1.0000e+000

1.1542e-001

1.0000e+000

4.5883e-001

1.0000e+000

3.9647e+000

1.0000e+000

1.7474e+001

4.9664e-001

1.9973e+001

1.0000e+000

5.2606e+001

2.4637e-001

9.4169e+001

5.6592e-001

2.2386e+002

7.0502e-001

Table 31.1: Frequen y and damping of the longitudinal

pitch rate command

10

symm. taileron

0

0

−10

symm. canard

−2

10

−10

−5

10

0

10

5

10

−4

−5

10

10

10

5

0

0

0

10

5

−4

−5

10

0

5

10

0

0

0

10

5

10

10

5

10

0

10

−5

−5

10

0

10

10

10

−5

−5

10 5

10

10

10

10

5

10

5

10

−2

10

10

0

10

10

−10

−5

10 5

−5

10 0

10

10

−10

10

10

10

10

10

−5

−5

10

Figure 31.3: Longitudinal

 

0

10

10

10

throttle

10

10

10

speed command

0

10

10

10

measured pitch rate

10

10

- ontroller

0

5

10

10

10

−5

10

0

10

5

10

- ontroller frequen y response

longitudinal:

15 states

lateral-dire tional:

16 states

This adds up to a total of 31 states for the

- ontrollers.

With other, more ad-

van ed model redu tion methods, this order ould possibly be redu ed further. To ope with some of the nonlinear ee ts, three additional lters were used in the ontroller adding 4 states. The total number of ontroller states is therefore

35. The longitudinal and lateral-dire tional

- ontrollers are

stable dynami

systems. Table 31.1 gives the frequen y and damping data of the longitudinal

- ontroller.

To further enhan e the visibility, the frequen y responses from

inputs to outputs are plotted in gure 31.3.

509

31.3 The Translation of the HIRM Design Criteria into -Obje tives In this se tion the HIRM design riteria as des ribed in se tion 27.3 will be translated into obje tives and riteria that an be used with an

H1 -, -

synthesis design pro edure. To in lude the handling qualities riteria into the

-design,

referen e models with ideal handling qualities are used. In se tion

31.3.1, these referen e models will be derived for both the longitudinal and the lateral-dire tional air raft motions. Se tion 31.3.2 deals with the way in whi h the robustness requirements are in orporated into the design. In se tion 31.3.3, the general inter onne tion stru ture in luding all the dynami s, lters and weighting fun tions needed to perform the design, is onstru ted.

31.3.1 Handling qualities riteria The longitudinal referen e model The ontrol strategy for the longitudinal air raft motions of the HIRM alls for a Pit h Rate Demand (PRD) system, ommanded by longitudinal sti k dee tion and a velo ity ve tor airspeed demand system, ommanded by throttle lever.

The longitudinal- ontroller to be designed will make the losed-loop

response, from pilot pit h rate ommands to resulting pit h rate, follow the referen e model response as losely as possible. The PRD referen e model used is a onstant speed equations of motion, se ond order transfer fun tion:

q 1 + q s = q 1 + 2!sp s + !s22 sp sp This leaves three parameters to be hosen to onstru t the referen e model:

 q = 1 s. !sp = 1.67 rad/s.  sp = 0.7

pit h mode time onstant: short period natural frequen y: short period damping:

These parameters were hosen using the following HIRM design riteria for

losed-loop response:





Pit h attitude frequen y response riterion: The plot lies within the level 1 boundary for good handling qualities, as depi ted in gure 27.13.

Dropba k riterion: The ratio of dropba k to steady state pit h rate, following a step input should lie between 0 and 0.25 se onds. The relationship for dropba k is:

db = q qss

2sp !sp

For the given referen e model a dropba k of 0.16 se onds was a hieved. For the speed demand system there are two riteria:



Speed ontrol time onstant riterion: For small amplitude speed ommands the time onstant should be in the range 0.75 to 1.5 se onds.

510



Speed ontrol overshoot riterion: overshoot should be less than 3%.

To fulll these requirements, a rst order ideal speed ommand referen e model with a time onstant of 1 se ond is hosen.

V 1 1 = = V 1 + V s 1 + s By diagonally augmenting the ideal Pit h Rate Demand system and speed

ommand system, one longitudinal referen e model is generated. This model will be used in the set-up of the longitudinal inter onne tion stru ture used for the

H1 -, -synthesis design pro edure.

The lateral-dire tional referen e model The ontrol strategy for the lateral-dire tional air raft motions of the HIRM

all for a velo ity ve tor Roll Rate Demand (RRD) system, ommanded by lateral sti k dee tion and a sideslip demand system, ommanded by rudder pedal dee tion. The HIRM riteria on the Roll Rate Demand system are:





Roll mode time onstant riterion: This time onstant should be

p

 0.4

se .

Roll attitude frequen y response riterion: This riterion onsists of a Ni hols plot from roll rate ommand (or lateral sti k dee tion) to roll angle. The resulting graph should lie in the region labelled with good response in gure 27.14.

To omply with the rst riterion a rst order model with the spe ied time

onstant is hosen.

The rst order model was unable to generate su ient

high frequen y lag to stay within the boundaries. For this reason, a rst order lowpass lter was added to provide the phase lag. The result is the following roll mode referen e model:

1 1 1 1 p = = p 1 + p s 1 + lag s 1 + 0:4s 1 + 0:07s For the sideslip demand system there is only one riterion:



Sideslip step response riterion: The response of sideslip to a step input in sideslip ommand should lie within the boundaries of gure 27.16.

The model used for the sideslip demand system is that of a simple rst order transfer fun tion:

1 1 = = 1 +  s 1 + 0:5774s

The two lateral-dire tional referen e models are diagonally augmented to one another, yielding one lateral-dire tional referen e model. This model is then used in the set-up of the inter onne tion stru ture used for the design pro edure.

31.3.2 Robustness onsiderations The design ight envelope for the HIRM model is:

511

H1 -, -synthesis

 

Ma h: 0.15 to 0.5 Altitude: 100 to 20000 ft.

In order to take the model variations over the ight envelope into a

ount, the linear system dynami s in various operating points are parameterized as a fun tion of the dynami pressure. The method used in this design, is to model the HIRM as a Linear Parametri ally Varying (LPV) system.

Although the

linear representations of the nonlinear HIRM model are fun tions of several variables, making them dependent on ight ondition and ight envelope, the

q = 1/2V 2 . All aerodynami varilinearly with q . With the onstru tion

leading parameter is the dynami pressure ables in the state-spa e model, vary

of the HIRM LPV-model, this linear dependen e is modelled via additional inputs and outputs on the HIRM model. This means that in the

-synthesis,

the variations of the model through the ight envelope are taken into a

ount through this varying parameter. For a more extensive review of the LPV model and the onstru tion of this model in the HIRM design see the design report [163℄. In the HIRM manual, two additional kinds of modelling errors are des ribed: errors on the aerodynami moment derivatives and errors on the total moment

oe ients. All these modelling errors are in orporated into the LPV HIRM model as un ertainties on the individual matrix elements.

These parametri

un ertainties are modelled using a real, stru tured, additive un ertainty representation. As a result of this, the models are augmented with additional inputs and outputs through whi h these un ertainties a t. The gain and phase requirements di tate that the losed-loop system should not be ome unstable when adding additional gain and phase osets at the input of ea h a tuator demand. The denition of these requirements exa tly ts the multipli ative un ertainty blo k used in this design, be ause the


m -blo k a ts

at the same point in the loop and adds its gain and phase osets. See gure 31.4.

31.3.3 Des ription of the inter onne tion stru ture The inter onne tion stru ture is what needs to be dened prior to an a tual

-design.

The layout of this stru ture, as well as the denition of the elements

within it, determines the su

ess of a

-design.

The stru ture used in this

HIRM design is given in gure 31.4 and is valid for both the longitudinal and the lateral-dire tional designs, but the exa t numeri al values of the elements and fun tions will of ourse dier between the two. The inter onne tion stru ture onsists of the following elements:

STICK This element represents the sti k lter. It is used to narrow down the bandwidth of the pilot to more realisti values.

ID

This element represents the referen e model with ideal handling qualities. The synthesis pro edure will try to shape the losed-loop response to t this referen e model.

K

The

H1 -, -synthesis ontroller.

ontroller is not yet present.

Before the design pro edure, the

Only the pla e in the inter onne tion

512

ID wm

STICK

K

wa

)qs

Wacts

Wdel cmd

+

e1

zm

)m

_

u

+

+

ACTS

)a

Wp

e2

za

HIRM

y noise

Wnoise

+

+

SENS Figure 31.4: Inter onne tion stru ture

stru ture and the number and type of inputs and outputs is known.

Wdel

The multipli ative robustness weight. This weighting fun tion s ales the multipli ative perturbation-matrix


m .

If Gnom is the nominal

plant, the set of plants hara terized by this un ertainty representation is given by:

G
= Gnom (I +
m Wdel ) At any frequen y

!, the magnitude of Wdel (!) an be interpreted as

the per entage of un ertainty in the model at that frequen y.

For

example: a Wdel value of 0.5 represents a 50% modelling error at that


a

frequen y. The additive perturbation matrix


a is used to model the real, stru -

tured, additive un ertainties to the individual state-spa e matrix ele-


m
qs

ments. The multipli ative perturbation matrix

ertainty in the model. The perturbation matrix


qs


m

represents the input un-

is used to model the Linear Paramet-

ri ally Varying (LPV) nature of the HIRM-system. It introdu es the linear dependen e of the A- and B -matrix elements on the dynami pressure into the system.

ACTS This element represents the a tuator dynami s as des ribed in the HIRM problem denition. The input is a tuator demand and the outputs are a tuator dee tion and rate. In the ase of multiple a tuators, these are diagonally augmented to obtain one a tuator system.

Wa ts

The a tuator weight.

This weighting fun tion s ales the admissible

a tuator dee tions and rates to unity.

In the simplest ase, Wa ts

would onsist of the inverse of the maximum a tuator outputs, resulting in an error (output of Wa ts ) of unity, if the maximum dee tion was rea hed.

HIRM The linear, state-spa e model of the air raft.

513

Wp The performan e weight. This weighting fun tion s ales the performan e error to unity. The input of Wp is the tra king error between

the response of the referen e model and the losed-loop system. This weighting an be seen as the denition of the allowable deviations from the referen e model response.

Be ause the output errors are

s aled to unity, these deviations have to stay below the value of the inverse of Wp at every frequen y. If at a ertain frequen y Wp =50, this means that the losed-loop response may deviate 1/50 = 2% from the referen e model.

SENS This element represents the dynami s of the sensors used. Also the anti-aliasing, not h and averaging lters des ribed in the HIRM problem denition, are in orporated into SENS . As with the a tuator dynami s, multiple systems are augmented diagonally into SENS .

Wnoise The noise weight. This weighting fun tion s ales the unity-intensity noise entering the system. Be ause it represents high frequen y sensor noise, high frequen y lters are used in Wnoise . Breaking the loops at the ontroller K and at the perturbation


-blo ks and

olle ting the remaining systems into one system results in the generalized plant P . The inputs and outputs of this generalized plant are depi ted in gure 31.5. The information of all systems and fun tions and the inter onne tions between them, present in the generalized plant, is used by

H1 -, -synthesis to

design a robust ontroller. The robustness and performan e properties of this

ontroller are related dire tly to the weighting fun tions used in the inter onne tion stru ture.

{ {

w

wa

za

wm

zm

cmd

d

noise

P

u

e1 e2

} }

z

e

y

Figure 31.5: Generalized plant

31.4 The Des ription of the Design Cy le 31.4.1 The general design y le In the H1 -, -synthesis design for the HIRM design hallenge, several steps need to be taken to ome to a ontroller that satises the design goals and

spe i ations. These steps will be referred to as the general design y le. The larger part of this general design y le will be performed twi e, for the longitudinal and lateral-dire tional designs separately.

514

The general design y le onsists of the following steps: 1) Dene the design problem: the design problem denition states the overall design problem to be solved.

In this ase it onsists of the design of a

ontrol augmentation system that provides a high angle of atta k ghter air raft with satisfa tory handling qualities. This design problem denition is treated in hapter 27. 2) Translate the design problem into a general inter onne tion stru ture. This stru ture denes the basi layout of plant, ontroller and ideal model inter onne tion. 3) Formulate riteria: a set of numeri ally dened riteria is used as spe i ation of the overall design goals and to evaluate the resulting design. These

riteria are dened in se tion 27.4. 4) Translate handling qualities riteria into referen e models: for air raft ontroller design problems in whi h handling qualities play a major role, like the HIRM, a referen e model is an ideal method of translating the time domain riteria into riteria that an be used in an

H1 -, -synthesis design.

The set-up of the referen e models with ideal handling qualities is treated in se tion 31.3.1. 5) Translate robustness riteria into system perturbations: the ontroller has to provide su ient stability and performan e over a range of ight onditions and has to be robust against several model un ertainties of the plant. To a

ommodate these riteria, the ontroller will be designed for a set of plants. This set onsists of a nominal plant with several perturbations a ting on it, to model the various un ertainties. The onstru tion of these plant models and perturbations for the HIRM is treated in se tion 31.3.2. 6) Extend the general inter onne tion stru ture: several riteria an be implemented dire tly into the design pro ess by adding elements, weighting fun tions or inputs and outputs to the general inter onne tion stru ture. The result of this pro ess an be found in gure 31.4. 7) Choose the weighting fun tions: after the inter onne tion stru ture is dened, the only variable elements are the weighting fun tions.

The hoi e

and onstru tion of these fun tions is the greater part of the design pro ess. Usually numerous redesigns are ne essary to nd a ombination of weighting fun tions that result in a ontroller whi h omplies with the riteria. Weighting fun tions are in a sense, the knobs the designer an turn to shape the resulting ontroller and losed-loop response. 8) Constru t the a tual inter onne tion stru ture and resulting generalized plant: this step involves writing a omputer program that implements the

onne tion of all the models, lters and weighting fun tions. The output of this program is a linear model of the generalized plant with the input/output denition of gure 31.5.

-synthesis: using the generalized plant resulting from the previ-synthesis pro edure designs the a tual ontroller. It onsists iteration s heme whi h involves an H1 -synthesis, a -analysis and

9) Perform a

ous step, the of a D-K

a tting of s aling fun tions.

10) Analyse the losed-loop system: using frequen y domain te hniques su h as

515

Singular Value analysis and

-analysis, the losed-loop system an be ana-

lyzed. Also, a variety of time domain te hniques an be used to investigate

omplian e of the resulting ontroller with the design riteria. If the design

riteria, frequen y or time domain, are not met, one has to go ba k to step 7), hange the weighting fun tions and redesign the ontroller. This pro ess

ontinues until all riteria are met or no more improvement is a hieved. 11) Redu e the order of the ontroller: the synthesized ontroller an have quite a large number of states. To redu e the order of the ontroller a balan ed realization of the ontroller is reated. This balan es the observability and

ontrollability grammians and orders the states a

ordingly [50℄.

Next a

trun ation an be used to remove (nearly) unobservable and un ontrollable modes. In this design, a number of high frequen y modes that were in ompatible with the HIRM's assumed 80 Hz Flight Control Computer (FCC) iteration rate were removed also. 12) Analyse the losed-loop system: using the same te hniques as in step 10), the losed-loop system with the redu ed ontroller an be analyzed. If redu tion deteriorates the ontroller performan e, one has to go ba k to step 11) and trun ate less states of the original ontroller. If the trun ated ontroller exhibits no performan e dieren es with the original ontroller, one

an go ba k to 11) and try to trun ate the ontroller further. The ontroller order redu tion ends when this balan e between ontroller performan e and

ontroller order has been found. A number of these analysis results for the resulting ontrollers will be presented in se tion 31.5. 13) Constru t the ontroller ar hite ture: the

- ontrollers are part of the om-

plete ontroller blo k. This blo k also onsists of input and output onne tions, signal additions and subtra tions and additional lters and fun tions. In order to ope with nonlinearities in the model or in the riteria, some nonlinear and limiting blo ks may also be present. The set-up of this ontroller ar hite ture is treated in se tion 31.2. 14) Test omplian e of the ontroller with the design riteria stated in 3): using the full nonlinear model, the automated evaluation software he ks if the

ontroller meets these riteria. This nal analysis in ludes several nonlinear time simulations, robustness and performan e he ks and other tests. Some results of the evaluation an be found in se tion 31.6.

31.4.2 Pra ti al onsiderations of the design y le In the numeri al exe ution of this design y le the matrix manipulation software pa kage Matlab is used.

In parti ular the add-on toolbox for analysis

and synthesis of robust ontrol systems

-tools (mutools) is used extensively.

Besides the fun tions provided by mutools, additional programs written by the author in Matlabs s ripting language, are used to implement ertain elements of the design y le. There is little intermediate analysis and redesign eort. Be ause the analysis part is in orporated into the design y le, it requires no extra eort. Both frequen y and time domain analysis te hniques are used in this design y le.

516

The frequen y domain analysis is strongly related to the used design methods:

H1 -, -synthesis.

These analysis te hniques an therefore be used to

see whether the design obje tives in terms of minimizing losed-loop gains are a hieved. The linear time domain analysis has more onne tions with the a tual riteria, in parti ular, the riteria on erning handling qualities.

If the

hoi e of the weighting fun tions and the set-up of the inter onne tion stru ture is done ideally, the frequen y and time domain analysis results oin ide. Unfortunately this may prove to be very di ult to a hieve. The most reliable method of testing omplian e with the design obje tives and riteria uses nonlinear time simulations. termediate linear

On e the ontroller ar hite ture is present, in-

- ontrollers an be

in orporated into this ar hite ture and

analyzed this way. For a redesign after an air raft design hange, the part of the design y le starting with step 7) has to be reperformed.

The hoi e of new weighting

fun tions results in the loop from step 7) to step 10).

After the riteria are

met, the order of the new ontroller an be redu ed with step 11) and 12). The eort related to this redesign depends on the s ale of the hange. Certain weighting fun tions may remain the same, while others may need onsiderable

hanging. If the air raft dynami s are hanged, the set-up of the HIRM model in step 5) also needs to be reperformed resulting in a new LPV HIRM model. 0

3

10

10

pitch angle weight

airspeed weight

2

10 −1

10

pitch rate weight

1

gain

gain

10

−2

10

0

10

−1

10 taileron weight

−3

10

−2

10

−4

10

−3

−4

10

−2

10

0

10 frequency [rad/s]

2

10

10

4

10

−4

10

Figure 31.6: Robustness weight

−2

10

0

10 frequency [rad/s]

2

4

10

10

Figure 31.7: Performan e weight

31.4.3 Weighting fun tion sele tion As stated earlier, the larger part of the design y le onsists of hoosing the weighting fun tions, designing a ontroller, evaluating the resulting system and redesigning the weighting fun tions; step 7) through 10). Be ause these weighting fun tions are responsible for the hara teristi s of the resulting ontroller, the su

ess of the design depends upon their hoi e.

Design dilemmas and

trade-os, su h as robustness versus performan e and performan e versus ontrol a tivity are therefore ree ted in the weighting fun tions hoi e. In the design y le used in the HIRM design, the design obje tives were to minimize the robust performan e level as indi ated by the frequen y

517

-plot.

A

ording to the theory, if robust performan e is satised, Nominal Stability, robust stability and nominal performan e are also satised. robust performan e is a hieved if the

-plot stays below unity over the entire frequen y range.

It

is not un ommon to nd that in omplex designs this goal annot be a hieved. With its many dierent and demanding riteria, design ight envelope and several model variations, this is also the ase with the HIRM design. This does not mean that the design will result in bad ontrollers. Remember that the a

eptability of the ontrol solution does not depend on

 alone.

Nevertheless,

the dis repan y in the theoreti al denition of a hieving design obje tives with a

-value below unity and the possibility of not a hieving this value is felt to -theory. One riterion that had to be a hieved however,

be a weak point of

was that of robust stability. The ontrollers had to guarantee stability for the

omplete set of model perturbations, as indi ated by a robust stability value below unity at all frequen ies. As an example of the use of weighting fun tions gures 31.6 and 31.7 give a robustness and performan e weight of the longitudinal HIRM design.

Wdel For the taileron demand signal, the robustness weight is a rst order lter with a 0.1% un ertainty at low frequen ies and 10% un ertainty at high frequen ies. The motivation behind this hoi e is that the un ertainty of the HIRM model is expe ted to in rease with frequen y.

Wp The longitudinal performan e weight onsists of a system with three diagonal entries: a weighting fun tion for the pit h rate error, for the pit h angle error and for the speed error. These three weights are performan e weights be ause they s ale the error between the losed-loop response and the ideal referen e model response. The plot of the weighting fun tions in gure 31.7 an be interpreted as the inverse of the allowable error. The pit h rate error and pit h angle error weighting fun tions are rst order low-pass lters. This puts the emphasis on steady state tra king and limits the bandwidth in whi h the losed-loop system has to follow the ideal models. The q -weight emphasizes that at low frequen ies errors may not ex eed 10%. To a hieve some form of pit h attitude hold and pit h attitude tra king in the nal losed-loop system, the pit h angle error is also used as a performan e output. The pit h angle weight limits the pit h angle error to 0.4%. The airspeed weighting fun tion has a large value of 200 at high frequen y, limiting errors to 0.5%. This is to a hieve good tra king of the transient part of the rst order ideal model. The weight is in reased to 1000 at low frequen ies to ensure steady state airspeed hold with minimal errors.

31.5 Analysis of the Resulting Controller 31.5.1 The frequen y domain analysis tests The designs were analyzed using H1 -, -theory frequen y

domain analysis

tools. First, the inter onne tion stru ture of gure 31.4 is onstru ted, resulting

in the generalized plant of gure 31.5. Closing the lower loop with the ontroller

518

7

8 7

Robust Performance: µ

6 Robust Performance: µ

6

5

5

4

Nominal Performance

4

3 3

1 0 -3 10

2

Nominal Performance

2

1

Robust Stability

Robust Stability

-2

10

Figure

-1

10

31.8:

longitudinal

0

1

10

10

frequency [rad/s]

Frequen y

- ontroller

2

10

0 -3 10

3

10

analysis

-2

-1

10

Figure

10

31.9:

0

10

1

2

10

10

frequency [rad/s]

Frequen y

lateral-dire tional

3

10

analysis

- ontroller

K yields the losed-loop system. All analysis te hniques used in this se tion will result in frequen y plots in whi h a value below unity indi ates a hievement of the pertinent design goals. The meaning of the frequen y analysis plots is as follows (see hapter 8 for the exa t denitions):





Robust Stability:

robust stability indi ates whether the system remains

stable for a given set of perturbations to the nominal model. Nominal Performan e: indi ates omplian e of the nominal, unperturbed system with the design requirements for losed-loop performan e.

These

requirements are implemented by the weighting fun tions in the inter on-



ne tion stru ture of gure 31.4. Robust Performan e: test if the losed-loop system meets performan e requirements and remains stable for the given set of perturbations to the nominal model. This test is performed by a

- al ulation on the omplete

losed-loop system. The robust stability, nominal performan e and robust performan e plots of the longitudinal system an be found in gure 31.8. Robust stability is a hieved, indi ating that the ontroller stabilizes the omplete set of plant systems. With the demanding set of requirements and the large plant perturbations, obtaining nominal performan e over the omplete frequen y range was impossible. With the weighting fun tions used, this was the best obtainable result. Nevertheless, the hoi e of weighting fun tions is stringent enough to give good performan e even with nominal performan e values above unity.

The performan e does

not deteriorate mu h if the un ertainty and ight envelope perturbations are introdu ed. This is indi ated by the robust performan e

-plot whi h is only

slightly larger than the nominal performan e plot. An interesting feature of this last plot is that be ause of the numeri al di ulty in the

- al ulation

with

the real valued perturbations used in this analysis, the upper and lower bounds of

 show a small gap.

Similar results were a hieved in the lateral-dire tional

design, as an be seen in gure 31.9.

519

31.5.2 Linear time domain analysis Figures 31.10 and 31.11 show the pit h and roll rate time responses to a step input at t=1 se ond. To he k losed-loop system tra king of the ideal referen e model, responses of both the ideal and the nominal system are plotted in the same gure, together with the error or dieren e between them. Besides the nominal system response, the response of the perturbed system with full parametri un ertainties on the matrix elements is plotted also. This gives an indi ation of the robustness of the resulting ontroller. Be ause the hoi e of the dynami pressure variations explained in se tion 31.2, the nominal system

orresponds to a dynami pressure value of

q = 6900.

Both gures show good tra king of the ideal response with little deterioration when un ertainty perturbations are added. This gives an indi ation of the robustness qualities of the ontrollers.

Only at the onset of the step re-

sponse a small dieren e between the ideal and the a tual response exists, but steady-state tra king is a hieved in both ases. 1.2

1.4

1.2

pilot 1

dotted:

ideal pitch rate response

solid:

nominal system

dashed:

perturbed system

pilot 1

command

command 0.8

0.8

dotted:

ideal roll rate response

solid:

nominal system

dashed:

perturbed system

0.6

0.6

0.4

0.4

0.2

0.2

q-error 0

p-error 0 -0.2

-0.4 0

1

2

3

4

5

6

7

8

9

-0.2 0

10

1

2

3

time [sec]

4

5

6

7

8

9

10

time [sec]

Figure 31.10: Pit h ommand lin-

Figure 31.11: Roll ommand linear

ear time response

time response

31.6 Results of the Automated Evaluation In the automated evaluation pro edure, the ontroller was subje ted to a series of nonlinear simulations to verify omplian e with the HIRM requirements. The overall impression was that the ontroller showed good performan e and robustness hara teristi s. The ontroller satised the riteria on erning disturban e reje tion, stru tural oupling and limited noise and turbulen e ontrol a tivity. Gibson riteria and departure prote tion also looked good. Only the lateral hannel multivariable un ertainty riterion was not satised ompletely. As an example two gures from the nonlinear simulation assessment manoeuvres are plotted. In gure 31.12 the response is given for a pit h rate demand, at M = 0.3 and h = 5000 ft. This plot shows that the pit h response a hieved orresponds

520

to the ideal handling qualities referen e model. The airspeed varies be ause of the in reased pit h attitude while the ontroller tries to ompensate for that with an in rease in thrust. In gure 31.13 the response is given for a roll rate demand, at M = 0.5 and h = 15000 ft with parametri un ertainties and measurement errors present. Despite the added un ertainties the roll rate response follows the ideal roll model. The transients in the roll rate and sideslip responses of this gure are

aused by the

2Æ

measurement error on

.

As a result of the instantaneous

appli ation of this error at t=0 se onds, the ontroller generates fast ontrol surfa e dee tions to try to for e the measured sideslip ba k to zero.

31.7 Con lusions and Lessons Learned H1 -, -synthesis, it was pos- ontrollers for the realisti ontrol augmentation

Using the robust ontrol design methodology of sible to design and analyse

design problem stated in the HIRM design ben hmark. These ontrollers were embedded in a nonlinear ontroller blo k ontaining dynami pressure ontroller output s heduling and several nonlinear and limiting elements.

This

resulting ontroller exhibits good performan e hara teristi s in terms of handling qualities and departure prevention, while maintaining robustness against model un ertainty and parameter variations. The strong points of



 

H1 -, -theory are:

Multivariable design problems an be solved in a fully multivariable design and analysis environment. The same framework is used for both synthesis and analysis of ontrollers and losed-loop systems. Robustness of the resulting ontroller against model un ertainty, unmodelled dynami s and parameter variations an be in orporated into the design

 

pro ess via a well-dened set of plant models around a nominal model. Handling qualities requirements an easily be implemented into the design pro ess with the use of an ideal referen e model. A powerful set of dedi ated software tools exists. The Matlab mutools toolbox provides the designer with all fun tions, subroutines and data stru tures



needed for the engineering appli ation of

H1 -, -synthesis.

The design y le an be largely automated.

During this design, several points of onsideration and lessons learned on erning the use of

H1 -, -synthesis were identied, some general and some spe i

to this design problem. These are:





A thorough understanding of the theory is needed to ee tively design and analyse ontrollers. A large and di ult part of the design y le onsists of hoosing and hanging the weighting fun tions. This eort an be redu ed by experien e and a good physi al understanding of the possibilities and limitations of the



systems at hand. The resulting ontrollers may have a large number of states needing good

521



model redu tion te hniques. It is not always feasible to keep on tuning the weighting fun tions to for e the

-value below unity for all frequen ies.

Often, design obje tives are too

ompli ated to be expressed exa tly as rst or even se ond order weighting



fun tions and good designs an be made with larger in a ontroller whi h satises robust stability. one of the strong points of



-values.

It is however, sensible to sear h for a set of weighting fun tions that results This way, use is made of

H1 -, -theory, namely: to be able to guarantee

stability for a given set of plant models. The linear ontrollers designed with

H1 -, -synthesis are not able to ope

with large nonlinearities in the model and the design requirements. A limited number of additional lters, fun tions and nonlinear blo ks an be added to the ontroller ar hite ture to deal with these nonlinearities. All in all, this design has learly demonstrated that

H1 -, -synthesis has apa-

bilities whi h an be used to solve a realisti nonlinear air raft robust ontrol law design problem. It is expe ted that

H1 -,  -theory will prove to be a valu-

able tool to the European Air raft Manufa tures for the design and analysis of Flight Control Systems.

522

x 10

pw

q

(deg/s)

(deg/s)

5 0 −5

0

5 time (s)

−5

5

−10

0 −5

10

0

5 time (s)

Va

5 time (s)

10

10

0

5 time (s)

2

5 time (s)

−15

10

0

(deg)

(deg)

0 −2

5 time (s)

−10

10

x 10

10

dtd

2 0

0

5 time (s)

x 10

−2

10

dcd

0

5 time (s) −6

6

x 10

10

dr

4

−5

0

5 time (s)

4

−7

5

0

−6

6

(deg)

0

dcs

2 0

0

thr1

5 time (s)

10

−2

0

5 time (s)

10

thr2 100

(%)

100

(%)

−15

−10

2

50

0

10

(deg)

−5

(deg)

(deg/s)

4

beta

−5

dts 0

x 10

10

−10

q cmd 6

5 time (s)

0

20 15

0

0

−7

5

(deg)

(deg)

(m/s)

100 99.5

−4

−20

10

25

100.5

0

−15

alpha

101

99

az

10

(m/s^2)

−7

10

0

5 time (s)

10

50

0

0

5 time (s)

10

Assessment manoeuvre: pitch rate demand, Mach 0.3, 5000 ft

Figure 31.12: Responses to a pit h rate ommand at Ma h 0.3 altitude 5000 ft.

523

pw

q

0

2

4

0

0 −1 −2

6

5

(m/s^2)

50

0

0

2

4

161 2

4

−2

6

0

2

p cmd

4

−2

6

(deg)

5

(deg)

−7 −8 −9

−5

0

−10

−10

0

2

time (s)

4

6

dcs

dcd

2

0 (deg)

0 (deg)

5

−2

0 −2

4

−4

6

6

4

6

dr

4

−5 −10

0

2

time (s)

4

6

−15

time (s)

thr1

0

2 time (s)

thr2 10

(%)

10

5

0

2 time (s)

1

2

0

time (s)

−1

4

0

20 6

6

dtd

60

0

2

dts 10

−3

0

time (s)

−6

4

4

0

80

2

6

2

time (s)

40

4

beta

2

time (s)

0

2

4

0 0

0

time (s)

(deg)

162 161.5

(deg/s)

−15

6

alpha 6

(deg)

(m/s)

Va

(deg)

4 time (s)

162.5

(%)

−5 −10

time (s)

160.5

az

1

(deg/s)

(deg/s)

100

0

2

4 time (s)

6

5

0

0

2

4

6

time (s)

Assessm. man.: roll rate demand, Mach 0.5, 15000 ft, param. unc.

Figure 31.13: Responses to a roll rate ommand at Ma h 0.5 altitude 15000 ft. with parametri un ertainties and measurement errors

524

32.

Nonlinear Dynami Inversion and

LQ Te hniques

Béatri e Es ande

1

Abstra t.

The ontribution presented here for the HIRM design

hallenge onsists of an expli it model-following te hnique, using a

ombination of non-linear inversion (NDI) and LQ synthesis. This te hnique allows us to over a wide ight envelope without the problem of gain s heduling.

It also aims at better re-usability of

the ontroller on dierent airframes. Here, it is tested on a set of typi al manoeuvres at dierent operating points.

32.1 Introdu tion The present hapter on erns the presentation of ONERA's parti ipation in the HIRM design hallenge. The ontrol law design whi h is here under study is based upon an expli it model-following te hnique using a ombination of nonlinear dynami inversion (NDI) and LQ synthesis, leading to a two-level ontroller stru ture:



a feedforward ontroller (NDI) whi h addresses the problems of performan e;



a feedba k ontroller (LQ) whi h addresses the problems of stability and robustness.

The two methods are presented in the tutorial part of the book (see hapters 9 and 5). Their implementation for the HIRM ase are des ribed in the following se tions.

32.2 Controller Stru ture The design of the feedforward ontroller uses a nonlinear model of the air raft and the NDI theory to treat the nonlinear aerodynami terms as well as the nonlinearities oming from kinemati and gyros opi ouplings. The design of the feedba k ontroller uses the linear quadrati ontrol theory to treat the deviations (whi h are assumed to be small) of the air raft outputs 1

xr from the model states xrm .

The presen e of an integral term on the

O e National d'Etudes et de Re her hes Aerospatiales (ONERA), BA701, 13661 Salon

de Proven e Air

525

ontrolled variables

z

makes the deviations tend asymptoti ally towards zero.

The overall stru ture of the ontroller is displayed in gure 32.1.

uffwd zp

Feedforward Controller

pilot input

xrm − zm

+

+

Kx

+

+

u control vector

+

xr

aircraft reduced state vector

−

z

aircraft controlled vector

Kz

+ Feedback Controller

Figure 32.1: Overall ontroller stru ture

32.2.1 Feedforward ontroller The formulation of the feedforward ontrol law is based upon the for e and moment equations. The ontrolled variables are in the present ase:



as

 

q  ,

pit h rate

q

or

or angle-of-atta k

velo ity ve tor roll rate and sideslip demand

 (depending on the value of ),

dened

pW , dened as pW

, dened as .

Airspeed is also one of the ontrolled variables, but it is treated separately of the others. The general s heme of the feedforward ontroller is given in gure 32.2 The equations for lift and lateral for es and for velo ity ve tor roll rate are written as follows :

g _ = q (p os  + r sin ) tan + (nz + os os )= os V g _ = p sin  r os  + (ny + os sin ) V g pW = (p os  + r sin )= os n (+ os os ) tan V z ny and nz are the load fa tors in stability axes,  the bank angle.

where and

526

(32.1)

the ight path angle

pilot inputs

fast dynamics

slow dynamics pc qc rc βc

pwc qc βc

load factors ny nz

φm θm αm βm Vm

uffwd . pd . qd . rd

pm qm rm βm

inversion

xm

u 1 1+Ts Non−Linear Model Aircraft modelled state vector xm

(on−board model)

Figure 32.2: Feedforward ontroller stru ture

The moment equations for pit h, roll and yaw about the body axes are given by:

Ixx p_ Ixz r_ = LA + LT + (Iyy Izz )qr + Ixz pq Iyy q_ = MA + MT + (Izz Ixx )pr Ixz (p2 r2 ) (32.2) Ixz p_ + Izz r_ = NA + NT + (Ixx Iyy )pq Ixz qr where LA , MA , NA are the omponents of the aerodynami moment and LT , MT , NT the omponents of the moment due to the thrust. The feedforward ontrol law is based on a dynami model of the air raft whi h is formulated by the following dierential equations:

x_ = F (x; u) z = h(x)

(32.3)

F and h are non-linear fun tions of state ve tor x (and of input ve tor u for F ), z being the ve tor of ontrolled variables.

In general (see equations 32.3)

The following assumptions are made:



it is assumed that the dynami s of angular rates are faster than those of angle-of-atta k and sideslip,



the inuen e of the ontrol surfa es dee tions on the translational dynami s is negligible.

From these assumptions, the feedforward ontroller an thus be separated into two steps: the rst one is related to the slow dynami s and the se ond one is related to the fast dynami s.

 with ommanded angular velo ities, by inversion of the for es equations

These assumptions allow the repla ement of the ommanded values on and

527

system 32.1. Then, the input ve tor

u is derived expli itly through a rst-order

dierentiation of these angular velo ities, whi h allows a dire t appli ation of the NDI te hnique. By linearizing the rotational dynami s about a trimmed

u of the input, the nal system to be inverted (moment equations, with

_ = (p;_ q;_ r_)) is written as :

value

_ = f (xm ) + g(xm )(uffwd

u)

(32.4)

In this equation the state ve tor used to ompute the dynami s is a mod-

elled state ve tor

xm , that is omputed by an on-board model.

The on-board

model is here identi al to the real air raft model (nonlinear hirmex ode) and is in luded in the feedforward system. It uses as its input ve tor the feedforward

ommands

uffwd,

linearization value

and delivers as output the modelled state ve tor

u is a ltered value of the input ve tor uffwd.

xm .

The

The basi obje tive for the dynami inversion is to ompute the input ve tor

uffwd

as a fun tion of the desired dynami s

ontrol law:

uffwd = u + g 1[ _ d

_ d .

This gives the feedforward

f (xm )℄

(32.5)

The dierent steps leading to equation 32.5 are detailed in the following se tions.

Slow dynami s As was mentioned above, it was hosen to perform the inversion in two su

essive steps orresponding to slow and fast dynami states. The system alled

slow dynami s treats the equations of for es and the kinemati equation for velo ity roll rate, whi h were written in system 32.1. It is represented in gure 32.3. The purpose of this inversion is to deliver ommanded angular velo ities

(p ; q ; r )

from the pilot ommands whi h are here

noti ed that the ommanded pit h rate

q

(pW ; q ; ).

It an be

is dire tly input from by the pilot.

As angular velo ities are the fast variables, they are used as inputs in the for e equations.

As

response for

:

_

and

pW

are dependent on in

p

and

r,

dening a model of

_d = ! ( d ) and inverting 32.1 gives:

p = p? os os  r? sin  r = p? os sin  + r? os  with the variables:

g sin (nz + os os ) V g (n + os sin ) r? = ! ( ) V y

p? = pW os +

528

(32.6)

φm θm αm βm Vm

coord. transformation ny nz (A/C)

γ µ nya nza

pilot input: qc α−limiter

pilot inputs: pwc βc

slow dynamics

pc qc rc

pc rc

Figure 32.3: Slow ontroller

!

where

is the desired bandwidth hosen for



in a

ordan e with handling

qualities requirements. If we onsider an angle-of-atta k limiter, the resultant ommanded pit h

qmax dened below  dynami s equation and a bandwidth ! , i.e.: g = ! (max ) + (p os  + r sin ) tan (n + os os )= os V z

rate is taken as the minimum of the pilot ommand and of in equation 32.7, based on the

qmax

(32.7)

Fast dynami s The system alled fast dynami s onsists of an inversion of the moment equations. The purpose of this inversion is to deliver a ontrol surfa es dee tions ve tor

uffwd for a given set of ommanded angular rates p ; q and r .

At this stage, the problem of ontrol allo ation is posed as the ase (like for the HIRM model) where the air raft model has more inputs than the ontrolled outputs, and when the use of the lassi al inputs does not allow su ient ontrol power. The angular a

elerations, expressed as in equation 32.4, are shaped to the desired dynami s:

_ d = (p_d ; q_d; r_d )t

whi h is of rst order in roll and yaw and of se ond order in pit h :

p_d = !p (p pd ) 529

q_d = 2q !q (q r_d = !r (r Here again, the parameters

Z

qd ) + !q2 (q qd ))

rd ))

!p , !q , q , !r

are hosen in order to fulll the

handling qualities requirements. The ontrol law

uffwd

is then omputed via a generalized inverse of the

linearized moment ontrol matrix:

uffwd = u +
[g(xm )
℄+ ( _d f (xm )) where

[:℄+

denotes the pseudo-inverse, and

(32.8)


a diagonal weighting matrix.

The pseudo-inverse orresponds to a minimization of the normalized ontrol surfa es dee tions.

Other solutions exist for ontrol allo ation, like daisy

haining, dire t allo ation, other generalized inverse, but they are still at the resear h level [29℄ [69℄.

32.2.2 Robust feedba k ontroller The purpose of the feedba k ontroller is to minimise the deviations between the real air raft state and the referen e model (on-board model). It is done by a linear quadrati method, known for its robustness, using only a redu ed state ve tor

xr , dened as a subset of the state ve tor x: xr = (v; w; p; q; r)t

The referen e model states are also represented here by a redu ed ve tor

xrm ,

whi h is:

xrm = (v; w; p; q; r)tm The ve tor of ontrolled variables z orresponds to the angular variables om-

manded by the pilot:

z = (pW ; q; )t

and it is ompared to the variables dedu ed from the referen e model ontained in ve tor

zm :

zm = (pW ; q; )tm

The ontrol law

ufb k

is expressed by the following expression, in whi h

the integral term on the ommanded variables

z

ensures a good asymptoti

behaviour [169℄:

ufb k = Kx (xr

Z

xrm ) + Kz (z zm )

(32.9)

where the gain matri es are obtained by minimization of a quadrati index:

J=

1

Z

0

(Æx_r t QxÆx_r + Æz tQz Æz + Æu_ t Qu Æu_ )dt 530

where

The weighting matri es

Æxr = xr xrm Æz = z zm Æu = u uffwd (Qx ; Qz and Qu ) of this

index are tuned in order to

meet robustness requirements.

32.2.3 Autothrottle The airspeed ommand is here treated separately from the other pilot ommands, by a spe i ontrol law alled autothrottle. The reason for this separate treatment is for pra ti al simpli ity, and it is justied by the fa t that the response time is onsiderably longer than for the other ommands. It omputes the total thrust required as a fun tion of the ommanded airspeed, by inversion of the drag equation.

The ommanded airspeed is on-

verted into a ommanded a

eleration by imposing a se ond order response. The damping and frequen y of this pres ribed response have to be tuned a

ording to performan e requirements. The autothrottle system is represented in gure 32.4.

1 s

ω2

commanded airspeed

Vc

+

+ 2ζω −

measured airspeed

+

m

+

Inversion of thrust model

cosα cosβ

throttle position

+

+

V

γ computed flight path angle

sin (.)

g trimmed thrust Figure 32.4: Autothrottle

32.2.4 Resulting ontroller omplexity and implementation issues The ontroller presented here, based on the NDI-approa h for the rst level and the LQ-method for the se ond level, is of relatively high omplexity and of high dynami order. This hoi e has been made due to the in reasing omputing power that will be available on modern air raft. The order ould be redu ed but the resulting performan e would also be ae ted.

One reason for this

omplexity is that the rst level outputs are ontrol surfa es dee tions. The air raft model whi h is in luded in the feedforward an be simplied by taking a simpler aerodynami model (e.g. analyti instead of based on look-up tables). Here again, the required level of realism of this plant model is one of

531

the main hoi es of this approa h. A su iently realisti plant model permits to a hieve optimal use of the ontrol power of the air raft. The present model (HIRM) with its asso iated ontroller, was implemented with su

ess in a real-time omputing environment.

This demonstrates that

despite its relative omplexity this method is not ex essively time- onsuming for a real air raft.

32.3 Design Des ription 32.3.1 Design obje tives The rst design obje tive is to get a general ontroller stru ture that an be easily re-used for any air raft mode [71℄. Typi ally, on e the ontroller is designed for one air raft, it should be possible to adapt it to another one with a limited number of hanges. When the hanges on ern only the aerodynami parameter tables of a given air raft (for example for the study of dierent aerodynami designs), the ontroller an be re-used without any modi ation. In parti ular, the tuning of the desired dynami s remains valid independently of the model onsidered. In some ases ontrol saturations an o

ur, e.g.

in the ase of dierent

ontrol powers, and require some readjustment of the dynami parameters and of the weighting matrix


whi h ontrols the distribution among the dierent

ontrol surfa es. Other solutions an be used, su h as pilot input ltering. This possibility enabled us to take into a

ount the su

essive versions of the air raft model used for this hallenge. The se ond obje tive to be mentioned is to separate the tuning parameters of the dynami s relative to the pilot inputs (performan e aspe ts) and to the turbulen e inputs (robustness aspe ts). Another of our obje tives is to have a ontroller whi h is able to treat and evaluate the new ontrol te hnologies, like thrust ve toring, strakes, su tion,et . In parti ular, with the NDI approa h, the problem of ontrol allo ation an be studied in a quite general framework.

32.3.2 Design y le des ription On e the ontroller stru ture exists, i.e. the inversion of the equations is formulated, the design y le onsists of the following steps:



tuning the bandwidth parameters in order to a hieve the performan e requirements, 1.

!q

and

q

hara terize the ideal pit h response and are tuned by

onsidering the dropba k riterion;

2. 3.

!p is dire tly related to the roll mode time onstant by !p = 1=T ; !r and ! are hosen by xing the dut h roll hara teristi s in frequen y and damping; the riterion on the sideslip response an here be he ked;

532



adjusting the weighting matri es of the feedba k ontroller. This tuning is done starting from an initial guess whi h is a diagonal matrix formed with the inverses of the squares of the variables onsidered (for ea h variable the value taken here is lose to the maximum value a

eptable). For example, the

Qz

matrix is initially taken as:

0

Qz = B 

1 0 0 0 z1 0 0 0 1

z12

2 2

1 C A

z32



performing robustness tests and if ne essary modifying the feedba k matri es.

32.3.3 The translation of HIRM design riteria into method dependent obje tives The NDI approa h presented here takes dire tly into a

ount the design riteria if they are expressed in terms of modes. For example, the riterion on roll mode time onstant (0.4s) is dire tly translated by:

!p = 1=0:4.

The other performan e requirements in the time domain are not dire tly introdu ed in the parameters of the desired dynami s. But, as there is a physi al meaning for the adjustable parameters, their tuning is done by means of time simulations, with elementary pilot inputs. The parameters are:

 !r ; ! ; !q ; q

orresponding respe tively to the rst order dynami s in

yaw and sideslip, and to the se ond order dynami s in pit h,

 ! , asso iated to the dynami s in  in luded in the angle-of-atta k limiter,

 !f ; f , orresponding to the se ond order dynami s for the air speed. The robustness is he ked independantly and an ne essitate some iterations in the design y le. The stability of the system being reliable, the tuning of weighting matri es of the se ond level still requires some experien e.

32.3.4 Comments on the eort required The hoi e of the parameters of the rst level dynami s are independant of the airframe model. That means that a given set of parameters an be used with the same validity provided that the air raft models of the real and on-board are identi al and that the ontrols are not saturated. This aspe t is one of the potential time-saving qualities of the inverse dynami s approa h. The adjustment of the se ond-level matri es an require iterations if the robustness requirements are not satisfa tory.

533

32.4 Analysis of the Resulting Controller The basis of this analysis are the design requirements listed in hapter 27.

32.4.1 Typi al time responses The riteria on dropba k and sideslip response are plotted in gures 32.5 and 32.6. It is shown that the ontroller gives here satisfa tory results, although the dropba k value is very small.

Dropback criterion

2.5

THETA (deg)

2

1.5

1

0.5

0

0

0.5

1

1.5

2

2.5

t (s)

Figure 32.5: Dropba k riterion

Typi al time responses are also useful in order to analyse the ontroller's behaviour and eventually rene the tuning of the rst-level bandwidth parameters, and of the diagonal weighting matrix surfa es.




between the dierent ontrol

The simulations are performed with the rst level alone, without

the se ond level orre tion. In the following gures (32.7 to 32.11), the time responses to simple step inputs are shown. In the ight ase 2001, strong aerodynami ouplings are present in the HIRM model. The results show that with the NDI method, the air raft's response is identi al to the pres ribed dynami s, in the ase where there are neither modelling errors nor saturated ommands.

534

Step sideslip requirement Upper boundary 1.2

Normalised sideslip angle

1

0.8

Lower boundary

0.6

0.4

0.2

0 0

1

2

3

4

5 6 Time(seconds)

7

8

9

10

Figure 32.6: Sideslip response riterion

32.4.2 Frequen y responses in the linear domain at nominal ight ondition Performan es The ompatibility of the rst-level pres ribed dynami s (1st order for velo ity ve tor roll rate, 2nd order for pit h rate) with the Gibson riteria an be

he ked, as is shown in gures 32.12 and 32.13.

Robustness The robustness of the feedba k ontroller is evaluated by plotting the open loop Ni hols graphs. In that ase, no feedforward ontroller is onsidered. The results are obtained with the standard Matlab linearisation tools (gures 32.14 and 32.15).

535

VV roll demand at flight condition 4010

VV roll demand at flight condition 4010 100 1

10

15

10

0

5

−5 0

10

10

166

0 −5

10

1.5

164

−15 0

5

160 0

10

1.5

0

5

10

−1 0

5

4450 0

5

6

BETA deg

ALPHA deg

Hm 4500

4

DTright deg

0

−20 0

10

−5

2

4

6

8

4

6

8

10

2

4

6

8

10

2

4

6

8

10

2

4

6

8

10

0

−10

−20 0

10

10

0

5

10

0.5

−10

0 −0.5

−30 0

2

4

6

8

−1 0

10

0.5

2

2

0

1.5

1.5

−1

5

10

−20

−0.5

2 0 0

10

2

1

−15 0

10

8

4550

5

−10

−0.5

−1 0

0 10

20

0.5

4600

10

−10

0

AZ m/s2

0 −0.5

8

5

1

AY m/s2

1 0.5

6

SUCTION

5

4

162

−10

−50 0

2

−1.5 0

5

THROTTL2

0

5

−20

10

DCleft deg

50

0 −10

−30

0

10

168

V m/s

THETA deg

100

5

DCright deg

−3 0

10

−20 −30

DR deg

5

150

AX m/s2

5

−2

−100 0

PHI deg

−1

−10

THROTTL1

0 −50

10

R deg/s

Q deg/s

Pw deg/s

0

DTleft deg

0 50

1 0.5 0 0

10

2

4

6

8

1 0.5 0 0

10

Figure 32.7: Roll rate response (Ma h 0.4, 10000ft)

Pitch demand at flight condition 2001, with AOA limiting 10

2 0 −2 0

15

0.3

40

0.2

35

0.1 0 −0.1 0

5

10

−10 −20 −30

0.1 5

10

0 0

15

0 5

10

15

70

0 −10 −20 −30

5

10

15

0

10

5

10

15

5

10

15

5

10

15

5

10

15

10

75

30

0

−10

0

−10

65

25 20 0

15

0.2

10

DCright deg

10

0.3

V m/s

5

THETA deg

PHI deg

−0.05 0

DTleft deg

4

DCleft deg

Q deg/s

Pw deg/s

0

0.4

R deg/s

6 0.05

0

0.5

DTright deg

Pitch demand at flight condition 2001, with AOA limiting 0.1 8

5

10

60 0

15

−20 0 5

10

5

10

−20 0

15

15

1

0.2

4 0 0

15

5

10

15

35

ALPHA deg

380

Hm

10

−10 −11

360 340 320 5

10

15

−10

5

10

−30 0

15

0.04

25 20

0.02 0 −0.02

5

10

15

−0.04 0

5

10

15

0.5 0 −0.5

−20

0.06

30

15 0

−13 0

0

THROTTL1

10

400

300 0

20

−9

−12 5

BETA deg

2 0

−8

SUCTION

6

AZ m/s2

AY m/s2

AX m/s2

8

0.4

5

10

−1 0

15

2

2

1.5

1.5

THROTTL2

0.6

10

DR deg

12

1 0.5 0 0

5

10

15

1 0.5 0 0

Figure 32.8: Pit h rate response (Ma h 0.2, 1000ft)

536

−7 at flight condition 4010 Pitch demand x 10 2 8

5

0 −0.5

5

−10 −20 −30

−1 0

10

5

0

10

−7

30

THETA deg

PHI deg

2 1 0

25

160

20 15

5

5 0

10

−20 −30

2

4

6

8

10

0

158

2

4

6

8

10

2

4

6

8

10

2

4

6

8

10

2

4

6

8

10

2

4

6

8

10

2

4

6

8

10

2

4

6

8

10

2

4

6

8

10

10

0

−10

0

−10

156

10

−1 0

0 −10

10

162

DCleft deg

x 10

V m/s

3

DTright deg

2

−2 0

10

10

DCright deg

4

0 −1 0

DTleft deg

0.5

R deg/s

Q deg/s

Pw deg/s

0

10

x 10

0

6 1

Pitch demand at flight condition 4010

−7

1

5

154 0

10

5

−20 0

10

2

4

6

8

−20 0

10

−7

8

1

6

0.5

x 10

1

−5

2

−0.5

0 0

5

−20

−1 0

10

−15

5

10 0 −10

5

−30 0

10

0 −0.5

−20

−25 0

10

0.5

SUCTION

0

DR deg

4

AZ m/s2

AY m/s2

AX m/s2

20 −10

2

4

6

8

−1 0

10

−8

ALPHA deg

4700 4600

BETA deg

14

4800

Hm

4

12 10

5

6 0

10

2 0 −2

8

4500 0

x 10

5

−4 0

10

5

2

2

1.5

1.5

1 0.5 0 0

10

THROTTL2

16

THROTTL1

4900

2

4

6

8

1 0.5 0 0

10

Figure 32.9: Pit h rate response (Ma h 0.4, 10000ft)

sideslip demand at flight condition 4010 10

0.5

R deg/s

Q deg/s

Pw deg/s

0 −0.1

0 −1

−3 0

10

0 −0.5

−2 5

0

DTleft deg

1

0.1

−0.2 0

1

5

−1 0

10

10

DTright deg

sideslip demand at flight condition 4010 x 10−3 0.2 2

−10 −20 −30 0

5

0 −10 −20 −30

2

4

6

8

10

0

10

5

10

7.116 7.115 0

1.3

−0.1

4571.9

5

10

7.12

0

DCright deg 4

6

8

−20 0

10

10 0 −10

10

−9.74 0

5

10

10

0

2

4

6

8

−1 0

10

1.5

2

2

1

1.5

1.5

0.5 0

5

−30 0

0.5

−0.5

−20 5

7.11 4571.85 0

DCleft deg

−9.735

BETA deg

Hm

4571.95

2

1

−9.73

7.13

ALPHA deg

4572

−20 0

10

20

0

−0.3 0

5

−9.725

AZ m/s2

AX m/s2

AY m/s2 10

0

−10

−9.72

−0.2 5

161 0

10

0.1

1.2

0

−10

161.05 5

0.2

1.25

1.15 0

161.1

SUCTION

−0.3 0

161.15

−0.5 0

5

10

THROTTL2

−0.2

161.2

7.117

DR deg

7.118

THROTTL1

0 −0.1

10

161.25

V m/s

7.119

THETA deg

PHI deg

10 0.1

1 0.5 0 0

2

4

6

8

10

1 0.5 0 0

Figure 32.10: Sideslip response (Ma h 0.4, 10000ft)

537

speed demand at flight condition 2001 10

speed demand at flight condition 2001 0.02 0.04

10

0.1

−0.01 −0.02 −0.03 0

0 −0.02

0.05

40

60

−0.06 0

−10 −20 −30

20

40

60

0 −10 −20

0

−0.04 20

DTright deg

0

R deg/s

0.02

Q deg/s

Pw deg/s

0.01

DTleft deg

0

−0.05 0

−30

0 20

40

20

40

60

0

−0.2 −0.3 0

20

40

60

10

21.42 21.4 21.38 0

20

40

100

60 0

60

0

−10

20

40

−20 0

60

20

40

−20 0

60

40

60

−0.05 0

4500

20

40

60

Hm 3500

BETA deg

ALPHA deg

25 20 15

20

40

60

5 0

20

40

60

20

40

60

20

40

SUCTION

−10

20

40

60

0 −0.5

−30 0

20

40

−1 0

60

0.01

2

0.005

1.5

1.5

0

60

0.5

2

−0.005

10

−0.01 0

20

40

60

1 0.5 0 0

20

40

60

1 0.5 0 0

Figure 32.11: Speed response (Ma h 0.2, 1000ft)

25

20

15

10

Open−Loop Gain (db)

3000 0

−11 0

0

−20

−10

30

4000

−9

10

THROTTL1

20

−8

THROTTL2

0

DR deg

AZ m/s2

AY m/s2

AX m/s2

0 0

60

1

−7

0.05

2

40

−10

20

4

20

0

−6

0.1

6

60

80

0.15

8

40

10

DCright deg

120

−0.1

DCleft deg

140

21.44

V m/s

21.46

0

THETA deg

PHI deg

10 0.1

20

60

5

0

3

−5

−10 1

1 Sluggish Response

−15 2

PIO prone 2 Good Response

−20

3 Quick, Ratcheting −25 −270

−180 Open−Loop Phase (deg)

−90

Figure 32.12: Linear roll Gibson riterion for 1st order roll dynami s

538

20

15

Open−Loop Gain (db)

10

5

L2

L1

L2

L3 0 L2

−5 L1

−10

−180

−90 Open−Loop Phase (deg)

Figure 32.13: Linear pit h Gibson riterion for 2nd order pit h dynami s

539

DTs loop open

20

0 db 0.25 db 0.5 db 1 db 3 db 6 db

0

DTd loop open 40

−1 db −3 db −6 db −12 db −20 db

−20

−40 −360

−270 −180 −90 Open−Loop Phase (deg)

Open−Loop Gain (db)

Open−Loop Gain (db)

40

−40 db 0

20

3 db 6 db 0

3 db 6 db 0

−40 −360

−1 db −3 db −6 db

−20 db

−270 −180 −90 Open−Loop Phase (deg)

−270 −180 −90 Open−Loop Phase (deg)

−40 db 0

20

0 db 0.25 db 0.5 db 1 db 3 db 6 db

0

Open−Loop Gain (db)

20

0 db 0.25 db 0.5 db 1 db 3 db 6 db

0

−40 −360

−40 −360

−270 −180 −90 Open−Loop Phase (deg)

−1 db −3 db −6 db

−20 db

−270 −180 −90 Open−Loop Phase (deg)

−40 db 0

Figure 32.14: Ni hols plots with su

essive loops open

540

−3 db −6 db

−20 db

−20

−12 db −20

−1 db

−12 db

DR loop open 40

−40 db 0

DCd loop open 40

−12 db −20

−3 db −6 db

−20 db

−20

−40 −360

Open−Loop Gain (db)

Open−Loop Gain (db)

20

0 db 0.25 db 0.5 db 1 db

−1 db

−12 db

DCs loop open 40

0 db 0.25 db 0.5 db 1 db

−40 db 0

DTs loop open

20

0 db 0.25 db 0.5 db 1 db 3 db 6 db

0

DTd loop open 40

−1 db −3 db −6 db −12 db −20 db

−20

−40 −360

−270 −180 −90 Open−Loop Phase (deg)

Open−Loop Gain (db)

Open−Loop Gain (db)

40

−40 db 0

20

3 db 6 db 0

3 db 6 db 0

−40 −360

−1 db −3 db −6 db

−20 db

−270 −180 −90 Open−Loop Phase (deg)

−270 −180 −90 Open−Loop Phase (deg)

−40 db 0

DCd loop open 40

−12 db −20

−3 db −6 db

−20 db

−20

−40 −360

Open−Loop Gain (db)

Open−Loop Gain (db)

20

0 db 0.25 db 0.5 db 1 db

−1 db

−12 db

DCs loop open 40

0 db 0.25 db 0.5 db 1 db

−40 db 0

20

0 db 0.25 db 0.5 db 1 db 3 db 6 db

0

−1 db −3 db −6 db −12 db −20 db

−20

−40 −360

−270 −180 −90 Open−Loop Phase (deg)

−40 db 0

DR loop open

Open−Loop Gain (db)

40

20

0 db 0.25 db 0.5 db 1 db 3 db 6 db

0

−1 db −3 db −6 db −12 db −20 db

−20

−40 −360

−270 −180 −90 Open−Loop Phase (deg)

−40 db 0

Figure 32.15: Ni hols plots with su

essive loops open, with parametri un ertainties

541

32.5 Con lusions The NDI/LQ approa h has been applied to the problem of robust ight ontrol of a ghter air raft model, the HIRM. The main advantages that hara terize the approa h are:



its exibility in terms of its potential appli ation to a wide range of vehi les,



its rapid implementation and easy a

omodation of desired handling qualities,



the good a

ura y of the rst level, that allows, for a given level of turbulen e, to have more manoeuvrability power,



the independant tuning of performan e and robustness.

In parti ular, it ould be possible to use new ontrol means su h as nose su tion without important hanges in the ontroller (a few hours of work). But it also has some drawba ks:

  

it does not work in ase of saturations, it does not take expli itely into a

ount the robustness riteria, the feedforward ontroller is rather omplex.

542

33.

The Robust Inverse Dynami s

Estimation Approah

Ewan Muir

1

Abstra t.

A Robust Inverse Dynami s Estimation ontrol system

has been designed su

essfully for the HIRM design problem. The method has proved simple to use. Within the onstrained design time, it has produ ed a ontroller, with a re ognisable stru ture and physi ally meaningful gain matri es, whi h largely satises the design riteria.

33.1 Introdu tion Robust Inverse Dynami s Estimation (RIDE, see Chapter 10) is a design method whi h has developed from two previous ontrol te hniques: the Salford Singular Perturbation Method [115℄ and Pseudo-Derivative Feedba k [14℄. The method

an be viewed as a form of realisable Non-linear Dynami Inversion (NDI) due to the use of a dynami inverse input whi h inverts the model with respe t to the ontrolled outputs, thus providing output de oupling, and Proportional plus Integral feedba k loops whi h ompensate for errors in the al ulation of the dynami inverse input and assign the desired losed-loop response hara teristi s. The dynami inverse input is also similar to the equivalent ontrol input used in Variable Stru ture Control.

33.2 Sele tion of the Controller Ar hite ture for the HIRM Problem 33.2.1 Control law stru ture As spe ied in the HIRM design do ument, the ontrol law inputs are pit h rate, whi h in ludes positive limiting, velo ity ve tor roll rate and sideslip demands. A speed ontroller is also implemented. Normal a

eleration and negative angle of atta k limiting fun tions are missing due to la k of design time, but implementation would be similar to the positive

 limiting.

The ontrol law drives the tailplane, whi h is dee ted both symmetri ally and dierentially, and the rudder. Symmetri anard dee tions ould also be 1

Defen e Resear h Agen y, Flight Dynami s and Simulation Department, Bedford, MK41

6AE, UK

543

in luded relatively easily to minimise trim drag and to provide additional pit h

ontrol when tailplane dee tion is limited.

This has not been done in this

study due to time onstraints. Symmetri thrust demands have been used to

ontrol air speed. The pilot inputs and ontrol surfa e demands listed above are built into the

omplete ontrol law stru ture shown in Fig. 33.1, the omponents of whi h are des ribed in further detail in se tions 33.2.2 to 33.2.6 below.

-

VT

dem

K

PT

+

ρ0 ρ

thrust demand

(CB)-1 T

thrust table look-up demand thrust to corrected throttle for altitude throttle demand

feedforward

h

symm. canard

0

ax, az

ax, az

q

dem

α

limit

vvdem

βdem α, β

p

dem

diff. canard

0

T

K

-1

V 2

2(s+20) (s+40)2

Mach +

K +

-

I

+

+

+ +

2

2(s+20) (s+40)2

symm. tail

Aircraft Model

diff. tail rudder

Mach

Velocity vector rdem roll / β demand

-

Demux

p

+

Mux

α

Pitch rate demand / qdem AoA limiter

K

P

Mach

VT p, q, r α, β

u^ di Ku

di

u, v, w

Calculate u,v and w based on VT, α and β

Mach

Core RIDE controller

Figure 33.1: Stru ture of RIDE ontrol law for HIRM

33.2.2

Core RIDE ontroller

The ore RIDE ontroller, identied in Fig. 33.1 above, onsists of the following omponents. The dynami inverse input

u^di de ouples the outputs, i.e.

the

body axis rates, from ea h other and the other air raft states by using moment

an ellation. The proportional loop with gain

KP

a ts to stabilise the air raft

and is added in this way to provide pseudo-derivative feedba k. The integral path on the ommands with gain

KI provides robustness to errors in the estiu^di . The PI ontroller enables the designer

mate of the dynami inverse input

to spe ify the losed-loop dynami s whi h are se ond order of the form

y = (s2 I + 2Zd n s + n 2 ) 1 n 2 y

(33.1)

The gain matrix al ulations are elementary:

KP = (CB ) 1 2Zd n

(33.2)

KI = (CB ) 1 n 2

(33.3)

544

KV = (CB ) 1 M

(33.4)

(CB ) 1 CA diagonal. Zd denes

Kudi = where

Zd, n

and

M

are all

(33.5) the damping and

frequen y of the losed-loop system response, while

M

n

the

spe ies the overall

gain of the feedforward input (the time onstant of the feedforward washout is set by

T ).

33.2.3 Feedforward The feedforward is a washout whi h provides an additional degree of freedom in tuning the initial air raft response to a pilot demand by adding the following input to the integral path

uff where

= TKsV+sI y

(33.6)

= (CB ) 1 M

KV 1 As both M and T are diagonal matri es, ea h of the demands an be viewed

separately with the feedforward inputs being distributed to the ontrol surfa es by the inverse motivator ee tiveness matrix

(CB ) 1 .

33.2.4 Pit h rate demand / AoA limiter The pit h rate demand system is provided by al ulating a body axis pit h rate using equation 33.7.

qdem = os qdem + sin pvvdem

(33.7)

To provide an angle of atta k limiting fun tion, the pit h rate required to y the air raft at the limit AoA is al ulated from equation 33.8.

qdem = _ dem + (p tan where

_ dem

a anz ) os  + (r tan + nx ) sin  VA os VA os

is al ulated from equation 33.9

_ dem = (limit where

bw

(33.8)

)bw

(33.9)

is the bandwidth of the AoA ontrol loop.

The pit h rate demand used to al ulate the ontrol surfa e dee tions is the minimum of the onventional pit h rate demand, al ulated from equation 33.7 and the pit h rate required to limit the angle of atta k to the maximum value, al ulated from equation 33.8.

Note that to date, this has only been

developed to limit the positive angle of atta k. introdu ed for limiting on the negative AoA limit.

545

A similar s heme ould be

33.2.5

Velo ity ve tor roll / sideslip demand

Body axis roll and yaw rates are al ulated from the demanded velo ity ve tor roll rate and sideslip demands using equations 33.10 to 33.13. The body axis roll rate demand of the RIDE ontroller is given by

pdem = os  os pvvdem + sin ( _dem

rvv )

(33.10)

The body axis yaw rate demand of the RIDE ontroller is given by

rdem = sin  os pvvdem os ( _dem rvv ) where the velo ity ve tor yaw rate, rvv , is al ulated from rvv = os r sin p and the demanded rate of hange of sideslip _ dem _dem = ( dem where

33.2.6

bw

(33.11)

(33.12) is given by

)bw

(33.13)

is the bandwidth of the sideslip ontrol loop.

Air speed ontroller

The speed demand is me hanised as a simple error driven proportional ontroller. The error demands a thrust whi h is translated ba k into a throttle setting.

33.2.7

Filters

During the design, a lter was in luded on the stabilator demand, both symmetri and dierential, to improve the phase margins. The lter used is

4(s+20) (s+40)

2

2

. The hoi e of this lter is in no sense optimal and was hosen by the rapid trial of lters of the same order but with varying time onstants and observing their ee t on Ni hols plots. The results from se tion 33.5 indi ate that this solution is adequate and satises the majority of the riteria. However, it does in rease the high frequen y ontrol law gain from pit h rate to tailplane. In retrospe t, a redu tion in the gains

n

may have produ ed adequate gain

margins without re ourse to these lters.

33.2.8 Measurement signals As the ontroller regulates prin ipally the body axis rates, the primary feedba k signals are

p, q

and

r.

Air speed is obviously required for the speed

ontroller. The body axis rates are also required in the al ulation of the dynami inverse input, together with body axis velo ities. These velo ities have been supplied based on the total air speed resolved into body axes using and

.

Measurements of

 and



are also used to al ulate the body axis roll

and yaw rate demands ne essary to provide the velo ity ve tor roll and sideslip

546

demand s hemes.

The



limiter needs AoA, air speed and longitudinal and

normal a

eleration feedba k. Ma h number is used for s heduling for onvenien e as it is readily available as an output from the model. Ideally, dynami pressure would be used as the s heduling variable.

33.2.9 Resulting ontroller omplexity and implementation issues The ontrol law diagram in Fig. 33.1 shows that the omplete RIDE ontroller

(CB ) 1 and 1 ((CB ) CA) need to be implemented as look-up tables s heduled with Ma h is simple. For the body axis rate demand system, only the matri es

number.

In this ase, the 5 design points are those for whi h the linearised

models are available, i.e. M0.2, 1k ft; M0.24, 20k ft; M0.3, 5k ft; M0.4, 10k ft; M0.5, 15k ft. The gain matrix equations 33.2 to 33.5 are simple and the

omputational load involved in both generating and running a RIDE ontroller is therefore minimal. The dynami elements are restri ted to the six integrators and the two se ond order lters used to improve the phase margins. The speed demand system only has one s alar gain with three one-dimensional look-up tables. The rest of the ontroller, whi h translates pilot demands into body axis rates and provides limiting fun tions, will be of a similar order of omplexity irrespe tive of the design method used.

33.2.10 Further extensions The urrent ontroller omits the logi required to handle ontrol surfa e position and rate limiting.

This logi is simple and would not impose a signi ant

omputational load.

Position limiting is handled by holding the integrators

on the integral and feedforward portions of the ontroller and rate limiting is handled by initialising the value of the integrator to give the maximum in remental ontrol surfa e dee tion possible given the rate limit.

33.3 Translation of Design Criteria into Method Dependent Obje tives The only riteria whi h map onto the RIDE design method are those for bandwidth whi h are dire tly governed by the sele tion of the matrix of damping,

M

and

T

Zd , has been set to unity.

n

matri es.

The

In addition, the sele tion of the

matri es also ae ts the rise time initially.

They have been used

to tune the pit h rate response to provide the required dropba k and and to ensure that the roll and yaw rate responses have a

eptable rise times. The method ontains no me hanism for expli itly guaranteeing robustness and thus any robustness analysis would need to be done separately using an additional tool.

547

33.4 Des ription of the Design Cy le 33.4.1 Extra t CB and CA information at sele ted trim points Look-up tables are onstru ted with an adequate spread a ross the ight envelope. Sele tion of the s heduling points relies on engineering judgement. Past experien e [176℄ shows that s heduling with Ma h number and/or dynami pressure is su ient. In the HIRM design, the

A

and

B

matri es were taken

from the 5 linearised models identied in se tion 2.6.

33.4.2 Initial gain sele tion The natural frequen y,

n , and damping, Zd , are sele ted rst. For the HIRM Zd = I3x3 . The

example, damping in ea h of the hannels was set to 1, i.e. frequen ies

n

were sele ted to give approximately the orre t bandwidths of

response. The pit h frequen y was hosen to be 5 r/s and the roll frequen y was set to 4 r/s.

A yaw frequen y of 4 r/s was also sele ted to ensure that

onsistent bandwidths are used for the velo ity ve tor rolls where a mix of body axis roll rate and yaw rate is required. One limitation whi h needs to be onsidered is that as the gain matri es

KP , KI

and

KV

are fun tions of

(CB ) 1 , as shown in equations 33.2 to 33.4,

n , will need to be redu ed for

the demanded response bandwidth, spe ied by

ight onditions where the motivator ee tiveness is greatly redu ed, in order to avoid una

eptably large gains.

33.4.3 Control law tuning Having sele ted the feedba k gain hara teristi s and he ked the initial response of the basi losed-loop system, the feedforward gains,

T 1

and

M,

are then hosen to give the orre t amount of dropba k in pit h and a suitably rapid onset of response in roll and yaw. the feedforward input washes out and

M

T 1

modies the rate at whi h

hanges the gain on the feedforward

input. The nal values arrived at were

2

3

3 0 0 T 1 = 4 0 10 0 5 0 0 5 for the

2

and

3

3 0 0 M = 4 0 10 0 5 0 0 1

p, q and r demands.

The proportional gain on the speed ontroller was hosen by in reasing its value until a response with an adequate rise time was a hieved with no overshoot in speed.

A value of 0.35 was eventually sele ted.

optimum bandwidths on the angle of atta k demand, on the sideslip demand

bw

Similarly, the

bw , (set to 5 rad/s) and

(set to 2 rad/s) were sele ted by in reasing the

bandwidths to provide the maximum performan e with adequate damping.

548

Trade os an be made between the magnitude of the inner loop gains, spe ied by

n

(whi h will ae t how lose the response is to that predi ted

theoreti ally but whi h will be limited due to stability onsiderations), against the feedforward gains (whi h boost the initial response but may lead to an overshoot). For the HIRM design, tuning was performed by sele ting values of frequen y and damping whi h would give an adequate speed of response but not introdu e instabilities when the sensors and a tuators were in luded. If the initial response speed was inadequate, the feedforward gains were adjusted to give the required speed of response and overshoot.

33.4.4 General omments on synthesis pro edure The ore HIRM ontroller an be produ ed simply; the gain matri es an easily be al ulated by hand.

De oupling is taken are of automati ally.

Tuning

of the losed-loop responses is performed based on the re ognisable ee t of ea h of the parameters. On e the basi RIDE ontrol law has been developed, produ tion of the rest of the ontroller is method independent. The me hanisms for handling ontrol surfa e position and rate limiting would be straightforward to implement.

33.5

Analysis of the Resulting Controller in Terms of the Applied Methodology

33.5.1 Information on time and frequen y response plots The time responses were produ ed using the omplete ontroller applied to the non-linear model with the full, detailed sensor representation. The frequen y responses were produ ed by linearising the non-linear losed-loop system and then analysing it. It should be noted that all of the time responses are plotted against time in se onds. In the time responses, the pit h rate, velo ity ve tor roll rate and sideslip demands are plotted using dashed lines.

The symmetri al tailplane

dee tion is plotted as a solid line in the ontrol dee tions subplot, the dierential tailplane dee tion is plotted as a dashed line and the rudder dee tion as a dotted line. Frequen ies marked on the frequen y responses are in rad/s. In Figs.

33.9 to 33.11 and 33.15 to 33.17, the frequen y responses with

perturbations are plotted as dot-dashed lines to allow the ee t of the perturbations to be observed.

33.5.2 Responses using non-linear air raft model Small amplitude non-linear time responses (Ma h 0.4, 10k ft) The small amplitude pit h rate response shown in Fig.

33.2 shows that the

air raft behaves in a broadly a

eptable manner although a slight os illation is

549

noti eable on the overshoot on the pit h rate response. The os illation is due to the se ond order lter used to improve the phase margin.

−9

10

vv roll rate (deg/s)

pitch rate (deg/s)

1.5 1 0.5 0 −0.5 0

2

4

x 10

5

0

−5 0

6

2

4

6

2

4

6

2

4

6

2 4 Time (secs)

6

−10

10

bank angle (deg)

AoA (deg)

10.5 10 9.5 9 8.5 0

2

4

x 10

5

0

−5 0

6

2

0.4

1

sideslip (deg)

Mach number

−9

0.405

0.395 0.39 0.385 0

2

4

x 10

0 −1 −2 0

6

4

control deflections (deg)

altitude (ft)

1.006

x 10

1.004 1.002 1 0.998 0

2 4 Time (secs)

6

5

0

−5

−10 0

Figure 33.2: Small amplitude pit h rate response (Ma h 0.4, 10k ft) Despite the os illations, the attitude response, shown in Fig.

33.3, is

smooth. Hen e, this os illation may be imper eptible to the pilot. The peak stabilator dee tion is a hieved in 0.11s and so the peak pit h a

eleration should also be rea hed in a similar time s ale, whi h meets the 0.15s riterion spe ied in the design do ument. The pit h attitude response in Fig. 33.3 shows that the dropba k is nearly zero and satises the relevant design riterion. The small amplitude roll rate response plotted in Fig. 33.4 shows that the

ontrol system satises the roll ommand with a time onstant of approximately 0.35s as spe ied. There is little oupling into either pit h rate or sideslip and the ontrol a tivity is small. The air raft response to a sideslip demand, shown in Fig. 33.5, is good. The sideslip response lies within the boundaries of Fig. 27.16 of hapter 27. The manoeuvre is a

ompanied by very little roll and no pit h rate. The ontrol a tivity is also moderate.

Large amplitude non-linear time responses (Ma h 0.4, 10k ft) The large amplitude pit h rate response shown in Fig. with the angle of atta k being limited at

30o.

33.6 is well behaved

The stabilator dee tions used

to produ e the manoeuvre are a

eptable although some os illations do appear

550

12

11.5

pitch attitude (deg)

11

10.5

10

9.5

9

8.5 0

1

2

3 Time (sec)

4

5

6

Figure 33.3: Pit h attitude response to al ulate dropba k (Ma h 0.4, 10k ft)

1.5

vv roll rate (deg/s)

pitch rate (deg/s)

0.4 0.3 0.2 0.1 0 −0.1 0

2

4

1 0.5 0 −0.5 0

6

AoA (deg)

8.85 8.8 8.75 8.7 8.65 0

2

4

sideslip (deg)

Mach number

0.4

0.396

2

4

6

2

4

6

2

4

6

2 4 Time (secs)

6

2 1

0.1 0.05 0 −0.05 0

control deflections (deg)

10000.5 10000

altitude (ft)

6

0.15

0.398

9999.5 9999 9998.5 9998 0

4

3

0 0

6

0.402

0.394 0

2

4

bank angle (deg)

8.9

2 4 Time (secs)

6

2 0 −2 −4 −6 −8 0

Figure 33.4: Small amplitude vv roll rate response (Ma h 0.4, 10k ft)

551

0.4

vv roll rate (deg/s)

pitch rate (deg/s)

0.4 0.3 0.2 0.1 0 −0.1 0

2

4

bank angle (deg)

AoA (deg)

8.8 8.75 8.7 2

4

0.4

sideslip (deg)

Mach number

6

2

4

6

2

4

6

2

4

6

0 −0.2

1.5

0.398 0.396

2

4

6

1 0.5 0 −0.5 0

control deflections (deg)

10000.5 10000

altitude (ft)

4

0.2

−0.4 0

6

0.402

9999.5 9999 9998.5 9998 0

2

0.4

8.85

0.394 0

0 −0.2 −0.4 0

6

8.9

8.65 0

0.2

2

4

6

2 0 −2 −4 −6 −8 0

Figure 33.5: Response to small amplitude sideslip demand (Ma h 0.4, 10k ft)

after 7 se onds. The reason for the de rease in stability is due entirely to the use of Ma h number instead of dynami pressure for s heduling. For this response, the Ma h number has redu ed to around M0.25, hen e the gain look-up tables will be using gains based on those for the M0.24, 20k ft where the air density is approximately 0.6. However the air raft is only at 10k ft (air density 0.9) and so the gains will be 50% too large, leading to the redu tion in stability. S heduling with dynami pressure would learly have been preferable.

The

oupling into roll and sideslip is minimal. The large amplitude roll response in Fig. well behaved during the

360o

roll.

33.7 shows that the air raft is

However there is a moderate amount of

oupling into pit h whi h is unexpe ted. This is due to gravity ee ts whi h are not being a

ounted for. the roll is

3o

Also, the maximum sideslip ex ursion during

whi h is in ex ess of that permitted. S ope for redu ing this is

limited however, as an in rease in the bandwidth of the demand loop leads to instability. The large amplitude sideslip demand produ es little oupling into roll and negligible oupling into pit h. The demand is satised with a response similar to the small amplitude one shown in Fig. 33.5.

Frequen y responses from linearised non-linear model (Ma h 0.4, 10k ft) Ea h of the frequen y responses shown in Figs. 33.9 to 33.11 avoids the ex lusion zone. It an also be seen that the perturbations have little ee t.

552

1

vv roll rate (deg/s)

pitch rate (deg/s)

20

10

0

−10 0

2

4

6

0.5 0 −0.5 −1 −1.5 0

8

AoA (deg)

25 20 15 10 5 0

4

6

8

2

4

6

2

4

6

8

2

4

6

8

2

4

6

8

0.5 0 −0.5 −1 0

8

0.45

0.5

0.4

sideslip (deg)

Mach number

2

1

bank angle (deg)

30

0.35 0.3 0.25 0.2 0

2

4

6

0

−0.5

−1 0

8

4

control deflections (deg)

altitude (ft)

1.06

x 10

1.04 1.02 1 0.98 0

2

4

6

8

10 5 0 −5 −10 −15 0

Figure 33.6: Response to large amplitude pit h rate demands with

 limiting

(Ma h 0.4, 10k ft)

80

vv roll rate (deg/s)

pitch rate (deg/s)

4 2 0 −2 −4 −6 0

2

4

6

bank angle (deg)

AoA (deg)

0

5

0

6

8

2

4

6

2

4

6

8

2

4

6

8

2

4

6

8

100 0 0

8

4

0.42

2

0.41 0.4

2

4

6

control deflections (deg)

9800 9600

2

4

6

Figure 33.7: Large amplitude

8

0 −2 −4 0

8

10000

9400 0

4

200

0.43

0.39 0

2

300

sideslip (deg)

Mach number

20

400

10200

altitude (ft)

40

−20 0

8

10

−5 0

60

10

0

−10

−20 0

360o roll at 70o/s (Ma h 0.4, 10k ft) 553

4

vv roll rate (deg/s)

pitch rate (deg/s)

0.4 0.3 0.2 0.1 0 −0.1 0

2

4

6

2

4

6

8

2

4

6

8

2

4

6

8

2

4

6

8

bank angle (deg)

1

9

AoA (deg)

0 −2 −4 0

8

9.1

8.9 8.8 8.7 8.6 0

2

4

6

0 −1 −2 −3 0

8

15

sideslip (deg)

0.405

Mach number

2

0.4

0.395

0.39 0

2

4

6

10 5 0 −5 0

8

4

x 10

control deflections (deg)

1.0004

altitude (ft)

1.0003 1.0002 1.0001 1 0.9999 0

2

4

6

8

15 10 5 0 −5 −10 0

Figure 33.8: Response to large amplitude sideslip demand (Ma h 0.4, 10k ft)

Small amplitude non-linear time responses (Ma h 0.24, 20k ft) From the small amplitude pit h rate response in Fig. 33.12, it an be seen that there is a small de rease in the damping of the system but the demands are still tra ked well and there is little oupling into roll or sideslip. The low damping apparent in the rst se ond of the responses is due to impre ise initialisation and so should be disregarded, as it is not apparent elsewhere in the responses. The small amplitude roll rate response in Fig. 33.13 shows similar hara teristi s to the previous plot in Fig. 33.4 at M0.4 although there is an in rease in the amount of oupling into both pit h rate and sideslip. Comparing the sideslip responses in Fig.

33.14 with those in Fig.

33.5,

shows that, despite the hange in ight ondition, the air raft exhibits a similar response to the demand.

However it an be seen that de rease in the

ontrol surfa e ee tiveness at the higher angle of atta k and altitude results in signi antly greater ontrol surfa e dee tions. Large amplitude demands have not been made for the M0.24 ase as the

ontrol law does not have onditioning of the integrators, in the event of ontrol surfa e position and rate limiting, and large demands invariably lead to both types of saturation on at least one of the surfa es.

554

40 0.2382 30 1.351

0.5672

20 3.218

10

7.663

0 18.25 −10 43.47 −20

−30

−40 −350

−300

−250

−200

−150

−100

−50

0

Figure 33.9: Symmetri tailplane freq. resp.

0.1

40

0.2382 30 0.5672 20

1.351

10 3.218 0

7.663 18.25

−10 43.47 −20

−30

−40 −350

−300

−250

−200

−150

−100

−50

0

Figure 33.10: Dierential tailplane freq. resp.

Frequen y responses from linearised non-linear model (Ma h 0.24, 20k ft) The symmetri and dierential tailplane frequen y responses both avoid the ex lusion zone, although only marginally in the symmetri tailplane ase. The rudder response does violate the zone.

Rudder was the one surfa e not to

have any additional ltering added to improve the phase margin. Su h ltering should be able to re tify this. As for the Ma h 0.4 ase, it an be seen that the perturbations have little ee t, although they do ae t the response at higher frequen ies than before.

555

40 0.1 30

0.2382 0.5672 1.351

20 3.218

10

7.663

0 18.25

−10

−20

43.47

−30

−40 −350

−300

−250

−200

−150

−100

−50

0

Figure 33.11: Rudder frequen y response

0.2

vv roll rate (deg/s)

pitch rate (deg/s)

0.5 0 −0.5 −1 −1.5 0

2

4

0.1 0 −0.1 −0.2 0

6

29 28 27

4

6

26 0

2

4

2

4

6

2

4

6

2

4

6

0.1 0 −0.1 −0.2 0

6

0.255

0.2

0.25

sideslip (deg)

Mach number

2

0.2

bank angle (deg)

AoA (deg)

30

0.245 0.24 0.235 0

2

4

6

0.1

0

−0.1 0

4

x 10

control deflections (deg)

altitude (ft)

2.0005 2 1.9995 1.999

1.9985 0

2

4

6

10 5 0 −5 −10 −15 0

Figure 33.12: Response to small amplitude pit h rate demand (Ma h 0.24, 20k ft)

556

1.5

vv roll rate (deg/s)

pitch rate (deg/s)

0.2 0 −0.2 −0.4 −0.6 −0.8 0

2

4

1 0.5 0 −0.5 0

6

29.5 29 28.5 28 0

2

4

4

6

2

4

6

2

4

6

2

4

6

3 2 1 0 −1 0

6

0.25

0.4 0.3

sideslip (deg)

Mach number

2

4

bank angle (deg)

AoA (deg)

30

0.245

0.24

0.235 0

2

4

0.2 0.1 0 −0.1 0

6

4

x 10

control deflections (deg)

altitude (ft)

2.0005

2

1.9995

1.999 0

2

4

6

10 5 0 −5 −10 0

Figure 33.13: Response to small amplitude vv roll rate demand (Ma h 0.24, 20k ft)

0.4

vv roll rate (deg/s)

pitch rate (deg/s)

0.2 0 −0.2 −0.4 −0.6 −0.8 0

2

4

0.2 0 −0.2 −0.4 0

6

29.5 29 28.5 28 0

4

6

2

4

2

4

6

2

4

6

2

4

6

0 −0.1 −0.2 −0.3 −0.4 0

6

0.25

1.5 1

sideslip (deg)

Mach number

2

0.1

bank angle (deg)

AoA (deg)

30

0.245

0.24

0.235 0

2

4

6

0.5 0 −0.5 0

4

x 10

control deflections (deg)

altitude (ft)

2.0005

2

1.9995

1.999 0

2

4

6

15 10 5 0 −5 −10 0

Figure 33.14: Response to small amplitude sideslip demand (Ma h 0.24, 20k ft)

557

0.1 0.2382 0.5672

40

30 1.351 20 3.218

10

7.663

0 18.25 −10 43.47 −20

−30

−40 −350

−300

−250

−200

−150

−100

−50

0

Figure 33.15: Symmetri tailplane freq. resp.

40

30 1.351

0.5672 0.2382

0.1

20 3.218

10

0

7.663 18.25

−10 43.47 −20

−30

−40 −350

−300

−250

−200

−150

−100

−50

0

Figure 33.16: Dierential tailplane freq. resp.

40

30

0.1

0.2382 0.5672

20

1.351

10 3.218 0

7.663 18.25

−10

−20

43.47

−30

−40 −350

−300

−250

−200

−150

−100

−50

0

Figure 33.17: Rudder frequen y response

558

33.5.3 Responses from sti k to attitudes (Ma h 0.4, 10k ft) Gibson riteria Figures 33.18 and 33.19 show that, given a suitable sti k s aling, the frequen y response from sti k to attitude satises the Gibson riteria for the Ma h 0.4, 10k ft ase.

Gain and phase rate riteria

180o phase is examined, based on the frequen y responses from sti k to attitude, it is apparent that the design does not satisfy the <-16

When the gain at

dB riterion as the gains for the system are as follows: Pit h: -13.8dB at Roll:

-10.1dB at

180o phase 180o phase

However, the phase rate for ea h system is a

eptable and both systems are level 1* as the frequen y at Pit h: 2.32Hz at 1.02Hz at

180o phase with a phase rate of 44:6o/Hz 180o phase with a phase rate of 310/Hz

20

15

10

Gain (db)

Roll:

180o phase and the phase rate are:

5

L1

0

−5 L1 −10 −180

−160

−140

−120 −100 Phase (deg)

−80

−60

−40

Figure 33.18: Sti k to pit h attitude response

559

25 20 15 10

Gain (db)

5 0 Sluggish PIO

−5

Oscillation −10 −15 Good −20 −25 −350

−300

−250

−200 −150 Phase (deg)

−100

−50

0

Figure 33.19: Sti k to roll angle response

560

33.5.4 Stability analysis (M0.4, 10k ft and M0.24, 20k ft) The frequen y responses of Figs. 33.20 to 33.22 have been produ ed as follows. For Figs.

33.20 and 33.22, a gain has been introdu ed on the tailplane and

rudder demand paths.

This gain has been set to 1.68 (4.5dB) on ea h of

the tailplane and rudder demand paths simultaneously. A time response has then been run with a small impulse of 0.1 deg/s amplitude and 0.1 se onds duration, being made at 0.5 se onds applied to the pit h rate, vv roll and sideslip demands. The time response has then been he ked to ensure that the system has remained stable. For Fig.

33.21, the gain was set to 1.33 (2.5dB). The natural frequen y

of the pit h rate response in Fig. 33.20 was measured, pit h rate response having been hosen as this was learly the least well damped and most prone to instability. would give

Based on the natural frequen y (3.6Hz), a time delay whi h

30o phase lag at this frequen y (23 mse ) was then inserted on ea h

of the tailplane and rudder demand loops simultaneously in addition to the 2.5dB in rease in gain. Figs. 33.20 and 33.21 show that at M0.4, the losed-loop system remains stable despite the simultaneous gain and phase additions. 0.5

0.4

Body axis rates (deg/s)

0.3

0.2

0.1

0

−0.1

−0.2

−0.3 0

0.5

1

1.5

2

2.5

3

3.5

4

Figure 33.20: Additional 4.5dB gain at the M0.4, 10k ft ight ondition For the Ma h 0.24 ase, Fig. 33.22 shows that the system remains just stable with an additional gain of 4.5dB. The stability riterion is therefore met. It was impossible to he k whether the system remained either stable or neutrally stable with simultaneous gain and phase osets as the ontrol surfa es started to rate limit.

As integrator onditioning in the event of ontrol surfa e rate

limiting has not yet been added this led to instability and han e masking the ee t of the additional gain and phase.

33.5.5 Response of speed ontroller Fig. 33.23 shows the response to a large amplitude demand on speed. It an be seen that the engine thrust in reases to the maximum level to provide a rapid

561

0.8

0.6

Body axis rates (deg/s)

0.4

0.2

0

−0.2

−0.4

−0.6 0

0.5

1

1.5

2

Figure 33.21: Additional 2.5dB gain and

2.5

30o

3

3.5

4

phase at the M0.4, 10k ft ight

ondition

0.4

0.2

Body axis rates (deg/s)

0

−0.2

−0.4

−0.6

−0.8

−1

−1.2 0

0.5

1

1.5

2

2.5

3

3.5

4

Figure 33.22: Additional 4.5dB gain at the M0.24, 20k ft ight ondition

562

a

eleration. A small steady-state oset, from the demanded 90m/s air speed, is observed. In pra ti e, this should not be a problem. 380

85

360

altitude (m)

air speed (m/s)

90

80 75

340 320

70 65 0

5

300 0

10

5

10

5

10

4

8

x 10

22 20

AoA (deg)

thrust (N)

6 4 2

18 16 14

0 0

5

10

12 0

Figure 33.23: Response to large amplitude speed hange demand

33.6 Results of the Automated Evaluation Pro edure The results from the omplete set of evaluations are as follows (see [177℄ for the meaning of test ase numbers). - Tests 11, 13 and 14 passed, agreeing with the results from Figs. 33.9 to 33.11 and 33.15 to 33.17. - Test 12 failed, disagreeing with the results shown in Figs. 33.20 to 33.22. - In test 21, the following values of RMS disturban e amplitude were re orded due to moderate turbulen e: 0.5002/s in roll rate 0.0236/s in pit h rate 0.9901/s in yaw rate 0.0938g in normal a

eleration.

- The responses for test 22 were impossible to interpret but this riterion is

overed at one ight ondition by the a

eptable responses in Figs. 33.18 and 33.19. - Some of the responses to the assessment manoeuvres were una

eptable as the demands aused ontrol surfa e rate limiting whi h is urrently not handled in the RIDE ontroller. Time responses are already shown in Figs. 33.2 to 33.8, Figs. 33.12 to 33.14 and Fig. 33.23. - Test 31 failed as g and negative

 limiting have not yet been implemented.

- Test 32 passed as shown in Figs. 33.24 and 33.25.

563

Structural coupling, pitch rate to taileron (−) and to canard (−−)

5

10

0

Gain

10

−5

10

−10

10

−15

10

−2

10

−1

10

0

1

10 10 Frequency (rad/sec)

2

10

3

10

Figure 33.24: Control law gain - pit h rate to ontrol surfa e

Structural coupling, normal acc. to taileron (−) and to canard (−−)

0

10

−5

Gain

10

−10

10

−15

10

−20

10

−2

10

−1

10

0

1

10 10 Frequency (rad/sec)

2

10

3

10

Figure 33.25: Control law gain - normal a

eleration to ontrol surfa e

564

- Test 41 did not produ e sensible results as the inputs aused the ontrol surfa es to rate limit. - Test 42 produ ed the following results. Control a tivity due to turbulen e:

0.2807 RMS on symmetri taileron, 0.8031 RMS on dierential taileron, 0.5507 RMS on rudder.

Control a tivity due to noise was:

13.1636 RMS on symmetri taileron, 2.1729 RMS on dierential taileron, 13.1636 RMS on rudder.

33.7 Con lusions and Lessons Learned A ontroller has been designed using Robust Inverse Dynami s Estimation (RIDE) for the HIRM design hallenge. The RIDE method uses an estimate of the inverse dynami s of the outputs to de ouple them. A Proportional plus Integral (PI) ontroller is then added to assign the desired dynami s.

The

proportional part of the PI ontroller provides stability and the integral part

ompensates for errors in the estimate of the inverse dynami s. A feedforward element is in luded to shape the transient response of the losed-loop system. The key advantages of the method are: (i) the simpli ity of the ontrol law synthesis and theory, whi h only requires a knowledge of elementary matrix algebra to fully understand, (ii) an identiable ontrol law stru ture, (iii) physi ally meaningful gain matri es and (iv) the apability of handling motivator position and rate limiting in a very simple manner.

Motivator redundan y an be handled separately from

the ontroller design. The overall simpli ity of both theory and design allows a rapid understanding of the method, fast design and redesign, with modi ation being simple to designers familiar with lassi al design methods. The simpli ity of the design method and the resultant ontroller stru ture does mean that only a subset of the design riteria an be in luded in the

ontroller synthesis.

Only the response shape and bandwidth riteria map

onto the design parameters. Also, robustness guarantees are not in luded and must be addressed separately using other tools. An initial version of the ontrol law for the HIRM design hallenge was synthesised very rapidly and with little modi ation, produ ed a

eptable responses.

A minimum amount of s heduling based on Ma h number proved

adequate and a minimum number of sensor signals were used. However before denitive on lusions an be made about the suitability of RIDE, a more omprehensive ontroller needs to be designed and tested. Su h designs and tests are already planned by DRA Bedford.

565

566

Part IV

Con luding part

567

34.

The Industrial View

C. Fielding and R. Lu kner 1

Abstra t.

2

The results from the RCAM and HIRM design hal-

lenges whi h have been des ribed in the previous hapters were evaluated in relation to industrial ight ontrol law design and development pro esses.

Experien ed evaluation teams were estab-

lished, led by Daimler-Benz Aerospa e Airbus for the ivil air raft design results (RCAM) and by British Aerospa e Military Air raft for the military air raft design results (HIRM). This hapter des ribes the ommon evaluation pro ess used and the interpretation of the results presented in the design teams' reports, in relation to industrial experien e and pra ti es.

34.1 The Evaluation Pro ess 34.1.1 Fa tors ae ting the design of ontrol laws In order for a team of designers to a hieve su

essful ight ontrol laws design and implementation, several fa tors will play a part, with the design te hnique used being just one su h fa tor to be onsidered. In no parti ular order, some of these fa tors are:



The design team's experien e and knowledge of ight me hani s, ight

ontrol and ontrol theory.

    

The tools and omputing fa ilities available to the design team. The time available to the design team and the level of support available. The adequa y and interpretation of the design requirements. The pro edures used for a hieving and ontrolling the design denition. The design team's experien e with the te hnique and the level of design automation used.



The visibility and larity of denition of the resulting design's fun tional spe i ation.

1

British Aerospa e Military Air raft, Aerodynami s Department, Warton Aerodrome,

Preston PR4 1AX, UK 2

Daimler-Benz Aerospa e Airbus, Flight Guidan e and Control, Flight Me hani s Depart-

ment, P.O.Box 950109, D21111 Hamburg, Germany

569

These are the main fa tors but there may well be others. Clearly it is always going to be a di ult task to lter out the benets whi h an be attributed to the design te hniques, from a set of results produ ed by dierent design teams from dierent organisations. This was re ognised before evaluation ommen ed and an evaluation pro edure established, aimed at produ ing the learest possible pi ture from the information available. This involved the development of a questionnaire, to apture the signi ant information from evaluations of the design entries (whi h were presented to the evaluators as a series of design reports). The resulting questionnaire was ommon to RCAM and HIRM designs and is summarised in the next se tion, so that readers an appre iate those aspe ts whi h are important to industry and the type of questions that were asked, and are likely to be asked in the future.

34.1.2 The Evaluation Questionnaire Experien ed individuals from industry and resear h establishments were asked to s reen and to evaluate the te hni al reports that were provided by the design teams. Their evaluation was stru tured by a questionnaire whi h addressed the following important topi areas:

  

The eort to learn, to implement and to apply the method. The appli ability of the design method to ight ontrol laws design. The omplexity of the resulting ontroller, its implementation and erti ation issues.



The robustness and performan e of the designed ontroller.

Spe i ally, the following questions were asked, ea h supplied with a detailed des ription to over the s ope and intention of the question, and a des riptive rating (ex ept for the last three questions) s aled between 1 and 5: 1 (Question 1 was related to the quality of the design do ument) 2a What level of understanding of the methodology is needed to obtain satisfa tory results ? 2b How do you rate the learning urve asso iated with using this method, i.e. how easy is the method to grasp ? 2 How easy would it be for you to take over the design and arry out a re-design, with the same method ? 2d Does the method support all ight ontrol law stru tures that you might possibly want to design ? 2e Is it possible to translate all of your design requirements into the design method syntax ?

570

3a Do you onsider the ontroller stru ture presented to have good visibility in terms of its fun tionality ? 3b How do you rate the omplexity of the design, in relation to the design problem omplexity ? 3 How suitable is the design for implementation in an air raft's on-board ight ontrol omputer ? 3d How suitable is the design for omplian e with your quali ation and

erti ation pro edures ? 4a Do you have any omments regarding the robustness of the design that has been a hieved ? 4b Do you have any omments regarding the performan e of the design that has been a hieved ? 4 Do you have any omments regarding the ontrol surfa e a tivity asso iated with the design ? In order to help to resolve any dieren es in judgement between the evaluators and designers, all designers arried out a self-evaluation of their own design entry, by using the same questionnaire. Additionally, the questionnaire identied the evaluators' and designers' levels of experien e in key areas of responsibility within the ight ontrol laws development pro ess. This would allow the ratings about dierent aspe ts of a design to be orrelated with the experien e levels, and hopefully, to explain any areas of disagreement. Spe i ally, the following areas of expertise were assessed:

     

Flight me hani s / Air raft stability Flight Control laws design Piloted simulation / handling qualities FCC implementation and testing Flight learan e / erti ation Flight test / post-ight analysis

In performing a design entry evaluation, the evaluators' judgements would not only vary with their levels of experien e in these areas, but also on their knowledge and experien e with the parti ular te hnique used to a hieve the design. This aspe t was also addressed by the questionnaire. Finally, it was planned to arry out an independent automati evaluation of the various ontrollers by using software-based tools to give quantitative results. Unfortunately this information was not available in time for the evaluation pro ess. The results of the automati evaluations will however, be available in the updated design reports produ ed be ea h of the teams.

571

34.1.3 The Design Entries In the following se tions, the design entries are often identied using a simple notation: Notation

Organisation

Design Method

Appli ation Report/ Chapter

-Synthesis

MS-11

DUT

EA-12

CERT-ONERA Eigenstru ture Assignment

RCAM [25℄/Ÿ22 RCAM [55℄/Ÿ19

CC-13

CUN

RCAM [88℄/Ÿ15

Classi al Control

LY-14

LAAS-CNRS

Lyapunov

RCAM [49℄/Ÿ20

FL-15

DUT

Fuzzy Logi Control

RCAM [207℄/Ÿ26 RCAM [130℄/Ÿ16

MO-16

DLR

Multi-Obje tive Parameter Synthesis

EA-18

LUT

Eigenstru ture Assignment

MS-19

NLR

-Synthesis

PC-20

UCAM

Predi tive Control

HI-21

ULES

EA-22

UNED

Eigenstru ture Assignment

MF-25

DLR

Model Following Control

RCAM [67℄/Ÿ24

CC-24

ALN

Classi al Control

HIRM

[89℄

LQ-26

CIRA

Linear Quadrati Control

HIRM

[9℄/Ÿ28

MO-27

DLR

Multi-Obje tive Parameter Synthesis

HIRM

DI-28

DRA

Robust Inverse Dynami s Estimation

MS-29

DUT

RCAM [77℄/Ÿ17 RCAM [208℄/Ÿ23 RCAM [119℄/Ÿ25

H1 Mixed Sensitivity

RCAM [242℄/Ÿ21 RCAM [52℄/Ÿ18

-Synthesis

DI-30

ONERA

MS-31

SMA

HI-32

UCAM/CCL

HI-33

ULES

Nonlinear Dynami Inversion/LQ

-Synthesis H1 Loop-Shaping H1 Loop-Shaping

HIRM

[73℄/Ÿ33

HIRM

[163℄/Ÿ31

HIRM

[72℄/Ÿ32

HIRM

[225℄/Ÿ30

HIRM

[191℄/Ÿ29

HIRM

[217℄

Where the two hara ter alphabeti identier indi ates the design method and the two hara ter numeri al identier refers to the GARTEUR te hni al publi ation produ ed for the Robust Flight Control proje t; e.g.

report

GARTEUR/TP-088-11 was produ ed for design entry MS-11 (any missing numbers are asso iated with reports whi h were not overing design entries). The expanded titles of the organisations an be found at page iii.

34.2 Civil Air raft Manufa turer's View

3

34.2.1 Introdu tion The aeronauti al industry's motivation to look for new robust ight ontrol

4

design methods , is driven by e onomi al needs.

The design pro ess has to

be ome more ee tive ( heaper, faster) and less risky (meet deadlines, be within budget) while guaranteeing the same high safety level. 3 4

Prepared by Robert Lu kner on behalf of the RCAM Evaluation Team

Design method stands for the overall design pro edure. 572

5

The introdu tion of new ontroller synthesis methods

into the industrial

development pro ess of ight ontrollers is done onservatively be ause the

urrent design methods represent a riti al apital for industry (trained people, software tools, et .). New methods will be adopted, only if they are mature and if benets have been demonstrated or if dynami ight ontrol problems that

annot be solved by existing methods, are driving te hnology. That is why, for su h appli ations, it takes, typi ally 10 to 20 years for a new ontrol synthesis method from its rst publi ation to its routine servi e in an industrial design pro ess. During the last twenty years, a faster appli ation was aggravated by the fa t that design o es of ivil air raft manufa turers were o

upied by the introdu tion and exploitation of digital te hniques.

Most of their resour es

have been absorbed to master the omplexity of digital ight ontrol systems and their software. The design pro ess stands in the foreground and the sear h for better ontrol synthesis methods was pushed into the ba kground as most of the design problems ould be solved more or less satisfa tory by the broad experien e and the extensive suites of lassi al design tools that have been available. There were of ourse, improvements (e.g. appli ation of optimization te hniques, transition from ontinuous to dis rete design methods) but this was, more or less, an automation or a onversion of existing te hniques. The tremendous development osts of digital ight ontrol systems and the modi ation osts that are aused by hanges, even for only slightly dierent air raft versions (e.g.

dierent engine types, stret hed versions et .), brings

up new requirements whi h are not only physi ally, but also e onomi ally reasoned. In this ontext, the idea of robust ontrol is very attra tive. System simpli ations or a redu tion of the number of dierent versions seem to be feasible. If su h benets are foreseeable, traditional design pro esses should be re onsidered and the introdu tion of new ontroller synthesis methods would be justied. The evaluation of new methods for ight ontrol appli ations is di ult and ostly.

Many dierent riteria have to be onsidered.

Possible benets

have to be weighed against the additional eort, osts and risks. In the past, de isions on the implementation of a new design method were often based on the intuitive judgement of experien ed engineers. But as the omplexity of the design pro ess has in reased and as the onsequen es of a de ision in today's development pro esses an be dramati , a systemati , obje tive approa h is needed. The main obje tive of the RCAM design hallenge is the demonstration of modern and robust ontrol synthesis te hniques to the European air raft manufa turers, oering help for their de ision making. Twelve design teams have demonstrated the appli ation of dierent design methods to the RCAM design hallenge in the previous hapters. The RCAM problem is a realisti , though simplied, ight ontrol example of industrial relevan e. The teams have presented their individual ontroller stru ture, and 5

Synthesis method is the mathemati al method that H1 , Eigenstru ture Assignment, et .).

(LQG,

573

is used to synthesize a ontroller

the results they have a hieved. Twelve of these methods are alled new as they have emerged during the last three de ades and are only rarely used by the ight ontrol o es of the aeronauti al industry. One team used a traditional

lassi al approa h, whi h an be taken to a t as a referen e. The pro edure for the evaluation of the dierent ontrol design and synthesis methods was set up, as previously des ribed. The relevant riteria for a design method assessment were addressed systemati ally and evaluators from industry and resear h establishments were asked for ratings and omments. A pre ondition for the evaluation of dierent ontrol design and synthesis methods was that all reports were written in a standardized format. This was a hieved by des ribing the RCAM problem in a manual, dening the RCAM model in MATLAB syntax with a standard nomen lature, and by pres ribing a standard lay-out for the design reports. The evaluation results, whi h are given and dis ussed below, show benets and drawba ks of the dierent methods. They allow the formulation of wellfounded re ommendations, in order to ease the de ision making for industry's design o es. But of ourse, the nal de ision to apply a new method has to be made by the design o e itself. It has to be noted that the RCAM hallenge serves only the above mentioned obje tives. It is not a ompetition between the design teams or their organizations. There will be no winner and no loser.

34.2.2 RCAM Evaluation Team The RCAM evaluation team onsisted of 20 evaluators, 12 from industry (Alenia, AVRO, BAe-D, BAe-MA, DASA) and 8 from resear h establishments (CERT, DLR, NLR). It was led by DASA. All evaluators are experien ed aerospa e ight ontrol professionals. Colle tively, they represent a high level of ompeten e in ight me hani s / air raft stability, ight ontrol laws design, piloted simulation / handling qualities and they had experien e in FCC implementation and testing, ight learan e / erti ation and ight test / post ight analysis. As it was the suitability of a design method for the industrial pro ess that was to be evaluated, an evaluator had not ne essarily to be familiar with the method he assessed. The questions of the questionnaire ould be answered without this knowledge. In ase of doubt, questions ould be left unanswered. The evaluation was done on a voluntary basis. As a thorough evaluation of a design entry takes one or two days, it was ne essary to divide the work between multiple evaluators. Therefore, most evaluators evaluated one or two reports and only one evaluator has evaluated all design reports. This has been

onsidered when the evaluation results were ompared.

34.2.3 Design Entries The twelve teams that have presented their designs in the previous hapters are from resear h establishments (CERT-ONERA, DLR Institute for Flight

574

Me hani s, DLR Institute for Roboti s and System Dynami s, NLR, LAAS) and from universities (CUN, DUT Aerospa e Engineering, DUT Ele tri al Engineering, LUT, ULES, UNED, UCAM). They have used methods that dire tly

ope with robustness against dened un ertainties, i.e. the plant model used for design ontains a nominal plant model plus expli it un ertainty modelling: MS

-synthesis

MS-11, MS-19;

and they have used methods whi h, for the same a

ura y of un ertainty modelling, inherently provide some robustness measures (e.g. gain and phase margins, eigenvalue sensitivity) but where robustness has to be proven in an analysis, starting an iteration y le:

H1

HI-21,

eigenstru ture assignment

EA-12, EA-18, EA-22,

CC

lassi al ontrol design

CC-13,

FL

fuzzy logi ontrol

FL-15,

MF

model-following

MF-25,

LY

Lyapunov

LY-14,

MO

multi-obje tive parameter synthesis

MO-16,

PC

predi tive ontrol

PC-20, PC-23.

HI EA

The evaluation is based on the se ond revision of the design reports from whi h the tutorial hapters and the design hapters of this book are derived. Due to the tight time s hedule, an evaluation of LY-14 and PC-20 ould not be in luded in this hapter.

Designer's Ba kground The prin ipal designer's level of experien e in dierent areas of ight ontrol system development was determined by the self-assessment. The self ratings state his level of experien e at the beginning of the design and do not des ribe the ompeten e of the resear h establishment or university and they do not in lude the support that a designer had available within his organization. All design teams knew the synthesis method they have applied quite well. They rated their knowledge and experien e as medium (CC-13, FL-15, EA-18, MS-19, HI-21) and high (MS-11, EA-12, MO-16). Only the designer of MF-25 used the method for the rst time. Three design teams (FL-15, MO-16, HI-21) rated their level of expertise in ight dynami s as low, i.e. very little to basi knowledge. This expertise is important for the sele tion of the ontroller stru ture, for weighting design requirements (whi h are often in ontradi tion) and for understanding and interpretation of the dynami behaviour of the ontroller. It is not surprising that the knowledge on theoreti al subje ts su h as ight me hani s, ight ontrol laws design and simulation is mu h higher than on pra ti al subje ts as implementation, testing, and erti ation.

This an be

even an advantage, as too mu h knowledge of possible implementation and

erti ation problems involves the danger of being too onservative and too relu tant in applying new methods.

575

Control Strategy and Ar hite ture A lot of the ontroller performan e is predetermined when the strategy and the ar hite ture is dened. Tables 34.1 and 34.2 give an overview of the ontroller stru tures hosen by the design teams. Basi ally, traditional ight ontrol ar hite tures have been hosen, i.e.:



De oupling of the ontrollers for longitudinal and lateral motion, sometimes with oupling terms (e.g. for turns). An ex eption is design HI-21 whi h ombines longitudinal and lateral inner ontrol loop into one dynami blo k.

This is a quite un onventional stru ture but it promotes

very interesting alternatives.



Hierar hi al stru tures with an inner loop for stabilization and attitude

ontrol and an outer loop for guidan e.

Ex eptions are MS-11, MO-

16, and EA-18 whi h ombine inner and outer loops in the longitudinal

ontroller. It is well known, as many evaluators have emphasised, that a suitable hoi e of the ontroller ar hite ture, based on ight physi s and mission demands is mu h more relevant than the applied synthesis method. To establish su h an ar hite ture, a lot of experien e and ight dynami s knowledge is ne essary. For designers with very little or only basi knowledge in ight dynami s (FL15, MO-16, HI-21), the sele tion of signals and ar hite ture was a very steep hurdle that they had to over ome. They managed it with some iteration and with the help of ight ontrol textbooks. The RCAM manual oers diverse referen e signals and measurement signals as ontroller inputs. The designers sele ted the input signals for their ontroller, individually oming to dierent solutions. Some ommonly used measurements, that are available from the inertial referen e systems, are not oered, e.g.

.

 and

Estimation of missing measurement signals, as done by some designers, is

onsidered as a part of the ontroller. This aspe t ould have been assigned to the RCAM air raft model. Thrust, elevator, aileron, and rudder were available for air raft ontrol and all of them were used by all designers. Most of the designs suer from the inability to formulate referen e values, whi h are onsistent with ight physi s and mission requirements. The 9 refer-

x , y , z , uV , vV , wV , V ,
y , _ ) dene a non-yable ight

en e signals (

path for the RCAM mission: entry and exit of turn requires innite roll rates and the transition from level ight to des ent requires an innite verti al a

eleration. A few teams used models (MS-19 and MF-25) or prelters (MS-11, MO-16) to generate a yable path, others introdu ed mode swit hes (CC-13, EA-18) and limiters (CC-13, FL-15, MS-19), feedforward (EA-18). While the denition of the overall (high level) stru ture was nearly independent of the sele ted synthesis method (ex ept MF-25), only the low-level 6

inner loop order + outer loop order = total order

7

The 26th order inner loop ontrol blo k is used for longitudinal and lateral ontrol

576

long. outer long. outer

MS-11

MS-19 HI-21 FL-15 EA-12

long. inner

long. inner

loop

loop

loop

loop

ommand

feedba k

ommand

feedba k

h, VA

outer and inner loop are ombined;

z , wV

z , wV

z h, h_ , VA z

z h, h_ , V , nx z

EA-22

z , wV , VA z , wV , VA , q , nz z , wV z

CC-13

z , wV , VA z , wV , VA

EA-18

MF-25

MO-16

z , wV , VA z , wV , VA z , VA

h, h_ , VA , V_ A , q ,  wV , VA , q wV , VA , q

feedba k:

z , VA , q , nx , nz , wV

wV , VA , DT H R

wV , VA , q q, 

remarks

long.

ontroller order

prelter:

h, h_ ; VA , V_ A

24

model, rate

4+16=20

limiter mode swit h

5+(26)

limiter

R

wV , V , wV , R wV , (zg , wV;g ) ! (RwV -wV; ), V , V , R wV; (V -V ), nz , q (RwV -wV; ), q oord R (V -VR ) VA , z no inner loop wV , VA  , VA , q oord uB , w B , u_ B , w_ B , q oord

R

wV , R wV , VA , VA , nx , nz , q q,  R

q, q

no inner loop

6

7

2

2+4=6

2

mode swit h 3

limiter, mode

4+1=5

sele tion model

8+1=9

mode swit h

4

Table 34.1: Overview of Longitudinal Controller Stru tures

stru ture was inuen ed by the sele ted synthesis method.

Evaluation of Design Method and Controller Stru ture The evaluators' ratings and the omments on the questions on erning the eort ne essary for learning and appli ation of the design method (question 2a to 2e), on erning the omplexity of the ontrol solution (question 3a to 3d) and the general omments, have been used for the following dis ussion of the design method and ontroller stru ture. More details are given in the RCAM evaluation report [103℄.

H1 and -Synthesis

H1 design (HI-21) -analysis (MS-11, MS-19) are grouped as they lead to similar evaluation

The three approa hes to robust design that are based on and 8

inner loop order + outer loop order = total order

9

The 26th order inner loop ontrol blo k is used for longitudinal and lateral ontrol

577

lat. outer lat. outer loop

loop

ommand feedba k

MS-11 MS-19 HI-21 FL-15 EA-12

EA-18 EA-22 CC-13

lat. inner

lat. inner

loop

loop

ommand

feedba k


y
y ,  , , , p, r uV , vV , uV , vV , , p, , , p, _ ,
y r , vV r , vV Vx , Vy ,
y ,   , , p, r
y , 
y ,  , , p, r R 
y ,  ,  , R ( - ), , , p, r , (- ) _
y , , , p, no inner loop r, y , 
y ,  , , , p, r, 
y ,   , p (roll), , , r (yaw)

MF-25


y , _

MO-16


y , _

remarks

lat.

ontroller 8

order prelter:

8+24=32

model

3+15=18

model,

2+3+(26)

9

mode swit h

Rlimiter R

,

R

R

, vB

R

R

, 

limiter,

4 3

3

2

2+3=5

mode sele tion

R, , p, r, , p_, r_ 

, R, p, r, , , p, r

model

14+1=15

prelter,

3+2=5

mode swit h

Table 34.2: Overview of Lateral Controller Stru tures

results. HI and MS rely strongly on modern mathemati al ontrol theory.

As a

normal ight ontrol engineer in industry is not familiar with HI and MS, it will take him time and eort to gain the understanding and the experien e whi h is ne essary for a professional appli ation (6 to 12 months). Knowledge in

lassi al frequen y-domain te hniques, that an be presumed, will ease learning. Another positive aspe t is that ommer ial software is available. All three design teams used ommer ial produ ts (MATLAB), without undue di ulty. Experien e is ne essary in how to translate design requirements into frequen y dependent weighting fun tions.

The designers used model-following

te hniques to in orporate most of the design requirements into ideal models. This approa h left only a few requirements for translation into HI or MS syntax (i.e. weighting fun tions). A model-following approa h is not inherent or parti ular to HI and MS; it has been ommonly and su

essfully used by industry, e.g. for autopilot designs. HI and MS in lude dened plant un ertainties by an un ertainty model  guaranteeing robust stability and robust performan e . Note that the term performan e in the method's sense, does not over all the RCAM performan e requirements, nor typi al industrial performan e requirements given in a ight

ontrol system spe i ation. The un ertainty model is a relevant improvement

578

with respe t to a systemati design, but un ertain parameter ranges are to be spe ied expli itly, and therefore they have to be known in advan e. An ex eedan e of the dened and modelled un ertainty range, might deteriorate

ontroller performan e drasti ally. So, the un ertainty modelling has to be done with are and physi al understanding.

Only limiting values that orrespond

with physi al boundaries or that must not be ex eeded in air raft operation (e.g. maximum weight) an be dened easily. Also some aution on erning the delity of the un ertainty modelling is advisable. The un ertainty model represents a set of linear models and annot

over spe i nonlinearities of the plant.

Although this does not present a

major drawba k for many ight ontrol design tasks, it does require additional testing, as there is no impli it proof for stability of the nonlinear system. The presented HI and MS design pro esses onsist of multiple steps, in luding analysis steps with linear and nonlinear models. As not all requirements

an be formulated in a straightforward manner, iterative tuning is ne essary. The tuning was done manually without support of an optimizer. The presented ontrollers are of high order.

The ontroller dynami s of

MS-11 turned out to be ill- onditioned during the design pro ess (fast poles, unstable poles), but an improvement was possible with some extra eort. Unstable ontrollers are una

eptable in general, and marginally damped and high frequen y poles ause problems during transformation of the ontinuous design into a dis rete ontrol algorithm. Order redu tion and residualisation has been used as a remedy. The evaluators ould not assess the ontrollers' dynami s, as the eigenvalues of the ontrollers are not given in the design reports. Only a part of the ontroller, but a big and the most important blo k, has been designed by HI and MS ontrol te hniques.

It seems to be impossible

to gain physi al insight into the dynami s of this blo k  espe ially if order redu tion te hniques are applied. This blo k has to be looked at as a bla k box.

Its missing visibility and its high omplexity presents a risk for the

10 .

industrial learan e and erti ation pro ess. Solutions need to be found

Another risk is the possibility of an abrupt ontroller performan e deterioration, if the assumed un ertainty range is ex eeded. A ight ontroller for a modern transport air raft will be embedded into 1000 to 2000 air raft whi h will operate 100 million ight hours over 50 years.

History has shown that

they will fa e many unpredi table events and situations.

The ight ontrol

system should behave a

eptably, as lassi al low gain designs typi ally do. If ontroller's deterioration an be predi ted or dete ted  preferably without additional sensors, a re onguration an be initiated. Results of the HI and MS design entries are amongst the best. One fa tor is the higher potential of the high-order ontroller, another fa tor is that the methods are well tailored to handle the RCAM model un ertainties. The fa t that ontrollers of signi antly lower order have rea hed similar results, leads to the assumption that drasti order redu tion is possible. 10

From a mathemati al point of view the omplexity of this blo k is low but the resulting

algorithm for the embedded software and the extensive testing that is ne essary leads to the

erti ation issues.

579

The on lusion is: HI and MS are more powerful te hniques than the lassi al ones.

They oer an interesting potential  whi h has to be paid for.

Weighing the pros and ons of HI and MS design for an industrial ight ontrol task has to be done ase by ase, e.g.

additional eort might be justied if

sensors an be saved or the number of operating modes an be redu ed.

Fuzzy Logi Control Evaluation results are based on entry FL-15. FL belongs to the group of intelligent ontrol te hniques and was introdu ed in 1965 by Zadeh.

The power of FL is to design ontrollers in an intuitive

way. Design is done by linguisti formulations and reasoning. A

eptable but not ne essarily optimally tuned solutions are the out ome. A designer has to understand the general idea of FL whi h is easy to grasp and he needs exer ise in appli ation of FL. Good knowledge of ight physi s and piloting te hniques is ne essary but an be taken as granted for an average ight ontrol engineer. Therefore, the eort for learning has been rated moderate to easy.

As the

method yields a nonlinear gain ontroller, theoreti al knowledge in nonlinear dynami systems ould be useful. A variety of ommer ial software tools are available. In the presented design, the author used a MATLAB toolbox that was developed at Delft University. FL seems to have its potential in improving those parts of a ight ontroller, where a ontrol strategy an be opied from piloting te hniques. So it is quite logi al, that a hybrid design te hnique was hosen: the outer loops were designed by means of FL, while for the inner loops lassi al design te hniques have been applied. It is very easy to formulate linguisti riteria in FL syntax, but it is not lear how to expli itly introdu e, quantitative frequen y-domain riteria. Furthermore, the FL method does not address robustness dire tly. Robustness an be addressed indire tly: if the ontroller uses low gains and if the behaviour of a human ontroller is mimi ked, as a human ontroller is normally adaptive and robust. But it has to be proven by analysis that robustness is a hieved, bringing the FL design methods to the typi al multi-step, iterative design pro ess with trial and error. A signi ant advantage of FL is that linearization of the ight me hani s equations is not required for the design, but linearization was ne essary to nd gains for the lassi al design of the inner loop ontroller. The fuzzy part of the ontroller is a stati , nonlinear, deterministi fun tion that an be realized as a lookup table. The ontroller stru ture is visible and the order is very low. No implementation and erti ation problems are to be expe ted. For the implementation of FL ontroller algorithms into a ight ontrol system by using a nonlinear look-up table, erti ation should be no problem. If fuzzi ation and defuzzi ation software or fuzzy opro essors are used instead, questions regarding erti ation need to be answered. The FL design te hnique aims at dynami systems whi h are di ult to model and that are hara terized by un ertainty and impre ision. This does

580

not apply to ivil transport air raft dynami s. They are a

urately modelled (from the FL point of view) with mu h eort and the models whi h are used for design and pilot training purposes are improved after ight testing by parameter identi ation te hniques. The situations where un ertainties an o

ur happen during ight testing and in rare failure ases. A bene ial future appli ation of FL ould be in the area of ight guidan e, where piloting te hniques an be opied and where pre ision requirements are not too high, e.g. relaxed altitude ontrol during ruise in order to take are of the engines.

Eigenstru ture Assignment Evaluation results are based on entries EA-12 (Modal Multi-Model Control Approa h), EA-18 and EA-22 (Eigenstru ture Assignment). EA is an extension of the well-known pole-pla ement te hniques.

EA al-

lows the designer to assign the losed-loop eigenvalues (poles) and additionally, to assign the eigenve tors or parts of them, within ertain limits.

Desirable

pole lo ations are spe ied in the military ying quality spe i ations (MIL-F8785C). The military spe i ations are often used as guidelines for ivil transport air raft, though their fullment is not ne essary for erti ation. By the assignment of eigenve tors, the zeros of the transfer fun tions an be inuen ed and oupling and de oupling of states and modes an be addressed dire tly. Though EA an be learned with a

eptable eort, very good knowledge of the pro ess and its eigenstru ture is needed. EA is available in ommer ial standard software pa kages.

All three de-

sign teams used MATLAB. EA-12 used its CERT in-house developed MODAL CONTROL TOOLBOX, EA-18 and EA-22 used their own MATLAB M-les. EA methods an design state feedba k or output feedba k, with onstant gains or with dynami feedba k. This allows a great variety of possible ontroller stru tures. EA-18 and EA-22 designed stati feedba k ontroller. EA-12 used stati gains for the lateral ontroller and a fourth order dynami ontroller for the longitudinal axis. EA is strongly linked to linear systems.

A potential is seen for exible

air raft where the aeroelasti behaviour is typi ally des ribed in terms of modes. De oupling of aeroelasti modes or minimum damping values are important requirements.

The design task is aggravated by marginal ontrollability of

ertain modes by the standard aerodynami ontrol surfa es. It seems to be impossible to dire tly formulate riteria su h as a tuator a tivity or passenger omfort as onstraints on variables; it is however, possible to do this indire tly, e.g. by hoosing losed-loop eigenvalues to ensure roll-o in order to inuen e ontrol surfa e a tivity. Work-arounds were used, ending with an iterative multi-step sear h pro edure. Guidelines for the denition of referen e eigenve tors ould be worked out, possibly by analyzing other good designs. The few hints given in the reports on how the eigenstru ture should be modied in order to a hieve those problemati al requirements are not su ient. In order to address all design riteria, it may be ne essary to supplement EA with other te hniques, as demonstrated by the design teams.

581

The time ve tor method of Doets h [54℄, where the solutions of dierential equations for for e and moment equilibrium are plotted for ea h one of the eigenve tors, ould be a possibility to ombine mathemati s and physi s in order to gain better insight  but some arduous work has to be expe ted. The three teams dened low-order ontrollers that have a rather onventional stru ture with good visibility, i.e. implementation and erti ation an be done by standard pro edures. Robustness was a hieved by using low feedba k gains (whi h gave adequate performan e) whi h were then he ked in a multi-model analysis.

Classi al Approa h Evaluation results are based on entry CC-13. CC is well known in the aeronauti al industry and has been applied sin e the beginning of automati ight ontrol.

Chara terized by su

essive loop

losure, the lassi al design is guided by understanding of the physi s of ight and a good deal of intuition and experien e that assists in sele ting ontroller stru ture. Physi al insight is ru ial and inherent to the method. CC is not a robust ontrol design method. Nevertheless, the presented design shows that robustness in the sense of the RCAM design hallenge an be a hieved with

lassi al design methods. A wide variety of ommer ial software tools are available. In the presented design, the author used MATLAB. This design entry is very useful as a ben hmark for the other entries using modern methods. It demonstrates that an a

eptable solution is possible by some engineering eort. The presented ontroller stru ture is visible and the order is low. No implementation and erti ation problems are to be expe ted. If CC leads to satisfa tory results, one might ask: "why look for modern robust methods?". There are at least two reasons:



The eort to apply CC (SISO te hniques) in reases progressively with the

omplexity of the plant model. CC omes to its limits when ontrollers for MIMO systems with high internal oupling are to be designed. That is why MIMO ontroller synthesis te hniques (e.g. in industry.

LQG) are applied

For future demanding design tasks, su h as an 800-seater

air raft or a new supersoni ommer ial transport, powerful and robust te hniques are required.



The di ulty to implement CC into a systemati and highly automated industrial design pro ess, as the designer and his experien e is an integral part of the method. This is underlined by the statement of one evaluator: a CC re-design ould be a nightmare, if ... the rationale behind the design is not available. This situation presents an in al ulable and potential risk for a modern industrial design pro ess.

Model-Following Approa h Evaluation results are based on entry MF-25.

582

The MF approa h onsists of three fundamental blo ks that hara terize the overall ontroller stru ture:



A ommand blo k (CB) whi h denes the required dynami behaviour (model) and whi h generates ideal state variables and their time derivatives as ommand signals.



A feedforward ontrol blo k (FFC) whi h ontains an inverse air raft model.

The FFC blo k omputes ontrol surfa e dee tions from the

CB's ommand signals. With an a

urate inverse model and no disturban es, the air raft (plant) would exa tly follow the ommanded signals and all requirements that are inherently dened into these signals would be fullled dire tly.



A feedba k ontrol blo k (FBC) whi h ompensates for dieren es between a tual and ommanded signals aused by disturban es (wind, sensor noise, et ), and un onsidered plant un ertainties.

The CB is asso iated with the outer loop, the FFC and FBC blo ks belong to the inner loop. To use MF for in-ight simulation seems to be very reasonable and has been done su

essfully for many years. Appli ations of the MF ontrol system design as presented in MF-25 are not known for automati or manual ight

ontrol, though it is urrent pra ti e to use model-following for partial tasks of autopilots (e.g. altitude a quire mode or are mode). The lear task sharing and the underlying philosophy makes it easy to grasp the general MF design on ept for any ight ontrol engineer with a ba kground in ight me hani s. The eort for learning MF was rated moderate. The CB synthesis an be straightforward, but experien e in the denition of a dynami system that fulls design requirements is needed. Re-use of a suitable CB is possible, i.e. it an be used for any air raft that has to full a set of existing requirements. In the same way, previous designs with desired performan e an be used for the CB. The authors took advantage of this feature: they based

11

their CB on the ontrolled RCAM model from CC-13

 whi h fulls the

design requirements. The prin iple, to put all knowledge on system dynami s into the feedforward path (FFC), is taken into a

ount by inversion of the nominal linear plant model. The matrix algebra involved in this al ulation is only at a basi level and not di ult. The authors used simpli ations that rely on the assumption of ideal model-following. The question, what happens if this assumption is not valid, has not been answered theoreti ally. 11

MF-25 took advantage from its late entry time, where rst results of other teams were

available. As the design of the CB an be seen as an isolated task, this approa h helped in meeting the tight time s hedule.

This is paid for by a predened (higher than ne essary)

omplexity of the CB and the a hievable results whi h has to be the same as for CC-13. The results for CC-13 and MF-25 omplied with requirements.

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A simple time-domain optimization te hnique is used to design the low-order feedba k ontroller. The authors point out that more sophisti ated synthesis methods an be used, if required. The author used MATLAB for his design, omplemented by an optimization subroutine. The MF approa h splits the design requirements into 2 ategories and adds a third: 1. Requirements that an be designed into the CB. These requirements on ern the desired ommand behaviour and are fullled by means of the CB and FFC blo k. 2. Requirements that have to be fullled by means of feedba k, e.g. stability,

ontrol a tivity, passenger omfort and disturban e reje tion riteria. 3. The additional MF requirement that the FFC should provide deviations to be as small as possible between ommanded and a tual response. Robustness in the sense of robust ontrol theory is related to feedba k. Therefore the question of robustness arises only for the requirements under 2. Many of the RCAM design requirements (e.g. ight path tra king) belong to ategory 1. They have been dened dire tly into the CB and the expe ted robust fullment has been a hieved and demonstrated. The question of robustness be omes relevant for FBC design. The synthesis method used does not onsider robustness expli itly. An additional time delay has been introdu ed, probably to keep gains and frequen ies low. Robustness is demonstrated by validation in worst- ase simuations. The ontroller appears to be generally robust. In some o-design ases, the damping of higher frequen y modes tends to de rease. Complexity of the MF-25 design, in terms of order, was rated moderate though it is higher than ne essary, be ause the ontrolled RCAM model from CC-13 has been used in the CB. A simpler CB stru ture seems to be feasible. Feedforward omplexity is determined by the omplexity of real air raft dynami s. The inversion of the nominal linear air raft model produ ed a stati FFC blo k of low omplexity. The longitudinal and lateral FBC blo ks ea h have a low order of unity. The high level of visibility and lassi al nature of this design should lead to straightforward erti ation. No problems are expe ted. The a hieved ight path tra king results are similar to CC-13. As CC-13 was sele ted as the model in the CB this is not surprising. The importan e of the exa t knowledge of the air raft dynami s for the FFC design is emphasised. Therefore, it would be interesting to see what will happen, if the FFC blo k does not exa tly mat h the real air raft dynami s (failure ases, unmodelled aeroelasti modes, operation at o-design points, et .).

Are more sophisti ated (nonlinear) inverse models needed?

Then the

omplexity of the design an in rease signi antly. MF proved to ope easily with the design hallenge. The lear strategy and philosophy makes it very attra tive. Some questions on erning the robustness

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apabilities and the limitations of MF in more demanding design tasks have to be answered.

Multi-Obje tive Parameter Synthesis Evaluation results are based on entry MO-16. The MO approa h onsists of a omputer-aided sear h pro edure, where the designer determines his best ompromise between a number of spe ied riteria. An ee tive division of work is organized, where the design engineer takes

are of all work that requires human intelligen e (i.e. to make de isions and to be reative, e.g. in sele ting the ontroller stru ture, in hoosing synthesis te hniques, omposing the analysis hain, and de iding whi h ompromise is best), and the omputer takes are of routine work (parameter optimization a

ording to given ost fun tions, data management, et .). The approa h is independent of the ontrol synthesis te hnique or the sele ted ontroller stru ture. MO an take any requirement into a

ount that an be expressed mathemati ally by a

ost fun tion. The learning eort is di ult to rate. Greater knowledge of the supporting software environment ompared to the other design entries seems to be ne essary, espe ially to implement ontrol design methods, to introdu e ost fun tions and to organize the automated design pro ess. A division of labour an be organized between a omputer software spe ialist, who sets up the software environment and adapts it to the design task, and the ight ontrol spe ialist who is doing the ontroller design. This will relieve the workload of the ontrol design engineer signi antly, making best use of his spe ialized apabilities. The author used the DLR-proprietary design environment ANDECS with a database fa ility. Commer ial ontrol software su h as MATLAB and

MAT RIXX

an be oupled via pro ess ommuni ation. A good CAE software environment in ombination with a powerful omputer (PC or workstation) is absolutely essential for this design approa h. The authors used lassi al and LQR design as low-level synthesis te hniques. The presented ontroller stru ture is visible and its order is low. No implementation and erti ation problems are to be expe ted. Mode swit hing e.g. between the altitude mode and the limb/des ent mode has been used to over the varying operating onditions. The ee ts of dierent dynami al designs an be seen in the time histories. Robustness is a hieved by a worst- ase multi-model approa h. The MO approa h appears to be very attra tive for industry, as it ree ts and automates typi al design pro edures.

The freedom to in orporate any

riterion and any low-level synthesis te hnique is a big advantage and allows design of all stru tures a designer wants to design. This is ree ted in the high ratings for questions 2d and 2e. Depending on the omplexity of the models and the ontrollers, omputing time for parameter optimization may be ome a relevant fa tor. The graphi al user interfa e that supports the designer has a signi ant inuen e on the e ien y of the design pro ess. A good presentation of the relevant requirements is essential for his de ision task.

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Flight ontrol systems are very omplex, and multiple requirements have to be visible for weighting them simultaneously. The dimension and resolution of a omputer s reen are limited. Therefore, the set-up of the graphi al user interfa e has to be organized e iently, whi h will require some additional eort. Summing it up, MO has a great potential to help stru turing and automating the industrial pro ess for ight ontrol design.

34.2.4 Dis ussion of the Evaluation Results Design Methods The term design method has not always been used by the design teams in the same sense: (MS,

sometimes it was limited to the ontrol synthesis method

H1 , EA, LQR, et .)

and sometimes it was used for the overall design

method that omprises multiple design steps. Due to this linguisti ina

ura y, the presented methods are named after the underlying mathemati al method (design entries MS-11, EA-12, FL-15, EA-18, MS-19, HI-21 and EA-22). Only three are labelled after hara terizing features (design entries CC-13, MO-16 and MF-25). The design teams used three dierent approa hes to assemble the design method:

1. Control theory approa h (MS-11, EA-12, FL-15, EA-18, MS-19, HI-21, EA-22): In the entre stands the ontrol synthesis method  preferably a powerful one. The ontroller stru ture has to omply with the onstraints of the method. Flight dynami s and mission requirements have to be translated into the method syntax  often a di ult task.

2. Flight physi al approa h (CC-13, MF-25): Flight dynami s, mission requirements and physi al reasoning are entral.

Controller stru tures are build up in blo ks that orrespond to

physi al fun tions or pilot tasks. Control te hniques are hosen that an be easily adapted to the design task. Optimization te hniques are used, half-automated tuning and trial and error pro edures, too.

3. Pro ess oriented approa h (MO-16): This approa h fo uses on the design engineer and the design pro ess, making use of the automation potential that is provided by today's powerful

omputers (PC and workstations) and software. An ee tive division of work is organized, where the design engineer takes are of all work that requires human intelligen e (i.e. to make de isions and to be reative), and the omputer takes are of routine parameter sear h work.

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Air raft Model for Flight Control Design The RCAM Model is a nonlinear model dened by nonlinear dierential equations. All designers had to linearize it, be ause their design te hnique is appli able to linear models only; the FL design method is the only ex eption. The nonlinear model was used for analysis and tuning. The MS design method requires the expli it modelling of the un ertainties. This additional eort is honoured with guaranteed robustness on erning stability, within the bounds of the design envelope and the assumed parameter un ertainties. Methods like MS and HI, that design high-order ontrollers whi h may a hieve better performan e, often design on the point, i.e. performan e deteriorates in o-design ases (i.e. failure modes, negle ted dynami ee ts et ). This requires high-delity modelling of the air raft dynami s and exa t knowledge of un ertainties. It is quite ommon that models of the air raft dynami s have to be orre ted during ight testing, whi h ould lead to re-design and modi ation of the ight ontroller in a late proje t phase.

Subsequent ad-

ditional testing implies a potential risk in terms of budget and time for an industrial programme. Certainly, this fa t applies to all methods, but the risk has to be rated higher for the high-performan e MS and HI ontroller designs than for less apable, but often well-behaved, lassi al designs.

Evaluation Questionnaire (Overall View of Entries) The following dis ussion is based on the mean values and the distribution of the ratings. In general, the results of the independent evaluators show good agreement, varying opinions ould always be explained by the dierent interpretation of a question. Only the key points are addressed below.

Eort Ne essary for Appli ation Level of understanding (Question 2a). The eort to gain the ne essary level of understanding was estimated to be medium for EA and FL. Higher eort is required for HI and MS to understand the mathemati al ba kground and to gain experien e in the method-dependent denition of requirements. MS needs additional knowledge to model (stru tured) un ertainties by LFT. The ne essary level of understanding for MO depends strongly on the sele ted synthesis te hnique (e.g. LQR for longitudinal ontroller and CC for lateral ontroller). The additional eort to ongure the software tool for MO is dependent on its user-friendliness and ould not be rated. CC requires detailed knowledge, but it an be presumed that mu h experien e from previous designs is available in industry.

Learning urve (Question 2b). Similar results were obtained for the learning

urve.

FL and EA an be understood with moderate and a

eptable eort,

while MS and HI are more di ult to grasp.

Re-design (Question 2 ). A re-design with a method you are not familiar with is always likely to be problemati al. It is di ult to estimate how many problems

587

you will en ounter without starting the work. This holds for lassi al and new methods, and seems to be the reason why all design methods are rated, more or less, in a small band around normal eort.

Flight ontrol law stru tures (Question 2d).

The denition of the ontroller

stru ture and the denition of its internal fun tional blo ks (ight path generator, inner and outer loops, de oupling of longitudinal and lateral axis) is essentially based on the interpretation of the requirements, physi al insight and experien e in ight ontrol design, rather than on the method. The hoi e of the synthesis method often poses unwanted onstraints on the sele tion of the

ontroller stru ture. Therefore, EA, FL, MS, and HI are used only for ertain parts of the overall stru ture. Controller parts where these synthesis methods were not appli able for design, have been synthesized by other methods. MO is appli able for every stru ture that an be des ribed mathemati ally; this is why it re eived the highest ratings. Entry HI-21 proposes an interesting ombination of longitudinal and lateral inner loops, leaving oupling and de oupling to the synthesis method. It would be interesting to ompare the high-order ontroller to a onventional approa h where oupling and de oupling are dened by stru tural blo ks (e.g. turn ompensation).

Design requirements (Question 2e). All methods have shown their power when design obje tives and method t; when otherwise, they have degraded to simple trial and error. In MO every requirement an be used that an be expressed mathemati ally by a ost fun tion. That is why it re eived the highest ratings.

Complexity of the Control Solution Visibility (Question 3a). The visibility of all approa hes is a

eptable or better. The MS methods have been rated signi antly lower, at the limit of being unsatisfa tory, be ause of their high-order dynami blo k for whi h physi al reasoning seems to be impossible.

Complexity (Question 3b). Parts of the stru ture that make it omplex (nonlinearities, swit hes, lters, et .) are method independent. Other parts, su h as high-order dynami blo ks are learly related to MS and HI. That is why the omplexity ratings for MS and HI are at the limit of being unsatisfa tory. Performan e omparisons showed, that omparable results have been a hieved with ontrollers of mu h lower order.

Implementation (Question 3 ). The experien e of the design teams in implementation of ight ontrollers into a ight ontrol omputer was very low. Not surprisingly, implementation issues have not been dis ussed in depth. Therefore, evaluators' omments are more general. All designs are ontinuous-time ontrollers. Typi al sample times for ivil ight ontrol systems are 25 ms for the inner loops and 50 to 150 ms for the outer loops.

This poses restri tions on ontroller bandwidth.

Furthermore,

dis retization of high-order ontrollers (HI and MS) an ause di ulties: high

omputing load or numeri al integrity of the dis rete algorithm, espe ially when the ontroller has lowly-damped modes. Disturban es of measurement signals,

588

su h as bias or noise, were not onsidered. Smoothing lters were not used.

1

The use of ideal integrators ( s ) will ause una

eptable drifts in real world

systems. Most of these implementation issues an be solved, but they may lead to additional iteration steps.

Certi ation (Question 3d). Certi ation issues have been addressed in general terms, for the same reason as for the implementation issues.

Problems are

expe ted with erti ation of the high-order dynami MS and HI ontrollers. As long as FL ontrollers are realized by nonlinear gain maps instead of FL algorithms, problems are not expe ted. If a real performan e benet an be demonstrated it is probable that methods for quali ation and erti ation would follow.

Controller Performan e Controller performan e ould be evaluated only qualitatively as the quantitative results of the assessment software were not available during the evaluation phase. Therefore, a more general dis ussion follows.

Fullment of Requirements All entries fullled, more or less, the minimum requirements, as judged from the plots in the design reports. If fullment is not omplete, it ould have been a hieved probably with a little bit more tuning eort or slight modi ations of the ontroller stru ture. The design teams used dierent approa hes to generate the ight path ommands, whi h are used as ontroller inputs. This inevitably leads to methodindependent performan e dieren es and it be omes ne essary to lter out this ee t. This variation is aused by the missing denition of a ight path generator in the RCAM des ription. The hoi e of stru ture and order was left to the design teams' dis retion. Obviously, the more omplex a ontroller stru ture is (dynami s, lters, logi , nonlinearities), the better are the a hievable results. Controller order lies between 5 (EA-12) and 56 (MS-11). Comparably good results  as far as ould be judged from the evaluated results  have been a hieved with 9th and 36th order

ontrollers. It is fair to assume therefore, that the high-order HI/MS ontrollers

an be onsiderably redu ed in order, without ompromising performan e. If

onrmed, order redu tion would be mandatory before implementation into the ight ontrol omputer.

Robustness There is no universal ontroller design te hnique that an over all design requirements, in luding the robustness requirements, in one step. Therefore, all teams built up design methods around their basi synthesis method. Iterative, multi-step design pro edures with synthesis and analysis steps are the out ome. Controller tuning is performed by means of sear h methods, i.e. optimization te hniques or simple trial and error.

589

If high-order ontrollers are designed for optimal performan e, ontroller transfer fun tions with not h lter hara teristi s and/or high gains in small frequen y bands are often the result. Su h a ontroller an deteriorate abruptly in o-design onditions. Be ause of this reason, it is important that high-order HI and MS ontrollers are not operated outside the range of their dened plant un ertainties. If ontrollability of the air raft an be lost, the probability has to be less than

10 9 o

urren es per ight hour.

MF-25 splits requirements into a ategory that an be dealt with by feedforward ontrol and a ategory that is handled by feedba k ontrol. As robustness is related to feedba k, robustness ae ts only the se ond ategory, whi h eases the design task signi antly. Robustness he ks by a worst- ase analysis or a

-analysis test, whi h would

allow judgement of the a hieved stability performan e of all ontrollers, was not available for the evaluators. The results of a

-analysis test

are dis ussed in

hapter 35.

34.2.5 Dis ussion and Lessons Learned The Robust Flight Control design hallenge has been a unique han e to ben hmark new design te hniques. The high number of RCAM design entries (thirteen) ree ts the high interest of universities and resear h establishments in ight ontrol problems.

Mu h eort was invested into the denition of the

RCAM problem formulation, the RCAM model and the evaluation pro edure  with a onsiderable result.

RCAM Problem Denition The air raft model is a simplied model of a ivil transport air raft that has well-behaved natural dynami s. In parti ular, the aerodynami model, engine model, a tuator models and sensor models are simplied.

The design mis-

sion, an automati ILS approa h without are, represents a small but relevant sequen e of a typi al transport air raft mission.

Air raft handling is simpli-

ed, i.e. no slat, ap, and landing gear extension. Operational onditions are moderate: no rosswind on the nal approa h, moderate levels of turbulen e and wind shear. No failures are onsidered, ex ept an engine failure during a non- riti al ight phase. A lot of performan e, safety, passenger omfort and

ontrol surfa e requirements have been dened, representing a typi al extra t of industrial requirements. More demanding are the robustness requirements that ontrollers should be robust against: variations in mass, enter-of-gravity lo ation, time delay, speed variations and engine failure.

The design task is

mu h more realisti than typi al a ademi ight ontrol problems. A design, from the study of the problem denition through to a presentable design report, will require between three and nine months, depending on the experien e and tools of the designer. The eort to make the ontroller ight-worthy is mu h higher.

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Evaluation Questionnaire  RCAM Experien e The evaluation questionnaire proved to be an ee tive tool for evaluating the dierent design methods and ontroller stru tures in a fair and omprehensive manner. The missing results of the assessment software turned out to be an advantage, as the evaluators fo used on the method and the ontroller stru ture. Some minor improvements should be made to the questionnaire in order to avoid misunderstanding if it is to be used again.

RCAM Design and Evaluation The teams have demonstrated modern design methods that emphasize robust designs. The hoi e of synthesis methods was unrestri ted, resulting in a on entration around EA and HI/MS. This aused dupli ation of eort, repetition of results and the impression of missing balan e. Other robust methods, for example, quantitative feedba k theory (QFT), are not overed. In the future, the appli ation of other methods should be en ouraged. The evaluation results of ten design teams are dis ussed in detail above. All designs have a hieved some level of su

ess. But before any of the ontrollers

an be onsidered for implementation into a ight ontrol system, more detailed evaluations are needed (investigation of disturban es, investigation of system failure ases, system safety analysis, real-time ight simulation, et .). The evaluation shows learly, that a lot of expert knowledge is needed for ight ontrol design, not only in relation to the method, but also and foremost, in relation to the pro ess (here: ight me hani s) and the requirements (here: ying qualities and ight mission). Design methods that rely only on good ontrol te hniques annot gain satisfa tory results. On the other hand, designers that repeat traditional approa hes will be ome unable to introdu e real new ideas. Modern methods have the potential to lead to new approa hes. Finally, it should be said, that the RCAM design hallenge has ontributed substantially to the progress of robust ight ontrol design. The GARTEUR A tion Group has brought together the ompeten e from industry, resear h establishments and universities from all over Europe. The ollaboration proved to be bene ial for all parti ipants. Industry was provided with a lot of useful information on robust ontrol te hniques. Resear hers be ame aware of industrial requirements on robust ight ontroller design and beyond. The dialogue has started promising ollaboration whi h should be ontinued. The RCAM ben hmark problem is a lasting out ome of the a tion group.

34.2.6 Re ommendations for Future Work With thirteen designs performed, the RCAM model an be ome the ben hmark

problem for demonstrating new robust ight ontrol design methods to the ivil air raft manufa turers.

The RCAM model should be improved, taking into

a

ount the lessons learned.

Realisti but more demanding missions should

be dened in order to allow powerful methods a better demonstration of their

591

potential. Where possible, su h improvements should aim to keep ompatibility with the existing work. The RCAM model an be used to ben hmark other mu h more futuristi  design te hniques, e.g. arti ial intelligen e methods su h as neural networks and geneti algorithms, that have emerged quite re ently.

These te hniques

have been applied with some su

ess to omplex, poorly understood problems. Although ight ontrollers are normally designed with high understanding of ight dynami s, these novel methods might have some usable potential for the treatment of failure onditions, where ight dynami s may hange dramati ally. The presented ontinuous-time designs need an extra design step: the dis retization of the ontroller algorithm. Robust methods that dire tly design dis rete ontroller algorithms would be very favourable, espe ially if high-order

ontrollers are the out ome (MS and HI). Resear h in this area is en ouraged. It an be on luded that e onomi stresses are for ing the industry to redu e

ost and time to market. Engineers are asked to design more qui kly, ever more

omplex systems. This will stay as a running demand, as industry restru tures and automates design pro esses to make them more e ient.

Even design

methods for ight ontrollers are ae ted by su h pra ti al demands. Methods and te hniques, su h as MO, that support an e ient automated ontrol design pro ess, are needed and should attra t resear hers' attention.

A knowledgement The author wants to thank the RCAM evaluation team: W. Alles (DASA), R. Bro khaus (DLR), J. Breeman (NLR), J. Bos (NLR), C. Fielding (BAe-MA), N. Foster (BAe-D), P. van der Geest (NLR), C. van Gelder (NLR), G. Game (BAe-D), G. Grübel (DLR), M. Hut hinson (AVRO), H.D. Joos (DLR), M. Horton (BAe-D), D. Laidlaw (AVRO), J.F. Magni (CERT), A. Nieuwpoort (NLR), M. S hifaudo (ALN), D. Vorley (BAe-D), K. Weise (DASA), J. Winter (DASA). Spe ial thanks are due to Prof. Bro khaus who evaluated all design entries.

34.3 Military Air raft Manufa turer's View

12

34.3.1 Introdu tion From an industrial viewpoint, in designing military air raft, the overall requirement is to be able to design a vehi le whi h meets the ustomer's operational performan e requirements, is safe to operate and whi h an be designed and built at an a

eptable ost, within an agreed times ale. As an essential part to a hieving the air raft's performan e and safety targets, the ight ontrol system needs to be arefully designed, taking into a

ount the requirements and onstraints that will be imposed through airframe and system hardware physi al limitations.

The air raft's ight ontrol laws provide the basis for

a hieving the desired performan e hara teristi s and oer great potential for 12

Prepared by Chris Fielding on behalf of the HIRM Evaluation Team

592

operational exibility in terms of the possible pilot-sele table and automati modes. At an early stage of the ight ontrol laws design pro ess, primarily based on experien e from other proje ts, it is possible to ease the later implementation, testing and ight learan e aspe ts, by making appropriate assumptions. This will involve hoosing the best ar hite ture for the ontrol laws, even before any design te hnique or pro edure has been onsidered. It is essential to use feedba k signals of su ient performan e and integrity, with multiplexing of signals in order to a hieve overall system integrity targets. Additionally, ba kup modes may need to be designed to over for omplete loss of any sensor information. On e the ontrol laws ar hite ture has been established, the gains, lters and nonlinear fun tions have to be designed. There are a wide range of (mainly linear) te hniques for determining the appropriate parameter values. It is the aim of the HIRM design hallenge and evaluation to identify the strengths and weaknesses of a range of su h te hniques, in order to highlight the methods whi h are best suited to support the ight ontrol laws design pro ess and whi h are apable of produ ing a robust design.

34.3.2 Industrial Considerations for Appli ation of Design Te hniques In pra ti e, it is usual to design ight ontrol systems to be apable of providing good air raft handling qualities over a wide range of operating onditions and to over the arriage of a range of internal and/or external stores. Additionally, new moding (e.g.

autopilot modes) might be added to the baseline

system at a later date. At any stage of the design pro ess, a ompetent engineer, other than the original designer, might inherit design responsibility and must be able to take over the design without undue di ulty. From a wider perspe tive, it must also be re ognised that there is a range of spe ialists and managers who need to work with the ontrol laws at ea h stage of the total design pro ess [123℄.

These might be simulation engineers and pilots, ight

ontrol omputer implementers and testers, and ight learan e authorities. The level of visibility of the fun tionality is important to these people, to help them to gain su ient understanding to arry out their tasks. As an essential part of the ontrol laws, the ontroller algorithms need to be exe uted in real time in the air raft's ight ontrol omputer.

Sin e this

apability is limited, it is important that ontrol algorithms are e ient and do not lead to real-time pro essing problems.

High order ontrollers, multi-

dimensional look-up tables and ompli ated nonlinear fun tions all add to omputer throughput requirements. There are also other aspe ts whi h ould ause implementation di ulties. For example, the ontrol algorithm's numeri al a

ura y requirements and its potential for gain s heduling, are all important with respe t to implementation in the air raft's ight ontrol omputer. The level of investment that industry will need to make in terms of developing design software to the appropriate quality standards and for training of

593

its designers to use a new method is also of interest; this needs to over the

ost of getting into a position of being able to use (and if ne essary, modify) the method, su h that satisfa tory results an be a hieved, leading through a straightforward ight learan e, to a su

essful ight test programme.

34.3.3 HIRM Evaluation Team An evaluation team was established, based on experien ed industrial aerospa e ight ontrol systems designers, who themselves might be regarded as potential  ustomers for the range of design te hniques available.

The industrial

ommunity have a genuine interest in the methods in terms of redu ed design

y le-time and asso iated osts, and improved produ ts. The use of industrialists on the evaluation team also supported a major obje tive of the urrent exer ise, in transferring the knowledge and experien e from the GARTEUR A tion Group into industry. The evaluation team was led by British Aerospa e Military Air raft and

omprised experien ed ight ontrol law designers from the following organisations:

  

British Aerospa e Military Air raft SAAB Military Air raft British Aerospa e Dynami s

Colle tively, this team has great experien e in the overall design of ight

ontrol systems and is highly apable of addressing all aspe ts of the design in relation to the ight ontrol laws design pro ess. Additionally, the evaluation team has olle tive experien e with appli ation of most of the proposed methods, pla ing it in a strong position to perform the evaluation. Ea h organisation was provided with the HIRM design reports and its ight ontrol spe ialists then evaluated the design entries and ompleted a set of questionnaires (as dened in the earlier se tion). The olle tive views and main points whi h were made are dis ussed below, for ea h of the design entries.

34.3.4 Evaluation of the Design Entries Linear Quadrati Approa h (LQ-26) This design entry (Chapter 28) fo used on the Linear Quadrati approa h. This well-established method is onsidered to be relatively easy to use, but requires a good understanding of optimal ontrol theory. In terms of the learning urve for ight ontrol appli ations, it was judged to be relatively easy to grasp. Obtaining the desired results by sele tion of weighting fun tions will probably provide the greatest di ulty for a new user. Some iteration may be ne essary if the frequen y response hara teristi s are not satisfa tory. The methodology results in a ontrol law ar hite ture similar to that for a lassi al design, but whi h has a multivariable stru ture. Being essentially

594

a time-domain method, the transient response hara teristi s an be tuned to some extent. The frequen y response hara teristi s have to be determined as a se ond step and therefore there is likely to be some iteration in the design (although this ould be automated). The resulting ontroller is of low order, although for this parti ular design, it was desirable to redu e the number of integrators. Implementation of the

ontroller should not be a problem, from the information provided and system quali ation should be straightforward. With the proposed ar hite ture, there is high potential for integrators winding up, espe ially sin e some of the

ontrolled variables are related through kinemati relationships. Although an anti-wind-up te hnique is proposed for the parti ular ontrol s heme, the ve integrators used in the design ould still give erti ation problems; this is a feature of the ar hite ture hosen and not the design te hnique itself. The ontrol law ar hite ture also ontains an undesirable swit hing logi based on the pilot's roll ommand.

This was an attempt to ompensate for

the ee ts of the gravity ve tor, during rolling manoeuvres. In pra ti e, the required ompensation an be expli itly in luded in the ontrol laws ar hite ture, prior to designing any ontrollers. The time histories presented were generally satisfa tory but there are low stability levels indi ated for some ases. Overall this was a very good eort, taking into a

ount the limited ba kground of the authors in ight ontrol problems.

Nonlinear Dynami Inversion and LQG (DI-30) This design entry (Chapter 32) presents an interesting ombination of the Nonlinear Dynami Inversion and Linear Quadrati methods. Being su h a ombination, it obviously needs a greater level of understanding than ea h of the methods individually. The most di ult part would appear to be asso iated with the inversion of the air raft equations of motion, although on e established, these equations would reside in the ontrol law framework. The method does not appear to involve any di ult mathemati s. The approa h onsidered handles the time response requirements well, by shaping the feedforward ommands and then using feedba k to provide tight tra king of these ommands (essentially a model following type of approa h). Robustness requirements are not addressed dire tly but are assessed following design, possibly leading to design iterations. The robustness aspe ts ould be addressed by introdu ing an automated pro edure. The results presented indi ate good robustness and performan e, with tight tra king of the feedforward terms. Control a tivity is as to be expe ted, but some spikey responses and rate limiting were noted. The high frequen y gains in the system (from inertial sensors through to a tuation ommands were not visible in the design as presented  noting that the HIRM design hallenge onstrains these gains to ensure a pra ti al design and avoid undue airframe/FCS aeroservoelasti ity problems). In general, this was another good design entry whi h in the NDI part of the

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design made good use of ight me hani s knowledge.

Robust Inverse Dynami s Estimation Approa h (DI-28) This design entry (Chapter 33) addresses the Robust Inverse Dynami s Estimation approa h to ontrol laws design and ontains relatively straightforward mathemati s. The method indire tly attempts to provide the appropriate air raft for es and moments to trim the air raft and de ouple the outputs, while proportional plus integral loops provide the desired set of air raft response

hara teristi s. The method is related to Nonlinear Dynami Inversion. The method gives perfe t results for the nominal model in the absen e of hardware dynami s; for this relatively trivial ase, the nominal design is very easy to a hieve. As hardware models and data un ertainty are introdu ed, more eort will be needed and iterations required for tuning the ontroller to meet robustness riteria. The eort needed will depend on the degree to whi h the nominal design is representative of the full high-order modelling assumptions, in luding the un ertainties. The method is good for shaping the time histories but does not handle robustness requirements dire tly (it is similar to Eigenstru ture Assignment in this respe t). Although robustness is measured after the initial ontroller has been designed and this may result in manual tuning, it should be possible to automate the iteration pro ess, to in lude robustness riteria. In terms of the stru ture of the design a hieved, a short oming is asso iated with the ontroller's high frequen y end-to-end gains whi h should have been onstrained to meet the HIRM's design spe i ation, in order to provide a realisti level of attenuation at airframe stru tural mode frequen ies. For implementation purposes the omplexity of the design is onsidered to be a

eptable, with a ontroller of low order. The Ni hols plots presented indi ate the design to be robust (it is essentially three de oupled single loop designs). The ontroller's performan e also appears to be satisfa tory, with the time histories meeting their spe ied values for bandwidth and handling qualities riteria. Overall ontrol surfa e a tivity seems to be a

eptable. This design entry provided an interesting and intuitive approa h whi h was easy to follow due to the links to ight me hani s whi h were taken into a

ount in the ontrol law ar hite ture. The method is relatively new and oers potential for rapid prototyping of initial designs. Further work needs to be arried out to develop the method for pra ti al appli ations, in luding implementation.

H1 , -Synthesis Approa h (MS-29) This design entry (Chapter 31) presented a very detailed appli ation of the

-synthesis

approa h.

In order to use this approa h for ight ontrol laws

design, a thorough understanding of the theory is needed to ee tively design and analyse ontrollers. The method is onsidered to be relatively di ult to grasp, due to the mathemati s involved and any ontrol engineer without a strong mathemati al ba kground is likely to struggle to learn this method.

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The design formulation presented allowed almost all of the design requirements to be in luded. Good use was made of the relationships between air raft physi al parameters in relation to the time domain handling qualities metri s. In ommon with some of the other design entries, the gravity ve tor terms seemed to have aused undue problems in the design. These should be in luded before designing the ontrollers , in order to provide some feedforward to help to minimise sideslip during rolling manoeuvres. Additionally the absen e of velo ity ve tor rolling meant that the the

- ontroller

inherently in luded

an equivalent term to suppress sideslip, and may have resulted in higher gains than ne essary. The lateral design was somewhat unusual in that it did not in lude a yaw rate feedba k term. The resulting ontroller's total order, with 35 states (longitudinal and lateral/dire tional), is far higher than might be expe ted for this type of ontrol problem (e.g. a lassi al design would be about 10th order). It is not lear what benets this large number of additional states oer, in terms of improved robustness and/or performan e.

Real time exe ution is likely to be a problem

and therefore further order redu tion is onsidered to be essential. In terms of erti ation, the method results in bla k box ontrollers whi h la k the visibility whi h is essential for ight erti ation purposes. For this parti ular design entry, the ontroller had some visibility, in that tables of

ontroller eigenvalues and an array of ontroller frequen y response plots were in luded (the original design had some eigenvalues whi h ould not be realised within the HIRM's assumed 80 Hertz digital omputing design onstraint; these were later designed out). A further impli ation of the high order ontrollers is that gain s heduling to over a wider envelope may also lead to timing di ulties for real-time implementation, although it is a knowledged that pro essing power is always improving and therefore, although this is a valid on ern it may not be justied. The robustness of the design appears to be good, whi h is to be expe ted for this method.

It is not lear how mu h of the a hieved robustness is due

to the high order of the ontroller and what would be lost if the order were redu ed. The performan e of the design also looks to be good, with indi ations of good handling qualities. The ontrol surfa e dee tions used for manoeuvring are as might be expe ted but the symmetri al anard, rudder and throttle RMS values due to moderate turbulen e seem to be high (note that a ben hmark value is not available for omparison purposes). Overall this was a very good eort, with a high level of omplian e with all aspe ts of the HIRM design hallenge do umentation.

A -Synthesis Approa h (MS-31) This was a further design entry (Chapter 30) on the

-synthesis loop shaping

approa h and not surprisingly, some of the ndings from the evaluations were therefore similar to those dis ussed above. As noted from the previous entry, the designer would need to have an up-to-

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date familiarity with the mathemati s involved, in order to be able to gain an understanding of the te hnique. For the design there is a step-by-step pro ess, although a detailed understanding appears to be ne essary for su

essful sele tion of weighting fun tions. It should however, be possible for engineers who

an work in terms of magnitudes and frequen ies to arry out a

-synthesis

design, without knowing the internal details of the method. For this method to gain a

eptan e from pra tising ight ontrol engineers, a visible and intuitive explanation of the te hnique is essential. The method was judged to be a bla k box approa h, whi h would undoubtedly ause di ulties from a erti ation point of view due to its la k of visibility. Some visibility would be gained if the state-spa e ontrollers were split into lters, giving the transfer fun tions from the dierent feedba k signals to the ontroller output signals. Also in ommon with the previous entry, the design, for what is a fairly straightforward problem, has resulted in a signi antly higher order ontroller than would be expe ted using a more  onventional approa h  although it does redu e the requirement for air data s heduling (at least for the ight envelope onsidered). The issue of the di ulties in gain s heduling of the high order ontrollers was also raised during evaluation. As the results were for an an initial design, it is expe ted that the order of the ontroller ould be redu ed by using model redu tion te hniques. The results a hieved demonstrate satisfa tory hara teristi s for most ases but learly showed that the xed ontrollers would have benetted from gain s heduling, to normalise the ee ts of dynami pressure. Overall, it was onsidered that this was another good design entry with a high emphasis on the ight ontrol aspe ts, with a well balan ed set of on lusions. The strong point of this entry was the manner in whi h the HIRM design problem was interfa ed with the

-synthesis

pro edure and in parti -

ular, how the design requirements for robustness and handling qualities were a

ommodated.

H1 Loop Shaping Approa h (HI-32) This was a further entry (Chapter 29) in the 

H1 / ategory, whi h serves

to emphasise the urrent level of interest in this methodology. As with the two previous entries, some ommon ndings were made during the evaluations. It was again found that some of the mathemati s was too detailed for industrial ight ontrol engineers and ould not be easily understood by them, unless they had undertaken signi ant resear h in this area (or perhaps, had re ently graduated with a Masters degree). However, the authors laim that a detailed understanding of the mathemati s is not a pre-requisite for applying the method. It was evident that the method involved a good read-a ross from the singleinput single-output prin iples asso iated with lassi al ontrol. The user does need to understand singular values quite thoroughly though, in luding their physi al interpretation.

It was judged that this method is more di ult to

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grasp than some of the other te hniques available. Some interesting alternative metri s for measuring robustness were introdu ed. The method overs all frequen y domain requirements but annot handle time domain requirements dire tly, although this was not seen to be a problem, sin e these an usually be a hieved by lassi al ommand path lter shaping. An alternative approa h is to onvert the time domain handling requirements into frequen y domain spe i ations. The ontroller has a favourable level of visibility in terms of its overall ar hite ture but this be omes lost within the state-spa e blo ks.

As a result,

it is not possible to he k the ontrollers hara teristi s without resorting to

omputer-based te hniques. If the resulting high order ontrollers have to be s heduled with several parameters, there will be a signi ant omputing requirement.

The la k of the state-spa e blo k's visibility is likely to lead to

additional erti ation issues. This method should a hieve the most robust ontroller possible (although other methods might possibly have a hieved a similar design). From the few time histories presented, the performan e looks good, but the bandwidths have been maximised and there is no visibility that the design onstraint on the allowable feedba k gains had been satised. Indeed, some of the time histories show signi ant rate-limiting and spikes in the ontrol surfa e responses, whi h is indi ative of ex essive high frequen y gains. This was a good eort overall, with some good rationale and assumptions made for handling the ight ontrol aspe ts. This helps the reader understand the ar hite ture hosen.

34.3.5 Dis ussion and lessons learned HIRM Design Challenge: Problem Denition It is rstly important to note that the HIRM design hallenge poses a relatively straightforward task, ompared with urrent industrial design problems, parti ularly for air raft with high levels of inherent instability (the HIRM is approximately neutrally stable). Additionally, the HIRM being ee tively a low Ma h number subsoni air raft, did not require onsideration of air ompressibility ee ts. However, it is onsidered that given the limited time available to the design teams, the problem as dened was suitable for its purpose. Indeed, the HIRM design hallenge has posed a more realisti problem than that whi h might usually be onsidered for ight ontrol resear h and has provided a good introdu tion to the ight ontrol design problem for the organisations whi h had no previous involvement in this area; it has given them a basis, and hopefully the motivation, for arrying out further work on ight ontrol. The HIRM problem denition was probably too ompli ated in some areas and la king enough detail in others. In parti ular, a wider ight envelope would have enfor ed gain s heduling to show the true omplexity of some of the ontrollers proposed. The air raft model should have in luded the ee ts of moving the air raft's entre of gravity fore and aft, sin e this would have provided a good robustness test with an extremely simple physi al interpretation.

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The HIRM design problem was, to some extent, an unknown element for the designers and the evaluators.

It was based on a DRA drop model high

in iden e aerodynami dataset, augmented with representative FCS hardware assumptions and design riteria provided by BAe-MA. Sin e the vehi le was not a known air raft, no ben hmark design existed, against whi h relative merits of alternative designs ould be assessed. With hindsight, it would have been preferable to have based the exer ise on a known air raft.

Evaluation Questionnaire  HIRM Experien e The use of the questionnaire for extra ting relevant information was su essful, sin e mu h information was gathered in a onsistent format. This made evaluation relatively easy and provided a good basis for the information presented in this hapter. From the responses to some of the questions, the questionnaire

ould have been improved slightly; it would have been bene ial to arry out more detailed advan e testing of the questionnaire, had proje t times ales been less demanding.

HIRM Design Challenge: Evaluation of Results Overall, the results and omments produ ed by independent evaluations showed good agreement; without wishing to repeat the details from the previous se tions, some more general points regarding the results were made by the evaluators. At the time when the HIRM Evaluation Team was a tive, the quantitative Automati Evaluation results were not available and any omparisons made are therefore, mainly of a qualitative nature. Very good agreement was a hieved for the level of understanding and learning urve for su

essful appli ation of the methods, the handling of design requirements by the methods and the resulting design omplexity. All evaluators, in luding the designers (as self-evaluators), had good experien e in these areas. Greater variations in evaluation results were seen for estimated re-design eort, ontrol law stru tures whi h ould be a

ommodated by the method,

ontroller visibility and ontroller implementation.

The variations in ratings

were mainly asso iated with variations in experien e of the evaluators and designers. Greatest variation in evaluation results was seen for ight learan e and quali ation aspe ts, where the evaluators' experien e varied onsiderably and the designers' experien e was minimal. Overall, the balan e of the design entries was heavily on entrated around

H1 / and this led to some dupli ation of eort and repetition of results. With

hindsight, it would have been preferable to limit the number of design entries

in ea h ategory, to en ourage a wider assessment of the te hniques available. It was learly established, that for ee tive design of ight ontrol laws, the designer needs to have, as an absolute minimum, a good understanding of ight me hani s and ight ontrol, in order to establish a suitable ontrol laws ar hite ture. An inappropriate layout will inevitably lead to a poor design and the benets due to using any design te hnique are likely to be devalued

600

or even lost in the pro ess. The ontroller design te hnique itself was seen as a small, but signi ant part of the total design pro ess, when we take into a

ount all the nonlinear aspe ts of ight ontrol whi h have to be addressed. Some of the designs suered from a la k of knowledge of ight me hani s and in parti ular, gravity ve tor ompensation, velo ity ve tor rolling and dynami pressure ee ts were not always in luded. Additional signi ant fa tors whi h required due onsideration are: inertial oupling, integrator onditioning, rate limiting and angle of atta k and airframe loading limiting fun tions. For all the methods it was judged that any minor re-design should be straightforward, provided that established tools, weighting fun tions and/or inter onne tion stru ture ould be used.

However, for any major re-design,

perhaps due to a signi ant hange in air raft hara teristi s, re-design would be more di ult; for example, if new weighting fun tions have to be determined. The level of eort required will, in general, be parti ular to the design problem. In pra ti e, robustness has to be demonstrated for all possibilities, in order to a hieve ight erti ation. For the so- alled bla k box ontrollers this was seen to be a potential problem.

Additionally, it was noted that the urrent

pro edures are based on past and urrent methods and would probably need to be developed in order to be ompatible with some of the proposed methods. All the designs whi h have been produ ed for the HIRM have a hieved some level of su

ess, but from the results available, more work would be needed to develop these to a ight-worthy standard; this to be expe ted from an initial design. Further work would also be required to enable relative levels of a hievement to be reliably judged  ea h design would need to be fully optimised with respe t to the design riteria. The evaluation exer ise would have benetted from a

-analysis for omparing the robustness of the ontrollers produ ed,

parti ularly for the designs with multiple feedba k loops. Although the robustness tests that were arried out were onsidered to be adequate, a

-analysis test

would have been simple and onsistent; the other

tests would have been used as supporting information, to aid interpretation. To a hieve the obje tives of the robust ight ontrol design hallenge, it was of vital importan e that results showing good performan e were presented. Despite a good design eort by very apable design teams, some of the results were onsidered to be of an a ademi  nature and were not onvin ing enough for the aeronauti al industry, and tended to raise many unanswered questions. In many ases, the visibility of what had really been a hieved did not be ome apparent until the nonlinear time histories were presented and in some ases, these did not always support the linear robustness results (possibly due to rate limiting ee ts). Finally, in fairness to the design teams, they did not really have the time to arry out a omplete design of ontrol laws and some were starting with a very limited knowledge of ight ontrol. As noted earlier in des ribing the evaluation pro ess, the su

essful design of ontrol laws is dependent on many fa tors and owes mu h to the designers, their working environment and the pro edures used to provide a framework for the design method used. It is onsidered that, given the right onditions, most

601

(or even all) the methods an be used su

essfully. The results from the HIRM design hallenge have onrmed that the methods an all be made to work, with ea h method having its own strengths and weaknesses. The la k of a ben hmark design for the HIRM did not allow any benets from the various design methodologies to be quantied.

However, from the

method des riptions and the results presented, the benets (over a lassi al design approa h) are asso iated with establishing a logi al framework for managing the linear design, whi h expli itly in ludes both design spe i ations and modelling un ertainties. A further advantage is oered by using optimisation algorithms to establish an e ient route to a hieving a design. Alternative measures of robustness were proposed and these should be bene ial over lassi al measures, provided they an be alibrated in terms of physi al hara teristi s. Disadvantages observed were asso iated with design visibility and omplexity, parti ularly if the ontrollers were of ex essively high order and presented in state-spa e format. It is onsidered that these aspe ts ould be addressed by improved presentation and do umentation of the ontrollers to provide total visibility and by further developments in model redu tion te hniques to redu e

omplexity.

General Dis ussion The ight ontrol resear h ommunity that is developing and applying new methods to air raft models is relatively large, ompared to the numbers of industrial pra titioners or resear h s ientists a tively designing ontrol laws for implementation and ight testing. Arguably there are many design te hniques

hasing relatively few real-life ight ontrol appli ations. It is re ognised however, that the te hniques have potential value for an extremely wide range of

ontrol engineering appli ations and development should therefore be en ouraged; ight ontrol provides a relevant, interesting and hallenging platform for the development of su h te hniques. If a new design te hnique is to be a

epted by any industrial ight ontrol systems design ommunity, it is essential that it an be made to be ompatible with their existing design pro ess (whi h itself is ontinually being reviewed and developed) and that pra ti al aspe ts, su h as implementation in the air raft's ight ontrol omputer, have not been ignored. If the new method annot be a

ommodated easily, then there will inevitably be low interest and a natural relu tan e to adopt the method. From the individual designer's viewpoint, he will tend to use the method with whi h he feels omfortable and an a hieve his obje tives, within the available times ale.

Usually, this results in the designer using the method

with whi h he has most familiarity and any hange in approa h is naturally di ult (perhaps seen as a risk), parti ularly if a high level of su

ess has been previously a hieved. For the individual, a hange an only really be justied if a satisfa tory design is not possible with his existing approa h (although this

ould, of ourse, be due to design problem physi al limitations) or if a qui ker and/or more ost-ee tive means of a hieving an a

eptably similar design an

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be demonstrated. For an organisation, there may be dierent reasons to hange to a dierent method su h as, the lower level of skill needed to use some of the newer methods due to their inherent automation (for example, brought about by optimisation). For su h methods, the designers need to be aware that any automation, whi h has been introdu ed into the design pro ess, potentially removes them from the design loop, possibly at the expense of losing some design insight (e.g. they might lose the feel for any design trade-os, if the omputer has performed this a tivity for them). Irrespe tive of the method, su h automation needs to be introdu ed arefully, in order to provide the designer with a summary of what route has been taken to a hieve the design, and the results at ea h stage (i.e. an audit trail). Engineering skills must not be sa ri ed in the pro ess; it is essential that the ontrol law designer retains authority and that this is not given to the omputer and design tools used for the design. An aim of the GARTEUR A tion Group was to provide greater awareness in industry of the use of the new methods. Although this has been a hieved, it has been a two-way ex hange, with mu h useful information owing from industry ba k into the external resear h ommunity. This has highlighted some of the pra ti al aspe ts that industry has to onsider, beyond robustness and performan e. This should enable the resear hers to expand their urrent design drivers, to onsider the pra ti alities of implementation and ight erti ation. Quite learly, ollaboration and ex hange of information is the best way to fo us method developments, if industrial appli ation is the target. Ultimately, the only way for an industrial organisation to truly determine the benets of any method in relation to their design problems, is to arry out a detailed assessment using suitably experien ed personnel. Su h hands-on experien e will allow the benets to be evaluated in a ontrolled environment. The HIRM design hallenge has indi ated what might be expe ted from the te hniques and should en ourage greater interest in the range of methods available.

34.3.6 Re ommendations for future work For ontrol methods development, it is important that the benets, whi h an be attributed to the use of a parti ular design te hnique, are demonstrated and quantied in a fair and meaningful manner.

Therefore resear hers need

to make every eort to demonstrate the benets of the new te hniques, in order to promote their methodology within industry. It is essential that good visibility and physi al links are established to allow theoreti al measures to be applied and related to ight ontrol problems, in order to provide engineering insight and results that are relevant and easily interpreted by the aeronauti al engineer.

In general, the status of many of the available tools needs to be

elevated from resear h tools to engineering tools whi h interfa e with the industrial design pro ess and whi h meet the appropriate software standards. In terms of spe i developments of the methods, it is onsidered that further developments in the following areas would be of benet:

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Controller visualisation te hniques. Controller order redu tion te hniques. Means of measuring design method benets. Development and alibration of robustness measures. Nonlinear analysis development (e.g. to over rate limiting ee ts).

To omplement the methods development work, industry needs to prepare further hallenging and realisti ben hmark problems, preferably based on existing air raft, to enable the ight ontrol resear hers to address genuine industrial appli ations. There needs to be more dis ussion between resear h design teams and industry to ensure that design requirements and onstraints are fully understood and in luded in design studies. A follow-on GARTEUR a tivity on the ight ontrol subje t, possibly using the HIRM, should be onsidered. The team whi h has been established on the

urrent exer ise, has shown that it an work well together with a high level of ommitment.

Industry has proposed some interesting and relevant topi s

for possible further studies. A GARTEUR exploratory group now needs to be established, in order to determine the level of interest in greater detail.

A knowledgement This hapter would not have been possible without the support of the HIRM Evaluation Team.

Design evaluations were arried out by the following per-

sonnel: Mike Walker (BAe-MA), Jonathan Irving (BAe-MA), Steve Barratt (BAe-MA), Karin Ståhl-Gunnarsson (SAAB-MA), Robert Hillgren (SAABMA), Per-Olov Elg rona (SAAB-MA), Kenneth Eriksson (SAAB-MA), Lars Rundquist (SAAB-MA) and George Game (BAe-D). Finally, the work of three other HIRM design teams is a knowledged  led by Alex Smerlas (ULES) for HI-33, Aldo Tonon (Alenia) for CC-24 and Jürgen A kermann (DLR) for MO-27.

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35.

Another View on the Design

Challenge A hievements

Georg Grübel Abstra t:

1

The value of the Design Challenge as a atalyst for

further advan ing robust ontrol theory, and for stratifying the ontrol design pro ess, is reviewed. Based on the RCAM robust ontrol design-entries data base, a need is re ognized for improved robust synthesis tuning, and for a post-design stability-robustness assessment based on linear (LFT) system models, whi h are automati ally generated from parameterized nonlinear air raft dynami s models.

35.1 Introdu tion A valuable a hievement of the GARTEUR Design Challenge on Robust Flight Control is a oherent data base of air raft ight ontrol designs, gained by dierent design teams using dierent design methods. Although this represents no omplete overage of state-of-the-art ontrol design methods, - notably the QFT method [116℄ is missing -, the ompiled results and the experien e gained in the design pro ess, allow some general ndings by asking:



Is ight ontrol relevant to robust ontrol theory? Meaning, does ight

ontrol impose a hallenge in further developments of robust ontrol theory, algorithms, and software tools?



Is robust ontrol theory relevant to ight ontrol design methodology? Meaning, does robust ontrol theory stratify a 'best-pra ti e' design pro ess?

A view on these two aspe ts may be seen as a methods-oriented evaluation of the Design Challenge a hievements, whi h is omplementary to the industrial view of the previous hapter. The ndings are partly based on an independent post-design stability robustness assessment of the nal RCAM design entries. However, it is believed that the same arguments hold for the HIRM ben hmark as well. 1

DLR German Aerospa e Resear h Establishment, Institute for Roboti s and System

Dynami s, Control Design Engineering, D-82234 Wessling, E-mail: georg.gruebeldlr.de

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35.2 Relevan e to Robust Control Theory A

ording to industrial evaluation, all nal design entries ould handle, - better or worse -, the posed ight- ontrol ben hmark requirements. The design entries dier in their ontrol stru tures.

They use either a

'ight-me hani s dened' ontrol stru ture, or an 'analyti ally dened' ontrol stru ture (e.g.

state feedba k), or a 'pilot-dened' nonlinear-gain feed-

ba k stru ture whi h results from a rule base of desired ight behaviour within a fuzzy-logi ontrol approa h.

The design methods dier in the way, how

parameter un ertainties are modelled in the design synthesis pro ess: either parameter un ertainty is dire tly modelled as a parameter interval (MS-19,

2

LY-14) , or parameter un ertainty is dealt with in a multi-model approa h (MO-16, EA-12), or the ee ts of parameter un ertainties are redu ed to plant gain- and phase intervals or to high-frequen y onstraints on sensitivity fun tions (MS-11, EA-22, HI-21), or there is no un ertainty modelling at all (FL-15, CC-13, MF-25). If a ontrol design method is hara terized to be a robust ontrol design method by the fa t that synthesis model un ertainties are expli itly dealt with in the ontrol-synthesis formalism, some methods (FL-15, CC-13, and MF-25) do not qualify as 'robust' methods, although they have been su

essfully applied in the Design Challenge. On the other hand, expli it un ertainty modelling in a suited synthesis formalism does not lead to a robust design per se, without a skilful synthesis-parameter tuning.

This was experien ed by some design

teams. Flight ontrol laws are quite omplex. For reasons of ' ontroller visibility', they are usually designed sequentially for longitudinal/lateral and inner/outer loop ontrol, possibly applying dierent ontrol stru tures and dierent synthesis methods for the dierent ontroller loops. For example, a lassi al inner-loop feedba k has been ombined with a fuzzy-logi outer-loop design. Or, a statefeedba k longitudinal ontroller has been ombined with a 'ight-me hani s dened' lateral ontroller with wash-out lters et . Hen e, un ertainty modelling whi h is used in a parti ular synthesis step, may not be adequate and general enough for an overall robustness assessment. Rather, a synthesis-independent

'post-design' stability-robustness assessment is required. Flight ontrol, in this respe t, poses a hallenge for robust ontrol theory be ause of the rather omplex operational deviations and dynami s un ertainties to be dealt with. 'Robust ontrol tools' [32℄ are needed to readily apply numeri ally e ient robustness analysis te hniques.

Parameterized LFT (Linear Fra tional

Transformation) models are a general system-des ription form for des ribing parameter-un ertain linear system models, and thereby an be used as a standard form for whi h su h tools should be appli able. Be ause air raft dynami s models are highly parameterized, there is a need for an automated generation of

LFT models from the given physi al un ertainty des ription. Proper CACSD tools (see next se tion) an redu e this tedious task from days to minutes; but there remain theoreti al and omputational issues to be solved, for example 2

abbreviations see Table 35.1

606

the automated derivation of minimum blo k-size LFT des riptions. An independent post-design robustness assessment has been performed [158℄ for the RCAM ben hmark, having just the ontroller Matlab S-fun tions of ten design entries available no matter how they were synthesised. For all (linearizable) ontrollers and over all un ertainty parameters, two approa hes of robustness analysis have been arried out. A

 stru tured singular value robustness

analysis was performed for the given nominal airspeed.

This approa h with

symboli linearization of the air raft equations did not allow to over airspeed deviations properly. Therefore, in addition, a worst- ase un ertainty-parameter sear h for eigenvalue relative stability (minimum damping of all eigenvalues) has been performed within the whole range of the ight envelope. The worst ase eigenvalue analysis was omputed by a numeri al optimization loop over the un ertain parameters, involving numeri al trimming of the air raft and system linearization. The results are ompiled in Table 35.1. Table 35.1 shows, that not all ontrollers are stability-robust for all possible un ertainty-parameter ombinations within the ight envelope. In part, this is due to the fa t that this post-design robustness analysis has been performed for

3

a larger deviation in verti al enter-of-gravity position

4 deviations ,

and also for airspeed

whi h both had not been spe ied as a design requirement. Hen e

the designers must not be blamed for any orresponding ontroller de ien ies.

But, most results in Table 35.1 demonstrate, that good overall robustness is indeed possible. Table 35.1 shows that low-order ontrollers may exhibit better stabilityrobustness properties than ontrollers with higher-order dynami s. mainly a question of tuning the synthesis parameters (e.g.

This is

weighting oe-

ients and weighting lters, or ontroller parameters) in order to make best use of the intrinsi apabilities of the orresponding synthesis method, or the hosen ontroller stru ture. Hen e, systemati tuning methods for robust ontrol

synthesis should be developed. Su h synthesis-tuning methods are likely to be based on intera tive numeri al optimization su h as the Methodof Inequalities (MOI) [255℄, or the method of Multi-Obje tive Parameter Synthesis (MOPS) 2 used in 16. The general need for optimization-based synthesis tuning in air raft ight ontrol design is also re ognized in [253℄. Sin e ontroller-order redu tion is part of the synthesis pro edure, 'optimal' tuning is also required for this task.

Synthesizing the lowest-order ro-

bust ontroller for a given appli ation is a question of multi-obje tive performan e/robustness ompromising. It is still an issue in air raft ight ontrol, due to the limited throughput of the available Ele troni Flight Control System

omputers. This hallenge was not su iently emphasized by most of the design teams. The large range of a hieved ontroller-dynami s order is interesting to note in Table 35.1. 3

A modelling error in dealing with the verti al enter-of-gravity position was dis overed

too late to be taken are of properly by the designers. 4

The requirement to he k robustness w.r.t. deviations from nominal airspeed has been

added late in the proje t.

607

35.3 Relevan e to the Control Design Pro ess Control design is an (iterative) a tivity within the triangle of the plant physi al system, design goals, and the algorithmi ontroller apabilities. This is visualized by Figure 35.1. The required ontrol theory for the inner design-triangle is supported by CACSD (Computer Aided Control System Design) algorithms, tools, and frameworks, e.g. [97℄. Commonly available toolboxes support (linear) plant analysis and ontrol synthesis. Online data-base support is required for a transparent and intera tive logging of a multi- riteria set up, whi h is the formal link between design goals and the analysis and synthesis a tivities. The CACSD design triangle is substantiated by ight- ontrol system evaluation and validation. This is supported by nonlinear simulation experimenting, in luding rapid prototyping with iron-bird and in-ight simulation, as well as hardware-in-the-loop man/ma hine interfa e validation by using o kpit ightsimulators.

DESIGN GOALS

tun sis syn

ics

am

the

dyn

control synthesis control code generation

plant analysis

s

tor

ad

ica

ap

ind

CACSD

ting

ting

lua

multi criteria set-up

eva

ers

mathematical modelling of performance & robustness

(linear) synthesis model

simulation experimenting

physical system- & uncertainty modelling

nonlinear closed-loop system models rapid prototyping

H/W-in-the loop

ALGOR. CONTROLLER

PHYSICAL PLANT

Figure 35.1: The ontrol-design inner and outer 'a tivity triangle'.

Commonly, ontrol-dynami s design is seen as an a tivity on two strata, namely (linear) plant analysis/ ontrol synthesis, and design validation by (nonlinear) simulation. Robust ontrol theory makes a third design stratum evident.

Mathemati al modelling of robustness riteria in addition to mathemati ally dened performan e riteria, leads to a multi- riteria problem, formalized by a multi- riteria set up, see Figure 35.1.

Using the stratum of a multi- riteria

608

set up allows omputer-aided synthesis tuning on the basis of an expli it omparison of a hieved results against a set of design requirements. This is more systemati than manual tuning, having multi- riteria requirements in one's mind only. In 16 it is shown how a multi- riteria set up an be dire tly used for adapting synthesis tuners in an automated way. It is also shown there, how most of the design goals an be mapped onto 're-usable' standard mathemati al

riteria des riptions to be evaluated via standard dynami s indi ators, su h as eigenvalues, time responses, and frequen y responses. The basi requirement of robust ontrol, namely, that real-world design un ertainties are to be expli itly taken into a

ount, yields the requirement that the inner design-triangle of Figure 35.1 is to be expli itly linked to the outer 'real-world' triangle. This impa ts the ight ontrol design pro ess by the need for modelling both the physi s and the un ertainty behaviour. For both kinds of modelling, the use of standard forms is advisable as a 'best-pra ti e approa h' be ause the 're-usability' of standard forms makes standardized simulation-, analysis- and synthesis tools readily appli able. Two kinds of standard model forms for des ribing a parameter-un ertain physi al system are demonstrated in the design hallenge. These are, (i), reusable omponent lass libraries for nonlinear, physi ally parameterized obje toriented air raft-system dynami s modelling [145, 177℄, and, (ii), linear, physi ally parameterized LFT models whi h are (semi-)automati ally derived from (i). The basis for that is symboli /numeri pro essing of omponent-equation type system models.

bodyfixed

Trafo veh.carried

experimental Trafo bodyfixed

airspeed

bodyLong

COG

COG

const. gravity

wind

u

atmos

RCAM

air wind

HIRM

Engine

body6DOF

kinetic

gust

Earth

aeroRCAM

1dim gravity Earth

engine

const.

1D

atmosphere

atmos1D

Figure 35.2: The RCAM/HIRM air raft dynami s omponent library [145, 177℄ for graphi al 'pi k & drag' system-dynami s aggregation. The usefulness of formalized symboli system modelling, Figure 35.3, is demonstrated by the fa t, that any nonlinear ight dynami s simulation ode used by the RCAM and HIRM Design Challenge teams (i.e. Matlab mex ode for Simulink and neutral DSblo k Fortran- and C- ode) was derived by automati ode generation from a 'single-sour e' symboli model, whi h itself is soft programmed in an intera tive 'pi k and drag' system modelling environment using respe tive omponent- lass libraries, Figure 35.2. From the symboli nonlinear air raft dynami s model, a standard LFT model an be automati ally derived [245℄ via symboli linearization and us-

609

graphical ’pick & drag’ system aggregation

physical system model

component class libraries

mathematical (symbolic) model

Matlab / Simulink

automatically generated from physical system objects aggregation & class libraries via

S-function (cmex)

modelling environment Dymola

Simulation - model

symbolic linearisation

LFT - model

neutral DSblock Fortran-, C-Code

.

via Maple

automatically generated

[ x = A (p) x + B (p) u ]

by PUM

Figure 35.3: Formalized symboli /numeri system modelling for parameterized system analysis and simulation.

ing the Matlab toolbox PUM (Parametri Un ertainty Modelling) [239℄. This was demonstrated for the RCAM ben hmark and served for the independent post-design



analysis as referred to in the previous se tion. Sin e an LFT-

model is a general type of model des ription, a system-theoreti onne tion of two LFT system models is again an LFT system model. This allows LFT models derived from signal-ow diagrams [261℄ to be ombined with LFT models derived from parameterized physi al models as des ribed above. In on lusion, robust ontrol theory per se, does not yield basi ally new air raft ight ontrol system ar hite tures. In fa t, as demonstrated in the Design Challenge, su ient experien e in ight me hani s and ight ontrol allows you to design quite robust ontrol laws on the basis of any 'method you know best', - provided that enough engineering time is spent for proper robustness tuning and re-iterative post-design robustness analysis. The required engineering time for robustness tuning and post-design robustness analysis, however, an be signi antly redu ed by a omputer-automated design pro ess whi h is stratied taking into a

ount the basi issues of robust ontrol.

Essentially, these are

the multi- riteria nature of synthesis-parameter tuning, and the availability of numeri ally reliable algorithms for stability robustness analysis.

610

type-

dynami



V =

entry no.

order

up bnd

unstable

Vnom

min = nom

min

due to:

V  unstable due to:

FL-15

5 (nl)

0.44

EA-12

9

0.36

0.27

0.47

EA-22

9

0.39

0.15

0.23

CC-13

11

0.51

0.04

0.17

MO-16

12

0.35

0.21

0.52

MS-19

35

1.36

-0.13

-0.26

MF-25

36

0.65

HI-21

36

1.53

LY-14

39

0.57

0.11

0.19

MS-11

62

0.49

-0.05

-0.27

1)

Tdel

2)

0.05

0.17

-0.18

-0.98

1) 2) v

3)

Type: FL = Fuzzy Logi , MM = Modal Multi-Model Synthesis, EA = Eigenstru ture Assignment, CC = Classi al Control, MO = MOPS, MS =



Synthesis,

1 , LY = Lyapunov

MF = Model Following, HI = H

Changing parameters: [m; delx; delz; Tdel; v℄ mass: m

.o.g.-x: delx

.o.g.-z: delz time delay: Tdel air speed: v

gain/phase instability parameters:

 m  150 000 kg 0.15  delx  0.31 0.0  delz  0.21 0.05s  Tdel  0.1s (padé approx.) 1.23 V (m)Stall  V  90 m/s

100 000 kg

Vnom 0

= 80 m/s

 k
k1  0.01


added in 

(

1) [100; 0.15; 0.21; 0.1; 90.0℄ 2) [150; 0.31; 0.21; 0.1; 71.8℄ 3) [100; 0.15; 0.21; 0.1; 58.2℄

analysis)

Table 35.1: Results of RCAM independent post-design robustness assessment.

611

36.

Con luding Remarks

Samir Bennani , Jean-François Magni and Jan Terlouw 1

2

3

The main purpose of the Design Challenge was to demonstrate, by means of design and omputer simulations, how twelve advan ed te hniques an be applied to design robust ontrollers for two fairly realisti ight ontrol ben hmark problems. The hosen level of omplexity of the RCAM and HIRM problem denitions was a trade-o between industrial relevan y and feasibility for the design teams to nd solutions in a limited period of time.

In order not to

hamper the teams in showing the potentialities of the applied methods, the idea was to leave them as mu h freedom as possible: freedom in the trade-o between a hieving good robustness and performan e and freedom in the hoi e of the ar hite ture. This has led to designs whi h speak for themselves and we have not ompared the results here in any detail. We believe that this book has signi ant tutorial value, while the two evaluation hapters give a broad and dierentiated view on the potential of the methods. To some extent, the Design Challenge has proven that modern te hniques

an be used to design ontrollers for realisti problems. Additionally, it has onrmed that requirements for industrial appli ation of new te hniques are quite severe. From an industrial point of view, desirable features of any te hnique an be assumed to be: transparen y, simpli ity, quality, a

ura y, delity, reliability, implementability, predi tability and generality. Even though the presented methods have mu h potential in the eld of improved robustness, better performan e, de- oupled ontrol and simpli ation of the design pro ess, some of them do not yet have the maturity required for industrialisation.

Even ma-

ture methods need to be arefully integrated into the industrial design pro ess to fully address the omplexities asso iated with modern air raft. One of the main problems en ountered remains the omplexity of the proposed ontrol solutions, whi h is partly driven by the hoi e of the ontrol ar hite ture. This is a ru ial a tivity in the design pro ess, whi h is not yet taken into a

ount su iently by the theoreti al ommunity.

1

Delft University of Te hnology, Fa ulty of Aerospa e Engineering, Stability and Control

Group, Kluyverweg 1, 2629 HS Delft, The Netherlands, E-mail: s.bennanilr.tudelft.nl 2

CERT ONERA, Département d'études et Re her hes en Automatique, BP 4025, F31055

Toulouse Cedex, Fran e 3

National Aerospa e Laboratory NLR, Flight Me hani s Department, Anthony Fokker-

weg 2, 1059 CM Amsterdam, The Netherlands.

612

Future resear h Several software tools and do uments have been developed by GARTEUR A tion Group FM(AG08) and used su

essfully throughout the proje t:



Manuals des ribing the RCAM and HIRM ben hmark denitions, in luding the air raft dynami s models, the environmental models (atmospheri turbulen e, windshear, et .), a traje tory generator for RCAM, the design requirements, an evaluation questionnaire and guidelines for the preparation of design reports.



Matlab/Simulink RCAM and HIRM design environments and evaluation tools, with whi h designs an be ompared with the requirements.

 

A ommon nomen lature. Twenty-one RCAM and HIRM design reports, whi h ontain more details than the design hapters of Parts II and III.

To en ourage broader parti ipation in the Design Challenge problems, the RCAM manual [145℄, the a

ompanying software and most of the design reports are made available and an be obtained from the NLR homepage

(http:\\www.nlr.nl).

Information about HIRM an be found at this omputer lo ation as well. The information listed above an be used for edu ational, resear h and appli ation purposes at universities, resear h establishments and in industry. It may serve as an introdu tion for people who are interested in how parti ular advan ed ontrol te hniques work.

Furthermore, in general, the ben hmark

problems an be used for a riti al review of design methods. Improvements and extensions of the ben hmarks are within rea h.

For

HIRM, for example, it would be interesting to allow a variation in the lo ation of the entre of gravity to introdu e relaxed air raft stability.

Design spe i-

 ations for both problems an be further improved and it may be advisable to propose ontroller ar hite tures for a next Design Challenge.

This would

ease the omparison of design methods. Extensions of the ight envelopes for RCAM and HIRM would make gain s heduling ne essary in most ases.

It

would be interesting to see how advan ed te hniques ope with this additional

omplexity. It is likely that in Europe aeronauti al resear h and te hnology development proje ts will be arried out more and more on a multi-national s ale.

The

work arried out by GARTEUR A tion Group FM(AG08) has benetted from two positive aspe ts. Firstly, results were obtained in only two years be ause twenty-three organisations worked very well together, in parallel but as a team. Se ondly, the lose ollaboration between many people from industry and the s ienti ommunity seems to have a

elerated the pro ess of bringing ( ontrol) theory to pra ti e. It is re ommended to ombine these two aspe ts again in future European resear h proje ts.

613

Appendix A. Used Nomen lature

Anders Helmersson and Karin Ståhl Gunnarsson 1

2

A.1 Introdu tion This hapter denes the re ommended nomen lature to be used within FM(AG 08). In the hapter, nomen lature regarding oordinate frames, transformations between oordinate frames, variables des ribing the air raft as well as some mathemati al quantities is proposed. Generally, ve tor notation has not been used. The user of the hapter is free to dene ve tor notation if that is preferred. The alphanumeri des riptors are aimed to apply in software. Throughout this hapter apital letters are given, but lower ase letters an used if preferred. If the software is restri ted to only have short variable names, for instan e with only six letters, as in F77 standard Fortran, the proposed alphanumeri des riptors an be shortened in a suitable way. ISO units shall be used in al ulations, while industrial units, su h as degrees, Hz and dB, an given in presentations.

A.2 Denition of Referen e Frames This se tion deals with the denition of referen e frames. The referen e frames that are dened are the earth-xed referen e frame, the vehi le- arried verti al frame, the body-xed referen e frame, the wind axis referen e frame, the stability axes referen e frame and the measurement referen e frame. All referen e frames are right-handed orthogonal.

A.2.1 Earth-Fixed Referen e Frame FE , (OE XE YE ZE ) The Earth-xed referen e frame is dened a

ording to Figure A.1.

FE is an earth-surfa e frame, with origin OE near the vehi le. ZE is positive XE is positive north and YE is positive east.

downward,

A.2.2 Vehi le-Carried Verti al Frame FV , (OV XV YV ZV ) The vehi le- arried verti al frame is dened a

ording to Figure A.2. 1

Linköping University, S-581 83

2

Saab Military Air raft, S-581 88 Linköping, Sweden

Linköping, Sweden

614

YE XE OE

ZE

Figure A.1: The earth-xed referen e frame

YV

(East)

OR

OV

(North)

XV

C.G. g

Ref. point aerodata

ZV

Figure A.2: The vehi le- arried verti al frame

This frame is moving with the vehi le and is parallel to the earth-xed frame. The origin

OV

is atta hed to the vehi le at the enter of gravity.

positive verti ally downward along the g ve tor.

XV

ZV is YV

is positive north and

is positive east.

A.2.3 Body Axis Referen e Frame FB (OB XB YB ZB ) The body axis referen e frame is dened a

ording to Figure A.3.

ZB

The origin

OB

is at the vehi le's enter of gravity.

is positive downward and

YB

XB

is positive forward,

is positive to the right.

For the transformation from vehi le arried verti al frame to body axes, the vehi le- arried verti al frame is rst rotated by the heading angle rotation is by the pit h angle

. The next

 and the last rotation is by the roll angle . 615

YB

OR

OB

C.G.

Ref. point aerodata

XB

ZB

Figure A.3: The body axis referen e frame

A.2.4 Wind Axis Referen e Frame FW ( OW XW YW ZW ) The wind axis referen e frame is dened a

ording to Figure A.4. This frame has the origin

OW

atta hed to the vehi le at the enter of gravity.

positive pointing forward along the velo ity ve tor V. and

YW

ZW

XW

is

is positive downward,

is positive to the right.

For the transformation from body axis referen e frame to wind axis referen e frame, the body axis system is rst rotated by the angle angle

.

The denitions of angle of atta k,



and then by the

, and sideslip, , are:

w  = ar tan a ua v = ar sin a VA where

ua , va and wa are the x-, y- and z- omponents of the ve tor airspeed Va VA is total velo ity

expressed in body axes, and

p VA = (u2a + va2 + wa2 ):

A.2.5 Stability Axis Referen e Frame FS (OS XS YS ZS ) The stability axis referen e frame is dened a

ording to Figure A.5.

OS of this frame is at the enter of gravity of the vehi le. XS forward, ZS is positive downward and YS is positive to the right. origin

The

is positive

For the transformation from body axes to stability axes, the body axis system is rotated by the angle

.

616

YB

YW β

XB α

Ref. point aerodata

OR β

XW

OB O

W

V C.G.

ZB

α

ZW

Figure A.4: The wind axis referen e frame

A.2.6 Measurement Referen e Frame FM (OM XM YM ZM ) The measurement referen e frame is dened a

ording to Figure A.6. origin

OM

The

of this frame is at the leading edge of the mean aerodynami hord.

XM is positive pointing ba kwards, YM ZM is positive pointing up.

is positive pointing to the right and

A.3 Coordinate Transformations The transformations asso iated with single rotations about the three oordinate axes as shown in Figure A.7 are now dened.

2

1 0 0 R1 (X1 ) = 4 0 os X1 sin X1 0 sin X1 os X1 2

os X2 0 sin X2 R2 (X2 ) = 4 0 1 0 sin X2 0 os X2 2

R3 (X3 ) = 4

3 5

3 5

3

os X3 sin X3 0 sin X3 os X3 0 5 0 0 1 617

YB Y S XB

Ref. point aerodata

OR

α

OB O

XS

S

V C.G.

ZB

α

ZS

Figure A.5: The stability axis referen e frame ZM

XM

YM

OM

c

OR

Ref. point aerodata

Figure A.6: The measurement referen e frame

A.3.1 Vehi le- arried verti al frame to body axes RBV onsists of the sequen e of rotations RBV = R1 ()R2 ()R3 ( )

The transformation matrix

The result of multiplying the three matri es is

2

3

os  os

os  sin sin  RBV = 4sin  sin  os os  sin sin  sin  sin + os  os sin  os 5

os  sin  os +sin  sin os  sin  sin sin  os os  os 

A.3.2 Body Axis to Vehi le-Carried Verti al Frame RVB onsists of the sequen e of rotations RVB = R3 ( )R2 ( )R1 ( )

The transformation matrix

618

The result of multiplying the three matri es is

2

3

os  os os sin  sin  sin os sin sin  + os sin  os  RVB = 4sin os  os os  +sin sin  sin  sin sin  os  os sin 5 sin 

os  sin 

os  os Note that

T . RV B = RBV

A.3.3 Body Axes to Wind Axes The transformation matrix

RW B

onsists of the sequen e of rotations

RW B = R3 ( )R2 ( ) The resulting matrix is

2

RW B = 4

os  os sin sin  os

os  sin os sin  sin sin  0

os 

3 5

A.3.4 Body Axes to Stability Axes The transformation matrix

RSB

onsists of the sequen e of rotations

RSB = R2 ( ) The resulting matrix is

2

RSB = 4

3

os  0 sin  0 1 0 5 sin  0 os 

A.3.5 Wind Axes to Body Axes The transformation matrix

RBW

onsists of the sequen e of rotations

RBW = R2 ()R3 ( ) The resulting matrix is

3

2

os  os RBW = 4 sin sin  os Note that

os  sin sin 

os 0 5 sin  sin os 

T . RBW = RW B 619

A.3.6 Stability Axes to Body Axes The transformation matrix

RBS

onsists of the sequen e of rotations

RBS = R2 () The resulting matrix is

3

2

os  0 sin  RBS = 4 0 1 0 5 sin  0 os  Note that

T . RBS = RSB

xa3

xb3

xa1

xb1

xb3

xb2 X2

X1

xa3

O

xa2

O a)

b)

xa2

xb2

xb1 X3 xa1

O c)

Rotation about a) xa1 b) xa2 c) xa3

Figure A.7: The basi rotations

A.3.7 Quaternion Attitude Representation An alternative to the use of Euler angles for dening attitudes, is to use the quaternion representation. This is a four- omponent entity whi h enjoys some advantages ompared to the Euler angles: it is numeri ally e ient and it has no singularities.

620

The quaternion parameters

q = (q0 ; q1 ; q2 ; q3 )T full the following ondition

q02 + q12 + q22 + q32 = 1 and are related to the Euler angles by

q0 q1 q2 q3

= = = =

( os(=2) os(=2) os( (sin(=2) os(=2) os( ( os(=2) sin(=2) os( ( os(=2) os(=2) sin(

=2) + sin(=2) sin(=2) sin( =2) os(=2) sin(=2) sin( =2) + sin(=2) os(=2) sin( =2) sin(=2) sin(=2) os(

=2)) =2)) =2)) =2))

Whi hever sign is hosen in the above equations must be applied onsistently a ross all equations. The equations above an be used to al ulate the initial values of the quaternions parameters given the initial values of the Euler angles. The derivatives of the quaternions with respe t to time an be expressed as

2

q_0 6 q_1 6 4 q_2 q_3 where

3

32

2

q0 0 p q r 7 6 q1 6p 0 7 1 r q 76 7= 6 5 2 4 q r 0 p 5 4 q2 r q p 0 q3

p, q and r are the body-axes angular rates.

621

3 7 7 5

A.4 Aeronauti al Variable Symbols and Alphanumeri Des riptors A.4.1 Air raft Related Quantities Name

Symbol Alphanumeri Unit

Wingspan Mean aerodynami hord Air raft inertia tensor x body axis moment of inertia x-y body axis produ t of inertia x-z body axis produ t of inertia y body axis moment of inertia y-z body axis produ t of inertia z body axis moment of inertia

FB Radius of gyration, y dire tion in FB Radius of gyration, z dire tion in FB Centre of gravity x position in FB Centre of gravity y position in FB Centre of gravity z position in FB Radius of gyration, x dire tion in

Lo ation of aerodynami entre x position in

FB

Lo ation of aerodynami entre y position in

FB

Lo ation of aerodynami entre z position in

FB

b

 I Ix Ixy Ixz Iy Iyz Iz rx ry rz X g Y g Z g Xa

YCG

m

ZCG

m

XAC

m

Ya

YAC

m

Za

ZAC

m

LEN

m

MASS

kg

l m S W St lt

Generalized length Air raft total mass Wing planform area Vehi le weight Tail unit planform area Longitudinal distan e between the

B

m

CBAR

kgm2 kgm2 kgm2 kgm2 kgm2 kgm2 kgm2

IAC IX IXY IXZ IY IYZ IZ

m

RX

m

RY

m

RZ

m

XCG

m

S W

m2 N

STAIL

m2

LTAIL

m


r

DELR

m


x

DELX

m


y

DELY

m


z

DELZ

m

aerodynami entre of the wing and body and the aerodynami entre of the tail unit Displa ement of aerodynami entre from entre of gravity Displa ement of aerodynami entre from entre of gravity along x body axis Displa ement of aerodynami entre from entre of gravity along y body axis Displa ement of aerodynami entre from entre of gravity along z body axis

622

Name

Symbol Alphanumeri Unit

Downwash angle Angle from the thrust axis of engine to the x-y body axis plane Angle between x body-axis and prin ipal axis of inertia Angle from the proje tion of engine thrust

Fp ) onto the engine x-y plane to

 T

EPSILON

rad

EPSILONT

rad

p

EPSILONP

rad



XSI

rad



ETA

rad

ve tor (

the lo al x-axis

Fp ) to

Angle from engine thrust ve tor ( the engine x-y plane

A.4.2 Engine Quantities Name

Symbol Alphanumeri Unit

X position of entre of gravity of engine 1 in

FB

Y position of entre of gravity of engine 1 in

FB

Z position of entre of gravity of engine 1 in

FB

Engine thrust ve tor at entre of gravity X position of point of appli ation of thrust of engine 1 w.r.t. entre of gravity in

FB

Y position of point of appli ation of thrust of engine 1 w.r.t. entre of gravity in

FB

Z position of point of appli ation of thrust of engine 1 w.r.t. entre of gravity in

FB

Rotational inertia of the engine Angular momentum of engine rotor Mass of engine Moment from engine about the

entre of gravity

XCGE1

XCGE1

m

YCGE1

YCGE1

m

ZCGE1

ZCGE1

m

Fp XAT P 1

FP

N

XATP1

m

YAT P 1

YATP1

m

ZAT P 1

ZATP1

m

IE

Ie he mE TE TEg !E

Gyros opi torque from engine Engine angular velo ity

623

HE

kgm2 kgm2 =s

MASSE

kg

TE

Nm

TEG

Nm

OMEGAE

rad/s

A.4.3 Air Data Quantities Name

Symbol

Alphanumeri Unit G

m=s2

Speed of sound in air

A

m/s

Pressure

PRESSURE

g

A

eleration due to gravity

g = 9:80665 m2 =s

a p Ambient pressure pa Total pressure pt Dynami pressure q Impa t pressure q Reynolds number Re Reynolds number per unit length Re0 Ambient temperature T Total temperature Tt Coe ient of dynami vis osity  Density of air  Airspeed ve tor in FB Va x- oordinate of Va in FB ua y- oordinate of Va in FB va z- oordinate of Va in FB wa p Total airspeed VA = (u2a + va2 + wa2 ) VA Calibrated airspeed V True airspeed VT AS Equivalent airspeed Ve Ma h number M S ale of turbulen e Lu ; Lv ; Lw Mean square value of gust velo ity u v w

QC

N=m2 N=m2 N=m2 N=m2 N=m2

RE

-

REPRIME

1/m

TAMB

K

TTOT

K

PA PT QBAR

RHO

kg=ms kg=m3

VAVEC

m/s

UA

m/s

VA

m/s

CVISC

WA

m/s

VAIR

m/s

VC

knots

VTAS

knots

VEAS

knots

MACH

-

LU, LV, LW

m

SIGMAU

m

SIGMAV

m

SIGMAW

m

A.4.4 Energy Related Quantities Name

Symbol

Es Ps

Spe i energy Spe i power

624

Alphanumeri Unit ES

m

PS

m/s

A.4.5 Variables Name

Symbol Alphanumeri Unit

Altitude (relative to mean sea level, Roll rate Pit h rate Yaw rate Total inertial velo ity

FB Inertial velo ity in FB Inertial velo ity in FB

Inertial velo ity in

x dire tion y dire tion z dire tion

Angle of atta k Angle of sideslip Pit h angle Roll angle

h = z) h p q r V u v w   

Heading angle Bank angle (=aerodynami roll angle)

FE FE z position of entre of gravity in FE x position of entre of gravity in

y position of entre of gravity in

Quaternion parameter Quaternion parameter Quaternion parameter Quaternion parameter

FE Velo ity of wind in FE Velo ity of wind in FE Velo ity of wind in FB Velo ity of wind in FB Velo ity of wind in FB Velo ity of wind in

x dire tion y dire tion z dire tion x dire tion y dire tion z dire tion

625

 x y z q0 q1 q2 q3 WXE WY E WZE WXB WY B WZB

H

m

P

rad/s

Q

rad/s

R

rad/s

V

m/s

UB

m/s

VB

m/s

WB

m/s

ALPHA

rad

BETA

rad

THETA

rad

PHI

rad

PSI

rad

BANK

rad

X

m

Y

m

Z

m

Q0

-

Q1

-

Q2

-

Q3

-

WXE

m/s

WYE

m/s

WZE

m/s

WXE

m/s

WYE

m/s

WZE

m/s

A.4.6 Derivatives of Variables Name

Symbol Alphanumeri

h_ p_ q_ r_ V_

Time rate of hange of altitude Time rate of hange of roll rate Time rate of hange of pit h rate Time rate of hange of yaw rate Time rate of hange of total inertial velo ity Time rate of hange of velo ity Time rate of hange of velo ity Time rate of hange of

u_

UBDOT

m=s2

FB

y dire tion

v_

VBDOT

m=s2

FB

z dire tion

w_

WBDOT

m=s2

ALPHADOT

rad/s

BETADOT

rad/s

THETADOT

rad/s

PHIDOT

rad/s

PSIDOT

rad/s

BANKDOT

rad/s

XDOT

m/s

y_

YDOT

m/s

z_

ZDOT

m/s

Time rate of hange of angle of sideslip Time rate of hange of pit h angle Time rate of hange of roll angle Time rate of hange of heading angle Time rate of hange of bank angle Time rate of hange of x position of

FE

Time rate of hange of y position of

FE

Time rate of hange of z position of

entre of gravity in

VDOT

QDOT RDOT

x dire tion

Time rate of hange of angle of atta k

entre of gravity in

m/s

rad=s2 rad=s2 rad=s2 m=s2

PDOT

FB

velo ity

entre of gravity in

HDOT

Unit

FE

_ _ _ _ _ _ x_

q_0 q_1 q_2 q_3

Time rate of hange of quaternion Time rate of hange of quaternion Time rate of hange of quaternion Time rate of hange of quaternion

Q0DOT

-

Q1DOT

-

Q2DOT

-

Q3DOT

-

A.4.7 Flight Path Related Quantities Name

Symbol Alphanumeri

Flight path angle Time rate of hange of ight path angle Flight path a

eleration Tra k angle Time rate of hange of tra k angle Verti al a

eleration

626

_

  _ h

Unit

GAMMA

rad

GAMMADOT

rad/s

GAMMADDOT

rad=s2

CHI

rad

CHIDOT

rad/s

HDDOT

m=s2

A.4.8 Control Surfa e Dee tions Name

Symbol

ÆA ÆCD ÆCS ÆE ÆR ÆSB ÆT ÆT D ÆT S ÆT H 1 ÆT H 1 su tion

Aileron dee tion Dierential anard dee tion Symmetri anard dee tion Elevator dee tion Rudder dee tion Speed brake dee tion Tailplane dee tion Dierential taileron dee tion Symmetri taileron dee tion Throttle position of engine 1 Throttle position of engine 2 Su tion

Alphanumeri

Unit

DA

rad

DCD

rad

DCS

rad

DE

rad

DR

rad

DSB

rad

DT

rad

DTD

rad

DTS

rad

THROTTLE1

-

THROTTLE2

-

SUCTION

-

A.4.9 A

elerations and Load Fa tors Name

Symbol Alphanumeri

Normal a

eleration at entre of gravity in

FB

Normal a

eleration, not at entre of gravity, in

FB

x body axis a

elerometer output at

entre of gravity, in

FB

y body axis a

elerometer output at

entre of gravity, in

FB

z body axis a

elerometer output at

entre of gravity, in

FB

x body axis a

elerometer output, not at entre of gravity, in

FB

y body axis a

elerometer output, not at entre of gravity, in

FB

z body axis a

elerometer output, not

FB FB , n = mgZ

an

AN

ANI

m=s2

Unit

m=s2

an;i anx

ANX

m=s2

any

ANY

m=s2

anz

ANZ

m=s2

anx;i

ANXI

m=s2

any;i

ANYI

m=s2

anz;i

ANZI

m=s2

at entre of gravity, in

n Fx n FB , nx = mg x Fy Lateral load fa tor in FB , ny = n y mg Fz Normal load fa tor in FB , nz = n z mg

Load fa tor in

Longitudinal load fa tor in

627

N

g

NX

g

NY

g

NZ

g

A.4.10 For e Quantities Name Drag for e, positive ba kward, in

FW Lift for e, positive upward, in fW

FW

Side for e, positive right, in

Longitudinal for e, positive ba kward, in

FB

Lateral for e, positive right, in

FB

Normal for e, positive upward, in

FB

Symbol Alphanumeri

D YW L T Y N

Unit

DRAG

N

YWFORCE

N

LIFT

N

TFORCE

N

YFORCE

N

NFORCE

N

A.4.11 Body Axis Quantities Name

Symbol Alphanumeri

ax ay az Fx Fy Fz FxA FyA FzA FxT FyT FzT ÆX ÆY ÆZ

A

eleration along the x-axis A

eleration along the y-axis A

eleration along the z-axis Total for e along the x-axis Total for e along the y-axis Total for e along the z-axis Aerodynami for e along the x-axis Aerodynami for e along the y-axis Aerodynami for e along the z-axis Thrust along the x-axis Thrust along the y-axis Thrust along the z-axis In remental x-axis for e In remental y-axis for e In remental z-axis for e

L

Total rolling moment Total pit hing moment Total yawing moment Aerodynami rolling moment Aerodynami pit hing moment Aerodynami yawing moment Thrust ontribution to rolling moment Thrust ontribution to pit hing moment Thrust ontribution to yawing moment In remental rolling moment In remental pit hing moment In remental yawing moment

628

M N LA MA NA LT MT NT ÆL ÆM ÆN

AX AY AZ

Unit

m=s2 m=s2 m=s2

FX

N

FY

N

FZ

N

FXA

N

FYA

N

FZA

N

FXT

N

FYT

N

FZT

N

DFX

N

DFY

N

DFZ

N

LTOT

Nm

MTOT

Nm

NTOT

Nm

LA

Nm

MA

Nm

NA

Nm

LT

Nm

MT

Nm

NT

Nm

DL

Nm

DM

Nm

DN

Nm

A.4.12 Coe ients of For es and Moments For the denition of oe ients like tions are used:

q =

q  2V

p =

pb 2V

r =

rb 2V

Cm_ , Cmq , Cl _ , et ., the following onven_  _ = 2V _ b _ = 2V

Name

Symbol Alphanumeri

Coe ient of drag Coe ient of drag at zero angle of atta k Coe ient of drag due to angle of atta k Coe ient of drag due to angle of atta k rate Coe ient of drag due to elevator dee tion Coe ient of drag due to symmetri

anard dee tion Coe ient of drag due to symmetri taileron dee tion Coe ient of drag due to speed brake dee tion Coe ient of axial for e Coe ient of sidefor e Coe ient of normal for e Coe ient of sidefor e due to angle of sideslip Coe ient of sidefor e due to angle of sideslip rate Coe ient of sidefor e due to roll rate Coe ient of sidefor e due to yaw rate Coe ient of sidefor e due to aileron dee tion Coe ient of sidefor e due to rudder dee tion Coe ient of sidefor e due to dierential

anard dee tion Coe ient of sidefor e due to dierential taileron dee tion

629

CD CD 0 CD CD_

Unit

CD

-

CD0

-

CDA

-

CDAD

-

CDÆE

CDDE

-

CDÆCS

CDDCS

-

CDÆT S

CDDTS

-

CDÆSB

CDSB

-

CX CY CZ CY

CX

-

CY

-

CZ

-

CYB

-

CY _

CYBD

-

CY p CY r C Y ÆA

CYP

-

CYR

-

CYDA

-

C Y ÆR

CYDR

-

CY ÆCD

CYDCD

-

C Y ÆT D

CYDTD

-

Name

Symbol Alphanumeri

Coe ient of lift Coe ient of lift at zero angle of atta k Coe ient of lift due to angle of atta k Coe ient of lift due to angle of atta k rate Coe ient of lift due to pit h rate Coe ient of lift due to elevator dee tion Coe ient of lift due to symmetri anard dee tion Coe ient of lift due to symmetri taileron dee tion Coe ient of lift due to speed brake dee tion Coe ient of longitudinal for e Coe ient of longitudinal for e at zero angle of atta k Coe ient of longitudinal for e due to angle of atta k Coe ient of longitudinal for e due to angle of atta k rate Coe ient of longitudinal for e due to pit h rate Coe ient of longitudinal for e due to elevator dee tion Coe ient of longitudinal for e due to symmetri anard dee tion Coe ient of longitudinal for e due to symmetri taileron dee tion Coe ient of longitudinal for e due to speed brake dee tion Coe ient of normal for e Coe ient of normal for e at zero angle of atta k Coe ient of normal for e due to angle of atta k Coe ient of normal for e due to angle of atta k rate Coe ient of normal for e due to pit h rate Coe ient of normal for e due to elevator dee tion Coe ient of normal for e due to symmetri anard dee tion Coe ient of normal for e due to symmetri taileron dee tion 630

CL CL0 CL CL_ CLq CLÆE CLÆCS

CLFT

Unit -

CLFT0

-

CLFTA

-

CLFTAD

-

CLFTQ

-

CLFTDE

-

CLFTDCS

-

CLÆT S

CLFTDTS

-

CLÆSB

CLFTDSB

-

CT CT 0

CT

-

CT0

-

CT 

CTA

-

CT _

CTAD

-

CT q

CTQ

-

CT ÆE

CTDE

-

CT ÆCS

CTDCS

-

CT ÆT S

CTDTS

-

CT ÆSB

CTDSB

-

CN

-

CN0

-

CN

CNA

-

CN _

CNAD

-

CN CN 0

CNq CNÆE

CNQ

-

CNDE

-

CNÆCS

CNDCS

-

CNÆT S

CNDTS

-

Name

Symbol Alphanumeri

Coe ient of normal for e due to speed brake dee tion Coe ient of rolling moment Coe ient of rolling moment due to angle of sideslip Coe ient of rolling moment due to anglee of sideslip rat Coe ient of rolling moment due to roll rate Coe ient of rolling moment due to yaw rate Coe ient of rolling moment due to aileron dee tion Coe ient of rolling moment due to dierential anard dee tion Coe ient of rolling moment due to dierential taileron dee tion Coe ient of rolling moment due to rudder dee tion Coe ient of pit hing moment Coe ient of pit hing moment at zero angle of atta k Coe ient of pit hing moment due to angle of atta k Coe ient of pit hing moment due to angle of atta k rate Coe ient of pit hing moment due to pit h rate Coe ient of pit hing moment due to elevator dee tion Coe ient of pit hing moment due to symmetri anard dee tions Coe ient of pit hing moment due to symmetri taileron dee tion Coe ient of pit hing moment due to speed brake dee tion

631

Unit

CNÆSB

CNDSB

Cl Cl

CL

-

CLB

-

Cl _

CLBD

-

Clp

CLP

-

Clr

CLR

-

ClÆA

CLDA

-

ClÆCD

CLDCD

-

ClÆT D

CLDTD

-

ClÆR

CLDR

-

Cm Cm0

-

CM

-

CM0

-

Cm

CMA

-

Cm_

CMAD

-

Cmq

CMQ

-

CmÆE

CMDE

-

CmÆCS

CMDCS

-

CmÆT S

CMDTS

-

CmÆSB

CMDSB

-

Name

Symbol Alphanumeri

Cn Cn

Coe ient of yawing moment Coe ient of yawing moment due to angle of sideslip Coe ient of yawing moment due to angle of sideslip rate Coe ient of yawing moment due to roll rate Coe ient of yawing moment due to yaw rate Coe ient of yawing moment due to aileron dee tion Coe ient of yawing moment due to rudder dee tion Coe ient of yawing moment due to dierential anard dee tion Coe ient of yawing moment due to dierential taileron dee tion

632

Unit

CN

-

CNB

-

Cn _

CNBD

-

Cnp

CNP

-

Cnr

CNR

-

CnÆA

CNDA

-

CnÆR

CNDR

-

CnÆCD

CNDCD

-

CnÆT D

CNDTD

-

A.5 Mathemati al Quantities, Symbols and Alphanumeri Des riptors A.5.1 Matri es and Norms Name

Symbol

smallest singular value

I In j x xH xT xy [X ℄ij or xij det(X ) tr(X ) i (X ) max (X ) min (X ) p (X ) (G) (X ) i (X )  

Name

Symbol

Unit matrix Unit matrix of dimension n Square root of -1 Complex onjugate Tranpose of omplex onjugate Transpose Pseudo-inverse The

i; j

element of matrix

Determinant Tra e The

ith eigenvalue

Largest and smallest eigenvalue Perron-Frobenius eigenvalue Stru tured singular value Spe tral radius The

ith singular value

Largest and

1-norm,

maxj i jxij j

Frobenius norm Spe tral norm

H2 -norm

Hankel norm Innity norm

-norm

A sto hasti pro ess Auto ovarian e fun tion Power spe tral density Krone ker delta

jjxjj1 jjxjjF jjxjjS jjGjj2 jjGjjH jjGjj1 jjGjj

x(t) xx ( ) xx(!) Æjk 633

Alphanumeri

Unit

EYE

-

EYEN

-

JAY

-

XBAR

-

XH

-

XTP

-

XPSINV

-

XIJ

-

DETX

-

TRX

-

EIGXI

-

EIGXMIN

-

EIGXMAX

-

EIGPFX

-

SSVG

-

SRX

-

SVXI

-

SVXMAX

-

SVXMIN

-

Alphanumeri

Unit

NORM1X

-

NORMFX

-

NORMSX

-

NORMH2X

-

NORMHANKX

-

NORMINFX

-

NORMMUX

-

X

-

COVX

-

SPECX

-

KRONDELTAJK

-

A.6 System Des riptive Symbols and Alphanumeri Des riptors A.6.1 System Des ription Name

Symbol Alphanumeri

A Bu Disturban es matrix of the state equation Bw State matrix of the output equation C Control matrix of the output equation Du Disturban es matrix of the output equa- Dw

Unit

State matrix of the state equation

A

-

Control matrix of the state equation

BU

-

BW

-

C

-

DU

-

DW

-

State ve tor

X

-

Derivative of state ve tor

XDOT

-

R

-

tion

x x_ Referen e signal r Error signal e Control signal u Measurement error m Measured variables y Exogenous input signal w Regulated output signal z Transfer fun tion of plant G(s) Transfer fun tion of ontroller K (s) Transfer fun tion of pre- ompensator P (s) Transfer fun tion of feedforward ompen- F (s)

E

-

U

-

M

-

Y

-

W

-

Z

-

G

-

K

-

P

-

F

-

PZW

-

PZU

-

sator of disturban es

w to z Pzw Open-loop transfer fun tion from u to z Pzu Open-loop transfer fun tion from

634

Name

Symbol Alphanumeri

w to y Pyw Open-loop transfer fun tion from u to y Pyu Closed-loop transfer fun tion from w to z Tzw Return dieren e Fi (s), Fo (s) Sensitivity fun tion S (s) Complementary sensitivity T (s) Additive perturbation
a (s) Input multipli ative perturbation
i (s) Output multipli ative perturbation
o (s) 1 1 ~ s) Normalized un ertainty, Wi
Wj
( Weighting fun tions W (s) Crossover frequen y ! Bandwidth ( 3 dB) !b Zeros zi Poles pi 1 Gain margin ( Am jGK180o j ) Phase margin m Lapla e operator s Undamped natural frequen y !n Damping ratio  Spatial frequen y

Cir ular frequen y ! Open-loop transfer fun tion from

PYW

Unit -

PYU

-

TZW

-

FI, FO

-

S

-

T

-

DELTAA

-

DELTAI

-

DELTAO

-

DELTANORM W

-

WC

rad/s

WB

rad/s

ZEROI

-

POLEI

-

GM

-

PM

rad

S

rad/s

WN

rad/s

ZETA

-

OMEGA

rad/m

WFREQ

rad/s

A.6.2 Subs ripts Name

Symbol Alphanumeri

B CG E S V W M wb t

Body axis Centre of gravity Earth-xed referen e frame Stability axis Vehi le- arried verti al frame Wind axis Measurement referen e frame Wing/body onguration Tail Command

635

Unit

B

-

CG

-

E

-

S

-

V

-

W

-

M

-

WB

-

T

-

C

-

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