A Practical Approach to Robustness Analysis with Aeronautical Applications

A Practical Approach to Robustness Analysis with Aeronautical Applications

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A Practical Approach to Robustness Analysis with Aeronautical Applications

GILLES FERRERES ONERA-CERT Toulouse, France

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

eBook ISBN: Print ISBN:

0-306-46973-1 0-306-46283-4

©2002 Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow Print ©1999 Kluwer Academic / Plenum Publishers New York All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Kluwer Online at: and Kluwer's eBookstore at:

http://kluweronline.com http://ebooks.kluweronline.com

Acknowledgments

I would like to thank all of the people with whom I worked on robustness analysis. First, very special thanks to Vincent Fromion for our long collaboration; he even shared the authorship of the first version of this book. Thanks to Gérard Scorletti, Jean Marc Biannic, Jean François Magni, Béatrice Madelaine, and Yann Le Gorrec. Last but not least, thanks to Carsten Döll and Stéphane Font.

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Contents

List of Figures List of Tables Introduction 1. 2. 3. 4. Part I

Motivation The applicative problems Organization Glossary

xiii xvii xix xix XX

xxii XXV

Preliminaries

1. INTRODUCTION TO AND LFTS 1. Computation of the standard interconnection structure 1.1 LFTs 1.2 The case of a single block of neglected dynamics 1.3 Two neglected dynamics 1.4 Parametric uncertainties 1.5 The general case 2. An introduction to the s.s.v. 2.1 Parametric uncertainties 2.2 The general case 2.3 Choice of weights on the model uncertainties 3. Measures of performance 3.1 stability 3.2 performance 4. A formal introduction to the framework 4.1 Notations and definitions 4.2 The skewed s.s.v. 4.3 Specific structures of model perturbations 4.4 Back to the robust performance problem 4.5 Sensitivities 4.6 Difficulties of the approach vii

3 4 4 5 7 8 9 10 11 13 13 14 15 16 16 17 18 19 20 23 26

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

2. APPLICATIVE EXAMPLES 1. A transport aircraft

2.

3.

1.1 The aerodynamic model 1.2 Design of a rigid control law 1.3 The flexible model 1.4 Design of a flexible controller A missile problem 2.1 The nonlinear model 2.2 The linearized model 2.3 Design of the autopilot The telescope mock-up

29 29 30 31 32 33 34 35 35 37 38

Part II Computation of a standard LFT structure 3. REALIZATION OF UNCERTAIN SYSTEMS UNDER AN LFT FORM 1. Introduction 2. Affine uncertainties 3. The general case 3.1 Interconnection of LFTs 3.2 Cheng and De Moor’s method 4. A simple method for physical models 4. APPLICATIONS 1. The missile 1.1 A physical representation 1.2 Morton’s method 2. The transport aircraft 2.1 The rigid LFT model 2.2 An alternative method for introducing uncertainties in the flexible model 2.3 The flexible LFT model Part III

43 43 44 47 47 48 50 53 53 53 54 57 57 57 58

Applications

5. COMPUTATION OF BOUNDS 1. Computation of the real s.s.v. 1.1 A solution based on the Mapping Theorem 1.2 Dailey’s method 1.3 Jones’ method 2. Computation of the mixed s.s.v. 2.1 Introduction to LMIs 2.2 A complex upper bound 2.3 A first mixed upper bound 2.4 A second mixed upper bound 2.5 A mixed lower bound

63 63 63 65 66 66 66 67 68 70 70

3.

4.

Contents

ix

A real lower bound 3.1 Background 3.2 Regularization of the problem 3.3 Migration of the closed loop poles through the imaginary axis: a first simple method 3.4 Migration of the closed loop poles through the imaginary axis: an LP method Summary

72 72 73

6. APPLICATIONS OF THE TOOLS 1. The missile autopilot 1.1 Robust stability (P1) 1.2 Robust performance: pole location (P2) 1.3 Robust performance: sensitivity function (P3) 1.4 Conclusion 2. The transport aircraft 2.1 Robust stability (P4) 2.2 Robust performance (P5) 2.3 An additional comparison Part IV Skewed

73

75 77 81 81 81 84 86 88 89 89 90 92

applications

7. SKEWED PROBLEMS IN ROBUSTNESS ANALYSIS 1. Checking a small gain condition despite model uncertainties 2. Nonlinear analysis in the face of parametric uncertainties 3. Direct computation of the maximal s.s.v. 4. Parametric robustness analysis 5. Gain-scheduled and adaptive robust control

95 96 97 99 100 101

8. COMPUTATION OF SKEWED BOUNDS 1. A first upper bound 2. A second upper bound 2.1 Checking a small gain condition 2.2 The case of structured uncertainties 2.3 An augmented problem 2.4 Computation of scaling matrices D1 and G1 3. A skewed lower bound 4. Summary

103 103 104 104 106 107 108 109 112

9. APPLICATION OF THE SKEWED TOOLS 1. Introduction 2. Robust stability 3. Robust performance 4. A further study of the upper bounds

115 115 115 117 118

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

Part V Nonstandard

applications

10. ROBUSTNESS ANALYSIS OF FLEXIBLE STRUCTURES 1. Introduction 2. Relationship between both formulations of the upper bound 3. Computation of D, G scaling matrices at two frequencies 3.1 The LMI problem 3.2 A theoretical justification of the approach 3.3 Conclusion 4. Checking a posteriori the validity of D, G scaling matrices on a frequency interval 4.1 A technical result 4.2 Introduction of the frequency 4.3 Checking the validity of D, G scaling matrices 5. An algorithm 5.1 The basic algorithm 5.2 An improvement 6. Application to the flexible aircraft 6.1 Application of the basic algorithm

7.

6.2 Application of the improved algorithm 6.3 A physical interpretation of the lower bound 6.4 Computation of a lower bound of The telescope mock-up

123 123 125 126 126 126 127 127 128 129 130 132 132 132 134 135 136 136 138 139

11. ROBUSTNESS ANALYSIS IN THE PRESENCE OF TIME DELAYS 1. Introduction to the problem 1.1 A one-sided skewed problem 1.2 A generic example 1.3 Computation of the robust delay margin 2. A detailed algorithm 2.1 Computation of bounds 2.2 The detailed method 3. An alternative small gain approach 3.1 An unstructured approach 3.2 A structured approach 3.3 Introduction of model uncertainties 3.4 An improved algorithm 4. Application to the missile 5. Conclusion

143 143 143 144 145 148 148 149 151 151 153 154 156 158 161

12. NONLINEAR ANALYSIS IN THE PRESENCE OF PARAMETRIC UNCERTAINTIES 1. Introduction 2. A graphical method 2.1 A classical method 2.2 An extension 3. A first based method 3.1 An LFT formulation of the problem

163 163 164 165 166 167 167

Contents

4.

5. 6.

3.2 A preliminary technical result 3.3 A generalized condition of oscillation 3.4 An extension 3.5 The use of bounds A second based method 4.1 A sufficient condition of non oscillation 4.2 Extension to the case of parametric uncertainties 4.3 A skewed problem An application Conclusion

xi

168 168 171 173 174 174 176 177 177 180

Conclusion

183

Appendices A– Numerical data 1. The airplane problem 1.1 The aerodynamic model 1.2 The flexible model 2. The missile problem B– Proofs 1. Lemma 2..4 (chapter 8) 2. Proposition 3.. 2 (chapter 8)

187 187 187 187 188 190 193 193 194

References

197

Index

203

About the author

205

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List of Figures

1.1 1.2 1.3 1.4 1.5

1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 2.1 2.2 2.3 2.4 2.5 2.6 2.7

Standard interconnection structure. Lower and upper LFTs. Introduction of neglected dynamics in the closed loop. Introduction of neglected dynamics at the plant inputs and outputs. Realization of a parametrically uncertain system as an LFT Computation of the standard interconnection structure. Computation of the standard interconnection structure. Stability domains in the space of parametric uncertainties. Robustness of a pole placement. Augmented problem for robust performance analysis. Standard interconnection structure. Augmented problem for robust performance analysis. Augmented problem for robust performance analysis. Introduction of a scalar weight in the interconnection structure. Introduction of a scalar weight in the interconnection structure. Sensitivities. Example of plot as a function of frequency in the case of flexible control systems. Lateral flight control system for a civil aircraft. Computation of a complete model of the aircraft. Closed loop missile system with uncertainties. Principle of loop shaping control. Elevation transfer. Flexible modes. Augmented model. xiii

4 5 6 7

8 9 10 14 15 16 17 21 21 23 24 25 28

31 32 36 37 38 38 39

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

3.1 3.2 3.3 3.4 3.5 4.1 4.2 4.3 5.1

5.2 5.3 5.4 6.1 6.2 6.3

6.4 6.5 6.6 6.7 6.8 6.9 7.1 7.2 7.3 7.4 7.5 7.6 8.1 9.1 9.2 9.3 10.1

Standard interconnection structure. The interconnection of LFTs is an LFT. Star product. Cheng and De Moor’s method. A physical method for introducing uncertainties in a plant model. Computation of an LFT model of the missile. Computation of the LFT missile model with Morton’s method. Computation of a complete LFT model of the aircraft. Geometrical interpretation of the m.s.m. in the complex plane. Zadeh and Desoer’s result and definition of a lower bound on Introduction of D scales in the standard interconnection structure. Migration of the poles through the imaginary axis. Missile autopilot - robust stability (P1). Missile autopilot - robust stability (P1). Missile autopilot - robust stability inside a truncated sector (P2). Missile autopilot - robust stability inside a truncated sector (P2). Missile autopilot - template for robust performance. Missile autopilot - robust performance (P3). Transport aircraft - robust stability (P4). Transport aircraft - robust stability inside a truncated sector (P5). Transport aircraft - robust stability (P4). Interconnection structure for analysis. Detection of limit-cycles in the presence of LTI parametric uncertainties. Detection of limit-cycles in the presence of LTI parametric uncertainties. Stability domain in the space of Stability domain in the space of Analysis of gain-scheduled and adaptive controllers. Standard interconnection structures for analysis. Robust stability. Robust performance. New problem of robust performance. Interconnection structure.

43 47 48 49 51 54 56 59 64 65 67 76 82 83 84 85 86 87 89 90 91 96 97 98 101 101 102 105 116 117 119 128

10.2 10.3 10.4 10.5 10.6 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 11.10 11.11 11.12 11.13 12.1 12.2 12.3

12.4 12.5 12.6 12.7 12.8 12.9 12.10 12.11 12.12

List of Figures

xv

Introduction of frequency as a parametric uncertainty. Plot for the flexible aircraft. Plot for the flexible aircraft. Root locus associated to the destabilizing model perturbation. Plot for the telescope mock-up. Introduction of delays in the closed loop system. Representation of the circle in the complex plane. Realization of the transfer Computation of the new interconnection structure. Introduction of neglected dynamics at the plant inputs. Standard interconnection structure. Introduction of neglected dynamics at the plant inputs. Standard interconnection structure. Robust delay margin for the missile example. Robust delay margin for the missile example. Robust delay margin for the missile example. Robust delay margin for the missile example. Robust delay margin for the missile example. The problem without parametric uncertainties. Detection of a limit-cycle - a graphical method. Detection of a limit-cycle in the presence of parametric uncertainties. Interconnection structures for skewed analysis. The case of a saturation. A graphical method for checking the absence of limit-cycles. Small gain test for checking the absence of limit-cycles. Small gain test for checking the absence of limitcycles - extension to parametric uncertainties. An example. Computation of the augmented plant. The robustness margin as a function of X and The new limit-cycle.

129 134 135 137 140 144 146 146 148 151 151 153 154 157 158 159 160 161 165 166 167 168 172 174 175 176 177 178 179 180

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List of Tables

5.1 5.2 6.1

10.1

Methods for computing a lower bound. Characteristics of the computational techniques. Maximal real s.s.v. obtained with the computational methods. Lower bounds.

xvii

77 78 88 138

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Introduction

1.

MOTIVATION

In many physical situations, a plant model is often provided with a qualitative or quantitative measure of associated model uncertainties. On the one hand, the validity of the model is guaranteed only inside a frequency band, so that nearly nothing can be said about the behavior of the real plant at high frequencies. On the other hand, if the model is derived on the basis of physical equations, it can be parameterized as a function of a few physical parameters, which are usually not perfectly known in practice. This is e.g. the case in aeronautical systems: as an example, the aerodynamic model of an airplane is derived from the flight mechanics equations. When synthesizing the aircraft control law, it is then necessary to take into account uncertainties in the values of the stability derivatives, which correspond to the physical coefficients of the aerodynamic model. Moreover, this airplane model does not perfectly represent the behavior of the real aircraft. As a simple example, the flight control system or the autopilot are usually synthesized just using the aerodynamic model, thus without accounting for the flexible mechanical structure: the corresponding dynamics are indeed considered as high frequency neglected dynamics, with respect to the dynamics of the rigid model1. Summarizing, a model never perfectly represents the real plant to be controlled, and it is necessary to deal with associated model uncertainties. These correspond, either to uncertainties in the physical parameters of the plant (and more generally model perturbations inside the control bandwidth), or to high frequency unmodeled or neglected dynamics (un1 As mentioned in chapter 2, this assumption can not be done in the case of future large dimension transport aircrafts.

xix

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

certainties beyond the control bandwidth). However, many control design procedures only use the nominal model of the plant, and treat uncertainties in an incomplete or heuristic way. synthesis schemes exist, which account for the available information on the nature and structure of these uncertainties: even if a great deal

of work has been devoted to this subject, these design methods remain difficult to use, and it is not easy to control the order and the structure of the resulting controller. Note moreover that a properly designed control law can be robust in the face of uncertainties, even if these ones were not explicitly taken into account during the design process. Engineers often use their physical knowledge of the plant to design in an heuristic way control laws, which appear a posteriori sufficiently robust. In this context, the issue is rather to validate a control law by analyzing its robust stability and performance properties. Various methods are available for solving this problem, depending on the nature and structure of the uncertainties. We focus here on the structured singular value (s.s.v.) approach. The first reason is that the s.s.v. provides a general framework to robustness analysis problems. As a second justification of this choice, the approach has been successfully applied to industrial problems. Two general issues arise when applying this method, which have motivated a great deal of work since the beginning of the Eighties. The first one is to put a specific control problem into a standard form, which is called an LFT (Linear Fractional Transformation). When applying the tools to this standard LFT form, the second problem concerns the computational requirement, which must remain reasonable even for large dimension problems. More precisely, the s.s.v. is to be computed at each point of a frequency gridding, and lower and upper bounds are computed instead of the exact value of 2 . Methods for computing these bounds can be divided into two large categories, namely the exponential and polynomial time ones: the computational requirement of exponential (resp. polynomial) time methods increases exponentially (resp. only polynomially) with the size of the problem.

2.

THE APPLICATIVE PROBLEMS

Three applicative examples are treated. The first one is a longitudinal missile autopilot. The linearization of a nonlinear missile model at a trim point is considered, with parametric uncertainties in the stability

2

since the problem of computing the exact value of

is known to be NP hard.

INTRODUCTION

xxi

derivatives and unmodeled high frequency bending modes. The second applicative example is the lateral flight control system of a civil transport aircraft. Depending on the problem to be solved, we consider either the single rigid aerodynamic model (with parametric uncertainties in the stability derivatives), or a more complete model including a flexible structural model (with damping ratio about 1 % for the bending modes). The third application is a telescope mock-up used to study high accuracy pointing systems. The mock-up, which is composed of a two axis gimbal system mounted on Bendix flexural pivots, is representative of very flexible plants (with damping ratio about 0.1 % for the bending modes). The following problems will be solved in parts 3, 4 and 5:

1. Application of classical tools to the robustness analysis of a rigid aircraft or missile (chapter 6): the idea is to evaluate existing methods for computing bounds of the s.s.v. on realistic examples. The two examples are complementary: because of the small number of parametric uncertainties in the missile model, exponential time methods for computing bounds of the s.s.v. can be applied. Conversely, since there is a large number of parametric uncertainties in the aircraft model, only polynomial time methods can be applied.

2. Advanced robustness analysis of the missile (chapter 9): the robust stability and performance properties of the autopilot are studied in the presence of uncertain stability derivatives and unmodeled bending modes. This example also emphasizes the usefulness of the skewed tools for some classes of practical problems.

3. Computation of the robustness margin in the special case of flexible structures (chapter 10): the s.s.v. is to be computed as a function of The robustness margin is then obtained as the inverse of the maximal s.s.v. over the frequency range. As said above, is usually computed at each point of a frequency gridding, and the robustness margin is deduced as the inverse of the maximal s.s.v. over this gridding. In some special cases such as the control of flexible structures, the plot (corresponding to the value of the s.s.v. as a function of ) may present narrow and high peaks, so that it is possible to miss the critical frequency (i.e. the frequency which corresponds to the maximal s.s.v. over the frequency range), even when using a very fine frequency gridding. If this critical frequency is

missed, the robustness margin is overevaluated, i.e. the result is too optimistic. Chapter 10 proposes a method which computes an estimate of the s.s.v. as a function of and which gives a reliable

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

value of the robustness margin. The method is first applied to the flexible transport aircraft, and then to the telescope mock-up.

4. Computation of a robust delay margin (chapter 11): the issue is to analyze the robustness properties of a closed loop in the face of classical model uncertainties (uncertain parameters and neglected dynamics) and uncertain time delays. This difficult problem presents a great practical interest. Indeed, when embedding control laws on a real-time computer, delays are to be considered simultaneously at

the plant inputs (because of the time needed to compute the value of the plant input signal as a function of the plant output signal) and outputs (because of the sensors measuring the plant output signal). The method is applied to the missile. 5. Detection of limit-cycles in a nonlinear parametrically uncertain closed loop (chapter 12): as an extension of the famous Lur’e problem, consider the interconnection of a Linear Time Invariant (LTI) system (subject to LTI parametric uncertainties) with autonomous separable nonlinearities (e.g. saturations). The first issue is to detect the pres-

ence of a limit-cycle inside this closed loop with a necessary condition of oscillation. The second issue is to guarantee the absence of limitcycles despite parametric uncertainties, with a sufficient condition of non-oscillation. The and skewed tools provide a solution to this interesting nonlinear analysis problem.

3.

ORGANIZATION

The book is organized as follows: Chapter 1 (Part 1): the framework is presented. The general case of a closed loop system subject to both parametric uncertainties and neglected dynamics is considered. The first step is to transform this uncertain closed loop into a standard interconnection structure. The s.s.v. is then introduced as a tool for studying the robustness properties of this interconnection structure, and thus equivalently those of the original uncertain closed loop system. Beyond its mathematical definition, a physical interpretation of the s.s.v. is especially given. Chapter 2 (Part 1): the airplane and missile examples are explained in details. The plant model is described, as well as the method for synthesizing the control law. The way to introduce uncertainties in the model is also presented. Chapters 3 and 4 (Part 2): the issue is to transform the original uncertain closed loop into the standard interconnection structure. Chapter 3 is devoted to the problem of parametric uncertainties

INTRODUCTION

xxiii

entering the open loop plant model, since this appears as the key issue to be solved. The idea is to realize this parametrically uncertain system as an LFT transfer, in which the uncertainties appear as an internal feedback. To a large extent, this difficult problem remains open from a theoretical point of view. A simple solution is presented here for the case of physical systems. As an illustration of the techniques of chapter 3, the standard interconnection structures for the airplane and missile problems are obtained in chapter 4.

Chapter 5 (Part 3): different ways to compute (bounds of) the classical s.s.v. are presented. Are considered the particular case of real parametric uncertainties and the general case of mixed uncertainties (i.e. a model perturbation simultaneously containing real parametric uncertainties and neglected dynamics). Chapter 6 (Part 3): the aim is twofold. On the one hand, the tools developed in chapter 5 are evaluated on the aeronautical examples. On the other hand, the resolution of the first physical problem, which is considered in this book, is detailed (point (1) of the previous section). Chapter 7 (Part 4): through the presentation of some of the physically motivated problems, which are solved in this book, this chapter illustrates in a rather qualitative way that the skewed s.s.v. can solve a large set of engineering problems. It is emphasized that many problems, which are encountered in practice, appear to be skewed problems.

Chapter 8 (Part 4): because of the practical importance of the skewed s.s.v. (see chapter 7), it is interesting to develop specific tools for computing skewed bounds: this is done in chapter 8. Chapter 9 (Part 4): the aim is twofold. The skewed tools developed in chapter 8 are evaluated on the missile example. On the other hand, the resolution of the physical problem, which corresponds to point (2) of the previous section, is detailed. Note that the skewed tools of chapter 8 will also be used in chapters 11 and 12. Chapters 10, 11 and 12 (Part 5): these chapters present the resolution of the problems, which correspond to points (3) to (5) of the previous section.

This book is organized in a particular way, from the simplest topics to the most technical ones. The first part introduces the framework and presents the applicative examples. The second part focuses on the way

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

to transform a parametrically uncertain plant into an LFT form. The third part focuses on the application of classical tools. All these three parts are expected to be readable by a large audience, with the exception of chapter 5, which presents methods for computing bounds of the s.s.v.. The reading of this chapter can be nevertheless avoided, since a summary is done at the end of this chapter. The fourth part is devoted to skewed problems. To a large extent, this part is here again expected to be readable by a large audience, except chapter 8 which presents computational methods: a summary is nevertheless done at the end of this chapter. The last part is the most technical. Unlike what is done before, the theoretical and practical results are presented inside a same chapter. Moreover, when compared to the problems of parts 3 and 4, the problems of part 5 are more sophisticated. Nevertheless, part 5 presents new solutions to difficult engineering problems.

INTRODUCTION

4.

GLOSSARY

LFT: Linear Fractional Transformation LMI: Linear Matrix Inequality LP: Linear Programming LTI: Linear Time Invariant MIMO: Multi Inputs Multi Outputs m.s.m.: multiloop stability margin SIDF: Sinusoidal Input Describing Function SISO: Single Input Single Output s.s.v.: structured singular value skewed s.s.v.: skewed structured singular value

xxv

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I PRELIMINARIES

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Chapter 1 INTRODUCTION TO

AND LFTS

The chapter is organized as follows. In the first section, simple examples illustrate how to transform a specific control problem into the standard interconnection structure. This qualitative presentation will be completed by a technical presentation of the associated methods in chapter 3. These techniques will be moreover applied to the aeronautical examples in chapter 4. The s.s.v. is then introduced in a qualitative way in the second section. More precisely, we first focus on the problem of robust stability inside the left half plane. The s.s.v. provides a solution to this problem, first in the context of parametric uncertainties, and then in the general context of mixed uncertainties (i.e. parametric uncertainties and neglected dynamics). The third section considers robust performance problems, and shows that the s.s.v. provides a general framework for analyzing the robustness properties of a closed loop subject to model uncertainties. The framework is finally introduced in a formal way in the last section. The (skewed) s.s.v. is first defined. The Main Loop Theorem, which plays a key role in the approach, is then introduced, before being interpreted in a qualitative way in the specific context of robust performance problems. Some difficulties of the approach are finally briefly indicated, and especially the need to compute bounds of the s.s.v. instead of the exact value. These difficulties will be further studied in the next chapters. 3

4

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

1.

COMPUTATION OF THE STANDARD INTERCONNECTION STRUCTURE

Consider an LTI closed loop subject to parametric uncertainties and neglected dynamics. Simple examples illustrate in the following subsections that it is most generally possible to transform a specific uncertain closed loop into the standard interconnection structure of Figure 1.1: the transfer matrix M(s) contains the dynamics of the nominal closed loop (i.e. the closed loop without any model uncertainty) and the way the various model perturbations enter the closed loop. On the other hand, all model perturbations are gathered in the uncertain transfer matrix As a preliminary, the notion of LFT, which plays a key role in the framework (and more generally in robust control and robustness analysis), is defined.

1.1

LFTS

Lower and upper LFTs are defined in this subsection in a generic way. Let denote either transfer matrices or complex matrices: in Figure 1.2, the lower LFT is the transfer between and while the upper LFT is the transfer between and Partitioning P compatibly with the as:

the LFTs

and

can be written as:

Introduction to µ and LFTs

1.2

THE CASE OF A SINGLE BLOCK OF NEGLECTED DYNAMICS

•

A single block of neglected dynamics is introduced in the closed loop. Consider the example of Figure 1.3: the plant model G(s), the controller K(s) and the weighting function W(s) are known transfer matrices, while the normalized neglected dynamics correspond to an uncertain transfer matrix, which is just known to satisfy the inequality:

By definition of the

norm, this relation becomes at frequency

As a first point, the model uncertainty is said to be additive in the context of Figure 1.3, since the true plant is modeled as This additive representation of the model uncertainties is especially used in the context of flexible structures, to represent an uncertainty in the bending modes dynamics. As a second point, the true uncertainty in the plant dynamics is so that the weighting function W(s) is used to introduce our knowledge of this uncertainty in the plant dynamics (see below). can now come back to the original problem, which is to transform •theWeuncertain closed loop of Figure 1.3 into the standard interconnection

structure M(s) – ∆(s) of Figure 1.1. This is very easy, since M(s) is simply the transfer matrix, which is seen by the model perturbation on Figure 1.3 (i.e. the transfer between and z).

5

6

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

the nominal closed loop (i.e. the one corresponding to • Assume isthatasymptotically stable. The issue is to determine the max-

imal amount of neglected dynamics, for which the closed loop remains stable. A simple solution is provided by the small gain theorem, which gives a necessary and sufficient condition of stability of the closed loop at frequency

possible choice of the weighting function W(s) is finally illustrated in •theA context of a simple example. Consider the case of a large dimension

plant model Gcomplete(s). In e.g. control, the order of the controller is equal to the order of the augmented plant, so that Gcomplete(s) is usually to be reduced into a lower dimension plant model G(s). A simple and yet generally conservative solution for taking into account the model uncertainty, which is induced by the reduction of the plant model, is to write that the plant is with: The template

is thus defined in the frequency domain as:

If is a MIMO transfer matrix with m inputs and outputs, can then be chosen as: As expected, if the normalized neglected dynamics satisfy equation (1.4), the model uncertainty which is induced by the reduction of the plant model, satisfies inequality (1.6).

Introduction to

1.3

and LFTs

7

TWO NEGLECTED DYNAMICS

•

Two neglected dynamics and are now introduced at the plant inputs and outputs. For the sake of simplicity, associated weighting functions Wi(s) are not accounted for. These neglected dynamics may especially enable to handle simultaneously uncertainties in the actuators and sensors dynamics. The aim is here again to transform the uncertain closed loop of Figure 1.4 into the standard interconnection structure As in the previous subsection, M(s) is the transfer matrix between the inputs and outputs whereas the structured model perturbation is:

In the previous subsection, was an unstructured model perturbation, since it corresponded to any transfer matrix satisfying the inequality (1.3). In the above equation, both and are assumed to satisfy the inequality (1.3). There’s however an additional structural information in equation (1.9). Remark: an unstructured model perturbation becomes at a complex matrix without specific structure, which simply satisfies is said to be a full complex block.

More generally, if the closed loop contains m neglected dynamics it can be transformed into the standard interconnection structure with:

The unit ball

is introduced in the space of transfer matrices

8

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

At frequency

this unit ball becomes:

Because of the block diagonal structure of ∆, the inequalities 1 and reduce to and block of neglected dynamics.

for each

As in the previous subsection, the aim is to compute the maximal amount of neglected dynamics, for which the closed loop remains stable. Assuming that the nominal closed loop is asymptotically stable, the robustness margin is the maximal value of k, for which the closed loop remains stable in the presence of neglected dynamics satisfying for It is still possible to apply the small gain theorem to the above problem. The result is nevertheless conservative, i.e. only a lower bound of the robustness margin is obtained. Indeed, when applying the small gain theorem to the interconnection structure the block diagonal structure of the model perturbation is not taken into account, i.e. is considered as a full block of neglected dynamics.

1.4

PARAMETRIC UNCERTAINTIES

Assume as an example that the parametric uncertainties affine way the state-space equations of the plant l:

enter in an

1 T h e parametric uncertainties may enter in practice in numerous other ways the (statespace or input/output) plant model. See chapter 3 for more details.

Introduction to

and LFTs

9

represents the normalized variation of the ith uncertain parameter. Using Morton’s method (Morton and McAfoos, 1985; Morton, 1985) (see chapter 3), the uncertain plant can be transformed into an where u and y are the physical inputs and outputs of the plant (see equation (1.13) and Figure 1.5). is a diagonal matrix of the form:

The scalar is consequently repeated qi times, where the augmented matrix

is the rank of

The idea is thus to add fictitious inputs and outputs and z, so as to introduce then the uncertainties as an internal feedback (see Figure 1.5).

the LFT model of the uncertain plant. It suffices •to Let connect the plant inputs and outputs u and y (see Figure 1.5) with the inputs and outputs of a controller K(s) (see Figure 1.6) to obtain the standard interconnection structure of Figure 1.1. M(s) corresponds in Figure 1.6 to the transfer matrix, which is seen by the model perturbation i.e. to the transfer between and z. Note finally that is no more a dynamic transfer matrix, unlike in the previous subsections. It is just a matrix gain containing real parametric uncertainties.

1.5

THE GENERAL CASE

We now consider the general case of a closed loop, which is simultaneously subject to parametric uncertainties and neglected dynamics.

10

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

The issue is first to realize the parametrically uncertain plant model as an (see subsection 1.4). The control law is then connected with the plant and the neglected dynamics are finally added at various locations of the closed loop. The standard interconnection structure is obtained, by noting that M(s) is the transfer matrix seen by the structured model perturbation

The example of Figure 1.7 combines the examples of figures 1.6 and 1.4. The parametric uncertainties are gathered in has the following structure:

whereas M(s) is the transfer matrix between

2.

and outputs

AN INTRODUCTION TO THE S.S.V.

The aim is to introduce in a rather qualitative way the s.s.v. as a tool for the study of the stability of a closed loop in the presence of a structured model perturbation. One focuses in this section on the stability property inside the left half plane. Other regions of the complex plane will be considered in the following section. For the sake of clarity, the first subsection presents the special case of a closed loop subject to parametric uncertainties. The general case of parametric uncertainties and neglected dynamics is treated in the second subsection. The third

subsection is devoted to a technical, but practically important issue, namely the choice of the weights on the model uncertainties.

Introduction to

2.1

and LFTs

11

PARAMETRIC UNCERTAINTIES

We come back to the problem of subsection 1.4, namely the stability of the interconnection structure when only contains real parametric uncertainties The problem reduces to the computation of the s.s.v. along the imaginary axis, i.e. for s = with

2.1.1 A PRELIMINARY RESULT For the sake of simplicity, M(s) is assumed to be a strictly proper transfer matrix. Let (A, B, C, 0) a state-space model of M(s). Noting that may be considered as a feedback constant matrix, the state-space matrix of the closed loop and its characteristic polynomial is:

where

is the vector of parametric uncertainties associated to With the classical properties det(XY) = det(YX) and det(I – XY) = det(I – YX), it is straightforward to rewrite the above equation as:

Since

and

represents the

open loop characteristic polynomial (i.e. the one associated to and thus to the nominal closed loop M(s)), the following result is obtained:

The above equation is the specialization to the case of the interconnection structure of a well known result, which is the basis of the multivariable Nyquist theorem.

2.1.2

A FREQUENCY DOMAIN STABILITY CRITERION The unit hypercube D is introduced as:

Remember that the real model perturbation For the sake of simplicity, we note with some abuse of notation this should be understood as Assume that the nominal closed loop is asymptotically stable, which is

12

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

equivalent to the assumption of an asymptotically stable transfer matrix

M(s) (i.e. all eigenvalues of the state-matrix A are strictly inside the left half plane). The problem can then be formulated, either as a robustness test ("Is the closed loop of Figure 1.1 stable for all parametric uncertainties inside the unit hypercube D ?"), or as the computation of a robustness measure ("What is the maximal value of k for which the closed loop of Figure 1.1 is stable for all parametric uncertainties inside the hypercube kD ?"). This last problem reduces to look for the smallest value of k, for which the closed loop becomes marginally stable (i.e. one or more poles on the imaginary axis and all other poles strictly inside the left half plane) for a parametric uncertainty inside kD. Assume indeed that and let k increase from the zero value: since the nominal closed loop is asymptotically stable, the closed loop becomes marginally stable for a value of inside kD before being unstable (because of the continuity of the roots of the polynomial as a function of the vector of uncertain parameters). On the other hand, equation (1.19) emphasizes the link between the singularity of the matrix and the presence of a closed loop pole on the imaginary axis at As a consequence of the above discussion, the s.s.v. is introduced as follows. The complex matrix M in the following definition may be understood as the value of the transfer matrix D EFINITION 2..1 The s.s.v. and to a real model perturbation

if no

associated to a complex matrix M is defined as:

satisfies

The idea is thus to find the minimal size model perturbation (or equivalently which renders singular the matrix the s.s.v. is defined as the inverse of the size of this model perturbation. 2.1.3

COMPUTATION OF THE ROBUSTNESS MARGIN

In the context of a robust stability problem,

represents

the size of the smallest parametric uncertainty which brings one closed loop pole on the imaginary axis at The robust stability margin is obtained by computing the s.s.v. along the imaginary axis:

Introduction to

and LFTs

13

The principle is thus to detect the crossing of one of the closed loop poles through the imaginary axis. corresponds to the size of the smallest parametric uncertainty which brings one closed loop pole on the imaginary axis.

Remark: several reasons exist for handling the s.s.v. rather than its inverse the multiloop stability margin (m.s.m.). As a first point, the s.s.v. can not take an infinite value, since the nominal closed loop is asymptotically stable, whereas the m.s.m. may be infinite (if no structured model perturbation exists, which destabilizes the closed loop). On the other hand, the s.s.v. can be considered as an extension of classical algebraic notions, namely the spectral radius and the maximal singular value of a matrix (i.e. its spectral norm - see below).

2.2

THE GENERAL CASE

The problem of extending the approach of subsection 2.1 to the case of neglected dynamics seems a priori more complex, since is now a dynamic transfer matrix instead of a simple gain matrix. Nevertheless, assume that a complex matrix was found, which satisfies at frequency It suffices then to find a transfer matrix with When applying to the interconnection structure, a closed loop pole is obtained on the imaginary axis at

2.3

CHOICE OF WEIGHTS ON THE MODEL UNCERTAINTIES

An important practical issue is the choice of weights on model uncertainties. It was seen in subsection 1.2 that a template is to be determined for each block of neglected dynamics. Consider now the case of parametric uncertainties Each is assumed to belong to an interval The normalized parametric uncertainties are then introduced as so that leads to Let the robustness margin: the parametrically uncertain closed loop remains asymptotically stable for each or equivalently for each The choice of the initial intervals is critical, as illustrated by the example of Figure 1.8, with only two parametric uncertainties and The zero point corresponds to the nominal values of the uncertain parameters: remember that the nominal closed loop system is asymptotically stable. The space of parametric uncertainties and can then be split into two subdomains. In the first one, which contains the zero point, the parametrically

14

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

uncertain closed loop remains asymptotically stable. In the second one, this closed loop is unstable. At the limit between the two subdomains, the closed loop is marginally stable.

Generally speaking, the approach provides the largest hypercube in the space of the normalized parametric uncertainties inside which the closed loop stability is guaranteed. This hypercube becomes a box in the space of the initial parametric uncertainties What illustrates Figure 1.8 is that various stability boxes are obtained depending on the choice of the initial intervals Obviously, the above discussion can be extended to the general case of a structured model perturbation containing parametric uncertainties and neglected dynamics: various stability domains are obtained depending on the choice of the weights on the parametric uncertainties and neglected dynamics. We will come back to the study of this important issue in the following.

3.

MEASURES OF PERFORMANCE

Performance can be defined in two different ways. In the case of a real model perturbation, a first solution is to study the robustness of the location of the closed loop poles despite parametric uncertainties. In the general context of a mixed model perturbation, a second and more classical solution consists in checking whether a frequency domain template on a closed loop transfer matrix remains satisfied despite model uncertainties. In the first case, performance is rather defined in the time domain, whereas it is defined in the frequency domain in the second one.

Introduction to

3.1

and LFTs

15

STABILITY

The special case of a real model perturbation is considered. As explained in section 2.1, the aim of analysis is to detect the crossing of one of the closed loop poles trough the imaginary axis. Assuming that the nominal closed loop poles lie inside the left half plane, the idea is to compute the s.s.v. on the border of this left half plane (namely the imaginary axis), and to compute the robustness margin with equation (1.22). More generally, the singularity of the matrix is equivalent to the presence of a closed loop pole at the point of the complex plane. As a consequence, robust stability inside generic regions of the complex plane can be studied: assuming that the nominal closed loop poles belong to it suffices to compute the s.s.v. along the border of to find the minimal size real model perturbation, which shifts one closed loop pole on this border. may be the unit disc in the case of discrete time systems, or a truncated sector in the continuous-time one (see Figure 1.9). Performance is indeed defined in this context by minimal values and for the damping ratio and the degree of stability To some extent, these specifications correspond to requirements on the rise time and overshoot of the closed loop step response, or on the time needed to reject an unmeasured disturbance or a non zero initial condition.

2

the degree of stability of a state-space matrix A is defined as is an eigenvalue of A.

where

16

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

3.2

PERFORMANCE

In the spirit of control, performance is achieved if a closed loop transfer matrix T(s) satisfies a frequency domain template at all frequencies Assume now the presence of uncertainties in the closed loop, so that

T(s) is now an (i.e. the transfer between w and z in Figure 1.10). is most generally a mixed model perturbation, containing parametric uncertainties and neglected dynamics. The nominal closed loop is assumed to satisfy the performance property at frequency

i.e. :

The robust performance problem consists in computing the maximal amount of uncertainties, for which closed loop performance is still achieved. The issue is thus to compute the maximal size of the mixed model perturbation for which the following relation holds true:

It will be seen in subsection 4.4 that this robust performance problem can be equivalently transformed into an augmented robust stability problem, involving an additional fictitious full complex block (which is called a fictitious performance block). Chapter 7 will moreover illustrate that the robust performance problem is a skewed problem rather than a classical

4.

problem.

A FORMAL INTRODUCTION TO THE

FRAMEWORK The notion of mixed structured model perturbation and the are formally introduced in the first subsection. The skewed is defined

Introduction to

and LFTs

17

in the second one. The can be considered as an extension of classical algebraic notions, namely the spectral radius and the spectral norm (third subsection). The fourth subsection introduces the Main Loop Theorem in the specific context of robust performance problems. The notion of sensitivity is defined in the fifth subsection. The last subsection presents in a qualitative way some difficulties arising when using the framework.

4.1

NOTATIONS AND DEFINITIONS

The dependence is left out in the following: in the new interconnection structure of Figure 1.11, the complex matrix M represents the value of the transfer matrix M(s) at while the model perturbation is an uncertain complex matrix, which also represents the value of the uncertain transfer matrix Remember especially that each block of neglected dynamics becomes a full complex block at A mixed structured perturbation is a free complex matrix with the following specific structure:

With classical notations (Fan et al., 1991), contains real scalars (which represent the parametric uncertainties), complex scalars and full complex blocks (which represent the neglected dynamics). The integers and define the structure of the perturbation. A real scalar (resp. a complex scalar ) is said to be repeated if the integer (resp. ) is strictly greater than unity. is said to be a complex model perturbation if it only contains complex scalars and full complex blocks. Conversely, is a real model perturbation if it only contains real scalars. is finally a mixed model perturbation when it simultaneously contains real and complex uncertainties.

18

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

Remark: for the sake of completeness, the model perturbation above can also contain repeated complex scalars. Nevertheless, in many control problems, only contains real (possibly repeated) scalars and full complex blocks. Specific problems however require the introduction of repeated complex scalars: see e.g. (Doyle and Packard, 1987) 3. Note finally that a non repeated complex scalar can also be considered as a one dimensional full complex block. The unit ball tion

is introduced in the space of the structured perturba-

Note that the relation can be rewritten as and The s.s.v. is defined as:

Remark: the notation emphasizes that this value simultaneously depends on the complex matrix M and on the structure of the model perturbation For the sake of simplicity, we will often drop out the dependence, i.e. simply note

4.2

THE SKEWED S.S.V.

is now split as where structured perturbations. The skewed

and are two mixed is defined as:

When computing the unit ball is expanded (or shrunk) by factor k until the matrix becomes singular for a structured perturbation inside When computing the unit ball (in the space of perturbations ) is expanded (or shrunk) by factor k, but the structured perturbation remains now inside its unit ball Proposition 4..1 illustrates that it is possible under mild conditions to 3

An other reason for handling repeated complex scalars is historical: during the Eighties, the real repeated scalars were often assumed to be complex, because the real nature of parametric uncertainties could not be taken into account by existing computational algorithms.

Introduction to

and LFTs

19

compute the exact value of (resp. of ) by computing recursively the exact value of (resp. of ). When computing the exact value of it is moreover possible to use either a fixed point or a dichotomy search.

P ROPOSITION 4..1 Let

the dimension of matrix then is the unique limit of the fixed point

iteration:

b/ if function:

then v(M) is the unique zero of the monotonous

c/ if point iteration:

then

is the unique limit of the fixed

Proof: see (Fan and Tits, 1992) for points a/ and c/ (with some technical arrangement because of the potential discontinuity of the mixed s.s.v.). Point b/ is deduced from point a/ using

Note however that Remark: takes an infinite value if and only if (Fan and Tits, 1992). It is indeed easily remarked that if then there exists a perturbation which renders the matrix singular and satisfies so that When applying analysis to the standard interconnection structure the nominal closed loop is assumed to be asymptotically stable, so that the s.s.v. can only take finite values. On the contrary, an infinite value can be obtained for the measure.

4.3

SPECIFIC STRUCTURES OF MODEL PERTURBATIONS

In the general case of a mixed model perturbation no analytical expression of the s.s.v. is available. Nevertheless, in the special

20

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

case of a single full complex block (resp. a single repeated real scalar), the s.s.v. coincides with the maximal singular value (resp. the real spectral radius When is a full complex block, The small gain theorem provides indeed a necessary and sufficient condition of stability in the context of an unstructured model perturbation. In an alternative way, the following result can be used: if A is a complex matrix, the size of the smallest unstructured complex matrix which renders the matrix singular, is In the context of the initial problem, matrix M is assumed to be invertible for the sake of simplicity. The singularity of is then equivalent to the singularity of As a consequence, the size of the smallest unstructured complex matrix which renders the matrix singular, is:

Consider now the case of a single real repeated scalar Then where the real spectral radius is the magnitude of the largest real eigenvalue of M: is zero if M has no real eigenvalue. To prove that it suffices to note that the singularity of the matrix implies the existence of a non zero vector satisfying:

which can be rewritten as:

and are thus an eigenvalue and eigenvector of real scalar, whose size is to be minimized, so that of the largest real eigenvalue of M.

is moreover a is the magnitude

Remark: in the same way, it can be proved that the s.s.v. coincides with the spectral radius (defined as the magnitude of the largest eigenvalue of M) when the model perturbation is a repeated complex scalar.

4.4

BACK TO THE ROBUST PERFORMANCE PROBLEM

The Main Loop Theorem is introduced in the particular context of a robust performance problem. Note that this Theorem has been widely

Introduction to

and LFTs

21

used in many other contexts, and that it is the basis of many existing results.

4.4.1

MAIN LOOP THEOREM

The standard interconnection structure of Figure 1.11 is here again considered. The mixed model perturbation is split into two mixed model perturbations and

so that the standard interconnection structure of Figure 1.11 can be equivalently rewritten as the interconnection structure of Figure 1.12. The complex matrix M is partitioned compatibly with equation (1.38) as:

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

22

Let and the transfer between and z on Figure 1.13. The following result is known as the Main Loop Theorem:

4.4.2

CHECKING A FREQUENCY DOMAIN TEMPLATE Performance can be defined as the minimization of the weighted norm of a closed loop transfer matrix. Let e.g. the sensitivity function S, i.e. the transfer between the reference input and the tracking error. A template is defined to reflect the design specifications. Nominal performance is achieved if:

In an alternative way, performance can be defined as the minimization of an error signal z(t) which is the response to an exogenous input signal

Here again, if M11 denotes the transfer between

and z, nominal

performance is ensured if:

A structured model perturbation is introduced in the closed loop (see Figure 1.13). The transfer between w and z now corresponds to the LFT

where M is partitioned as in equation (1.39). Frequency is fixed. Performance is guaranteed at this frequency despite the model uncertainty inside the unit ball if and only if: (i) Robust stability of the closed loop is ensured:

(ii) The transfer between

and z remains lower than 1 despite the

model uncertainty

A fictitious full complex block is added (see Figure 1.12). With reference to subsection 4.3, it can be remarked that:

Introduction to

and LFTs

23

Let the augmented model perturbation of equation (1.38). Using the Main Loop Theorem, it can be claimed that the two above conditions are satisfied if and only if: As a consequence, the robust performance problem reduces to an augmented robust stability problem, in which a fictitious performance block is added (Doyle, 1985). Note that this block is possibly structured: if signals and z are decomposed as and and if we are just interested in the transfer functions

between scalar signals and the performance block as a complex diagonal model perturbation.

will be chosen

Remark: equation (1.47) corresponds to a test ("Is robust performance guaranteed inside the unit ball "). As proved later, the computation of the robustness margin (i.e. the maximal value of k, such that robust performance is guaranteed inside ) is a skewed problem.

4.5

SENSITIVITIES

Let where the are most generally mixed structured model perturbations. Without loss of generality, the aim is to analyze the sensitivity of the s.s.v. with respect to the first model perturbation Let the complex matrix M be partitioned as in equation (1.39), so that corresponds to and to In the same way as in subsection 4.2, the model perturbation is weighted by a scalar i.e. the model perturbation inside the interconnection structure becomes (see Figure 1.14):

24

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

The weighting matrix

in Figure 1.14 is transfered from

to

M (see Figure 1.15), so that M becomes:

Note that the weighting scalar can be introduced in three different ways in the interconnection structure. The first one is equation (1.48). The second one is:

and the associated value of

is:

A third solution is:

and the associated value of

is:

Introduction to

and LFTs

25

Even if these 3 solutions are equivalent, the third one is often chosen for numerical reasons (Braatz and Morari, 1991). The sensitivity with respect to is defined as:

It is easily proved that is a non decreasing function of so that is necessarily non negative. Moreover, it can be proved that the sensitivities are well-defined and equal to the corresponding full derivatives, almost everywhere on any interval of variation of (Braatz

and Morari, 1991).

Indeed, can be interpreted as the derivative of with respect to (with the restrictions above). As a consequence, the higher the value of the more critical the model perturbation in the value of Consider as an illustration the case where this means that has no influence on the value of Figure 1.16 illustrates this point. Let where the are real scalars. Figure 1.16 represents the space of the Each point of the curve C corresponds to a value of the for which the matrix is singular. Conversely, all points strictly inside the domain, whose border is C, correspond to values of the for which the matrix is nonsingular. In an obvious way, the zero point (which corresponds to and belongs to this domain of nonsingularity. represents the size of the largest square in the space of the which is centered on the zero point and inside which the matrix is guaranteed to be nonsingular. In Figure 1.16, the sensitivity with

26

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

respect to is strictly positive, whereas the sensitivity with respect to is zero. Remember finally that bounds are computed in practice instead of the exact value of see especially (Douglas and Athans, 1995) for the computation of the sensitivity of the upper bound by (Fan et al., 1991).

4.6

DIFFICULTIES OF THE

4.6.1

COMPUTATION OF

APPROACH

BOUNDS

Computing the exact value of the s.s.v. is an NP hard problem (Braatz et al., 1994), so that the computational burden of the algorithms, which compute the exact value of is necessarily an exponential function of the size of the problem. It is consequently impossible to compute the exact value of for large dimension problems. A usual solution is to compute upper and lower bounds instead of the exact value. The associated algorithms can be exponential time (like the algorithms which compute the exact value of ), or more interestingly polynomial time. Even if the gap between the bounds can not be guaranteed a priori when using polynomial time algorithms, good results can be obtained in realistic examples: this will be illustrated in the following.

From a computational point of view, a (skewed) upper bound is typically obtained as the solution of a convex or quasi-convex optimization problem, namely a Linear Matrix Inequality problem (Boyd et al., 1994; Gahinet et al., 1995). Conversely, methods which compute a lower bound are generally heuristic, and the computational burden is required to be low. The most classical solution consists in solving in an heuristic way a non convex optimization problem: a lower bound corresponds indeed to a local optimum of this non convex optimization problem, whereas the exact value of corresponds to the global optimum of this optimization problem. The idea is more precisely to obtain the lower bound as the limit of a fixed point iteration which is obtained by rewriting the necessary conditions of optimality as (Packard et al., 1988; Young and Doyle, 1990). Note however that the associated power algorithms are not guaranteed to converge, and that the final result depends on the initialization of the fixed point iteration. This will be further detailed in the following. The usefulness of bounds is now explained. For the sake of clarity, we restrict our attention to the case of a real model perturbation

Introduction to

and LFTs

27

Remember the unit hypercube D is defined in equation (1.20). An upper bound of gives a sufficient condition of nonsingularity of the matrix which is thus guaranteed to be nonsingular for all parametric uncertainties inside

Note also that an upper bound of the s.s.v. becomes a lower bound of the m.s.m., so that a lower bound of the robustness margin is finally obtained as:

In the context of a robust stability problem in the presence of parametric uncertainties, closed loop stability can thus be guaranteed inside the hypercube in the space of uncertain parameters. Conversely, a lower bound of gives a sufficient condition of singularity of the matrix i.e. there exists a real model perturbation with singular (in the context of a robust stability problem, is a destabilizing model perturbation). The usefulness of a lower bound is twofold. As a first point, gives a measure of the conservatism of the upper bound by examining the tightness of the interval which contains the exact value of As a second point, an associated worst-case model perturbation is usually provided with by the computational algorithm.

4.6.2 THE SPECIAL CASE OF FLEXIBLE SYSTEMS A common practice is to compute the s.s.v. at each point of a frequency gridding The robustness margin is deduced as:

When choosing a sufficiently fine frequency gridding, good results are obtained in many practical examples. A specific problem however arises in the context of flexible systems: indeed, narrow and high peaks may be obtained on the plot of as a function of frequency (see Figure 1.17). The use of a frequency gridding is unreliable in such a

case: the risk is to miss a peak on the plot, and to overevaluate the robustness properties of the closed loop (by underevaluating the value of the maximal s.s.v. over the frequency range). In the context of this new and difficult problem, chapter 10 proposes a method for computing a reliable estimate of as a function of

28

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

Chapter 2 APPLICATIVE EXAMPLES

The first section describes the rigid and flexible models of the transport aircraft, while the second section describes the nonlinear and linearized missile models. These two sections also present the design of associated controllers. The third section describes the telescope mock-up, which complements the flexible aircraft example since it is even more flexible (about 0.1 % for the damping ratio of the bending modes for the telescope, about 1 % for the aircraft).

1.

A TRANSPORT AIRCRAFT

Two different control problems are considered. The first classical one consists in designing a flight control system for a rigid airplane. In the second one, which is far less standard, a flexible airplane is considered. The aerodynamic model is presented in the first subsection. A simple static control law is synthesized for this rigid model in the second subsection. The third subsection presents the flexible model, which is to be added to the aerodynamic model in order to obtain a complete aircraft model. The fourth subsection presents the design of the new flight control system.

Glossary sideslip angle. roll angle. yaw rotational rate. roll rotational rate. acceleration output. aileron deflection. 29

30

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS rudder deflection. angle of attack. pitch angle. inertial speed. acceleration due to gravity.

1.1

THE AERODYNAMIC MODEL

Let the state vector, whereas the output vector is denotes the control input vector. The lateral state-space equations of the aircraft are (see appendix A for numerical data):

This model is obtained as the linearization of a nonlinear model at the trim value The acceleration at the center of gravity is:

At a point of coordinates x and z (with respect to the center of gravity), the acceleration is:

Uncertainties are introduced in the 14 coefficients which characterize the aerodynamic model, namely the stability derivatives and As an example, the coefficient is rewritten as:

represents the nominal value of the coefficient. The constant scalar weights the uncertainty in this coefficient, with respect to uncertainties in the other coefficients. which is assumed to belong to the interval [1,1], finally represents the normalized parametric uncertainty in The scalar is called in the following "the weight" in the coefficient The weights in the 14 stability derivatives are chosen in the following as 10 %. A second order actuator is added at the control input

Applicative examples

31

whereas a third order actuator is added at the control input

Remarks: (i) All quantities p, r and are expressed in degree or degree/s. The acceleration output is expressed in (ii) For the sake of simplicity, the acceleration is measured in the following at the center of gravity.

1.2

DESIGN OF A RIGID CONTROL LAW

A static output feedback K assigns the closed loop eigenstructure, and a static feedforward H introduces the pilot demands in the closed loop, which is summarized in Figure 2.1. The gain matrix K is synthesized with a classical modal approach with decoupling objectives. The poles of the open loop aerodynamic model (i.e. the rigid model without actuators) are:

The corresponding closed loop poles are chosen as:

32

1.3

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

THE FLEXIBLE MODEL

A complete model of the aircraft is obtained by adding its rigid and flexible models at the outputs (see Figure 2.2). A real modal statespace representation of the flexible part is given in appendix A. The corresponding flexible modes are:

In chapter 4, uncertainties in the natural frequencies of the bending modes above will be introduced in the aircraft model.

Applicative examples

1.4

33

DESIGN OF A FLEXIBLE CONTROLLER

An observed state feedback controller is synthesized. On the one hand, the rigid closed loop poles are placed in the same way as in subsection 1.2. On the other hand, some of the flexible modes are moved into the left half plane in order to increase their damping ratio. Following the previous subsection, the design model is built by adding the rigid and flexible models at the outputs. Actuators are then introduced at the aircraft inputs (see Figure 2.2). Second order transfer functions are finally added at the inputs, with and These filters increase the roll-off properties of the controller, and thus its robustness with respect to unmodeled flexible modes. Notice that the maximal frequency of the bending modes in the previous subsection is about 15 rad/s. A state feedback controller is first synthesized for this design model. The actuators and filters poles are not to be moved. The four rigid closed loop poles are chosen as in subsection 1.2. The 6 closed loop flexible modes are chosen as:

The damping ratios of the 4 critical flexible modes are increased up to 30 %. The two other flexible modes are left unchanged. Summarizing, the closed loop poles, corresponding to the state-feedback controller, are:

34

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

An observer is now to be built. Since the states of the filters are available, the new design model is the same as the one for the state feedback, except that it does not include the filters. The observer gain places the closed loop poles roughly in the same way as the state feedback gain. When applying the observed state feedback controller to the complete aircraft model (which thus includes the rigid model, the flexible model and the actuators), the closed loop poles are classically the poles of the statefeedback controller and those of the observer.

2.

A MISSILE PROBLEM

The nonlinear aerodynamic model is described in the first subsection, while the linearized model is presented in the second one. The autopilot is synthesized in the third subsection.

Glossary angle of attack. q: pitch rotational rate. actual tail deflection angle. commanded tail deflection angle. commanded acceleration. actual acceleration. M: Mach number. V: missile velocity. Mass: missile mass. pitch moment of inertia. Q: dynamic pressure. S: reference area. d: missile diameter.

Applicative examples

2.1

35

THE NONLINEAR MODEL

A nonlinear longitudinal missile model is extracted from (Reichert, 1992). The control input is the tail deflection while the outputs used by the autopilot are the acceleration and rate outputs The state vector is where is the angle of attack. The missile behavior can be described by the following nonlinear equations:

whereas the normal acceleration output

is given by:

This model is essentially parameterized by the Mach number M (between 2 and 4). The dynamic pressure Q, the reference area S, the diameter d, the pitch moment of inertia and the missile mass are indeed constant, while the missile velocity V is proportional to M. and depend on M, and

See (Reichert, 1992) for numerical data. The main nonlinearities are the variation of and as a function of since these quantities are third order polynomials of whose range of variation is 20degrees. The actuator is finally a second order transfer function:

with

2.2

and

THE LINEARIZED MODEL

The above model is linearized at a medium value angle of attack. The following LTI model is obtained:

of the

36

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

This model is essentially parameterized by the 4 stability derivatives and For the sake of simplicity, it was assumed that in equation (2.7), so that and correspond to the linearization of the nonlinear functions and with respect to On the other hand, and coincide with the quantities and in equation (2.9).

The issue in the following will be to analyze the local stability and performance properties of the closed loop missile in the presence of parametric uncertainties in these 4 stability derivatives and in the face of neglected dynamics, namely a high frequency bending mode. To this aim, the parametrically uncertain missile model will be first transformed into a standard LFT structure where gathers the uncertainties in the stability derivatives. On the other hand, the bending mode is represented by an additive model perturbation and its template 1/W(s) is extracted from (Reichert, 1992; Balas and Packard, 1992). The uncertain closed loop missile is presented in Figure 2.3, where K(s) represents the missile autopilot. Note that the weights in the stability derivatives are chosen as 5 %. In the context of control, the frequency domain performance is defined through the sensitivity function S, i.e. the transfer between the commanded acceleration and the tracking error

Applicative examples

2.3

DESIGN OF THE

2.3.1

PRINCIPLE OF LOOP SHAPING

37

AUTOPILOT CONTROL

In the case of a SISO system, the very classical Bode method shapes the magnitude and phase of the open loop frequency response where K(s) and G(s) respectively represent the controller and plant model. As an extension to the case of a MIMO plant G(s), the singular values of the open loop transfer or can be shaped (McFarlane and Glover, 1990; McFarlane and Glover, 1992). To this aim, pre- and post-compensators and are added at the plant inputs and outputs. These transfer matrices are chosen so that the singular values reflect the design objectives.

A specific robust stabilization procedure is then applied to the augmented plant (see Figure 2.4). A controller is obtained, which stabilizes the augmented plant. The final controller is K(s) = Note that the above robust stabilization procedure presents special properties, since it tries as much as possible to keep the desired open loop shaping while enforcing a stability closed loop constraint.

2.3.2

APPLICATION

The choice of the compensators is not detailed, the interested reader is referred to (Ferreres and M’Saad, 1996) for a related work. The values of the pre- and post-compensators are:

38

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

where

and

The design model is the linearized missile model at with the second order actuator.

3.

THE TELESCOPE MOCK-UP

This mock-up, which is used to study high accuracy pointing systems, is composed of a two axis gimbal system mounted on Bendix flexural

Applicative examples

39

pivots. We more precisely focus on the elevation axis of the telescope. A 40th order identified model is available. The frequency response of the transfer function between the commanded torque and the acceleration measured on the telescope main body, is depicted in Figure 2.5. The characteristics of the 20 poorly damped modes of the model are presented in Figure 2.6. The main control design objective is the rejection of supporting vehicle disturbances. To achieve this goal, an augmented model is built on the basis of the identified model As usually in or synthesis, the control design objective is expressed under the standard form of Figure 2.7, as the minimization of the transfer from the disturbances (position and velocity of the supporting vehicle and ) to the controlled outputs (position and velocity of the telescope and ).

C represents the control input. Three measurements are available, which add sensor dynamics to the system : (inertial elevation angle and angular acceleration) and (relative elevation

40

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

angle). Bendix pivots are represented by feedbacks of stiffness and friction and The order of the augmented system is equal to N + 6, where N is the order of (40 if the full order identified model is used) and 6 represents the 2 Bendix states and the 4 sensor states. See (Madelaine and Alazard, 1998; Alazard et al., 1996; Madelaine, 1998) for the design of the control law. Note simply that the order of the controller is 13, so that when applying this controller to the 46th full order plant model of Figure 2.7, a 59th closed loop flexible system is obtained. In this example, the issue will be to analyze the robustness properties of this controller with respect to uncertainties in the natural frequencies of the bending modes (see chapter 10).

II COMPUTATION OF A STANDARD LFT STRUCTURE

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Chapter 3 REALIZATION OF UNCERTAIN SYSTEMS UNDER AN LFT FORM

1.

INTRODUCTION

The issue is to transform a closed loop subject to model uncertainties (parametric uncertainties and neglected dynamics) into the standard interconnection structure of Figure 3.1. As illustrated in chapter 1, the key issue is to take into account the parametric uncertainties entering the open loop plant model: the uncertain transfer matrix (where

into an LFT ation.

is a vector of uncertain parameters) is to be transformed

where

is a real model perturb-

We first consider the simple case of parametric uncertainties entering in an affine way the state-space model of the plant. A simple method (Morton and McAfoos, 1985; Morton, 1985) is indeed available for this special case, which is often encountered in practice. The general problem is then considered in the third section. It is proved in e.g. (Belcastro and Chang, 1992; Lambrechts et al., 1993; Cheng and DeMoor, 1994) 43

44

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

that an LFT model can be obtained in the following very general case: the coefficients of the state space model or transfer matrix are rational functions of the parametric uncertainties. This covers most of the engin-

eering examples. The problem is however the potential non minimality of the computed LFT model: see also (Font, 1995). This is an important problem from a practical point of view. Consider a simple example with two parametric uncertainties and Assume that an LFT model of the transfer matrix was computed with:

The LFT model is non minimal if an other LFT model could be found, which equivalently models with a simpler structure for the real model perturbation, e.g. :

The model perturbation (3.2) is more attractive than the one of equation (3.1) for two reasons. When applying the tools to the interconnection structure the computational amount is an increasing function of the complexity of the model perturbation As a second reason, when computing e.g. the classical upper bound of (Fan et al., 1991), the result is a priori more conservative with the model perturbation (3.1). It is indeed observed in practice that the more repeated a scalar, the more conservative the upper bound (Packard and Doyle, 1988). Nevertheless, note that an LFT model can be reduced a posteriori with various heuristic methods (Beck et al., 1996). As a final point, a simple method is proposed for transforming an uncertain physical plant model into an LFT form (section 4.). The next chapter will apply this method, as well as the method by (Morton and McAfoos, 1985; Morton, 1985), to the two aeronautical examples.

2.

AFFINE UNCERTAINTIES

Consider the case of parametric uncertainties way the state-space model of the plant:

entering in an affine

Realization of uncertain systems under an LFT form with dim x = n, dim be rewritten as:

and dim

45

The above equations can

where

The idea is to introduce additional fictitious inputs and outputs w and z, so that the uncertainties appear as an internal feedback with To this aim, the following augmented model is introduced:

Matrices represent the nominal plant model (i.e. the one corresponding to The issue is to look for matrices and for a structured model perturbation satisfying

The LFT

with:

can be written as:

46

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

Because of the affinity of the equations as a function of the parameters Matrices et must satisfy:

Let

the rank of matrix

which can be factorized as:

with sequence:

and

As a con-

It is then deduced that:

With reference to the above equation and to relations (3.11) and (3.12), one obtains by identifying each term:

The issue is finally to come back to an input/output framework. Equation (3.3) corresponds to a transfer matrix where as equation (3.6) corresponds to the augmented transfer matrix:

As expected, when using the feedback

one obtains:

Realization of uncertain systems under an LFT form

3. 3.1

47

THE GENERAL CASE INTERCONNECTION OF LFTS

• It is first illustrated that the interconnection of various elementary LFTs is also an LFT. Consider the example of Figure 3.2, which represents the way a complete LFT model of the transport aircraft will be

obtained in the next chapter. Remember that a complete aircraft model is obtained by adding at the physical outputs the rigid and flexible models (see chapter 2). An LFT model was separately obtained for both rigid and flexible models. This means that the rigid LFT model contains the physical inputs and outputs and additional inputs and outputs with the fictitious feedback (see Figure 3.2). In the same way, the flexible LFT model contains the physical inputs and outputs and additional inputs and outputs with the fictitious feedback The complete LFT model of the aircraft, is computed as an interconnection of the two above LFTs. The augmented model perturbation is while H(s) is the transfer matrix between inputs u, and outputs y, and

• The star product S(Q, M) of Q and M is defined in Figure 3.3 as a specific interconnection of two LFTs. It corresponds to the transfer

48

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

between inputs

and outputs

where Q and M are partitioned as:

3.2 CHENG AND DE MOOR’S METHOD As said at the beginning of this chapter, an LFT model can be computed when the coefficients of the state space model or transfer matrix are most generally rational functions of the parametric uncertainties. The aim of this subsection is to illustrate this by detailing the method by (Cheng and DeMoor, 1994). This technique uses the fact that an interconnection of LFTs is also an LFT. The idea is to realize separately the linear and nonlinear parts of the model. Let:

where the coefficients of the matrices and are rational functions of the vector of parametric uncertainties. As in the previous section, the above equations can be rewritten as:

Realization of uncertain systems under an LFT form

49

where:

is rewritten as:

where the

are scalar rational functions of

Consider then the

fictitious matrix

Using the method of the previous section, it is easy to realize the matrix as an LFT where and is the rank of matrix

On the other hand, the scalar rational functions are realized as elementary LFTs where is a diagonal matrix containing possibly repeated real scalars (the number of repetitions of the scalar in the model perturbation obviously depends on the structure of the scalar rational function To

50

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

illustrate this in a simple way, let:

and let

Then:

Any rational first order function can thus be realized under the above LFT form. Remember now that so that an LFT model of the matrix can be computed as the interconnection of the LFT with elementary LFTs see the example of Figure 3.4, in which two scalar rational functions and are considered. The method is thus conceptually simple, even if a large number of elementary LFTs is possibly to be handled in practice. The approach

moreover proves that an LFT model can be obtained in the very general case of coefficients of the state space model or of the transfer matrix, which are rational functions of the parametric uncertainties. Nevertheless, the method does not give (at least a priori) the minimal size LFT model.

4.

A SIMPLE METHOD FOR PHYSICAL MODELS

The aim of this section is to illustrate through a simple example a straightforward method, for introducing uncertainties in the plant model and obtaining the associated LFT model (see also the next chapter). Consider a real modal state-space model of a flexible plant. When focusing on a single flexible mode with damping ratio and natural frequency one obtains:

Where the are row vectors, while C is a matrix. An uncertainty in the natural frequency is introduced as:

Realization of uncertain systems under an LFT form

where denotes the nominal value and state-space model above becomes:

51

the relative variation. The

Fictitious input (vector) w and output (vector) z are introduced:

Let When applying the fictitious feedback the issue is to determine the integer n and the state-space matrices and for which the state-space model of equation (3.31) reduces to the

state-space model of equation (3.30).

52

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

This problem can be solved in an analytical way. However, an alternative very simple solution uses the physical representation of the

state-space model (3.30): see Figure 3.5. The idea is to compute the augmented plant H(s) of Figure 3.5, with inputs u, and and outputs y, and (and

Let The LFT transfer coincides with the transfer (3.30) between input u and output y, and this LFT realization is minimal.

Chapter 4 APPLICATIONS

The computation of the LFT models of the missile and transport aircraft is detailed in this chapter.

1. 1.1

THE MISSILE A PHYSICAL REPRESENTATION

In the same way as at the end of the previous chapter, the parametric uncertainties in the stability derivatives are directly introduced in the physical missile model. Figure 4.1 represents equations (2.11) of the linearized missile model, with multiplicative uncertainties being introduced in the stability derivatives and (i.e. is e.g. replaced by Note that the tail deflection input is renamed as u in this figure, in order to avoid any ambiguity with the vector of parametric uncertainties. The LFT model of the missile is obtained as:

where and H(s) is the transfer in Figure 4.1 between inputs and outputs This LFT model is minimal. For the sake of simplicity, only uncertainties in the 4 stability derivatives were introduced above. Nevertheless, it would be possible to introduce uncertainties in the other physical parameters in Figure 4.1. It is worth emphasizing that the missile equations (2.11) are affine with respect to uncertainties in the 4 stability derivatives However, they are no more affine when considering additional uncertain53

54

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

ties in the physical parameters QS, d,

or

since products

such as now appear. Thus, Morton’s method can not be applied in this new context. On the contrary, it is straightforward to introduce additional uncertainties in QS, d, or in the missile model of Figure 4.1.

1.2

MORTON’S METHOD

This method is illustrated in the specific context of the missile example. The same result as in the previous subsection is obtained. The idea is to rewrite equations (2.11) of the linearized missile model as:

with:

Applications

As in the previous subsection, the tail deflection input u. Let then:

Equation (4.2) is rewritten as:

with:

and:

55

is renamed as

56

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

The above equations can thus be rewritten as:

with

and:

Let the augmented plant with additional fictitious inputs w and outputs z:

When applying the fictitious feedback the initial plant of equation (4.8) is recovered. The LFT model is given in Figure 4.2 (with

Applications

2. 2.1

57

THE TRANSPORT AIRCRAFT THE RIGID LFT MODEL

We come back to the problem of chapter 2 (subsection 1.1), and note first that the 14 stability derivatives enter in an affine way the state-space equations (2.1), (2.2) and (2.3) of the aerodynamic aircraft model. The LFT model can thus be computed with Morton’s method. Note that the parametric uncertainties in the stability derivatives could be directly introduced in the physical model, as in the missile case. Nevertheless, it is simpler here to apply Morton’s method, because of the complexity of the aerodynamic equations (2.1), (2.2) and (2.3). It is moreover interesting to emphasize that the 14 stability derivatives enter as rank-one model perturbations the state-space aerodynamic model, even when these coefficients simultaneously enter the state matrix A and output matrix C (this is e.g. the case of ). When considering the jth uncertain parameter, this means that the corresponding matrix in equation (3.5) is a rank one matrix, so that the associated real scalar is non repeated. As a consequence, the LFT model of the rigid transport aircraft contains 16 inputs (2 physical inputs and 14 fictitious inputs) and 18 outputs (4 physical outputs and 14 fictitious outputs). The associated model perturbation contains 14 non repeated real scalars.This LFT model is minimal.

2.2

AN ALTERNATIVE METHOD FOR INTRODUCING UNCERTAINTIES IN THE FLEXIBLE MODEL

An alternative method to the approach of chapter 3 (section 4.) is presented for introducing uncertainties in the frequency of the bending modes. Let and the natural frequency and damping ratio of the flexible mode. When reducing the real modal state-space representation to this single mode, one obtains :

An uncertainty in the natural frequency

is introduced as:

where denotes the nominal value and the associated relative variation. Assuming that the damping ratio and frequency variation are

58

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

small (i.e. and proximated as follows:

), the dynamic matrix can then be ap-

can thus be rewritten as

with :

is a rank one matrix, which is factorized under the form The uncertainty can hence be expressed as a feedback of gain between the scalar fictitious input and output :

A possible choice for

2.3

and

is :

THE FLEXIBLE LFT MODEL

Uncertainties are introduced in the natural frequencies of the 6 flexible modes. A real modal state-space representation of the flexible model is used. If the method of chapter 3 (section 4.) is applied, the flexible LFT model contains 6 twice repeated real scalars. Otherwise, if the method of subsection 2.2 is used, the LFT contains 6 non repeated real scalars. The complete LFT model is simply computed by adding at the physical outputs and the rigid and flexible LFT models (see Figure 4.3): remember indeed that the interconnection of LFTs is an LFT. where is a real model perturbation containing 14 real non repeated scalars (corresponding to uncertainties in the 14 stability derivatives). is a real model perturbation containing, either 6 real twice repeated scalars, or 6 real non repeated scalars (corresponding to uncertainties in the frequencies of the 6 bending modes).

Applications

59

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III APPLICATIONS

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Chapter 5 COMPUTATION OF

BOUNDS

Methods are described in this chapter for computing bounds of the real, complex or mixed Remember that M denotes in the following a complex matrix, which may correspond to the value of the transfer matrix M(s) at

1.

COMPUTATION OF THE REAL S.S.V.

The specific case of a model perturbation which only contains real non repeated scalars is considered in this section.

1.1

A SOLUTION BASED ON THE MAPPING THEOREM

The definition of the unit hypercube D is first recalled:

The s.s.v. or its inverse the multiloop stability margin (m.s.m.) is defined as:

The m.s.m. can thus be interpreted as the lowest value of k, such that the image of the unit hypercube D by the operator det(I – contains the origin of the complex plane (see Figure 5.1 this image is called a value set). However, building this image is generally a computationally infeasible task. Nevertheless, in the special case of a model perturbation containing only non repeated real scalars the operator det(I – kdiag M) 63

64

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

is a complex multilinear function of vector It is thus possible to use a result in chapter 9 of (Zadeh and Desoer, 1963), which claims that the

image of the unit hypercube D by the operator det(I – kdiag M) is contained in the convex hull of the images of the vertices of D (Mapping Theorem). A lower bound of the m.s.m. can thus be obtained as the smallest value of k, for which the convex hull contains the origin (see Figure 5.2).

Remarks: (i) The approach is exponential time, since an hypercube has vertices when n parametric uncertainties are considered. (ii) Like Zadeh and Desoer’s result, Kharitonov’s and edge theorems (Barmish and Kang, 1993; Bartlett et al., 1988) can also be interpreted in a frequency domain approach as Zero Exclusion Tests (Barmish and Kang, 1993). The difference between these three theorems is the way the uncertain parameters enter the plant model. (iii) In the special case of a real matrix M (this especially occurs at the zero frequency, i.e. when M is the DC gain of the transfer matrix M(s)), the result by Zadeh and Desoer provides the exact value of the m.s.m. The convex hull coincides indeed with the image of the unit hypercube D by the operator det(I – km diag M), since both sets lie in the real axis.

The result by Zadeh and Desoer was used in (DeGaston and Safonov, 1988) as the basis of a computational algorithm, which provides the exact value of real using a branch and bound scheme. The idea is to partition the unit hypercube into a set of smaller hyperrectangles, and to apply Zadeh and Desoer’s result on each of these hyperrectangles, in order to decrease the conservatism of the lower bound of An upper bound of

(i.e. a

lower bound) is also computed to stop the partitioning

Computation of

bounds

65

process, when the gap between the bounds is sufficiently small: see (DeGaston and Safonov, 1988; Ferreres et al., 1996a) for further details.

Remark: the algorithm in (DeGaston and Safonov, 1988) can be summarized in 3 operations, computing the lower bound, computing the upper bound and partitioning. Other algorithms with the same branch and bound structure were developed in (Chang et al., 1991; Balakrishnan et al., 1991; Newlin and Young, 1992). However, the aim of the algorithm in (Newlin and Young, 1992) is not to compute the exact value of mixed µ, but rather to reduce the gap between the bounds to an acceptable value (typically 20 %), thus limiting as much as possible the exponential growth of computation with the number of uncertainty blocks (remember the problem of computing the exact value of is NP hard).

1.2

DAILEY’S METHOD

This method, which provides a real lower bound, is based on a conjecture about the structure of the minimum norm parameter perturbation (Dailey, 1990). Assume that all but two components and of the minimum norm parameter perturbation achieve the maximal magnitude k:

Rather than solving the general equation with respect to the vector parametric uncertainties:

of

it suffices now to solve it with respect to the two free components and and this reduces to finding the real roots of a set of two quadratic

66

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

equations in and remember indeed that M is complex, so that the above equation has generally a single solution in and If the absolute values of the obtained and are less or equal to k , a destabilizing parameter perturbation with norm k has been obtained. Relation (5.3) defines a two dimensional face of the hypercube kD, so that the above operation has to be done on each such face of the hypercube. The method, which is thus exponential time because of the exponential growth of the number of two dimensional faces as a function of the number of parametric uncertainties, is nevertheless easy to implement. The technique provides a lower bound of real since the counterexamples in (Holohan and Safonov, 1993; Ackermann, 1992) prove that it can not be assumed in the general case that more than one component of the minimum norm parameter perturbation achieves the maximal magnitude.

1.3

JONES’ METHOD

This method, which provides a real upper bound, is based on linear algebraic manipulations and on the properties of the determinant (Jones, 1987). An upper bound of is obtained as:

where A* denotes the complex conjugate transpose of A, is the largest eigenvalue of A, D is a real diagonal scaling matrix and is a permutation matrix, defined as Since there are matrices to be considered in the case of an n dimensional vector of parametric uncertainties, the method is here again exponential time. This upper bound is proved in all cases to be less conservative than the upper bound of the complex s.s.v. of subsection 2.2. Note that the permutation matrix may indicate a direction in the space of uncertainties which leads to instability.

2. 2.1

COMPUTATION OF THE MIXED S.S.V. INTRODUCTION TO LMIs

A Linear Matrix Inequality (LMI) is an inequality constraint A(x) < 0, where A(x) is an hermitian square matrix (A(x) = A*(x)) depending linearly on the vector x of parameters:

Computation of

bounds

67

Matrices and are fixed. Various techniques and softwares (Boyd et al., 1994; Gahinet et al., 1995) are available for solving the feasibility problem: does there exist a value of vector x for which the relation A(x) < 0 holds true ? Let two hermitian matrices A and B, with the quantities and are defined as:

where denotes the maximal eigenvalue of a matrix M. is almost always equal to the maximal generalized eigenvalue except possibly when (see Proposition 5.1.c of (Fan and Tits, 1992)). Let A(x) and B(x) matrices which linearly depend on vector x. The minimization of is a quasi-convex optimization problem, which can be recast as the minimization of scalar under the LMI constraint A first simple solution is a dichotomy search over for a given value of the feasibility problem "does there exist a value of vector x satisfying is to be solved. More sophisticated solutions, involving the direct minimization of are also available (Gahinet et al., 1995).

2.2

A COMPLEX

UPPER BOUND

The standard interconnection structure is considered. A upper bound at frequency the small gain theorem

of Figure 5.3.a is directly provided by

This theorem does not account for the structure of the model perturbation (s). To this aim, a scaling transfer matrix D(s) is used to decrease

68

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

the conservatism of the µ upper bound (5.8) (Safonov, 1982). D(s) must commute with (s), i.e.:

D(s) can thus be introduced into the interconnection structure without modifying the stability properties of the closed loop (see Figure 5.3.b). Frequency is fixed: let then and Remember that the structure of the model perturbation is (see equation (1.26)):

where and are real and complex scalars, while is a full complex block. Because of equation (5.9), the scaling matrix D must belong to: with The new

upper bound is:

The issue is thus to minimize the conservatism of this upper bound by finding the optimal value of D, which minimizes This convex and non differentiable optimization problem can be solved in numerous ways: see (Doyle, 1982; Safonov, 1982; Packard and Doyle, 1993) and included references. Remark: this

upper bound can be rewritten as1:

This is an LMI problem involving the minimization of a maximal generalized eigenvalue.

2.3

A FIRST MIXED

UPPER BOUND

The upper bound above does not account for the real nature of the parametric uncertainties, so that this upper bound is called "a complex upper bound": real and complex scalars are indeed treated in the same way in the structure of the scaling matrix D. An additional scaling 1

More precisely,

noting that

Computation of

matrix G can take into account the specificity of scalars

bounds

69

G must

belong to:

with The new

upper bound is (Fan et al., 1991):

The complex upper bound of the previous subsection is recovered with G = 0 (see equations (5.12) and (5.13)). On the other hand, the new quasi convex and non differentiable optimization problem can be solved, either using a general LMI solver (the issue is to minimize a maximal generalized eigenvalue), or using the specific structure of the optimization problem. Indeed, an alternative formulation of the above mixed µ upper bound was proposed in (Young et al., 1995). The following sets of scaling matrices are associated to

with with

is consequently a positive definite matrix, unlike is an hermitian block diagonal matrix unlike which is a real diagonal matrix. The following Lemma is extracted from (Young et al., 1995). L EMMA 2..1 If there exist scaling matrices

and

such that (F =

then

Remark: roughly speaking, the upper bound proposed in the above Lemma involves the minimization of a maximal singular value, whereas the upper bound of (Fan et al., 1991) involves the minimization of a maximal generalized eigenvalue. These bounds are nevertheless identical (Young et al., 1995): see Lemma 2..1 of chapter 10.

70

2.4

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

A SECOND MIXED

UPPER BOUND

In the general case of mixed uncertainties, (Fan et al., 1991) first formulate the problem of computing the exact value of as a smooth constrained optimization problem. The result is then used to derive a computable upper bound, in which a maximal generalized eigenvalue is to be minimized with respect to the sets and of scaling matrices (see the previous subsection). An other approach for computing a upper bound is proposed by (Safonov and Lee, 1993). This method basically uses the positivity theorem: the interconnection structure is first transformed, so that the new feedback uncertainty block becomes positive (using a bilinear transformation). The conservatism of the positivity theorem is then minimized with multipliers. The optimal value of these multipliers is here again obtained as the solution of an LMI problem. Both methods in (Fan et al., 1991; Safonov and Lee, 1993) solve in a more efficient way the problem of (Doyle, 1985), in which a µ upper bound was proposed for the case of non repeated real scalars. As noted in (Safonov and Lee, 1993), the two upper bounds in (Fan et al.,

1991; Safonov and Lee, 1993) are equivalent for this case. The equivalence can be further characterized using a result in (Haddad et al., 1992) (Corollary 2, p. 2819).

2.5

A MIXED

LOWER BOUND

The exact value of the mixed s.s.v. can be obtained as the solution of a non convex optimization problem, whose global maximum coincides with the exact value of The idea of the power algorithms by (Packard et al., 1988; Young and Doyle, 1990) is to write the necessary conditions of optimality as f (x) = x. Power iterations x(k + 1) = f¨(x(k)) are then used to asymptotically solve f (x) = x. Note that this algorithm only provides (at least a priori) a local maximum of the non convex optimization problem, i.e. a lower bound. The optimization problem is first detailed in the following two propositions. As a preliminary, the real spectral radius of a complex matrix A is defined as the magnitude of the largest real eigenvalue of A :

is zero if matrix A has no real eigenvalue.

PROPOSITION 2..2

Computation of

Proof: a structured model perturbation exists a non-zero vector x satisfying:

bounds

71

is searched, for which there

can be rewritten as:

Equation (5.20) can then be equivalently rewritten as:

so that

is an eigenvalue of matrix

Since the smallest size

destabilizing model perturbation is searched, the maximal value of is thus to be obtained, i.e. the maximal value of the real spectral radius of matrix over P ROPOSITION 2..3 Let (see equation (5.10)):

and

Then:

It is worth emphasizing the coherence between the definition of

and

the counterexamples of (Ackermann, 1992; Holohan and Safonov, 1993). When considering complex uncertainties, it suffices to search the destabilizing model perturbation over the unit sphere. However, as proved in (Ackermann, 1992; Holohan and Safonov, 1993), no assumption can be

made about the real perturbations

which are thus simply assumed to

lie inside their unit ball (i.e.

The issue is now to solve the optimization problem, noting as a preliminary that a local minimum will be necessarily obtained in the general case (since the problem is non convex) and that the computational burden should remain minimal. As said above, a classical solution (Packard et al., 1988; Young and Doyle, 1990) is to rewrite the necessary conditions of optimality under the form f (x) = x, where f is a vectorial function of vector x. A fixed point method finds a limit x* of the series When the series converges, x* satisfies indeed the necessary conditions of optimality and a local minimum has been obtained, i.e. a lower bound. Nevertheless, the value of x* depends on the initial condition of the series, because of the non convexity of the problem. On

72

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

the other hand, the series does not necessarily converge: a limit-cycle may especially appear inside the power algorithm. In practice, the algorithm by (Young and Doyle, 1990) generally presents good convergence properties. The computational burden is low and the quality of the lower bound is usually good. However, it is worth emphasizing that this power algorithm exhibits poor convergence properties in the case of a purely real model perturbation (see below). Note finally that the power algorithm by (Young and Doyle, 1990) is not detailed here, since a generalization of this power algorithm to the skewed problem will be presented in section 3. of chapter 8.

3. 3.1

A REAL

LOWER BOUND

BACKGROUND

The aim of this section is to compute a reliable upper bound of the robustness margin (i.e. a lower bound of the maximal s.s.v. over the frequency range) in the context of a purely real model perturbation, which contains a large number of parametric uncertainties (Ferreres and Biannic, 1998b). Note indeed that the primary aim is to compute an interval of the robustness margin, rather than an interval of the s.s.v.

as a function of This is not an easy problem, since the real lower bound of (Dailey, 1990) is exponential time. On the other hand, the mixed lower bound of (Young and Doyle, 1990) is polynomial time, but the convergence properties of the power algorithm are very poor in the context of a purely real model perturbation. The result provided by this power algorithm is consequently not reliable in this specific context. A state-space approach was proposed in (Magni and Döll, 1997): consider the interconnection structure M(s) and let (A, B, C, 0) a statespace representation of M(s), which is assumed to be strictly proper just for the ease of notation. The idea is to interpret the model perturbation as a fictitious feedback gain which moves the poles of the closed loop, whose state matrix is A + from the left half plane through the imaginary axis. The norm of is to be minimized during the process of migration of the poles towards the imaginary axis. The algorithm provides a model perturbation which brings one closed loop pole on the imaginary axis at An upper bound of the robustness margin is thus obtained as the size of and its inverse is a lower bound of the s.s.v. (and thus a lower bound of the

maximal s.s.v. over the frequency range). Good results are generally obtained with this technique. There are however some technical difficulties. A key issue is especially to choose

Computation of

bounds

73

which poles of the nominal closed loop (i.e. which poles of the state matrix A) are to be moved towards the imaginary axis.

3.2

REGULARIZATION OF THE PROBLEM

A well-known solution for improving the convergence properties of the power algorithm by (Young and Doyle, 1990) is to regularize the real problem, by adding a small amount of complex uncertainty to each real

uncertainty (Packard and Pandey, 1993). Let

and

the original

data of the real µ problem, which thus consists in computing (a lower bound of) A model perturbation is introduced, with the same structure as except that the real scalars become complex. Let then the augmented model perturbation and let:

The issue is now to compute a lower bound of The higher the value of the larger the amount of complex uncertainties, and the better the convergence properties of the power algorithm, which is applied to this regularized problem. is classically chosen as 5 % or 10 %. However, the lower bound obtained for this regularized problem is not a lower bound for the original real

problem. The power algorithm provides indeed a critical model perturbation which renders the matrix singular. No model perturbation was found, which would render the matrix singular. It is especially worth emphasizing that the matrix is not a priori singular.

3.3

MIGRATION OF THE CLOSED LOOP POLES THROUGH THE IMAGINARY AXIS: A FIRST SIMPLE METHOD

Nevertheless, it can be remarked that the real part

of

is prob-

ably a good initial guess in the search of a model perturbation which renders the matrix singular. There is however a technical difficulty, which is to decide when a matrix X is singular. An obvious

solution is to compute the magnitude of the determinant det(X), the minimal singular value or the condition number and to decide that X is singular when one of these quantities is lower than a given value. The value of this threshold is however difficult to determine, and it may depend on the problem data. If the aim is to compute a lower bound of the maximal s.s.v. over the

74

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

frequency range, rather than a lower bound of at a fixed frequency a natural solution is to remember that is the value of the transfer matrix M(s) at Let (A, B, C, 0) a state-space representation of M(s), which is here again assumed to be strictly proper just for the ease of notation. It can be expected that one pole of A + is close to the point of the imaginary axis, if a lower bound of the regularized s.s.v. was computed. A very simple solution is thus to increase the size of the model perturbation until one pole of A + crosses the imaginary axis at the point Since one pole of A + is expected to be close to the point of the imaginary axis, the value of for which a pole of A + crosses the imaginary axis, is possibly close to 1. In an obvious way, a lower bound of was computed instead of a lower bound of But more importantly, a model perturbation was obtained, which brings one closed loop pole on the imaginary axis. The inverse of the size of is thus a lower bound of the maximal s.s.v. over the frequency range. The method is thus the following. A frequency gridding is first chosen, as usually in analysis. A regularized lower bound is computed at each point of this gridding, and the real part of the augmented model perturbation, which is provided by the power algorithm, is extracted. The value of is then increased from the initial value of 1 until one pole of A+ crosses the imaginary axis. At each point of the frequency gridding, a lower bound of the maximal s.s.v. over the frequency range was thus obtained as the inverse of the

size of the model perturbation The best estimate of the maximal s.s.v. over the frequency range is finally chosen as the highest value of this lower bound over the frequency gridding. Remarks:

(i) The problem is slightly more complex in practice. There exist indeed two ways of optimizing In the above approach, is increased from its nominal value of 1, until the target pole crosses the imaginary axis (i.e. the pole of A + which is the closest to the point of the imaginary axis, if a lower bound of the regularized s.s.v. was computed). It is however possible that some other poles of were found to be strictly unstable. In this case, an alternative is to decrease until all unstable poles cross the imaginary axis. This approach provides a smaller destabilizing perturbation, but the associated

frequency is no longer guaranteed to be close to the initial value (ii) The method can be readily extended to the problem of robust stability inside a region

of the complex plane (typically a truncated sector).

Computation of

3.4

bounds

75

MIGRATION OF THE CLOSED LOOP

POLES THROUGH THE IMAGINARY AXIS: AN LP METHOD More sophisticated methods can be used to move the poles of A + through the imaginary axis. In the spirit of (Magni and Döll, 1997), a solution is to introduce an additional model perturbation The aim is then to find the minimal size model perturbation which moves one pole of A + through the imaginary axis. This problem can be easily recast as a simple Linear Programming problem. First remember that the model perturbation is diagonal:

Let the closed loop pole, which is the closest to the imaginary axis. Using well-known results on eigenvalues derivatives (Kato, 1980), and assuming that the magnitude of the additional parametric uncertainties is sufficiently small, the first order variation of is computed as an affine function of the

In order to impose that is moved onto the imaginary axis, one imposes Remember then that the infinity norm of the model perturbation is to be minimized, so that the problem reduces to the minimization of scalar under the constraints:

It may be necessary in practice to modify the above problem, when the eigenvalue is not sufficiently close to the imaginary axis. In this case indeed, the accuracy of the first order development may not be sufficient, to directly move the pole onto the imaginary axis. A solution consists in partitioning the real segment between and the imaginary axis, and to iteratively perform the migration on each sub-segment. At the end of this process, the method of the previous subsection is applied, to ensure an exact pole placement on the imaginary axis.

76

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

Remarks: (i) In the same way as in (Magni and Döll, 1997), the Frobenius norm of the model perturbation could be minimized, instead of the infinity norm. The results are however more conservative than those obtained with the LP method above, since the s.s.v. handles the infinity norm. (ii) It would be possible to impose that the imaginary axis is crossed at a given point in order to compute a lower bound of at a givenfrequency In this case, two equality constraints 0 and instead of a single one are to be considered. Nevertheless, the approach becomes numerically sensitive, since it appears difficult to move the closed loop poles onto some regions of the imaginary axis (especially at very low and very high frequencies). (iii) It is worth emphasizing that the aim is not to compute a lower bound of at a given frequency, but to detect the peak values on the plot, in order to directly compute a lower bound of the maximal real s.s.v. over the frequency range. Assume that a µ lower bound was computed for the regularized problem at a frequency which is far from the critical frequency Using especially the LP method above, the imaginary axis can be crossed at a frequency which is very close to (see Figure 5.4). Indeed, the imaginary axis is not constrained to be crossed at a given point and the norm of the model perturbation is to be minimized. As a consequence, it is generally observed that the imaginary axis is crossed at frequencies which corres-

Computation of pond to the peak values on the

plot (remember that

bounds

77

is homogeneous

to the inverse of the robustness margin, so that the peak values on the plot correspond to model perturbations which are of minimal size). The method consequently detects the critical frequencies on the plot: see also chapter 10 (subsection 6.4). (iv) The following table summarizes the three methods presented above.

4.

SUMMARY

A great deal of work was devoted to the problem of computing the s.s.v.. Existing computational algorithms can be sorted following various criteria:

Nature of the structured model perturbation: two large categories of methods can be considered, corresponding to the special case of a real model perturbation and to the general case of a mixed model perturbation 2.

Nature of the result: the algorithm provides the exact value of lower bound or a

a

upper bound.

Computational requirement: in polynomial time algorithms (resp. exponential time algorithms), the computational amount is a polynomial (resp. exponential) function of the size of the problem. Nevertheless, it must be noted that the computational amount can also largely differ for two methods inside the same category. Remember finally

that an algorithm, which computes the exact value of

is necessarily

exponential time, since this problem is NP hard.

2

For historical reasons, the special case of a complex model perturbation can be included in

the general case of a mixed model perturbation.

78

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

As an illustration, the characteristics of the computational methods, which were presented in this chapter, are summarized in the above table. In practice, a trade-off is to be achieved between the accuracy of the result, the computational requirement and the size of the problem. As a simple example, the exact value of can be computed in the case of small size problems, without an excessive amount of computation. However, computing the exact value of in the case of medium or large size problems would require an excessive computational amount. More generally, in the case of small dimension problems, exponen-

tial time algorithms can be used. However, it is necessary for large size problems to compute an interval of the s.s.v. with polynomial-time algorithms. Even if the gap between the bounds can not be guaranteed a priori, good results can be nevertheless obtained in realistic examples, as illustrated in chapter 6. Finally, concerning more specifically the computational methods, the following two points are recalled: The mixed upper bound by Fan et al is obtained as the solution of a quasi-convex optimization problem, namely an LMI problem.

Conversely, the mixed lower bound by Young and Doyle is obtained as the solution of a non convex optimization problem: the idea is more precisely to obtain the lower bound as the limit of a fixed point iteration However, the associated power algorithm is not guaranteed to converge, and the final result depends on the initialization of the fixed point iteration. Nevertheless, good results are generally obtained, except in the case of a purely real

Computation of

bounds

79

model perturbation. In this specific context, it is more interesting to use Dailey’s method if the size of the problem is sufficiently small (this method is indeed exponential-time). Otherwise, section 3. proposes a polynomial-time method, which directly computes a lower bound of the maximal s.s.v. over the frequency range.

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Chapter 6 APPLICATIONS OF THE

TOOLS

The aim of this chapter is to evaluate the applicability of the computational methods, which were presented in the previous chapter. See (Ferreres et al., 1996a) for a related work in the context of a single-axis and three-axes missile autopilot. See also chapter 5 (section 4.) for a summary of the computational methods. The first section evaluates the robust stability and performance properties of the longitudinal missile autopilot, robust performance being defined either as robust pole location inside a truncated sector, or as robust shaping of the sensitivity function. The lateral flight control system is analyzed in the same way in the second section. Since this controller was synthesized with a modal approach, robust performance is defined in this context as robust pole location inside a truncated sector.

1.

THE MISSILE AUTOPILOT

The aim of this section is to analyze the local stability and performance properties of the autopilot in the presence of parametric uncertainties in the 4 stability derivatives The high frequency bending mode is not taken into account (see chapter 9), so that the model perturbation only contains 4 non repeated real scalars. The weights in the stability derivatives are chosen as 5 %. See chapters 2 (subsection 2.2) and 4 (section 1.) for details concerning the description of the linearized missile model and the building of the interconnection structure.

1.1

ROBUST STABILITY (P1)

5 methods of chapter 5 are used, namely the mixed upper bound by Fan et al (subsection 2.3), the classical complex upper bound (subsec81

82

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

tion 2.2), the real upper bound by Zadeh and Desoer (subsection 1.1), the real upper bound by Jones (subsection 1.3), and the real lower bound by Dailey (subsection 1.2). Note first that all these methods give nearly the same result (around 0.13) at

Figure 6.1 presents the real upper bound by Zadeh and Desoer (dashed line) and the real lower bound by Dailey (solid line). Note the good accuracy of the interval at nearly all frequencies. The maximal

value of the s.s.v. is obtained as 0.13 at The result is non conservative. The corresponding uncertainties in the stability derivatives are This means that the closed loop missile is stable despite simultaneous uncertainties of in the four stability derivatives. Note also the discontinuity of the bounds by Zadeh and Desoer and by Dailey at at while at very low

Applications of the

tools

83

frequencies. This means that the exact value of the real s.s.v. is also discontinuous at the zero frequency, as already mentioned in (Ferreres et al., 1996a).

Figure 6.2 presents the results obtained with the four upper bounds (classical complex upper bound in solid line, real upper bound by Jones in dashed line, mixed upper bound by Fan et al in dash-dotted line, real upper bound by Zadeh and Desoer in dotted line). This Figure illustrates the decreasing conservatism of these bounds. The best results are obtained with Zadeh and Desoer’s and Fan et al’s methods. More precisely, better results are obtained with Zadeh and Desoer’s method at low frequencies, equivalent results are obtained at medium frequencies, and better results are finally obtained with Fan et al’s method at high

84

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

frequencies. The classical complex upper bound, the real upper bound by Jones and the mixed upper bound by Fan et al are continuous at unlike the real upper bound by Zadeh and Desoer.

1.2

ROBUST PERFORMANCE: POLE LOCATION (P2)

Robust stability inside a truncated sector is studied. The minimal degree of stability (resp. the minimal damping ratio) is chosen as 0.3 (resp. 0.4). The nominal degree of stability (resp. the nominal damping ratio) is 0.49 (resp. 0.62).

We proceed in the same way as in the previous subsection. The same 5 methods are used. Here again, all these methods give nearly the same result (around 0.12) at We then focus on the real upper bound

Applications of the

tools

85

by Zadeh and Desoer (dashed line - see Figure 6.3) and the real lower bound by Dailey (solid line). Note here again the good accuracy of the interval at nearly all frequencies, and the discontinuity of the bounds by Zadeh and Desoer and by Dailey at at while at very low frequencies. The real s.s.v. is thus discontinuous at the zero frequency.

The maximal value of the upper bound by Zadeh and Desoer is 0.25 at while the maximal value of the lower bound by Dailey is 0.24 at rad/s. The accuracy of the estimate of the robustness margin is thus very good (less than 5 %). The corresponding uncertainties in the stability derivatives are 5/0.25 = 20%. The missile autopilot consequently presents good robust performance properties in

86

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

the presence of uncertainties in the aerodynamic model. When analyzing robust stability inside the left half plane, two peaks were obtained at (the real was discontinuous at this frequency) and at medium frequencies In the same way, when analyzing robust stability inside a truncated sector, two peaks are here again obtained at and at medium frequencies As a final point, Figure 6.4 presents the results obtained with the four upper bounds (classical complex upper bound in solid line, real upper bound by Jones in dashed line, mixed upper bound by Fan et al in dash-dotted line, real upper bound by Zadeh and Desoer in dotted line). The same comments can be done as for Figure 6.2.

1.3

ROBUST PERFORMANCE: SENSITIVITY FUNCTION (P3)

Applications of the

tools

87

Robust performance is analyzed in the frequency domain, using the sensitivity function S (see section 2. of chapter 2 for the definition of S). A fictitious performance block is added to the model perturbation, which now contains 4 non repeated real scalars and a full complex block.

The template on the sensitivity function S is essentially the same as the one in (Balas and Packard, 1992), except that the low frequency performance is relaxed and that the template is multiplied by factor 0.9 (see Figure 6.5). The low frequency performance is relaxed in order to focus on the performance at medium frequencies. On the other hand, the template is multiplied by 0.9, so as to choose the worst allowable performance as and (the bandwidth of the nominal sensitivity function S is thus leading to a nominal closed loop rise time

88

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

Figure 6.6 presents the mixed lower and upper bounds in solid and dashed lines and the complex upper bound in dash-dotted line. The mixed lower and upper bounds nearly coincide at all frequencies. The maximal value of the mixed upper bound is 0.87 at (the result is nearly non conservative). The corresponding uncertainty in the stability derivatives is thus Note finally that a peak at medium frequency is here again obtained (around 11 rad/s). This strongly suggests that the degradation of the stability and performance properties is essentially due to a decrease of the damping ratios of some closed loop poles at medium frequencies.

1.4

CONCLUSION

The missile autopilot was proved to exhibit good robust stability and performance properties in the presence of uncertainties in the aerodynamic model. The robustness margin obtained when analyzing the robust stability property inside the left half plane is very good, as well as the robustness margin corresponding to the robust stability property inside a truncated sector. The results obtained when defining the performance in the frequency domain (subsection 1.3) appear disappointing. However, as indicated in chapter 9, the poor quality of these results is due to the use of classical tools: it will come out that the results obtained when defining the performance in the frequency domain and when using skewed tools appear very close to those obtained when defining the performance through a truncated sector. Concerning the methods, see first the table above for a comparison of the estimates of the robustness margins, obtained with the different computational methods. In the case of the missile problem, whose dimension is not too large, the best results are obtained by combining the

upper bound by Zadeh and Desoer and the lower bound by Dailey. The mixed

upper bound by Fan et al also gives good results.

Applications of the

2.

tools

89

THE TRANSPORT AIRCRAFT

With reference to chapter 2, the rigid aircraft of subsection 1.1 is considered, and the robustness properties of the static output feedback controller of subsection 1.2 are studied. The case of the flexible aircraft will be considered in chapter 10. The model perturbation consequently contains 14 real non repeated scalars, corresponding to uncertainties in the stability derivatives. The weights in these stability derivatives are chosen as 10 %. Because of the large number of uncertainties, only polynomial time methods can be applied to this problem. See table 6.1 for a summary of numerical results.

2.1

ROBUST STABILITY (P4)

Figure 6.7 presents the mixed upper bound as a function of frequency The maximal value is 0.229 at The corresponding

90

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

uncertainties in the stability derivatives are The robust stability properties of the flight control system are thus very good. The star on Figure 6.7 represents the

lower bound of section 3. of

chapter 5 (method # 1 - see chapter 10 for a comparison of methods #1 to #3 in the case of the flexible aircraft). This lower bound is obtained as 0.184 at The gap between the bounds of the maximal s.s.v. over the frequency range is about 19 %, which is acceptable. The circle on this same Figure represents the lower bound by (Magni and Dö ll, 1997), which is obtained as 0.177 at The two lower bounds give thus a rather equivalent result in the case of this example.

2.2

ROBUST PERFORMANCE (P5)

Robust stability inside a truncated sector is now studied. The minimal degree of stability (resp. the minimal damping ratio) is chosen as 0.4

Applications of the

tools

91

(resp. 0.5). The nominal degree of stability (resp. the nominal damping ratio) is 0.7 (resp. 0.61). The results of Figure 6.8 are of the same type as those obtained in the previous subsection. The maximal value of the mixed upper bound is 0.648 at 0.79 rad/s. The corresponding uncertainties in the stability derivat-

ives are The robust performance properties of the flight control system are thus quite satisfactory.

The star on Figure 6.8 represents the lower bound of section 3. of chapter 5. When computing this lower bound as a function of frequency, two different peaks are obtained: the first one is 0.579 at (the corresponding value of the upper bound is 0.58 at while the second one is 0.564 at The lower bound of

92

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

the maximal s.s.v. over the frequency range is thus obtained as 0.579 at while the upper bound is obtained as 0.648 at The gap between the bounds of the maximal s.s.v. over the frequency

range is thus about 10.6 %, which is quite reasonable.

2.3

AN ADDITIONAL COMPARISON

As remarked in chapter 5, the mixed upper bounds of subsections 2.3 and 2.4 are equivalent. This can be checked on the example of Figure 6.9, which corresponds to the robust stability problem (P4). The 4 plots represent the complex upper bound, the mixed upper bound of subsection 2.4 and the mixed upper bound of subsection 2.3 (computed using the Analysis and Synthesis Toolbox and the LMI Control Toolbox). As expected, the results obtained with the 3 mixed upper bounds are essentially equivalent. The result provided by the LMI Control Toolbox is globally more accurate than the one provided by the Analysis and Synthesis Toolbox. Nevertheless, it should be noted that the algorithm of the Analysis and Synthesis Toolbox is designed in order to minimize the computational burden for large dimension problems. It is moreover possible to use various options, so as to achieve a balance between the accuracy of the result (i.e. the degree of optimality of the result) and the computational requirement.

IV

SKEWED

APPLICATIONS

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Chapter 7 SKEWED PROBLEMS IN ROBUSTNESS ANALYSIS

The aim of this chapter is twofold. The first one is to detail some of the physical problems, which are solved in the rest of the book. The second one is to illustrate the usefulness of the skewed (i.e. ) approach for engineering problems. To this aim, a large class of important practical problems is considered, which requires the skewed tool rather than the classical tool. See also chapter 11 for the presentation of a specific problem, which uses an extension of the and skewed tools, namely the problem of computing a robust delay margin in the presence of model uncertainties. Sections 1. and 2. give two examples, in which the problem of checking the robustness properties of a closed loop reduces to the problem of checking a small gain condition despite model uncertainties: as proved in chapter 8 (section 2.), this is a skewed problem involving an augmented model uncertainty. See also chapter 11 (subsection 3.3) for an other example of problem which reduces to checking a small gain condition despite model uncertainties. The difficult QFT problem of translating closed loop frequency domain specifications into open loop specifications on the MIMO controller frequency response can also be solved in an approximate way with the skewed tool. Here again, the problem reduces to the issue of checking a small gain condition despite a model uncertainty in the controller frequency response: see (Ferreres and LeGorrec, 1999) for further details (in a controller reduction context). Three other examples of skewed problems are given in sections 3. to 5., namely the direct computation of the maximal s.s.v. over a frequency interval, the maximization of the domain of the allowable model uncertainties and the analysis of gain-scheduled or robust adaptive controllers. 95

96

1.

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

CHECKING A SMALL GAIN CONDITION DESPITE MODEL UNCERTAINTIES

• Let the interconnection structure of Figure 7.1.a, in which the full complex block represents either a block of neglected dynamics, or a fictitious performance block (see section 4.4 of chapter 1). contains the closed loop dynamics and the weighting function, the latter corresponding either to the template on the neglected dynamics, or to the performance requirement (i.e. the template on the closed loop transfer matrix of interest). In the rest of the subsection, the complex matrix typically represents the value of the transfer matrix (the same remark can be applied to the complex matrices and ). On the one hand, if represents a block of neglected dynamics, this one is assumed to be maintained inside its unit ball (i.e. 1). The small gain theorem gives then a condition of stability of the interconnection structure, namely:

On the other hand, if represents a fictitious performance block, nominal performance is achieved if equation (7.1) is satisfied.

• A structured model perturbation is introduced in the closed loop, so that the transfer seen by the full complex block becomes an (see the new interconnection structure of Figure 7.1.b).

Skewed

problems in robustness analysis

Partitioning M compatibly with the

The transfer can be computed as:

between

97

as:

and z (with

on Figure 7.1.b)

If represents a block of neglected dynamics which is to be maintained inside its unit ball, the small gain theorem gives then a condition of stability of the interconnection structure of Figure 7.1.b, namely l :

On the other hand, if represents a fictitious performance block, robust performance (in the face of the model perturbation ) is achieved if equation (7.4) is satisfied. The issue is to find the maximal size of the structured model perturbation which still satisfies, either the robust stability property in the presence of a given amount of neglected dynamics or the robust performance property (if is a performance block). This maximal size

is given by

2.

NONLINEAR ANALYSIS IN THE FACE OF PARAMETRIC UNCERTAINTIES

As an other example, consider the closed loop system of Figure 7.2. represents a nonlinearity, whereas the represents a 1

The LFT of equation (7.3) is assumed to be well-posed despite the uncertainties in

98

A PRACTICAL APPROACH ROBUSTNESS ANALYSIS

parametrically uncertain transfer matrix, i.e. is a real model perturbation. The Sinusoidal Input Describing Function (SIDF) is intro-

duced for the nonlinearity For the sake of simplicity, is defined for a SISO nonlinearity. The definition is however readily ex-

tendible to the general case of a MIMO nonlinearity. A sinusoidal input is applied to The output of have odd symmetry, is written as:

which is supposed to

where contains the super harmonic part of signal the complex gain:

The SIDF is

In the context of the first harmonic approximation, the signal assumed to be filtered by the low-pass transfer matrix

is

The MIMO nonlinearity is now replaced by in the closed loop system of Figure 7.2 (see Figure 7.3.a). Roughly speaking, is a block of neglected dynamics, which takes into account the super harmonic part of the signal (Katebi and Zhang, 1995; Ferreres and Fromion, 1998). is only known by the relation:

where

is a known function of the magnitude X and frequency

of the limit-cycle.

Skewed

problems in robustness analysis

99

Figure 7.3.a is then reshaped into Figure 7.3.b. A sufficient condition for the absence of limit-cycles in the nonlinear closed loop is given by:

As in the previous section, for given values of X and the problem consequently reduces to the issue of checking a small gain condition in the presence of a real model perturbation The aim is indeed to compute the maximal amount of parametric uncertainties, for which the

sufficient condition (7.8) for the absence of limit-cycles remains satisfied.

3.

DIRECT COMPUTATION OF THE MAXIMAL S.S.V.

When applying analysis to the standard interconnection structure the s.s.v. is classically computed as a function of using a frequency gridding, and the robustness margin is deduced as:

However, especially in the case of flexible systems, narrow and high peaks may appear on the plot, so that only a prohibitively fine frequency gridding could find them (Freudenberg and Morton, 1992). An attractive solution in this case is to directly compute the maximal s.s.v. over the frequency range, or - more interestingly - over small frequency intervals (Ferreres and Fromion, 1997). As illustrated briefly below, this maximal s.s.v. can be computed as the solution of an augmented skewed problem. The idea is to treat the frequency as an additional uncertainty. A possible solution is provided by the following Lemma: see (Ferreres and Fromion, 1997) and included references, see also (Doyle and Packard, 1987; Helmersson, 1995) for alternative methods.

LEMMA 3..1 Let (A,B,C,D) a minimal state-space model of the asymptotically stable transfer matrix M(s). There exists a frequency and a perturbation satisfying if and only if there exists an augmented perturbation satisfying with:

100

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

and

Let

the maximal s.s.v. over the frequency interval The following Proposition claims that the computation of is a skewed problem: see also (Ferreres et al., 1996b) for an alternative skewed problem, in which the frequency is treated as a one-sided uncertainty 2.

PROPOSITION 3..2

is thus maintained inside the interval [0,1], while the initial model uncertainty belongs to the expanded or shrunk ball In the context of the robustness analysis of a flexible system, a first solution is thus to handle an augmented skewed problem in order to directly compute the maximal s.s.v. over a frequency interval. Nevertheless, we focus in this book on an alternative approach, which is computationally more efficient and which gives yet good results in practical examples: see chapter 10 and (Ferreres and Biannic, 1998a).

4.

PARAMETRIC ROBUSTNESS ANALYSIS

It is briefly illustrated that the tool, the sensitivities (see subsection 4.5 of chapter 1) and the skewed tool can be combined in order to maximize the stability domain of the closed loop in the space of a structured model perturbation: see (Ferreres and M’Saad, 1996) for an application to a missile example. For the sake of clarity, let a closed loop subject to just two parametric uncertainties and Figure 7.4 shows the stability domain in the space of and which correspond to normalized uncertainties in the stability derivatives and of a missile (Ferreres and M’Saad, 1996). The zero point corresponds to the nominal closed loop system, which is by assumption asymptotically stable. analysis provides the 2

See also chapter 11 (subsection 1.1) for the definition of a one-sided uncertainty.

Skewed

problems in robustness analysis

101

largest square in the parameter space, inside which closed loop stability is guaranteed. The tool however provides the largest rectangle in the parameter space, inside which closed loop stability is guaranteed.

Nevertheless, the s.s.v. and the measure can provide the same stability domain, e.g. in the case of Figure 7.5 ( and now correspond to normalized uncertainties in the stability derivatives and ). In practice, analysis is first applied. sensitivities detect then whether the situation of Figure 7.4 or 7.5 is encountered. In the situation of Figure 7.4, the guaranteed domain of stability in the space of model uncertainties is further extended with the tool.

5.

GAIN-SCHEDULED AND ADAPTIVE ROBUST CONTROL

Local properties of a gain-scheduled control system can be analyzed with the and tools. Let the vector of scheduling parameters: the

102

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

plant and controller can generally be expressed as LFTs involving a diagonal matrix which contains possibly repeated parameters (Ferreres et al., 1995). The augmented plant of Figure 7.6 is the closed loop obtained by connecting both LFTs. A structured perturbation is added to account for model uncertainties and performance blocks. Since the range of variation of the scheduling parameters is generally known a priori, a skewed problem is obtained, i.e. is maintained inside its prescribed range of variation, while the size of is free. An other skewed problem is obtained by introducing uncertainties in the scheduling parameters the maximal allowable value of is then computed, such that local stability or performance of the control system is guaranteed for all belonging to a prespecified set. Note finally that adaptive robust controllers can be analyzed in the same way (Ferreres et al., 1995).

Chapter 8 COMPUTATION OF SKEWED

Two mixed upper bounds and a mixed in this chapter.

1.

A FIRST

BOUNDS

lower bound are proposed

UPPER BOUND

Let a mixed structured perturbation (see equation (1.26)). Remember from chapter 5 that the sets and of scaling matrices D and G are associated to

Scaling matrix D must thus satisfy Let where and are mixed structured perturbations. Let the dimension of Scaling matrices associated to perturbations are introduced, with and are then defined as:

so that is a scaling matrix associated to perturbation Proposition 1..1 presents a first mixed upper bound (see subsection 2.1 of chapter 5 for the definition of the quantities and P ROPOSITION 1..1 (Ferreres and Fromion, 1997)

103

104

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

Remarks: (i) The optimal values of scaling matrices G in Proposition 1..1 can be computed with recent methods for solving LMIs: see chapter 5 (subsection 2.1). (ii) The upper bound of (Fan and Tits, 1992) is recovered when the model perturbation only contains full complex blocks. (in) The classical upper bound of (Fan et al., 1991) can be obtained in Proposition 1..1 by taking empty, so that It is possible under mild conditions to compute the exact value of by computing recursively the exact value of (see subsection 4.2 of chapter 1). Analogously, Proposition 1..2 claims that it is possible to compute the upper bound of Proposition 1..1 by computing recursively the upper bound of (Fan et al., 1991).

PROPOSITION 1..2 (Ferreres and Fromion, 1997) Let:

If iteration

2.

then

is the unique limit of the fixed point where h is defined as:

A SECOND

UPPER BOUND

An alternative mixed upper bound is proposed in this section (Ferreres and Fromion, 1999). The approach, which consists in transforming the problem into an augmented problem, basically uses the Main Loop Theorem.

2.1

CHECKING A SMALL GAIN CONDITION DESPITE MODEL UNCERTAINTIES IS A SKEWED PROBLEM

• First reshape the standard interconnection structure of Figure 8.1.a (with ) into the structure of Figure 8.1.b and partition M compatibly with the as:

Computation of skewed Remember finally in Figure 8.1.b that the LFT transfer w and z (with ) can be computed as:

bounds

105

between

is to be maintained inside its unit ball, while the size of is free. is moreover supposed to contain only real (repeated) scalars.

• Lemma 2..1 is essentially a skewed version of the Main Loop Theorem (see subsection 4.4 of chapter 1).

L EMMA 2..1

Proof of Lemma 2..1: the Lemma is essentially a scaled version of the Main Loop Theorem:

with:

106

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

Remark: condition means that the LFT is well-posed for all If is a fictitious complex block corresponding to a performance requirement, it is worth pointing out that is significant of the robust stability property in the face of uncertainties in Remember indeed that generally represents at frequency margin is computed as:

and that the associated robust stability

The rest of this subsection considers the case of a full complex block the general case of a mixed model perturbation will be considered in the following subsection. The following Lemma is essentially a restatement of Lemma 2..1, since in the case of a full complex block

L EMMA 2..2 If

then

and:

Checking a small gain condition despite model uncertainties is thus a skewed problem. It suffices indeed to compute an upper bound of v(M) so as to compute an upper bound of represents the maximal size of the model uncertainty the small gain condition is satisfied.

2.2

for which

THE CASE OF STRUCTURED UNCERTAINTIES

is now a mixed model perturbation. The following Lemma is essentially a combination of Lemmas 2..1 (chapter 5) and 2..1. L EMMA 2..3 Let and some (D,G) scaling matrices associated to (see equation (5.16)). Let If and:

then Proof of Lemma 2..3: Using Lemma 2..1 of chapter 5, simply note that equation (8.12) implies:

so that Lemma 2..1 can be applied.

Computation of skewed

2.3

AN AUGMENTED

bounds

107

PROBLEM

• As a preliminary, using equation (8.6), it is first easy to prove that:

with:

Note especially that • Lemma 2..4, which is inspired by a previous work in (Sideris and Pena, 1990), transforms the original skewed problem into an augmented problem (see appendix B for the proof).

L EMMA 2..4 Let:

Assume that

and let

Then:

if and only if:

Remarks: (i) Matrix X is invertible in the above Proposition because of the assumption (ii) Lemmas 2..1, 2..2 and 2..3 do not require the assumption of a real

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

diagonal model perturbation above Lemma.

This one is however necessary in the

• The mixed upper bound is presented in the following proposition. Its proof is a straightforward application of Lemmas 2..3 and 2..4.

PROPOSITION 2..5 Let M a complex matrix. Let and some (D, G) scaling matrices associated to the model perturbation . Let and H the complex matrix of equation (8.15). Let the s.s.v. of equations (8.16) and (8.17). Assume that and Then is an upper bound of

Remarks: (i) The following subsection presents simple methods for computing scaling matrices and (ii) For fixed values of scaling matrices and the simplest solution for obtaining a upper bound with Proposition 2..5 is to compute an upper bound of typically the mixed upper bound of (Fan et al., 1991). Since this one is directly available in the

Analysis and

Synthesis Toolbox or in the LMI Control Toolbox, the implementation of a computational algorithm is rather easy. (iii) Nevertheless, a technical difficulty is to check the condition when bounds are computed instead of the exact values. A solution is to check that where and mean lower and upper bounds.

2.4

COMPUTATION OF SCALING MATRICES AND

The main advantage of the upper bound above is its ease of implementation, so that we look for a simple method of computation of (sub) optimal scaling matrices and which minimize to some extent In the special but significant case of a full complex block the scaling matrices are simply and This is especially the case of a robust performance problem in the face of parametric uncertainties. Concerning the computation of a suboptimal diagonal scaling matrix which minimizes to some extent a first method uses the Perron eigenvector approach (Safonov, 1982). An associated routine is available in the Robust Control Toolbox of Matlab. The method is computationally efficient.

Computation of skewed

bounds

109

A second method for computing a suboptimal diagonal scaling matrix is to minimize the Frobenius norm instead of the 2-norm of see the classical Osborne’s method and its variations: see especially (Beck and Doyle, 1992), which proposes an efficient implementation of the method, and included references. Here again, an

algorithm is available in the Robust Control Toolbox of Matlab. Concerning the computation of a not necessarily diagonal scaling matrix see also the routines in the Analysis and Synthesis Toolbox.

Concerning the computation of a scaling matrix a simple suboptimal method is proposed in (Young et al., 1995). Loosely speaking, the idea is simply to cancel with the skewed hermitian part of the blocks of which correspond to the real parametric uncertainties. See also chapter 10 (subsection 5.2) for an other treatment of the same problem. See finally chapter 9 (section 4.) for a first evaluation of these methods on the missile example.

3.

A SKEWED

LOWER BOUND

Like the exact value of the exact value of can be obtained as the global maximum of a non convex optimization problem. In the spirit of (Packard et al., 1988; Young and Doyle, 1990), a power algorithm is proposed for solving this problem. The necessary conditions of optimality are rewritten as and a lower bound of corresponds to a limit of the sequence Even if the sequence does not necessarily converge and if the result depends on the initial value this algorithm

usually exhibits good convergence properties and a low computational burden. The optimization problem is first introduced in Proposition 3..1, which is an extension of Proposition 6.2.c of (Fan and Tits, 1992). The power algorithm is then proposed in propositions 3..2 (see appendix B for the proof) and 3..3.

PROPOSITION 3..1

where

and mi is the dimension of

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

is called the generalized spectral radius of matrices A and B. The rest of the exposition is simplified by considering a block structure with just a real repeated scalar, a complex repeated scalar and a full complex block. The general case of a mixed uncertainty is easily obtained by duplicating the appropriate formulae for each block.

PROPOSITION 3..2 Let lower bound

with

A

of v(M) must satisfy:

for some real scalar

with:

denotes the Euclidean norm. The complex vectors b, a, z and w are partitioned compatibly with the uncertainty structure as:

with e.g.

Remark: the above Proposition is an extension of Theorem 4 of (Young and Doyle, 1990). It is applicable under technical non-degeneracy conditions exposed in this reference.

Computation of skewed

bounds

111

A solution to this system of equations can be found via a power iteration method.

PROPOSITION 3..3 Consider the following power iteration:

where

and

evolve as:

if

then

else

if

then

else

and are chosen positive real so that If the algorithm converges to some equilibrium point, then a lower bound for

is

Remarks: (i) An alternative to equation (8.33) is to use a first order filter where If necessary, a large value of N can be chosen to recover the convergence properties of the standard power algorithm of (Young and Doyle, 1990). (ii) The power algorithms of (Packard et al., 1988; Young and Doyle,

112

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

1990) usually exhibit good convergence properties and a low computational burden. Nevertheless, when considering complex model perturbations, "limit-cycles can occur, and seem to occur more often when there

are large repeated scalar blocks" (Packard et al., 1988). In the same way, "significantly poorer convergence properties" are obtained in the case of

a real model perturbation (Young and Doyle, 1990). In this last case, note that the extension to skewed problems of the method of chapter 5 (section 3.) is essentially straightforward.

4.

SUMMARY Two mixed

upper bounds and a mixed

lower bound were proposed

in this chapter. All these bounds can be computed in polynomial time. The exact value of can be obtained by computing recursively the exact value of in the same way, the first mixed upper bound can be computed either directly, or recursively using the mixed upper bound of (Fan et al., 1991). A first solution for computing this mixed upper bound is thus to recursively compute the classical mixed upper bound of (Fan et al., 1991): this one is available in e.g. the Analysis and Synthesis Toolbox or in the LMI Control Toolbox. A more computationally efficient solution is to solve the quasi-convex LMI problem associated to this upper bound (using e.g. the LMI Control Toolbox). In this context,

when comparing the LMI problems associated to the mixed and upper bounds, it is worth pointing out that the computational complexity is the same. Concerning the second

problem into an augmented

upper bound, the idea is to transform the

problem. When splitting the model per-

turbation as remember as a preliminary that is to be maintained inside its unit ball while the size of is free. The main

advantage of this second upper bound is that it is easier to implement than the first one. Its computation is indeed done in two steps. The first one consists in computing and scaling matrices for the model perturbation

in the special but practically important case of a full

complex block and are simply chosen. Otherwise, simple suboptimal methods for computing and were proposed. The second step consists in a single computation of the upper bound of (Fan et al., 1991), applied to an augmented problem: this second step is straightforward, since the upper bound of (Fan et al., 1991) is directly available in standard Matlab softwares. The main drawback of this second upper bound is that can only contain real (possibly repeated) scalars (unlike the first mixed upper bound, which can be applied to a generic problem of robustness analysis) . Skewed problems with such a specific structure are nevertheless

Computation of skewed

bounds

113

encountered in practice. Chapter 7 especially illustrated the practical interest of the problem of checking a small gain condition despite parametric uncertainties (i.e. is a full complex block while contains the parametric uncertainties). Such a problem can be especially encountered when analyzing the robust performance properties of a closed loop in the presence of parametric uncertainties. As an other physical example, in a linear closed loop containing nonlinearities (e.g. saturations), a sufficient condition for the absence of limit-cycles corresponds to a small gain condition, which is to be satisfied despite LTI parametric uncertainties in the linear part of the closed loop. A power algorithm in the spirit of (Packard et al., 1988; Young and Doyle, 1990) is finally proposed for the computation of a mixed lower bound.

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Chapter 9 APPLICATION OF THE SKEWED

TOOLS

The aim of this chapter is threefold. The first one is to analyze the robust stability and performance properties of the missile autopilot in the presence of aerodynamic uncertainties and high frequency bending modes: see also (Ferreres and Fromion, 1999). The second purpose is to illustrate the usefulness of the skewed tool in the context of a realistic application. The applicability of the computational methods, which were presented in chapter 8 (see section 4. for a summary), is finally evaluated trough this example.

1.

INTRODUCTION

The reader is first referred to chapters 2 (subsection 2.2) and 4 (section 1.) for the description of the linearized missile model and the building of the interconnection structure. The local stability and performance properties of the autopilot are to be analyzed in the presence of parametric uncertainties (in the 4 stability derivatives and neglected dynamics (a high frequency bending mode). The model perturbation consequently contains 4 non repeated real scalars and a single full complex block. The weights in the stability derivatives are chosen as 5 %.

2.

ROBUST STABILITY

Robust stability is first analyzed with the tool (see Figure 9.1). The maximal value of the mixed upper bound is 0.24 at 228 rad/s, so that the maximal uncertainty in the stability derivatives is However, in the context of (Balas and Packard, 1992), the controller must tolerate a given amount of uncertainty in the bending mode, as defined 115

116

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

by the template. It is thus logical to maintain this uncertainty inside its unit ball in the analysis problem. The maximal value of the first mixed upper bound is obtained as 0.13 at so that the maximal uncertainty in the stability derivatives becomes (see Figure 9.1).

Remarks: (i) In figures 9.1 to 9.3, "mixed upper bound" should be understood as the upper bound by (Fan et al., 1991), whereas the 'mixed lower bound" should be understood as the lower bound by (Young and Doyle,

1990). The "complex upper bound" corresponds to the specialization of the mixed upper bound to the case of complex uncertainties (see subsection 2.2 of chapter 5 - in the context of the missile example, the real nature of the parametric uncertainties is not accounted for in the

Application of the skewed µ tools

117

computation of the complex upper bound). The "first mixed upper bound", the "second mixed upper bound" and the "mixed lower bound" are defined in chapter 8. (ii) Neither the mixed lower bound, nor its skewed version are presented in Figure 9.1, since the power algorithms did not converge. The problem at low and medium frequencies is indeed too close to a real problem.

3.

ROBUST PERFORMANCE

Robust performance is analyzed. A fictitious performance block is added to the model perturbation, which thus contains 4 non repeated real scalars and two full complex blocks. The template on the sensitivity function S is the same as in chapter 6 (subsection 1.3).

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

When applying the tools to this robust performance problem l , the maximal s.s.v. is 0.86 at (see Figure 9.2 - the result is nearly non conservative). The corresponding uncertainty in the stability derivatives is thus However, if the performance block and the bending mode are maintained inside their unit balls, the maximal uncertainty in the stability derivatives becomes (see the bottom diagram of Figure 9.2). This illustrates that a brute application of analysis can provide an overly conservative result in the context of a robust performance problem.

The skewed s.s.v. presents two peaks at and 12 rad/s. When comparing the upper and lower bounds at these frequencies, the result is nearly non conservative for the first peak and reasonably conservative (6 % between the two bounds) for the second one.

4.

A FURTHER STUDY OF THE BOUNDS

UPPER

•

When comparing the two upper bounds in figures 9.1 and 9.2, the second upper bound appears more conservative (at least in this example). Nevertheless, this conservatism remains quite reasonable: note

especially that the two upper bounds are nearly identical around the peaks on the plots, so that the robustness margin obtained with both upper bounds is nearly the same. This justifies to a large extent this second upper bound, since it is easier to implement than the first upper bound.

•

In the example of Figure 9.2, the model perturbation (which is to be maintained inside its unit ball) contained two one-dimensional complex blocks, namely a fictitious performance block and a block of neglected dynamics. The scaling matrix was consequently chosen as while the scaling matrix was computed with the Perron eigenvector approach. To further evaluate the methods of section 2.4 (chapter 8), a new mixed model perturbation is chosen, which contains the fictitious performance block, the neglected dynamics and two parametric uncertainties. Remember there are 4 parametric uncertainties in and In the lower subfigure of Figure 9.3, and are included in the model perturbation i.e. the associated parametric uncertainties are 1

This problem is very close to the problem of subsection 1.3 (chapter 6), the only difference being the full complex block which corresponds to the high frequency bending mode and which is not taken into account in chapter 6 (subsection 1.3).

Application of the skewed

tools 119

to be maintained within Conversely, in the upper subfigure of Figure 9.3, and are included in

The following quantities are computed:  The first mixed

 The second mixed

upper bound: this one is used as a reference.

upper bound with scaling matrices and the structure of the model perturbation is not accounted for. This upper bound is not presented in the upper subfigure of Figure 9.3, because of the bad results which were obtained: the assumption was not satisfied on many points of the fre-

120

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

quency gridding, so that the value of the upper bound was chosen as on these frequency points.  The second mixed

upper bound with suboptimal scaling matrices and matrix was computed with the suboptimal method by (Young et al., 1995), whereas matrix was computed using either the Perron eigenvector approach or Osborne’s method. The results provided by both methods are most generally equivalent. When comparing with the first mixed upper bound, the second mixed upper bound with suboptimal scaling matrices and appears more conservative, but this additional conservatism is quite reasonable.

V

NONSTANDARD

APPLICATIONS

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Chapter 10 ROBUSTNESS ANALYSIS OF FLEXIBLE STRUCTURES

1.

INTRODUCTION

The previous chapters emphasized the usefulness and efficiency of the analysis techniques. Nevertheless, as remarked in chapter 7 (section 3.), the application of these techniques is unreliable in specific fields, such as the control of flexible structures. Remember indeed that the principle of analysis is to compute the s.s.v. as a function of frequency the robustness margin is then deduced as the inverse of the maximal s.s.v. over the frequency range. In practice, the s.s.v. is usually computed at each point of a frequency gridding. This technique is however unreliable in the case of narrow and high peaks on the plot, since it becomes possible to miss the critical frequency (i.e. the frequency for which the maximal s.s.v. is obtained may lie between two points of the gridding), and thus to overevaluate the robustness margin. This problem especially arises in the case of flexible systems (Freudenberg and Morton, 1992). As a consequence, there has been a regain of interest for robustness analysis techniques, which do not use a frequency gridding: see e.g. (Ly et al., 1994). The solution proposed in chapter 7 (section 3.) consists in transforming a classical frequency dependent analysis problem into an augmented skewed problem, in which the frequency appears as an additional uncertainty (namely a real repeated scalar). It becomes then possible to directly compute the maximal s.s.v. over a frequency interval Two problems however arise: Because of the NP hard characteristics of the problem, an upper bound is computed instead of the exact value of In order to reduce the conservatism of this upper bound, which is calculated for the 123

124

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

augmented intervals

problem, it is more interesting to consider a set of small instead of a large frequency interval

The size of the augmented problem (which is to compute the max-imal s.s.v. over an interval can be much larger than the size of the original problem (the computation of the s.s.v. at a fixed frequency ). With reference to subsection 2.3 of chapter 5, remember that the problem of computing the upper bound of (Fan et al., 1991) reduces to an LMI problem, in which the optimization

parameters are scaling matrices D and G. The computation of the optimal D,G scaling matrices for the augmented problem can be a computationally (very) involving task: more precisely, with reference to Lemma 3..1 of chapter 7, the model perturbation corresponding to the augmented problem contains the initial model uncertainties and the frequency. The associated D,G scaling matrices can thus be split into the scaling matrices, corresponding to the uncertain fre-quency, and the scaling matrices corresponding to the initial model uncertainties. The computational burden of the augmented problem is often very heavy, because the scaling matrices contain numerous optimization parameters of the LMI problem: the uncertain frequency appears indeed as a repeated real scalar where m denotes the order of the state-space model of M(s) in the standard interconnection structure

An alternative solution was proposed in e.g. (Feron, 1997; Gahinet et al., 1995). The idea is to synthesize D,G scaling matrices, which simultaneously work at two neighboring frequencies and Since this method does not use the augmented problem above, the computational burden is expected to be much lower. Moreover, this heuristic technique can be theoretically justified: it is proved indeed that when the frequencies and are sufficiently close (in a sense which will be precisely defined in the following), the D,G scaling matrices work on the whole segment, defined by its two extremal frequencies and Even when the frequencies and are not a priori sufficiently close, D,G scaling matrices can be synthesized, which simultaneously work at these two frequencies, and it is checked a posteriori whether the D,G scaling matrices work indeed on the whole segment. This chapter especially proposes an easy and yet rigorous method, for solving this last problem of checking the validity of D,G scaling matrices on the whole segment. This method essentially relies on the alternative formulation of the mixed upper bound of (Fan et al., 1991), which was proposed in (Young et al., 1995). The technique is finally applied to the flexible airplane problem (Fer-

Robustness analysis of flexible structures

125

reres and Biannic, 1998b) and to the telescope mock-up (Ferreres et al., 1998). See also (Magni et al., 1999) for an alternative technique, which is complementary to the one proposed here. The chapter is organized as follows. The relationship between both formulations of the mixed upper bound of (Fan et al., 1991; Young et al., 1995) is clarified in the second section. The basis of the method is presented in sections 3 and 4. The algorithm is summarized in section 5. The application is finally done in section 6.

2.

RELATIONSHIP BETWEEN BOTH

FORMULATIONS OF THE BOUND

UPPER

The reader is referred to chapter 5 (subsection 2.3) for a presentation of both formulations of the mixed upper bound of (Fan et al., 1991; Young et al., 1995). The following Lemma is extracted from (Young et al., 1995). It is worth pointing out that the transformations in the Lemma below (from (D,G) to and from to (D,G)) do not depend on the complex matrix M.

LEMMA 2..1 Let:

with (10.1). Let:

Assume that

and

satisfy equation

where the unitary matrix U and the hermitian positive definite matrix P result from the polar decomposition:

Then that

and and

Then

and

satisfy equation (10.2). Conversely, assume satisfy equation (10.2). Let:

satisfy equation (10.1).

126

3.

3.1

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

COMPUTATION OF D, G SCALING MATRICES AT TWO FREQUENCIES THE LMI PROBLEM

Let the complex matrices and Let a structured model perturbation. The issue is to find the minimal positive value of scalar for which there exist scaling matrices and satisfying:

This is a well posed generalized eigenvalue problem, which consists in finding the minimal value of satisfying with:

and:

Remark: when comparing the above LMI problem with the one associated to the mixed upper bound of (Fan et al., 1991) (see equation (10.1)), the number of constraints is increased, but the number of optimization parameters is the same.

3.2

A THEORETICAL JUSTIFICATION OF THE APPROACH

Let D and G some scaling matrices satisfying equation (10.6) at two frequencies and Let and the corresponding scaling matrices, computed with Lemma 2..1. and thus satisfy:

with Assume that a first order approximation of The spectral norm

and are sufficiently close, so that is valid:

is convex, i.e. the following relation is satisfied for

Robustness analysis of flexible structures

where and implies that

127

are any fixed matrices. Equation (10.9) consequently

As a consequence, (D,G) scaling matrices, synthesized on frequencies and are valid on the whole segment when a first order approximation of is valid on this frequency interval.

3.3

CONCLUSION

A first solution is to choose a sufficiently fine frequency gridding so that the first order condition of subsection 3.2 is satisfied at all points of the frequency gridding:

(D,G) scaling matrices are then computed, which simultaneously work at frequencies and in order to compute (D,G) scaling matrices, which are valid on the whole interval Nevertheless, this approach will often need an unnecessarily fine frequency gridding, since the above equation is just a sufficient condition for guaranteeing the property that (D,G) scaling matrices, which simultaneously work at frequencies and also work on the corresponding interval. An alternative solution is to choose a looser frequency gridding (D,G) scaling matrices, which simultaneously work at frequencies and are here again computed, and the validity of these D,G scaling matrices on the frequency segment is checked only a posteriori. In this new context, a solution is to use Proposition 4..2 of the following section.

4.

CHECKING A POSTERIORI THE VALIDITY OF D,G SCALING MATRICES ON A FREQUENCY INTERVAL

A technical Lemma, which is the basis of this section and coincides with Lemma 2..4 of chapter 8, is presented in the first subsection. The second subsection proposes a method for introducing the frequency as an additional uncertainty (see section 3. of chapter 7 for alternative methods). The technique for checking the validity of D,G scaling matrices on a frequency interval is finally presented in the third subsection.

128

4.1

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

A TECHNICAL RESULT

Let the LFT transfer between and z in Figure 10.1. is a real diagonal model perturbation, whereas the complex matrix M is partitioned as:

The transfer between

and z is thus:

L EMMA 4..1 Let:

Assume that

if and only if:

Then:

Robustness analysis of flexible structures

4.2

129

INTRODUCTION OF THE FREQUENCY

In order to apply the previous Lemma, an LFT model is to be derived for the dynamic system M(s), in which the frequency appears as a real parameter. The issue is more precisely to determine a complex matrix H such that, for a given strictly positive frequency

Let (A,B,C,D) a state-space representation of M ( s ) , and m the dimension of matrix A. It is well known that (see Figure 10.2 with

with:

Further note that:

with:

As a consequence:

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

where chapter 3):

is the star product of

Interestingly, the LFT

and T (see subsection 3.1 of

remains well-posed for

Equation (10.19) is thus valid for all

4.3

CHECKING THE VALIDITY OF D,G SCALING MATRICES

Consider

scaling matrices, which satisfy at frequency

with is thus an upper bound of The issue of this subsection is to compute the maximal size interval containing and on which the above inequality holds true. is then an upper bound of inside the interval The following Proposition is essentially an application of Lemma 4..1.

PROPOSITION 4..2 If:

then the inequality:

holds true for where Let (see also equation (10.25)):

and:

and

are computed as follows.

Robustness analysis of flexible structures

131

and:

Let then:

Let

the real negative eigenvalue of

Let then Then:

of maximal magnitude. Then:

the real positive eigenvalue of

of maximal magnitude.

Proof: z Note as a preliminary that (see equation (10.31)):

and:

As a consequence:

Condition (10.28) in the above Proposition can be simply rewritten as which is equivalent to the condition in Lemma 4..1. z Let The assumption in Lemma 4..1 means that the LFT (or equivalently since ) is well-posed: see the discussion in subsection 4.2. z Lemma 4..1 can thus be applied. Equation can be rewritten as which means that the satisfying this last equation are the inverses of the eigenvalues of z

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

5. 5.1

AN ALGORITHM THE BASIC ALGORITHM

The issue is to compute the maximal s.s.v. frequency interval Let:

over a large

The basic algorithm consists in the following steps: is split into a set of small frequency intervals

z z The

method of subsection 3.1 is applied to each interval

The

validity of the computed D,G scaling matrices on the interval is checked with the method of subsection 4.3. If the scaling matrices are indeed valid on a upper bound is found on this interval, and nothing more is to be done inside Otherwise, the interval is split into

two smaller intervals

and

and

the method of subsection 3.1 is applied to each of these two intervals. This process of branching on the frequency is repeated until a upper bound is found on each frequency interval whose union

gives the large initial frequency interval z A reliable upper bound of is the maximal value of the Either the state-space method by (Magni and Döll, 1997), or the frequencydomain method of chapter 5 (section 3.) provides a lower bound of If the gap between the bounds is not sufficiently small, it is possible to further branch on the frequency, in order to decrease the value of the upper bound of

5.2 of

AN IMPROVEMENT

In the above algorithm, if the aim is just to compute an upper bound it is unnecessary to apply the method of subsection 3.1 to all

intervals Assume indeed that a lower bound of was a priori computed (i.e. before applying the algorithm of the previous subsection). If the s.s.v. is proved to be less than on a frequency interval the critical frequency (which corresponds to the maximal s.s.v. over the frequency range) does not belong to this interval. It is thus useless to apply the algorithm of the previous subsection to this

interval. The issue is to find a computationally cheap method for checking whether the s.s.v. is less than on a frequency interval

Robustness analysis of flexible structures

133

A solution is to proceed as follows (see also (Magni et al., 1999) for an alternative method): Let a frequency point which may belong to the gridding in the algorithm of the previous subsection. Find (sub)optimal values

of

scaling matrices, which minimize the singular value:

with

Remember that

is fixed.

• If the singular value above is found to be less than unity, apply the method of subsection 4.3 to compute a frequency interval inside which the s.s.v. is guaranteed to be less than The desire to minimize the singular value in equation (10.38) should be

understood as an heuristic way to maximize the size of the interval The issue is thus to minimize to some extent the singular value in equation (10.38). In the aircraft example of the following section, the minimization was done as follows (see also subsection 2.4 of chapter 8): • Computation of a suboptimal diagonal

scaling matrix, which min-

imizes to a large extent with the Perron eigenvector approach: see (Safonov, 1982). The method is computationally efficient and yet accurate. A routine is moreover available in the Robust Control Toolbox of Matlab.

• Computation of an initial value of the

scaling matrix with the idea

of (Young et al., 1995). Loosely speaking, the idea is simply to cancel with the skewed hermitian part of the blocks of correspond to the real parametric uncertainties.

which

• A simple gradient method further minimizes the quantity in equation (10.38) with respect to Note that this quantity is not necessarily differentiable, and that the optimization problem is seemingly non convex with respect to It is thus important to have a good initial

guess for Despite its roughness, the method above gave quite good results in the example. Nevertheless, more sophisticated methods could be investigated, with the constraint to remain computationally cheap. Otherwise, the best solution would be to apply the basic algorithm of the previous

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

subsection. Remark: the idea of reducing the computational burden by eliminating frequency intervals, which can not contain the critical frequency, can be traced back at least to (Ferreres et al., 1996b). Nevertheless, this reference uses the augmented problem, in which the frequency is treated as an additional uncertainty. The approach proposed here is computationally much cheaper.

6.

APPLICATION TO THE FLEXIBLE AIRCRAFT

Following chapter 4 (section 2.), two methods are available for introducing uncertainties in the frequencies of the bending modes. Each uncertainty is represented, either by a non repeated real scalar, or by a

Robustness analysis of flexible structures

135

twice repeated real scalar. Both representations give nearly equivalent results in the following: we consequently present the results obtained with the simplest representation, i.e. the one with non repeated real scalars.

6.1

APPLICATION OF THE BASIC ALGORITHM

As explained in chapter 4 (section 2.), uncertainties are simultaneously introduced in the 14 stability derivatives and in the natural frequencies of the 6 bending modes, whose natural frequencies are 7.35, 8.62, 12.5, 13.5, 14.1 and 14.3 rad/s. The weights in the stability derivatives are chosen as 10 %, while the weights in the frequencies are chosen as 20 %. The model perturbation contains 20 real non repeated scalars. The basic algorithm of subsection 5.1 is first applied. 50 points were

136

A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

chosen for the initial gridding, namely 25 points between 0.01 and 100 rad/s and 25 points between 5 and 20 rad/s. The results are presented in figures 10.3 and 10.4. The value of the upper bound of is 1.97 (between 9.93 and 10.00 rad/s): the flight control system can thus tolerate an uncertainty of 5.1 % in the stability derivatives and an uncertainty of 10.2 % in the frequencies of the bending modes. The value of the lower bound provided by (Magni and Döll, 1997) is 1.94 at 9.97 rad/s (this lower bound is represented as a "*" in figures 10.3 and 10.4). The gap between the bounds is consequently less than 2 %. The computations were done in 1700 s (about half an hour) on an efficient Sun SPARCstation. A lower bound of the maximal s.s.v. over the frequency range is computed with the method of section 3. of chapter 5 (see subsection 6.4 for details). The frequency gridding is simply chosen as 20 points between 9.93 and 10.00 rad/s, since the maximal value of the upper bound of was found inside this interval. The value of the lower bound is 1.966 at 9.95 rad/s (this lower bound is represented as a "o" in figures 10.3 and 10.4). The result is thus even more accurate than the one provided by (Magni and Döll, 1997).

6.2

APPLICATION OF THE IMPROVED ALGORITHM

The improved algorithm of subsection 5.2 is applied. Most of the frequency intervals, which correspond to the 50 points of the initial frequency gridding, were eliminated, and the basic algorithm of subsection 5.1 is applied only to 8 intervals. The same result is obtained for The computations were done in only 474 s (about 8 minutes). This improved solution appears thus especially computationally efficient. Moreover, the result is nearly non conservative.

6.3

A PHYSICAL INTERPRETATION OF THE LOWER BOUND

The aim of this subsection is to illustrate the usefulness of a µ lower bound. Let (A, B, C, 0) a state-space representation of the transfer matrix M(s) in the interconnection structure A destabilizing perturbation is provided with the lower bound. Remember that the value of this lower bound is 1.94 at 9.97 rad/s with the method by (Magni and Döll, 1997). This means that the norm of is 1/1.94 and that the closed loop state matrix has a pole on the imaginary axis at This is confirmed by the root locus of Figure 10.5. This root locus

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137

was obtained by plotting the eigenvalues of when varies between 0 and 1. The star "*" represents the initial poles (associated to while the circle "o" represents the final poles (associated to The closed loop rigid poles at the bottom of the Figure are not especially moved by the application of the destabilizing perturbation

The 2 closed loop flexible modes, which are not actively controlled (see subsection 1.4 of chapter 2), are neither especially moved by the application of Note that two poles are associated to each flexible mode in the figure: one for the state-feedback controller and one for the observer. These two poles do not coincide, even if they are close, because the state-feedback and observer gains were computed with two different methods. The 4 closed loop flexible modes, which are actively controlled, are

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

moved indeed by the application of

The closed loop flexible pole,

which crosses the imaginary axis, corresponds to the state-feedback closed loop pole:

and thus to the open loop flexible mode:

This suggests that the robustness of the observed state-feedback controller could be increased by improving the robustness with respect to an uncertainty in the characteristics of this critical bending mode.

6.4

COMPUTATION OF A LOWER BOUND OF

This subsection compares the results obtained with the three methods proposed in chapter 5 (section 3.). A rough frequency gridding is first considered. The three methods are then evaluated around the critical frequency. 6.4.1

USE OF A ROUGH FREQUENCY GRIDDING

The frequency gridding is chosen as a set of 30 points logarithmically spaced between 0.01 rad/s and 100 rad/s. The results are presented in the above table. It is especially worth emphasizing that methods (1) and (2) only detect a single peak, while the third method, which is based on Linear Programming, detects the two most critical peaks (these ones also appear in Figure 10.4).

Robustness analysis of flexible structures

6.4.2

139

COMPUTATION AROUND THE CRITICAL FREQUENCY

Let the critical frequency interval, which correspond to the maximal value of the upper bound (namely [9.93 rad/s, 10.00 rad/s]). When choosing as a frequency gridding a few points between and it is interesting to point out that the same result is obtained with all three methods. The lower bound is 1.966 and the critical frequency is 9.95 rad/s. 6.4.3

CONCLUSION

Two different cases are to be considered: if the method of subsection 5.1 has been already applied or if our physical knowledge of the problem enables to predict the frequency domains corresponding to the peaks on the plot, methods (1) or (2) are the most suitable ones, since they give good results while being very simple to implement. Otherwise, if no guess of the critical frequency domains is available, method (3) is especially attractive, since it seemingly enables to detect the peaks on the plot, even when using a rough frequency gridding (at least in the case of the example). Moreover, remember that the improved algorithm of subsection 5.2 assumes that a lower bound of good quality is a priori available. When applying method (3) on a rough frequency gridding, such a lower bound may be obtained, and the associated computational burden remains reasonable.

7.

THE TELESCOPE MOCK-UP

The issue is to check the validity of the 13th order controller on the full 46th order plant. Uncertainties are introduced in the natural frequencies of the 20 bending modes contained in the identified telescope model: see chapter 2 (section 3.) for the presentation of the mock-up and

chapter 4 (section 2.2) for the computation of the LFT model. A model perturbation with 20 non repeated real scalars is to be handled. The basic algorithm of subsection 5.1 is first applied. Results are shown on Figure 10.6, which displays the upper bound as a function of frequency. Note first the very fine peaks on the plot. An upper bound of the maximal s.s.v. over the frequency, interval [0, 1600 rad/s] is obtained as 46.80, between 1101.75 rad/s and 1103.95 rad/s. The controller can thus tolerate an uncertainty of only in the frequencies of the bending modes. The star on Figure 10.6 corresponds to the lower bound by (Magni and Döll, 1997), namely 44.94 at 1103.04 rad/s. The gap between the bounds is less than 4 % , the result is nearly non conservative.

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

The associated computations were achieved in 14776s (about 4 hours) on an efficient Sun SPARCstation. If the aim is just to compute an upper bound of the maximal s.s.v., the improved algorithm of subsection 5.2 can be applied. All frequency intervals are eliminated, except the one between 1097.34 rad/s and 1114.98 rad/s. The basic algorithm is then applied to this single interval, and the same value is obtained for the upper bound of the maximal s.s.v., namely 46.80 between 1101.75 rad/s and 1103.95 rad/s. The associated computations were achieved in 3210 s (less than an hour).

For both algorithms, the computational requirement is thus not negligible in the case of this example. Nevertheless, the computed value of the robustness margin is reliable, in the sense that it can be guaranteed that a critical peak on the plot was not missed. The result is moreover

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141

nearly non conservative, and the telescope problem is quite challenging: the more flexible the example, the heavier the computational burden. In the case of very lightly damped flexible modes, the frequency intervals have to be chosen very small to detect very narrow and high peaks on the plot. In practice, the computational burden simultaneously depends on the structure of the model perturbation (this burden increases with the complexity of this structure - 20 real scalars in the case of this example), on the values of the damping ratios of the bending modes (roughly between 0.2 % and 2 % in the case of this example) and on the number of these bending modes (20 modes in the case of this example).

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Chapter 11 ROBUSTNESS ANALYSIS IN THE PRESENCE OF TIME DELAYS

The problem of analyzing the robustness properties of a closed loop in the presence of classical model uncertainties (parametric uncertainties and neglected dynamics) and uncertain time delays presents a great practical interest. Indeed, when embedding control laws on a real-time computer, time delays are to be accounted for at the plant inputs (because of the time needed for computing the value of the plant input signal as a function of the plant output signal) and outputs (because of the sensors which measure the plant output signal). The aim of this chapter is to provide an algorithm for computing a robust delay margin: see (Ferreres and Scorletti, 1998) for a complete theoretical justification. The method is presented in a qualitative way in the first section. The second section details the computational algorithm. An alternative small gain approach is presented in the third section. Both methods are compared on the missile problem in the fourth section: time delays are added at the input and outputs and uncertainties in the stability derivatives are considered.

1.

INTRODUCTION TO THE PROBLEM

A frequency dependent test is introduced, whose resolution at each frequency essentially reduces to a problem with a specific structure (namely a one-sided skewed problem). If the exact value of could be computed, the exact value of the robust delay margin could be obtained.

1.1

A ONE-SIDED SKEWED

PROBLEM

As an extension of the classical the one-sided skewed of a complex matrix M is defined (Ferreres et al., 1996b). In the stand143

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

ard interconnection structure

the uncertainty is split as diag is a mixed perturbation, while contains real (possibly repeated) scalars

Let the one-sided unit ball

D EFINITION 1..1

1.2

A GENERIC EXAMPLE

For the sake of clarity, the method is introduced with the following generic example. Let the uncertain plant model, i.e. a MIMO transfer matrix which depends on a vector of parametric uncertainties. Following chapter 3, can be transformed into an LFT where:

is a model perturbation with real possibly repeated scalars Controller K(s) is connected with and a model perturbation is added at the plant inputs (see Figure 11.1). represents the uncertain time delays:

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145

This uncertain closed loop is finally transformed into the standard interconnection structure M(s) M(s) is simply the transfer matrix seen by the model perturbation in the closed loop of Figure 11.1.

Remark: other examples could be considered, since other sources of parametric uncertainty or neglected dynamics could be introduced at various points of the closed loop (see chapter 1). Time delays could be moreover introduced at the plant outputs, in order to take into account delays in the sensors. The key issue is to obtain the standard interconnection structure M(s) with is a mixed model perturbation gathering all classical model uncertainties (parametric uncertainties and neglected dynamics), whereas the model perturbation contains all uncertain time delays. The robust delay margin is now defined in the specific context of the example. Assume that each parametric uncertainty is normalized, so that the vector belongs to the unit hypercube:

A specific value of is applied to the closed loop of Figure 11.1. Let the MIMO delay margin associated to this closed loop: this means that the robust stability of the closed loop is guaranteed for all values of time delays inside the interval The robust MIMO delay margin is then defined as the minimal value of over the unit 1 hypercube :

1.3

COMPUTATION OF THE ROBUST

DELAY MARGIN The aim of this subsection is to present the principle of the method in a qualitative way: see the following section for a detailed algorithm.

• Let M(s) -

the standard interconnection structure, with diag Remember is a mixed model perturbation gathering all classical model uncertainties, whereas gathers all uncertain time delays (see equation (11.4)). 1 It could also be interesting to consider the alternative problem, which is to compute the maximal size of the model uncertainties, for which a robust delay margin is guaranteed.

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

Assuming that the nominal closed loop (i.e. the closed loop without time delays and model uncertainties) is asymptotically stable and with reference to the Nyquist stability criterion (Ferreres and Scorletti, 1998), the key point is to note that the analysis of the robust stability property of the closed loop reduces to the problem of detecting at each frequency the singularity of the matrix i.e.:

The frequency is fixed in the following, and the dependence is drop out, so that M and mean the values of M(s) and

• The model perturbation

is redefined as:

where the

are non repeated complex scalars. These are rewritten as so as to introduce uncertain phases or time delays into each SISO channel of the closed loop.

At this stage, classical tools can be applied in order to compute conservative values of the MIMO phase and delay margins: see (Safonov, 1982) and section 3.. The problem is that this method does not account for the equality norm constraint

it just imposes an inequality

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147

constraint This explains why conservative results are obtained with this method, even if the exact value of is computed. In order to fully take into account this information and with reference to the bilinear transformation, the idea is simply to rewrite the complex

scalars

as:

where and the real scalar belongs to the interval (see Figure 11.2). The magnitude of is now constrained to be 1, and a real scalar is to be handled. is thus (see also Figure 11.3 for an LFT realization):

• We come back to the initial interconnection structure complex scalar

with With reference to equations (11.8) and (11.10), each is rewritten as an LFT with:

In the same way, the model perturbation where:

is rewritten as

As a consequence, the initial interconnection structure of Figure 11.4.a can be transformed into the new equivalent interconnection structure of Figure 11.4.b, where and (see subsection 3.1 of chapter 3 for the definition of the star product The interconnection of LFTs is indeed an LFT.

• The robust delay margin is to be computed, i.e. the minimal value of the MIMO delay margin when the model perturbation belongs to its unit ball (see the end of subsection 1.2). With reference to equation (11.7), this reduces to the computation at each frequency of is indeed to be maintained inside its unit ball while the in are to be expanded in Note indeed that is equivalent to in the relation:

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

Remark: consider the problem of checking whether a template on a closed loop transfer function remains satisfied despite classical model uncertainties (parametric uncertainties and neglected dynamics). Following chapter 1 (section 4.4), this classical robust performance problem can be converted into an augmented stability problem, in which a full complex block is added to the structured model perturbation. As a consequence, the method described in this section can be extended to the problem of checking whether a template on a closed loop transfer function remains satisfied despite parametric uncertainties, neglected dynamics and delays.

2. 2.1

A DETAILED ALGORITHM COMPUTATION OF BOUNDS

The methods of subsections 1.1 and 1.2 (chapter 5) are briefly extended to the computation of a bound of where M is a complex matrix and contains only real non repeated scalars. The idea is simply to note that vertices (resp. two-dimensional faces) are handled in the method of subsection 1.1 (resp. 1.2). Let a vector of parametric uncertainties. Two cases are to be considered: 1.

is to be expanded inside its one-sided unit ball

2.

is to be maintained inside its unit ball (i.e. the unit hypercube D).

In the context of the method of subsection 1.1 (chapter 5), remember that all scalar components of a vertex take an extreme value. In the first case above, this means that is chosen as or In

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149

the second case, In the context of the method of subsection 1.2 (chapter 5), belongs to a two dimensional face of the hypercube if all its components take an extreme value, except two ones. The extreme values are chosen in the same way as above. The two other components and are obtained as in subsection 1.2 (chapter 5). In the first case, is an acceptable solution if and belong to the interval [0, k] . In the second case, is acceptable when and are inside [–1, 1].

Remark: both methods above are exponential time, and they are applicable to problems with a few parametric uncertainties and delays, which is the case of the missile example of section 4.. See (Ferreres et al., 1996b) for the computation of a polynomial time upper bound of

2.2 THE DETAILED METHOD • The difficulty of the method of subsection 1.3 is that uncertain

scalars are to be handled, which vary between and If methods exist for deciding whether is less than a given strictly positive value at a fixed frequency (see e.g. the previous subsection), it is not so obvious to decide whether is zero, especially when bounds of are computed instead of the exact value. On the other hand, it would be more convenient to represent the frequency response with a real non repeated scalar. As a first reason, the computational methods of the previous subsection are difficult to extend to the case of real repeated scalars. As a second and more important reason, the gap between the bounds tends to increase with repeated scalars. Nevertheless, this requirement imposes to split the circle corresponding to into two half circles (see Figure 11.2): the bottom half circle (between 1 and -1) and the top half circle (between -1 and 1).

• In this context, the uncertain scalar

so that corresponds to circle. As a consequence:

is rewritten as:

i.e. to the bottom half

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

with:

As in subsection 1.3, the model perturbation be rewritten as an LFT where:

can

Here again, the initial interconnection structure of Figure 11.4.a is transformed into the new equivalent interconnection structure of Figure 11.4.b, where and An upper bound of is now to be computed. If this upper bound is strictly greater than 1, a lower bound of the robust delay margin can be deduced. An upper bound of this margin can also be obtained with a lower bound of

• If the upper bound of is less than 1 at the previous step, no destabilizing value of the delays can be found on the bottom half circle. The delay frequency response is thus rewritten as:

with:

The top half circle (between -1 and 1) is obtained when As above, a new interconnection structure is computed, in which the model perturbation contains the original mixed model perturbation and the real non repeated scalars An upper bound of is calculated. If it is less than 1, an infinite robust delay margin is obtained. Otherwise, a lower bound of this margin is deduced. In this case, an upper bound of the margin can also be obtained with a lower bound of Remark: the interest of the above method is that it is geometrically very simple. See nevertheless (Scorletti, 1997) for a more complex, but potentially richer method.

Introduction of time delays

3.

151

AN ALTERNATIVE SMALL GAIN APPROACH

A classical method is first presented for computing a MIMO phase or delay margin, in the case of a closed loop without any other model uncertainty than the uncertain time delays: a conservative value of the MIMO phase or delay margin is computed with the small gain theorem (first subsection), possibly with D scaling matrices (second subsection). The method is extended to the case of a closed loop subject to uncertain time delays and to classical model uncertainties (third subsection).

3.1

AN UNSTRUCTURED APPROACH

Let the closed loop of Figure 11.5, where K(s) and G(s) are the controller and the plant. A block of neglected dynamics is added at the plant inputs. This closed loop is first equivalently transformed into the standard interconnection structure of Figure 11.6.a. M(s) is the transfer matrix seen by the model perturbation in Figure 11.5. A necessary and sufficient condition of stability is provided by the small gain theorem at frequency

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

It was assumed above that is an unstructured model perturbation. Nevertheless, a specific structure can be given a posteriori to

or:

The idea is thus to introduce phases in order to compute a MIMO phase margin with equation (11.20). When combining equations (11.20) and (11.22), one obtains:

Let:

A MIMO phase margin at frequency

is:

If the following structure is now given to

and using

a MIMO delay margin at frequency

Remarks: (i) if the MIMO phase margin is delay margin is infinite.

is:

degrees, and the MIMO

(ii) MIMO phase or delay margins can also be computed with an inverse model perturbation at the plant inputs (see Figure 11.7). In this case, the delays are to be introduced as:

i.e.:

Introduction of time delays

153

Direct model perturbations rather account for high frequency model uncertainties, whereas inverse model perturbations rather represent low frequency model uncertainties (Doyle et al., 1982). As a consequence, better estimates of the delay margin can be expected with an inverse (resp. direct) model perturbation at low (resp. high) frequencies (see section 4.).

3.2

A STRUCTURED APPROACH

In the previous subsection, a specific structure was given to the model perturbation only a posteriori, so that equations (11.25) and (11.27) are just sufficient conditions of stability of the closed loops of figures 11.5 and 11.6.a. It is nevertheless possible to introduce scaling matrices in order to re-

duce the conservatism of the approach. The model perturbation diagonal (see equation (11.22)), so that diagonal scales commute with the model perturbation, i.e.:

is

These scales can thus be introduced in the closed loop of Figure 11.6.a without modifying its stability properties (see Figure 11.6.b). The small gain theorem gives a new sufficient condition of stability at frequency :

Noting:

less conservative values of the MIMO phase and delay margins can be computed with equations (11.25) and (11.27). The issue is to minimize at frequency the quantity:

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

with respect to invertible and diagonal real and positive scaling matrices This is the same problem as in subsection 2.2 of chapter 5 (computation of acomplex upper bound). (Sub)optimal solutions to this optimization problem can be found in the Analysis and Synthesis Toolbox

or in the Robust Control Toolbox (the routine psv.m of this last Toolbox is especially efficient).

3.3 3.3.1

INTRODUCTION OF MODEL UNCERTAINTIES INTRODUCTION

Consider the general case of a standard interconnection structure M(s) with is a mixed model perturbation gathering all classical model uncertainties, whereas gathers all uncertain time delays (see equation (11.4)). As proved in the following, the issue of computing a lower bound of the robust delay margin essentially reduces to a skewed problem, in which the mixed model perturbation is to be maintained inside its unit ball, while the size of the model perturbation is free. The problem is however more complex than it may appear at a first glance: when the only uncertainties in the closed loop are the time delays, it was indeed remarked in the previous subsections that the small gain approach provides a conservative value of the MIMO delay margin, even if the exact value of the s.s.v. is computed. In other words, the small gain theorem, with or without scaling matrices, provides just a sufficient condition of stability. In the same way, even if the problem of computing a lower bound of the robust delay margin essentially reduces to a skewed problem, this does not mean that the exact value of the robust delay margin would be obtained if the exact value of the skewed s.s.v. could be computed.

Introduction of time delays

155

3.3.2 COMPUTATIONAL METHOD The following Lemma is introduced as a preliminary. It can be proved in the same way as Lemma 2..1 of chapter 8.

L EMMA 3..1 Let M a complex matrix, a mixed model perturbation which is to be maintained inside its unit ball, and a mixed model perturbation whose size is free. Then:

We now come back to the initial problem. Remember that is a mixed model perturbation which gathers the parametric uncertainties and neglected dynamics, while is a structured complex model perturbation corresponding to time delays. The associated interconnection structure with is presented in

Figure 11.8. Note that matrices M and matrices M(s) and

at

are the values of the transfer

Frequency

is fixed in the following.

The idea is simply to use the first upper bound of chapter 8 in order to compute a lower bound of the robust delay margin. is to be maintained inside its unit ball, while the size of is free. The difficulty is to prove that a lower bound of the robust delay margin is indeed obtained with this approach. Let the dimension of In the same way as in chapter 8 (section 1.), scaling matrices associated to perturbations are introduced and are then defined as:

so that bound of

is a scaling matrix associated to if the following LMI is satisfied:

is an upper

When multiplying the above inequality on the left and on the right by:

The above LMI becomes:

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

where:

Let:

Equation (11.38) can be rewritten as:

The key point is to note that the above LMI implies that upper bound of

is also an

with:

and is a full complex block with the same dimensions as As a consequence, using Lemma 3..1 and the fact that for any complex matrix H, it can be claimed that:

Noting that rewritten as:

Remember that

the above property is

is the transfer matrix seen by the complex

structured perturbation which models the time delays (see Figure 11.8). Following subsections 3.1 and 3.2, the above equation means that the sufficient condition of robust stability, with respect to uncertain

time delays, is satisfied for all can thus be used in equations (11.25) and (11.27) to compute lower bounds of the robust MIMO phase and delay margins.

3.4

AN IMPROVED ALGORITHM

The issue is to compute the robust delay margin, i.e. the minimal value of the MIMO delay margin when belongs to its unit ball. The following algorithm, which is to be applied at each frequency will be used in the following section:

Introduction of time delays

157

1. Computation of an upper bound of typically the mixed upper bound of (Fan et al., 1991). If it is greater than 1, stop. It is indeed impossible to guarantee that the closed loop without time delays is stable when the classical model uncertainties in

belong

to the unit ball. 2. Computation of a lower bound of the robust delay margin using the small gain approach of section 3. and a direct model perturbation. If an infinite margin is obtained, stop. 3. Computation of a lower bound of the robust delay margin using the small gain approach of section 3. and an inverse model perturbation. If an infinite margin is obtained, stop. 4. The method of section 2. is applied. Remark: Steps 2 and 3 are performed before Step 4 because they are computationally less involving. Their purpose is to eliminate the frequency points, for which the margin is infinite.

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

4.

APPLICATION TO THE MISSILE

As a preliminary, when analyzing the robustness of a control law in the presence of time delays, three frequency bands are classically to be distinguished, namely the low, medium and high frequency bands. In the low frequency band, even if the MIMO phase margin is not especially high, the associated MIMO delay margin is generally high because of the low values of In the high frequency band, the rolloff properties of the controller generally ensure an infinite delay margin. The lowest values of the MIMO delay margin are consequently obtained at medium frequencies.

The issue in this section is to compute the robust delay margin in the presence of aerodynamic uncertainties in the 4 stability derivatives and 3 time delays are introduced at the missile input and at the 2 outputs. A frequency gridding is used, with 100 points between 0.01 rad/s and 200 rad/s. With the exception of the last two points (181 rad/s and 200 rad/s), a finite robust delay margin is obtained with the small gain approach and a direct or inverse model perturbation. Figure 11.9 shows the results obtained with the small gain approach and

Introduction of time delays

159

with the approach of section 2.: in an obvious way, this approach gives

better results than the small gain one. More precisely, let (resp. the lower bound of the robust delay margin obtained with the small gain approach and a direct

(resp. inverse) model perturbation. Let then the lower bound of the robust delay margin obtained with the approach of section 2.. On Figure 11.9, the solid line corresponds to corresponds to max

while the dashed line

The reduced conservatism of the approach of section 2. can be further illustrated on Figure 11.10, which represents on a logarithmic scale (solid line) and (dashed line): i.e. the margin provided by the approach of section 2., is greater than the one provided by the small gain approach at all frequencies. Note moreover on Figure 11.10 that the result provided by the small gain approach and an inverse (resp. direct) model perturbation is indeed better at low (resp. high) frequencies. The minimal value of over the fre-

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

quency range is computed as 5.8 ms at rad/s (lower bound of the robust delay margin obtained with the small gain approach - see Figure 11.11).

One then focuses on the results obtained with the approach of section 2.. The upper bound of the robust delay margin is infinite, except in two frequency intervals [6 rad/s, 7.4 rad/s] and [34.3 rad/s, 44.7 rad/s]. These two intervals are consequently emphasized in figures 11.12 and 11.13. Note the small gap between the lower and upper bounds of the robust delay margin obtained with the approach of section 2., at least near the critical frequency. When computing the minimal value of these lower and upper bounds over the frequency range, an accurate estimate of the robust delay margin is obtained as [8.8 ms, 9.0 ms] at Note in Figure 11.13 that the result provided by the small gain approach is nearly non conservative at the critical frequency However, the lower bound of the robust delay margin obtained with the

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161

small gain approach still decreases for so that the minimal value over the frequency range of the lower bound provided by the small gain approach is obtained at Much better results are thus obtained with the approach of section 2..

5.

CONCLUSION

The approaches of sections 2. and 3. are compared: At least in the missile example, better results are obtained with the approach of section 2.. This is logical, since a much finer description of the delay frequency response is used.

The small gain approach of section 3. only provides a lower bound of the robust delay margin, while the approach of section 2. provides an interval of this margin. A destabilizing value is moreover obtained for the mixed model perturbation and time delays. The interest of the

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

method of section 2. is also to provide a measure of the conservatism of a lower bound of the robust delay margin.

Nevertheless, real (resp. complex) scalars represent the uncertainty in the delay frequency response in section 2. (resp. section 3.). The presence of complex scalars improve the regularity of the plot as a function of frequency As a consequence, the peaks on the plot are more narrow with the approach of section 2., so that a finer frequency gridding is to be used. The small gain approach of section 3. is easier to implement, since it essentially reduces to a skewed problem. The associated algorithm is polynomial time. Conversely, the approach of section 2. is possibly less easy to implement, but better results are expected and an interval of the robust delay margin is obtained instead of just a lower bound. Note that

the algorithm in this book is exponential time, but that a polynomial time version could be derived (Ferreres et al., 1996b).

Chapter 12 NONLINEAR ANALYSIS IN THE PRESENCE OF PARAMETRIC UNCERTAINTIES

The aim of this chapter is to study the existence of limit-cycles in a closed loop, which simultaneously contains nonlinearities and parametric uncertainties (Ferreres and Fromion, 1998). Three methods are presented. First, the issue of detecting a limit-cycle with a necessary condition of oscillation is considered: a graphical method and a based method are proposed. A second based method is then proposed, which uses a sufficient condition of non oscillation, i.e. the issue is now to check the absence of limit-cycles despite parametric uncertainties. An example is finally presented: the necessary condition of oscillation is used to synthesize a controller which modifies the characteristics (magnitude and frequency) of the limit-cycle.

1.

INTRODUCTION

The analysis of nonlinear control systems remains a challenging problem, despite numerous years of extensive research. As the starting point of this chapter, a classical problem is considered, namely a closed loop system which simultaneously contains an LTI transfer function and a separable autonomous nonlinear element: If no a priori knowledge of the nonlinearity is available, this one is simply assumed to belong to a sector. Closed loop stability is checked with the circle or Popov criteria (Desoer and Vidyasagar, 1975). Conversely, if the characteristics of the nonlinear element are a priori known, a solution is to replace this element by its Sinusoidal Input Describing Function (SIDF): see e.g. (Gray and Nakhla, 1981; Katebi and Zhang, 1995; Khalil, 1992) and included references. The harmonic linearization method is then applied, either for detecting the 163

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

presence of limit-cycles in the closed loop (see e.g. (Chin and Fu,

1994; Anderson and Page, 1995) for realistic applications of this classical nonlinear analysis method), or for checking their absence. It would be interesting to extend the above classical problems to the case of a closed loop, which is also subject to LTI parametric uncertainties. This was partially done for the case of a sector-type non linearity: see e.g. (Chapellat et al., 1991; Gahinet et al., 1995). The aim of this chapter is to extend the classical harmonic linearization method to the case of parametric uncertainties. Existing results in robustness analysis (i.e. analysis of the robustness properties of an LTI closed loop system, subject to LTI model uncertainties) are used as the basis 1 . Two different problems are considered. The first one is to detect a limit-cycle with a necessary condition of oscillation: a graphical method is first proposed, which can be considered as an extension of the classical method (section 2.). A based method is then proposed in section 3.. As a second problem, the problem of checking the absence of limitcycles despite parametric uncertainties is studied. An alternative based method is proposed, which uses a sufficient condition of non oscillation (section 4.). An example is presented in section 5.. Concluding remarks end the chapter. Note that the starting point of this chapter is to remark that the necessary condition of oscillation leads to a problem of detecting the singularity of a matrix, which depends on the parametric uncertainties, whereas the sufficient condition of non oscillation leads to a problem of checking a small gain condition despite parametric uncertainties. On the other hand, the problem of checking the presence or the absence of limit-cycles in the face of parametric uncertainties has primarily an engineering interest. The main purpose of this chapter is consequently to show that this difficult nonlinear problem can be solved efficiently with the s.s.v. Because many tools are already available, this chapter, in the same spirit as (Katebi and Zhang, 1995), has primarily a practical interest, beyond the necessary theoretical justifications.

2.

A GRAPHICAL METHOD

The case of a single SISO nonlinearity is considered. In the first subsection, a classical graphical method for detecting limit-cycles (without parametric uncertainties) is recalled. Subsection 2.2 then proposes an 1

The interest of using linear tools (namely control, analysis and synthesis) in a nonlinear control problem was already emphasized by (Katebi and Zhang, 1995) in a different context.

Nonlinear analysis

165

extension of this method, for detecting limit-cycles in the presence of parametric uncertainties.

2.1

A CLASSICAL METHOD

Let the closed loop system of Figure 12.1, where represents a SISO nonlinearity and G(s) a transfer function. Assume that where X is a positive real scalar. The output of the nonlinearity, which is supposed to have odd symmetry, can be written as:

where contains the super harmonic part of signal u. R and S can be computed as:

The SIDF is introduced as:

It is assumed in this section that the signal is essentially filtered by the low-pass transfer function G(s) (first harmonic approximation), so that a necessary condition of oscillation of the closed loop is:

Remark: as a first point, the above equation represents a necessary condition for the existence of a limit-cycle, if the first harmonic approximation is valid. As a second point, when considering a magnitude X0 and a frequency satisfying equation (12.4), the corresponding limitcycle can be stable or unstable.

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

In practice, generally does not depend on so that equation (12.4) can be solved by looking for the intersection(s) of the plots of and 1/N(X) in the complex plane (see Figure 12.2).

2.2

AN EXTENSION

Parametric uncertainties are introduced in the closed loop system of Figure 12.1: the transfer function now depends on LTI uncertain parameters For the sake of convenience, the vector of parametric uncertainties is assumed to belong to the unit hypercube D: The necessary condition for the existence of a limit-cycle becomes:

Defining the generalized value set

in the complex plane as:

and assuming as in the previous subsection that the issue is to find the intersection (s) of the generalized value set with the plot of 1 / N ( X ) in the complex plane. The problem is thus to build the generalized value set as a function of This is not an easy problem, which is only solved for special structures of parametric uncertainties. This is especially the case of interval plants:

Nonlinear analysis

167

where uncertain coefficients ai and bj belong to intervals. See e.g. (Kheel and Bhattacharyya, 1994) and included references.

3.

A FIRST

BASED METHOD

This section and the following one consider the general case of a MIMO nonlinearity. As a preliminary, the first subsection illustrates that the general problem of detecting a limit-cycle in the presence of parametric uncertainties can be recast into an LFT framework. A technical result is then presented in the second subsection. It is proved in the third subsection that the issue of detecting a limit-cycle can be made equivalent to the issue of detecting the singularity of a matrix, which depends on parametric uncertainties: this problem can thus be treated in the framework. An extension of the method is presented in the fourth subsection. The use of

bounds is finally discussed in the last subsection.

3.1

AN LFT FORMULATION OF THE PROBLEM

Consider the general case of a MIMO transfer matrix where is a vector of parametric uncertainties Following chapter 3, can be expressed as an LFT where is a diagonal matrix:

In the generalized problem of Figure 12.3 (to be compared with Figure 12.1), the aim is twofold:

If no limit-cycle exists for the nominal closed loop system the minimal amount of parametric uncertainties is to be found, for which a limit-cycle is obtained in the closed loop system of Figure 12.3. If a limit-cycle exists in the nominal closed loop system, it is interesting to visualize the movement of this limit-cycle (i.e. the variation

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

of its magnitude X and frequency uncertainties.

3.2

as a function of parametric

A PRELIMINARY TECHNICAL RESULT

A technical Lemma is needed, which forms the basis of the Main Loop Theorem (see subsection 4.4 of chapter 1). Let P a complex matrix and and two mixed model perturbations. The two interconnection structures of Figure 12.4 are equivalent, if Partition then compatibly P as:

LEMMA 3..1 If matrices

and

are invertible, then:

Proof: A classical property of the determinant is used:

where D is invertible.

3.3

A GENERALIZED CONDITION OF OSCILLATION

Nonlinearity is replaced in Figure 12.3 by its SIDF It is then straightforward to transform Figure 12.3 into Figure 12.4.a, with replaced by Assume a fixed value for matrix (and thus for the associated vector

of parametric uncertainties) in Figure 12.3.

Nonlinear analysis In the case of a SISO nonlinearity existence of a limit-cycle is:

169

the necessary condition for the

In the general case of a MIMO nonlinearity, the necessary condition becomes (Gray and Nakhla, 1981):

X and are fixed in this subsection, so that is a constant matrix. As a consequence, the matrix (i.e. the transfer matrix at frequency seen by the real model perturbation in figure 12.4.a) can be computed a priori. Proposition 3..3 introduces a method for computing the robustness margin which measures the size of the smallest parametric uncertainty for which the necessary condition of oscillation in equation (12.14) is satisfied. Remember as a preliminary that is a robustness margin, while the s.s.v. is homogeneous to the inverse of a robustness margin.

DEFINITION 3..2 with if the nominal closed loop (obtained with satisfies equation (12.14). Conversely, if no satisfies equation (12.14).

PROPOSITION 3..3 If:

Proof: Lemma 3..1 is used. let and the first assumption of the Proposition, it can be claimed that:

Using

Using then the second assumption, it is easy to see that the assumptions of Lemma 3..1 are satisfied, so that:

thus coincides with Remark: the second assumption in Proposition 3..3 means that the

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

necessary condition of oscillation is not satisfied at inal closed loop system, since

for the nomwhen

Concerning the first assumption in Proposition 3..3, it is worth pointing out that measures the robust stability property of the transfer matrix (i. e. the transfer matrix between u and y in Figure 12.3). Assume indeed that P(s) is asymptotically stable. Define then the stability margin as the maximal value of k, such that the transfer matrix is asymptotically stable for all can be computed as:

As a final point, the following Corollary proposes to compute a lower bound of the robustness margin when the first assumption in Proposition 3..3 is not satisfied.

COROLLARY 3..4 If

then:

Proof:

• If the assumptions of Proposition 3..3 are satisfied and the above Corollary reduces to this Proposition (Inequation (12.19) becomes an equality). • If let and By definition of Using then the assumption of the Corollary, it is straightforward to see that the assumptions of Lemma 3..1 are here again satisfied, so that:

Noting then that that:

it can be claimed

As a consequence:

The necessary condition of oscillation of equation (12.14) is not satisfied, and k is consequently a lower bound of the robustness margin 2

Alternatively, the singularity of the matrix is not well-posed.

means that the LFT

Nonlinear analysis

3.4

171

AN EXTENSION

In the above method, is to be computed at each point of a gridding. The robustness margin is then visualized as a function of X and on a 3D plot (see section 5.). Nevertheless, a peak value of may be missed with such a gridding (i.e. when this peak value lies between two points of the gridding), and the robustness properties of the closed loop would be overevaluated. A simple method is proposed here for avoiding a gridding of the magnitude X, by treating this magnitude (more precisely the SIDF as an additional (fictitious) uncertainty. Frequency is fixed. An augmented skewed problem is obtained. Even if the method is not applicable to all kinds of nonlinearities, it can be applied to a large class of usual (especially memoryless) ones.

• The aim of this subsection is to compute the robustness margin

Lemma 3..5 provides an alternative definition of when using Proposition 3..3.

The proof is trivial

LEMMA 3..5 If: then:

• Consider first a classical nonlinearity, namely a saturation

The associated SIDF N(X), which does not depend on frequency which is a real scalar, can be computed as (see also Figure 12.5):

and

With respect to Figure 12.5 and equation (12.27), it is obvious that leads to

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

• Consider now the generic case of a memoryless SISO nonlinearity: here again, the SIDF N(X) is a real scalar which does not depend on frequency can thus be translated into The idea is thus to rewrite N(X) as:

where the

are chosen so that

leads to

• In the general case of a diagonal MIMO memoryless nonlinearity the SIDF can be written as:

where the are fixed diagonal matrices and X denotes now a vector of magnitudes while N(X) is a diagonal matrix. Here again, leads to • Applying now equation (12.29) to Figure 12.3, this Figure can be transformed into Figure 12.4, where:

the fictitious real model perturbation contains the uncertainties in the magnitude vector X, or equivalently in the SIDF N(X). the real model perturbation tainties.

contains the true parametric uncer-

Nonlinear analysis Matrices

173

are incorporated in the transfer matrix P(s).

The following Proposition presents a method for computing

PROPOSITION 3..6 Let the interconnection structure of Figure 12.4. If: then:

is the skewed s.s.v. associated to the complex matrix and to the real model perturbation is to be maintained inside its unit hypercube

Remarks: (i) The assumptions of this Proposition are essentially extensions of the assumptions in Proposition 3..3. Note especially that the second assump-

tion in Proposition 3..6 means that the necessary condition of oscillation is not satisfied by the nominal closed loop system, for frequency

and

for a vector of magnitudes (ii) A frequency gridding can be avoided, by treating as an additional uncertainty in an augmented problem (see section 3. of chapter 7). (iii) The above method can be applied to nonlinearities, which are not necessary memoryless: The SIDF may depend on and it may take complex values. For a fixed value of the key point is to be able

to reparameterize that the trajectory of imposing and

in an affine way as so in the complex plane is the same when

Proof: this Proposition is an extension of Proposition 3..3 and Lemma 3..5 The issue is to compute the maximal value of

over

The fictitious parametric uncertainties associated to N(X) must consequently remain inside the unit hypercube while the true parametric uncertainties in are to be expanded, until the necessary condition of oscillation in equation (12.14) is satisfied.

3.5

THE USE OF

BOUNDS

A necessary condition of oscillation is used in the above method. With reference to the end of section 3.1, two different cases are to be considered: If no limit-cycle is obtained for the nominal closed loop system, the

aim is to find the minimal amount of parametric uncertainties, for which a limit-cycle is obtained. It is interesting in this context to

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

compute both upper and lower bounds, since the

lower bound (resp.

the upper bound) gives a size of the parametric uncertainties, for which the necessary condition of oscillation is satisfied (resp. not satisfied). The lower bound moreover gives a model perturbation for which the necessary condition of oscillation is satisfied. It is then interesting to apply to the closed loop, in order to check whether the first harmonic approximation is valid and whether the corresponding limit-cycle is stable or unstable (see section 5.).

If a limit-cycle is already obtained for the nominal closed loop system, the idea is rather to visualize the movement of this limit-cycle (i.e. the variation of its amplitude and frequency) as a function of the parametric uncertainties. In this new context, it is more interesting to obtain a model perturbation which satisfies indeed the necessary condition of oscillation. A lower bound is thus more interesting than a upper bound.

4.

A SECOND

BASED METHOD

The sufficient condition of non oscillation is introduced in the first subsection. The method is extended to the case of parametric uncertainties in the second subsection. The third subsection shows that a skewed problem with a special structure is to be solved. The problem of using (skewed) bounds is discussed in the last subsection.

4.1

A SUFFICIENT CONDITION OF NON OSCILLATION

• Some results in (Katebi and Zhang, 1995) are recalled here. See this reference for details. We come back to the problem of subsection 2.1 (see also Figure 12.1). The issue is to take into account the super har-

Nonlinear analysis

175

monic part of the signal u(t), at the output of the nonlinearity To this aim, the frequency response of the nonlinearity is rewritten as where represents the error induced by the SIDF approximation.

• In the case of a SISO nonlinearity,

is known by the relation:

where is a function of X and With respect to Figure 12.1, a sufficient condition of non oscillation is then that the inverse of the frequency response of the nonlinear element does not inter-

sect the frequency response of the linear part G(s) of the closed loop. As in section 2.1, when

and

do not depend on frequency

Figure 12.6 suggests a graphical method for checking the absence of limit-cycles.

• In the case of a MIMO nonlinearity,

is known by the relation:

Figure 12.1 is transformed into Figure 12.7.a, by replacing the nonlinearity

by its frequency response

formed into Figure 12.7.b, where

Figure 12.7.a is then trans-

is the transfer seen by

in

Figure 12.7.a. Applying finally the small gain theorem to Figure 12.7.b, a sufficient condition of non oscillation is obtained as:

where satisfies equation (12.32). The sufficient condition of non oscillation thus becomes:

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

4.2

EXTENSION TO THE CASE OF PARAMETRIC UNCERTAINTIES

• As a preliminary, it is straightforward to extend the graphical method in the previous subsection (see also Figure 12.6) to the case of a transfer function which depends on some parametric uncertainties : see section 2.2.

• The general case of a MIMO nonlinearity is considered. As in section 3.1, parametric uncertainties are introduced in the closed loop of Figure 12.1 by rewriting as an LFT where is a real model perturbation (see Figure 12.3). Replacing then by its frequency response Figure 12.3 is transformed into figures 12.8.a and 12.8.b. The sufficient condition of non oscillation is:

The robustness margin is the maximal size of parametric uncertainties, for which the sufficient condition above is satisfied.

DEFINITION 4..1

if the nominal closed loop (obtained with does not satisfy equation (12.34). Remark: in the case of a diagonal MIMO nonlinearity, diagonal scaling matrices can be introduced in the small gain test, in order to reduce the conservatism of the sufficient condition of non oscillation (i.e. to consider instead of in definition 4..1).

Nonlinear analysis

4.3

A SKEWED

177

PROBLEM

• With reference to chapter 8 (subsection 2.1), the computation of

at a point is a skewed problem, which involves a mixed uncertainty is here a fictitious full complex block). It is consequently possible to use the first mixed skewed bound of chapter 8 (section 1.). Otherwise, the special structure of the problem can be accounted for, and this specific skewed problem can be transformed into an augmented problem: see chapter 8 (section 2.). This solution is easier to implement, since standard tools are directly available in Matlab Toolboxes (such as the Analysis and Synthesis Toolbox or the

LMI Control Toolbox). • A sufficient condition of non-oscillation is used in this section. A skewed upper bound is thus more attractive than a skewed lower bound, whose interest is only to measure the conservatism of the upper bound. Indeed, even if a model perturbation is obtained, which does not satisfy the sufficient condition of non oscillation, this does not mean that a limit-cycle is obtained when applying to the closed loop. On the contrary, the upper bound gives a maximal size of the parametric uncertainties, for which the absence of limit-cycle is guaranteed.

5.

AN APPLICATION

The necessary condition of oscillation is used to synthesize a controller, which modifies the characteristics (magnitude and frequency) of a limitcycle. This is an interesting engineering problem, since a limit-cycle is not necessarily a trouble in a nonlinear closed loop: a high frequency limit-cycle can linearize in an approximate way the nonlinear closed loop. Within the framework, the following example illustrates that a troublesome limit-cycle (with a large magnitude and a low frequency) can be moved into a suitable limit-cycle (with a smaller magnitude and a higher frequency). For the sake of clarity, this example is chosen willingly simple: it is nevertheless straightforward to extend the method to the more general case of a MIMO nonlinearity, with a more complex parameterization of the controller.

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

Let the closed loop system of Figure 12.9, where G(s) is the plant model and K(s) the controller:

with

and

A saturation is present at the plant input (see equation (12.26)). A limit-cycle is obtained, whose predicted characteristics are a magnitude X = 2.46 and a frequency (using the method of subsection 2.1). The characteristics obtained in simulation are X = 2.56 and these values are thus very close to the predicted ones, which means that the first harmonic assumption is valid. The limit-cycle is moreover stable. This limit-cycle is to be moved by retuning the controller K(s). The tuning parameters are K, and (more precisely The augmented plant of Figure 12.3 is simply obtained by directly introducing "uncertainties" into the physical model of K(s) (see Figure 12.10 N(X) is the SIDF of the saturation). The real model perturbation is The quantity is presented as a function of X and on Figure 12.11 (see Corollary 3..4) 3. Remember that there is a limit-cycle in the nominal closed loop system, at 3

The lower bound by (Dailey, 1990) is used in this section. Because the real model uncertainty only contains 3 parametric uncertainties, this lower bound coincides with the

exact value of

Nonlinear analysis

X = 2.46 and point, and

As a consequence, increases around this critical point.

179

at this

The limit-cycle is to be moved at X = 1.2 and (i.e. the frequency of the limit-cycle is roughly multiplied by a factor 2, while the magnitude is divided by the same factor). A lower bound is computed for

at this new point

associated model perturbation

namely 1.22. An

is provided as:

This model perturbation is applied to K(s), which becomes:

Here again, a stable limit-cycle is obtained in simulation, with X = 1.207 and (see Figure 12.12).

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

Remark: remember first that represents the minimal size of the model perturbation for which a limit-cycle is obtained at (see

definition 3..2). In the context of the above design problem, a limit-cycle is obtained for the nominal value of the controller, and this controller is retuned in order to move the limit-cycle at a point it is thus interesting to use the minimal size model perturbation associated to Indeed, the corresponding controller can be considered as the controller, which is the closest to the nominal one, and which moves the limit-cycle at

6.

CONCLUSION

The general aim of this chapter was to check the existence or the absence of limit-cycles in a closed loop subject to parametric uncertainties.

The aim was more precisely to show that this nonlinear problem can be solved with existing tools of linear robustness analysis (i.e. analysis of

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181

the robustness properties of an LTI closed loop system, subject to LTI model uncertainties): Computation of the frequency response of a parametrically uncertain transfer function (Kheel and Bhattacharyya, 1994). Use of the LFT framework to transform the original problem into an equivalent standard interconnection structure.

Computation of upper and lower bounds of the (skewed) structured singular value.

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Conclusion

The general aim of this book was to emphasize the usefulness of the approach for practical robustness analysis problems. The and skewed tools were first introduced. Their efficiency was then illustrated through realistic examples, namely a flight control system for a civil aircraft, a missile autopilot and a telescope mock-up. Consider an LTI plant subject to LTI model uncertainties. These ones can be divided into two main categories, namely the parametric uncertainties (which more generally represent the uncertainties in the plant dynamics inside the control bandwidth) and the neglected dynamics (which represent the uncertainties outside the control bandwidth). Assuming that a controller was synthesized with the nominal model of the plant, the issue is to study the effect of these uncertainties on the closed loop stability and performance properties. It was shown in chapters 1 and 7 that the s.s.v. and its skewed version provide a general framework for treating most of the robust stability and performance problems. The only requirement is to transform the closed loop with all its uncertainties into a standard interconnection structure: in this context, the main problem is to account for the parametric uncertainties, which may enter the open loop plant model in numerous ways. It is most generally possible to transform the plant model with its parametric uncertainties into a standard form, namely an LFT (part 2). In chapter 7, it was emphasized that large classes of problems, such as robust performance analysis, are skewed problems, i.e. a single application of the tool provides in these cases just a robustness test. A recursive application of the tool is thus necessary to compute the robustness margin. Because of the practical usefulness of these skewed problems, a much more efficient solution is to directly compute a bound 183

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

of the skewed s.s.v. Various methods were thus presented in chapter 8 for computing bounds of the skewed s.s.v. while chapter 5 described methods for computing bounds of the classical s.s.v. These techniques can be classified according to 3 different criteria:

nature of the result: upper and lower bounds of the (skewed) s.s.v. are generally computed instead of the exact value, the main reason being the computational requirement. nature of the model uncertainties: when the problem only contains

parametric uncertainties, the associated model perturbation is said to be real. On the contrary, this model perturbation is complex if there are only neglected dynamics. In the general case of parametric uncertainties and neglected dynamics, the associated model perturbation is said to be mixed. A great deal of work has been devoted to the computation of (a bound of) the real s.s.v.. On the other hand, methods for computing bounds of the complex s.s.v. have been lately extended to the mixed case.

Computational requirement: the algorithms are either polynomial time, or exponential time, i.e. the computational burden is a polynomial or exponential function of the size of the problem.

All methods in chapters 5 and 8 were evaluated on the aeronautical examples, and it was illustrated that the robustness margin can be computed with a satisfactory accuracy, even for large dimension problems: in the context of a problem involving a large number of uncertainties, our approach was to compute lower and upper bounds of this margin using polynomial time algorithms. Even if the gap between these bounds can not be guaranteed a priori, it appears reasonable in our realistic examples. As a final point, the usefulness of the and skewed tools was briefly illustrated in part 5 for three non standard problems: A computationally efficient method was proposed, for directly computing an interval of the maximal s.s.v. over a frequency interval. Good results were obtained in the examples, and an accurate estimate of the robustness margin was computed.

The issue was then to analyze the robustness properties of a closed loop in the face of parametric uncertainties, neglected dynamics and uncertain time delays. This is not a classical problem, since delays do not correspond to classical LTI finite-dimensional transfer functions.

Nonlinear analysis

185

It is nevertheless possible to parameterize the frequency response of these delays in order to reduce the problem to a standard problem. Let a linear closed loop system with additional separable nonlinearities (e.g. saturations). Classical methods exist for detecting the presence or the absence of limit-cycles in this closed loop. The approach enables to extend these classical methods to the case of parametric uncertainties entering the linear part of the closed loop.

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Appendix A Numerical data

1. 1.1

THE AIRPLANE PROBLEM THE AERODYNAMIC MODEL

The numerical data for the aerodynamic model are as follows (see equations (2.1) and (2.2)):

A state-space representation (A, B,C, D) of the aerodynamic model is: 187

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

Remember that the two inputs of the above model are the four outputs are ny, p, r and

1.2

and

while

THE FLEXIBLE MODEL

A real modal state-space representation (A, B, C, D) of the flexible part of the aircraft is:

Appendix A: Numerical data

189

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

Here again, the two inputs of the above model are four outputs are ny, p, r and

2.

and

while the

THE MISSILE PROBLEM

The missile model is linearized at The first input and first two outputs correspond to the physical ones, whereas the four additional inputs and outputs correspond to uncertainties in and A state-space representation (A, B, C, D) of the LFT model is given by:

Appendix A: Numerical data

191

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Appendix B Proofs

1.

LEMMA 2..4 (CHAPTER 8)

The proof essentially follows the one of Theorem 3.1 in (Sideris and Pena, 1990). Assume •property:

that

One proves in the following that the

is satisfied if and only if the matrix first that the assumption •tionNote(B.1) is satisfied for (B.1) is satisfied for all

_

is nonsingular for all enables to claim that equaIt can be noted then that equation if and only if:

Noting A the matrix in the above equation, A can be rewritten as:

where X denotes the matrix:

193

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

X is invertible because of the assumption equation (B.3) essentially requires the following property:

Obtaining

• Let:

Remember indeed that only contains real (repeated) scalars. Matrix can be written under the form (to be compared with equation (B.3)):

is invertible because of the assumption erties and can be rewritten as:

Using the prop-

As a consequence:

It can be finally checked that

2.

PROPOSITION 3..2 (CHAPTER 8)

• Considering theorem 4 of (Young and Doyle, 1990), a must satisfy:

lower bound

where a, b, z and satisfy equations (8.26,8.28,8.29-8.31). On the other hand, it can be remarked that if a destabilizing perturbation is found for the scaled matrix

the perturbation

is then destabilizing for matrix

consequently correspond to a destabilizing perturbation for M with:

Appendix B: Proofs is thus a lower bound for • Very simply, when choosing

equations (B.10,B.11) become:

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Index

Computational aspects, xiv, 26, 44, 63, 76, 91, 100, 103, 112, 124, 132, 133, 136, 139, 140, 143, 162, 184 Convex hull, 64

Loop shaping, 36 Norm, 5 Performance, 16, 22, 36 Adaptive control, 102 Aircraft Analysis, 88, 134 Controller, 31, 33, 138 LFT model, 57, 134 Model, 30, 32 Numerical data, 187 Augmented problem, 23, 72, 99, 104,

Discrete time, 15 Exact value of 18, 26, 64, 69, 70, 77, 143, 178 Fixed point, 19, 71, 104 Flexible structure, xiii, xv, 5, 27, 29, 50,

112, 124, 148, 171, 173, 177

57, 58, 99, 123, 188

Augmented plant, 6, 37, 39, 45, 51, 56, 102, 178

Frequency gridding, 27, 73, 123, 127, 136, 138, 158, 162, 171

Frobenius norm, 75, 109 Bilinear transformation, 69, 147 Bounds lower bound, 26, 64, 65, 70, 71, 77, 80, 116, 132, 136, 138, 139, 174, 178, 179 upper bound, 26, 44, 63, 66–69, 77, 80, 116, 124, 174 One-sided skewed lower bound, 148, 158 One-sided skewed upper bound, 148, 158 Skewed lower bound, 109, 112,

Gain scheduling, 101 Generalized eigenvalue, 67 Interconnection structure, 4, 17, 21, 23, 43, 67, 69, 72, 96, 104, 129, 145, 147, 150, 167, 168, 173

LFT Computation, 43, 53, 129, 147 Definition, 4, 9 Interconnection, 46, 147, 167 Limit-cycle, xvi, 71, 98, 111, 163 Linear Matrix Inequality, 66, 69, 104, 124,

115, 177

Skewed

upper bound, 103, 104,

112, 115, 155, 158, 177

126, 155

Branch and bound, 65

Linear Programming, 74, 138

Characteristic polynomial, 11 Circle criterion, 163 Closed loop poles, 12, 15 lower bound, 72 Damping ratio, 15 Degree of stability, 15

Main Loop Theorem, 20, 104, 105, 168 Mapping Theorem, 64 Margin Delay, 143 Performance, 23 Stability, 8, 12, 13

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A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS

Minimality of an LFT model, 44, 50, 52, 53, 57 Missile Analysis, 81, 115, 158 Autopilot, 37 Bending mode, 36, 115

Design specifications, 87, 115 LFT model, 53, 190

Model, 34 Numerical data, 35, 190 Modal state-space representation, 32, 50, 57, 58, 188

Multilinear, 64 Multiloop stability margin, 13, 27, 63

Multiplier, 70 Non repeated scalars, 18, 57, 58, 63, 70, 81, 89, 115, 134, 139, 148, 149 Nonlinearity sector-bounded, 163 SIDF, 165 NP hard, 26 Nyquist stability criterion, 11, 146

function, 22, 36, 87 Singularity, nonsingularity, 12, 15, 20, 25, 27, 73, 146, 164, 170

Skewed Computation and application, 95, 103, 115, 123, 154, 173, 174

Definition, 18 Small gain Despite model uncertainties, 95, 106, 176

Standard, 6, 8, 20, 151, 175 With scaling matrices, 67, 153 Spectral radius, 13, 20, 70, 110 Stability (asymptotic or marginal), 12 Star product, 47, 130, 147 State-space lower bound, 72, 138

test, 99, 129, 134 upper bound, 124 Structured singular value Application, 81 Computation, 63 Definition, 18

Interpretation, 12, 13, 15, 136 Observer-based controller, 34, 137 One-sided skewed

Application, 100, 150, 158 Computation, 148 Definition, 143 Osborne’s method, 109, 120 Performance Frequency-domain, 16, 22, 36, 87, 96, 106, 113, 117, 148 Time-domain, 15, 84, 87, 90 Perron eigenvector method, 108, 120, 133 Popov criterion, 163 Positivity, 70 Power algorithm, 70, 73, 74, 109, 117

Telescope Analysis, 139

Controller, 40 Model, 38 Template, 6, 13, 16, 22, 36, 87, 96, 116, 117, 148 Time delay, 143, 184 Toolbox, 92, 108, 112, 133, 154

Uncertainty Affine parametric uncertainty, 44,

54, 57, 58 Mixed, 17

Natural frequencies of flexible modes, 50, 57, 134 Neglected dynamics, 5 Parametric uncertainty, 8, 43

QFT, 95

Reduction, 6, 95 Regularized problem, 73 Repeated scalars, 9, 17, 20, 44, 49, 58, 102, 105, 110, 112, 124, 135,

149 Rise time, 15, 87 Roll-off, 33, 158 Root locus, 136 Saturation, 171, 178 Scaling matrices, 66–68, 103, 108, 112, 118, 124, 126, 127, 132, 133, 153, 176 Sensitivity 23, 100 eigenvalue, 75

Rational parametric uncertainty, 48 Uncertain frequency, 99, 129

Unstructured uncertainty Additive, 5, 36 Direct and inverse, 153, 159 Value set, 63, 166 Weight on the parametric uncertainties, 13, 30, 36 Weighting function, 5, 6, 96 Well-posed, 97, 106, 130, 131, 170 Worst case model perturbation, 27, 71, 73, 136, 161 Zero Exclusion Test, 64

About the Author

Gilles Ferreres was graduated in 1990 from the engineering school Ecole Supérieure d’Electricité (Supélec - France). He did a PhD thesis on "Robustness tools for adaptive control", with application to the synthesis of missile autopilots. This was done in collaboration with the Laboratoire d’Automatique de Grenoble (a French CNRS laboratory specialized in Automatic Control) and the French industrial company AerospatialeMissiles. He joined then the Office National d’Etudes et de Recherches Aerospatiales (ONERA), the French research agency specialized in aeronautics and space. His topics of interest cover the range from theory to application and include robustness analysis, robust control, adaptive control and set membership identification.

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