Nonlinear Control in the Year 2000, Volume 2 (Lecture Notes in Control and Information Sciences)

Lecture Notes in Control and Information Sciences Editor: M. Thoma 259 Springer London Berlin Heidelberg New York Bar...

Author: Alberto Isidori | Francoise Lamnabhi-Lagarrigue | Witold Respondek

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Lecture Notes in Control and Information Sciences Editor: M. Thoma

259

Springer London Berlin Heidelberg New York Barcelona Hong Kong Milan Paris Singapore Tokyo

AlbertoIsidori,FranqoiseLamnabhi-Lagarrigue and WitoldRespondek(Eds)

Nonlinear Control in the Year 2000 Volume 2 With 88 Figures

~

Springer

Series

Advisory

Board

A. B e n s o u s s a n • M.J. G r i m b l e • P. K o k o t o v i c H. K w a k e r n a a k • J.L. M a s s e y • M. M o r a r i

• A.B. K u r z h a n s k i

•

Editors A l b e r t o Isidori, P r o f e s s o r

Dipartimento di Informatica e Sistemistica, Universita di Roma," La Sapienza", 00184 R o m e , I t a l y Franqoise Lamnabhi-Lagarrigue, Docteur D'~tat L a b o r a t o i r e d e s S i g n a u x et S y s t e m s , CNRS SUPELEC, 91192 G i f - s u r - Y v e t t e , F r a n c e Witold Respondek, Professor L a b o r a t o i r e d e M a t h d m a t i q u e et I n f o r m a t i q u e , INSA d e R o u e n , 76131 M o n t S a i n t A i g n a n , F r a n c e

ISBN 1-85233-364-2 Springer-Verlag London Berlin Heidelberg British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data Nonlinear control in the year 2000 / Alberto Isidori, Franqoise Lamnabhi-Lagarrigue and Witold RespondeL p. cm - (Lecture notes in control and information sciences, ISSN 0170-8643 ; 258-259) Includes bibliographical references. ISBN 1-85233-363-4 (v. 1 : acid-free paper) - ISBN 1-85233-364-2 (V. 2 : acid-free paper) 1. Nonlinear control theory. I. Isidori, Alberto IL Lamnabhi-Lagarrigue, F. (Franqoise), 1953- III. Respondek, W. IV. Series. QA402.35 .N66 2001 003.5--dc21 00-045600 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. © Springer-Vedag London Limited 2001 Printed in Great Britain The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Typesetting: Camera ready by editors Printed and bound at the Athenaeum Press Ltd., Gateshead, Tyne & Wear 6913830-543210 Printed on acid-free paper SPIN 10775940

Contents

Volume 1 Subanalyticlty

of distance and spheres in S-R geometry

......

1

Andrei Agrachev, Jean-Paul Gauthier Principal invariants of Jacobi curves ..........................

9

Andrei Agrachev, Igor Zelenko The De Casteljau

algorithm

on SE(3) .........................

23

Claudio Altafini Trajectory tracking by cascaded backstepping control for a second-order nonholonomic mechanical system ................

35

Nnaedozie P.I. Aneke, Henk Nijmeijer, Abraham G. de Jager Static output feedback stabilization: from linear to nonlinear and back .....................................................

49

Alessal~dro Astolfi, Patrizio Cola~leri Semi-linear diffusive representations for nonlinear differential systems ...........................................

fractional 73

Jacques Audounet, De~is Matig~o~, Gdrard Mo~tsel~g Controllability properties of a class of control systems on Lie groups ........................................................

83

Victor Ayala, Luiz A. B. San Martin Stability analysis to parametric uncertainty extension multivariable case ............................................

to the 93

Miguel Ayala Botto, Ton va~ den Boom, Josg Sd da Costa External

stability and continuous

Liapunov

functions

.........

103

machine

113

AT~drea Bacciotti Optimal

control with harmonic

Iyad Batloul, Maze~ Atamir

rejection of induction

VI Nonlinear QFT synthesis based on harmonic balance and multiplier theory .............................................

123

Alfonso Ba~os, Antonio Barreiro, Francisco Gordillo, Javier Aracil Group invariance and symmetries in nonlinear control and estimation ....................................................

137

Johll S. Baras A g l o b a l i z a t i o n p r o c e d u r e f o r l o c a l l y s t a b i l i z i n g c o n t r o l l e r s . . . 171

Joche~z Behre~s, Fabian Wirth Optimal control and implicit Hamiltonian systems ............

185

Guido Blankenstem, Arian van der Schaft Robust absolute stability of delay systems ....................

207

Pierre-Alexmzdre Bliman Stratification du secteur anormal dans la sph6re de Martinet de petit rayon ................................................

239

Bernard Botmard, Emmanuel Trdlat N o n e q u i l i b r i u m t h e o r y for n o n l i n e a r s y s t e m s . . . . . . . . . . . . . . . . .

253

Christopher I. Byrnes A regularization of Zubov's equation for robust domains of attraction .....................................................

277

Fabio Camilli, Lars Grib~e, Fabian Wirth A remark on Ryan's generalization of Brockett's condition to discontinuous stabilizability ...................................

291

Francesca Ceragioli A p p l i c a t i o n s o f Lie a l g e b r o i d s in i n e c h a n i c s a n d c o n t r o l t h e o r y 299

Jesds Clemente-Gallardo O b s e r v e r design for locally o b s e r v a b l e a n a l y t i c s y s t e m s : convergence and separation property .........................

315

Fabio Conticelli, A~tm~io Bicchi A n H o 0 - s u b o p t i m a l f a u l t d e t e c t i o n f i l t e r f o r b i l i n e a r s y s t e m s .. 331

Claudio De Persis, Alberto Isidori

Vii

A d a p t i v e control of feedback linearizable systems by orthogonal approximation functions ...........................

341

Domitilla Del Vecchio, Riccardo Marino, Patrizio Tomei S a m p l e d - d a t a low-galn integral control of linear systems with actuator and sensor nonlinearities ............................

355

Thomas Fliegner, Hartmut Logemann, Eugene P. Ryan State feedbacks without asymptotic observers and generalized PID regulators ...............................................

367

Michel Fliess, Richard Marquez~, Emmanuel Delaleau E i g e n s t r u c t u r e of nonlinear Hankel o p e r a t o r s . . . . . . . . . . . . . . . . .

385

Kenji Fujimoto, Jacquelien M.A. Scherpen D i s t r i b u t e d a r c h i t e c t u r e for t e l e o p e r a t i o n o v e r t h e i n t e r n e t ... 399

Denis Gillet, Christophe Salzmann, Pierre Huguenin S i n g u l a r L-Q p r o b l e m s a n d t h e D i r a c - B e r g m a n n t h e o r y o f constraints ....................................................

409

Manuel Guerra R o b u s t tracking of multi-variable linear systems u n d e r parametric uncertainty .......................................

423

Veit Hagenmeyer F l a t n e s s - b a s e d c o n t r o l o f t h e s e p a r a t e l y e x c i t e d D C d r i v e . . . . . 439

Veit Hagenmeyer, Philipp Kohlrausch, Emmanuel Delaleau S t a t e d e t e c t i o n a n d s t a b i l i t y for u n c e r t a i n d y n a m i c a l s y s t e m s

453

Mohamed All Hammami C o n t r o l l a b i l i t y p r o p e r t i e s o f n u m e r i c a l e i g e n v a l u e a l g o r i t h m s . 467

Uwe Helmke, Fabian Wirth On the discretization of sliding-mode-like controllers . . . . . . . . . .

481

Guido Herrmann, Sarah K. Spurgeon, Christopher Edwards N o n l i n e a r a d a p t i v e s t a t e s p a c e c o n t r o l for a class o f n o n l i n e a r systems with unknown parameters ............................

Christian Hintz, Martin Ran, Dierk SchrSder

497

VIII

AI1 o b s e r v e r v i e w o n s y n c h r o n i z a t i o n

.........................

509

Henri J.C. Huijberts, Henk Nijmeijer Regularity

of the sub-Riemannian

distance and cut locus .....

521

S6bastien Jacquet Industrial sensorless control of induction

motors

..............

535

Feedback invariants and critical trajectories; Hamiltonian formalism for feedback equivalence

..............

545

Fabrice Jadot, Philippe Martin, Pierre Rouchon

Bronistaw Jakubezyk Paths in sub-Riemannlan

geometry

...........................

569

Frgddric Jean Observability of C~-systems

for LC%single-inputs .............

575

Philippe Jouan Robust

control of a synchronous

power generator

Matei Kelemen, Aired Francis Okou, Ouassima Akhrif, Louis-A. Dessaint

.............

583

IX

Volume 2 Control of a reduced size model of US navy crane using only motor position sensors ........................................

1

Bdlint Kiss, Jean Lgvme, Philippe Mullhaupt Algorithms for identification of continuous time nonlinear systems: a passivity approach .................................

13

Ioan D. Landau, B. D. O. Anderson, F. De Bruyne Flatness-based boundary control of a nonlinear parabolic equation modelling a tubular reactor .........................

45

Alan F. Lynch, Joachim Rudolph Dynamic feedback transformations of controllable linear timevarying systems ...............................................

55

Franfois Malrait, Philippe Martin, Pierre Rouchon Asymptotic controllability implies continuous-discrete feedback stabilizability ........................................

time 63

Nicolas Marchand, Mazen Alamir Stabilisation of nonlinear systems by discontinuous state feedback ................................................

dynamic 81

Nicolas Marchand, Mazen Alamir, Iyad Balloul On the stabilization of a class of uncertain systems by bounded control .......................................................

95

Lorenzo Marconi, Alberto Isidori Adaptive

nonlinear

control of synchronous

generators

........

107

Riccardo Marino, Gilney Datum, Franfoise Lamnabhi-Lagarrigue Nonlinear observers of time derivatives from noisy measurements of periodic signals ......................................

123

Riccardo Marino, Giovanni L. Santosuosso Hamiltonian representation of distributed parameter with boundary energy flow ...................................

Bernhard M. Maschke, Arjan van der Schaft

systems 137

X D i f f e r e n t i a b l e L y a p u n o v f u n c t i o n a n d c e n t e r m a n i f o l d t h e o r y . 143

Frdddric Mazenc C o n t r o l l i n g s e l f - s i m i l a r traffic a n d s h a p i n g t e c h n i q u e s . . . . . . . .

149

Radl J. Mondragdn C, David K. Arrowsmith, Jonathan Pitts Diffusive r e p r e s e n t a t i o n for p s e u d o - d i f f e r e n t i a l l y d a m p e d nonlinear systems .............................................

163

Gdrard Montseny, Jacques Audounet, Denis Matignon Euler~s d i s c r e t i z a t i o n a n d d y n a m i c e q u i v a l e n c e of Nonlinear Control Systems .................................

183

Ewa Pawtuszewicz, Zbigniew Bartosiewicz S i n g u l a r s y s t e m s in d i m e n s i o n 3: c u s p i d a l case a n d t a n g e n t elliptic flat case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

193

Mich~le Pelletier Flatness of nonlinear control systems and exterior differential systems .......................................................

205

Paulo Sdrgio Pereira da Silva M o t i o n p l a n n i n g for h e a v y c h a i n s y s t e m s . . . . . . . . . . . . . . . . . . . . .

229

Nicolas Petit, Pierre Rouchon C o n t r o l o f a n i n d u s t r i a l p o l y m e r i z a t i o n r e a c t o r u s i n g f l a t n e s s . 237

Nicolas Petit, Pierre Rouchon, Jean-Michel Boueilh, Frdddric Gudrin, Philippe Pinvidic C o n t r o l l a b i l i t y o f n o n l i n e a r m u l t i d i m e n s i o n a l c o n t r o l s y s t e m s . 245

Jean-Francois Pommaret S t a b i l i z a t i o n o f a series D C m o t o r b y d y n a m i c o u t p u t f e e d b a c k 2 5 7

Richard Pothin, Claude H. Moog, X. Xia S t a b i l i z a t i o n o f n o n l i n e a r s y s t e m s via F o r w a r d i n g mod{LgV} . . 265

Laurent Praly, Romeo Ortega, Georgia Kaliora A robust globally asymptotically stabilizing Feedback: T h e e x a m p l e o f t h e A r t s t e i n ' s circles . . . . . . . . . . . . . . . . . . . . . . . . .

Christophe Prieur

279

XI R o b u s t s t a b i l i z a t i o n for t h e n o n l i n e a r b e n c h m a r k p r o b l e m (TORA) using neural nets and evolution strategies ...........

301

Cesdreo Raimdndez O n c o n v e x i t y in s t a b i l i z a t i o n o f n o n l i n e a r s y s t e m s . . . . . . . . . . . .

315

Anders Rantzer Extended Goursat normal form:

a

geometric characterization

.

323

Witold Respondek, William Pasillas-L@ine T r a j e c t o r y t r a c k i n g f o r 7r-flat n o n l i n e a r d e l a y s y s t e m s w i t h a motor example ...............................................

339

Joachim Rudolph, Hugues Mounier N e u r o - g e n e t i c r o b u s t r e g u l a t i o n design for n o n l i n e a r parameter dependent systems ................................

353

Giovanni L. Santosuosso Stability criteria for time-periodic systems via high-order averaging techniques ..........................................

365

Audrey Sarychev Control of nonlinear descriptor systems, a computer algebra based approach ...............................................

379

Kurt Schlacher', Andreas Kugi Vibrational control of singularly perturbed

systems ...........

397

[(laus R. Schneider R e c e n t a d v a n c e s in o u t p u t r e g u l a t i o n o f n o n l i n e a r s y s t e m s

. . . 409

Andrea Serrani, Alberto Isidori, Cristopher I. Byrnes, Lorenzo Marconi Sliding mode control of the PPR mobile robot with a flexible joint ..... : ....................................................

421

Hebertt Sira-Ram{rez T h e I S S p h i l o s o p h y as a u n i f y i n g f r a m e w o r k f o r s t a b i l i t y - l i k e behavior ......................................................

443

Eduardo D. Sontag Control design of a crane for offshore lilting operations .......

Michael P. Spathopoulos, Dimosthenis Fragopoulos

469

Xll Set-valued differentials and the maximum principle of optimal control .......................................................

487

Hdctor J. Sussmanr~ Transforming a single-lnput nonlinear system to a strict feedforward form via feedback ................................

527

Issa Amadou Tall, Witold Respondek E x t e n d e d a c t i v e - p a s s i v e d e c o m p o s i t i o n o f c h a o t i c s y s t e m s . . . . 543

RaT~jaT~ Vepa O n c a n o n i c a l d e c o m p o s i t i o n o f n o n l i n e a r d y n a m i c s y s t e m s . . . . 555

Alexey N. Zhirabok N e w d e v e l o p m e n t s in d y n a m i c a l a d a p t i v e backstepping control ..........................................

565

Ala~ S.I. Zmober, Julie C. Scarratt, Russell E. Mills, All Jafari Koshkouei List of participants at the 2nd NCN Workshop

...............

623

Control of a R e d u c e d Size M o d e l of US N a v y Crane Using Only M o t o r Position Sensors* Bs

Kiss 1, Jean LSvine 1, and Philippe Mullhaupt 1

Centre Automatique et SystSmes Ecole des Mines de Paris 35, rue Saint-Honor~ F-77305 Fontainebleau, France {kiss, levine ,mulhaupt }@cas. ensmp, fr

A b s t r a c t . Two control problems related to a particular underactuated mechanical system, the reduced size US Navy crane, are addressed. The open-loop motion planning problem is solved by showing that the model of the crane is differentially fiat with a fiat output comprising the coordinates of the load as its first components. The closed-loop global asymptotic stabilization of equilibria is achieved using an output feedback regulator of proportional-derivative type. The extension of this approach to tracking is analyzed based on simulation results.

1

Introduction

Cranes constitute good examples of nonlinear oscillating pendulum- like systems with challenging industrial applications. Their control has been approached by various techniques, linear [1,7,8,15] and nonlinear [6,9,14]. Cranes present two interesting properties from the control engineering point of view. They are underactuated, i.e. the number of actuators is less than the number of degrees of freedom (see [2]). Moreover, only partial information can be used for closed-loop control, i.e. measurement of the whole state is unavailable (especially as far as the rope angles or the load position are concerned) (see [13]). In this paper the particular example of the reduced size US Navy crane is studied. (All presented methods can be extended to a large class of similar equipment [10].) Two control problems are addressed: open-loop real-time motion planning and closed-loop stabilization. The solution presented to the open-loop motion planning problem allows to calculate the necessary control input as to move the load along any (sufficiently smooth) trajectory in the working space using the flatness property [3-5] of the system. The second control problem is * Research supported by the Nonlinear Control Network, European Commission's Training and Mobihty of Researchers (TMR) Contract # ERBFMRX-CT970137

http: / /www.supelec.fr /lss/N CN / crane.html# P DP

2

B. Kiss, J. L~vine, and Ph. Mullhaupt

the c l o s e d - l o o p stabilization of both an equilibrium and a trajectory. Since the only measurements available are the motor positions (recall that the load position or the rope angles are not measured) this problem can not be solved using full state feedback. Instead, a classical PD output regulation is proposed. Global closed-loop stability of equilibria can be proved using LaSalle's invariance principle [12] and the particular structure of the crane dynamics. Note that this result is of particular practical interest to reduce the time to d a m p the oscillations of the load during harbor operations [16]. Simulation results show that the same regulator m a y also be used for tracking. Based on experimental considerations, it appears that our PD regulator together with flatness based trajectory planning outperforms the globally stabilizing regulator, though no proof is presented herein. The remaining part of the paper is organized as follows. The next section is devoted to the general description of the experimental setup. Modeling equations are given in Section 3. The solution of the o p e n - l o o p motion planning problem is presented in Section 4 based on the flatness property of the model. Asymptotic global stability of equilibria in c l o s e d - l o o p using output feedback regulators of proportional-derivative type is studied in Section 5. Simulation results of an extension of the same controller with open-loop trajectory planning for tracking are presented in Section 6.

2

General

description

of the

experimental

setup

The reduced scale (1:80 size) model 1 of the US-Navy crane is depicted in Figure 1. Four DC motors (three of them winching ropes) are mounted on the structure allowing to manipulate the load in a three dimensional workspace. The control objective is to move the load swiftly from an initial position to a desired final position without sway and avoiding obstacles. Since the accelerations of the motors tend to create oscillations of the load, simultaneously fast and swayless displacements are hard to realize. The reduced size model comprises: 9 A load (maximal nominal mass: 800g) 9 A mobile pulley guiding the rope which hoists the load. 9 A rotate platform actuated by the DC motor no.4 using a synchronous belt transmission. 9 A hoisting system mounted on the rotate platform comprising three ropes winched by three DC motors (motors no.l, 2, 3): - A h o r i z o n t a l rope attached to the mobile pulley and ending up on the winch of motor 1. - a v e r t i c a l rope attached to the mobile pulley and ending up on the winch of motor 2. This rope prevents the mobile pulley from falling. i the reduced scale model was made by Walter Rumsey, Paris, France

Control of a Reduced Size Model of US Navy Crane

3

- A r o p e a t t a c h e d to t h e load passing through t h e m o b i l e pulley and e n d i n g u p on the winch o f m o t o r 3. ,, A power electronics unit S. It receives sensor signals from t h e i n c r e m e n t a l encoders m o u n t e d on the m o t o r axes a n d t r a n s m i t s t h e m to a c o m p u t e r . It also provides the necessary power a m p l i f i c a t i o n to the DC m o t o r s .

*~A$ ZOO0

~

ntermediatepulleys fixed on the boom ,/'~.

mobile~,

load

motors with i sensorsand winches i!. power electronics i:~-

~pratt orm

::;~/:;~i ~'+ ;~

F i g . 1. The reduced (1:80) size US Navy crane in the authors' lab.

T h e control a l g o r i t h m is i m p l e m e n t e d on a personal c o m p u t e r e q u i p p e d with a s t a n d a r d d a t a acquisition card. T h e m e a s u r e d signals are t h e r o t a t i o n angles of the m o t o r axes which allow to calculate directly t h e rope lengths and the r o t a t i o n angle of the p l a t f o r m . T h e c o r r e s p o n d i n g velocities are c a l c u l a t e d using n u m e r i c a l derivation.

3

Model Equations

Figure 2 gives the schematic r e p r e s e n t a t i o n of the crane. T h e electronic t i m e c o n s t a n t s are negligible w.r.t, the mechanical t i m e c o n s t a n t s . Consequently, the i n p u t variables of the m o d e l are the torques T1, T2,T3,T4 delivered by the m o t o r s no.1 - 4 respectively. 2 the power electronics unit was made by the Institut d'Automatique of the t~cole Polytechnique Fgdgrale de Lausanne, Switzerland

4

B. Kiss, J. Ldvine, and Ph. MuUhaupt

Observe t h a t along each rope there is an i n t e r m e d i a t e pulley fixed to the boom. Since the length of the rope sections between these pulleys a n d t h e winches of the m o t o r s are constant, we consider t h a t the m o t o r s drive d i r e c t l y the axis of the i n t e r m e d i a t e pulleys a n d we reduce all r o t a t i n g i n e r t i a s along each rope on these axes. T h e s a m e simplification is m a d e concerning the belt transmission, i.e. we consider t h a t m o t o r 4 drives directly the axis o f the r o t a t e p l a t f o r m and we reduce all r o t a t i n g inertias to this axis. All ropes are considered to be rigid.

T2'm2~ _ T3'm3~ L,

(x21'x22'x23) (x31'x32'x33)

i i

(XI,X2,X 3) 0 m

Tl'ml

,M

F i g . 2. Simplified representation of the 3D US Navy crane

T h e following variables and i n e r t i a p a r a m e t e r s are introduced: * x l , x2, x3: position of the load, 9 z01, x02, x03: position of the m o b i l e pulley, 9 x l l , x~2, x13: position of the pulley winding the horizontal rope a t t a c h e d to the mobile pulley, 9 x21, x~2, x23: p o s i t i o n of the pulley winding the vertical rope a t t a c h e d to the mobile pulley, 9 x31, xa2, x3z: position of the pulley hoisting the load, 9 Lo: length of the rope section between the pulley hoisting the load and the mobile pulley, 9 L1 : length of the horizontal rope a t t a c h e d to the mobile pulley, 9 L2: length of the vertical rope a t t a c h e d to the mobile pulley, 9 L3: length of the rope a t t a c h e d to the load,

Control of a Reduced Size Model of US Navy Crane

5

9 m: mass of the load, 9 m0: mass of the mobile pulley, 9 rnl, rn2, rna: rotating inertias reduced to the intermediate pulleys' axis, 9 M : rotating inertia reduced to the axis of the rotate platform The construction of the crane is such that the three intermediate pulleys are aligned. Thus we introduce two geometric parameters a s and a3 such that xlj = a i x U for i = 2, 3 and j = 1, 2, 3. Observe also that xi3 is constant for i = 1,2,3. Denote by q = ( q l , . . . ,q12) T = (xl, x2, x3, xol, xo~, x03, x11, x12, L0, L1, L~, L3) T the vector of system variables. Geometric constraints are present due to the various cable distances that need to be compatible with each other. Theorem

1. The dynamics associated to the US Navy crane are

m ~ l = A l ( x l - xol)

m ~ =AI (x2 - zo2) m x 3 : A 1 (x3 - x03) - rng

mo~o~ = - ~ ( ~ - ~ o ~ ) + ~ ( ~ o ~ - ~ ) + ~ ( ~ o ~ - ~ 9 ~~)+~ (~o~- ~ )

mo~o~= - ~ (~-~o~)+~ (~o~- ~ ) + ~ (~o~ ~ ) + ~ (~o~- ~ 3 ~ ) ~o~o~=-~ (~-~o~)+~ (~ o~- 9~ ) + ~ (~o3-~ 9 ~)+~4 (~o~- ~ ~3)-mog 0 = A I ( L 3 - Lo) - A4Lo rnlLl=--A2L1 + T1

(1)

m2L2-=-A3L2 -4- T2 rn3L3~--A1 (L3 - Lo) 4-T3 M Y 11 = --,~2 (xo 1 - x i i )--0~2 ,~3 (Xo 1-- ot 2 x 11)--~3,~4 (~01-- a3.~ 11 )-~-~5.~' 1 1 - - ~ x 12

M x12=-A2( xo2-x12 )--o;2 ~3( xo2-0t2z12 )---ot3A4( zo2-0L3x12)-l-~sz12-~T4x11 subject to the constraints C1=

~

(xi - xoi) 2 - ( L a -

C2 = ~

xoi - z l i ) ~ - L

Lo) 2

=0

= 0

ki=l

c~ = ~

(xo~ - ~ / ~

- i

= 0

\i=I C4 =

~

x0i - a3xli)

2 -

\i----1

C ~ = ~1 ( ~ 1 + ~ -

r~32)

=o

L

= 0

(2)

6

B. Kiss, J. L~vine, and Ph. Mullhaupt

The multipliers A1,... , A5 are associated to the constraints C 1 , . . . , C5. Recall that the kinetic and the potential energy of the system read

Wk= 2

(m:~ + mo~.~i) + E M~.~i + E m,L~ i=1

i=1

i=1

Wp =mgxa + mogz0a, hence the Lagrangian is defined b y / : = Wk -- Wp. A proof of this result can be found in [10]. 4 4.1

Flatness

and

Motion

Planning

Flatness

A flat output is given by Y = (Y1,-.-, Y4)T = (xt, x2, xa, x03) T, the coordinates of the load and the height of the mobile pulley. As long as we omit freefall reference trajectories, i.e. ~a r the first three equations of (1) give At, z01 and z02 as functions of Y and Y. Equations 5 to 8 of (1) and Constraints C1, C4 and C5 of (2) can then be used to express {A2, A3, A4, Xll, x12, La, L0} as expressions of A1, xl, x2, xa, x01., x02, x03 and derivatives up to order 2 (thus giving expressions involving Y, Y, Y, y(a) and y(4)). Next, the constraints C2 and Ca give Lt and L2. The other equations of (1) give the inputs Tt, T2, T3 and T4 (T4 is obtained after solving the last two equations for T4 and As). The inputs are expressions of Y and its derivatives up to order 6. 4.2

Motion planning

Assume that the position, velocity, acceleration, jerk and all derivatives up to 6th order of the flat output (including the position of the load) at the start time t, are given by (YI,~'I,YI,...,Y/(S),]I/(6)) and the desired final configuration of the flat output at time ~F is (YF, YF, YF, " ' ' , y(5) F , y(6)~ F / " We can construct 13 th degree polynomials,

Y~i(t)=Y1~ + (YF, - Yzi) E aji j=t

\tF -- tl ]

(3)

where the coefficients aji, j = 1 , . . . , 13 and i = 1 , . . . ,3, are computed by solving linear equations, whose entries are combinations of the initial and final conditions. In particular, motion planning between two different equilibria YI and Y---Fcan be obtained simply by setting Yt = Yt, I~i = YI = . . . = Yt(5) = y]6) = 0 and YF = "YF, YF = ~'F = - . . = YF(5) = YF(6) = 0. The input to be applied that results in the above trajectories is then computed using the flatness property as described in the previous subsection.

Control of a Reduced Size Model of US Navy Crane 5

Output

Feedback

Regulation

We wish to stabilize the crane at a given equilibrium point of the load (xl, 5r2, x3) and at a given height of the mobile pulley s Using the constraints and the dynamic equations at equilibrium, we find the equilibrium values of the remaining variables: x11, x12, L0, L1, L2, L3 and the corresponding input torques to be applied: T1, T2, T3 and T4. (Observe that T4 = 0 for all equilibria). Define the error variables as eqi = qi -- qi where qi stands for ith component of q. Additionally define ~ = arctan(~-m), the rotation angle of the rotate platform. Then the corresponding error variable is e~ = ~ - ~. Recall that the measured variables are: L1, L2, L3 and ~. T h e o r e m 2. The four PD controllers,

711 = T1 + kdleL1 -{- kpleL1 7"2 = T2 +

kd2eL~ +

kp2eL~

(4)

T a = T 3 + kdadL3 + kpzeL3 T4=kd4er162

applied to the crane dynamics (1) with Constraints (2) assure closed-loop global stability of the equilibrium (xl, s x3, x03). The proof relies on two lemmas as in [11]. Let us define the following energylike function:

(5)

W = Wk + Wp + Wa,l, with

we,,, = ~

k,,4, + kp,~ \i=1

+

T, eL,

(6)

i=1

the "potential" energy stored in the controllers. L e m m a 1. The derivative of W along closed-loop trajectories is given by: r 2

W = -k~4,

.

- k ~ 4 2 - k~34~ - k~,~e~

(7)

L e m m a 2. The only invariant trajectory compatible with 1~ = 0 is the equilibrium trajectory, i.e. xl(t) = 5:1, x~(t) = x2, xz(t) _~ x3 and x03(t) = x03.

8

6

B. Kiss, J. L~vine, and Ph. Mullhaupt

E x t e n s i o n to Tracking

Assume that a reference trajectory is constructed so as to steer the load from an idle point to another idle point with obstacle avoidance. This can be done using polynomials as described in Section 4. Denote the polynomial reference trajectory of the flat output by Yc. Based on flatness, one can calculate the reference trajectory of all other variables in the system as functions of Ye, ~'c, Y c , . . . , y(6). Denote by qc = ( q l c , . . . , ql2c) T "-- (Xlc, XXc, X3c, XOlc, X02c, X03c, Xllc, Xl2c, Loc, Llc, L2c, L3c) T the reference trajectory of all system variables and by Tic, T2c, T3c, T4c the reference inputs. We investigate in this section the closed-loop behavior of the system using the same PD regulator as before but fed by the above references. This modified controller is referred to as the tracking controller. Define eq, - qic - qi where qi is the ith component of the vector q and qic is the ith component of qc. The tracking PD controller is given by: T1 = Tic -~- kdleL1 Jr kpleL~ T2 = T2c -~- kd2eL~ "}- kp2eL2 T3 = T3c + kd3eL3 + kpseL3 7"4 = T4c + kn4~ + kp4e~.

(8)

Note that for equilibrium trajectories we get the same PD regulator as before. T h e o r e m 3. Let the final point qc(tF) of the reference trajectory be an equilibrium of the system and suppose that all derivatives alon 9 the reference trajectory are bounded. Then qc(tF) is asymptotically stable in closed-loop using the tracking P D controller. The stabilization property of the tracking controller given by (8) has been validated using simulation. Comparison of the closed-loop behavior of the two controllers during point to point steering is undertaken. The global stabilizing controller is fed by the equilibrium reference of the desired final point and the tracking controller is fed by the reference trajectory obtained by flatnessbased motion planning. Two reference trajectories connecting the same initial and final points with transit time of 2.5 seconds are envisaged. The first trajectory (Figures 36) is a horizontal displacement. The second one (Figures 7-9) is a parabola in the vertical plane determined by the two points. The globally stabilizing controller produces the same behavior in both cases with damped oscillations, while the tracking controller stabilizes the desired reference and arrives at the equilibrium faster and with less oscillations. The same gains kdi, kpi (i = 1 , . . . , 4) are used for both controllers.

Control of a Reduced Size Model of US Navy Crane

9

Notice t h a t the tracking controller o u t p e r f o r m s t h e g l o b a l one, hence decreasing b o t h the residual sway a n d the reaching t i m e . moron in the verlical pMne y-z

motion in the horizontal plane x-y

/ i1"

a~

(m)

(m) 0'

F i g . 3. Closed-loop tracking behavior under PD control. Trajectories of the load in the horizontal and vertical planes: i) global stabilizing equilibrium controller (hashed line); ii) tracking controller with motion planning; iii) reference to steer to equilibrium along a straight line (dotted).

Hodzo~tal rope length LI

~

ol

t

,6

z

21

lime (sec)

3

~

total l e i t h L of fi'm main rope

,

4,

,

o

ol

I

Ii

2

26

~

s

.

,

.,

i

(see)

F i g . 4. Closed-loop tracking behavior, rope lengths: i) global stabilizing equilibrium controller (hashed line); ii) tracking controller with motion planning; iii) reference to steer to equilibrium along a straight fine (dotted).

References 1. G. Corriga, A. Giua, and G. Usai. An implicit gain-scheduling controller for cranes. IEEE Transactions on Control Systems Technology, 6(1):15-20, January 1998. 2. B. D'Andr~a and J. L~vine. Modelling and nonlinear control of an overhead crane. In J.H. van Schuppen M.A. Kashoek and A.C.M. Ran, editors, Progress in Systems and Control Theory 4, Robust Control o/ Linear Systems and Non1990. linear Control, Proc. MTNS'91, Vol. II, pages 523-529. Birkhs

10

B. Kiss, J. L6vine, and Ph. Mullhaupt

II~k 0~-,)

"

. . . .

Ilml 1 ~ 1

I~e (~c)

Fig. 5. Closed-loop tracking behavior, angles: i) global stabilizing equilibrium controller (hashed line); ii) tracking controller with motion planning; iii) reference to steer to equilibrium along a straight line (dotted).

Fig. 6. Closed-loop tracking behavior, motor tensions: i) global stabilizing equilibrium controller (hashed line); ii) tracking controller with motion planning; iii) reference to steer to equilibrium along a straight line (dotted).

3. M. Fliess, J. L~vine, Ph. Martin, and P. Rouchon. Lin$arisation par bouclage dynamique et transformations de Lie-B/icklund. C.R. Acad. Sci. Paris, 1317:981-986, 1993. 4. M. Fliess, J. L~vine, Ph. Martin, and P. Rouchon. Flatness and defect of nonlinear systems: introductory theory and examples. International Journal of Control, 61(6):1327-1361, 1995. 5. M. Fliess, J. Ldvine, Ph. Martin, and P. Rouchon. A Lie-B/icklund approach to equivalence and flatness of nonlinear systems. IEEE Transactions on Automatic Control, 38:700-716, 1999. 6. M. Fliess, J. L~vine, and P. Rouchon. A generalised state variable representation for a simplified crane description. International Journal of Control, 58:277-283, 1993. 7. D. Fragopoulos, M.P. Spathopouios, and Y. Zheng. A pendulation control system for offshore lifting operations. In Proceedings of the 14th IFA C Triennial World Congress, pages 465-470, Beijing, P.R. China, 1999. 8. T. Gustafsson. On the design and implementation of a rotary crane controller. European Journal of Control, 2(3):166--175, March 1996. 9. K.S Hong, J.H. Kim, and K.I. Lee. Control of a container crane: Fast traversing, and residual sway control from the perspective of controlling an underactuated

Control of a Reduced Size Model of US Navy Crane motion b Iho ~ a k ~ l i ~ n e y-z

motk~ k~the horizon~l phme x.y

az

11

~N

'3,,: :1 (m)

(m)

F i g . 7. Closed-loop tracking behavior under PD control. Trajectories of the load in the horizontal and vertical planes: i) global stabilizing equilibrium controller (hashed line); ii) tracking controller with motion planning; iii) reference to steer to equilibrium along a parabola (dotted). ~lal

*,

o am

i

9

o

a*

~ a l km~h L M Ihe rn~n rol~

j-~ I

1

rope lenglh LI

I

1.

"

2

z*

~

tirne(~x:)

~l

*

,,

*

. . . . . .

.m;'(.r162

. . . . . .

F i g . 8. Closed-loop tracking behavior, rope lengths: i) global stabilizing equilibrium controller (hashed line); ii) tracking controller with motion planning; iii) reference to steer to equilibrium along a parabola (dotted). system. In Proceedings of the ACC, pages 1294-1298, Philadelphia, PA, June 1998. 10. B Kiss, J. LSvine, and Ph. Mullhaupt. Modeling and motion planning for a class of weight handling equipments. In Proceedings of the 1$th International Conference on Systems Engineering, Coventry, UK, September 2000. 11. B Kiss, J. LSvine, and Ph. Mullhaupt. A simple output feedback PD controller for nonlinear cranes, submitted to: 39th IEEE Conference on Decision and Control, Sydney, Australia, December 2000. 12. J. La Salle and S. Lefschetz. Stability by Liapunov's Direct Method With Applications. Mathematics in Science and Engineering. Academic Press, New York, London, 1961. 13. J. L~vine. Are there new industrial perspectives in the control of mechanical systems ? In Paul M. Frank, editor, Advances in Control, pages 195-226. Springer-Verlag, London, 1999. 14. J. L~vine, P. Rouchon, G. Yuan, C. Grebogi, B.R. Hunt, E. Kostelich, E. Ott, and J. Yorke. On the control of US Navy cranes. In Proceedings of the European

12

B. Kiss, J. L~vine, and Ph. Mullhaupt

,,+

.

.

m,,we d ~ + , , + ~ I+ retool+ + f + ~ , ~ t ~ ~

.

.

.

I

.

.

.

.

.

.

.

.

.

.

......

~,,,~.<~ . . . . . . . . . . . . .

.,,~.~ . . . . . .

Fig. 9. Closed-loop tracking behavior, motor tensions: i) global stabilizing equilibrium controller (hashed line); ii) tracking controller with motion planning; iii) reference to steer to equilibrium along a parabola (dotted).

Control Conference, pages 213-217, Brussels, Belgium, July 1997. 15. A. Marttinen, J. Virklomen, and R.T. Salminen. Control study with a pilot crane. IEEE Trans. Edu., 33:298-305, 1990. 16. R.H. Overton. Anti-sway control system for cantilever cranes. United States Patent, (5,526,946), June 1996.

Algorithms for Identification of Continuous Time Nonlinear Systems: a Passivity Approach. Part I: Identification in Open-loop Operation Part II: Identification in Llosed-loop Operation I. D. Landau 1, B. D. O. Anderson 2, and F. De Bruyne 2,3 I Laboratoire d'Automatique de Grenoble (CNRS-INPG-UJF) ENSIEG, BP 46 38402 Saint Martin d'H~res, France landau@lag, ensieg, inpg. fr 2 Department of Systems Engineering and Cooperative Research Centre for Robust and Adaptive Systems, RSISE The Australian National University Canberra A C T 0200, Australia 3 Siemens EIT ES (Advanced Process Control Group) Building 15/0+, Demeurslaan 132 B-1654 Huizingen, Belgium K e y w o r d s : Recursive Identification, Nonlinear Systems, Adaptive Systems, O u t p u t Error A b s t r a c t . Algorithms for the identification of continuous time nonlinear plants operating in open-loop and in closed-loop are presented. An adjustable output error type predictor is used in open-loop operation. An adjustable output error type predictor parametrized in terms of the existing controller and the estimated plant model is used in closed-loop operation. The algorithms are derived from stability considerations in the absence of noise and assuming that the plant model is in the model set. Some convergence results based on passivity concepts are presented. Subsequently the algorithms are analyzed in the presence of noise and when the plant model is not in the model set. Examples illustrate the use of the various algorithms.

General introduction O u t p u t error identification algorithms for linear systems have been known for a long time to offer excellent performances in the presence of output noise. In particular they do not require a d y n a m i c model for the noise, the only requirements relate to independence with respect to the excitation signal and its boundedness [7,12].

14

I.D. Landau et al.

It seems therefore interesting to extend this type of algorithm to the identification of continuous time nonlinear systems operating in open-loop. In the present paper we focus on the recursive identification of nonlinear plants whose outputs cannot be expressed linearly in terms of the unknown parameters (i.e. y # 00re where y is the output, 80 is the vector of parameters and r is a vector of nonlinear functions of various variables). Therefore the first part of the paper will be dedicated to this subject. The development of algorithms for plant model identification in closed-loop has been an important line of research in the last few years. This line of research has been motivated by several factors: 1) the fact that in a number of situations identification in open-loop is difficult or simply not feasible (unstable plants, drift), 2) the presence of a controller in the loop (which has to be re-tuned), 3) the possibility of capturing the dynamic characteristics of the plant model which are critical for control design. In the context of linear models, recursive and batch algorithms for plant model identification in closed-loop have been proposed, analyzed and evaluated experimentally [18,10,8,17,13,9]. Such algorithms have already moved towards standard use in industry. One of the successful ways to develop algorithms for identification in closedloop is to consider "closed-loop output error" schemes. [8,17]. The problem of closed-loop identification of nonlinear time-varying systems in the presence of a linear or a nonlinear controller has been discussed in [3,10] using the Hansen scheme and in [18] using a gradient approach. The convergence of the algorithms is not discussed. In the present paper we focus on the recursive identification of nonlinear plants operating in closed-loop with a nonlinear controller using a closedloop output error identification scheme. Preliminary results can be found in

[191. Passivity properties of various linear time-varying input-output operators play an important role in assessing the convergence properties for the various algorithms. The paper is organized as follows. Part I is dedicated to open-loop identification. Part II is dedicated to closedloop identification. An example of identification of an open-loop unstable nonlinear plant model in closed-loop operation is given in Section 2.4. The concept of strong strict passivity and related properties are used extensively in this paper and outlined in Appendix A.

Algorithms for identification of continuous time nonlinear systems Part

I. I d e n t i f i c a t i o n

1.1

in open-loop

15

operation

O p e n - l o o p o u t p u t e r r o r i d e n t i f i c a t i o n . T h e basic e q u a t i o n s and problem setting

The objective is to estimate the parameters of a single input single output (SISO) nonlinear time invariant system described by

s:

v = P0(u,v)

(1)

where P0 is an unknown causal nonlinear operator, u is the control input signal, y is the achieved output signal and v is the disturbance signal allowed to enter the system nonlinearly. It is not assumed that the output y can be expressed linearly in terms of some parameter vector 90. For ease of notation the time argument will be omitted when there are no ambiguities. It is required that the system P0 is Bounded Input Bounded Output (BIBO) stable. In the sequel we often make use of linearizations of some nonlinear operators around their operating trajectories. We therefore require that the plant and the model (to be defined subsequently) are smooth functions of the input signal, the output signal and the disturbance signal. This means that if the operator is linearized around any (stable) trajectory, the resulting linear (time-varying) system is BIBO stable. See [5] for more details. We consider the following adjustable model for the system defined by (1)

v(o) = P(o, u)

(2)

where P(O, u) defines the adjustable plant model, predictor and u is the plant model input.

u

P

y(O) is

the output of the

Y

_

~(0)

Fig. 1. Open-Loop Output Error (OLOE) identification scheme The output error is defined as = y - y(0).

The following assumptions wilt be made until further notice:

(3)

16

I.D. Landau et al.

(i) BOo such that P(Oo,U) = P0(u,0) for all u E / : 2 , and v - 0 (subsequently in the case v _-- 0 the a r g u m e n t v will be deleted) (ii) N o t a t i o n : The partial derivative of P(6, u) with respect to 6j is denoted by P ~ (6, u) for j = 1,. 9 d where d is the dimension of the p a r a m e t e r vector ~. The operator P~. (6, u) and its time derivatives exist and are n o r m - b o u n d e d V2 along the trajectories of the pre&ctor which r e q m r e s / t to be bounded. This assumption is not particularly restrictive as P and P(6) are assumed to be smooth operators. (iii) The input u and the stochastic disturbance v are independent. Assumption (i) means that at least for 6 = 60 and in the absence of noise, the plant is in the model set. (The case when this is not true will be discussed separately in Section 1.4). The generic p a r a m e t e r adaptation algorithm (PAA) which will be used throughout the paper is the continuous time version of the general PAA used in [9]:

~(t) = F(t)r F - l ( t ) = - [1 - Al(t)] F-l(t) + A2(t)r162 0 < Al(t) < 1,

(4) (5)

0 < A2(t) < 2, F(0) > 0, F-l(t) > ~F-I(O), 0 < c~ < cr

where 6(t) is the estimated p a r a m e t e r vector, r is the open-loop o u t p u t error (defined above), r is the observation vector, F(t) is the a d a p t a t i o n gain matrix, A1 (t) is a time-varying forgetting factor and A2 (t) allows one to weight the rate of decrease of the a d a p t a t i o n gain. T h e two functions A1 (t) and A~(t) allow one to have different laws of evolution of the a d a p t a t i o n gain. Some of the typical cases are: 1. 2. 3. 4.

Al(t) Al(t) Al(t) Al(t)

= <

1; A~(t) -- 0;/6(t) = 0; F(t) = F(0) (the gradient algorithm); 1; A2(t) - 1 (recursive least squares type algorithm); eonst < 1; A2(t) -- 1 (least squares with forgetting factor); 1; l i m t ~ Al(t) = 1; A~(t) -- 1 (variable forgetting factor).

We will consider subsequently that the assumptions (i) through (iii) are valid and furthermore, for some analysis, that: (iv) v = 0 (v) The higher order terms in the Taylor series involving expansions in powers of (y - y(6)) and (6o - 6) along the trajectories of the system can be neglected This will allow us to implement the appropriate p a r a m e t e r estimation algorithm to begin with (i.e. it allows us to find the observation vector r and

Algorithms for identification of continuous time nonlinear systems

17

to analyze its asymptotic properties. In the first stage we will use several expansions in Taylor series for the expression of the plant o u t p u t and predictor output and we will neglect the terms of power higher or equal to 2. A subsequent analysis will discuss the case when these terms are not neglected. It will also treat the presence of disturbances and unmodeled dynamics, requiring Assumption (iii).

1.2

Nonlinear open-loop output error algorithms

One has the following result (the N L O L O E algorithm): 1. Under the assumptions (i) through (ii), (iv) and (v) one has

Theorem Ior

r

=

[P'(O,u)] T = [P~l(o,

~)

...

P~.(o,~)]r

(6)

that lim e(t) = 0

(7)

t--+ o o

and lim CT(t)(0(t) -- 60) = 0.

(8)

t-.o~

Remark

I.l:

1. For the particular case when one can write

y(6) -= P(6, u) = cT (t)O where r has

is a vector of linear or nonlinear functions of y(O) and u one [P'(0, u)] T = r

2. The condition (8) assures that the estimated parameter vector 6, converges to a set defined as Vc = { 6 : c T ( t ) ( 6 - - 60) = 0}-

(9)

If cT(t) (O -- 60) = 0

(10)

has a unique solution O = 60, the parameter vector will converge toward this value. In fact this condition is a "persistence of excitation" condition for the nonlinear case.

18

I.D. Landau et al.

P r o o f o f T h e o r e m 1: The proof will be done in several steps. Step I: Establishing the expression e = f(80 - 8(t)) From (3) one has: = P(8o, u) - P(8, u)

(11)

Using a series expansion around 8, one has = P(8o, u) - P(8, u) = P(8, u) + P'(8, u) (80 - 8) - P(8, u) = P'(8, u) (00 - 8),

(12)

neglecting higher order terms in ( 8 0 - 8). Therefore (11) becomes e = P'(8,

u) (80 - 8)

(13)

Step II: (Stability proof) With r given by (6), (12) together with the P.A.A. given by (4) and (5) define an equivalent feedback system characterized by the following equations: g :

Yl

=

( - P ' ( 8 , u) 8(t)) = ul = - y a

O(t) = F ( t ) [ P ' ( O , u)lT e = F ( t ) [ P ' ( O , u)] T us

y= = P'(O, u) "~(t)

(14)

(15) (16)

where

(17)

~(t) = 8(t) - 0o

and u j, yj, j = 1,2 define the inputs and outputs of the equivalent feedforward and feedback blocks, respectively. The feedforward block is characterized by a unit gain. Refer to Figure 2. In the general case with F(t) time-varying, the feedback path is not provably passive and we have to use an extension of the passivity theorem, given in Appendix (Theorem 6) as well as the definitions of the systems belonging to the class L(A) (excess of passivity) and N ( F ) (lack of passivity) (see Appendix, Definitions 2 and 3). Consider the equations (15) and (16) together with (17). Equations (15) and (16) correspond to a state space representation considered in Lemma 3 (appendix) with A = O, B = F(t)r

C = cT(t),

~: = "g, u = E = us,

# = y~ = P ' ( 8 , u) "g(t).

D = 0 (lS)

Algorithms for identification of continuous time nonlinear systems

19

( 00

T

~ U2

'(0, u)] ~ Fig. 2. Equivalent feedback representation of the identification scheme Taking P(t) = F-l(t) in Lemma 2 one gets using (5), (A.11) [1 - Al(t)]

F-l(t) - A2(t)r162 (t) = Q(t)

(19)

also,

S(t) -- 0 and R(t) = O.

(20)

Notice that the positive semidefiniteness of (167) is not being claimed. Now using Lemma 3 one has --

P'(•, u) ~s dr

0

1

> 2 ~r(t)F-l(t)

lZ:

- ~

~(t) -

1 b~ (t0)F_ l(t0 ) ~(t0)

,~(,-) II~'r(r)r

2 dr

= 12 ~r(t)F-l(t) ~(t) - 1 b~ (t0)F_ 1(r ~(t0) - ~l ft ] A2(-) Ily2(r)ll 2 dr.

(21)

Therefore, it follows from Definition 3 that the equivalent feedback block belongs to the class N(F) with F = )~2(t) (i.e. it falls short of being provably passive). The feedforward block belongs to the class L(A(t)) with A(t) > A2(t) and applying Theorem 6, (7) and (8) result. 1.3

A n a l y s i s of t h e A l g o r i t h m s in t h e P r e s e n c e o f N o i s e

In the following analysis we will make the following assumptions:

20

I.D. Landau et al.

9 T h e noise signal v m a y enter nonlinearly. However, we show later t h a t one can only prove convergence w.p.1 if the noise is additive. 9 T h e signal to noise ratio (SNR) is high. 9 T h e noise is zero mean, finite power and i n d e p e n d e n t of the e x t e r n a l excitation r. 9 T h e higher order terms in certain Taylor series expansions a r o u n d the n o m i n a l t r a j e c t o r y are neglected (i.e. one assumes t h a t they are small c o m p a r e d to the noise level). 9 OPo~ (u, O) = OP~ (00, u, 0) is assumed to be a B I B O ( a s y m p t o t i c a l l y ) stable operator. Here OPo~ (u, 0) denotes the linearization of P0 in response to a p e r t u r b a t i o n in v a r o u n d the t r a j e c t o r y u and v = 0. Denote by Y = P0 (u, 0)

(22)

the values of the p l a n t o u t p u t o b t a i n e d in the absence of noise (i.e. v - 0). Denote by

= Po(u, v)

(23)

the values of the p l a n t o u t p u t o b t a i n e d in the presence of noise. Define = y+ w

(24)

where w is the p e r t u r b a t i o n coming from the noise v. Then

9 = P(u,O) + OPo,(u,O)v,

(25)

i.e. one has

w = OPo. (u, O) v.

(26)

Therefore the effect of the noise can be considered to be additive in a s m a l l noise situation. We will also assume t h a t w(t) is zero m e a n (if it is not the case, the m e a n value can be removed). Note t h a t for the case of an a d d i t i v e noise, one has w -- v. If the noise is not additive, one can not g u a r a n t e e u and w to be independent and therefore w.p.1 convergence of the p a r a m e t e r s can not be assured in this case. T h e analysis will be done in the context of a decreasing a d a p t a t i o n gain a l g o r i t h m (i.e. Al(t) - 1 and As(t) > 0 Vt). Since the continuous t i m e P.A.A. a l g o r i t h m given in (4) and (5) is in this case a l i m i t i n g case of a discrete t i m e least squares t y p e algorithm, one can use an averaging techniques for large t and in p a r t i c u l a r the O.D.E. a p p r o a c h developed by Ljung [9,12]. We will prove the following theorem which establish the w.p.1, convergence conditions in the presence of noise for the a l g o r i t h m presented in Section 1.2.

Algorithms for identification of continuous time nortlinear systems

21

T h e o r e m 2. Consider the P . A . A O(t) = F(t) r

F-~(t) = ~2 r

e(t)

cr(t);

(27)

~2 > 0; F(0) > 0

where F(t) is the adaptation gain matrix, r e(t) is the output error.

(28) is the observation vector and

,, Assume that the stationary processes r 0) and s(t, O) can be defined for O(t) =-_0 (i.e. 0 is assumed to be constant). 9 Assume that O(t) generated by the algorithm (27), (28) belongs infinitely often to the domain 7) s for which the stationary processes r O) and s(t, O) can be defined. ,, Define the convergence domain D c as : D c = {O: r 9 Assume that r

(29) O) can be expressed as

e(t, O) = H (Oo, o)r (t, O)[O0 -- O(t)] + w(t, Oo, O)

(30)

where H(Oo, O) is a linear time varying causal operator:' having the structure H(Oo, O) = h(Oo, O) I, where h(Oo, O) is a scalar operator. 9 Assume that r O) and w(t, 0o, O) are independent. Then Prob {trim 0(t) E :De} = 1

(31)

/f H'(Oo,O) = H ( O o , O ) - 2-~-LI;

A(t) > A2 Vt > to

(32)

is a strong strictly passive operator z for all O. C o r o l l a r y 1. With the same hypotheses as for Theorem 2, if c r ( t , 0)(0 -- 00) = 0

(33)

has a unique solution 0 = 0o, then the condition that H'(Oo, O) be a stron 9 strictly passive operator implies that Prob

O(t) = e} = 1.

(34)

1 The structural assumption on H can be omitted at the cost of more complicated calculations. 2 The definition of a strictly strongly passive operator is given in appendix.

22

I.D. Landau et al.

For the algorithms presented in Section 1.2, the equations of the output error for a fixed value 0 takes the form:

~(t, o) = Cr(t)(00 - e) + ~(t, 00)

(35)

i.e. H(Oo, O) = 1 and therefore condition (32) is automatically satisfied. As can be observed, the stationary process r 0) for all the algorithms will depend only on the external excitation u (it will not be affected by the noise v). On the other hand, if v(t) is an additive noise, w(t) -- v(t)) and it is independent of u. As a consequence w(t) and u(t) are independent. Therefore the w.p.1, convergence of the parameters in the stochastic case is assured. Proof of Theorem 2 Define

n(t)

=

71 f_l(t)"

(36)

Using (5) one gets

h(t) = 71 [A2 r

r

-- R(t)].

(37)

The ordinary differential equation associated with the algorithm (27), (28) takes the form [9,12] 0(r) = R -~ (r) f(O(r))

(38)

R(v) = A~ G (O(r)) - R(r)

(39)

where

f(O) = E{r

R)e(t, 0)}

(40)

G(0) = E{r

o)r

(41)

and 0)}.

Using (30) one gets for 9 -- const,

f(O) = E{r But r E{r

-- 0) + E{r

(42)

0) for 9 = const and w(t) are assumed to be independent. Therefore •) w(t, 0o, g)} = 0

(43)

The ODE defined by (40) and (41) becomes:

o(~) = - n - ~ ( ~ ) f{r 0) H(Oo, O)CT(t, 0)} = - R - l ( r ) G(O) O(r) k(~) = ~ E{r 0) cT(t, 0)} -- n(~)

~(~) (44)

(45)

Algorithms for identification of continuous time nonlinear systems

23

where

= 0 - e0.

(46)

The stationary (equilibrium) points of the ODE which correspond to the possible convergence points of the algorithm are given by Vc = { 0 :

G(t~)(0-~;)--0}

(47)

or equivalently by :De = {0: cT (t , 0) (0 -- 00) = 0}.

(48)

If there exist 0 and r 0) such that the condition (48) has a unique solution then one has a single equilibrium point ~ -- t~0, which is the only possible convergence point of the algorithm. The next step is to establish the stability properties of the equilibrium points of the ODE which will give the w.p.1, convergence property for the algorithm. Define the candidate Lyapunov function V(0, R) = ~T(v) n(v) 0(v).

(49)

r(t)

Since G(8) r 0 (because of the implicit assumption that is not identically null) one has R(v) > 0 and therefore, Y(~', R) is a positive definite radially unbounded function outside :Dc for all t > to. Along the trajectories of (4.24) one gets

• dv

v(~, R) = - ~

[~(e) + ~T(e) - ~ G(e)] ~ - ~ R(T) ~

(50)

and for concluding on the stability of the equilibrium points it is sufficient to show that

[5(e) + ST(e) - ~2G(e)] ~> 0 v ~

0

(51)

HT -- A2 [] cT(t, 0)} [H-~ I] r

(52)

or equivalently that G(~) = E {r

~) [H +

= 2E {r

(53)

is a positive definite matrix function (one takes into account that H is a diagonal matrix H = where h is a scalar operator). To prove this it is enough to show that

hi

too~ t

24

I.D. Landau et al.

for any constant vector l E R u where d is the dimension of the parameter vector 0. But the integral (54) can be viewed as the input output product for a scalar operator [h-~---~] with input u =

cT(t,O)l

and output y =

[ h - ~__22]

Cr(t, 0)z. Using condition (32) of Theorem 2, (54) can be rewritten as

lim l [~otlT r t.-+c~ t

[h- A(--~)]r (55)

since A(t) - A2 > 0 Vt, h - T is a strictly passive operator (since H -- hi), and one takes into account the fact that the effect of the initial conditions vanishes as t --+ cx) because of division by t. 9

1.4

Robustness Analysis

It is important to analyze the robustness of the identification schemes when the plant is not in the model set, when the output is affected by a disturbance that is allowed to enter the system nonlinearly and when the higher terms in the Taylor series expansion around the nominal trajectory cannot be neglected. The objective of the analysis is to show that norm boundedness and mean square boundedness of all signals is assured for a certain type of characterization of the mismatch between the model and the plant and of the terms of higher order in the Taylor series expansion. The plant will be described by

y = Po(u,v) +AP(u,v)

(56)

where Po(u,v) is the "reduced" order plant, v(t) is a zero mean bounded disturbance, and zSP(u, v) is a BIBO operator that is due to the unmodeled part of the system. Note that the BIBO assumption might be unnecessarily restrictive. The estimated model is assumed to be represented by:

y(O) = P(O, u) with the property that

(57)

Po(u, O) = P(Oo,u).

Algorithms for identification of continuous time nonlinear systems

25

To start with, we show that the effect of the noise and the unmodeled dynamics upon the system can be considered to be additive. Denote by

y = P(Oo, u) = P0(u, 0)

(58)

the values of the output obtained for the reduced order plant in the absence of noise. Denote by

fl -- Po(u, v) + AP(u, v)

(59)

the values of the plant output in the presence of noise and with the unmodeled dynamics. Define = y + yp

(60)

where yp is the perturbation coming from the noise v and the unmodeled plant dynamics. Then

9 = P(Oo, u) + OPo,(u, O) v + AP(u, O) + OAP~(u, O) v.

(61)

Here, cgPo~(u, 0) denotes the linearization of P0 in response to a perturbation in v around the trajectory u and v = 0. Note that terms of order higher than one in the Taylor series expansion have been neglected; these are taken care of subsequently. Also, O.4P~ (u, 0) denotes the linearization of A p in response to a perturbation in v around the trajectory u and v = 0. Therefore

yp = [OP0,(u, o) + OAP~(u, 0)] v + AP(u, 0)

(62)

It is assumed that OPo~(u, 0), AP(u, 0) and OAP,,(u, 0) are BIBO operators leading therefore to a bounded yp. Therefore the equation of the output error will take the form

e(t) = r

O)T [00 -- 0(t)] "~- W(I)

(63)

with w(t) = y~(t) + O(00 - 0) where yp reflects the perturbation due to the unmodeled part of the plant and the possible bounded output disturbances, and O(90 - 8 ) reflects the effect of the high order terms in all Taylor series expansions. One has the following result:

26

I.D. Landau et al. 9 Assume that the external excitation u(t), the noise v(t) are norm bounded

T h e o r e m 3.

lim f t

uS(T) dv < aS;

a s < o%

(64)

vS(r) dr < ~2;

Zs < ~ .

(65)

t--+co Y ' r = o

lim

* Assume that 0 ( 0 o - 9) is n o r m bounded. * Assume that the true system is stable. . Assume that the P A A of (4) and (5) with A1 (t) =_ 1 is used. Then, the output errors(t) and the predicted output y(9, t) are norm bounded. C o r o l l a r y 2. If the external excitation u(t) and the noise v(t) are mean square bounded lim t -4.cc,

lim t ---~oo

uS(r) d r < a 2 t + k ~ , ; =0

//

as<eo;

O
(66)

~2 < cx~;

O < kv <(x~

(67)

--

vS(r) dr < ~ t + kv;

=0

and 0(90 - 9) is mean square bounded, then e(t) and y( 9, t) are mean square bounded. Proof." Under the hypotheses of Theorem 3, yp will be norm bounded and therefore w(t) will be bounded. The straightforward application of Theorem 5 in Part II (taking in account (63)) leads to the desired result. 1.5

Example

We refer to Section 2.4 for an example comparing NonLinear Open-Loop Output Error (NLOLOE) and NonLinear Closed-Loop Output Error (NLCLOE) identification.

Part II. Identification in closed-loop operation 2.1

Closed-loop output error identification. The basic equations and problem setting

The objective is to estimate the parameters of a single input single output (SISO nonlinear time invariant system described as in (1) by s

y=p0(u,v)

(6s)

Algorithms for identification of continuous time nonlinear systems

U

27

Y

+t

ccL(O)

Fig. 3. Closed-loop output error identification scheme

We will assume that the plant P0 is a BIBO operator. We refer to [4] for a theory based on kernel representations which allows the recursive closed-loop identification of unstable nonlinear plants. The plant is operated in closedloop with a known nonlinear controller, i.e.

C:

u=-e(y,r)

(69)

where r is an external reference which is assumed to be quasi-stationary and uncorrelated with v. The controller C is a causal BIBO nonlinear operator of both r and y. The closed-loop operator from the measured reference signal r to the measured output signal y, as defined in Figure 3 is denoted by

y = To(r, v).

(70)

It is required that the closed-loop system is Bounded Input Bounded Output (BIBO) stable. In the sequel we often make use of linearizations of some nonlinear operators around their operating trajectories. We therefore require that the plant, the model (to be defined subsequently), the controller and all closed-loop operators are smooth functions of the reference signal, the input signal, the output signal and the disturbance signal. This means that if the closed-loop operator is linearized around any (stable) trajectory, the resulting linear (time-varying) system is BIBO stable. See [5] for more details. We consider Che following adjustable model for the closed-loop system defined by (68) and (69) (See also Figure 3) y(O) = P(/9, u(/~)) u(O) = -C(y(O), r)

(71) (72)

where P(O, u) defines the adjustable plant model, y(O) is the output of the closed-loop predictor and u(O) is the plant model input.

28

I.D. Landau et al.

The closed-loop output error is defined as

r

= Y -- y(O).

(73)

The following assumptions will be made until further notice: (i) 300 such that P(Oo,u) = Po(u,O) for all u E Z~e and v -- 0 (subsequently in the case v _ 0 the argument v will be deleted) (ii) N o t a t i o n : The operator OPu (0, u) is the linearization of P(O, u) in response to a perturbation in u along the input trajectory u. The operator OCy(r, y) is the linearization of C(y, r) in response to a perturbation in y along the trajectories produced r and y. It is assumed that OPt(O, u) and OCu(r, y) exist for all allowable u, y and r. They are linear time-varying operators along the trajectories of the closed-loop system. (iii) N o t a t i o n : The partial derivative of P(O, u) with respect to Oj is denoted by P ~ (0, u) for j = 1,. 9 d where d is the dimension of the parameter vector b. The operator P~ (0, u) and its time derivatives exist and are norm-bounded

V3 along the trajectorms of the closed-loop predictor which requires r to be bounded 9 This assumption is not particularly restrictive as P and P(O) are assumed to be smooth operators. (iv) Let us define the operator

PcL(O) = [I + OPu (0, u(O))OCy (r, y(0))]. (74) It is assumed that PCL = PcL(Oo) and its inverse Pc~ exist along every trajectory of the closed-loop system encountered during the identification process. Both operators are linear time-varying operators and P e t is BIBO by assumption (smoothness of the closed-loop). (v) The reference r and the stochastic disturbance v are independent. Assumption (i) means that at least for 0 = 00 and in the absence of noise, the plant is in the model set. (The case when this is not true will be discussed separately in Section 2.3). The generic parameter adaptation algorithm (PAA) which will be used for identification in closed-loop is of the same form as the one given in (4) and (5) except that the open-loop output error e(t) will be replaced by the closed-loop output error eCL(t):

~(t) = F(t)r

(75)

where O(t) is the estimated parameter vector, eeL (t) is the closed-loop output error, r is the observation vector, F(t) is the adaptation gain matrix. We will consider subsequently that the assumptions (i) through (iv) are valid and furthermore, for some analysis, that:

Algorithms for identification of continuous time nonlinear systems

29

(vi) v = 0 (vii) The higher order terms in the Taylor series involving expansions in powers of (u-u(O)), (y-y(O)) and (00-0) along the trajectories of the system can be neglected

2.2

Nonlinear

closed-loop output

error algorithms

In this section, we present the derivations of the algorithm and we provide a stability analysis in a deterministic environment assuming that the system can be modeled exactly and that one can neglect terms of power higher than one in certain Taylor series expansions. T h e results in this section heavily rely on concepts of strong strict passivity outlined in the appendix. One has the following result (the N L C L O E algorithm): Theorem

4. Under the assumptions (i) through (iv), (vi) and (vii) one has

for = [P'(O, u(O))] T = [P~,(O, u(O))

r

...

P~(O, u(O))] T

(76)

that lim gcL(t) = 0

(77)

t--+ o o

if the linear time-varying operator H = Pc~

A(t) ~ - I;

A(t) > As(t),

Vt

(78)

is strongly strictly passive 3. If furthermore Pc~ has a finite-dimensional description as in (162)-(163) one has also lim r

-- 00) = 0.

(79)

t - +c<)

Remark

II.l:

1. T h e condition (78) assures t h a t the closed-loop output error goes a s y m p totically to zero, and that the estimated p a r a m e t e r vector 0, converges to a set defined as ~c = {0: r

00) = 0}"

(80)

It is assumed here that H has the form (155)-(156). See Definition 1 in the appendix for a definition of strong strict passivity.

30

I.D. Landau et hi. If cT(t) (0 -- 00) ----0

(81)

has a unique solution 0 = 00, the p a r a m e t e r vector will converge toward this value. In fact this condition is a "persistence of excitation" condition for the nonlinear case. 2. T h e passivity condition of T h e o r e m 4 can be relaxed by m a k i n g other choices for r as will be indicated later. Note that passivity conditions occur also in the linear case. Proof of Theorem

4: The proof will be done in several steps.

S t e p I: Establishing the expression eCL : f(00 - O(t)) One has the following lemma: 1. Under the assumptions (i) through by), (vi) and (vii) the closedloop output error is given by

Lemma

e e l -- Pc~ P'(O, u(O)) [00 - O(t)].

(82)

Proof." From (68) with v -- 0 one gets

y = P(Oo, u) = P (0o, u(O)) + [P (0o, u) - P (0o,

u(0))]

(83)

and using a series expansion around u while neglecting higher order t e r m s in (u - u(e)) one gets

P (0o, u) - P (0o, u(O)) = - O P , (0o, u) [C(y, r) - C(y(O),

r)].

(84)

On the other hand IV(Y, r) - C (y(O), r)] can be expressed as

IV(y, r) - c (v(0), r)] = oc~(r, y) (y - y(o) )

(85)

(neglecting higher order terms in (y - y(0))) and therefore

P (00, u) - P (00, u(O)) = -OPu (00, u)OCy(r, y) (y - y(O)).

(86)

Using the definition of eCL given in (73), (86) can be re-written as

y = P (0o, u(O)) - OP~ (00, u)OCu(r, y) e e l

(87)

Subtract now (71) from (87) and use (73). One gets

e c L = P (0o, u(O) ) - P (0, u(O) ) - OP~(Oo, u)OCy(r, y) e e l

(88)

Using a series expansion around O, one has

P (0o, u(e)) - P (0, u(e)) = P(e, u(O)) + P'(O, u(e)) (0o - 0) - P(e, u(O)) = P'(e, u(O)) (00 - 0), (89)

Algorithms for identification of continuous time nonlinear systems

31

neglecting higher order terms in (80 - 8). Here P'(8, u(8)) has to be read as P'(8, u)l~=~(e ). Therefore (88)becomes eeL = P'(8, u(8)) (80 - 8) - aPu(8o, u)aC~(r, y) e e l

(90)

from which one obtains [I + OP,(8o, u)aC~(r, Y)]~CL = P'(8, u(8) ) (80 - 8)

(91)

from which (82) results using the definition of PCL given in (74). S t e p II: (Stability proof) With r given by (76), (82) together with the P.A.A. given by (75) and (5) define an equivalent feedback system characterized by the following equations: eCL = Yl = Pc~ ( - P ' ( 8 , u(O)) ~(t)) = P c ~ Ul = - P c ~ Y2

(92)

8(t) = f ( t ) [ P ' ( 8 , u(8))] T eCL = F(t)[P'(8, u(8))] T u2

(93)

Y2 = P'(8, u(8)) O(t)

(94)

where = e(t) - 80

(95)

and uj, yj, j = 1,2 define the inputs and outputs of the equivalent feedforward and feedback blocks, respectively. Refer to Figure 2 for a similar equivalent feedback system (e is replaced by eCL and u by u(8)). Consider the equations (93) and (94) together with (95). Equations (93) and (94) correspond to a state space representation considered in Lemma 3 with A = O, B = F ( t ) r C = cT(t), D = 0 x = O, u = ~CL = US, y = YS = P'(8, u(8)) ~(t). The also that (i.e.

(96)

system (96) has the same structure as the system (18). Therefore it will satisfy an inequality of the type (21). It follows then from Definition 3 the equivalent feedback block belongs to the class N ( F ) w i t h / ' = As(t) it falls short of being provably passive).

By hypothesis, Pc~ belongs to class A(A(t)) with A(t) > A2(t). It now follows by a straightforward application of Theorem 6 that ul E /:2, xl E Z:oo, 8 E Z:oo and limt-+oo ~l(t) = 0. By hypothesis (see Assumption (iii)), r (given by (76)) and all its time derivative are bounded; this implies that ul = -Ys = - r T~ E s The boundedness of us = Yl follows from the boundedness of Xl and ul and Equation (163). It is now straightforward to see that ul E s Indeed,

32

I.D. Landau et al.

and both term on the right hand side of the equality sign are individually in /~oo. By Barbalat's lemma (see [14], Corollary 2.9, pg 86), ul E Z:2, ul E / : ~ and ~i1 E s imply that l i m t _ ~ ul(t) = O. 9

Relaxation o f t h e s t r o n g s t r i c t l y passive condition

Algorithm AFNLCLOE Neglecting the swapping correction terms which anyway become negligible when one uses decreasing adaptation gains (As(t) :> 0, limt~oo Al(t) -- 1), (82) can be also written as

= PciPoL(e)(P8

(O)P'(0,

u(O))[80

-

(97)

where the time-varying operator PCL(9) is defined in (75). In this case, following the same procedure as for the NLCLOE algorithm one has to choose

r

= Pc~(9) P'(9, u(9)).

(98)

In this case one filters P'(9, u(9)) through a linear time-varying closed-loop system which depends upon the current estimate 8. r can also be viewed as an approximation of the gradient of a quadratic criterion in terms of e e l around 80 (to). The corresponding strongly strictly passive condition will become ,~(t)

H:-Pc~PcL(O)--TI;

A(t) :>A2(t), Vt > t 0

(99)

should be strongly strictly passive. Clearly in the vicinity of 00, this condition is much more likely to be satisfied, than condition (78) for NLCLOE. This of course requires that at each instant Pc~(9) derived by (74) is stable. If this is not the case, then as in the identification of linear models (e.g. recursive maximum likelihood, adaptative filtered closed-loop output error) one uses the last stable estimated filter Pc~(8)-

2.3

R o b u s t n e s s Analysis

The robustness analysis will be done along the same lines as for the open loop case. The plant will be described by

y = Po(u, v) + AP(u, v)

(100)

Algorithms for identification of continuous time nonlinear systems

33

where Po(u,v) is the "reduced" order plant, v(t) is a zero mean bounded disturbance, and AP(u, v) is a BIBO operator that is due to the unmodeled part of the system. Note that the BIBO assumption might be unnecessarily restrictive. The estimated model is assumed to be represented by:

y(8) = P(8, u)

(101)

with the property that Po(u, O) = P(8o, u). The true input u and the estimated input u(8) are generated by (69) and (72) respectively. To start with, we show that the effect of the noise and the unmodeled dynamics upon the closed-loop system can be considered to be additive. Denote

by y = P(8o, u) = Po(u, O)

(102)

u = -C(y,r)

(103)

the values of the input and output obtained for the reduced order plant in the absence of noise. Denote by 9 -- P0(~, v) + Ap(~, v) = -c(o,~)

(104)

(105)

the values of the plant input and output, i.e. in the presence of noise and with the unmodeled dynamics. Define 0 = Y + Yp

(106)

= u + up

(107)

where yp a n d up are the perturbations coming from the noise v and the unmodeled plant dynamics. Then 9 = P(80, u) + c3P~(8o, u) up + cOPov(u, O) v

+AP(u,O) + cOAPu(u,O) up + cOAPv(u,O) v

(108) (109)

and

(t = - C ( y + yp,r) = - C ( y , r) - cOCy(r,y) yp.

(110)

Here, OPo~(u, 0) denotes the linearization of P0 in response to a perturbation in v around the trajectory u and v -- 0. Note that terms of order higher than

34

I.D. Landau et al.

one in the Taylor series expansion have been neglected; these are taken care of subsequently. Also, OAPu(u , 0) and OAPv(u,O) denote the linearization of AP, respectively, in response to a perturbation in u and v around the trajectory u and v = 0. Therefore

yp = [OP~(8o, u) + bAPu(u, 0)] up + [OPov(u, O) + OAF,(u, 0)] v +AP(u,O) up = - 8 C y ( r , y) yp

(111)

(112)

and combining (111) and (112) one gets Yp = Pc~ [(OPo.(U, O) + az~Pv(u, 0)) v + AP(u, 0)].

(113)

where tbc~ = [I + ((gP~(80, u, 0) + OzbP, (u, 0)) OCy(r, y)]-i is assumed to be a BIBO (asymptotically) stable I/O operator leading to a bounded yp . On the other hand the neglected terms in the developments leading to (82) for the closed-loop output error and (113) for the perturbation term have also to be taken into account. Therefore the equation of the closed-loop output error will take the form

eel = P~ r

T [80 - 8(t)] + w(t)

(114)

with w(t) = yp (t) + O(80 - 0) where yp reflects the perturbation due to the unmodeled part of the plant and the possible bounded output disturbances, and 0(80 - 8) reflects the effect of the high order terms in all Taylor series expansions. One has the following result T h e o r e m 5. Assume that the closed-loop output error is described by:

eeL = H cT(t) (80 -- 8(t)) + w(t)

(115)

where w(t) represents the combined effect of unmodeled dynamics, bounded disturbances and of the high order term in the Taylor expansions around the nominal trajectories. Here, H and r depend on the algorithm used. 9 9 9 9 9

Assume that H is a linear time-varying operator. Assume that the true closed-loop system is stable. Assume that C(y,r), OCu(r,y), A p and Pc~ are BIBO operators. Assume that the P.A.A. of (75), (5) with A1 (t) - 1 is used. Assume that the external excitation r(t) and the equivalent disturbance w(t) are norm bounded, i.e. lim i t t_~oojr=or~(r) dr<~; lim t---I. OO

[

=0

w~(r) dr < / ~ " --

)

a 2<~,

(116)

/~2 < oo.

(117)

Algorithms for identification of continuous time nonlinear systems

35

9 A s s u m e that 0(00 - 0) is norm bounded Then the closed-loop output error ecL(t), the predicted output y(O, t) and the predicted input u(O, t) are norm bounded if

A(t)

/t=H--~-I;

A(t) >A2(t) Vt

(llS)

is a strongly strictly passive linear time-varying operator.

C o r o l l a r y 3. Under the same condition (118) of Theorem 5, if the equivalent disturbance w(t) and the external excitation r(t) are mean square bounded, i.e. lim fr t t---+o~

lim

t-+co

r2(r)dr
a2
0
(119)

f12
0
(120)

----0

----0

w2(r) d r < / 3 2 t + k , o ;

then eeL, y(O), u(O) and r

are mean square bounded.

In fact this theorem says that even when one uses simplified nonlinear models, provided that the error between the true plant and a nominal reduced model is small in some sense, the boundedness of the signals is assured by the passivity conditions of Theorem 4, now evaluated for the nominal reduced model. The result is akin to those in Lyapunov stability theory in the presence of nonvanishing disturbances. P r o o f o f T h e o r e m 5: Defining 0(t) = e(t) - e0,

(121)

(115) can be written as eCL = --

H (~T(t) "~(t) Jr- W = H fi + w

(122)

where ,~

= --r

~'(t).

(123)

Defining

,x~(t) T (t) ~'(t) ~(t) = ecL(t) + --5--r

(124)

36

I.D. Landau et al.

one obtains from (122)

p(t) = -

{[H - ;11+ -~

}

[A(t) - A2(t)] cT(t) ~(t) + w

- --HCT(t) ~(t) + w = / ~ + w.

(125)

The operator H is not only strongly strictly passive but in addition it is input strictly passive since A(t) - A2(t) > 0 Vt. From the properties of input strictly passive systems one has using (125)

fi(r) p(r) dr > -Tg + 5

fi2(r) dr +

w(r) fi(r) dr;

for some 5 > 0, Vt > to.

(126)

On the other hand taking into account the input-output properties of the adaptation algorithm (75) and (5), one has from (21) (in closed-loop framework) with cT(/) = p,(0, u(0)) and taking into account (123) and (124)

-/(t(r) p(r) dr--

(0, u(/9)) (~(r) eeL (r) dr +

2(r) [I~T(r)r

,/to

_>-~1 ~:P(to) F-l(to) g(to)

(128)

and (126) becomes 5 ~i ~ ( r ) dr

< %~- J(tt w(r)

~(r) dr +

10r(to) F-l(to)~(to).

(129)

o

Expanding the inequality to to t one gets -2

fi~(1-)dr < ~p

-[ph(t)+w(t)] ~ <_0, wZ(r) dr +

p > 0, and integrating from

w(r) fi(r) dr.

(130)

Adding (129) and (130) one obtains

( 5 - 2 ) ~i ~(r)dr <-7~~

[jtlw~(r) dr+~ ~(t~176176

From the norm boundedness of w(t) and with p < 25 inequality (131) implies that fi(t) will also be norm bounded. The signal ~) = H fi

(132)

Algorithms for identification of continuous time nonlinear systems

37

is also n o r m b o u n d e d since H is a B I B O o p e r a t o r . It r e m a i n s to show t h a t are n o r m bounded.

eeL(t) and r

F r o m (122) and (132), eCL(t) = /)(t) + w(t) from which one concludes t h a t eeL(t) is n o r m bounded since ~(t) and w(t) are n o r m bounded. Since y(r is n o r m b o u n d e d by s t a b i l i t y of the true closed-loop s y s t e m it results also t h a t y(8, t) = y ( t ) - eeL(t)is norm bounded. If the controller C(y,r) and OCy(r, y) are b o u n d e d input bounded o u t p u t stable it results t h a t u(9, t) will also be bounded. T h e p r o o f extends s t r a i g h t f o r w a r d l y for m e a n square boundedness.

R e m a r k II.2: 9 Suppose that AP(u, v) : 0 for simplicity, i.e. the system can be modeled exactly. Then (113) reduces to

yp = [I A- OP~(~o, u, O)OCy(r, y ) ] - i OPv (~o, u, O) v.

(133)

Note t h a t if the noise is additive, OPv (~0, u, 0) = 1 in the equation above. 9 It follows from (113) t h a t w(t) depends on u a n d y and it results t h a t b o t h w(t) and r d e p e n d on the reference signal r. This shows t h a t w(t) and r ~) are not independent and this causes the N L C L O E a l g o r i t h m to produce biased estimates. 9 T h e s i t u a t i o n is different in the linear case where a consistent e s t i m a t e is o b t a i n e d when the system is in the m o d e l set and the reference a n d noise signal are independent; see e.g. [8]. Indeed, it follows t h a t (133) reduces to

yp = (I + PCy) -I v

(134)

which is independent of the reference signal r. In the linear case and with the system in the model set, w -- yp is therefore independent of r

2.4

An Example

Consider the open-loop unstable p l a n t m o d e l described by = u + ~0 z 2 y = x+ v

(135) (136)

with z, u, y in R 1. It is assumed t h a t ~0 < 0. Consider the controller u = - (y3 + by2) + r = - C ( y ) + r.

(137)

It is assumed that 60 is unknown but b is known. Several remarks can be made concerning this closed-loop system.

38

I.D. Landau et al.

1. For b = 00 and v _-__0, the closed-loop system equation becomes = -x 3 + r

and the closed-loop system is asymptotically stable. 2. For b ~ 00 the closed-loop system is BIBO The estimated plant model will be described by ~(0) -- u(O) -I-0x(0) ~ y(O) -- x(O)

(138) (139)

and the estimated control will be given by u(0) = -[y(0) 3 + by(0) ~] + r = -C(y(0)) + r

(140)

To apply and analyze the identification algorithm we need the following quantities(with p = ~t)

P'(O, u(O)) = (p- 20y(0)) -1 y(0) ~,

(141) (142)

aP~ (0, u(O)) = (p - 20 9 ( 0 ) ) - 1 ,

bP,,(Oo,u, O) = aP~,(Oo,u(Oo)) =

(p - 2 00 y) - 1

(143)

,

ocy(~, 9(0)) = 3 9~(0) + 2b 9(0),

(144)

oc~(~, 9) = oc~(~, 9(00)) = 3 9 ~ + 2b 9. One can express now

PCL(O) and

PS/Z(0):

PCL(O)= [1+ 3y2(O)+2by(O)] .

- 2o 9(o)

(145)

P+[3Ya(O)-2(O-b)Y(O)] =

-; = -~ ~

(146)

'

p - 20 y(0) P S t ( ~ = p + [3 92(0) - 2(0 - b)9(0)]

(147)

For this example the various algorithms will have the following forms:

NLOLOE The observation vector is r

= (p - 20 9(0)) -1 9(0) 2

(148)

with the Open-Loop Output Error predictor shown in Figure 1. Note that the open-loop plant is unstable but is maintained in an "open-loop stable region".

Algorithms for identification of continuous time nonlinear systems

39

NLCLOE The observation vector r is as in (148) with the Closed-Loop Output Error predictor shown in Figure 3. The convergence condition requires that Pc~(80) - -~ be strongly strictly passive where Pc~(9) is given by (147). In this example one should make the assumption that -200 y > 0 Vt i.e. with 00 < 0 this means y(t) > 0 must hold for all t > to, as well as the assumption that 3y ~ - 2(00 - b) y > 0 Vt > to. This can be achieved along a trajectory generated with r > 0 (for 0o > 0 the sign of r should be changed). Notice that i f y is a constant signal, with -200 y > 0 and 3y 2 - 2 ( 0 0 - b ) y > 0 for all t > 0, then POLl(00) -- -~ will be strongly strictly passive for small A. This suggests that low frequency signals should be used, as P ~ (00) - ~ is still likely to be strongly strictly passive along the associated trajectories. We apply the N L O L O E and the N L C L O E algorithms using the previous example with b = - 0 . 4 , r = 2 + 0.5 sin(0.1 t) and v zero mean white Gaussian noise with variance (7~. The parameter which is to be identified recursively is given by 00(t) -

-0.5 fort<315 -0.5-4- 0.25 sin(0.03 t) for t > 315,

(149)

i.e. the parameter 0o is first held constant and then allowed to vary sinusoidally. We adopt a least squares strategy with forgetting factor (At ----0.5, A2 = 1) and the algorithm is initialized with 0(0) = 0. Figure 4 shows the identification results in a noiseless situation. Both the N L O L O E and the NLCLOE algorithms allow a consistent identification of 00. The tracking results are better with the N L O L O E algorithm. Figure 5 shows (as can be expected) the appearance of a systematic bias on the estimate in a noisy situation with the N L C L O E algorithm. Note that the noise effect can be be reduced (at the expense of the tracking performance) by increasing the value of At. The advantage of the N L C L O E algorithm lies in the identification of unstable plants in a closed-loop situation; we refer to [4] for an example with a modified NLCL OE algorithm. Another advantage of the NLCL OE algorithm lies in the recursive identification of reduced complexity models in a low noise situation.

Conclusion The key contribution of this paper has been to show that the framework for a number of open loop and closed-loop output error identification algorithms can be pushed out from linear systems to nonlinear systems. Hence our results, not surprisingly, for the most part assume that the high order terms can be neglected in certain Taylor series expansions, or we assume that they

40

I . D . Landau et al.

-0

-0;

''

-0.~

",

',

-0.4

-0.5

~ -0.7

-0.8

i

100

i

200

i

300

i

400

I

500

I

600

i

700

i

800

L

9~0

1000

Fig. 4. Identification of Oo(t) ( - - ) in the noiseless case (a s = 0), respectively, using the NLCLOE ( - - ) and NLOLOE (...) algorithms.

0: -0.1 f -0.2

-0, i

~

-0.5

-00

-07

-0.9

l

l

,

I

l

t

,

l

,

1(30

2(X)

300

400

500

600

700

800

900

1(300

Fig. 5. Identification of Oo(t) ( - - ) in a noisy situation (a s = 0.01) using the NLCLOE ( - - ) and NLOLOE (...) algorithms with ~1 = 0.5 and ~2 = 1.

are at least small. Other than that, both the noisy and noiseless case are captured, as is the possibility that the true plant m a y not lie in the model set and that the parameters can be slowly time varying. Possible relationship with Extended K a l m a n Filters and nonlinear observers deserves to be studied in the future.

Algorithms for identification of continuous time nonlinear systems

41

A c k n o w l e d g e m e n t : The second author wish to acknowledge the funding of the US Army Research Office, Far East and the Office of Naval Research, Washington.

A

Appendix

Consider the system

y = Hu

(154)

and assume that it accepts a state space representation

= f(x, u, t) y = h(~,t)

(155) (156)

with x E R n, y E R m, u E R "~, f, h continuous in t and smooth in x. Suppose f(0, 0, t) = 0 and h(0, t) = 0 for all t > 0. D e f i n i t i o n 1. The system H is said to be s t r o n g l y s t r i c t l y p a s s i v e if there exist a positive definite (storage) function V(x, t) which satisfies *n

(Ixl) _< v ( x , t ) _< -y~ (Ixl)

V(O,t) -= O,

(157)

Yt >_ 0

(158)

where 71 ([x[) and "r~ (Ix]) are class ]Coo functions, and there exists a positive definite function (dissipation rate) r _> 73 (Ixl); 3'3(') G ]Coo such that

l

yT (r)u(r) d r >_ V (x(t), t) - V (x(to), to) + 0

fl

r

dr

(159)

Vt,to with t > to.

D e f i n i t i o n 2. A system S with input u, output y and state x (see (155) and (156)) is said to belong to the class L(A) if it is strongly strictly passive and in addition the following strengthened version of (159) holds yT(r) u(r) dr >_ V ( x ( t ) , t ) - V(xo, to) +

ill

+~

r

r) d r

uT(T)rA(, -) uO') d,-; A(O > 0 Vt _> to.

(160)

R e m a r k : The system S belonging to the class L(A) has an excess of passivity

42

I.D. Landau et al.

D e f i n i t i o n 3. A system S with input u, output y and state x (see (155) and (156)) is said to belong to the class N(F) if the integral of the input output product satisfies the following modified version of (159)

yr(T) u(,-) d,- _> V ( x ( t ) , t ) - U(xo,to)+

r

~I s' yr(r) r(,-) y(,-) d,-; /'(t)___O Vt___to(161) o

where V and r are non negative functions. Remarks: 1. The system N(F) has a lack of passivity.i 2. Note that there is no Kor property imposed on V and r in contrast to the L(A), and strong strict passivity does not follow from (161). We now turn to some generalizations of the Positive Real L e m m a [1] to timevarying systems [7,15]. Consider the linear time-varying multivariable system

= A(t) x(t) + B(t) u y = C(t) x(t) + D(t) u

(162) (163)

with x E R n, y E R m, u 9 R m and A(t), B(t), C(t) and D(t) continuous in t. L e m m a 2. ([15,7]) The system (162), (163) is passive if there exists a sym-

metric time-varying positive definite matrix function P(t) differentiable with respect to t, a symmetric time-varying semi-definite matrix Q(t) and matrices S(t) and R(t) such that P(t) + AT(t)P(t) + P(t)A(t) = -Q(t) BT (t)P(t) -- C(t) = sT(t) D(t) + DT(t) = R(t)

q(t) s(t)] ST(t) R(t) > 0

for all t > to.

(164) (165) (166) (167)

The following lemma is trivial to prove. L e m m a 3. If the matrices A(t), B(t), C(t), D(t) satisfy the set of equations

(164), (165) and (166) for some matrices P(t), Q(t), S(t), R(t) with appropriate dimension, the integral of the input-output product can be expressed as

yr(r) u(r) dr =

xT(t) P(t) x(t) - ~

Algorithms for identification of continuous time nonlinear systems

43

+ 2

+ uT(v) R(v) u ( r ) ] , Vt > to.

(168)

T h e o r e m 6. Consider the feedback connection of two systems S1 and S2

with state space realizations, containing state vectors xl and x2 respectively. Suppose that S1 is linear time-varying and belongs to the class L(A) and its storage function V1 and dissipation rate r are independent of x2. Suppose that the system $2 belongs to the class N ( F ) and its storage function I/2 and dissipation rate r are independent of xl. Suppose that V1 and V2 are differentiable. Suppose that no external excitation is acting on this feedback system. Then, if A(t) - F(t) > $

Vt > to and some ~ > O,

(169)

* the equilibrium state x T = IxT, x T] is globally uniformly stable (with xx (t) and z2(t) e s * Also, lim xl(t) = 0 and u~ ~

t-4OO

s

(170)

P r o o f : Follows the lines of [6]. See also [9].

References 1. B.D.O. Anderson. A system theory criterion for positive real matrices. SIAM Journal of Control, 5:171-182, 1967. 2. B.D.O Anderson and P.J. Maylan. Synthesis of linear time-varying passive networks. IEEE Transactions on Circuits and Systems, 21:678-687, 1974. 3. S. Dasgupta and B.D.O Anderson. A parametrization for the dosed-loop identification of nonlinear time-varying systems. Automatica, 32:1349-1360, 1996. 4. F. De Bruyne, B.D.O Anderson and I.D. Landau. Closed-loop output error identification of nonlinear plants using kernel representations Submitted for the Conference on Decision and Control, Sydney, 2000. 5. C.A. Desoer and M. Vidyasagar. Feedback Systems: Input and Output Properties. Electrical Science Series, Academic Press, New York, 1975. 6. M. Krstic, I. Kanellakoponlos, and P. Kokotovic. Nonlinear and Adaptive Control Design. Wiley, New York, 1995. 7. I.D. Landau. Adaptive Control - The Model Reference Approach. M. Dekker, New York, 1979. 8. I.D. Landau and A. Karimi. Recursive algorithms for identification in dosedloop: A unified approach and evaluation. Automatica, 33:1499-1523, 1997. 9. I.D. Landau, R. Lozano, and M. M'Saad. Adaptive Control. Springer Verlag, United Kingdom, 1997. 10. N. Linard, B.D.O. Anderson, and F. De Brnyne. Identification of a nonlinear plant under nonlinear feedback using left coprime fraction based representations. Automatica, 35:655-667, 1999.

44

I . D . Landau et al.

11. L. Ljtmg. System Identification: Theory for the User. Prentice-Hall, Englewood Cliffs, New Jersey, 1987. 12. L. Ljung and T. Soderstrom. Theory and Practice of Recursive Identification. MIT Press, Cambridge, Mass, 1983. 13. M. Gevers. Towards a joint design of identification and control ? Essays on control: perspectives in the theory and its applications, H.L. Trentelman and J.C. Willems Editors, Birkh~user, pages 111-151, 1993. 14. K.S. Narendra and A.M. Annaswamy. Stable adaptive systems. Prentice-Hall, Englewood Cliffs, New Jersey, 1989. 15. V.M. Popov. Hyperstability of Automatic Control Systems. Springer Verlag, Berlin, 1973. 16. E.D. Sontag. On the input-to-state stability property. European Journal of Control, 1:24-35, 1995. 17. E.T. Van Donkelaar and P.M.J. Van den Hof. Analysis of closed-loop identification with a tailor-made parametrization. Selected Topics in Identification, Modelling and control, 9:17-24, 1996. 18. F. De Bruyne, B.D.O. Anderson, N. Linard, and M. Gevers. Gradient expressions for a closed-loop identification scheme with a tailor-made parametrization. Automatica, 35, 1999. 19. I.D. Landau, B.D.O. Anderson, and F. De Bruyne. Closed loop output error identification algorithms for nonlinear plants. Proc. IEEE-CDC 1999, 606-611, Phoenix Arizona, USA.

F l a t n e s s - b a s e d B o u n d a r y Control of a N o n l i n e a r Parabolic Equation Modelling a Tubular R e a c t o r Alan F. Lynch and Joachim Rudolph Institut fllr Regelungs- und Steuerungstheorie TU Dresden, Mommsenstr. 13 01062 Dresden, Germany {alanl, rudolph}~erss I I. et. tu-dresden, de

A b s t r a c t . A nonlinear parabolic equation modelling an isothermal tubular reactor in one space dimension is considered. The control acts at the boundary of the inflow. It is shown that the system is "fiat" with the outflow concentration playing the role of a flat output. Hence, the concentration field throughout the reactor and the control can be parametrized using an infinite series expansion depending on the fiat output and its derivatives. This series is shown to have a non-zero radius of convergence provided the fiat output trajectory is chosen as a Gevrey-function of class two. A simulation result illustrates the usefulness of the approach in achieving finite-time transitions between stationary concentration profiles.

1

Introduction

Flatness-based control is useful in motion planning and trajectory tracking for finite-dimensional nonlinear systems [3,4,16,15]. The trajectories of differentially flat finite-dimensional systems are completely parametrized differentially by the trajectories of a "flat output" which can be freely assigned (at least from a mathematical point of view). The flatness-based control approach has recently been extended to infinite-dimensional systems. In the infinite-dimensional case, the type of PDE controlled determines how the flat output parametrizes the system trajectories. For example, in the first application of flatness to infinite-dimensional systems, a one-dimensional wave equation is reduced to a linear time-delay system. The resulting trajectory parametrization involves delay operators [17,7]. More recent work on other hyperbolic equations, the general linear telegraph equation [5] and the classical model for heat exchangers [20], uses parametrizations involving "distributed delay" operators. For parabolic or biharmonic equations, the parametrizations involve infinite power series expansions in the space variable. The series coefficients depend on time derivatives of the flat output up to an infinite order. In order to ensure convergence of these series, the flat output is taken as a Gevrey-function of class at most 2.

46

Alan F. Lynch and Joachim Rudolph

This approach has provided solutions to trajectory tracking problems for linear systems. For example, Euler-Bernoulli equations modelling flexible robot arms [8,1] and piezoelectric devices [13]. Similar work on linear parabolic equations includes the boundary control of the heat equation [14,16], concentration control in tubular reactors [10,9], and a cylindrical coordinate model of an electromagnetic valve [18]. Throughout much of the aforementioned work on linear PDEs, module theory provides an appropriate mathematical framework [6,11]. This paper extends results for linear parabolic equations, which are based on series expansions, to the nonlinear case. We consider a nonlinear parabolic equation which models a tubular reactor in one space dimension. The reactor has non-negligible axial dispersion and a quadratic reaction rate. This system shares the same flat output as a similar linear equation considered in [10]: the outflow concentration. The formulation of the infinite series solution and a sufficient condition for series convergence are our main contributions. The paper is structured as follows. Section 2 describes the control problem and how a flat output leads to its solution. In Section 3, the formal infinite series dependence of the concentration field on the flat output is derived. Section 4 provides sufficient conditions on the trajectory of the flat output to ensure series convergence. Section 5 presents a simulation result, and Section 6 provides some conclusions.

2

Tubular

reactor

model

InflOW

Outflow c~(0, t) = 0

c(-1, t) = =(t: t ....................Di.rection of flow ~ > 0

..........t

.:'-

I t ............. p-

X x=-O Fig. 1. Tubular reactor of unit length described by (1).

We consider the control of concentration in an isothermal tubular reactor, sketched in Fig. 1, where both convection and axial dispersion are modelled. In the reactor a single reactant is consumed in a second order reaction. With the usual assumption that radial and angular dependence of the concentration field can be neglected, the normalized model describing concentration in the

Flatness-based Control of a Nonlinear Parabolic Equation

47

reactor's axial direction is

ct(z,t)=c,:~(x,t)-~,c~(z,t)-e(c(x,t)) c~:(O,t) -- O, c ( - 1 , t ) = u(t), c(z,0) = ,/,(m:),

~,

xE(-1,

O),t>O

(la)

t > 0

(lb)

t > 0

(lc)

;r 9 [ - 1 , 0 ] .

(ld)

Here, c denotes the reactant concentration, subscripts of c denote partial Oc derivatives (e.g. c~ -- ~ ) , and u denotes the control applied at the inflow of the reactor. The constant scalar parameters are ~, ~ > 0, where v is proportional to the velocity of the reactant flow and ~ is proportional to the reaction rate. For non-zero ~ the P D E (1) is nonlinear due to the quadratic reaction term ~c 2. For simplicity, we assume the control is the inflow concentration. The more practically relevant flux-control, where u is proportional to the gradient of the inflow concentration, can be treated the same way. We denote the initial concentration profile as ~b and assume it is a stationary solution of ( l a ) - ( l c ) . Further examples and details on flatness-based control of chemical reactors, including tubular reactors, can be found in [19] and the references therein. The control problem considered is the following: perform a transition between two stationary profiles in a finite time T. To solve this problem, one must compute a (non-negative) control u that achieves a change in the outflow concentration c(O,t) from some constant value C1 -- ~b(0) for - c o < t < 0 to some other constant value C2 3> 0 for 0 < T < t < oo. Following [10], which treats the linear case with ~c replacing the term ~c 2 in (la), we solve this problem by introducing a fiat o u t p u t 9 as the outflow concentration: y(t) -- c(0, t). This makes it possible to derive a formal power series solution in x for the concentration field, with coefficients that depend on 9 and its derivatives (see the next section). This trajectory parametrization in terms of y allows us to treat the fiat o u t p u t trajectory as a design parameter which can be varied (subject to a series convergence constraint) to meet the control objective. Having found an appropriate fiat output trajectory, the entire concentration field can be computed. The control u(t) --- c ( - 1 , t) follows directly by evaluating the series at x -- - 1 . The numerical c o m p u t a t i o n s are based on truncated series. 3

Formal

series

solution

In this section we derive a formal series solution for the reactor concentration by expressing it as a power series in x with coefficients a~ (k > 0) depending on time: co

c(x,O =

xk

(2) k=O

48

Alan F. Lynch and Joachim Rudolph

Since the flat output is the concentration at x -- 0, and since the gradient of the concentration at x = 0 is zero, we have

ao(t) = c(O,t) = y(t) at(t) = c~(O,t) = 0.

(3a) (3b)

To obtain the recursion relation for the series coefficients we substitute the following series into (la) oo

k

k----O co

k

k--O

xk

oo

c, 0,, t) = ~ ] a~ (t) g k----O

oo

k

oo

X l.

E o,,(t) Z o (t) x k=0

j=0

oo

k

xk

= k=Oj=OE E aJ(t)ak-j(t)j!(k _ j)!

= ~

(Cauchy's product formula)

.j(t)a~_j(t) k~

k=0j=0

Equating coefficients of

'

ak = ak_ 2 + Yak-1

xk/k! gives

-1- ~

Z

adak_2_d,

j=o

k > 2.

(4)

J

Assuming the series converges, the control can be computed as

u(t)

,.,, ( - - 1 ) k = c(--1,t)

akK,)

:

k!

'

k=O

and the concentration at any point of the reactor is given by (2). From (3) and (4), we observe that when convection is absent (u = 0), all odd coefficients are zero. Further, notice that compared with finite-dimensional flat systems, where all system variables can be expressed in terms of a finite number of derivatives of the flat output [3,4], here an infinite number of derivatives are required to determine the concentration field.

Flatness-based Control of a Nonlinear Parabolic Equation 4

Convergence

of the

formal

series

49

solution

In the previous section we derived a formal dependence of the concentration field on the flat output in terms of an infinite series. This section provides sufficient conditions on y to ensure series convergence. T h e o r e m 1. Let y : ]R ---r]~ be a Gevrey-function of class a < 2, i.e., a C ~ function which satisfies suplu (O(t)l< te~

where

_

IV'

m 7--T,

Vl>O,a<2, _

(5)

_

m and 7 are constants. Then, the radius of convergence of the series co~cients given by (3) and (4) is gr~ate,- than

(2) with

4

u + ~ v 2 + 8~m + 167 -1"

(6)

Proof. Using induction, we will first show

sup la~~ tEl~

m M k (1 + k)!a

< 7---i - k,~------7 -

(7)

for all 1, k > 0, where M, 7, and m are constants and 1 < a < 2. Assumption (7) holds true for k = 0 and k = 1 as a consequence ofaz = 0 and the Gevrey assumption (5) on a0 = #. Differentiating the recursion formula (4) I times and using the triangle inequality yields la~0l < ate_ 2 -4- v l % _ l l + e

j

j

k--j-2"

Assuming (7) holds for k _> 2, we show it holds for k. Substituting the assumed bounds:

mM k-2 (l + k - 1)! a v m M k-1 (l + k - 1)! a suPtEgla(k0(t)l < t+-----T7 (k - 2) '~-1. + 7t (k - 1) '~-1.

+ •

8m2Mk-~k-2 t ( )(It) 7' ~ k-2 (l-r+j)!~ j=o r=0 J ~l.--gSY1 (r+k-j-2)! ~ (k-j-2)!

mMk(l+k)[a ( ( k , k - 1 ) ) a-1

< v--r- k!~ +

(8)

--1

\ 7-ff~-0u

(k)a

+ V47

u )

-ffZ

,m2Mk-2~(k-2~((l+k-1)'~ a 7' j } \ (~S_T~ ] j ! ( k - j - 2 ) ! j=o

(9)

50

Alan F. Lynch and Joachim Rudolph

<

7'

k!~-i

[ 7 M2k

+~

+~

=

1) (10)

mMk(l+k)'a[7_~ <

7t

v orn] + 2-M + ~ "

k! a - '

(11)

Going from (8) to (9) we used Lemmas 1 and 2 given in the appendix. Since for a < 2 the expression in square brackets in (10) is a bounded function of k, we equate its upper bound (the expression in square brackets in (11)) to 1 and obtain - ~M

-

+

= 0.

(12)

Having shown (7), we apply the Cauchy-Hadamard Formula to compute the radius of convergence. The radius of convergence R of a series )-~k CkZk is given by 1

R-

limk_~oo[ck[1/k .

(13)

Hence, using (13) and the inequalities (7) with I = 0, we obtain the following bound for the radius of convergence of series (2): R > limk-.oo inf ( k ~ ) tE~ I

I

1/k

1

= "M" []

Remark 1. Since the reactor has a length of one, we require a unit radius of convergence. Hence, v + ~m + 27 -1 < 2. Remark 2. From (6), we note that larger values of m, u, and L0lead to smaller lower bounds on the radius of convergence. Larger values of 7 lead to larger lower bounds on the radius of convergence. Remark 3. It is interesting to note that unlike in the linear case considered in [10] and [14], we have not shown an infinite radius of convergence for a < 2. However, the above proof can be applied to the linear case to show an infinite radius of convergence when a < 2 by multiplying the bounds (7) by k! ~/2-1.

5

Simulation result

From Theorem 1, series convergence is ensured by choosing y to be a Gevreyfunction of class two. In order to meet this condition and to ensure the control

Flatness-based Control of a Nonlinear Parabolic Equation

51

objective is met we make use of ~bo : IR -~ R defined as

J

O

d~o(t) "-

f J r exp(-t/(r(1-r))')dr

/ 1 f~ exp(-1/(rO-r))')d~

t
(14)

t > T

with the real parameter ~r _> 1. The function r162borrowed from [10], is strictly increasing from zero to one for 0 < t < T and all its derivatives are zero at z = 0 and z = T. Increasing the parameter ~ leads to an increased slope for the transition. Setting

y(t) = cx + (c2 - Ca) Co(t) allows us to achieve outflow concentration transitions between C1 for t _< 0 and C2 for t > T. The choice of y is not unique and other similar shaped Gevrey-functions could be used. We consider a transition from C1 = 0 to Cz = 1/2 in 20 units of time (T = 20 in the definition of y). Taking cr = 1 implies y is a Gevrey-function of class 1 + 1/~ = 2. The model parameters are taken as u = 2/3 and p = 6/4. Using Cauchy's Integral Formula it is possible to compute a 7 in inequality (5), and using (6) gives a radius of convergence greater than one. A 20 coefficient approximation of the concentration field is shown in Fig. 2.

6

Conclusion

This initial work provides a nonlinear extension of a recently developed flatness-based approach to boundary control of certain linear parabolic equations. Our approach provides means for computing the inflow concentration trajectory leading to a finite-time transition between stationary regimes. The method is based on an infinite power series parametrization of the concentration field in terms of a flat output. Other examples of nonlinear parabolic PDEs can be treated in an analogous manner. Acknowledgement. This work was supported by the Deutsche Forschungsgemeinschaft.

A

Appendix

Theorem 1 requires two lemmas which are provided in this section. The first lemma gives an identity involving factorials, and the second lemma recalls a result from [12] which we prove here for the reader's convenience.

52

Alan F. Lynch and Joachim Rudolph

..-' ...

.... .

9

9

....,...":'"''"""

9

.

.:: -.

.

.... 9

.

.

., 9

9

9

.9

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

20

X

-1

t F i g . 2. Concentration field for an outflow transition of amplitude 1/2. The control the concentration at x = - 1 .

is

Lemma

1.

i!j!(i+j+l+l),=~(lr)(J+r)!(i+l 1)!

(i+j+

_r)!,

i,j,l>O.

(15)

r=0

Proof. Let ~ : [0,1) --> g , i > 1, be defined as ~i(t) : ( l - t ) following general derivative f o r m u l a can be proven by induction:

-i-1.

The

~}k)(t ) _ (i+k)!~ (1-" t) - ' - k - I -- (i~!k)!~i+k(t) _ (i + k)!

i! Hence, for l > 0

(~i~j)(t)(t)

dl

= ~7((1 - t)-i-l(1 _

t)--J-l)

dl = ~

----~}~j+l(t)

((1 -- t) -i-j-2)

(~i~y)(')(O) = (i + j + l + I)! (i+j+l)!

(16)

Flatness-based Control of a Nonlinear Parabolic Equation

53

Using the p r o d u c t rule yields

r-----0

(17) r---~O

t-1

E q u a t i n g (16) and (17) we o b t a i n (15). Lemma

2 ([12]). For real numbers a > 1 and Lk > 0

(•176

L~ <

Lk

k=0

kk=0

,

t > o.

/

Proof. Since a > 1, for any integer i E [0, l] we have

(

Li l--

Ek=0Lk

<1,

or equivalently

nence~

ZL _- F o-I k ~k k=0

k=0

<_

Lj j=0

Lk = k=0

L~

.

k=0

References 1. Y. Aoustin, M. Fliess, H. Mounier, P. Rouchon, and J. Rudolph. Theory and practice in the motion planning and control of a flexible robot arm using Mikusiriski operators. Proc. 5th Symposium on Robot Control, Nantes, France, 287-293, 1997. 2. M. Fliess, J. L6vine, P. Martin, and P. Rouchon. Sur les syst~mes non lin6aires diff6rentiellement plats. C. R. Acad. Sci. Paris Sdr. I Math., 315:619-624, 1992. 3. M. Fliess, J. L6vine, P. Martin, and P. Rouchon. Flatness and defect of non-linear systems: Introductory theory and examples. Internat. J. Control, 61:1327-1361, 1995. 4. M. Fliess, J. L6vine, P. Martin, and P. Rouchon. A Lie-B/icklund approach to equivalence and flatness of nonlinear systems. IEEE Trans. Automat. Control, 44: 922-937, 1999.

54

Alan F. Lynch and Joachim Rudolph

5. M. Fliess, P. Martin, N. Petit, and P. Rouchon. Commande de l'~quation des t616graphistes et restauration active d'un signal, Traitement du Signal. Special, 15: 619-625, 1998. 6. M. Fhess and H. Mounier. Tracking control and 7r-freeness of infinite dimensional linear systems. In Dynamical Systems, Control, Coding, Computer Vision, and Discrete-event Systems, G. Picci and D. S. Gilliam (Ed.), Birkh/iuser, Boston, 45-68, 1999. 7. M. Fliess, H. Mounier, P. Rouchon, and J. Rudolph. Controllability and motion planning for linear delay systems with an application to a flexible rod. Proc. 3$th IEEE Conference on Decision and Control, New Orleans, LA, 2046-2051, 1995. 8. M. Fliess, H. Mounler, P. Rouchon, and J. Rudolph. Syst~mes lin~aires sur les op~rateurs de Mikusirlski et commande d'une poutre flexible. ESAIM Proe., 2, 183-193, (URL: http://www.emath.fr/proc/Vol.2/index.htm), 1997. 9. M. Fliess, H. Mounier, P. Rouchon, and J. Rudolph. A distributed parameter approach to the control of a tubular reactor: a multi-variable case. Proc. 37th IEEE Conference on Decision and Control, Tampa, FL, 439-442, 1998. 10. M. Fliess, H. Mounier, P. Rouchon, and J. Rudolph. Controlling the transient of a chemical reactor: a distributed parameter approach. Proc. Compu-

tational Engineering in Systems Application IMACS Multiconference, Hammamet, Tunisia, 1998. 11. M. Fliess, H. Mounier, P. Rouchon, and J. Rudolph. Gevrey functions, Mikusir~ski 's operational calculus and motion planning for some elementary distributed parameter systems. In Preparation. 12. M. Gevrey. La nature analytique des solutions des ~quations aux d~riv~es partielles. Ann. Sci. if:cole Norm. Sup., 25:129-190, 1918. 13. W. Haas and J. Rudolph. Steering the deflection of a piezoelectric bender. Proc. 5th European Control Conference, Karlsruhe, Germany, paper No. F1010-4, 1999. 14. B. Laroche, P. Martin, and P. Rouchon. Motion planning for a class of partial differential equations with boundary control. Proc. 37th IEEE Conference on Decision and Control, Tampa, FL, 736-741, 1998. 15. J. L~vine. Are there new industrial perspectives in the control of mechanical systems? In Advances in Control, P.M. Frank (Ed.), Springer, London, 197226, 1999. 16. P. Martin, R. Murray, and P. Rouchon. Flat systems. In Plenary Lectures and Mini-Courses, $th European Control Conference, Brussels, Belgium, 211-264,

1997. 17. H. Mounier, J. Rudolph, M. Petitot, and M. Fliess. A flexible rod as a linear delay system. Proc. 3rd European Control Conference, Rome, Italy, 3676-3681, 1995. 18. R. RothfuB, U. Becker, a n d J. Rudolph. Controlling a solenoid valve - a distributed parameter approach. Proc. 14th Int. Syrup. Math. Theory of Networks and Systems -- mtns'2000, Perpignan, France. To appear in June, 2000. 19. P. Rouchon and J. Rudolph, R~acteurs chimiques diff6rentiellements plats : planification et suivi de trajectoires. In Automatique et procddds chimiques, J.P. Corriou (]~d.), Hermes, Paris, Partie II, Chap. lll.f. To appear. 20. J. Rudolph. Randsteuerung von W~irmetauschern mit 5rtlich verteilten Parametern: ein flachheitsbasierter Zugang. at - Automatisierungstechnik. To appear in 2000.

Dynamic Feedback Transformations of Controllable Linear Time-varying Systems Franqois M a l r a i t , P h i l i p p e M a r t i n , a n d Pierre Rouchon Centre Automatique et SystSmes l~,cole des Mines de Paris 35 rue Saint-Honord 77305 Fontainebleau Cedex, France malrait@cas, ensmp, fr, mart in@cas, ensmp, fr, rouchon@cas, ensmp, fr

A b s t r a c t . We show that a linear time-varying single-input system of state dimension n which is N-controllable (N > n) can be transformed into a chain of integrators by an exogenous dynamic feedback of size N - n and a change of coordinates.

1

Introduction

T h e a n a l y t i c linear t i m e - v a r y i n g single-input s y s t e m

it = A ( t ) x + B(t)u,

x E ~n,

(1)

is controllable at to if and only if the c o n t r o l l a b i l i t y m a t r i x

Ck(t) := (Bo(to) Bl(to) . . . B ~ - l ( t o ) )

(2)

has rank n for some integer k (see, e . g . , [ l l ] ) ; we have set B0 : = B and, for each i > 0, B i + l ( t ) := A(t)Bi(t) -

dBi(t).

More precisely, we say the s y s t e m is N - c o n t r o l l a b l e at to, where N is the smallest integer such t h a t Cg(to) has r a n k n. Notice t h a t at a generic p o i n t to a controllable s y s t e m is n-controllable (i.e., N = n); the case N > n denotes a singularity. T h e goal of this p a p e r is to show t h a t if the s y s t e m (1) is N - c o n t r o l l a b l e at to, it can be t r a n s f o r m e d a r o u n d to into the chain of N integrators yl = y2

~]N-1 -~ Y N fIN ~ Y

56

Francois Malrait, Philippe Martin, and Pierre Rouchon

by a d y n a m i c feedback of size N - n and a coordinate change. Notice this feedback is always exogenous [7,2], i.e., it cannot be expressed as functions of z and the derivatives of u. The paper generalizes the generic case N = n, where this can be achieved with a static feedback(i.e., a dynamic feedback of size N - n = 0), and in particular the time-invariant case [9,12,1,8,4]). It also improves the result of [5] in the discrete-time setting, where (after a d a p t a t i o n to the continuous-time setting) a much bigger dynamic feedback of size n N would be required; see also [10] for a related result in the context of pseudolinearization of nonlinear systems. The interest of the result is twofold: it gives a parameterization of the trajectories of (1) without singularities, hence solves the motion planning problem; it allows to arbitrarily assign the poles, hence solves the tracking problem. As an example, consider the two-dimensional system ml ----mS

x2 = t~u.

Away from t = 0 the system is 2-controllable and can be obviously transformed into a chain of two integrators by the static feedback u = v / t ~. At t = 0, the system is only 4-controllable since C4(t)

=

-2t

2

"

T h a n k s to a dynamic compensator of size 2 = 4 - 2, the four-dimensional extended system

z~ = t~u lbl ~-- W2

zb~ = u

is 4-controllable since d,(t) =

t 2 0

0 0

"

Hence t = 0 is not a singular point for the extended system, which can be transformed into the chain of integrators yl = y ~ y~ = y3

y3=y4

y4=v

Dynamic Feedback Transformations of Controllable LTV Systems

57

by the change of coordinates

Yl = xl

+

2tx2 + 3t2wl

-

2taw2

y~ = 3x~ + 6twl - 3t~w2 Y3 = 6wl Y4 = 6w2 and the feedback u = v/6. As a consequence, every trajectory t ~-+ (x(t), u(t)) of the initial system can be parameterized by

t2 xl(t) = y(t) -- ~i~(t) + -~j(t) 1 t t~... x2(t) = ~iJ(t) -- f i ( t ) + ~ Y (t) u(t) = 6 y(4), where t ~-} y(t) is an arbitrary function.

2

Single

input

analytic

systems

T h e o r e m 1 Assume the analytic linear time-varying single-input system = A(t)x + B ( t ) u ,

x E ]~n,

(3)

is N-controllable at to. Then there exists an analytic dynamic compensator (v = R ( t ) x + S ( t ) w + T(t)u,

w E ]~'-n,

(4)

such that the extended system (3)-(4), which has dimension N , is N-controllable at to. As a consequence (3)-(4) can be put around to into a chain of N integrators by an analytic static feedback and an analytic change of coordinates. Proof. By assumption the controllability matrix s

= (Bo(to) Bl(to) . . . B N - I ( t o ) )

has rank n. Let :=

(B, o ...

B,._,)

be a n x n invertible submatrix made with the columns of CN. Such a matrix CN always exists since by assumption CN(to) has full rank n. Let also :=

(B,

...

58

Franqois Malrait, Philippe Martin, and Pierre Rouchon

be the submatrix made with the remaining columns of CN. We want to build a dynamic compensator of size

N

-

n,

(v = R ( t ) x + S(t)w + T(t)u, such that the extended system

(:)_ which is of dimension N, is N-controllable. In other words, we want the N x N controllability matrix CN := (B0 B1 ... B l v - l ~ To T1 T~-l ] " to be invertible at to; we have set To :-- T and, for each i > 0, Tk+l :--- RBk + STk - Tk.

(5)

We define

$ : = (To ... T,._,) ~-' := (Tl+io . . . Tl+i,_,)

r

$" :

(T,.... (TI+.

T,N_,) ... TI+,N_,)

;

with these notations and up to a reordering of the columns of CN,

and (5) for k = 0 . . . N - 1 can be written in matrix form as

We then define R , S , T by choosing 7 - : = O(N-n)xn, "~f :: I(N-n)x(N-n) and TN := O(u-n)• this implies by (6) R := "/-'tiN 1

s:=$"-M~, and yields (up to the same reordering) the invertible controllability matrix

Dynamic Feedback Transformations of Controllable LTV Systems

59

Notice that T is indeed defined by the choice of 7- and "i-, since by definition T := Tik such that ik = 0. We are thus left with the generic case where the system

is N-controllable at to [9,12,1,8,4]. T h a n k s to the change of coordinates

y :=

//1

x,

where H0 := (0 . . . 0 1) C~ 1

Hi+l : = n i A + [-Ii,

i >_ O,

the system is transformed into Yl =Y2

~]N-I = YN

v ~ = ( [ ~ - 1 + g~_~Ya)~ + u,

which completes the proof.

Example 1. We illustrate the proof on the system xl = tx2 x2 = tu. It is 4-controllable at t = 0 with controllability matrix -1

C 4 ( t ) ----

0

"

Using the notations in the proof, we take i0 = 1, il = 3, i2 = 0, ia = 2, ~- :-- 02• and 7- := 12• so that

!)

-3t --

~000O10

"

Franqois Malrait, Philippe Martin, and Pierre Rouchon

60 Hence,

( Ol)

(;)

yielding the dynamic compensator ~bt -- u (v~ = - z ~ + t W l .

The extended system is 4-controllable at t = 0 and can be transformed into a chain of four integrators using the change of coordinates

3

Va

-1

Y4

0

t 1

wl w2

"

Conclusion

The idea of using an exogenous dynamic compensator to remove a singularity can be adapted to other classes of systems. For example, the linearized mass-spring system controlled by the spring stiffhess ~ri -- z~

:~2 = - x l - sin (t) u

has a periodic singularity at t = krr, k E Z. It is everywhere 3-controllable with controllability matrix C3(t)

= (

0

\ -sin(t)

- s i n ( t ) 2cos(t)~ cos(t) 2 s i n ( t ) ] "

The dynamic compensator ~b = u yields an extended system with the everywhere invertible controllability matrix

2cos(t)~

t.;3(t) =

0 -sin(t) ) - s i n ( t ) cos(t) 2sin(t) 1 0 0

hence removes the periodic singularity. It is even possible to remove singularities of nonlinear systems, as illustrated by the mass-spring system z2

----

(o (0) q-

--Xl

= f(z) "k g(x)u.

u

Dynamic Feedback Transformations of Controllable LTV Systems

61

This system is generically static feedback linearizable but has a singularity at points of the form (0, ~2) [3]. Nevertheless, the "nonlinear controllability matrix"

l,

I, I ,ll/- (

has rank 2 around (0, ~2), ~2 5s 0 if the higher-order bracket [f, [f, a]] is taken into account. Thanks to the exogenous dynamic compensator w=w2+u-1, the extended system i- = ](x~ + 9(z)u is static feedback linearizable around (0, ~2), ~2 # 0. Indeed, [~, [f,9]] = - 2 9 and the "nonlinear controllability matrix" 0

(] I/,~1 t],I],~,]])

=

-11

-Xl

~2

2

/

o2

2w 2w 2 + 2 ]

has rank 3 around (0, x2, ~b), x2 # 0. Clearly, the change of coordinates Yi :----zi(x2 + XlW)

y2 := -x~ + (x2 + zlw) 2

and the feedback 2(-x~ + 3(x2 + xlw)2)(1 - w 2) + v

u :=

2 ( ~ + (~2 + ~1~) 2)

yields

~h=y2,

92=y3,

93=v.

Acknowledgements The paper is based on a remark of Bernard Malgrange [6]. It can be seen as an interpretation in the framework of control theory of the fact that a finite-type D-module with one independent variable which is torsion-free is stable-free.

References 1. P. Brunovsk3L A classification of linear controllable systems. Kybernetika, 6(3):173-188, 1970.

62

Franqois Malrait, Philippe Martin, and Pierre Rouchon

2. M. Fliess, J. L6vine, Ph. Martin, and P. Rouchon. A Lie-B/icklund approach to equivalence and flatness of nonlinear systems. IEEE Trans. Automat. Control, 44:922-937, 1999. 3. B. Jakubczyk and W. Respondek. On linearization of control systems. Bull. Aead. Pol. Sci. Set. Sci. Math., 28:517-522, 1980. 4. T. Kailath. Linear Systems. Prentice-Hall, 1980. 5. E. W. Kamen, P. P. Khargonekar, and K. R. Poolla. A transfer-function approach to linear time-varying discrete-time systems. S I A M J. Control ~ Opt., 23(4):550-565, 1985. 6. B. Malgrange. Lettre du 26 janvier. Communication persormelle. 7. Ph. Martin, R. Murray, and P. Rouchon. Flat systems. In Proc. of the 4th European Control Conf., pages 211-264, Brussels, 1997. Plenary lectures and Mini-courses. 8. A. S. Morse and L. M. Silverman. Structure of index-invariant systems. S I A M J. Control, 11(2):215-225, 1973. 9. L. M. Silverman and H. E. Meadows. Controllability and observability in timevariable linear systems. S I A M J. Control, 5:64-73, 1967. 10. E. D. Sontag. Controllability and linearized regulation. IEEE Trans. Automat. Control, AC-32(10):877-888, 1987. 11. E. D. Sontag. Mathematical Control Theory. Springer, 2nd edition, 1991. 12. W. A. Wolovich. On the stabilization of controllable systems. IEEE Trans. Automat. Control, AC-13:569-572, 1968.

Asymptotic Controllability Implies Continuous-discrete Time Feedback Stabilizability Nicolas M a r c h a n d t a n d Mazen A l a m i r 2 1 Laboratoire d'Automatique et de GSnie des Proc~d~s UCLB-Lyon 1, 43 bd du 11 Novembre 1918 69622 Villeurbanne, France marchand@lagep, cpe. fr

2 Laboratoire d'Automatique de Grenoble ENSIEG BP 46 38402 Saint Martin d'H~res Cedex, France

A b s t r a c t . In this paper, the relation between asymptotic controllability and feedback stabilizability of general nonlinear systems is investigated. It is proved that asymptotic controllability implies for any strictly positive sampling period of a stabilizing feedback in a continuous-discrete time framework. The proof uses receding horizon considerations to construct a stabilizing feedback.

1

Introduction

For linear systems, it is well known t h a t a system is (globally) a s y m p t o t i cally stabilizable by means of a s t a t e feedback if and only if it is (globally) a s y m p t o t i c a l l y null controllable. F u r t h e r m o r e , the stabilizing feedback m a y be chosen s m o o t h . Does an analogous p r o p e r t y exist for general nonlinear systems of the form (1) r e m a i n s an open question in the nonlinear control theory framework. = f ( x , u)

(1)

Brockett et al. [1] gave the following three necessary conditions for the existence of a stabilizing C 1 feedback: T h e o r e m 1 ( B r o c k e t t e t a l . [1]). If f is C ~ and the feedback u E C 1 is such that u(O) = 0 and the origin is asymptotically stable for ic = f ( x , u(x)), then:

i. there is no uncontrollable modes of the linearized system associated with eigenvalues with nonnegative real parts 2. the origin is attractive 3. f maps every neighbourhood of the origin onto a neighbourhood of the

origin

64

Nicolas Marchand and Mazen Alamir

Applied on a linear system k = Ax+Bu, it simply gives that [A : B] has to be full rank which follows from the asymptotic null controllability assumption. When only continuous feedbacks are considered, the first condition is no more necessary as shown by Kawski [6]. However, the third condition remains even if f is only continuous [14]. This last condition, often quoted in the literature as the Brockett's condition, shows that there is no hope to obtain a general relation between asymptotic controllability and feedback stabilizability if one imposes regularity assumptions on the feedback. Remained to explore the field of discontinuous feedback. Indeed, discontinuous feedbacks arise often in many areas of control theory as well as practice. Nevertheless, it immediately yields to the difficulty: how should be defined the solution of (1) when u(x) is discontinuous? The best known theoretical tool for this is the Filippov theory [4]. Unfortunately, it was shown in [10] that it also yields to the Brockett's necessary condition. Moreover, it is proved in [3] that for affine in controls systems, the existence of a stabilizing feedback in the Filippov sense implies the existence of a non stationary continuous feedback. In a recent paper, Clarke et al. took a slightly different approach [2]. Instead of considering the continuous time solution of system (1), which may even not be defined if no regularity is assumed, the solution used is the one of a continuous-discrete time system. With this approach, it can be established that asymptotic controllability implies a particular type of continuousdiscrete time feedback stabilizability (see Th. 2). This mainly follows from a theorem established in [9] and generalized in [11], that argues, roughly speaking, that for asymptotically controllable systems, there always exists a continuous Lyapunov function V that can be decreased by means of a control. Using a regularization theorem and Rademacher's theorem, it is proved that one can find a feedback that makes decrease a sufficiently precise lipschitz local approximation of V so that V also decreases. In this paper, the assumptions and the continuous-discrete time solutions considered are identical to Clarke's work [2]. With these assumptions, the asymptotic controllability is proved to imply the existence of a feedback that asymptotically stabilizes the system in continuous-discrete time, whatever the sampling period T > 0. The present result has the advantage of ensuring the asymptotic stability of the continuous-discrete time closed loop system, when only practical stability was obtained with Clarke's result. On the other hand, it is not possible (at least simply) to make the sampling period tend to the zero in order to get generalized solutions of the closed loop equation = f ( ~ , k ( x ) ) ( s e e [13]). The paper is organized as follows. After some preliminary definitions, the main result (that is Th. 3), is presented in Sec. 2. The last section is dedicated to its proof.

Asymptotic Controllability Implies Stabilizability 2

Problem

2.1

statement

Preliminary

and

main

65

result

definitions

The system considered in this paper is of the form (1) where f is assumed to be continuous and locally lipschitz in x uniformly w.r.t, to u. This assumption ensures the existence and the uniqueness, for any essentially bounded control u and initial condition x0, of a trajectory x(.; x0, u), solution of the initial valued problem {k = f(x,u(t)),x(O) = x0}. This will not be practically restrictive since only uniformly bounded controls will be considered in the following. The system is assumed to be globally asymptotically controllable, that is [12]: D e f i n i t i o n 1 ( g l o b a l a s y m p t o t i c c o n t r o l a b i l i t y ) . System (1) is said to be globally asymptotically controllable if there is a measurable function u : ~'~ • ~ + --~ ~P with for all x E ~n, u(x, .) E Z:~~ such that: 1. (attractivity) Vxo E A n, limt-~r162x(t; xo, u(xo, .)) = 0 2. (stability) VR > 0, 3r(R) > 0 such that Vxo E B(r(R)), one has x(t; xo, u(xo, .)) E B(R) for all t > 0. where s denotes the set of functions f : ~ -+ ~P, essentially bounded on every compact set [a, b] Furthermore, to rule out the case when an infinite control is required to bring the state of the system to the origin, one assumes that: 1. There exists a neighbourhood of the origin 12(0) C R n and a compact set H C NP such that for all xo E 12(0), there exists a function u satisfying the above definition such that u(xo, t) E H for almost all t.

Assumption

a.a.t

Some definitions that enables a proper definition of (a) solution(s) to the closed loop system are next given. Let a partition of ~ + be defined by: D e f i n i t i o n 2 ( p a r t i t i o n ) . Every series ~r = (ti)ie~ of positive real numbers such that to = 0, Vi, j E l~, ti < tj and l i m i ~ t i = + c r will be called a partition. Furthermore, let (when it makes sense): 9 d(Tr) := supie•(ti+l - / i ) be the u p p e r d i a m e t e r of 7r, 9 d(rr) := infie~(ti+l - ti) be the l o w e r d i a m e t e r of 7r. With the above definition, one can define the notion of ~r-trajectory that can be seen as a continuous-discrete time solution of (1). This is an intermediate between the classical continuous time approach x = f ( x , k(x)) and the Euler integration giving x = f(x(ti), k(x(ti), ti)).

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Nicolas Marchand and Mazen Alamir

D e f i n i t i o n 3 ( r - t r a j e c t o r y ) . T h e r - t r a j e c t o r y x~(.;xo, k) of system (1), associated with a partition rr = (ti)ier~, an initial condition x0 = x(to) and a control strategy k, is the time function obtained by solving successively for every i E N: = f ( x , k(x(t~),t))

t E [0,t;+~ - td

i = 0, 1, 2 , . . .

(2)

using as initial value the endpoint of the solution of the preceding interval. Eq. (2) reminds receding horizon. Indeed receding horizon consists in finding, at sampling time ti, an open-loop control t -+ k(x(ti),t) defined for t E [0, 7] (with T _> ~(zr) possibly infinite) and in applying it during the interval [ti, ti+l]. Repeating this scheme gives a control depending upon x(ti) and the time t E [0,ti+~ - ti] as in (2). These definitions are a slight generalization of some definitions originally introduced by Clarke et al. in [2] where u was independent of the time. 2.2

Existing result and main contribution

With the above definitions, the result obtained in [2] is the following: 2 ( C l a r k e et al. [2]). Assume that system (1) is globally asymptotically controllable and satisfies Ass. 1. Then, there exists a measurable function k : ~n _+ ~p such that for every real numbers R > r > O, there exists M ( R ) > O, T ( R , r) and 6(R, r) > 0 such that for every partition 7r such that dQr) < ~(R, r), one has: Theorem

1. (bounded trajectory) Vx0 e B(R), Vt _> 0, x~(t; ~0, k) e B ( / ( R ) ) 2 (attractivity) W0 E B(R), Vt > T(R, r), x~(t; x0, k) e B(r). 3. (stability) limR-~0 M ( R ) = O. The above result underlines a relation between global asymptotic stability and a kind of stability, called s-stability ("s" stands for sampling) in the original paper [2]. This concept of stabilisation enables the generalization of the concept of stabilisation well known in the continuous case. Indeed, if one takes an initial condition x0 and a sequence of partitions 7rt such that dQrt) -+ 0 as l -+ cxD, the functions x~,(.;xo,k) remain in a bounded set. Because f ( x , k ( x ) ) is also bounded on this set, these functions are equicontinuous hence, using ArzelaoAscoli's Theorem, there is a subsequence that converges to a function that we denote x(.; x0, k). Any limit x(.; x0, k) of such convergent subsequences can be considered as a generalized solution of the closed loop system & = f ( x , k(x)). These generalized solutions always exists t h o u g h it m a y not be unique and the system is globally asymptotically stable with respect to that definition of solution. Practically, these solutions are impossible calculate and one m a y prefer to keep the continuous-discrete time scheme, t h a t is to fix a partition. In that

Asymptotic Controllability Implies Stabifizability

67

case, the obtained stability is clearly not asymptotic since the upper diameter d(Tr) of the partition may have to tend to zero with 6(R, r) as R tends to zero. However, the 7r-trajectory of the system is guarantied to remain in the ball B(r) after some time T(R, r), which leads to practical stability of the closed loop system. The aim of this paper is to answer this problem: T h e o r e m 3. Assume that system (1) is asymptotically controllable and satisfies Ass. 1, then, for all 6 > O, there exists a measurable function k :

~n • jR+ _+ ]~e such that: 1. Vx E 1~,, k(x, .) E s 2. VR > r > O, there exists M ( R ) > 0 and T ( R , r ) > 0 such that for any partition 7r such that d(Tr) > 6, one has: (a) (bounded trajectory) Vx0 e B(R),Vt > O, x~(t; xo, K) E B ( M ( R ) ) , (b) (attractivity) Vxo E B(R), Vt > T(R, r), x,r(t; Xo, K) E B(r), (c) (stability) limn-~0 M ( R ) = O. Clearly, this theorem has the advantage of insuring the asymptotic stability of the closed-loop system and not only a practical stability, since it is not necessary to sample infinitely fast as the state comes to the origin. On the other hand and like every scheme based on sampling, fixing a priori the sampling schedule may induce problems due to blow-up in finite time. The proposed feedback in its present form is not an exception to this rule. Nevertheless, this can easily be cleared up by making the sampling period 6 depend dynamically upon the current state. This point won't be detailed here. Similarly as in [2], Th. 3 leads to the following definition of global asymptotic cd-stability (where "cd" stands for continuous-discrete): D e f i n i t i o n 4. An asymptotically controllable system satisfying Ass. 1 that admits for all 6 > 0 a function k as in Th. 3 will be said globally asymptotically

cd-stabilizable 3

Proof

of Theorem

3

The aim of this section is to prove that under Ass. 1 and an asymptotic controllability assumption, one can construct a feedback k : ~'~ • ~ + that asymptotically cd-stabilizes system (1). Let us first begin with the following definition that makes the reading of the sequel easier. D e f i n i t i o n 5 ( b o u n d e d c o n t r o l s t r a t e g y ) . Let denote by bounded control strategy any measurable function u : ]Rn • It~+ --+ ~P such that for all R > r > 0, there exists M ( R ) > 0 and T(R, r) > 0 satisfying: 1. (stability) limR~0 M ( R ) = O, 2. (bounded trajectory) Vx E 13(R), Vt > O,

x(t; x, u(x, .)) E B ( M ( R ) ) ,

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Nicolas Marchand and Mazen Alamir

3. (attractivity) Vx E B(R), Vt > T ( R , r), x(t; x , u ( x , .)) E B(r). 4. (bounded control) For all compact set X C ]Rn and x E X , u(x, .) belongs almost everywhere to a compact subset L / C 1RP. It should be emphasized that this notion concerns the open-loop trajectory contrary to Def. 4 of asymptotic stability where rr-trajectories, that is dosedloop trajectories, are considered. This difference is significant since, as underlined in [7], the existence of such open-loop controls is far from implying directly the asymptotic cd-stability. The proof of Th. 3 is based on receding horizon considerations and splits up into the following main parts that make up the three next subsections. 1. Firstly, under Ass. 1 and asymptotic controllability assumption, system (1) is proved to admit a bounded control strategy that enables the definition of a cost function like in the infinite horizon control framework. 2. Then, it will be established that, for every J > 0, there is a feedback law that decreases this cost function for every partition 7r such that _d(rr) _ J. 3. Finally, it will be proved that the above feedback globally asymptotically cd-stabilises system (1). 3.1

Definition of a cost function

The aim of this section is to introduce the following intermediate lemma, which proof is given in Appendix A. L e m m a 1. If system (1) is asymptotically controllable and satisfies Ass. 1, then there exists a bounded control strategy v, a function G : IR + --+ ]R+ of class C 1 and, for all R > O, a decreasing function AR : IR+ --+ ]R + such that:

I. G and its derivative g are of class K, 1 2. For every x E ]Rn, the below integral W ( x , v(x, .)) converges

w(x, v(x, .)):= 3. R I > R ~ > 0

~

G(llx(r;x,v(x,.))ll)d~

(3)

AR,(0)>.4R~(0),

~. limR-+0 AR(0) = 0,

5. vx ~ B(R), Vt _ 0, IIx(t;x,v(x,-))11 6. f+oo G ( A n ( r ) ) d r converges.

_< AR(t),

Clearly, items 5 and 6 imply item 2. W ( x , v(x, .)) is the cost associated with the initial state x and the open-loop control v ( x , . ) . It takes the receding horizon classical form when no weighting is put on the control. The proof of this lemma, detailed in Appendix A, splits up into in the following points: 1 following Hahn [5], any continuous strictly increasing functions f : R + --+ IR+ such that f(0) = 0 will be said of class/(7

Asymptotic Controllability Implies Stabifizability

69

1. In a first step, it is proved t h a t system (1) admits a b o u n d e d control strategy v as soon as it is asymptotically controllable. 2. This enables to define, for all R > 0, a "gauge" function An, fulfilling items 3, 4 and 5 of Lem. 1. 3. F i x i n g / ~ = R and using Massera's l e m m a [8, Lem. 12], gives a function G fulfilling item 1 of Lem. 1 and such that f:oo G(Ai~(v))d T converges. 4. Verifying that

3.2

f:oo G(A~(v))dv

converges for every R ends the proof.

Formulation o f the f e e d b a c k

Let C > 0 be a real constant and B(~c~-), k E Z, be concentric balls defining a subdivision of ~n. It follows from the asymptotic controllability assumption and from Lem. 1, that there exists for every k E ~ , a c o m p a c t s e t / g o such that for all

x E B(~), v(x,t)

belongs almost everywhere to H e .

For all

x E ~ n \ { 0 } , let us define: 9 n~ E Z be the larger relative integer n such that x E B ( c ) . to Lem. 1 and for all x E/R n, one has:

vt > 0,

IIx(t; x, v(x, .))11_

According

A~_=(t)

with in addition, v(x,t) E H c for almost every t > O. 9 ~tx E Z be the larger relative integer n such that for which there exists an open-loop control u : ~ + -+ ~P such that: (u(t) E H c k

vt ->- o,

almost everywhere II~(t;~,u)ll -<- A c (t)

(4)

According to the previous item, it is clear that:

~ > .~

(5)

Note also that along every open-loop trajectory x(.; x, u), one has:

fi~(t;~,u) >_ fi~

Vt > 0

(6)

Indeed, u(. + t) is an open-loop control belonging to H c . Hence, using (4) and the decrease of A_~;, one has for every t ~ ~ 0:

IIx(t'; x(t; z, u), u(.+t))]l--IIx(t'+t; 9 U~ C s

~:, u ) l l

_< A_~.= (t'+t) _< A~c (t'J,

denote the set 2 of open-loop control u fulfilling conditions (4).

2 by definition of fix, the set U~ can not be empty.

70

Nicolas Marchand and Mazen Alamir

Finally, let W(x) be the minimum cost associated with x3:

W(x) := inf{W(x,u); u E Uz}

(7)

L e m m a 2. For all J > O, there is a definite function E6(x) (that is ~6 (x) =

0 r

x = 0}, such that every feedback of the form (8} cd-stabilise (1}. K(x,t):=u(t)

withuEV~:= {u 9

W(x,u)<W(x)+st(x)}

(S)

The proof of this last lemma ends the proof of Th. 3.

3.3

Asymptotic c d - s t a b i l i t y o f t h e c l o s e d - l o o p s y s t e m

First of all, note that the definition of U~ and (6) give: Vt ~> O,

~z~(t;x,K)__>fix

along the 7r-trajectories

(9)

Items 1, 2(a) and 2(c) of Th. 3 are quite easy to verify: 1. By (8), for all x E ~n, one has It'(x, .) = u(.) E Va~ C Ux C/:~P. 2. For every partition rr = (t/)ier~, the r-trajectory of (1) with feedback (8) satisfies: (a) for all R > 0, all x e B(R) and all t > 0 (since A_~_ is decreasing):

xr(t;x,K) (Eg)B ( s u p A c (t)~ C B (A2_~7(O))emCa lB.3(A2c_ff~_(O)) \tE~+

~

]

and (5)

Hence, one has x~(t;x, K) e/~(A__f~_(0)) with nR := inf~eB(R ) nz. (c) Lemma 1.4 gives: limR~0 A2_,7~_(0) = 0 In order to conclude, it only remains to prove item 2(b) of Th. 3. Let R > r > 0 be two real numbers, x E D(r, R) 4 be the initial state of system (1) and x~(ti; x, K) the state of du system (1) at time ti of the partition, when the feedback K defined by (8) is applied. The aim is to prove that there exists a time T(r, R) such that for every partition zr of lower diameter d(a') > J and every x E :D(r, R), one has x~(t; x, K) E B(r), for all t >_T(r, R). This proof follows the three following steps: 1. First of all, it will be proved that for all R > r > 0, there is an integer N1 (r, R) such that, for all x E ]~n, for every partition zr, such that d(zr) > J, and for all instant t; of it such that x~(ti;x,K) E D(r,R), one has fi~.(t,+tr162 > fix,(t,;x,K). In other words, fi~.(t,;~,K) increments of one, at worst, every N1 (r, R) sampling period. This directly follows from 3 it does not necessary exist a control u E s such that W(x, u) = W(x). 4 where 9(r, R) denotes the closed disk of lower radius r and upper radius R

Asymptotic Controllability Implies Stabilizability

71

the choice of r that will be done in the following and that will insure the decrease of the cost function W at each sampling time ti. This will imply the decrease of Ilxr(ti; x, K)I I and hence, after some steps, the growth of

hx~(t,;r,K)" 2. It will follow quite easily from the previous item that for all R > r > 0, there exists an integer N(r,R) such that for every partition of lower diameter greater than 6 and every x E /)(r, R), one has x~(t; x, K) E 13(r), for all t >_ tN(~,R)- This last point is almost the objective of the present section with this slight difference that N(r, R) does not depend upon the partition contrary to tN(~,R). 3. Finally, the existence of a time T(r,R), independent of the partitions will be proved. This last point follows from the choice of the open-loop controls maid such that the corresponding trajectories remain below the gauge function AMt ). If for some i, ti+l - t l happens to be to large, the trajectory will naturally reach the ball B(r) "in open-loop'. Hence, it can be deduced that there is a time T(r,R) after which, even if tN(~,R) _> T(r, R), the trajectory will reach the ball B(r).

E x i s t e n c e o f N1 (r, R) Let 6 > 0 be a fixed real number and F denote the primitive of G vanishing at the origin. From (9) and for all partition rr, one has along the trajectories: hr.(t,+,;r,K) >_ fir,(t.;r,K). For the ease of the reader, let xi := xr(ti;x, K) denote the state of system (1) at time ti. At time ti+l, the state xi+l of the system is given by integrating system (1) between ti and ti+l with control K(xi,t) = ui(t) as defined by (8). Hence:

w ( * d <_ w ( . i , u i ) <_ w ( . d + ee(.~) Now, if hr,+, = hr,, ui(. -- (ti+l -- ti)) also belongs to U,,+I, giving: w(xi+l)

< w(~;+l,

u~(. - (ti+l - ti)))

< W(xi, ud -

i -f 0 t +1 t,

G(llx(w;

xi, ui)ll)dw

t , + l -- t i

< W(xi) + r

-

G(llx(r;

xi, ui)ll)dr

(10)

dO

Using ti+t -- ti >_ 6 and definingS: Vp > 0,

S(p) : : max(

sup [If(x, u)[I, 1) ~B(Ap(0)) u E/4p

5

C S(2-r~; ) is an upper bound to the time derivative of x(.;x, u) when u E U~

(11)

72

Nicolas Marchand and Mazen Alamir

it follows that6:

> 1 [F([[xi[[)-F([,x/[[-min([[xi[] - s([-r )

S(~.)6))]

(12)

Combining inequalities (10) and (12) gives:

w(.~+x) - w(~i) < ~ ( ~ i ) -

r(ll~,ll)-t(ll~ill-min(ll~dl,S( ~ )') ) s(-~<)

Hence, the aim here consist in doing an appropriate choice of r in order to force the right member of this last inequality to remain strictly negative. Let: e6(x) := min

(

1 S -~

(~-)

'

8 ) F(l,x,,) I1~11 2

(13)

With such a choice: 9 if Ilzill -< S(2C-g<,) ~, one has:

w(~+,) - w(~) 9 if II*ill > w(x,+,)

s(~C-~,)a, -

1

_<e~(~,)

S'-'t~-~7~,)r(ll~'ll) (,3) < -~6(~)

(14)

one has using (13):

w(x,) <-

~

*'"

-

F(,~,,)-F(,~,,-~(;~., )) "(;6r~, )

Since 0 < ~ < 1 and, G being increasing, its primitive F is convex, the second term of this inequality is negative or null. Hence: W(xi+,)-

W(xi) <_-r

In both cases (14) and (15), one has W(xi+z) - W(xi) defining for any R > r > 0, ~ ( r , R) := inf,~v(r,R ) r

(15)

<_-r

Moreover, it can be proved that

6 there is a "min" function in this expression since ~"-+ Ilxill - S(~-7~. )r vanishes before r = 6 if ~ < 8 "2~xl"1

Asymptotic Controllability Implies Stabilizability

73

~ ( r , R) > 0. Indeed, by (11), one necessarily has S(~-~-) _<_S(2.-~- ) _< s(~Cs ), with nR := inf~eB(t~ ) n~. Therefore, with the definition of e6, it follows:

m m \ S ( r ~-c ' - ) , II;'11 ~ 2

> m i n k 9s ( r ~ ' ) _

,

II-;'ll

(16)

T h e right m e m b e r of inequality (16) is clearly continuous with respect to x and hence one effectively has e__z(r,R) > 0, for all R > r > 0.

Remark 1. 6 can be chosen as small as needed, however, it can not taken null. Indeed, it would be then impossible to insure that ~ ( r , R) > 0 which precisely gives the convergence of the state to the origin. W i t h the two points below illustrated on Fig. 1, one can define NI (r, R) : 9 According to Lem. 1, every trajectory starting in the ball in the ball B(AR(t)) for all t > 0. In particular, one has: x E :D(r,

R) =r W(x) <_

B(R) remains

G(AR(v))dr

(17)

9 According to (11), the time derivative of the trajectory can be bounded on every c o m p a c t set. Hence if x E :D(r, R), the cost W(x) will necessarily be larger than a m i n i m u m cost Wmin(r) corresponding to the fastest decrease of the state:

W(x) < Wmi.(r) ::r x E B(r)

(18)

G(A,~(.))

C(n) C(II~II) G(,.)

I

Upper and lower bound of W(x) for x E

v(~, R) Consequently, choosing NI(r,R) > ~

Ni(r, R) as the smallest integer such that:

'(/0

)

G(AR(v))dr- Wmin(r)

(19)

For all x E T~(r, R), one has the following relation that ends the proof of the first points described above:

fix.(ti+Nx(.,R);J:,K) > hx,.(t,;z, K)

(20)

74

Nicolas Marchand and Mazen Alamir

Existence de

N (r, R)

The existence of N directly follows from the one of N1. Indeed, let re(r) := inf{m E 25; B ( M ( C ) ) C B(r)}. Clearly, re(r) is the smallest integer such that, for all x E B(2,-~--~), the rr-trajectory x~(.; x, K ) remains in B(r). Using this and since, for any x E ~ ( r , R ) , one has nx > nR, one can easily verc ify that, choosing for all R > r > 0, N ( r , R ) :-" Nl(2,--~,R)(m(r ) - nn) guarantees that, for any partition lr of lower diameter d(~-) > if, any x E /)(r, R) and any i > N(r, R), one has hx.(t,;~,K) < re(r). Therefore, for any i -> N(r,R), x r ( t i ; x , K ) E ]3( m--~), c giving for all t >_ tN(~,R),

x,(t; x, K) E B(M(~7i~) )) C B(r). This is exactly our second objective. E x i s t e n c e o f T(r, R)

It only remains to prove that T(r, R) can be chosen independently of the partition to conclude the p r o o f o f T h . 3. First of all, recall that, in the continuousdiscrete time scheme used here (see Def. 3), the system evolves in open-loop between two sampling instants. According to (8), the control u is chosen at each sampling t i m e (ti)ie[O,N(r,R)] in the set Vx.(ti;x,K) C Ux.(tl;x,K). According to (6), u E U., and hence using (4), one has: x(t; x, u) E B(AR(t)). Consequently, if the time between a sampling instant ti and the next one ti+l becomes too large, the trajectory will meet in open-loop the ball B(m(r)) in a time less than Tmaz(r,R). Since Tma~(r, R) depends only upon r and R one can conclude by taking T(r, R) := N(r, R)Tma~(r, R). T h a t way, for any partition zr of lower diameter d(Tr) >_ (f and for any x E :D(r, R), one has: Vt >_ T(r,R), x r ( t ; x , K ) E B(r). This last point ends the proof of Th. 3.

A

Proof

of Lemma

1

Recall that the following proof is m a d e up with four main points detailed in section 3.1. Existence of v According to Th. 2, there exists a feedback law tr such that, for any R > r > 0, there is M_(R) > 0, T(R, r) > O, tf(R, r) and a partition rr(R, r) of upper diameter d(n) < of(R, r), so that: * (bounded trajectory) Vx E B(R), Vt _> 0, * (attractivity) Vx E B(R), Vt > T(R, r), * (stability) lima-.0 M(R) = O.

xr(n,r)(t; x, ~r E B(M (R) ), *~(s,~)(*; x, ~) e t~(~),

Moreover, for any c o m p a c t X of IiU* and all x e X, x(x~(n,,)(t;x,x)) compact subset Np(x) of ~P depending only upon 2r [2].

is in a

Asymptotic Controllability Implies Stabilizability

75

In order to simplify the notations, let 7rz := ~(11~11, ~11~) be a partition such that d(~r~) < ~(11~11,11~) and let the control strategy w be defined by:

~(~,t) := ~ (x..(t,;~,~))

t e [t,,t,+d

(21)

For all x E 1R", one has x(t;x,w(x, .)) = x~=(t;x, ~) for all t > O. T h e openloop trajectory 7 x(t; x, w(x,.)) obtained by applying the control law w is clearly identical to the r - t r a j e c t o r y xr= (t; x, x). Hence, the system will meet the ball B ( l ~ ) in a time less t h a n T(IMI, lt~lt). In order to obtain a bounded control strategy in the sense of Def. 5, it only remains to prove the attractivity of the origin. This can be simply obtained by applying repetitively the control strategy w. For all x E ]Rn, let:

x0 := x

xk+l := x (tk(llxll); xk, w(xk,

tk(ll~ll) :=

with:

.)) (22)

T( 21~, ~ ! )

tk (llxll) is the

time s needed to go from a state of norm ~ to a state of norm less than 211~11 For all t > 0 and R > 0, let k~ and T~ be defined by: k+t 0 the unique integer k such that: t E ] Ej=0 k tj(n), V "k+l t.,(R)I ~j=0

k~ :=

T~ :=

if t < t0(R)

ift >

{o

k~

~=0

(23)

to(R)

i f t _< to(R)

(24)

if t > to(R)

tj (R)

and v be given by:

v(x,t)

:=

{w(/'tl~ w x

)

t

) f~176

~:ll;xk~l~ll,v(xk~l~n,.) , t - T i i x l I

-

f o r t >t0(llxll)

For all x E ]Rn, the open-loop control v(x,.) gives the generic trajectory profile x(.; x, v(x, .)) depicted on Fig. 2. It is then easy to verify that v is a bounded control strategy in the sense of Def. 5. This gives the first point of the proof: for all R > r > 0, there is M~(R) := M(R) > 0 and T~(R,r) so that: 1. lim My(R) = lim M(R) = O, R-~O

R-~O

2. Vx E B(R), Vt >_ O, x(t;x,v(x, .)) E B(Mv(R)) = B(M(R)), 7 w{x, .) is an open-loop control strategy independent of any partition, though it is deduced from a partition rr~. s tk{Hx[D has nothing to do with any partition

76

Nicolas Marchand and Mazen Alamir

M(llxll) I1~11"

~

~

M(~-J~)

-~V

~

8

to(ll~ll)

~ t's(llxll)

~ t~(ll~ll)

./=0

j-=o

Fig. 2. Generic state trajectory x(.; x, v(x, .))

3. Let nn be the smallest integer such t h a t M ( ~ R ) < r and, for all x E 13(R)\B(2R-~-~R), let nx be the smallest integer 9 such t h a t

X---,n~-1

M(~,xl~)2 _< r. Then, for all t -> z._,j=0 tj(llxll), one has

x(t;x,v(x,.)) E

B(M(~))

C B(r). Noticing that every trajectory with initial condition in B(~.---~) remains in B(r), it becomes clear t h a t it is sufficient to bound ~=0 t~(llxll) for x ~ B(R)\~(~--~) in order to conclude:

j=0

"--

2~ ' 2 ~ + ~ ] < ( n ~ - l ) T

(

R )=: <_ ( n n - 1) T ( R ,2-X-L~

,

) Tv(r,R) (26)

Hence, for all x e B(R) and t >_ T,(r,R), one has x(t;x,v(x, .)) E B(r) ~~ 4. For any c o m p a c t set X C ~n, the open-loop control v(x,.) satisfying the two previous points is in a c o m p a c t subset H C ~P almost everywhere since ~ also satisfies this property (see [2] for further details on the construction of to). D e f i n i t i o n o f An Figure 3 illustrates the construction that follows. Let 1~ > 0, be a fixed radius. Let AR : IR+ --+ ]R+ be the above defined function:

AR(t) :=

(Mv(/~) Mv(~)

i f t E [0, T v ( / ~ , ~ ) ] i f t E ]T~(/~, ~

(27) -

Since v is a bounded control strategy, for all x E B(/~) and t > 0, one has x(t; x, v(x, .)) e B(AR(t)). Function An : ~ + -+ ~ + defined below extends 9 note that x E B(R) implies that nx < hR. 10 Note that Tv (r, R) depends upon r through nn

Asymptotic Controllability Implies Stabilizability ...... -"M-~"(-R-!..... "~'--'T..........................

, II~ll

$

i

.

.

.

.

.

.

.

.

.

.

I I

LT~(R,R)JI

AR(t)

~ ( t - T,(R, ~-))

I

..... __~/_~ ....... J, .

7?

I I

I

-

t~

,~

I

I

t

Fig. 3. Illustration of function AR

this result for all R > 0 and x E B(R):

{Mr(R) AR(t) := ~R(t- Tr(R, 6)) min (Mr(R), ,~R(t)) For all R < / ~ and

x E B(R),

if R > / ~ and t < Tr(R, ifR>Randt>Tr(R,~) i f R _< k

6)

one knows that for all t _ 0,

9 (t; ~ , , ( ~ , .)) e e(~R(t)). By construction of v, one also knows t h a t x(t; z, v(x, .)) E Mr(R). These two points give that for all R < / ~ , all x E B(R) and all t ~ 0, one has

x(t; x, v(x, .)) E B(AR(t)). For all R > / ~ and x E / ) ( R , R), there is an integer k~ such t h a t kz

n . := ~,J(ll~ll) e [n(R,R),n(R, R)] j=O and

*(Tk. ;*, v(., .)) e ~(R) (with tk(ll~ll) defined by (22)). Now, by definition of v, one has for all t > 0: Ilx (t; x (T~=; z, v(x, hence, for all t ->

.)), v (x(Tk,;

Tr(R, T), ~

x, v(x, .)), .))ll = [Ix (t + T~, ; x,

v(x,

.))ll

one has using (27) and the decrease of A/~:

,.x (t; x, v(x, .) ),. < A.(t - T,=) <_AR (t - Tr (R, ~ ) )

(28)

Using (28) together with the fact that for all x E B(R) and all t >_ 0, x(t;x, v(x, .)) E B(M. (R)), one gets item 5 of Lem. 1, namely, for every R > 0, every x @B(R) and every t _~ 0, one has x(t; x, v(x, .)) E B(AR(t)).

78

Nicolas Marchand and Mazen Alamir

The decrease of function An and item 3 of Lem. 1 are clear. Item 4 directly follows from the construction of An: An(O) < M,,(R) with lim M~ (R) = 0. n..-r0

Obtaining G A/~ is a strictly positive function such that for every t > 0, limt-~oo An(t) ----0 (recall limR-.0 M(R) = 0). Using Massera's l e m m a [8, Lem. 12], there is a function G of class K with derivative 9 also of class K such that

f0+~176 G

converges.

Convergence of ]o

G(An(v))dr

In order to conclude, it only remains to verify that f o G(An(r))dr is convergent for all R > 0. This follows quite easily from the convergence of f + ~ G (An(r)) dr. For R < R, An(t) = An(t), and hence f+oo G (An(v)) dr is convergent. For R > / ~ , one has:

~o+CO T~(R,~) +oo G(An(r))dr= /jo G(An(v))dV + ~T~(n-,~)G(An(r)dT-n =

V (An(v)) dr +

Jo

f o G(An(r))dr

fo

V (Aa(v)) d r

is also convergent. This ends the proof of Lem. 1.

References 1. Brockett, R. W., Millman, R. S., and Susmann, H. S. (1983) Asymptotic stability and feedback stabilization. In: Differential Geometric Control Theory. Birkhs Boston-Basel-Stuttgart. 2. Clarke, F. H., Ledyaev, Y. S., Sontag, E. D., and Subbotin, A. (1997) Asymptotic controllability implies feedback stabilization. IEEE Trans. on Automatic Control, 42(10):1394-1407. 3. Coron, J. M. and Rosier, L. (1994) A relation between continuous time varying and discontinuous feedback stabilization. Journal of Mathematical Systems, Estimation and Control, 4(1):64-84. 4. Filippov, A. F. (1988) Differential equations with discontinuous righthand sides. Kluwer Academic Publishers, Dordrecht, Boston, London. 5. Hahn, W. (1967) Stability of motion. Springer Verlag, Berlin-Heidelberg. 6. Kawsld, M. (1990) Stabilization of nonlinear systems in the plane. Systems & Control Letters, 12(2):169-175. 7. Marchand, N. (2000) Commande s horizon fuyant : thdorie et mise en oeuvre. PhD Thesis, Lab. d'Automatique - INPG, Grenoble, France.

Asymptotic Controllability Implies Stabilizability

79

8. Massera, J. L. (1949) On Liapounoff's conditions of stability. Annals of Mathematics, 50(3):705-721. 9. Sontag, E. D. (1983) A Lyapunov-like characterization of asymptotic controllability. Siam Journal on Control and Optimization, 21:462-471. 10. Ryan, E. P. (1994) On Brockett's condition for smooth stabilizability and its necessity in a context of nonsmooth feedback. Siam Journal on Control and Optimization, 32(6):1597-1604. 11. Sontag, E. D. and Sussmann, H. J. (1995) Nonsmooth control-Lyapunov functions. In: Proc. of the IEEE conf. on Decision and Control. New Orleans, USA, 2799-2805. 12. Sontag, E. D. (1998) Mathematical control theory, deterministic finite dimensional systems. Springler Verlag, New York Berlin Heidelberg, second edition. 13. Sontag, E. D. (1999) Stability and stabilization: Discontinuities and the effect of disturbances. Nonlinear Analysis, Differential Equations, and Control. Kluwer. 551-598. 14. Zabczyk, J. (1989) Some comments on stabilizability. Appl. Math. Optim.,

19(1):1-9.

Stabilisation of Nonlinear Systems by Discontinuous Dynamic State Feedback Nicolas Marchand 1, Mazen A l a m i r 2, a n d Iyad Balloul ~ ] Laboratoire d'Automatique et de Gdnie des Procddds, UCLB-Lyon 1 43 bd du 11 Novembre 1918 69622 Villeurbanne, France marchand@lagep. Cl~. fr

2 Laboratoire d'Automatique de Grenoble, ENSIEG - BP 46 38402 Saint Martin d'H~res Cedex, France Nazen. Alamir@inpg. fr - Iyad. Balloul@inpg. fr

A b s t r a c t . In this paper, a dynamic state feedback that stabilizes general nonlinear affine in the control systems is proposed. It is deduced from the solution of an Hamilton-Jacobi like equality with two boundary conditions. The asymptotic stability of the closed loop system is proved and some examples axe given. An interesting feature of this method is that the so obtained feedback takes the form of a generalised sliding mode control, in the sense that it is a dynamic feedback on one half part of the space and a static feedback on the other half - with eventually a discontinuity on the surface dividing the space.

1

Introduction and problem statement

The stabilization problem is a widely studied subject a n d r e m a i n s one of the most challenging features as long as no special structure is assumed for the system to be stabilized. In the late eighties, works on the Brockett's necessary conditions [6,20,18] u n d e r l i n e d the fact that a regular 1 feedback may fail to stabilize regular systems. Based on this observation, a great effort has been done on the design of discontinuous a n d time v a r y i n g stabilizing control laws as it appeared as a possible solution for these delicate cases. T h e rigid spacecraft's a t t i t u d e control problem is an e m b l e m a t i c e x a m p l e of it (see [16,14] and the references therein). T h e approach proposed in this paper falls within the scope of the search of systematic m e t h o d s for the design of stabilizing control laws t h a t are applicable even to systems t h a t fail to satisfy the Brockett's conditions. The control law proposed in this paper is based on o p t i m a l control, which consist in applying at t i m e t, the control u(x(t)) t h a t m i n i m i z e s a cost function over a time horizon T ( t h a t m a y be infinite). In order to insure the 1 at least ~1

82

Nicolas Marchand et al.

closed loop stability of the receding horizon scheme, it is necessary to assume (among other assumptions) either that the horizon is infinite (which is often unrealistic in a non linear framework) or that the state satisfies a final equality constraint at the end of the horizon T (which m a y pose some problems of existence) [15]. In many cases, this is not entirely satisfactory. The main idea of this paper consist in the following proposal: it is reasonable to think that it is sufficient, in order to guarantee the stability of the closed loop, to be able to bring down some weighted norm of the state at the end of the horizon; and this, with eventually large (but bounded) intermediate excursions. Indeed, by repeating this strategy, it will be possible to steer the state to the origin. This naturally leads to the introduction of two internal states, namely the reduced weighted norm of the state (7[[x[[, where 7 E [0, 1D which will be the goal for the state and the remaining time (T - t). A quite similar idea has been previously used by Clarke et al. [7] in a more theoretical aim. In this paper, a control law based on the above scheme is properly formulated in term of a time-varying partial differential equation with two boundary conditions taking the classical form of an Hamilton-Jacobi-Bellman equation. The Hamilton-Jacobi-Bellman equation is known in the optimal control related literature to give a necessary condition to an optimal control problem [12]. This enables for the class of affine in the control systems: = f ( x ) -}- g(x)u x E ~n, u E ]~P and f and g continuous

(1)

to derive a stabilizing control law of the form: = ~(x,{)

whenever C(x,{) > 0

(2a)

= a(x)

whenever C ( x , ~ ) <_ 0

(2b)

u = K(x,~)

(2c)

E ~ m and the functions C and ~ are continuous Note that this can be seen as a generalised sliding mode control in the sense that C(x, ~) = 0 gives the sliding surface; (2a) and (2b) define the feedback on each side of the space. Note that, contrary to classical sliding mode control, the feedback is dynamic on one part of the surface. It can be shown with simple calculations that this class of feedback includes the usual discontinuous, dynamic and even time-varying feedbacks. An underlying important issue in all Hamilton-Jacobi-Bellman based approaches is the solution of the partial differential equation. This task in itself constitutes an active research area [9,4,5]. Despite some academic examples in which the PDE can approximatively or analytically be solved, the present work may legitimately suffer from the same criticisms that can be done to this field. On the other hand, it may profit by all future advances in the numerical or analytical solution of H J-like partial differential equations. However,

A Stabilizing Discontinuous Dynamic State Feedback

83

beside some more theoretical stabilization problem, the present work is a contribution to the almost unexplored field of the numerical design of dynamic discontinuous control laws for nonlinear systems. This paper is organized as follows. In the next section, some preliminary sufficient conditions for Lyapunov stability of the closed loop system are given. The expression of the proposed feedback and a proof of the Lyapunov asymptotic stability of the closed loop system are exposed in Sect. 3. Finally, the proposed scheme is applied on two nonlinear academical systems, including one failing to satisfy the Brockett's conditions.

2

Sufficient conditions

for Lyapunov

stability

The closed loop system (1-2) is assumed to be defined on a set D = ,~' x --- C ~ " x ~,n such that, for every initial condition in 1'2, a solution of the differential system (1-2) exists and remains in/2. The following definition will be used in the remaining of the paper to characterize the asymptotic stability of the augmented system (1-2): D e f i n i t i o n 1. System (1-2) will be said to be asymptotically stable if x is asymptotically stable (sometimes mentioned as the asymptotic stability of the set {0 • S}) and ~ remains bounded. The next proposition gives sufficient conditions for asymptotic stability of system (1-2). Recall that, following [10], a function f : ~ + --+ ~ + will be said of class K: if f(0) = 0, f is continuous and strictly increasing. P r o p o s i t i o n 1 (Sufficient c o n d i t i o n s f o r a s y m p t o t i c s t a b i l i t y ) . A system of the form (1-2) will be asymptotically stable in the sense of Def. 1 if, for every initial condition in [2, the part ~ of the augmented state vector remains bounded and there exists: 1. a function al of class IC such that:

{(x,~) :/= 0

and C(x,,~) < 0} ==~ { C ( x , a ( x ) ) >

o,,(llxll)}

. a function U : s --~ I~ + such that: (a) U is radially unbounded:

lim U(x,~) = +oo Ilxll~oo, :rex (b) U is positive definite with respect to the set {0 x ~ } : i. for all ~ E---, U ( O , ~ ) = 0 ii. there is a function ~ of class Ir such that:

x # 0 =~ u(x,~) > e(llxll) > 0

(3)

84

Nicolas Marchand et al. (c) there is a function a2 of class IC such that every trajectory of the system satisfies:

{C(x(t),~(t)) < 0} {V(x(t + O+),~(t + 0+)) - V(x(t),~(t)) < O}

(4a)

{C(x(t),~(t)) > O} :r

(x(t),~(t)) ___ -a2(llx(t)ll)

- ~ exists and satisfies

(45)

Note first that (3) avoids chattering (except at the origin), that is infinitely fast oscillations between the two possible states C(x, ~) _< 0 and C(x, ~) > 0. Indeed (3) implies that whenever C(x, ~) _~ 0, (25) instantaneously puts the system back in the situation where C(x,~) > ~x(llxll) and, when x # 0, ~x(llxID prevents c from vanishing right after. As a consequence, as long as x ~ 0, C can only be negative on a countable set of instants. A proof of the above proposition will not be given here but the reader may refer to [13]. Note that it simply follows from (4a)-(4b) that the Lyapunov function U is strictly decreasing for x ~ 0 except on a countable set of instants where it may be constant. 3

3.1

Expression

of the proposed

feedback

Assumptions

System (1) will be assumed to satisfy the following assumptions: A s s u m p t i o n 1. There exists a function r of class tC such that for all initial state xo E ~n, for all u : ]R -+ ~ m and for all t! > O:

IIx(tl)ll ~ +

f['

IIx(r)ll 2 + IluO-)ll2dr _> r

(

sup IIx(~)ll. \0~6_.tl /

(5)

This assumption is actually not very restricting and is satisfied by most systems. Indeed, it only conveys the idea that the system can not move infinitely fast to the origin with bounded controls. A s s u m p t i o n 2. There exists strictly positive real numbers T, Pl and P2, a C 1 function V : [ O , T ] x ~ n .+ ]~ and a real number 7 E [0,1[ such that the following conditions hold: I. V satisfies the following P D E with terminal condition for all (t, x) E

[0, T] • ov(t,~) Ot

"~-

ov(t,=~f{x ~ Ox ~, ] "~ p 2 x T x

V(T, x) = Ilxll 2

1-3- ~

-- 4pl

D~

OV(t'x)T : O~

0

(6a) (6b)

A Stabilizing Discontinuous Dynamic State Feedback

85

2. V satisfies the "boundary" condition for all x ~ 0 V(O, x) < 711xll~

(7)

Using classical arguments of the Hamilton-Jacobi related literature (see also the proof of Lemma 1 for further details), one can easily notice that Assumption 2 express the existence of an optimal control associated with the cost function:

§S

§

Based on this, it becomes clear that (7) imposes to that optimal trajectory, the additional property that the state ]lx(T)ll 2 at the end of the horizon is lower than 711x(0)ll 2. With the above assumptions, it seems possible to design a feedback as explained in the introduction. The generic behaviour of the closed loop system is depicted on Fig. 1.

IMI "rllxll!

7211~11 !

time

tlt+T

3.2

tl + T

Fig. 1. Generic behavior of the closed loop system

Main result

T h e o r e m 1. If system (1) satisfies Assumptions I and 2, then the closedloop system defined by: = f(x) + g(x)u(x,~) 5=(O1)

(8a)

wheneverC(x,,)>O

(8b)

~ = (ll;ll) wheneverC(~,~)
(8c)

is asymptotically stable in the sense of Def. 1 with: ( 1+7~ 1+7c C(x,~) := min I1~11- - ~ 1 , - - ~ g l -1

T x ovT

u(x,~) := ~ptg ( )ffx-x ( T - ~ 2 , x )

- V ( T - 52,x)

}

(9) (10)

86 3.3

Nicolas Marchand et al. Proof of Theorem

1

In order to prove T h e o r e m 1, the following l e m m a is needed: Lemma

V(T

Under the assumptions of Theorem 1, one has: ~, x) > min(pl, p~)r

1. -

(11)

P r o o f o f L e m m a 1: In this proof, t and consequently x(t) = x are fixed. Let u(.) designate an arbitrary profile of the control over the time interval It, oo[ and for all r > t where solutions make sense, let us briefly denote by x ( r ) the corresponding trajectory of the state having x(t) as initial condition. Using these notations, one can define the following cost function: tet-t-{2

J(t,x,~=,u)

:=

using the property gets:

IIx(t +~2)112 + J,

V(T, x)

P=llx(~)ll2+Pxllu(7.)ll=dT-

(12)

= Ilxll ~ to rewrite the terminal weight in (12) one

dV IIx(t +{2)112 = V ( T - { 2 , x ) + ft t+{~ d--~(7.+T-t-{2,x(7.))dT-

(13)

Using (6a) to rewrite the derivative ~-rV(7- + T - t - ~2, x(7-)) one gets:

dVdr(7.'' x(7-)) - OY(r',Otx(7-)) + OV(7-',Oxx(7-))[f(x(7-)) + g(x(7-))u(7.)] m

OV (r', 0 (7-))g(x(7-))u(7-)x _ p~ll~(7-)ll ~ +

4pli cqV(r~..~xx(7-))

cOV(7-',Ozx(7-))T

(14)

with 7-' := 7- + T - t - ~ . Combining (14) with (13) and putting it in (12) leads to:

J(t,x,{~,u) = V(T-{u,x)+ f,t+{~Plllu(r ) -u*(r',x(r))l]2dr where u*(x(7-), 7-') := --~-TplgT(x(7-)) o v T ,tT- ! , x(~-))

(15)

(16)

Note that u* depends upon ~2 through 7-'. Equation (15) enables to write (for all u):

V(T-~,x)

(17)

= g*(t,x,{~) > 0

where J*(t,x,~2) denotes the value of the cost function J(t,x,~2, u) when the control u(.) is taken equal to the "optimal" feedback strategy u*(x(.), ~) with ~ fixed. Therefore, one has: /.t+{~

V ( T - {2, x)

=

II~*(t + {~)ll2+Jt p~llx*(7.)ll = + Plllu*(x*(7-),{)ll~dT- (18)

A Stabilizing Discontinuous Dynamic State Feedback

87

where x*(v) is the solution of the closed loop system &(r) = f(x(7")) + g(x(v))u*(x(v), ~) where ~ is still a fixed parameter and u*(x(v), ~) is given by (16) with initial condition x(t) = x. Using (5), (18) gives:

V(T- ~ , x) >_ min(pl, p~)r

(19)

5 P r o o f o f T h e o r e m 1: According to Sect. 2, it is sufficient to prove that one can exhibit a Lyapunov U function for system (8) as required by Proposition 1. Let U defined below be a candidate Lyapunov function with A" := ~ n and

---:= {(~1,~) ~ ~+

x

[0, T]}:

u(x,~) = V ( T - ~ , x ) + ~

sup

,,_-1

V(0,~)

]

(20)

Existence and uniqueness of the solution: 9 The existence of a solution to the initial value problem given by (8) is a direct consequence of the property of the trajectory emphasized in the proof of Proposition 1. Indeed, according to (3) (that will be proved to be held) the discontinuities can only occur on a countable set of instants. Between these instants, the initial value problem is given by (Sa)-(8b) that admits, according to theory of continuous ODE [8], at least one solution. Solutions of (8) are classically piecewise defined on these intervals without calling for discontinuous ODE specific tools. 9 The uniqueness can not be insure without the following additional assumption: A s s u m p t i o n 3. For all initial condition (t, x(t)), the solution of the differential system below is unique o n [t, t + T]. =

[(x) + g ( x ) u ( x , ~ )

~= (O1) However, it should be emphasized that the uniqueness is not required and the results presented here remain true for each solution of the initial value problem. Asymptotic stability: 1. Let us first prove that every solution starting in $2 remains in ~2: Clearly, all that is needed be to established is that ~ remains in [0, T]. Indeed, x belongs to ~ n and ~1 clearly remains in ~ + since it is set to a positive value o f ~ when C(x,~) _< 0 and it remains constant when C(x,~) > O. According

88

NicolasMarchand et al.

to (8b) and (8c), one already knows that ~u _< T. A rapid look these two equations also inform us that, if ~u should become negative, there must exist a time t* > 0 such that ~2(t*) = 0. Then, according to (6b):

V(T-~2(t*), x(t*))= V(T,x(t*))= IIx(t*)ll 2 hence, it follows from (9):

C(x(t*),~(t*)) = min {Hx(t*)ll-L?-u ~l(t*), ~+2-~7-~l(t*)-Ilx(t*)ll} < 0 and ~(t*) is then set to T by (8c). Consequently, any solution starting in 12 remains in 12. 2. Let us now prove that system (8) satisfies the implication (3) with C defined by (9): Note that in the case of system (8), a(x) = (11~12). We have by definition of C and using (7): C(z,a(x)) = man { ~-~-211xll,~-[-~7 1 Ilxll-

v(0, x) }

_>rain { ~ _ . ~ [[x[[, -1-+- ~' r [[xl[ - 71[xl[} = 1 -27[]x[1 System (8) with C defined by (9) therefore satisfies the implication (3) with 3. Finally, let us prove that U defined by (20) is a Lyapunov function for system (8) in the sense of Proposition I:

9 U is definite positive with respect to {0 x 122}: According to (6b), V(T, 0) = 0. It follows from (11) and (7): 0 = rain(p1, p~)r

< V(0, 0) < 7110[[ = 0

Hence, according to the definition of U given by (20), one has: +co

v(o) = V(T,

O) + ~ IV(O,0)] = 0 k----1

It remains to prove that U(x,() > 0 if x r O. This simply follows from (11). Indeed, one has:

U(x,~) = V ( T - ~ , x )

+ y~

sup

V(O,~)

A Stabilizing Discontinuous Dynamic State Feedback

89

Using (11), one gets:

U(x,~) >_ min(pl, p2)r

p~) Z

r

(21)

k----1 It ensues that U verifies the second item of Proposition 1 taking ~(r) :-- min(pl,p2)r

9 U is radially unbounded with respect to x: It follows from (21) that lim U(x,~) = +cx~ Ilzll~+oo 9 u is U~icay decreasing when C(x, ~) > 0 (U satislies (40): Using the definition of U given by (20), and using the fact that according to (8b) ~l remains constant as long as C remains strictly positive, one has: dU __ dV(T-~a(t),x(t)) OV(T-~.2(t),x(t))~2(t ) + OV(T-~x(t),x(t)) x(t) --

dt

=

Ot

Using (8a) and (8b): dU _ -~ -

OV(T-r

at

+ OV(T-6.~(t),x(t)) Ox

[f(x(t)) +

5(t))]

g(x(t))u(x(t),

With (6a) and (10), it gives: dV

_

--p~llx(t)l[ ~

10V(T-~(t),x(t))glxlt~gT[xlt~

~,

o~

, , ,,

, ,,,

OV(T-E.~(t),x(t)) T

o~

Hence:

d--U-U< -p~llx(t)lf dt -

(22)

U therefore satisfies (4b) with as(r) = p~r 2. 9 u

is decreasing when C(x, ~) < 0 (U satis~es (4a)):

According to the behaviour of C imposed by Assumption (3), C vanishes after a non null period where it changes continuously and is strictly positive. Let t be this instant where C vanishes. One has at time t:

U(x(t),~(t)) = V ( T - ~ 2 ( t ) , x ( t ) ) +

Z

sup

V(0,~)

(23)

using (8c) and the continuity of the motion of x it becomes at time t + 0 + : U(x(t+O+),~(t+O+))

= V(O,x(t))+y]~

sup

V(O, s

(24)

According to the definition of C given by (9), C vanishes if and only if one has either [[x(t)[[-- l+2-~l(t ) or V ( T - ~2(t), x ( t ) ) = l+2-~l(t):

90

Nicolas Marchand et al. -

In the first case, subtracting (23) from (24) after having replaced ~1 (t) by ~--~II~(t)ll gives:

u(x(t +

0+),~(t

+ 0+)) -

= v(0, ~(t)) -

-

V(T -

u(x(~),~(t)) ~(t),

~(t)) -

sup V(0,

~) _<0

~5(llz(t)ll) since V(O, x(t)) < supeeB(ll~(t)ll)V(O, ~) and V ( T - ~ ( t ) , x(t)) > 0 (see (11)). In the second case, by integrating t --~ dW(T-~2(t),~(t)) on a small time dt interval [r, t] before C vanishes, one gets using (22): V ( T - ~2(t), x(t)) - V ( T - ~ ( r ) , x(r)) < 0 (25) Well, at instant v, C(v) > 0 hence, according to (9), one has V ( T ~ ( v ) , x(r)) < 1+--2-2~l(r ). Since at the given instant t where C vanishes V ( T - ~ 2 ( t ) , x ( t ) ) = 1+2-~-~1(t), (25) becomes:

1 +2~(t) < L-~7_~(r) which clearly contradicts the fact that ~1 remains constant as long as C remains strictly positive. 9 ~ remains bounded: ~ is clearly bounded since it belongs to the compact [0, T]. Furthermore, it ensues from (21) that lirn~,_~ U(x,~) = +cr Since U is radially unbounded with respect to x and since ~ remains bounded, for any initial condition (x0,~0) E 1"2, the set {(x,~) E/2; U(x,~) _< U(x0,~0)} is bounded. The decrease of U along the trajectories allows to conclude that ~l also remains bounded. 4

Examples

In this section, the proposed scheme is applied on two academic examples. The first one proposed by Kawski [11] fails to satisfy the Brockett's necessary conditions. The second one is the classical stabilization problem of the angular velocities of a rigid spacecraft in failure mode. The numerical resolution of the Hamilton-Jacobi equation were obtained using the NumSol routines [3]. 4.1

Kawski's example

Let us first consider the the academic system proposed in [11]: xl = u x~ = x ~ - xl3

(26a) (265)

At the origin, the linearized has an uncontrollable mode and consequently fails to be stabilizable by means of a C1 static feedback [6]. Figure 2 represents the evolution of the states of the system for an initial condition x0 := (1, 6). Note the joint evolution of HxH~ and ~1 and recall that this follows from the fact that ~1 is a goal for [Ix[Is that goes decreasing when reached.

A StabilizingDiscontinuous Dynamic State Feedback

91

W h e n c o m p a r e d with K a w s k i ' s feedback s, the p r o p o s e d m e t h o d seems to be m o r e efficient. Note t h a t K a w s k i ' s work has m a i n l y a m a t h e m a t i c a l interest a n d the efficiency of the feedback was n o t a priority. Comparative States

e v o l u t i o n

States

of It~112 and ~1 .o

.......

..........

, .....

s

....... i .......... .....

+~

o...

~

0.2

Control

-4soo

..........

i

i

9 9:

%

0.4

~...........

.....

0.2

..

oi

~

"

Control

. . . . . . .

..........

i ..........

!....

~..... o

-looo

:.

AAAA^^^^^ tVVVVVVVVVV]

~ ....

i . . . . . . . .

~a

0.2S o

........

.

.

.

.

.

.

. . . . . . . . . . . . .

o

Proposed feedback

0.2

0.4

Kawski's feedback [11]

F i g . 2. Stabilization of system (26) with the proposed method and comparison with Kawski's feedback (right)

4.2

Stabilisation of a rigid spacecraft in failure mode

T h e s t a b i l i z a t i o n of the a n g u l a r velocities of a rigid spacecraft using only two controls (instead of the three usually available) has been widely s t u d i e d in the literature (see e.g. [2,19,17]). This system is known to be s t a b i l i z a b l e by means of a s m o o t h feedback [6], and is a classical e x a m p l e of an u n d e r a c t u a t e d system, a class t h a t has received an increasing a t t e n t i o n in the last years. T h e d y n a m i c of the rigid spacecraft with two control torques can be d e s c r i b e d by the following equations: 0)l --

J2 - J3 j ~ - - w 2 ~ 3 + ul J3 - J1

r

- ""~wawl

r

J1 - J~ T ~ i ~

+ u~

(27a) (27b) (27c)

where w is the a n g u l a r velocities vector and J1 := 2 5 0 0 k g m ~, J~ := 6500kgrn 2 and J3 : = 8 5 0 0 k g m ~ are the inertia m o m e n t u m 3 of t h e three axis of the body.

u(x) = ( - x l + ~Ex~) + K(x~ - x?) with a choice of E = 500 and t( = 3 in order to have the "fastest" stabilization with a reasonable control 3 the numerical values axe from SPOT 4 spacecraft (see [14] and the references therein for further details)

92

Nicolas Marchand et al. Comparative evolution of Ilwll

Angular velocities

2

and ~1

3 t .s

. . . . . . . . .

9

?..................... i : : ......... ; .........

o.s o -o

. . . .

""t~ii,~ii" ::.........

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

........ I~1

~

J

? ..........

. . . . . . . . .

1 .Is 1

~ ..........

i ..........

i ..........

{ ..........

i .........

" ..........

i ..........

; ........

........

7

i ..........

97

7 "l~. . . . . . . . . .

!

16

20

i

O . S o ~ 7

s

11oi . . . . . . . . . .

..........

t

1|

Ui

~2

U2

i

! ! o

i$

lO

F i g . 3. Stabilization of system (27) with the proposed feedback If one c o m p a r e s the proposed m e t h o d with s o m e o t h e r feedbacks of the litera t u r e (Fig. 4), it seems to be m o r e efficient to b r i n g the s t a t e s to the origin. In return, the controls law is discontinuous where the others are s m o o t h . 2

w 2

1.:

........

!

....... ....

o.s

-o

.......

; ........

i .......

i .........

i . . . . . .

~ ........

1.:

........

0.6

-o

6

so

w

0)

! ........

............

o

lO

.....

2

! .......

9 .s

! . . . . . .

o.s

~--

20

-

......

.....

!

.......

: .......

i ......

~ ......

]

3o

-o

3o

u

u 3

:

......

U

i ........

!

_o

o

o

Io

2o

:

:~o

Proposed feedback Brockett's feedback [6] Aeyels feedback [1] F i g . 4. Comparison with some feedbacks proposed in the literature

References 1. Aeyels, D. (1985) Stabilization of a class of nonlinear systems by smooth feedback control. Systems & Control Letters, 5(5):289-294 2. Aeyels, D., Szafranski, M. (1988) Comments on the stabilizability of the angular velocity of a rigid body. Systems & Control Letters, 10(1):35-39 3. Balloul, I. (2000) Commande robuste des systSmes non lin~aires. PhD Thesis, Lab. d'Automatique - INPG, Grenoble, France 4. Beard, R.W., Saridis, G.N., Wen, J.T. (1997) Galerkin approximation of the generalized Hamilton-Jacobi-Bellman equation. Automatica, 33(12):2159-2177

A Stabilizing Discontinuous Dynamic State Feedback

93

5. Beard, R.W., Saridis, G.N., Wen, J.T. (1998) Approximate solutions to the time-invariant Hamilton-Jacobi-Bellman equation. Journal of Optimization Theory and Applications, 96(3):589-626 6. Brockett, R.W., Millmarm, R.S., Susmann, H.S. (1983) Asymptotic stability and feedback stabilization. In: Differential Geometric Control Theory. Birkhs Boston-Basel-Stuttgart 7. Clarke, F.H., Ledyaev, Y.S., Sontag, E.D., Subbotin, A. (1997) Asymptotic controllability implies feedback stabilization. IEEE Trans. on Automatic Control, 42(10):1394-1407 8. Filippov, A.F. (1988) Differential equations with discontinuous righthand sides. Kluwer Academic Publishers, Dordrecht-Boston-London 9. Georges, D. (1996) Solutions of nonlinear optimal regulator and Ho~ control problems via Galerkin methods. European Journal of Control, 2(3):211-226 10. Hahn, W. (1967) Stability of motion. Springer Verlag, Berlin-Heidelberg 11. Kawski, M. (1990) Stabilization of nonlinear systems in the plane. Systems K: Control Letters, 12(2):169-175. 12. Lee, E.B., Markus, L. (1967) Foundations of Optimal Control Theory. John Wiley & Sons, Inc., New-York, London, Sidney 13. Marchand, N. (2000) Commande & horizon fuyant : th6orie et mise en oeuvre. PhD Thesis, Lab. d'Automatique - INPG, Grenoble, France 14. Marchand, N., Alamir, M. (1998) Numerical stabilization of a rigid spacecraft with two actuators. In: Proc. of the IFAC Workshop on Motion Control, Grenoble, France, 81-86 15. Mayne, D.Q., Michalska, H. (1990) Receding horizon control of nonlinear systems. IEEE Trans. on Automatic Control, 35(7):814-824. 16. Morin, P., Samson, C., Pomet, J.-B., Jiang, Z.-P. (1995) Time-varying feedback stabilization of the attitude of a rigid spacecraft with two controls. Systems & Control Letters, 25(5):375-385 17. Outbib, R. (1994) On global feedback stabilization of the angular velocity of a rigid body. In: Proc. of the IEEE Conference on Decision and Control, NewYork, USA, 912-913 18. Ryan, E.P. (1994) On Brockett's condition for smooth stabilizability and its necessity in a context of nonsmooth feedback. Siam Journal on Control and Optimization, 32(6):1597-1604 19. Sontag, E.D., Sussmann, H.J. (1988) Further comments on the stabilizability of the angular velocity of a rigid body. Systems & Control Letters, 12(3):213-217 20. Zabczyk, J. (1989) Some comments on stabilizability. Appl. Math. Optim.,

19(1):1-9.

On t h e S tab ilizat ion o f a Class o f U n c e r t a i n S y s t e m s by B o u n d e d C o n t r o l Lorenzo Marconi 1 and Alberto Isidori 2 I Dipartimento di Elettronica Informatica e Sistemistica, University of Bologna, Via Risorgimento 2, 40136 Bologna-Italy Dipartimento di Informatica e Sistemistica, University of Rome, V. Eudossiana, 18, 00184 Rome - Italy and Department of Systems Science and Mathematics, Washington University, St.Louls, MO 63130 - USA A b s t r a c t . This paper deals with the problem of stabilizing an uncertain chain of integrators by means of bounded (arbitrary small) control. The problem is solved by designing a nonlinear state feedback which renders the uncertain chain of integrators input to state stable with respect to exogenous inputs affecting the integrators dynamics. The design methodology proposed can be also seen as a key tool to globally asymptotically stabilize a class of uncertain feedforward systems. The notion of input to state stability with restriction and the general version of the small gain theorem as introduced in [9] are employed to derive the result.

K e y w o r d s : feedforward systems, saturated control, input to state stability, nonlinear small gain theorem, robust control.

1

Introduction

In the last years a large research effort has been devoted to the problem of globally stabilizing feedforward systems. To this end several stabilizing techniques using bounded controls (i.e. control laws whose amplitude is bounded by explicit values which, in turn, can be arbitrarily small) have been proposed. An important impulse to the research activity in this direction has been given by the work [8], where the problem of stabilizing a chain of integrators by saturated (arbitrarily small) control laws has been solved. The design methodology proposed in [8] has then inspired several works dealing with the more general problem of globally stabilizing by state feedback linear and nonlinear systems with feedforward structure (see, a m o n g others, [1], [3], [4], [6], [9]). A detailed description of the main design methodologies which adopt small control for stabilizing feedforward systems can be found in [2]. A c o m m o n drawback of all these design methodologies is t h a t they are not able to cope with possibly uncertain parameters in the chain of integrations. Motivated by this, the problem of stabilizing an uncertain chain of integrators

96

Lorenzo Marconi and Alberto Isidori

by bounded control is the main goal of this paper. Specifically the attention is focused on the system ~ = Itlx~ + vl i:~ = It~x3 + v~

~gn--1 =" I t n - - l X n + V n - 1

i:n = It.u

(I)

where Pi, i -- 1 , . . . , n are uncertain (possibly time varying) real parameters which are assumed bounded from below and from above by known positive numbers, namely 0 < i t l L _< It i _< It i U

i=l,...,n,

v n - 1 represent exogenous inputs while u is the control input. The main result of this paper is the development of a design m e t h o d o l o g y for constructing a bounded control law able to render system (1) input to state stable with respect to the inputs v l , . . . , vn with non zero restrictions on the inputs (see [9]), and providing explicit formulas for the input to state (linear) gains and for the restrictions to which the exogenous inputs are constrained. The main idea upon which the stabilizing control law is derived, can be also successfully employed in order to stabilizing certain class of feedforward systems, namely in case the exogenous inputs vi are replaced by suitable nonlinear functions. For instance, as shown in [5], the same design procedure used to render input to state stable system (1), can be successfully used also for globally asymptotically stabilizing the feedforward system v l , v~, . . . ,

xl = Itlx~ + g l ( x ~ , . . . , ~ n , i t ) x2 -= It2X3 "4- if2 (it3 . . . . , Xn, t t)

x , , - i = It,~-lx,~ + gn-l(x,~,it) =

(2)

where gj(Jzi, 9.., ~:n, It), j = 1 , . . . , n - 1, are nonlinear functions vanishing at the origin which are assumed to be l o c a l l y L i p s c h i t z in their arguments. The design methodology draws inspiration by [1] and use the powerful definition of input to state stability with restrictions and the modified version of the small gain theorem as introduced in [9].

2

Main result

To our purposes, a saturation function is any differentiable function a : IR -+ h~ which enjoys the following properties:

On the Stabilization by Bounded Control 9 9 9 9

97

I~r'(s)l : - I d a ( s ) / d s I < 2 for all s. str(s) > 0 for all s # O, ~r(O) = O. a ( s ) = sgn(s) for Isl _> 1. I~(s)l > Isl for Isl < 1.

M o t i v a t e d by the results in [1], consider the new s t a t e variables Zi : = X l

Zj := Xj "q- A j _ t a ( I 4 ~ J ~ - ~"3-1') Aj_ 1

j = 2,...,n

and choose the control law u as

u = -An~(~)

(3)

where Ai, Ki, i -- I, 2,...,n, are positive design parameters. In this way, system (i) transforms as zl = -1~i

A ,KlZl,

iat---ff~l ) + p i z 2 +

K j zj

=

+

vi

"i- l z i Aj_I

+

9

) + Kn-la'( K,~-lzn-1

~,~ = - ~ . h n a (

hn-i

)~?n-x-

+ vj

(4)

T h e following l e m m a , which is a s t r a i g h t f o r w a r d extension of a s i m i l a r result in [1] (see L e m m a 1), shows t h a t if the h i ' s and K i ' s are p r o p e r l y chosen and if the a m p l i t u d e of the exogenous i n p u t s vj, j = 1 , . . . , n - 1 are sufficiently small, then the s t a t e of (4) enters in finite t i m e in the set

s~ := {z ~ ~ Lemma

:

hi Iz~l < ~//,

i = 1,...,n}.

(5)

1. Consider s y s t e m (4) and a s s u m e that

I1~(')11~ --- V~,M

i=

1,...,

~ -- 1

f o r s o m e ViM. I f hi and K i can be chosen so that the f o l l o w i n g inequalities are fulfilled hj h~+----L~< - (6) h'j+~ 2 f o r j = 1 , . . . , n -- 1 and

4 K~-I

u L h~ ~,_~ h~-1 < ~n

(7)

98

Lorenzo Marconi and Alberto Isidori

f o r j -- 2, . . . , n -- 2, then there exists a time 7"* such that z(t) E [2

f o r all t >_ T* .

Moreover [~j(t)[ _< 2 p ~ Aj

j = 1,...,n

(8)

f o r all t >_ T*. Proof. Consider the nested sets [2i:={zElR":

I z j [ < A--2 -

for all

j>i}

(9)

Kj'

a n d note that [2 = / 2 1 . Following the arguments of [1], the l e m m a is a consequence of the following two facts

(a) all the [2i's, i = 1, . . . , n, are positively invariant; (b) every trajectory starting in IRn \ [2n, enters in finite time the set [2n and every trajectory starting in Di \ [2i-1 for i -- 2, . . . , n, enters in finite time the set [2i-t. To prove claim (a), we show first that z E [2i

and

[zi[ = A i / K i ::~ s z e [2i ~

< 0

I~il < 2.~A~

(lO)

for all i : 1 , . . . , n . As a matter of fact, let i = 1 and consider the expression of ~?i .

,

KlZi

zi = --/al-~ICrt--~ i ) -l-//iz2 Jr Vl .

If [zl[---- A l / K 1 , since [z21 5 A2/K~, [vii _< Vl,M and in view of inequality (6) for j = i and of the first in (7), it turns out that zizi < 0. Moreover, so long as z E [21, we have that u A2 Now, assuming (10) true for i = g - 1, we prove t h a t it is true also for i - g. To this end, let z E [2t and consider the expression of ~?t Ktzt 1.~l ZlZl-1 )~?t-i + vt h = - m A t ~ ( - T - - ) + ~tzt+l + i{t_ 1o., ( Kt~At-- 1 at The term c r ' ( K t - l z t - 1 / A t - 1 ) ~ t - 1 can be bounded by 4 K t _ l p tU At-i In --1 fact, ~r'(s) = 0 for Is[ > I and therefore, if [zt-l[ > A t - 1 / K t - i then ( A t - 1 / I { t - l ) = 0. Otherwise z E [2t-t by definition and, using (10) as. ? 9 9 sumed true for i = g - 1 and the bound on a'(.), we obtain cr ( A t - 1 / I < t - 1 ) z t - 1 <

On the Stabilization by Bounded Control

99

4Kt-~l~t-tAt-1. In view of this and of the inequalities (6)-(7) for j = g, this proves the upper implication in (10) for i = g. Moreover, so long as z 9

At+i +

I~,1 _< mv A, + ~,~ ~

4 K t - 1 / ~ L 1 At-1 + Vt,M __ 2 . U A,.

i.e. the lower implication in (10) for i = L Hence, by induction, we conclude that (10) holds for all i = 1 , . . . , n. To show t h a t the sets/2i, i ---- 1 , . . . , n, are positively invariant note that if z 9 0/2i, then Izjl = A j / K j for s o m e j > i. Since/2/ C /2j for j > i, from the previous considerations we deduce that ~'jzj < 0. Hence trajectories starting in/2/ can not leave/2i. Claim (b) can be proved with the following arguments. Consider z(0) E llq." \ / 2 n and suppose that IZn(t)l > An/Kn for all t > 0. This means t h a t x , = -/JnAnsgn(zn(0)) and, in c a s e zn ( 0 ) > O,

x . ( t ) = x,,(o) - ~ . A . t . LFrom the definition of Zn it turns out that zn(t) = zn(0) - A n - l O ' ( K n - l z n - l ( 0 ) ) -~- An-lO'( I~'n-lzn-1 (t)) - ]In Ant

An-I

An-I

z . ( t ) < z . ( o ) - ~ . A . t + 2A._~ and this implies t h a t there exists a time T such that zn(T) = 0 which is a contradiction. A similar conclusion holds if Zn (0) < 0 and this proves that z enters 12n in finite time. Now suppose there exists { s u c h that z(t) C /2i \ / 2 i - t , namely ]zj(t)] <_ Aj/tt] for j = i , . . . , n but Izi-~(t)l > A i - 1 / K i - l , for all t _> [. This means that

x~_l

= z~ - A ~ _ ~ s g n ( ~ i _ l

(0)

9

1 implies Ai-i > A~/K~, it follows, in case

Since inequality (6) for j = i zi_l(t-) > 0, that x i - i _< - c

where

Ai c :-- Ai-t - - Ki

and

xi_,(t) < xi_~(~ -- e(t -- O . LFrom the definition of zi_ 1 it follows t h a t

z~_l(t) = z~_~(t3 - A,_~,~( z~_~(t) < z,_t(t-) + 2A,_~

-

Ki-~zi-2(~ )+ Ai_~ e(t - i)

Ai_2,r(Ki-zzi-2(t)

) - c(t - 0

100

Lorenzo Marconi and Alberto Isidori

and this implies the existence of a t i m e T such t h a t z i - l ( T ) = 0 which is a contradiction. A similar c o n t r a d i c t i o n can be reached in case zi-1 (t-) < 0 a n d this shows t h a t z enters D i - I in finite time. This concludes t h e p r o o f of the lemma. Now define

7/, := 24 p--iU

7 i : = 24/j/v

i= 1,2,...,n

- 1

(11)

and

t17 I~[i-I

~[ivTl

i : 2,...,n

j = 1,...,i-

1.

(12)

T h e next l e m m a states the c o n d i t i o n s under which s y s t e m (4) is i n p u t to s t a t e stable with respect to the i n p u t s vi a n d gives explicit expressions for the linear gains. In p a r t i c u l a r the notion of i n p u t to s t a t e s t a b i l i t y we will refer to, is t h a t proposed in [9] which generalizes the s t a n d a r d definition given in [7] introducing restrictions on the initial s t a t e a n d on the inputs. In p a r t i c u l a r we r e m i n d t h a t a nonlinear system x = f ( x , u) with s t a t e x = ( x i , . . . , x n ) E lR n and i n p u t s u -- ( u l , . . . , u m ) E IRm is said to be i n p u t to s t a t e stable with restrictions X on the s t a t e a n d A : = ( A t , . . . , Am) on the inputs if, for each x0 E X and each m e a s u r a b l e i n p u t s I]uil[co < A i , there exist class-/C functions i 3'0(') a n d 3q(') i = 1, ..., m such t h a t the s o l u t i o n of the s y s t e m with x(0) = x0 exists for all t E [0, oo) and satisfies

IIx, llco _< max{'1o(Ix(O)l), ,__~Xm{'Y,(llu, llco)} } lim sup I ~ ( t ) l <

t ---+o o

--

m a x { 7i( lim sup l u d t ) l ) }.

i=l,...,rn

t ---+c,:~

T h e functions 7i(') i = 1, ..., m are usually called the i n p u t to s t a t e gains of the system. 2. Assume, in addition to the inequalities (6), (7), that the K i ' s are such that

Lemma

Ki-i

7i_i < 1

(13)

i = 2,...,n.

Then system (4) is input to state stable with no restriction on the initial state and restriction Vj,M on the inputs vj, j = 1 , . . . , n 1, with linear gains. In particular ~i ~i l i m s u p Iz~(t)l _< m a X {Tv, lim sup Ivl(t)l, 9 99, 7v,_x limsup Iv,-l(t)l} t-+co

t-*co

t~co

1 A function 7(') : [0, a) --+ [ 0 , ~ ) is called class-K: function if it is continuous, strictly increasing and 7(0) = 0.

On the Stabilization by Bounded Control where %, ~1 = 21# L K~, %j ~1 = 27~j/K1, j = 2 , . . . , n -

6 Ki-1

i-1 Ki I~iL 7vj

5'~j :=

j=

101

1, and

1,...,i-1 (14)

3 (j-i+l)

j=i,...,n-1.

z//~_iK l for i = 2, . . . , n.

Proof. In the following we take advantage of the result of Lemma 1, and in particular of the fact that there exists a time T* such that z(t) E 12 for all t > 7*. The key idea is to use the small gain theorem in the general version introduced in [9] for systems that are input to state stable "with restrictions" (on initial state and inputs) showing how conditions (6), (7) together with the assumption z E /2 render the conditions regarding the restrictions automatically fulfilled. To begin with, consider the dynamics of zj, for j -- 2 , . . . , n - 1, described by the j-th equation in (4). It is easy to show that this system, regarded as system with inputs zj+l, vj and wj-1 where P.

w-1

:=

and state zj, is input to state stable without restriction on the initial state, nonzero restriction on the inputs and linear gains. As a matter of fact ]zj I is decreasing if

namely if ~(K~lz~l) > IzJ+~l Ivj) + )wj-~l Aj ~ + PjL Aj

(15)

The previous inequality can be fulfilled by some Izji if, for instance, the following restrictions are assumed on the inputs, Izj+~l 1 A--7 < ~

and

Iv~l + Iwj_l[ 1 #JLAj < -'2

(16)

By definition of saturation function, it turns out that (15) is satisfied, namely Izjl is decreasing, if

Iz~l > ~

+

K~#}~

Hence the system is input-to-state stable, with no restriction on the initial state, restrictions (16) on the inputs zj+l, vj, wj-1, and linear gains. In particular

llz llo _< max{ -~3 IIz~+xll~,~

,[yj [,a, ~

i.wj_ l ]Ia}

(17)

102

Lorenzo Marconi and Alberto Isidori

or, in view of the estimate

Iwr

3

IIzr < max{-;-. IIzr ~xj

~ 2K~-11~-11, 3

~Jxjl lpj vr

6Kj t

~ Ij/Jj I~-xlI~}

-

(18)

Similar conclusions can be drawn also for the system with state zl and inputs z~, vl and the system with state z . and input ~.-1 (see the first and the last in (4) respectively). In particular the above arguments can be used to show that the zl-system is input to state stable without restriction on the initial state and restrictions on the inputs z2, Vl given by Iz2l 1 A--'7 < 2

Ivll 1 A1/~L < 2

(19)

and linear gains as in

2 Ilzlllo < max{~llz211~,

zf 2K-"--~I1~,iI1~} "

(20)

Similarly the zn-system turns out to be input to state stable without restriction on the initial state, restriction on the input w . _ l given by

IWn-l"'-'-'~]< 1 and linear gain Eft - 1

IIz.lla < 7::c..L I1~.-111~. /Xn/~n

We study now system (4) by iterating a procedure which, at each step, considers the feedback interconnection (see Fig. 1) of the subsystem with state

r := ( z , , . . . , z~-l), inputs vl, v2, . . . , v j - 1 , zj and output kj-1 and the subsystem with state z j , inputs v j , z j - 1 , zj+l and output zj. At the generic step j, j = 2 , . . . , n , we assume that ( A 1 ) the upper subsystem satisfies an asymptotic bound with no restriction on the initial state, restrictions Vl,M, ..., vj-I,M on the inputs vl, . . . , v j - 1 , nonzero restriction Aj on the input zj, the asymptotic estimate

I1~-111~ _< max{'g-lll~Jll~, ~;-~llvlll~, .... , ~j_, "-1 Ilvj-~llo}

(21)

j - 1 , i = 1 , . . . , j - 1 defined as in (11), holds for output z j - 1 , with 7 j - l , 7v, (12). Moreover, for all t >_ 7"*, Izj(t)l < Aj (i.e. the restriction on the input zj is fulfilled). 9

( A 2 ) the lower subsystem satisfies an asymptotic bound with no restriction on the initial state, restriction Vj,M on the input v j , nonzero restrictions

On the Stabilization by Bounded Control

103

Fj-, F+ on inputs ~j_~, zj+l, the asymptotic estimate (18) holds for the output zj. Moreover, for all t > T*, I~j_l(t)] ~ Fj-" and ]zj+l(t)l < F+ (i.e. the restrictions on the inputs z.j-1, zj+l are fulfilled). Note that, by the previous calculations, assumption A2 is satisfied at each step. As a matter of fact, since z(t) 6 [2 for t > T*, we have that [zj+l[ < ~ and [zj-l[ < 2Py_IAj-I-Hence the condition (6) (for i - j) and -- K i + 1 the second in (7) guarantee that restrictions (16) are automatically fulfilled whenever Ivj] < Vj,M. This makes assumption A2 satisfied for all j = 2 , . . . , n.

v1 ...

I Zj

I vj-i

zj-1

Fig. 1. Feedback interconnection at step j. Moreover it is easy to show that assumption A1 holds for j = 2. As a matter of fact, since z(t) 6 /2 for t > T*, we have [z2[ _< - ~ and thus, in view of the inequality (6) (for j = 1) and the first in (7), it turns out that the restrictions (19) are fulfilled whenever [vl[ ~ Vl,M- Furthermore, by the first equation in (4), it turns out that

I~11 _< 2~[I<~1zll + ~[Iz~l + h i < max{@[Kllz~l, 3~[Iz~l, 3lv~l}. Recalling (20), we have that

I1~111~< max{12p U IIz~llo, 12 ~[ Ilvlllo} which, in turn, implies (21) for j : 2. We have proved in this way that the assumption A1 is satisfied for j = 2. We suppose now that the assumptions hold for some j > 2 and, by studying the intereonnection of Fig. 1, we show that they hold also for j -t- 1. To this end note that, in view of (18),(21), inequality (13) represents a small gain condition for the feedback of Fig. 1. By the small gain theorem (see

104

Lorenzo Marconi and Alberto Isidori

[9]) we conclude that the interconnection satisfies an a s y m p t o t i c bound with no restriction on the initial state, restrictions Vl,M, . . . , Vj,M on the inputs vl, . . . , vj, and some nonzero restriction Aj+ 1 on the input zj+l, which turns out to be fulfilled. T h e c o m p u t a t i o n of the linear gains between the inputs vt, . . . , vj and ~j (needed to iterate the procedure) can be done as follows. By the definition of ~j in (4) and by the definition of saturation function we have t h a t

<_ max{81~UKjlzj[, 4pUIzj+l[, 8 K j - l l ~ - x l , 41vii}

(22) and thus, by the estimation of

[]zjlla in (18),

I[zjlla _< max{24~u II/~+lllo, 24~-u-juIlvj[la,

Pj

48P. U--JLKj-1 I1~-1llo} " Pj

(23)

Now consider the estimation (21). By (18) and by the small gain condition (13), it is easy to show that 3

"-1

I[~j_l[]a < max{ ~ j 7 J

IIz~+lll~, ~;-1 IIvlllo, ..., j - i IIv~-lllo, ~v,_,

3

KjpL7 j - 1 II~Jllo} -

By embedding this estimation in (23) and by the small gain condition (13), it is easy to conclude that

II~Jll _< max{~llzj§

~{, II~llo, .-., ~{, II~llo}

where ~J, ~ , , ..., ~{, defined as in (11) -(12). Hence the procedure can be iterated. Finally, once the previous procedure has been carried out, the gains "~, between the inputs vi, i = 1, . . , n - 1 and the states zj can be c o m p u t e d by means of simple (but tedious) calculations. The previous result claims that if the p a r a m e t e r s Ai and Ki of the control law (3) can be tuned so as to fulfill the conditions (6)-(7) and (13) for some set of Vi,M'S, then the overall system (4), and thus also (1), is input to state stable with restrictions Vi,M on the inputs v~. R e m a r k a b l y the control law solving the problem is very simple as it is obtained by nesting s a t u r a t i o n functions of the states xi, namely u = -un with u0 = 0.

where

uj = Aja(h~

Xj "q- Uj--1

Aj

)

j = 1,..., ,,

On the Stabilization by Bounded Control

105

We conclude this paper by discussing how to tune the Ai's and K i ' s so that the conditions (6)-(7) and (13) are fulfilled. Before all note that an admissible choice for those p a r a m e t e r s is Ai

Ki =

-- Kici

i = 1, ..., n

Kg i

(24)

where g and cl are suitable positive n u m b e r s while K is an arbitrary positive number. As a m a t t e r of fact it is easy to realize that inequalities (6) are indeed fulfilled if the ci's are taken according to ci+ l

eta+ 1

- ~ --

2

c~

i --

1,...,

n -

1

(25)

for arbitrary positive g < I, ci > 0. Moreover,in view of (25), simple computations show that inequalities (7) are all satisfied for sufficientlysmall Vj,M if g is taken sufficientlylarge. Finally a large value of g makes also the small gain conditions (13) fulfilled. Furthermore, given a set of p a r a m e t e r s fulfilling the conditions in question, it is possible to construct a infinite family of similar parameters, n a m e l y infinite other control laws solving the problem of rendering system (1) input to state stable. For instance, it is easy to realize that, given an admissible choice {(A*, K*) : i = 1,..., n} which fulfills (6)-(7) and (13) for some {Vi,M : i = 1, ..., n}, then, for any 9 > 0, also (Ai, K i ) : = (e A*, If*)

i = 1, ..., n

(26)

fulfills (6)-(7) and (13) with Vi,M given by Yi,M

:~

9 V *i , M

i ~

1, . " , n "

Since lu(t)l < Am for all t > 0, the previous result claims that the a m p l i t u d e of the control law can be rendered arbitrarily small (by taking 9 small) provided that the exogenous inputs are bounded by sufficiently small positive numbers. Analogously note that, once v*i,M and A* for i = 1..., n are known and given the upper bounds of the inputs vj affecting the systems, the previous t r e a t m e n t allows for c o m p u t a t i o n of the m i n i m u m a m p l i t u d e of the control law (namely the m i n i m u m value of 9 needed to counteract the effect of the vi's, namely to have bounded states. Moreover, since the p a r a m e t e r Ki's are left unchanged in (26) (i.e. Ki = K~), the input to state gains given by (14) are not modified by the introduction of the p a r a m e t e r 9 As much simple to prove is the following result in which, besides the values of Ai's and Vi,M'S, also the Ki's are changed by the introduction of the parameter r -"'k

y-k

P r o p o s i t i o n 1. Suppose the sets { ( A * , K i ) : i = 1 , . . . , n } , { i,M : i = 1 , . . . , n -- 1} are such that (6)-(7) and (13) hold. Then, f o r any e > O, the choice (Ai, Ki) := (e i A~, e It'*)

i = 1,...,n

(27)

106

Lorenzo Marconi and Alberto lsidori

ful lts (6)-(7) and Vi,M : = r ~J*i,M

with v,,M gi e, bU i = 1,. .. , n - 1.

(28)

The result claimed in the previous proposition, which can be easily proved by substitution of (27) in (6)-(7) and (13), plays a key role in stabilizing feedforward systems described by (2) as shown in [5]. A key feature of the control law (3) which comes up by adopting the parameters (27) is that, in case e is sufficiently small, ,~i > 3~i+1, i = 1, ..., n - 1, namely the level of the outer saturation functions is lower than the level of the inner ones. This is an interesting feature, with respect to previous results on the subject, which indeed has been employed in [5] to globally asymptotically stabilize (2).

3

Conclusions

In this paper the problem of rendering input to state stable an uncertain chain of integrators with respect to exogenous inputs has been addressed. W i t h respect to previous works on the subject, the design methodology is able to cope with (possibly time-varying) uncertanties which affect the gains of the integrators. The approach proposed can be easily extended to the problem of globally asymptotically stabilizing a certain class of nonlinear feedforward systems.

References 1. Grog-nard F., Sepulchre R. & Bastin G. (1999) Global stabilization of feedforward systems with exponentially unstable Jacobian linearization. Systems fJ Control Letters, 37:107-115. 2. lsidori A. (1999) Nonlinear Control System II. Springer-Verlag, NewYork. 3. Lin W. & Li X. (1999), Synthesis of upper-triangular nonlinear systems with marginally unstable free dynamics usign state-dependent saturation. International Journal of Control, vol 72, 1078-1086. 4. Liu W., Chitour Y. & Sontag E. (1996) On finite gain stabilizability of linear systems subject to input saturations. S I A M Journal Control Optimization, July. 5. Marconi L. & Isidori A. (1999), Robust global stabilization of a class of feedforward nonlinear systems, accepted for publication to Systems fJ Control Letters. 6. Sussmann H. J., Sontag E. D. & Yang Y. (1994) A general result on the stabilization of linear systems using bounded controls. I E E E Transactions on Automatic Control, 39(12):2411-2425. 7. Sontag E. D. (1989) Smooth stabilization implies coprime factorization, I E E E Transactions on Automatic Control, 35:435-443. 8. Teel A. (1992) Global stabilization and restricted tracking for multiple integrators with bounded controls. Systems ffl Control Letters, 18:165-171. 9. Teel A. (1996) A nonlinear small gain theorem for the analysis of control systems with saturations. IEEE Transactions on Automatic Control, 41(9):1256-1270.

Adaptive Nonlinear Excitation Control of Synchronous Generators with U n k n o w n Mechanical Power Riccardo Marino 1, Gilney D a m m ~, and Franqoise Lamnabhi-Lagarrigue ~ I Dipartimento di Ingegneria Elettronica, UnlversitA di Roma Tor Vergata, via di Tor Vergata 110 00133 Rome, Italy marino@ing, uniroma2, it

2 Laboratoire des Siguaux et Syst6mes, CNRS Sup~lec, 3, rue Joliot-Curie 91192 Gif-sur-Yvette Cedex, France damm@iss, supelec, fr, lamnabhi@iss, supelec, fr

A b s t r a c t . A nonlinear adaptive excitation control is designed for a synchronous generator modeled by a standard third order model on the basis of the physically available measurements of relative angular speed, active electric power and terminal voltage. The power angle, which is a crucial variable for the excitation control, is not assumed to be available for feedback. The feedback control is supposed to achieve transient stabilization and voltage regulation when faults occur to the turbines so that the mechanical power may permanently take any (unknown) value within its physical bounds. Transient stabilization and voltage regulation are achieved by a nonlinear adaptive controller, which generates both converging estimates of the mechanical power and the new equilibrium point compatible with the required terminal voltage.

1

Introduction

The analysis of transient stability of a synchronous generator connected to an infinite bus when large and sudden faults occur is a classical power system problem, which has been addressed via Lyapunov techniques (see for instance [12], [10], [6] ) in which models of increasing complexity are used) in order to determine the critical clearing time, that is the time before which the fault has to be cleared so that the faulted trajectory still belongs to the stability region of the stable operating condition. Feedback linearization techniques were proposed in [5], [3], [14] to design stabilizing controls with the purpose of enlarging the stability region of the operating condition. Nonlinear adaptive controls are proposed in [1] which keep the machine in synchronism when short circuits occur in the transmission lines. Nonlinear adaptive controls are also proposed in [15] to improve damping without requiring the knowledge of the operating point. T h e nonlinear feedback control algorithms so far proposed in the literature make use

108

Riccardo Marino, Gilney Damm, and Frangoise Lamnabhi-Lagarrigue

of power angle measurements which are physically not available and have the difficulty of determining the faulted equilibrium value which is compatible with the required terminal voltage once the fault (mechanical or electrical failure) has occurred. In this paper we make use of the standard third order model used in [15] (see [2], [13] ) to show that the terminal voltage, the relative angular speed and the active electric power (which are actually measurable and available for feedback) are state variables in the physical region of the state space. Since the purpose of the excitation control is to regulate the terminal voltage without loosing the synchronism, we compute the zero dynamics of the system with respect to the terminal voltage and we obtain a highly nonlinear second order dynamics. We then design following [8] a nonlinear adaptive feedback control on the basis of physically available measurements (relative angular speed, active electric power and terminal voltage) which is adaptive with respect to the unknown mechanical power generated by the turbines and achieves transient stability for all physical faults affecting mechanic power generation. The mechanical power estimation quickly recovers the faulted value so that the faulted equilibrium point corresponding to the desired terminal voltage level can be determined and the synchronous generator can be smoothly transferred to the faulted equilibrium point with no loss of synchronism.

2

Dynamical model

Consider the simplified mechanical model expressed in per unit as

D =

+

ws

-

Pe)

(1)

where: 6(rad) is the power angle of the generator relative to the angle of the infinite bus rotating at synchronous speed ws; w(rad/s) is the angular speed of the generator relative to the synchronous speed ws i.e. w = wg - ws with wg being the generator angular speed; H(s) is the per unit inertia constant; D(p.u.) is the per unit damping constant; Pm(p.u.) is the per unit mechanical input power; Pe(p.u.) is the per unit active electric power delivered by the generator to the infinite bus. Note that the expression w~/Wg is simplified as w~s/Wg "' ws in the right-hand side of (1). The active and reactive powers are given by

p.

=

Xds

v,

)

Q = ~-Z. Eqcos(J) da

(2)

v} Xds

(3)

Adaptive Nonlinear Control of Synchronous Generators

109

where: Eq(p.u.) is the quadrature's EMF; V,(p.u.) is the voltage at the infinite bus; Xd, --" XT + 89 + Xd(p.u.) is the total reactance which takes into account Xd(p.u.), the generator direct axis reactance, XL(p.u.), the transmission line reactance, and XT(p.u.), the reactance of the transformer. The quadrature EMF, Eq, and the transient quadrature EMF, Eq, are related by

Xd, E'

Xa - X~ V~eos(~i)

E~ = x~-Z ~

(4)

x2

while the dynamics of Eq are given by

dE'q (5) dt = T~o(Kcu! - Eq) in which: X~, = XT + ~XL 1 +X~(p.u.) with X~ denoting the generator direct axis transient reactance; ul(p.u. ) is the input to the (SCR) amplifier of the generator; Kc is the gain of the excitation amplifier; Tdo(s) is the direct axis short circuit time constant. Substituting (2) into (1) and (4) into (5), we obtain the state space model

= -~

+ ~-

P m - .-=-, E:sinff).~,. + -XdTX~-, V} sinff)cosff)

1 (

X~,E,_{_X~:.X~;zcos(~i) )

(6)

in which (~, w, Eq) is the state and uf is the control input. Since Pe is measurable while E~ is not, it is convenient to express the state space model using (~,w, Pe) as states which are equivalent states as long as the power angle (f remains in the open set 0 < 5 < re. Taking account of the notation

T~0 = ~X~, T 60 da

where T~0 is the direct axis transient short circuit time constant, differentiating (2) with respect to time, and using (1)-(5), we obtain

D ~s : --~'~ - ~(P~ - Pro) ]he ----- ~-1, PeT~0

V,

+ T~o~((--~sin(~)[Keufa, + Td~

,

V,

.

--Xd)xh---~wsm((i)]

+ T~oPewcot(~)} (7)

110

Riccardo Marino, Gilney Datum, and Fran~oise Lamnabhi-Lagarrigue

which is valid provided t h a t 0 < J < 7r. Note t h a t when J is near 0 or near rr the effect of the i n p u t u ! on the overall d y n a m i c s is g r e a t l y reduced. T h e generator t e r m i n a l voltage is given by

Vte~ =

j X , E eJ(~ +6) + jXdVseJ-~ q jXds

where

Xs -~ S T -b X-''~'L 2 Xa, = Xa + X~ so t h a t its m o d u l u s is

v, =

~-L-~X2E~ + v } x ~ + 2X, X~EqV, cos(J))~

Xds ~

s

q

or in the new s t a t e variables

( X~P: XJV? 2X, Xd Pecot(5)~ 89 Vt = k V}sin~(j) + X$--7 + Xa, /

(8)

which is the o u t p u t of the system to be regulated to its reference value Vtr =

l(p.u.) If this is done, the zero d y n a m i c s will be

~to D

(o = - # t o +

to,

Xa V~ sin(5)cos(J) -ff(P~ + x , x ~ ,

v, sin(J) Xs

~ - -~-~ V,~sin~(J) Xds

which are very complex, and for some initial conditions or p a r a m e t e r s values m a y become u n s t a b l e as we m a y see in Fig. (1) where 6 goes o u t of the region 0 < 5 < 180 and to grows out of physical boundaries. T h e o p e r a t i n g conditions (J0,to0, Pc0) of the synchronous g e n e r a t o r m o d e l (7) are given by

Adaptive Nonlinear Control of Synchronous Generators

111

60

_o.~

70 4JO 60 40 ~0 ~0 10

--3 0 --10 ~ 0

o.~

--40

0.2

0.4

Flg.l. Zero Dynamics

wo=O

P~o = P,.

V, ~-r-Kc~s~i.(~)

-P~ +

= 0

Aa.d8

Note t h a t while w0 = 0, Pe0 = P,- are not affected by u j, from the third equaiv tion above we see that there are two operating conditions 5,, 5u, 0 < ~. < y, < 5. < 7r for constant inputs u s > (PmXd,)/(KeV,); (5,,O, Pm) is an asymptotically stable equilibrium point while (6u, O, Pro) is an unstable equilibrium point. The stable operating condition (Ss, 0, Pm) and the corresponding excitation constant input

h'cus~

Pm X as V, sin(5,)

are chosen so that the modulus of the generator terminal voltage

v, = ~

1

, , ~ 2 r.-2 2

~ , ~.c,,Io + v,2x~ + 2X, XaKcuso V, co~(,~,))

is equal to the prescribed value. The power angle is not measurable and is also not a physical variable to be regulated; the only physical variable to be regulated is the o u t p u t Vt, while (Vt, ca, Pe) are measured and are available for feedback action. As a m a t t e r of fact (Vt,w, Pe) is an equivalent state for the models (6) and (7) since (see appendix for the derivation of this formula)

5 = arccotg

V~

-

-

- d- ~ Xx~, +

v~2-

"'S

2

(9)

112

Riccardo Marino, Gilney Damm, and Frantjoise Lamnabhi-Lagarrigue

If the parameters (V,, Xs, Xd, Xds) are known, state measurements are available. From (9) it follows that in order to regulate the terminal voltage Vt to its reference value (Vtr = l(p.u.)) J should be regulated to

J, = a , . c c o t g

(

V8

(

Xa~

- x --7 +

~/

X~p3'~

,,,))

(10)

From a physical viewpoint the natural choice of state variables is (Vt, w, Pe) which are measurable. The state feedback control task is to make the stability region of the stable equilibrium point (Vtr, 0, Pro) as large as possible. In fact the parameter Pm may abruptly change to an unknown faulted value Pm/ due to turbine failures so that (Vtr, 0, Pro) may not belong to the region of attraction of the faulted equilibrium point (Vtr, 0, PraY). The state feedback control should be design so that typical turbine failures do not cause instabilities and consequently loss of synchronism and inability to achieve voltage regulation. A reduction from Pm to (Pm)I of the mechanical power generated by the turbine, changes the operating condition: the new operating condition (Js)I is the solution of (Pro)1 + Si"(J)S = o

P,.

8i.(J,)

and since (Pm)f is typically unknown, the corresponding new stable operating condition (J,)y is also unknown. 3

Robust

and adaptive

stabilization

The model (7) is rewritten as

~Od

D & = -~w

= -lpo ~}o

w~ - ~-(Pe - O) +

+

(Xd X d ) " XdsXtds

+ P~w cot(J)

(11)

in which O(t) is a possibly time-varying disturbance: the parameter 0 is assumed to be unknown and to belong to the known compact set [0m, 0M]: the lower and upper bounds On, 0M are known. Let Jr(t) be a smooth reference signal to be tracked. Define ()q > 0)

Adaptive Nonlinear Control of Synchronous Generators

113

~(t) = o(t) o~(t) (M" = -x~+g, = x - (M* = (M + A13 - ~

so that the first two equations in (11) are rewritten as

L

6 ---- -Ai(f + ab 9

D

= --~(M + (M('O(t)H

- Pe) - A ~ + Aiw - d~

Define (A2 > 0, k > 0) the reference signal for Pe as

p:=

HI

D

1

/(M,~2_]

(Ms

while/~ is an estimate of 0 = P m and

P, = P e - P * so that (11) m a y be rewritten as (0 -- 0 - 0)

-~+~ /5~= _ l_li_p,+ V_~T,sin(cl)K,u!+ (Xd- X'd)V,2wsin2(6)+P,wcot(6) "1~o Xa, do Xa, X'a,

H

-A~ + 1 + A1 -~

(M s

k

(~~+~,+~+~(~) , )~ __ 0+--DL+~g ( +A~+:~+~ 0"s

Defining (A3 > O)

038

114

Riccardo Marino, Gilney Damm, and Fran~oise LamnabhJ-Lagarrigue

~=0-0 TjoXn, uf _ V , ~ 6 ) 1

~o = T~---~P~

~o (Xd -- X~)

~TX~ V}~ si. ~(6) - Po~ r

H { ( - A ~ + I + A~D) (-A~$+~) +-~d s

~) + -( ~-+~1+~2+~~ (-~)~) o+b -~

-

+~+~2+~

g

_~. _~)}

Po---v~:-K~:-~;~+H~ 03~

03~

the closed loop system becomes

9

= -6-

e

~._

~(_~)~

~2~ - ~-P~ - ~

~.-

~ + ~-0

MS ~

-

-~

4 -

+.x~ +.,x~ + ~-

+A~+A2+~

/5

The adaptation law is (7 is a positive adaptation gain)

~:~o~ ((~ (~_~_ ~_ ~ where Proj(y, ~) is the smooth projection algorithm introduced in [11]

Proj(y, O) = y, if p(O) <_ 0 Proj(y, O) = y, if p(O) >_0 a.d (gra@(0), U) < 0 Proj(y, O) = [1 - p(O)[grad p(O)[], otherwise (12)

Adaptive Nonlinear Control of Synchronous Generators

115

with

p(O) =

2

J

for e an arbitrary positive constant which guarantees in particular that:

i) Om -- e < ~(t) < OM + e

ii) IProj(u,O)l <_ lYl iii) ( 0 - ~)Proj(y, 9) > ( 0 - 9)y Consider the function

w = ~(~ + ~ + P,~)

(13)

whose time derivative, according to (11), is

-

(

-

+~1+~2+~

~P~-~

-

+~1+~+~

Pe 2

Completing the squares, we obtain the inequality 2- 2

(14)

W _< - A I D - A2~ 2 - aa/5~2 + ~O

which guarantees arbitrary s robustness from the parameter error t~ to the tracking errors J, &, PeThe projection algorithms (12) guarantee that /} is bounded, and, by virtue of (13) and (14), that J, tb and /he are bounded. Therefore, 9 is bounded. Integrating (14), we have for every t 3> to > 0 --

t (Al~ 2 + A ~ w 2 + A3Pe ~ 2)dr" + ~2

02dr > W ( t ) - W ( t o ) Jto

Since W ( t ) > 0 and, by virtue of the projection algorithm (12), ~(t) < O M -- O,,,+

116

Riccardo Marino, Gilney Damm, and Franqoise Lamnabhi-Lagarrigue

it follows that - om + ~)~(t - ~o) J[ts (,~,~-2 + ~2co2)a, - <_ W ( t o ) + -~(oM 2 which, if W(t0) = 0 (i.e. to is a time before the occurrence of the fault), implies arbitrary .C~ attenuation (by a factor k) of the errors ~ and ~5 caused by the fault. To anMyze the asymptotic behavior of the adaptive control, we consider the function

v=

1(~+~ + p,)+ ~

The projection estimation algorithm (12) is designed so that the time derivative of V satisfies i~ < - A I J 2 - A2& 2 - A3Pe 2

(15)

Integrating (15), we have nmt_.,~

(,~1~ ~ + ~

+ , ~ 3 ~ ) a , - _< v ( 0 ) - v ( ~ )

< ~

,/tO

From the boundedness of ~},~ and/~e, and Barbalat's L e m m a (see [91, [71) it follows that

limt..,~

/~5(t) |

= 0

I.P~(t)J 4

Simulation

results

In this section some simulation results are given with reference to the eightmachine power system network reported in [3] with the following data: cv~ = 314.159 rad/s D = 5 p.u. H = 8s Trio = 6.9s Kc = 1 X d = 1.863 p.u. X~ = 0.257 p.u. X T = 0.127 p.u. XL = 0.4853 p.u. The operating point is 5, = 72 ~ P m = 0.9 p.u., w0 = 0 to which corresponds Vt = 1 p.u., with V, = 1 p.u.. It was considered a fast reduction of the mechanical input power, and simulated according to the following sequences 1. The system is in pre-faulted state.

Adaptive Nonlinear Control of Synchronous Generators

117

2. At t = 0.5s the mechanical input power begins to decrease. 3. At t = 1.5s the mechanical input power is 50% of the initial value. 4. At t = 2s the controller starts to drive the system to the new predicted equilibrium power angle. 5. At t = 4s the system finishes to drive the system to the new equilibrium. T h e simulations were carried out using as control parameters Ai=20 7=1

1
(I)

(1) aO

,

i

i

~v

05

1

i

i

,

15

i

2

i

~

1.~

r

. . 2.5. . . 3

3.5

4

(2)

i

45 i

,

~

r

0,5

1

1.5

2

F

,

~

0

01/

t

i

2.5

3

3.5

315 ~ ,15

4

4.5

(2)

0.5 0.4

r

1 0,96 0.96 0,94 0.9~ 0.9

0 . 8 ~

i

I

I

0.5 1

1.5 2

'

,

,

,

J

Z5

(3)

I

3

-"r-'-

15

i

4

i

4.5

4.4

-~

i

I

I

i

I

I

I

I

i

I

I

I

[

I

0.5

1.5

2

25

3

15

4

4.5

05

t

1,5

2

2.5

3

(~)

(b) Fig.2. al) (g (-), dr (--), (g,)I (--) a2) P,, (-),# (-.), P~ (- -) ~) ,o bl) Vt b2) Control signal Fig. 2a) shows that the predicted equilibrium power angle ((if s)y) is available since t = 1.5s i.e. from the m o m e n t when the mechanical power arrives at its final value. One m a y see that ~ follows perfectly its reference value cir. Actually, cfr m a y be very fast, but the magnitude of the control signal will grow as well. In these simulations, the control signal was kept lower than twice of the necessary to keep the system at its original equilibrium point. If larger controls are acceptable, the system m a y be driven faster to its equilibrium. In Fig. 2b) one m a y see that the estimation of the mechanical power is very fast and accurate, and that the electrical power is correctly driven to the

118

Riccardo Marino, Gilney Damm, and Fran~oise Lamnabhi-Lagarrigue

mechanical one. The electrical power just leaves its equilibrium value during the time that the system is driven for its new equilibrium point. One can see that betbre being driven to its final equilibrium value, ~ is kept at its original one (0.5 < t _< 2). The same may be observed in Fig. 2c) for the rotor velocity. Fig. 3a) shows how the output voltage drops during the fault, and goes to its correct value when the system is driven to the predicted equilibrium point. Note that during the time (1.5 < t < 2) the system is stable, but the output voltage is not anymore the correct one. Finally, one can see in Fig. 3b) that the control signal is very smooth and is kept inside the prescribed bounds.

5

Conclusions

In this work we have computed the zero dynamics of the system with respect to the terminal voltage having then obtained a highly nonlinear second order dynamics. We show by simulations that this zero dynamics may become unstable by a simple change of the input mechanical power. This is our motivation to control the power angle and the relative angular speed as well as the terminal voltage. We then show that the terminal voltage, the relative angular speed and the active electric power are an equivalent set of state variables tbr the system. They have the advantage of being measurable standard outputs from the system. We then present the relation between the terminal voltage and power angle which allows us to avoid measurement of the power angle as well as to compute the new equilibrium angle that produces the correct terminal voltage. Finally we present a nonlinear adaptive feedback control that stabilizes the system to the pre-faulted equilibrium point, recovers the correct value for the mechanical power and then, the correct value of the faulted equilibrium power angle, driving the system smoothly to this new equilibrium point. The system may be kept arbitrarily close to the original equilibrium point and may be driven arbitrarily fast to the new equilibrium point. The only restriction will be tile magnitude of the control signal. As a continuation of this research, the problem of transmission line failure will be addressed. We will as well make a deeper study on the behavior of the zero dynamics of the system with respect to the terminal voltage.

Adaptive Nonlinear Control of Synchronous Generators

6

119

Appendix

Substituting (2) into (3) we have that

q = P~cot(5)

v] Xd~

(16)

Then one will find that

Q2 _

P e2

sin2(5 )

2

Pg - 2Pecot(5) V /

V: + x~--:

where we used the relation 1

cot s = ~sin-= - 1 Then, it is easy to find that

X~ "-" 4- P}) 4- X~---~V}(X~ - X~) 4- 2 (X~ 4-xd~XSXd)p~ cot(5)

x,: P: v} x:+2~X.~p~cot(5) - v,~ sin~(~ + x.~ " ., = Vt2 where the last relation may be verified looking at (8). We then may rewrite

x~ v} ,,~ Vt~ = ~-i(Q2 4- P~ ) 4- -X~s tad - X~) § 2XsP~cot(5) Then, we have

Cot(a)-

1

2X~ P~ Vt2

1

X~(Q:+H)

2X~ P~ V2

1 v y , x ~ x~) 2 x , P~ x~----~~ ~ -

(17) Substituting (17) into (16) one will find that

2V2 Q ~ v~ r Q~ + x , + P: + --~,,~ - v~~)

=o

120

Riccardo Marino, Gilney Damm, and Fran~oise Lamnabhi-Lagarrigue

and consequently we may find its roots. By physical bounds on Q (it must assume positive and negative values while V~ and X~ are positive) we find that

v,/2 Q= - ~ +~ / ~

x~_2 -

v ~

and then we have that

02

=

v:,v2

x~.2~

-

v:vs /.2 x,x.#

- 2 - - - -

v t

-

x~_2

+

": (18)

Substituting (18) into (17) one will find that

V~

E

/

cot(~) = (x--~')(-x-Z" x~ + VV2

2

=(x--~')(-x-Z"x~+VV2

Ks2 A e !

and finally we have that

---Xa

,~ = arccot

+

lit2 -

w / P~

For the equilibrium value, we just replace Pc and Vt by its reference values P,, and Vtr and then we find

~s = arccot

~

- - -Xd8 X~+

V,~-

" ~s P ~2

A c k n o w l e d g m e n t s : T h e second a u t h o r would like to acknowledge the financial s u p p o r t of C A P E S Foundation.

References

1. Bazanella, A., Silva, A. S., Kokotovic, P. V., Lyapunov design of excitation control for synchronous machines, Proc. 36th IEEE - CDC, San Diego, CA, 1997.

Adaptive Nonlinear Control of Synchronous Generators

121

2. Bergen, A. R., Power Systems Analysis, Prentice Hall, Englewood Cliffs, N J, 1989. 3. Gao, L., Chen, L., Fan, Y. and Ma, H. , A nonlinear control design for power systems, Automatica, vol. 28, pp. 975-979, 1992. 4. Krstic, M. , Kanellakopoulos, I. and Kokotovic, P.V. , Nonlinear and Adaptive Control Design, J. Wiley, New York, 1995. 5. Marino, R. , An example of nonlinear regulator, IEEE Trans. Automatic Control, vol. 29, pp. 276-279, 1984. 6. Marino, R. and Nicosia, S., Hamiltonian-type Lyapunov functions, Int. J. of Control, vol. 19, pp. 817-826, 1974. 7. Marino, R. and Tomei, P. , Nonlinear Control Design - Geometric, Adaptive and Robust, Prentice Hall, Hemel Hempstead, 1995. 8. Marino, R. and Tomei, P. , Robust adaptive state-feedback tracking for nonlinear systems, IEEE Trans. Automatic Control, vol. 43, no. 1, pp. 84-89, Jan. 1998. 9. Narendra, K. S. and Annaswamy, A. M., Stable Adaptive Systems, Prentice Hall, Englewood Cliffs, N J, 1989. 10. Pal, M., A. and Rai, V. , Lyapunov-Popov stability analysis of a synchronous machine with flux decay and voltage regulator, Int. J. of Control, vol. 19, pp. 817-826, 1974. 11. Pomet, J. and Praly, L., Adaptive nonlinear regulation: estimation from the Lyaptmov equation, IEEE Trans. Automatic Control, vol. 37, pp. 729-740, 1992. 12. Siddiquee, M., W. , Transient stability of an a.c. generator by Lyapunov direct method, Int. J. of Control, vol. 8, pp. 131-144, 1968. 13. Wang, Y. and Hill, D. J., Robust nonlinear coordinated control of power systems, Automatica, vol. 32, pp. 611-618, 1996. 14. Wang, Y., Hill, D. J., Middleton, R. H. and Gao, L., Transient stability enhancement and voltage regulation of power systems, IEEE Trans. Power Systems, vol. 8, pp. 620-627, 1993. 15. Wang, Y., Hill, D. J., Middleton, R. H. and Gao, L., Transient stabilization of power systems with an adaptive control law, Automatica, vol. 30, pp. 1409-1413, 1994.

Nonlinear Observers of Time Derivatives from Noisy Measurements of Periodic Signals Riccardo Marino and Giovanni L. Santosuosso Dipartimento di Ingegneria Elettronica Universith di Roma Tor Vergata via di Tor Vergata 110 00133 Rome, Italy {Marino, Santosuosso}@ing.uniroma2.it

A b s t r a c t . The problem of estimating the time derivatives of a bounded measured periodic signal of known period T affected by an unknown bounded noise is addressed. As it is known when the measurements are not affected by noise, linear high-gain observers can provide arbitrary small estimation errors for the time derivatives by increasing the observer gains. In the presence of noise there is a limitation on the gains to be used beyond which increasing the gains increases the estimation error as well. We propose a robust adaptive observer which improves the performance of a linear high gain observer in the presence of measurement noise, by estimating a finite number of terms in the periodic signal Fourier expansion.

1

Introduction

Inaccuracy in signal m e a s u r e m e n t s is a critical issue when it is required on-line e s t i m a t i o n of the time derivatives of a m e a s u r e d signal. This arises in m a n y control applications, such as t a r g e t t r a c k i n g (see [2]), and P.I.D. r e g u l a t i o n . In the following we consider a s m o o t h scalar signal f ( t ) E C n+l b o u n d e d together with its derivatives, periodic with known period T, and affected by a m e a s u r e m e n t error w(t), with IIw(t)l I g WMAX and WMAX E ~ + for all t > to, and describe an on-line o b s e r v a t i o n s t r a t e g y for its t i m e derivatives. By setting ~1 = f ( t ) , (~ = f(l)(t) = did(1),... , (n = f(n-1)(t) = ~

,

the

vector ( = [C1,-.-, (hi T is the s t a t e of the s y s t e m

= Ac~ + bcf(n)(t) y = Cc~ + w(t)

(1)

where y(t) E ~ is the system o u t p u t , w(t) E ~ is a b o u n d e d noise, Ac, b~, and Cc are canonical matrices in B r u n o w s k y forms, i.e. A~ =

0

Co= [10...0].

'

=E00 " 11

124

Riccardo Marino and G. L. Santosuosso

System (1) can be seen as a linear system affected by the extended disturbance w , ( t ) = [f(n)(t), w(t)] T affecting both system dynamics and the output available for measurement. The problem of robust linear system observation in a continuous time setting has attracted considerable attention in the control community in recent years, and several criteria can inform the estimation strategy. The celebrated Kalman filtering approach is widely used and offers the optimal filtering algorithm when the power spectral density of the noise is known. In contrast with the conventional algorithms which minimize the variance of the estimation error, when the statistical noise properties are often unavailable, several advances in signal estimation have focused of Hc~ estimation methods (see [13]), which aim at minimizing the peak of the spectral density of the estimation error, as expressed in the context of L2 gain performance measure. Notice that if in system (1) SUpvE[t0 T] ]IW(7")II--~ 0, i.e. measurement error tends to zero, then it can be shown (see [16], [5]) that a suitable linear "high gain" observer can guarantee arbitrary small estimation error. To illustrate its application to system (1), consider the observer = Ac~ + k n S - l ( k ) H ( y

- cc~)

(2)

where k is a positive parameter,

5"(k) =

k '-10 0 0 k n-2 0

...0 ... 0

0 0

o o o liio " H = ["1,

and the positive constants c~i, 1 < i < n are chosen such that the roots of p(s) = s n +cq s n - I + a 2 s n - 2 + . . . + c ~ n have negative real part. If measurement errors cannot be neglected, then by setting ~ = S ( k ) ( ~ - ~ ) estimation error dynamics become = kfI~ + b J ( n ) (t) - k n H w ( t ) ,

the rescaled

(3)

where fI = Ac - H C c . Consider the Lyapunov function V(~) = ~ T p ~ , were p = f~o eATteAtdt. By computing the time derivative of V(~) along the trajectories of (3) and by completing the squares we obtain k

Nonlinear Observers from Noisy Measurements By (4), recalling that "-\~r

125

= S-l(k)~, it can be shown by using standard

Lyapunov arguments in [8] that there exist suitable functions 1~;(-,-) E ]CL i - - 1 , . . . n and positive reals tq,~2 such that for i = 1 , . . . n

(5) sup

I1~(~)11 -

to<_r<_t

If the measurement error is zero, then by inequality (5) it follows that linear high-gain observers can provide arbitrary small estimation errors for the time derivatives by increasing the observer gains, recalling that f(")(t) is bounded. In the presence of noise there is a limitation on the gains to be used beyond which increasing the gains the estimation error increases by a factor k i- 1 for the i - th derivative. Motivated by this arguments we propose a robust adaptive estimation observer which improves the performance of a linear high gain observer in the presence of measurement noise when in system (1) ~1 is a periodic signal with known period T. We consider the Fourier series expansion (see [4]) of the n-th time signal derivative and set

dnf(t) - cT(t)O + ~(t) di n where r r

(6)

e ~P, 0 E ~P, and for 1 < i < ~, = as, sin ( i ~ t )

0~i+1(t) = a 2 i + l c o s ( i ~ t )

e2i = 02i+1 =

~ a"y(t) sin ( i ~ t ) dt dt ~ 2 f [ d"d~t t cos(i~tldt" a21+lT '

a a, T

(7)

The vector 0 E ~P collects the coefficients of the Fourier expansion divided T by the entries of a given rescaling vector a = [ a l , a s . . . ( r p ] . Besides, by ortogonal properties of sinusoidal functions, there exist a positive real k s E ~+, such that the persistency of excitation condition

~

t+T r162

dr >_ k~Ip

for all t > to

(S)

is satisfied. The approximation error r can be made arbitrarily small by a suitable choice of p, and by Bessel inequality a known bound on the norm of the vector 0 can be found. We formalize now the robust adaptive observer properties as follows.

126

Riccardo Marino and G. L. Santosuosso

D e f i n i t i o n 1. Consider system (1). Set a"S(t) _- - r where r dt n ~P, 9 E ~P are defined as in (7). A robust adaptive observer for system

y = Cr + w(t)

E

(9)

is a finite dimensional system

~ = I21(X,y,t),

X E ~ ~, r >_ n,

driven by the input y(t), such that for any to E ~, any x(to) e ~r X(t) is globally bounded, and there exist n functions ~i(', ") E/CZ:, i = 1 , . . . n and positive real numbers 73, 74 such that for i = 1 , . . . n

r

,,-,0)

+73 sup II~(T)ll+Ta sup IIw(r)ll. t0
(io)

to
Since e(t) can be made arbitrarily small by a proper choice of p, (10) has the potential of improving the estimation error by increasing p with no increase of the control gains as in (5).

2

Main

result

We show in this section how to construct robust observers for system (9) according to Definition 1. Let's introduce now the filters (see [3], [10])

~j = - A j ~ j + ~ j + l ,

j = 1,...n-

2,

~j E ~P

(11)

where ~j (0) = 0, and Aj, j = 1 , . . . n - 1 are positive reals. The trajectories of the filters evolve on a bounded set, specifically ~MAX

II~j(t)ll ~ ~ _ ~ ... ~ j , if

IIr

j = n - 1,... 1

(12)

~ CMAX, with #)MAX E ~+ for all t > 0. Set now dn ~

(13)

bc

dj_1 = [Ac + ~j-11n] d~,

j = , - I,...2,

dl -- [I, d12, . . . , dln] T satisfies the equality s n-1 + d ~ s n-2 + . . . + d , . = (~ - ~ )

(s - A ~ ) . . . (~ - ~,_~).

Nonlinear Observers from Noisy Measurements

127

Introduce now the transformation n

z = ~ - Z dj~/_lO j=2

(14)

Note t h a t by (13)-(14), it can be shown proceeding as in [10] that system (9) becomes

= Acz + ditTo + bc~(t)

(15)

y = c~z + w ( t ) .

Consider now the observer z = A ~ + all(T0 + H ( y - Cc~)

Fq(t) (y - Cc~.)

(16)

where ~ E ~n, 0 E ~v, H is chosen so t h a t the triple [Cc, (A~ - HCc), dl] is strictly positive real (see [11]), and F is a s y m m e t r i c and positive definite matrix. Operator q(t) = [q~(t), q 2 ( t ) , . . . , qp(t)] 7" is defined as

{

~ti(t) if 0i
qi(t) =

~li(t) if IOil >ci and ( y - Cc~.)~li($)Oi<_ O

(1 - pi(Oi)) ~li(t)

(17)

otherwise,

where 1 < i < p, pi(O) = ~ $ , - c , , and pi is a positive scalar, ci is the known bound for the uncertain time varying p a r a m e t e r Oi(t). Note t h a t the projection operator (17), is reminiscent of the operator Proj [-,-] defined in [17] and adapted to our case. In particular q(t) is continuous and setting = y-Co5 and0=0-0, and a l l t > t 0

tii _< c~ + u Iqi(t)l < I~xi(t)l , OTq(t)~l > oT~I (t)~l.

w e h a v e (see [17]), if 0i(t0) < c i , for 1 to. By setting now 5 = z - ~, and/J = 0 - 0, by (15), (16), the error dynamics become z = AY. + dl~T(t)O(t) + 4~ew+ = -rq(t)Ce~ =

z(t) -}-

- Fq(t)

[0 1] we.

(21)

dj~T_ 10(t )

where A = Ae - HCc, we = [e(t), w(t)] T , ~e = [be, H ] . The triple (Ce, A, dl) is strictly positive real, so that by Meyer-Kalman-Yakubovic Lemma (see for instance [11]), there exists a symmetric positive definite matrix P satisfying A T p + P A = --ITl -- eQp

(22)

P d l = C cT

for a positive real e, a vector l, and a symmetric positive definite matrix Qp. Consider the Lyapunov function (23) Compute au along the trajectories of system (21), obtaining dV _-- 89 [ P A + AT p] 5 + ~TPq~ew, -Ti+0r~l (t)CeZ -- Orq(t)Ce~+O~r-~ [0 I~] w,.

(24)

By "completing the squares" z-Tpq~eWe < 1~o[[z- 2[1 + P2e ~ [[~e[[2 [[We[[2' where e is any positive real. By property (20), 0T~l(t)Ce~ -- OTq(I)C,5<_ O, which substituted together with previous inequality in (24) and rearranging terms, yields

(25) where dl > 0 if e is chosen sufficiently small, d2 > ~ II+e(t)ll ~ for all t > to, and 63 --- I]F-III . At this point, in order to complete the stability analysis, we need the following result. L e m m a 1. ( E x t e n d e d p e r s i s t e n c y o f e x c i t a t i o n L e m m a ) . Consider the system = A ( t ) x + B(t)O + C(t)we = D ( t ) x + E(t)we

(26)

Nonlinear Observers from Noisy Measurements

129

with bounded input We(t) E ~m, and bounded state (x(t)• t~(t)) e (Bz• B~), where (/3= x B~) C (~" x ~P). Assume that:

i) all matrices in (26) uniformly bounded, the time, with ~ ii) there exist a smooth

i.e. A(t), B(t), C(t), D(t), E ( t ) are continuous and and B(t) is uniformly differentiable with respect to also uniformly bounded; proper function V ( x , z , t ) such that

al (llxll2_t_ 0 2 ) < V ( x , ~ , t , < a 2 (llxll2.k. l0 [2) -

(27)

for all t >_ to, to E ~, and suitable positive reals al, a~, a3, a4, a5 E ~+ iil) there exist two positive reals T, k* E ~+, such that the persistency of excitation condition

ft

t+TBT(v)B(v)d7 - >_ k*Ip

(28)

for allt > to

is satisfied. Then system (~6) is input to state stable (see [14] ), i.e. setting A(t) =

-[xT(t), ~T]Texist-

suitable functions ~(., .) E ICE and r

E IC such that

[ \ IIA(t)ll ~ ,l(llA(to)ll , t - to)+ ~ ( sup I1~(~')111 / \to
(29)

for all t > to, and all A(to) E (B~• B~), and in particular x = O, ~ = O, is a globally exponentially stable equilibrium point when we(t) = O, Vt > to. Proof. Consider the class of radially unbounded functions w ( x ( t ) , ~ ( t ) , t ) = v(x(t),~(t),t) + p3 O(t)~(t) - B r ( t ) x ( t )

(30)

where P3 is a positive scalar parameter to be defined later, and Q(t) is generated following [9], [12], by the filter dQ(t) _ -Q(L) + BT(t)B(t), with Q(to) = e-Tk*I, where k* is the integer defined in (28). By virtue of assumption i i i ) Q ( t + T) >_ e-T f~+r B r ( r ) B ( r ) d r >_ e - r k * I > 0. If cm < [[B(t)[I _< CM for all t > to, where Cm, CM are suitable positive reals then it is straightforward to deduce that dt

--

C2MI >_ Q(t) > k*e-2TI, for all t > t0.

(31)

In view of hypothesis il) al (i]xll2_t_ 0 2) < W ( x , O , t ) < a 6 ( l l x H 2 w [ O i 2 )

(32)

130

Riccardo Marino and G. L. Santosuosso

where

a6 = a2 + 2p3c2Mmax {c~, 1 }.

By computing the time derivative of

By hypothesis

W(z(t), O(t), t) we have

ii)

dWdt_<--a3[Ix[I 2+a4[[w[I 2+a5[]0 I[w[[

By hypothesis i) there exist positive reals fl, f2 E ~+, such that

f >IQD-.T-B A-.TI ; >IIQE-BTC[I y for all t > to. By "completing the squares" we have

(34) q~_ Br~ 2+4p3f, II~II~

< P3 --~ QO- BT z 2 +4paf2IIwe[12(35)

2pa(QO-BTx)T(QE-BTC)we

a5 iiw~l12+~_~_ ~ 2

(37)

where e4 = 2a-(k*)2e-4T. Substituting 2a5 terms we obtain

(34)-(36) in (33) and rearranging

W < a2bt I1~11~ +a~b~ 0 ~ +p3b3 where b~ = ~ [-a~ + p~ (4I~ + d,)]

OO - B T x ,

b~ = _ ~

2 -4-tr

iiw41~ '

(k.)~ ~-4~ b~ =

--~,

/~2 =

a~ bi < 0, for i = 1,2,3. By a4+ ~~e 'r +4paf~. Note that for any p3 < 4]a+c~., setting x~ = mini=t&3 {Ib;I}, we can conclude that W _< - x l W + x2 IIw41 z Recalling (32) it is straightforward to deduce that the system is input to state stable (see [14]), and both 0(t) and x(t) are attracted in an open ball in (~n x NP) centered in the origin with radius

7"~=

tc~a~ (kto
(38)

Nonlinear Observers from Noisy Measurements

131

We are now ready to state the m a i n result of this note.

Proposition

1. Consider system (9), and consider the Fourier expansion of the periodic signal ~ according to (6)-(7). Then (16)-(17) is a robust observer according to Definition 1.

Proof. L e m m a 1 can be applied to the error system (21), by taking into account that the time derivative of (21) along the L y a p u n o v function (23) satisfy (25), and condition (28) in L e m m a 1 is satisfied for B(t) = dl~T(t). T h e persistency of excitation of ~1 (t) is implied (see [15], T h e o r e m 2.7.2) by the fact that ~l(t) is the output of the linear filters (ll)_driven by the input r which is persistently exciting. T h e trajectories of O(t) are a t t r a c t e d in an open ball in ~P centered in the origin with radius 7"d~ ----min {2Cl +/~, ~Ol(~MAX, WMAX) (EMAX dr WMAX)} ,

(39)

where the function P1 (~MAX, WMAX) can be determined by substituting the error system (21) parameters in the expression ~ in (38). By c o m p u t i n g the derivative of system 5 = A5 + dl~T(t)O(t) + 4~we along the L y a p u n o v function VI(~') -- = 89 it can be shown that the trajectories of 5(t) are attracted in an open ball in ~'~ centered in the origin with radius 7~2 = ~2T~w + ~3~MAX + y4WMAX, where ~ P3 Y4 are suitable constants t h a t are independent on ~MAX, WMAX. Recalling the definition of ~, we have t h a t ~i, 1 < i < n is attracted in a set around the origin bounded by

"~-6MAX~03"~ ~04WMAX]

= [(02 "~ 0 5 ) ' ~

(40)

where 05 is a suitable constant that is independent on WMAX, SMAX. By collecting together (39) and (40) it is easy to deduce that the e s t i m a t i o n error of the signal time derivatives is bounded by (10).

3

Simulations

In order to illustrate the proposed e s t i m a t i o n strategy, we have simulated the estimation of the second order time derivative of the signal sin 27r (t - 0.3)

f(t) = e

2

(41)

affected by a measurement error expressed as

w(t) = 0 . 2 5 , 1 0 -3 [(sin 44~-t) 2 - (sin 52zrt) 2 k

(+ (sin 687rt) ~ - (sin 1247rt) 2] . J

(42)

132

Riccardo Marino and G. L. Santosuosso

T h e signal that is actually available or measurement is y(t) --- f ( t ) d- w(t), and observer's task is to estimate

d2f(~)

dt 2 . In this example it has been chosen

a measurement error whose magnitude is relatively "small" with respect to the magnitude of the signal to be observed. In fact, the noise is such that supTe[t0 T] ][w(t)ll ~ 0.001 for all t ~ 0, while supre[to T] I[f(t)ll ~ 1.64 , lated the behavior of a robust adaptive observer (16) according to the strategy in this note, and compared a high gain observer (2). In the observer (16) (17) At = 2, A2 = 4, H = [12, 44, 48] r . The entries of tor [ c l , . . . , c14] T are chosen as follows:

(17) (11), constructed its performance with (11) it has been set vector r and vec-

r = [40 sin 2rrt, 40cos2rrt, 160sin4~rt, 160cos4~rt, 200 sin 67rt, 200cos6rrt, 200sin8rrt, 200cos8rrt, 300sin107rt, 300cos 10rrt, 600sin12rrt, 600cos 12rrt, 1000 sin147rt, 1000cos 14rrt] r , and cl = 20, c2 = 20, c3 = 10, c4 = 10, c~ = 2, c6 : 2, c7 = 0.2, cs = 0.2, e9 = 0.1, c10 = 0.1, cll : 0.1, cx2 = 0.1, c13 = 0.1, c14 = 0.1. The coefficients pi, 1 < i < 14, are chosen as pi = e__t The matrix F 2" has been chosen as a diagonal matrix with F(1, 1) = . . . . F ( 4 , 4 ) = 10, and F ( 5 , 5 ) = . . . . F(14, 14) = 20. In Figure 1 is reported the signal noise for 0 < t < 0.5 seconds. In Figure 2 are reported the signals to be estimated, namely df(t) d2f(t) ~1 = f ( t ) , ~ d t , ~3 d t ~ , the estimates produced by the robust adaptive observer, r

r

~3(t) and the estimation errors for 0 < t < 15. i

i

Notice that for t > 10 sec. 1~'3(t)- r < 0.1, while the observer approaches steady state condition at t = 30 sec. It has been simulated the performance of a high gain observer with H = [12, 44, 48] T, and k = 250, in two different situations. First, the high gain observer has been tested with input reference = f ( t ) with f ( t ) given by (41), t h a t is w i t h o u t measurement error. Then the high gain observer has been given the same input y(t) = f ( t ) + w ( t ) as the robust adaptive observer. In Figure 3 are reported in steady state condition, for 30 < t < 32, adaptive observer estimation errors compared with the high gain estimation errors, respectively without and with noise. The high gain observer yields an error of more than 300 % of the signal to estimated for a noisy input, while the adaptive observer steady state error is within 0.1 %. The poor performance of the linear high gain observer in the presence of noise, by recalling (5) is related to the fact that in the presence of noise there is a limitation on the gains to be used, beyond which increasing the gains increases the estimation error as well. This circumstance suggests t h a t a decrease in the value of k = 250 would be beneficial for a linear high gain observer. Notice however, as reported in Figure 3, t h a t even without noise the high gain observer error in the estimation of C3 is higher t h a t the adaptive observer, being around 3 % i.e. more than 30 times greater t h a n the adaptive observer relative error. A decrease in the value of the high gain k would !

I

Nonlinear Observers from Noisy Measurements

133

therefore yield an increase of the e s t i m a t i o n error of ~3 when the high gain observer e s t i m a t e s the signal w i t h o u t noise. Since e s t i m a t i o n errors in the presence of noise are greater t h a n errors in noiseless situations, this e x a m p l e clearly shows t h a t for signal (41) c o r r u p t e d by noise (42) the observation s t r a t e g y proposed in this note o u t p e r f o r m s any linear observer.

Ill

X I O•

J o

-1 *1s

oce

ol

ols

02

o2s os Tmvtm:l

o3e

04

o,6

os

Fig. 1. Signal noise for 0 < t < 0.5.

,..1.

~0

5

!

10

15

0

5

10

15

5O

0

1.5

!

~0

5

-

!

:

0

10

15

.0

5

10

15

10

15

0

5

10

15

10

15

0

5

10

15

5O

5

10

15

0

5

Fig. 2. On the left are reported ~l(t), On the center are reported ~l(t), r sponding errors, for 0 < t < 15.

r

r

r

from top to bottom respectively. On the right are reported the corre-

134

Riccardo Marino a n d G. L. Santosuosso

x 10"a

10~

x 10~ - ............. i ~ 2i.............

-1

30 xlO~

5

31

..........

32

i

31

32

30

31

32

32

30

31

32

32

30

31

32

- ............. i ~ i1.............

!. . . . . . . . . .

0

30 x 10~

.

-5

30

31

0

"

32

30

0.5

31

. . . . . .

i . . ~. . .

!. . . . . . . .

0:

-0, -0.05 30 Fig. if2 high with

31

32

30

31

3. O n the left are r e p o r t e d the steady s t a t e a d a p t i v e observer errors ~1 - ~1, ~2, ~3 - ~3 from top to b o t t o m respectively . O n t h e c e n t e r are r e p o r t e d the gain observer errors without noise, a n d on the right high gain observer errors noise, for 30 < t < 32.

Nonlinear Observers from Noisy Measurements

135

References 1. Aloliwi, B. and H. K. Khalil. "Robust Adaptive Output Feedback Control of Nonlinear Systems without Persistence of Excitation". Automatica, Vol. 33, No. 11, pp. 2025-2032, 1997. 2. Bar-Shalom Y. and Fortmann, T. E. Tracking and Data Association. Academic Press Boston, 1988. 3. Bastin, G. and M. Gevers, "Stable Adaptive Observers for Nonlinear Time Varying Systems". IEEE Trans. Automat. Contr., Vol. 33, N. 7, pp.1054-1058, July 1988. 4. Bachman G L. Narici and E. Beckenstein "Fourier and Wavelet Analysis." Springer Verlag, Berlin, 2000. 5. Esfandiari F. and H. K. Khalil. "Output Feedback Stabilization of Fully Linearizable Systems". Int J. Contr. Vol. 56, pp. 1007-1037, 1992. 6. Gautier J. P., H. Hammouri and S. Othmann. "A Simple Observer for Nonlinear Systems with Application to Bioreactors", IEEE Trans. Automat. Contr., Vol. 37, pp.875-880, 1992. 7. Isidori, A. "Nonlinear Control Systems". Springer Verlag, Berlin, 3-rd edition, 1995. 8. Khalil, H.K. Nonlinear Systems. Prentice Hall, Upper Saddle River, (N J), 2-nd edition, 1996. 9. Kreisselmeier, "Adaptive Observers with Exponential Rate of Convergence". [EEE Trans. Automat. Contr., Vol. 22, pp.2-8, 197"7. 10. Marino, R. and P. Tomei. "Global Adaptive Observers for Nonlinear Systems via Filtered Transformations". [EEE Trans. Automat. Contr., Vol. 37, N. 8, pp.1239-1245, Aug. 1992. 11. Marino, R., and P. Tomei. Nonlinear Control Design - Geometric, Adaptive and Robust. Prentice Hall, Hemel Hempstead, 1995. 12. Marino, R. and P. Tomei. "Adaptive Observers with Arbitrary Exponential Rate of Convergence for Nonlinear Systems". IEEE Trans. Automat. Contr., Vol. 40, N. 7, pp. 1300-1304, July 1995. 13. Shaked, U. "Ho~ Minimum Error State Estimation of Linear Stationary Process" IEEE Trans. Automat. Contr., Vol. 34, pp.832-847, 1989. 14. Sontag, E. D. "On the Input to State Stability Property". European J. Control, N.1, 1995. 15. Sastry, S. and M. Bodson Adaptive Control- Stability, Convergence and Robustness. Prentice Hall, Englewood Cliffs, N J, 1989. 16. Saberi, A. and A. Sannuti. "Observer Design for Loop Transfer Recovery and for Uncertain Dynamical Systems". IEEE Trans. Automat. Contr., Vol. 35, pp.878-897, 1990. 17. Pomet, J.B. and L. Praly "Adaptive Nonlinear Regulation: Estimation from the Lyaptmov Equation". IEEE Trans. Automat. Contr.,Vol. 37, N. 2, pp.729-740, June 1992.

Hamiltonian Representation of Distributed Parameter Systems with Boundary Energy Flow Bernhard M. Maschke 1,~ and Arian van der Schaft 1 1 Faculty of Mathematical Sciences Department of Systems, Signals and Control University of Twente, P.O.Box 217 7500 AE Enschede, The Netherlands 2 Laboratoire d'Automatisme lndustriel Conservatoire National des Arts et Mdtiers Paris, France {maschko, a. j. vanderschaft }@math .utwente .nl

Recently a port controlled Hamiltonian formulation of the dynamics of distributed parameter systems has been presented, which incorporates the energy flow through the boundary of the domain of the system, and which allows to represent the system as a boundary control Hamiltonian system. This port controlled Hamiltonian system is defined with respect to a Dirac structure associated with the exterior derivative and based on Stokes' theorem. The definition has already been shown to encompass the examples of the telegrapher's equations, Maxwell's equations, the vibrating string, and the one-dimensional compressible fluid. Abstract.

1

Introduction

The Hamiltonian formulation of classes of distributed parameter systems has been a challenging and fruitful area of research for quite some time, which has been very instrumental in proving all sorts of results on integrability, the existence of soliton solutions, stability, reduction, etc., and in unifying existing results. Recently, there has been also a surge of interest in the control of nonlinear distributed parameter systems, motivated by various applications. At the same time, for finite-dimensional nonlinear systems a satisfactory theory has been developed concerning the generalized Hamiltonian modelling of physical systems with external (input and output) variables. This has led to the theory of port-controlled Hamiltonian (PCH) systems [6], [1], [5], [7], aimed at applications in the consistent modelling and simulation of complex interconnected physical systems, and in the design and control of such systems, exploiting the Hamiltonian and passivity structure in a crucial way [7], [4],

[5].

138

B.M. Maschke and A.J. van der Schaft

Recently in [2] we have started to expand the research program on finitedimensional PCH systems to the distributed parameter (or, infinite-dimensional) case. The first idea for doing so is to try to extend the theory as for instance exposed in [3] to distributed parameter systems with external variables (inputs and outputs). However, a fundamental difficulty which arises is the treatment of boundary conditions. Indeed, from a control and interconnection point of view it is quite essential to describe a distributed parameter system with varying boundary conditions inducing energy exchange through the boundary, since in many applications the interaction of the system bwith the environment (e.g. actuation or measurement) will actually take place through its boundary. Obvious examples are the telegrapher's equations (describing the dynamics of a transmission line), where the boundary of the system is described by the voltages and currents at both ends of the transmission line, or the vibrating string (or, more generally, a flexible beam), where it is natural to consider the forces and velocities at one or both ends of the string as the external variables of the system. On the other hand, the treatment of infinite-dimensional Hamiltonian systems in the literature (see again [3]) seems mostly focussed on systems with infinite spatial domain, where the variables go to zero for the spatial variables tending to infinity, or on systems with boundary conditions such that the energy exchange through the boundary is zero. The problem is already illustrated by the Hamiltonian formulation of e.g. the Korteweg-de Vries equation. Here for zero boundary conditions a Poisson bracket can be formulated with the use of the differential operator since by integration by parts this operator is obviously skew-symmetric. dx ' However, for boundary conditions corresponding to non-zero energy flow the differential operator is not skew-symmetric anymore (since the remainders are not zero when integrating by parts). In [2] we have proposed a framework to overcome this fundamental problem, by defining a Dirac structure on certain spaces of differential forms on the spatial domain and its boundary. Then we employ the definition of a port-controlled Hamiltonian system with respect to a Dirac structure, as already given in previous papers (see e.g. [6]) for the finite-dimensional case, to describe implicit PCH systems, in order to formalize distributed parameter systems with boundary external variables as infinite-dimensional PCH systems.

2

Distributed parameter port controlled Hamiltonian systems

In this section we sketch our framework for describing distributed parameter systems as port controlled Hamiltonian systems. Basic ingredients are the identification of a suitable space of energy variables, closely connected to the geometry of the spatial variables of the distributed parameter system, and the definition of a suitable Divac structure on the space of energy variables.

Hamiltonian representation of distributed parameter systems

139

Let N be an n - d i m e n s i o n a l m a n i f o l d with b o u n d a r y ON (of d i m e n s i o n n 1), representing the space of spatial variables. W e d e n o t e by I l k ( N ) , k = 0 , 1 , . . . , n , the space of k-forms on N , and by Ilk(ON), k = 0 , 1 , . . . , n 1, the space of k-forms on ON. T h e d u a l linear space (ilk (N))* can be n a t u r a l l y identified with II"-k(N), by replacing the d u a l i t y p r o d u c t between J2k(N) and ( i l k ( N ) ) * by

(Zla):=f ~^~,

aEilk(N),

~EIln-k(N),

(1)

with A denoting the wedge p r o d u c t of differential forms. C o n s i d e r now as space of energy variables the linear space l,~ defined as follows: l; : = l i P ( N ) x Ilq(N)

•

Iln-q(ON)

(2)

for p and q positive integers satisfying p + q = n + 1.

(3)

By linearity Y is also the space of flows (the rate energy variables). T h e space 11 will be the carrier space for the c o n s t a n t Dirac s t r u c t u r e representing the interconnection structure of d i s t r i b u t e d p a r a m e t e r systems. Its dual space l]* can be identified as above with the linear space

12" ~_ Iln-P(N) x Iln-q(N) x Iln-P(ON)

(4)

(note t h a t (n - 1) - (n - q) = n - p). representing the space of efforts, or co-energy variables, of the system. For simplicity only the s y m m e t r i c case p = q = k is considered, in which case 2k = n + 1, whence it follows t h a t n is necessarily odd. In fact, the two cases of p r i m a r y interest for us will be n = 3, k = 2, a n d n = 1, k = 1. Using the identification above there exists on V x 12" the following n a t u r a l bilinear form (expressing a s y m m e t r i z e d form of power) 1 1 ( ( ( f ~1 , f~f, f~, e~, ~,, 1

1 (f~, ~ ~),

:= ./( (~k ^f~ + ~, ^f~, + ~ ^f~ + ~ , ^ f~,)

+[ (~^fg+~^f~) (e~)

(5)

JO N

with for i -- 1, 2 (Sb, f ~ ) e I l ~ ( N ) • I l k ( N )

f~

E Iln-k(ON)

( e ~ , e ~ t ) E Iln-k(N) x ff~-k(N) ", (ilk(N))* x (Ilk(N)) *

eib

E Iln-2-k(ON) ~-- (ilk-l(ON))*

(6)

140

B.M. Maschke and A.J. van der Schaft

The subscripts "E" and "M" here stand for two different energy domains ("electric" and "magnetic" in the examples of Maxwell's equations and the telegrapher's equations, or "kinetic" and "potential" in the vibrating string or in fluid dynamics), while the subscript "b" stands for "boundary". T h e o r e m 1. [2] Define the following linear subspace of l; x Y*

D - { ( f E , f M , fb,eE,eM,eb) e ]? X ]2" I

[;:] [0( ] [,:]= [o1:] L MIoNJ }

with d : OP(N) --~ F2p+I(N) the usual exterior derivative. Then :D C l; x 1;* is a constant Dirac structure, that is, D = D • with _l_ denoting the orthogonal complement with respect to the bilinear form ( ( , ) ) . The definition of a distributed parameter port-controlled Hamiltonian systems follows immediately. Consider a Hamiltonian density (energy per volume element)

H : a k ( N ) x Ok(N) x N ~ a n ( N )

(8)

resulting in the total energy 7/ := fN H, with gradient vector denoted as grad "H = (JEH, JMH) e (Ok(N) x Ok(N)) *

(9)

Using the above identification we thus obtain the co-energy variables JEH E 12n-k(N) and JMH E O'~-k(N). Now, consider time-functions

(aE(t),aM(t)) 9 Ok(N) x Ok(N),

t 9~

(10)

The k-forms o~n represent the (infinite-dimensional) generalized reOt , OaM Ot locities corresponding to the energy storage in N. They are connected to the ~ , f g = _ 0a_z.~ Dirac structure :D by setting fE = _ oot ot " (The minus sign is included in order to have a consistent energy flow direction.) Setting finally eE : t~Eg, eM = ~MH, in the Dirac structure one obtains D e f i n i t i o n 1. [2] The distributed parameter port controlled Hamiltonian system with manifold of spatial variables N, state space Ok(N) x Ok(N), Dirac structure :D on Ok(N) x Ok(N) x Ok-t(tgY) given by (7), and namiltonian density H , is given as

(_l)oO-kd]f E. ] [el2] = [0101] [6MIIJ[6EH] with fb, eb 9 Ok-t(ON) denoting the boundary variables.

(11)

Hamiltonian representation of distributed parameter systems

141

Note that (11) defines a (nonlinear) boundary control system, with inputs, say, fb, and outputs eb. It immediately follows from the power-conservation property of the Dirac structure that any distributed parameter port-controlled Hamiltonian system satisfies along its trajectories the energy-balance eb ^ h ,

-'~-=

s

(12)

expressing that the increase in internally stored energy in N equals the incoming power via the boundary ON. The definition can be easily extended to cover also distributed ports. Finally, energy dissipation can be incorporated into the framework by terminating some of the boundary or distributed ports by dissipative elements, leading to an inequality in (12).

Example 1 (Vibrating siring). Consider an elastic string subject to traction forces at its ends. The spatial variable z belong to some segment, for instance N = [0, L] of the real line II~. The dynamics of the string arise from the interaction of the elastic-potential energy and the kinetic energy of the string. Let us denote by u(z, t) the displacement of the string. The elastic potential energy is a function of the strain variable, the 1-form given as aE(t) = e(z, t)dz E 121 ([0, L])

(13)

where e(z, t) = b~Tu(z, t). The associated co-energy variable is the stress variable which is the 0-form (function) or(z) E IV~ L]), which is related to the strain variable aE using the Hodge star operator (associated with the canonical inner product on JR) and the characteristic elasticity modulus T as tr = T * a E . The kinetic energy is defined through the energy variable given as the kinetic m o m e n t u m , which is the 1-form

aM(t) = p(z,t)dz E $21 ([0, L]),

(14)

with the co-energy variable being the velocity v(z,t) = ~ u ( z , t ) at z, interpreted as a 0-form v E /'2~ L]), related to the kinetic m o m e n t u m by ~) - " •/1 * CrM, where p is the mass density of the string. The total energy density (sum of the elastic potential and kinetic energy density) is then the following one-form:

Hstring (ClE, C~M) = 1 (a A c~E + V A a M ) ,

(15)

which by definition of the Hodge star product m a y be expressed as a quadratic form on (~E and (~u- The total energy of the string is therefore ~'[string "-~ fN H,tring, the elastodynamic equations of the string may be expressed as

142

B.M. Maschke and A.J. van der Schaft

a d i s t r i b u t e d p o r t controlled H a m i l t o n i a n s y s t e m according to Definition 1 with power flow t h r o u g h the b o u n d a r y (iN = {0, L} o f N a n d b o u n d a r y p o r t variables being the stress at the t e r m i n a l p o i n t s of t h e line: fb = - - ~ l r g a n d the velocity: eb = vl6 N. T h e power balance (12) becomes -

eb ^/b

=

-

v(L)o'(L),

(16)

expressing t h a t the t i m e derivative of the t o t a l m e c h a n i c a l energy is equal to the balance of the mechanical work at the p o i n t s 0 a n d L.

3

Conclusions

In [2] the above framework has been further a p p l i e d to Maxwell's equations and the t e l e g r a p h e r ' s equations for a transmission line. C u r r e n t work concerns coverage of fluid d y n a m i c s models, and the a p p l i c a t i o n of the presented framework to (passivity-based) control of d i s t r i b u t e d p a r a m e t e r systems, e x t e n d i n g the results in e.g. [1,4,5,7]. Finally, a topic of present research is the consistent discretization of d i s t r i b u t e d p a r a m e t e r p o r t controlled H a m i l t o n i a n systems to finite-dimensional P C H systems, with a p p l i c a t i o n s to s i m u l a t i o n .

References 1. M. Dalsmo and A.J. van der Schaft. On representations and integrability of mathematical structures in energy-conserving physical systems. SL4M J. Cont. Opt., 37(1):54-91, 1999. 2. B.M. Maschke and A.J. van der Schaft. Port controlled Hamiltonian representation of distributed parameter systems. In N.E. Leonard and R. Ortega, editors, Workshop on Lagrangian and Hamiltonian methods for Nonlinear Control, pages 28-38. IFAC, 2000. 3. P.J. Olver. Applications of Lie Groups to Differential Equations. SpringerVerlag, second edition, 1993. 4. R. Ortega, A.J. van der Schaft, and B.M Maschke. Stabilization of port controlled Hamiltonian systems. In D.Aeyels, F.Lamnabhi-Lagarrigue, and A.J. van der Schaft, editors, Stability and Stabilization of Nonlinear Systems, volume 246 of LNCIS, pages 239-260. Springer, 1999. 5. R. Ortega, A.J. van der Schaft, B.M. Maschke, and G. Escobar. Stabilization of port-controlled Hamiltonian systems: Passivation and energy-balancing. Submitted to Automatica, University of Twente, 1999. 6. A.J. van der Schaft and B.M Maschke. The Hamiltonian formulation of energy conserving physical systems with external ports. A rchiv ]iir Elektronik und Ubertragungstechnik, 49:362-371, 1995. 7. A.J. van der Schaft. L2-Gain and Passivity Techniques in Nonlinear Control, 2nd revised edition, Springer-Verlag, Communications and Control Engineering Series, London, 2000.

Differentiable Lyapunov Function and Center Manifold Theory Frdddric Mazenc 1NRIA Lorraine, Projet CONGE ISGMP BAt A, Ile du Saulcy 57 045 Metz Cedex 01, France mazenc@cigale, lot ia. fr

Abstract. Continuously differentiable Lyapunov functions for nonlinear systems for which the asymptotic stability can be proved via the center manifold theory are constructed. They are of particular interest when some parameters of the systems are not exactly known.

1

Introduction

It is well known that when the linear approximation at the origin of a timeinvariant system is asymptotically stable, then the system is locally asymptotically stable. When this result cannot be applied to a system, it does not mean that this system is not locally stable. In this case, its local asymptotic stability property can be examined by using the center manifold theory. This theory, exposed in [2] and in [1], has proved its great usefulness in various fields of the nonlinear control theory. As a direct consequence of this theory, there are for instance the theory of singular perturbations and designs of stabilizing control laws for some nonholonomic systems (see [5]). As explained in [1], one of the great advantages of the center manifold theory is that it is entirely independent of one's ability to construct a Lyapunov function, which is a priori not an easy task. However, it is well-known that the knowledge of a Lyapunov function can be of great help when problems such as, for instance, robustness issues or the determination of a subset of the basin of attraction of an asymptotically stable system must be addressed. This consideration motivates the purpose of the present note which is to establish a link between the two major parts of the theory of stability of nonlinear systems that are the center manifold theory and the Lyapunov approach. The proof of the center manifold theorem given in [3] is based on the construction of a Lyapunov function. Unfortunately, the proposed Lyapunov function is not continuously differentiable. This lack of smoothness is a drawback for several reasons:

144

Fr~d~ficMazenc

* The Lyapunov function cannot be used as a tool in a backstepping or a forwarding context (these approaches require the knowledge of at least C 1 Lyapunov functions for some subsystems). 9 When some parameters of the system are not exactly known, the Lyapunov function cannot be constructed (see our example in Section 3). 9 Significant robustness properties cannot be inferred from it. The main result of our work consists of showing how by slightly modifying the Lyapunov function provided in [3] a smooth Lyapunov function is obtained. Observe that the derivative along the trajectories of the Lyapunov function we construct is definite negative only on a neighborhood of the origin. No globM result can be expected unless extremely restrictive assumptions are imposed. We borrow from [4] the technique of proof we adopt. The paper is organized as follows. In Section 2, the main result is stated and proved. An example in Section 3 illustrates the usefulness of our Lyapunov construction in the context of systems with parameters not exactly known. Some concluding remarks in Section 4 end the paper.

Preliminaries. 1. For a real valued C 1 function k(.), we denote by kl(.) its first derivative. 2. We assume throughout the paper that the functions encountered are sufficiently smooth. 3. A function 7(X) is of order one (resp. two) at the origin if for some c > 0, the inequality [7(X)I < cIX[ (resp. [7(X)[ < elXl ~) is satisfied on a neighborhood of the origin. 4. A function V(.) on ~n is positive definite if V(x) > 0 for all x :f: 0 and V(0) = 0. 5. A positive definite function V(.) on ~n is a strict Lyapunov function for the system :~ = ~(X) if

~x (x)~(x) < o, VX r o 2

(1)

M a i n result

Consider the nonlinear system

{ ~ = Mx + a(z,z)

(2)

= f ( x , z) where x E ~"~, z E ~"" are the components of the state, and introduce a set of assumptions.

Differentiable Lyapunov Function

145

A s s u m p t i o n A 1 . The matrix M is Hurwitz and ~z(0, 0) is a critically stable matrix i.e. all the eigenvalues of -~z (0, 0) are on the imaginary axis. Moreover ~(x, z) is a function of order two at the origin. A s s u m p t i o n A 2 . There exists a function h(z) of order two at the origin such that, on the one hand

z(Z)f(h(z),z)

= Mh(z) + a(h(z),z)

(3)

and, on the other hand, two positive definite functions V(-), W(.) such that on a neighborhood of the origin 0V

Oz ( z ) f ( h ( z ) , z ) <_ - W ( z )

(4)

are known. R e m a r k 1. According to Assumption A1, one can determine two positive definite symmetric matrices Q and R such that

QM + MrQ

= -R

(5)

R e m a r k 2. When the Assumptions A1 and A2 are satisfied, one can prove the local asymptotic stability of the system (2) by invoking the center manifold theory. Let us state the main result. 1. Assume that the system (2) satisfies the Assumptions A1 and A2. Then there exists a continuously differentiable function l(.) zero at zero with a definite positive first derivative such that the function

Theorem

U(x,z) -- l(V(z)) + ( x - h ( z ) ) T Q ( x - h(z))

(6)

is a strict Lyapunov function for the system (2). P r o o f . Let us introduce a new variable ~ = x - h(z). Its time derivative is: = M x + . ( x , z) -

z)

on = M~ + a ( h ( z ) , z ) - -5-;(z)f(h(z),z) + Mh(z)

(z)

+ a ( x , z ) - a ( h ( z ) , z ) + ~ ( z ) f ( h ( z ) , z ) - -$-;(z)f(x, oh z) Oh = m~ + a ( x , z ) - a ( h ( z ) , z ) + ~7(z)[f(h(z),z) - f(x,z)] Since the functions a(x, z) and h(z) are of order two at the origin, we deduce that there exists a function r(z, ~) of order one such t h a t

= M~ + r(z,~)~

(8)

146

Frdd4ric Mazenc

On the other hand, there exists a function

g(z,~) such that

= f(h(z),z) + 9(z,~)~

(9)

We construct now a Lyapunov function for the system (2) using the representation (8)(9). One can check readily that the derivative of the function (6) along the solutions of (8)(9) satisfies:

P(~, z) < - l ' ( v o ) ) W ( z ) + ~'(y(z))~(z)g(z,~)~ - C R~

(10)

+2CQr(z, ~)~

Since the function r(z, ~) is of order one, on a sufficiently small neighborhood of the origin the inequality

2~r Qr(z,~)~ _< l { ' r R~

(11)

holds. On the other hand, when a _>

:

WR-T[,

F(V(z))~7(z)g(z,~) ~ < 4Aa]~[~ + a [l'(V(z))~z (z)g(z,~)] 2 <_ 88 pg + . [~'(y(z))~(z)g(z,~)] ~

(12)

Combining (10), (11), (12), we obtain

~f(x,z) <_ - l ' ( V ( z ) ) W ( z ) + a

[

l'(V(z))

(z)g(z,~)

-

CPg

(13)

Since W(z) and V(z) are positive definite functions, one can determine continuously differentiable strictly increasing functions zero at zero ai(.), i -- 1 to 3 such that:

w(z) > m(Izl),

~OV (z ) < ~(1~1), y(z) < ~3(Izl)

(14)

According to (13) and (14) the inequality

gr(x,z) < - l l ' ( V ( z ) ) W ( z ) - 1 ( r Pff,

(15)

is satisfied if: ~l'(~(l~l))~2(Izl)21g(z,~)l

~ <

1~:(1~1)

(lO)

On any neighborhood of the origin, there exists F > 0 such that [g(z,~)l 2 < F. As a consequence, the previous inequality holds if, on a neighborhood of the origin,

I

~: (~:(s))

g(,) _< 2aFa2(a:l(s))2

(17)

Differentiable Lyapunov Function

147

Since the functions ai(-)'s are zero at zero, stricly increasing and continuously differentiable, a function l(.), zero at the zero, continuously differentiable, with a definite positive first derivative and such that the previous inequality is satisfied can be easily determined. This concludes the proof.

3

Example

In this section, we illustrate by an example how Theorem 1 can be used to construct a Lyapunov function when some parameters are not exactly known. Consider the following two-dimensional system [3, Example 4.15]

{ x -~ --x q- az ~

(18)

Z~ZZ

Our objective is the construction of a s m o o t h Lyapunov function in the case where a is approximately known. We suppose t h a t a = b-b ~(t) where ~(t) is an unknown continuously differentiable function such that ~(t) E [ - e , e ] and t h a t b <= 0 and e are known. To simplify, let b -- - 1 . In [3, Example 4.15], it is shown t h a t the solution of the center manifold equation

h'(z)[zh(z)] + h(z) - z 2 = 0

h(O) -~ h'(O) = 0

(19)

is h(z) = z ~ +O([z[3). This expression of h(z) leads us to perform the change of coordinate ~ = x -b z 2, which transforms (18) into

{ ~ =-~+-C(t)z~+2z2~-2z4-z 3 + z~

(20)

R e m a r k 3. LFrom the proof of [3, T h e o r e m 4.15] we deduce that the derivative of function

= 2z +

(21)

along the solutions of (20) is negative definite when E(t) = 0. But one can check that it is not so when ~'(t) r 0 for some t _> 0. Moreover, even if we assume that Y(t) is exactly known, no Lyapunov function can be deduced from [3, Theorem 4.15] since the sign and the size of ~(t) are unknown. By applying Theorem 1, one can prove t h a t the function

U(z,~) = ~ 2 + lz2

(22)

148

Fr6d6ric Mazenc

is a strict L y a p u n o v function for (18), when e is s m a l l e r t h a n a constant we will d e t e r m i n e . T h e derivative of V(z, ~) along the solutions of (20) satisfies (y(z,~) = _ ~ 2 _ ~4 + z ~

<-

-

+ -g(t)~z 2 + 2z2~2 _ 2~z 4 ( 89

4 +

-

2

(23)

z4

It follows t h a t when ~ < 89then U(z,~) < - ~

_

1 4 ~z

(24)

on the n e i g h b o r h o o d of the origin defined by ]{I < ~ , Iz] < 88

4

Conclusion

We have a d d e d a result to the collection of the c o n s t r u c t i o n s of L y a p u n o v functions available in the literature. We have e s t a b l i s h e d a link between the center m a n i f o l d theory and the L y a p u n o v a p p r o a c h . T i m e - v a r y i n g and discrete-time versions of T h e o r e m 1 can be proved.

References 1. D. Aeyels (1985) Stabilization of a class of nonlinear systems by a smooth feedback control. Systems & Control Letters, vol. 5, pp. 289-294, . 2. J. Carr (1981) Application of Centre Manifol Theory. Springer Verlag. 3. H. Khalil (1996) Nonlinear Systems. 2nd ed. Prentice Hall. 4. F. Mazenc (1998) Cascade of Unstable Nonlinear Systems, Local and Global Stabilization. Systems & Control Letters, vol. 35, pp 317-323. 5. A. Teel, R. Murray, G. Walsh (1992) Nonholonomic Control Systems: From Steering to Stabilization with Sinusoids. Proc. of the 31st Conference on Decision and Control. Tucson, Arizona.

Controlling Self-similar Traffic and Shaping Techniques Rafil J. Mondrag6n C 1, David K. Arrowsmith 2, and Jonathan Pitts 1 i Dept. of Electronic Engineering, 2 Mathematics Research Centre Queen Mary and Westfield College London E1 4NS, United Kingdom r . j . mondr agon@qmw, ac. uk

The paper considers shaping control of a two-queue network. It has been postulated that shaping has no effect on the network performance for long range dependent (LRD) traffic. The two-queue network has shaping in one queue of the two-queue network which vastly improves the overall performance without removing the LRD properties. In particular, a critical value of the shaping parameter is found which changes the decay of one of the queues from power law to exponential. Abstract.

1 Introduction Packet traffic that is self-similar and has long range dependence (LRD) can produce queue lengths that decay as an stretched exponential (Weibull law) [1,2] or as a power law [3,4]. For a small number of aggregated heavy-tailed ON traffic sources, the queue decay as a power law. As the number of sources tends to infinity the queue decays as an stretched exponential [5]. For power law decaying queues it has been shown that the LRD property is a necessary but not sufficient condition to create slow decaying queues. It has been noticed that some LRD traffic produces exponential queue length distributions similar to that of short range dependence traffic. In particular, if the ONsojourn time decays as a power law and the queue service is deterministic, the queue length decays as a power law [4,3,6]. This slow decay of the queue length probabilities (Weibullian or Pareto) has important implications in the development and implementation of a packet based network, for example, in the provision of buffer size [1]. The high buffer occupation is not necessarily cured by increasing the buffer size as self-similar traffic may still eventually saturate the buffers [1,7]. The large occupation of the queues produces waiting times that are unacceptable for time-sensitive traffic (voice, video) degrading their quality of service (QoS). There are several ways to modify the traffic to improve the QoS, for instance, traffic shaping, resource management, call admission control and priority control. These network operations do not remove the self-similar traffic properties [8] and so their effect on the QoS is limited. Furthermore, Erramilli,

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Radl MondragSn et al.

Narayan and Willinger [9], using flow balance in the Fourier domain, have shown that the queue length diverges for traffic with stretched exponential queue probabilities when the output from the shaper is less singularly long range dependent than the input. We consider here the case where the queue length probability decays as a power law which depends on the ON sojourn-time of the traffic and not on the LRD of the traffic. This is the case when aggregating a small number of heavy-tailed ON traffic sources. A possible way to reduce the waiting time that the traffic spends in the network is to reduce the ON sojourn time and consequently the buffer occupation. The control method that we propose is based on the modification of the ON sojourn time from a power law decay to an geometrical decay. The control modifies the outgoing traffic of the queues. We show that the ON-sojourn time of the output traffic can be modified without overflowing the queue. The control is based on the assumption that the network consists of several links that are joined by switches/buffers/queues. As the control is applied at the output of the queue this implies that the control has a negative effect on the first queue that can be nullified by gains in other queues of the network.

1.1

L o n g R a n g e D e p e n d e n t Traffic a n d P a c k e t T r a i n s

Long range dependent traffic can be represented with an O N / O F F where the O N / O F F sojourn time distributions decays as a power The aggregation of O N / O F F sources with long tail in the O N / O F F times produces self-similar traffic that tends to fractional Brownian (FBM) as the number of sources increases [5].

model, law [9]. sojourn motion

For one source, the traffic is modelled with an O N / O F F source described by the indicator random variable 01 Yn = y ( x . ) =

O < xn < d (OFF-no packet/cell) d < xn < 1 (ON-packet/cell)

(1)

where xn is a stochastic variable with specific probabilistic characteristics. These O N / O F F models are also known as packet trains [10]. The self-similar property and the long range dependence are studied by considering the batch variable

zL(j) = y ~ Yo-1)L+i, i----1

(2)

Controlling Self-similar Traffic

151

that is the cumulative number of packets/cells generated after L iterations. The average number of packets is E(ZL) = N-*~lim~ ~'=

i=l Y(j-1)L+I

)

= L E ( y ) = L)~

(3)

with variance var(zL) = N-*~lim~ ~'=

i=1 Y(j-1)L+I -- L~ L

L

---- L~(1 - L~) + 2 E Z

E(yjyi)

(4)

j = l i>j L--1

= L~(1 -- L ~ ) + 2)~ Z ( L

-- i)C(i).

i=1

The autocorrelation function decays as the power law, C(i) ~ a i -~ + ,~, if L -+ oe then var(L) L ~ L2_ ~ and the Hurst parameter is H = (2 - / 3 ) / 2 (here f ( t ) t - . ~ g(t) means that f ( t ) / g ( t ) -+ C as t -+ (x) and C constant). 1.2

T r a i n l e n g t h s and heavy tailed distributions

The ON/OFF packet trains can have ON- and/or OFF- periods that last for a very long time with non-negligible probability. Asymptotically, for large packet-trains, these distributions are power law distributions (Pareto law). More precisely, for the ON-events (O) the probability of generating a packet train of ONs of length L >> 0 is, [11-15],

~D(O __~L) L~flo L _ S

1 < ~ < 2.

(5)

1
(6)

For the OFF-events (Z) ~O(~.

_~ L) L ~

L_Z,

The above distributions are heavy tailed distributions. The auto-correlation is C(L)

(i-"

L

L -~

>/3 if/3 > a.

(7)

Hence the autocorrelation decays as the power law C(L) ~ ~ + a L - 7 , with 7 either a or fl, meaning that the traffic produced by the map is long range dependent. The Hurst parameter is H=

(2-a)/2 (2 /3)/2

if or_>/3 if / 3 > a . '

(8)

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Radl Mondrag6n et al.

The aggregation of traffic sources produces active/silent periods of traffic activity. An active period occurs when at least one of the sources is emitting and the silent period is when all the sources are not emitting. We do not consider the case when the number of aggregates tends to infinity and for which the silent periods disappear. The traffic statistics are then represented by a fractional Gaussian model. For N identical heavy tailed O N / O F F sources the decay of the probability of an active region of length L :>:>0 is 7~(A > L) L~c~ L_~.

(9)

for a silent region is [16] P ( S >_ L) L Z ~ L-OL -(N-1)(#-u)

(10)

As the traffic sources are independent and identical, the aggregated traffic decays as a power law and the Hurst parameter is the same as for the individual sources.

1.3

Queue dynamics and performance

We consider a deterministic queue with infinite capacity which is fed by N independent O N / O F F sources. The queue's dynamics is split into silent periods, where there are no arrivals, and active periods where at least one of the sources is active. These periods are obtained by the aggregation of N independent sources. The queue length dynamics, Qn, of a deterministic server is described by Q , + l = Q,~ - O(Qn) + M , + l

(11)

where Mn+l is the number of arrivals at time n + 1, O(Qn) is the Heaviside function O(Qn > 0) = 1 and O(Qn = 0) = 0 and it describes the queue departures. We are considering an infinite queue and so we take the equilibrium case where the average traffic queue output is equal to the cumulative average input traffic, that is )~0 = N)~. Moreover, the queue is a low pass filter and this implies that the high frequencies of the input traffic are modified by the queue. However, the long range dependent properties of the traffic are given by the low frequencies (long times) of the traffic which are not modified by the queue. Hence to first approximation, the output map has the same LRD properties as the aggregated traffic streams given by Eqs.(9, 10). The traffic and queue behaviour is modelled using chaotic maps (for a review of this approach, see [17]). An O N / O F F traffic source is modelled with the

Controlling Self-similar Traffic

153

map xn+, = Y'(xn; d) = {}'l(Xn) = xn + ((1 - d ) l d m ' ) x m' 9T 2 ( x n ) -- Xn - (d/(1 - d)m2)(1 - x n ) m=

O<xn

d<xn
(t2) where x n E (0, 1) and .Ti(x), i = 1, 2 are continuous functions and d is a parameter that is used to discriminate between the functions and its associated indicator r a n d o m variable that models the O N / O F F behaviour {~

O < Xn < d, d < xn < 1,

Yn = y ( X n ) =

O F F (no packet/cell) ON (packet/cell)

(13)

One of the most attractive features of using chaotic m a p s is the concise description of the complex packet traffic. In c o m m o n with measured packet traffic, the maps can generate self-similar traffic with long range dependence, where the Hurst parameter is a simple function of the maps parameters [11,18,16]. The maps can model traffic with short range dependence (Poissonlike) and also traffic with heavy tailed ON a n d / o r O F F periods. As the properties of the tails of the O N / O F F periods can be manipulated individually the maps can produce multi-fractal traffic [15] as well as mono-fractal traffic (like fractional Brownian models). Chaotic m a p s can also model aggregated traffic when several self-similar traffic streams combine at a single node. T h e aggregated traffic can then be modelled with two chaotic maps which preserve the traffic load and the long range dependence of the aggregates [19,15].

2

The Selector

Our control m e t h o d is based on the observation that the buffer length probability depends on the properties of the sojourn active time of the aggregated traffic and not on the properties of the decay of the auto-correlation function. If the behaviour of the sojourn time changes from a power law to a geometrical decay then the queue length changes from a power law to an exponential decay. Ql

heavy-tailed traffic

,. selector

Fig. 1. The selector, shown as a diamond, decides if a packet is retained or transmitted from the queue Qx. This decision is taken by the selector map s

154

Radl Mondrag6n et al.

I

L'

L I

I

I I I I I I I I I I I,_:~:,~,=~

IIIIIIIIII

(a)

I

nmT rm_,-,

(b)

empty slots/packets introduced by the selector

(c)

~mm

n

~m Fmf] mD m lb._heavy ,,-:oo, traffic []J

time

Fig. 2. The selector breaks a packet-train (a) of length L in a sequence of shorter trains. The length probability of the shorter trains decays geometrically; (b) If the selector splits the packet-train seldom then the small trains "travel together"; (c)lf the queue never empties, then the dynamics of the size of the small trains is fully determined by the selector The selector control modifies the traffic at the output of the queue and decides if a packet is transmitted or not. T h e decision in the selector is taken using the m a p s defined by

s, = s(x,)

=

01 0 < x , < p p < xn < 1

R E T A I N E D , packet/cell T R A N S M I T T E D , packct/cell,

(14)

The only parameter of the control is the ratio p : 1 - p between ret a i n e d / t r a n s m i t t e d traffic. Figure 1 shows the selector control. If the outgoing traffic is "heavy" tailed, the selector modifies the traffic from heavy tailed to exponential decay. Thus the selector changes the service mechanism of the queue from deterministic to be effectively " r a n d o m " . If the queue o u t p u t traffic is represented by the O N / O F F model

q" =

01 Q , = 0 Qn > 0

O F F , no packet ON, packet,

(15)

then the modified output traffic from the two-queue network is given by z. = qns..

(16)

Obviously, if the selector retains packets in the queue, then the average size of the queue length increases which can be arbitrarily large. T h e percentage number of "transmitted" packets is given by g = 1 - p and the percentage of "retained" packets is 1 - g = p

Controlling Serf-similar Traffic

2.1

155

Modified Train Packets and Long Range D e p e n d e n c e

In the case that the output traffic is a heavy tailed ON-train, the selector breaks the long train into shorter trains. The length of these smaller trains is distributed geometrically. If the ON-trains are seldomly broken then the smaller trains that are created "travel together" (see Fig. 2). In this case the modified traffic scales as follows: L--1

WL(j) = E

Z(J-t) L+i'

(17)

i----1 is the cumulative number of packets/cells generated after L iterations. The average number of packets is

Z(J-1)L+i

E(WL) ----N-~c~lim~ E "=

= L E ( w L ) = L2 = LA

(18)

\i=1

that is the same as the uncontrolled network, because we are considering infinite buffers. The variance

var(wL) = lim

1 ~_~ L_ 1

N--*oo'i

j=l

(i~=l z(j-I)L+i-L/k L

)2

(19)

L

= L~(1 - L~) + 2 ~

~

E(w~w,)

i>j

j=l

The function E(wjwi) is given by

E(wjwi) = P(wj -- 1,w{ = 1) = P ( w j -- 1)P(wi = llw j --- 1) = ~p(s,

= 11~5 =

1) p ( y ,

=

lly~ = 1)

(20)

where P ( A I B ) is the probability of A given B. If C(n) - P(yi+, = llyi = 1) then L-1

var(zL)

~,

LA(1 - LA) + 292A E ( L

- i)C(i)

(21)

i----1 where C(i) ~ A + JL -~. If p in (14) is zero then the variance is not modified and the traffic is long range dependent. If the load and p are small, the small trains then travel together and the autocorrelation still decays as a power law and the modified traffic is LRD (see fig. 2). If the queue never empties, because the selector is retaining too much traffic or because the traffic load is high, then the statistical properties of the output traffic is completely determined by the selector. In this case, the output traffic

156

Raft] MondragSn et al.

is effectively a random process and the size of the ON packet trains decay geometrically and the packet train lengths are uncorrelated (see Fig. 2c). The only drawback is that usually this happens when the queue is growing unboundedly. These two extremes of the output tail behaviour do not occur abruptly as the selector is switched on (see Fig. 3).

O 0.I

0

oe o + ++

v

QD

Q

0.01

O.O01

. . . . . . . .

'

I0

. . . . . . . .

R

I00

lOGO

log(L)

I0000

Fig. 3. Change of the autocorrelation function as a function of the selector parameter p for LRD traffic with an ON heavy-tailed, the autocorrelation decays as L -~ and traffic load of 0.2. If p is small (p ---- 0.01) (top graph) the decay of the autocorrelation is a power law L -~ . As p increases the correlation tends to a horizontal line. For p large (not shown in the graph) the autocorrelation is zero The advantage of introducing the selector is not observed in the performance of queue Q1 but in the following queues that the traffic visits.

3

A two-queue

network

and

its performance

We now consider the effects of the selector modified traffic in the queue Q1 of the two-queue network shown in Figure 4. The two traffic streams that arrive at queue Q2 are called the "shaped" traffic(output traffic of Q1) and the "independent"traffic respectively. The in-going traffic of Q~ is given by the aggregation of N (where N is small) ON heavy-tailed sources. The length of the second queue, Q2, depends on the properties of the in-going traffic at both queues. Thus, even though the selector creates a traffic output from queue Q1 with geometrical decaying ON sojourn time, the aggregated traffic that arrives at the second queue could be heavy tailed and then, in this case, Q2 decays as a power law. There are two possible scenarios for this: (a) the independent traffic has ON-train lengths that decay as a power law then the length of Q2 decays also as power law;

Controlling Self-similar Traffic

157

(b) the independent traffic has an ON-tail that decays geometrically but when aggregated with the shaped traffic, the aggregated traffic is heavy tailed and the Q2 length decays as a power law. To understand the case (b) we assume that the first queue is fed with several ON heavy-tailed sources, the shaped output of QI is LRD (p and NA are small) and the second queue is fed jointly by the shaped traffic and by the selector process. Q,

LRD traffic X2 "independent" traffic

Q 2

Fig. 4. Two-queue network

As was previously mentioned, the selector cuts the train of ONs into small trains that travel together. The selector will break the output traffic train of Q1 into small packet trains. If the length of the original train is L the selector introduces Lp empty packets such that there are Lp + 1 small trains traveling together. As the service rate of Ql is deterministic these small trains will pass the second queue in L(1 + p) units of time. In this period of time the independent traffic produces LA2 packets where X2 is the load of Q2. The condition LX2 > Lp means that, as both traffic merge at Q2, the empty spaces created by the selector are filled in by the independent traffic, creating again a large train that is LRD. If LX2 < Lp then the empty spaces created by the selector are not filled and the aggregated trace is still given by a set of trains, perhaps larger than the ones created by the selector, whose lengths are geometrically distributed. In this case the length of Q2 decays as an exponential. This approach gives a critical value to decide how often the selector should retain a packet in Q1. The selector on queue Q1 creates on average p < A2 spaces between traces and then the length of Q2 does not change. Thus it still decays as a power law. For the case p > A2, it is not possible to recover the ON sojourn times of the non-shaped traffic and the queue length decays are given by the properties of the independent traffic. This reasoning suggests a critical value is p = X2. An example of this transition is shown in Fig 5 where the first queue is fed by four independent traffic sources, modelled by four intermittency maps, with a power law decay in their ON sojourn time P ( O N = K) ~ a K -22~, where a is a constant, and geometrical OFF sojourn time. The average load

158

Raftl Mondrag6n et al.

of the aggregated traffic is ~1 = 0.5. The second queue is fed by the output of the first queue and the independent traffic that is given by a random traffic stream with load A2 = 0.2. Figure 5(a) shows the queue length distributions when the selector is off. The first queue (diamonds) and second queue (crosses) decay as :P(x = Q) Q-1,2s. The decay is the same because the first queue does not change the heavy-tailed properties of its in-going traffic and the second queue length behaviour is determined mainly by the properties of the outgoing traffic of the first queue. This is an example which shows that the queue, acting like a low pass filter, does not change the properties of the LRD traffic. Figure 5(b) shows the effects of the selector in the case p < As. The queue length probability of the first queue increases due to the retained traffic. The second queue still decays as a power law because the aggregated traffic is heavy-tailed. Figure 5(c) shows the case of the critical value p = A2. The numerical experiment shows large fluctuations when evaluating the queue length probability. Figure 5(d) shows the case p > As where the behaviour of the second queue has change from power law to exponential decay. Clearly the traffic that only uses Q~ notices a great improvement in the performance of the queue when p > Az. Its waiting time (average queue size) decreases drastically. Note that the increase of Qi as p changes is moderate. The probability that the queue length is 1000 changes from 10 -5 for p = 0 to 10 -4.5 for p = 0.25.

4

Conclusions

We have demonstrated that long range dependence of self-similar traffic is not a difficult restriction for improving the quality of service of time-sensitive packet traffic. We have shown that one of the major bottlenecks in packet traffic, i.e. the average queue lengths, can be modified by changing the duration time of traffic activity without destroying the long range dependence. The benefit of this modification is that the queue length probability distribution changes from a power law to an exponential, meaning that the average queue size decreases drastically. We have implemented a simple control mechanism, based on the modification of the outgoing traffic of a queue, which shows that it is possible to change the behaviour of a consecutive queue. The control is a simple mechanism which breaks the traffic streams using a chaotic map. The single parameter of the selector m a p is related to how often a traffic stream is broken into smaller pieces. There is a critical value of this control parameter where the queue length probability changes abruptly from a power law to an exponential.

Controlling Serf-similar Traffic

159

w

o ##++++

~-2

*7

0

I

0.5

1.5

2

2.5

2

2.5

~

21,

3

log(L) (a) 0

,-1-2

$ $r

&

4

05

l

I5

log(L) (b) O, -1 M

~-~ N4 -5

I

4 0

oi,

i

,i, log(L) (c)

* 2 ~,~,::+.

~,-2 II ~4

§

-5

+ -7

E

log(L) (d)

F i g . 5. Queue length probabilities for queues Q, (diamonds) a n d Q2 (crosses) w h e n (a) the selector is off p ---- O, (b) the case p -- 0.1 < A2, (c) t h e case p -- 0.2 = A2 a n d (d) p = 0.25 > A2

160

Ratll Mondrag6n et al.

Acknowledgments T h e authors gratefully acknowledge the s u p p o r t of the UK E P S R C u n d e r grant GR/L78659.

References 1. I. Norros. (1993) Studies on a model for connectionless traffic, based on fractional brownish motion. Conf. On Applied Probability in Engineering, Computer and Communication Sciences, Paris:16-18 2. O. Narayan. (1998) Exact asymptotic queue length distribution for fractional brownish traffic. Advances in Performance Analysis, 1 (1):39-63 3. O J. Boxma and J. W. Cohen. (1998) The m/g/1 queue with heavy-tailed service time distribution. IEEE Jou. on Selected Areas in Communications, 16, No.5:749-763 4. R. J. Mondragdn. (2000) Intermittency maps and queues: Modelling self-similar traffic and its performance, preprint 5. M. S. Taqqu, W. WiUinger, and R. Sherman. (1997) Proof of a fundamental result in serf-similar traffic modelling. Comp. Comm. Rev., 27, No. 2:5-23 6. H.P. Schwefel and L. Lipsky. (1999) Impact of aggregated, self-similar on/off traffic on delay in stationary queueing models. Spie Conference on Performance and Control of Network Systems III, Boston, Mass., 3841:184-195 7. C. Huang, M. Devetsikiotis, I. Lambadaris, and R. Kaye. (1995) Fast simulation for self-similar traffic in atm networks. IEEE ICC95 Seattle, Washington, pages 438-444 8. P. Pruthi and A. Popescu. (1997) Effect of controls on self-similar traffic, in Proceedings of the 5th IFIP A T M Workshop Bradford, UK 9. A. Erramilli, O. Naranyan, and W. Willinger. (1996) Experimental queueing analysis with long-range dependent packet traffic. I E E E / A C M Trans on Networking, Vol 4, No 2:209--223 10. R. Jain and S. A. Routhier. (1986) Packet trains: Measurements and a new model for computer network traffic. IEEE Journal on Selected Areas, 4:986-995 11. A. Erramilli, R. P. Singh, and P. Pruthi. (1994) Chaotic maps as models of packet traffic. In Proc. I T C 15, The Fundamental Role of Teletra]fic in the Evolution of Telecommunication Networks, pages 329-338 12. A. Erramilli, P. Pruthi, and W. Willinger. (1994) Modelling packet traffic with chaotic maps. I S R N K T H / I T / R - 9 4 / 1 8 - S E , Stockholm-Kista, Sweden 13. P. Pruthi and A. Erramilli. (1995) Heavy-tailed on/off source behaviour and serf-similar traffic. Proc ICC 95 14. H. G.Schuster. (1995) Deterministic Chaos An Introduction. 3rd. Ed. VCH Verlagsgesellschaft, Weinheim Germany 15. R. J. Mondragdn. (1999) A model of packet traffic using a random wall model. Int. Jou. of Bif. and Chaos, 9 (7):1381-1392 16. R. J. Mondragon, D. Nucinkis, and D.K. Arrowsmith. (2000) Aggregation of lrd traffic using chaotic maps. in preparation 17. A. Erramilli, R. P. Singh, and P. Pruthi. (1995) An application of deterministic chaotic maps to model packet traffic. Queueing Systems, 20:171-206

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18. L. G. Samuel, J. M. Pitts, R. J. Mondrag6n, and D. K. Arrowsmith. (1998) The maps control paradigm: using chaotic maps to control telecoms networks. In Broadband Communications, The Future of Telecommunications, Eds. Kiihn P. and Ulrich R., Chapman and Hall, London, pages 371-382 19. L. G. Samuel, J. M. Pitts, and R. J. Mondrag6n. (1997) Towards the control of communication networks by chaotic maps: Source aggregation. In 1Ioi

2b, Proc.ITC15, Teletraffic Contributions for the information age, eds V. Ramaswami and P.E. Wirth, Elsevier, Amsterdam, pages 1369-1378

Diffusive Representation for Pseudo-differentially Damped Nonlinear Systems G4rard Montseny 1, Jacques A u d o u n e t 2, and Denis M a t i g n o n 3 1 LAAS/CNRS 7 avenue du Colonel Roche 31077 Toulouse cedex 4, France, montseny@laas, fr

2 MIP/CNRS, Universit4 Paul Sabatier 118 route de Narbonne 31068 Toulouse cedex 4, France, audounet ~mip. ups-t ise. fr

3 ENST, URA CNRS 820, dpt TSI 46 rue Barrault 75634 Paris cedex 13, France, mat ignon~t s i. enst. fr

A b s t r a c t . A large class of visco-elastic and elasto-plastic systems, frequently encountered in physics, are based on causal pseudo-differential operators, which are hereditary: the whole past of the state is involved in the dynamic expression of the system evolution. This generally induces major technical difficulties. We consider a specific class of pseudo-differential damping operators, associated to the so-called diffusive representation which enables to built augmented state-space realizations without heredity. Dissipativity property is expressed in a straightforward and precise way. Thanks to state-space realizations, standard analysis and approximation methods as well as control-theory concepts may therefore be used.

1

Introduction

Visco-elasticity and elasto-plasticity are difficult to take into account in control theory: modelling is most of t i m e delicate and control of such s y s t e m s is quite an open problem. In p a r t i c u l a r the presence of discontinuous functions generates non regular trajectories and therefore implies the possible existence of fast or " a b n o r m a l " behaviors in the solutions. However, a c c u r a t e descriptions of such singular p h e n o m e n a are of great i m p o r t a n c e in m a n y concrete s i t u a t i o n s [6], [8], because they involve non negligible energy transfers and, due to non-linearities, have consequently significant effects on the s y s t e m evolution at slow time-scale. A large class of such systems, frequently encountered in physics, are based on causal pseudo-differential o p e r a t o r s , s o m e t i m e s with l o n g - m e m o r y characteristics: classical examples are fractional derivatives or integrals a n d various

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combinations of them [9]. Pseudo-differential operators are hereditary: the whole past of the state is involved in the dynamic expression of the system evolution. This generally induces major technical difficulties. Furthermore, from the thermodynamical point of view, consistence of the model is a difficult question in most cases. The pseudo-differential operators under consideration here are those which admit a so-called diffusive representation i.e. which can be simulated by using a non hereditary operator of diffusion type in a an augmented state space. Analysis and approximation as well as control of the related models are then performed in the context of this representation with classical tools of applied mathematics. The dissipativity of the models admits a quantitative evaluation by the way of internal (hidden) variables associated with the augmented state space. The paper is organized as follows. In section 2, we state the problem and the associated framework. The definition of pseudo-differential damping is given in section 3. The section 4 is devoted to a constructive approach of diffusive representations in the perspective of concrete analysis and numerical simulations. In section 5, we state and prove the main result of the paper. This result enables to transform the initial hereditary problem into a Cauchy one with infinitesimal generator and energy functional. In section 6, we apply this result to the problems introduced in section 2 and we exhibit the specific properties generated by the diffusive formulation, namely about asymptotic behaviors. Through numerical simulations, we finally treat an example of pseudodifferentially damped second order system in section 7, in order to illustrate the efficiency of the approach from the point of view of approximations.

2

Framework

Let g a real separable Hilbert space with scalar product (.[.)e, V a potential, and:

w/o (R+;E)

L,o% (R+;Z)

(1)

a causal and continuous hereditary (the whole past X[0,t] of X at time t is involved in 7-l(X)(t)) non linear operator [2]. We consider the following autonomous functional dynamical equation: X " + 7-/(X) + grad V(X) = 0,

(2)

with initial conditions: X(0) = X0 such that V(Xo) < +co, X'(O) = X~ E 8.

Diffusive Representation for Nonlinear Systems

165

We define the (mechanical) energy of (X, X') T by: 1

Era(t) = V(X(t)) -t- ~

IIX'(t)ll~.

(3)

If X E Wt~o;~ (R+; g) is solution of (2), then we have:

dEm(t)

d--7---

(7/(x)(t)lx'(t))~.

(4)

When 7i - 0, (2) is conservative and obviously, E,~(t) = Era(O). If (7/(X)(t)lX'(t)) ~ >_ 0 t-a.e,, then (2) is dissipative (on the trajectory

(x,x')r).

D e f i n i t i o n 1. The "position-force" relation defined by 7/(X) is said thermodynamically consistent if there exists a Hilbert space Jr and: 3 ~ : WToc~ (R+;S) -+ L~oc (R+;.T ") causal and continuous, 3 Qt > 0 a pseudo - potential on ~-, 3 P _> 0 anon - negative potential on ~, such that, for any x E W2o~ (R+;C) :

_ep (O(x)(t))

(7/(x)(t)[x'(t))e = Qt (r

+ dt

t - a.e.

(5)

In the decomposition (5) of the mechanical power (7/(x)(t)lx'(t))c, the first term is the (positive) dissipation rate and the second term is the derivative of the free-energy function P(O(x)(t)). Let

E(t) := Era(t) + P(k~(X)(t))

(6)

denote the energy of system (2), we easily deduce: P r o p o s i t i o n 1. If 7/(X) is thermodynamically consistent, then system (2)

is dissipative: for any (X, X') solution of (2), dE(t) d~

-

Qt

(kh(X)(t))

<_ 0 t -

(7)

a.e.

Remark 1. Controls may be considered, under the general form: X " + 7 / ( X ) + grad V(X) = u(t, X, X'). Classical viscous damping defined by B X ' , B positive, may also be added without difficulty. Examples. 1.

Viscous damping:

7/(X)(t) = B X ' ( t ) , x ' , O,(~) = (B~l~)c, e ( ~ ) = 0.

B >__ O, jz = g,

~P(X) =

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2. Coulomb dry friction [1]: s = R, 7 t ( X ) ( t ) =

ksign ( X ' ( t ) ) if X ' ( t ) r 0, k > 0 -W(X(t)) i f X ' ( t ) = O,

~" = n , v~(X) = x ' , Qt(~) = kl~,l,

P(~) =

o.

Note that, with function sign understood in the multivalued sense: sign (0) - [-1, 1], system (2) may be rewritten: X " + ksign (X') + V ' ( X ) 9 0 t - a.e. 3. Hysteresis damping [19].

3 3.1

Pseudo-differential damping Pseudo-dlfferentlal operators

The operators 7t under consideration in this paper involve pseudo-differential components [18]. They are causal and considered in a way parallel to the classical one using Laplace transform instead of the Fourier one. The analogy will not be emphasized here, essentially because these operators belong to a subclass which is more conveniently directly described by a class of symbols. For simplicity, we restrict the statement to scalar systems (8 = R); extension to the vector framework requires further technical adaptations (in particular in infinite-dimensional cases, such as PDEs). We denote by S~_ (R) the space of causal tempered distributions on R [16] and by s the Laplace transform defined by: (s = f : o o e_V~,u((r ) d~r. A complex valued function on R + • C will be defined as a symbol and, when this expression makes sense, to such a symbol H we associate the causal operator:

H(cr, 0a): S~.(R)--~ S~.(R) x

(8)

~-+z=H(a,O~)x=s

In the case of Volterra (singular) operators: When H ( a , . ) = s following is immediate:

Proposition

2. Let H a symbol such that For any ~r > O, H(a, .) = s Then H (a, Oa) is the Volterra operator:

(H((r, Oa) x) (a) =

/o

h(a, a - v)x(v) dr =

/o~h(a, v)x(v) dr.

.), the

.).

(9)

Rigorously speaking, the symbol of a causal Volterra operator is not unique: it is only defined up to an algebraic quotient. Indeed, it is easy to see that

Diffusive Representation for Nonlinear Systems

167

any h such that h(~, v) = h(a, 7-) on 0 < v < a defines the same operator, but the associated symbol H(cr, .) -- s .) may obviously be different. Note that in the convolutive case, h(er, v) -- h(v) and g(p) reduces to the classical transfer function. Note also that various regularity properties with respect to the a-variable may be considered, in accordance to the specific needs of the problem in which such operators are involved.

3.2

Pseudo-differential damping

We study the two following types of damping which are of particular interest in concrete situations: 9 linear pseudo-differential visco-elasticity [6] defined by:

~ ( X ) = H(t, a O x ' ,

(10)

9 pseudo-differential elasto-plasticity defined by [17]: s :=s(x)(t)

:=

f IX'ldr,

7t(X) = [H(s,O,) (X o S ( X ) - I ) '] oS(X).

(11)

(12)

The model (10), (11), both non-linear and hereditary and introduced by P.-A. Bliman and M. Sorine [1], defines S as an intrinsic clock such that relatively to the intrinsic time s = S(X(t)), the definition

Xs := X o S(X) -1

(13)

gives the linear law:

7"ls(Xs) = H(s, c3,)X~s.

(14)

Such (endochrone) phenomena are frequently encountered in hysteresis theories [19]. The main difficulties in the analysis of models of that type lie in their heredity: the expression of 7t(X(t)) involves the whole past (Xo<_r
168

4

G~rard Montseny et al.

Diffusive representation

In the following, we essentially consider the convolutive case, for simplicity. Most of results remain available in the general case which sometimes requires specific technical developments and will be presented in a further paper (see also [11]). We present here a simplified introduction to diffusive representations for pseudo-differential operators of diffusive type. These causal operators are in fact defined so as to admit a representation, using an augmented state space, by a diffusive system. As in the previous section, a denotes a time-variable (t or s, with ~d, -- IX'l). 4.1

T h e a l g e b r a z~' o f c o n v o l u t i v e d i f f u s i v e s y m b o l s

We first introduce the concept of diffusive symbol, on which are based the diffusive state-space realizations. Let H (a, (9o) an operator with symbol H (a, p). This operator is of diffusive type when there exists~(a,~) such that [10]:

~(a,~)

H(a,p) = f0 +~ P + (

d~, p = iaJ, w E R, a > 0;

(15)

The solution fi of (15), when it exists, is unique and called the diffusive symbol of H(cr, 0a). 1. ~ is solution of (15) if and only if the impulse response of H(a, Oa), denoted by h, is given by:

Theorem

h(a, .) = s

.).

(16)

Proof. (formal} From Laplace transform inversion formula and Fubini theorem, for any ~r > 0 and some a > 0: h(a,v) = 2i----~j,_i~~ eP'H(~r,p) dp= 2i---~ja_io~ \ 2~, j ~ _ ~

e Pr ~-~ dp

-~(~,~) a~ =

Jo

v+~

e -~T ~(~, ~) d~

= ( ~ ) (~, r). As a consequence of the analyticity of h(a, .), we have the so-called "pseudolocal property": C o r o l l a r y 1. If H(a, (9o) is of diffusive type, then: sing supp (H(g, Oo)z) C sing supp z

for any x.

(17)

Diffusive Representation for Nonlinear Systems

169

Examples. We consider the particular case of fractional integrators and derivators which are interesting due to their simplicity and popularity. The diffusive symbol of H(t, Or) = Ot a(t), ~e (a(t)) > 0, is expressed by1:

#(t,~) = sin(~ra(t)) fp~-~(t), ~ > O.

(18)

71"

2. The diffusive symbol of H(Ot) = 0 t 1 ln(0t) is given by2: ~(~) -- fp~ - 76(~);

(19)

the associated impulse response is h(t) = - ln(t) - 7. 3. The diffusive symbol of H(at) = ea~ (aat) is given bya: = e-~162

(20)

the associated impulse response is h(t) = t + 1a " 4. Any stable rational transfer function with real poles is the symbol of a diffusive operator. Many other examples can be found in [10]. Obviously, thanks to linearity, the space of diffusive symbols is isomorphic to a subspace of pseudo-differential operators. Let us now consider two convolutive operators H(ao), K(0a), with respective diffusive symbols #, 7. We have the following results [10]: T h e o r e m 2. The product operator H((9~,) o K(Oa) is also diffusive. Its diffusive symbol is defined by an internal product denoted by -fi~C-ff. When -fi,"ff are regular, this product is expressed: ~ ' ~ K = - - ~ ( ~ * p v ~ ) - - V ( ~ * pv~).

(21)

T h e o r e m 3. Equipped with product # , the space A' of convolutive diffusive symbols is a commutative algebra of causal tempered distributions on R(, with Frgchet topology. i fp f and pvf respectively denote the "finite part" and "principal value" distributions associated to non locally integrable functions f [16]. They may be viewed as the derivative of sufficiently high order in the sense of distributions, of some locally integrable functions. For example, pv~ is the (causal) derivative of ln(Ixl) and fp~ is the derivative of the causal function ln(x). 2 7 denotes the Euler constant. Ei(a) = du.

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G~rard Montseny et al.

This induces an isomorphic algebra of convolutive pseudo-differential operators. Thanks to closedness of A' and continuity of # , both algebraic and analytical developments4 can therefore be performed in A ~ which is the general mathematical framework fitted to diffusive representation. We do not describe here the topology of A ', we only give here-after a simply sufficient condition for -f E ,4' [10], [11]: P r o p o s i t i o n 3. If --E1+~ E LI(R+), then -fie A'. 4.2

Diffusive r e a l i z a t i o n s o f H(O~,)

S t a n d a r d r e a l i z a t i o n s We consider the following input-output equation (in a suitable Hilbert state-space):

{ a~r162 y(~) =

+co~r

= .(~), r -f(~,, ~) r ~) d~.

= 0, ~ > 0

(22)

If-f is the diffusive symbol of H ( a , 0o), then we have: T h e o r e m 4. The input-output correspondence x ~-~ y defined fies: y = H(a, Oa) x.

by (22) satis-

Proof. From (22), and Fubini theorem: y = f0 co-fi(cr, ~) ~0a e - ~ x ( a - v) dr d~ = =

/o~

-f((r,~)e-~Td~

) x(a-r)

dv=

Zo(/:-f) (or,v) x (at -

v) dr =

fo

ah(a, a - v) x(r) dv = (H(a, Oo)x) (cr).

D e f i n i t i o n 2. The input-output state equation (22) is called the standard diffusive realization of H(a, r Various other state-space realizations may be built (see [10]); in particular, by using Fourier transform with respect to 77, with ~ = 47r~2:

{ oav(~,~ -+coa}r ~) = .(~) ~(~), ~(o,A)= o, AER u(~) =

oo ~(~' A)~(~, A) d~.

(23)

Remark 2. This last formulation, which gives to diffusive pseudo-differential operators a physical meaning, is at the origin of the term "diffusive representation".

4 Namely numerical analysis.

Diffusive Representation for Nonlinear Systems

171

The following result will be fundamental in the sequel: L e m m a 1. The pseudo-differential operator O~ l H (Oo) has diffusive symbol

5#-~. Proof. It is sufficient to prove that 6 is the diffusive symbol of c9~-1. From the well-known property ~ 6(~) = 0:

0~

(o+00

f+oo

~r

f+00

= -Jo

f+oo

,~OaCd~=Jo ~(-~r f+00

+XJo 5 d ~ = x .

~6r

E x t e n d e d diffusive r e a l i z a t i o n s Extended realizations enable to take into account more general pseudo-differential operators. We consider here the following, which is well-adapted to visco-elastic and elasto-plastic phenomena. It consists in derivating the output, which obviously leads to the state-space realization of x ~ z = tg~H(cr, egG)x: {0or

+00

z=a~

r

+00

pCd~=

~ (-~r + x) d~.

(24)

Note that this last formulation is of the abstrm=t form:

- ~ = A X + Bx, X0 = 0 z = C ( X + Dx).

(25)

From lemma 1 and according the previously introduced notions and notations, the following result is obvious: T h e o r e m 5. The correspondence x ~-4 y = H(Oo)x realized by (22), is also realized by:

{

oo

Y = fo

=

= o,

'~g~ (-~ r + ~) d~.

> o

(26)

F i n i t e - d i m e n s i o n a l a p p r o x i m a t e diffusive r e a l i z a t i o n s They are obtained from discretization of the ~-variable in (24), following standard methods of partial differential equations and numerical analysis. We only give some indications, more details will be found in the referenced papers. Given a finite mesh X K = { ( k } l < k < K C R +, and /2K = {Ak(~)} a suitable set of interpolating functions, a finite-dimensional approximation of r defined

172

G@rardMontseny et al.

by (22) is obtained by: K

r

= E

r

(27)

Ak(~),

k=l

and an approximation of y (defined by (22)) is then deduced: ff(r

=

~ ~d~ =

~(o',~) ~Ak(~) d~

r k=l

= ~

(28)

~ (~) Ck(~)

k=l

Under simple and natural hypothesis on XK and J~K and according to fitted topologies, we may state: P r o p o s i t i o n 4. The finite-dimensional approximate realization o f x ~ y = H ( a , Oo)x :

= -~

Ck + z, Ck(0) = 0

= Z P k Ck

(29)

k=l

is convergent when K -+ +oo:

(30)

- H ( a , Oo)x --+ O.

C o r o l l a r y 2. The finite-dimensional approximate realization o f x ~

z =

OaH(a, aa)x :

{ a_~_~_= --~k Ck + z, Ck (0) = 0 dt

K

K

k----1

k=l

(31)

is convergent: - OaH(a, Oa)x --~ O.

(32)

From a different point of view, thanks to topological density of the space of measures in A ~, optimal K-dimensional diffusive realizations of the form (31) may easily be obtained by solving (15). Solutions are built in the pseudoinversion sense s, with ~ E A~K C A~, the K-dimensional space of Dirac measures with support XK. This requires Hilbertian formulations and is not presented here (see [10]). An example of optimal approximate diffusive realization is given in section 7. 50rthogonal projection.

Diffusive Representation for Nonlinear Systems 5

173

Main result

In order to built dynamical models for pseudo-differential visco-elasticity and elasto-plasticity, we prove the following result on which will be based the thermodynamical consistency of 7t. It gives a sufficient (and probably necessary) condition to get positiveness of operator H(0a). T h e o r e m 6. If the diffusive symbol-fi of O~lH(O~) is such that: 3/~,u e L/at(R+) O A', #,u_> 0, ~'= (f#/~ + u,

(33)

then we have the following balanced diffusive realization of z = H(O~)x: {Ooiv+fo~W=(vffi+v/~z, w(O,')=O,a>O +~o z= [(~-,/}~) iv + ~ ] d~.

(34)

Furthermore we have the estimate for any a > O: d 2 x(a)H(c%)x((r)= ~.+_~r~i IlivlIL2(R+)

.~+oo (k,~/ ~. , ~ p - x ( a ) ., / ~ / 2 d~. (35)

Proof. I. By change of function r = ~ z =

((#-@)r

= =

/0

=

~#~ (-~r f0+~176

g(-(r

and theorem 5,

+ x)d( =

#r

+ ~) d~ + 0o

/0

~r

~ ( - ~ r +~)d~ =

~ (-~r f0+~176

+ ~) d~ =

d( = O~O;~H(O~)~ = H(O~)x.

2. Furthermore,

=

= =

[~

iv - ~ v ' ~ ~ +

~ ~] d~ =

[-~iv~- +~ (v~+ v"ff) iv+ (~iv2- 2J-ffiv~ + ~ ' ) ] d~= fo +~

~[-~iv+~

( v ~ + eY~)] d~+

fo +~

(~iv-~

vg)~d~ =

Remark 3. 1. Property (35) is in fact much more precise than positiveness (fos xH(Oo)x do" _> 0) which, in the context of diffusive representation, appears as a simple corollary.

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G4rard Montseny et al.

2. Besides positiveness, relation (35) suggests the natural Hilbert energy state-space for (34): jv = L2(R~). As a consequence, we have in the particular case of fractional operators: C o r o l l a r y 3. For H(O~) = kl O~~ + k 2 0 S .2, kl,ks > O, 0 < al,C~s < 1, properties (3S), (34), (35) are ,eri~ed, with."

t. sin(Tr(1-al))

sin/w/1Ta~))

Proof. Obvious from (18) and lemma 1. 6 6.1

Application

to pseudo-differentially

Thermodynamical

damped

systems.

c o n s i s t e n c e o f 7/

The following results are then deduced from theorem 6, by ordinary computations: T h e o r e m 7. If H(Ot) satisfies hypothesis of theorem 6, then 7I(X)

H ( Ot) X' is thermodynamically consistent, by taking:

~x(t) = ~(t, .), ~ defined by (34), with x(t) := X'(t), 0 +~176 p(~)

(36)

2

=

89

s iI~[IL~(R+) 9

Proof. Obvious from theorem 6 and definition 1 with $ = R. T h e o r e m 8. If H(O,) satisfies hypothesis of theorem 6, then 7ls(Xs)

=

g (s, O~)X~s (see section 3.2) is thermodynamically consistent, by taking: s = j~Ot I x ' l dr,

(37) ~ox(t) = ~o(s(t), .), ~o definedby (34) with z(s) := ~ X ( t ( s ) ) ,

Q'(~) ~ J 0 =

P(~o) = 1

:

Diffusive Representation for Nonlinear Systems

175

Proof. Similar to theorem 7, with :

]o 6.2

Time-local s t a t e - s p a c e r e a l i z a t i o n s o f (2)

By coupling the diffusive realization of 7/ and the main state equation, we obtain suitable global models for pseudo-differentially damped systems, with existence of an infinitesimal generator (time-local system) :

Corollary 4. (concrete state-space realizations) Denoting: : - V / - ~ - V/(u(~) and Mt(~) := X / ~ +

M(r

r162

(38)

non hereditary global state-space realizations of (2) (Cauchy problems) are then explicitly built: visco-elastic model: o~x +

(39)

M t~ + e , x | ~] d~ + V'(X) = o

[ Ot~+~-OtX

|

{ IF O~X+

---0,

elasto-plastic model." [Mt~o+sign(OtX)|

90

(40)

Otto + ~ ~o[OtX I - OtX | M = O,

with initial condition (Xo, X~, ~Oo) and energy functional: E(t) = v(x(t))

+

(otx(t)) ~ + ~

lifo(t, .)IlL=m+),

(41)

such that: dE(t) - _ dt

Q, (~(t, .)) <_ o v t > o.

(42)

Proof. Obvious from: dX dX dt ds - d~-ds

Xt IX, I E sign(X')

(43)

and:

Ot~ = o~ - -ds ~ =O,~lX'l.

(44)

176

G6rard Montseny et al.

Remark ~. Dry friction dissipation is obtained by (40) with M = M t = 0, Under weak hypothesis, existence and uniqueness of the solution of (39), (40) in a fitted Hilbert state-space ~ can therefore be proved from classical energybased methods of partial differential equations (Galerkin method for example). Note that in the non Lipschitz case (40), existence of a priori energy estimates proves to be decisive in order to suppress mathematical ambiguousness inherent to such systems [13]. Furthermore, finite-dimensional convergent approximations of (39), (40) can efficiently be elaborated from energy error estimates. This enables to build finite-dimensional differential approximated models with arbitrary precision, of the form: ~d~ -- F(z), z(t) E R N. Finally, thanks to the existence of an infinitesimal generator for (39) and (40) (induced by (34)), classical tools of control theory may be employed. Note that the damping function, defined by the abstract form ~/(X) and equivalently by the concrete state-space realization (34), may also be considered as a pseudodifferential (closed-loop) control, constructed for example by minimization of a cost functional J ( ~ ) . Indeed, pseudo-differential diffusive controls have proved to be of particular interest for robustness purposes in linear control problems (see [3], [4], [5], [12]). From a slightly different point of view, such methodologies have also successfully been used in pseudo-differential passive control of linear infinite-dimensional systems in [14], [15], [3]. 6.3

Analysis of asymptotic behaviors

From (42), specific techniques like LaSalle invariance principle [7] then enable to find asymptotic equilibrium states. In (40), they systematically depend on the initial condition: V (X0, X~, 90) E ~, 3 ! (Xoo, 0,900) E ~, such t h a t :

E(t) J, Eoo = V(Xoo) + 1119ooll X(t) ---r Xoo, cgtX(t) ---+O, 9(t, .)

(45) ) 900 stronglyin L2(R~-),

with the following characteristic equation for equilibrium: 0 ~176 [Mt(~) 9oo(~) + a ~ ~(~)] d~ = -V/(Xoo),

(46)

aoo E sign (0) = [-1, 1]. Note that this last expression explicitly involves the diffusive realization of "H(X), through its characteristic parameters M and ~. Excepted in very simple cases (dry friction), such an explicit characterization is not accessible from initial formulation (2).

Diffusive Representation for Nonlinear Systems

177

A n example of numerical simulation

7 7.1

Problem statement

In order to highlight the efficiency of diffusive representation from the point of view of numerical simulations, we consider the second order oscillator with visco-elastic damping 7-/(X) := A b t ~ X ', 0 < a < 1, A > 0:

O~X = - A a ~ - " X - f ( X ) .

(47)

From corollary 4 and (18), model (47) is equivalently transformed into: Ot~X : --A

sin(a~) ,~. C d ~ - f ( X ) = 0

(48)

Otr = - ~ r + Ot X .

A K-dimensional optimal diffusive approximation of 0 t ~ has been performed (see section 4.2), with the following parameters6: a = 0.75 K=25 ~1 = 0.001 ~2~ = 50 000 ~k+~ = 2.093102 (Pk)~
model (48) can be rewritten under the form: { ~x_Kr _- - X 2 dt

"@t = - f ( X l ) - A C r

(49)

dd~t = A r + B X 2. System (49) has been simulated by classical Runge-Kutta method, with A : 2, in the linear: f ( X ) = X, and non-linear: f ( X ) = sin X cases. K = 25 has been chosen for high precision. Smaller values of K are generally sufficient in physical situations.

178 7.2

G~rard Montseny et al. Numerical results

The frequency response and the pole-zero m a p 7 of the approximation of Ot~ are given in figures 1, 2. Note that on 6 decades, phase is constant (67.5 ~ and magnitude decreases at rate of 0.75 x 20 dB/dec; these properties are characteristic of fractional integrators. Evolution of the linear system is shown in figures 3, 4, 5. Long memory visco-elastic behavior is clearly visible: after a few oscillations generated by the elastic component of 7/(X), X ( t ) slowly decreases to 0, involving both the viscous and elastic component of the pseudo-differential damping. In figures 6, 7, 8, non-linearity significantly affects the evolution: due to the elastic component of 7~(X), small overshoots appear at the beginning, while the visco-elastic counterpart considerably slackens the system. This is the consequence of the particular choice of initial conditions, near an unstable equilibrium point (sin(X0) _~ 0).More detailed simulations (namely in presence of elasto-plastic damping) will be presented in a further paper devoted to numerical approximation.

lOO

.

~

:

elo

~ ....

.

.

.

i .........

, ............

: .........

: .........

::. . . . . . . . . . . . . .

: ..............

................

: .....

i . . . . . . . . .

. . . . . . . . . . . . .,. . . . .

::. . . . . . . . . . . . ,. . . . . .

;: . . . . . . . . . . . .,. . . . . . . . . . . . . . . . . . . . . . .

o

J. J :

........

..~

1o ~

................

1 0 "~

1~

1~'

i

. . . . . . . . .

lo ~

1~

Fig. 1. Frequency response of the approximate 0 ~

7 Only the domain [-0.1,0] +/[-0.002,0.002] is visible in the figure.

Diffusive Representation for Nonlinear Systems x l o .=

~

1

o.e

t f .-o.

-1

-l.e

-O.Ol

-0.08

-o,o?

~ol

.QQ6

-0,04

-0.03

.~3,~

~.01

F i g , 2. Pole-zero (partial) map of the approximate 0 ~

evw-I~ ~ ~

-1

1 7,

r

_~'

~

,

y (-) aed ~

:

:

i

i

'

i

i

W (- -)

"

Fig. 3. Linear model O~X + )~O~+~X + X = 0

P~

-2'

15

~

2

{~r ~

25 mPy

Fig. 4. Linear model O~X + AO~+~ + X = 0

~.......

"

:

.

....

179

180

G~rard Montseny et al. I

OS

0

i

.as

:t

s

o

i i

...........

.o.---"

v

s

lo

is

~

m

I

~

........

~

~

,4

Fig. 5. Linear model ~ X + )~O~+~'X + X = 0

L~WAqm ~* m

.....

~. . . . .

:

............

y (-) a,~ p,mL~

W (- -}

i ........

)

~. . . . . . . . . .

::

L

>-.

Fig. 6. Non-linear model O~X + ,~O~+~'X + sin(X) = 0

O3

.

,

,

42

44

O2

~

0IS

|

o,

-o0 3~

3~

4

a,XNy

46

Fig. 7. Non-linear model Ot2X + AO]+~X + sin(X) - 0

4a

m

Diffusive Representation for Nonlinear Systems E~i~ll~l+ ~ lhl 4tlhi#~l cl+nzponm~l 14

+

1o

s+

~o

ti~ i

'~

m

m

Fig. 8. Non-linear model O•X + AO~+c~X+ sin(X) = 0

,m

181

182

G6rard Montseny et al.

References 1. P.-A. Bliman and M. Sorine. "Dry friction models for automatic control", In Proc. of Euromech. Colloquium 351: Systems with Coulomb friction, Vadstena (Sweden), August 5-7 1996. 2. H. Br~zis, Analyse fonctionnelle - Thdorie et applications, Masson 1983. 3. F.-A. Devy Vareta, D. Matignon, J. Audounet, G. Montseny, "Pseudo-invariance by matched scaling: application to robust control of a flexible beam", Second European Conference on Structural Control, Marne-la-Vall~e (France), July 2000. 4. F.-A. Devy Vareta, J. Audounet, G. Montseny, "pseudo-invariant diffusive control", MTNS 2000, june 19-23, 2000, Perpignan (France). 5. F.-A. Devy Vareta, P. Bidan, E. Irving, G. Montseny, "Pseudo-invariance by matched-scaling: a new concept for multivariable robust control", submitted to publication. 6. M. Fabrizio, A. Morro, Mathematical problems in linear visco-elasticity, SIAM Studies in Applied Mathematics, 1992. 7. A. Haraux, Syst~mes dynamiques dissipatifs et applications, Masson 1990. 8. M.A. Krasnoselskff and A.V. PokrovskiL Systems with hysteresis, SpringerVerlag, Berlin Heidelberg, 1989. 9. D. Matignon, G. Montseny (Ed.), Fractional differential systems: models, methods and applications, ESAIM: Proc. Vol. 5, December 1998, URL: www.emath.fr/Maths/Proc/Vol.5/index.htm. 10. G. Montseny, "Diffusive representations of pseudo-differential timeoperators", ESAIM: Proc. Vol 5, pp 159-175, December 1998, URL: www.emath.fr/Maths/Proc/Vol.5/index.ht m. 11. G. Montseny, "Representation diffusive: principes et extensions", actes du S~minaire Toulousain Reprgsentation Diffusive et Applications, LAAS/CNRS, No 1, Toulouse (France), sept. 2000 (to appear). 12. G. Montseny, "la commande CRONE r~-interpr6t6e et g6n~ralis~e", to be published. 13. G. Montseny, "Sur le comportement paradoxal de eertaJns systbmes difffirentiels ~tsecond membre discontinu: ~tude d'un exemple', Internal note, to be published. 14. G. Montseny, J. Audounet, D. Matignon, "Fractional integrodifferential boundary control of the Euier-Bernoulli beam", 36th IEEE CDC Conference, San Diego (USA), 1997, pp 4973-4978. 15. G. Montseny, J. Audounet, D. Matignon, "Perfectly absorbing boundary feedback control for wave equations: a diffusive formulation", 5th International Conference on Mathematical and Numerical Aspects of Waves Propagation Phenomena, Santiago de Compostela (Spain), July 2000 INRIA-SIAM. 16. L. Schwartz, Thdorie des distributions, Hermann 1973. 17. M. Sorine, D. Matignon, G. Montseny, "On a class of pseudo-differential hysteresis operators", submitted to publication. 18. M. E. Taylor, Pseudodifferential Operators, Princeton University Press, 1981. 19. A. Visintin. Mathematical models of hysteresis, Topics in nonsmooth analysis, Birkhafiser Verlag, Basel Boston Berlin, 1988.

Euler's Discretization and Dynamic Equivalence of Nonlinear Control Systems Ewa Pawtuszewicz* and Zbigniew Bartosiewicz** Technical University of Bialystok Wiejska 45 15-351 Bialystok, Poland epav@cksr, ac. bialyst ok. pl

A b s t r a c t . Euler's discretization transforms a nonlinear continuous-time system into a discrete-time one. It is shown that if two continuous-time systems are dy-

namicaUy feedback equivalent then their Euler's discretizations are dynamically feedback equivalent. Dynamical equivalence is characterized by isomorphism of differential or difference algebras associated to the systems. These algebras form two categories. Euler's discretization defines a covariant functor from the category of differential algebras to the category of difference algebras.

1

Introduction

In 1992 B. J a k u b c z y k introduced d y n a m i c equivalence of c o n t i n u o u s - t i m e nonlinear control systems [6,7]. D y n a m i c t r a n s f o r m a t i o n s , t h a t were proposed, d e p e n d e d on derivatives of states and controls. Moreover, with any s m o o t h or analytic control system there was associated a differential algebra, i.e. a function algebra with a differential o p e r a t o r . T h e definitions and results o b t a i n e d by J a k u b c z y k were carried over to nonlinear discrete-time systems [2]. In this case d y n a m i c t r a n s f o r m a t i o n s dep e n d e d on p a s t and future values of states and controls. I n s t e a d of differential algebra, difference a l g e b r a was considered. S i m i l a r l y as in the c o n t i n u o u s - t i m e case it was proved t h a t the difference algebra is the only invariant of d y n a m i c feedback equivalence. T h e results were global, i.e. the t r a n s f o r m a t i o n s of the systems were defined on the entire s t a t e space. In b o t h cases algebraic objects are in one-to-one correspondence with systems, so systems m a y be represented by differential or difference algebras. T h e y form two categories, with m o r p h i s m s preserving differential or difference structures. S i m i l a r language was used in [3-5,10]. We s t u d y the Euler's discretization of a nonlinear continuous-time system. D u r i n g this process the Supported by KBN under W/IMF/3/99 ** Supported by KBN under W/IMF/1/00

the Technical University of Bialystok grant the Technical University of Bialystok grant

184

Ewa Pawtuszewicz and Zbigniew Bartosiewicz

equation

=/(x, u) is t r a n s f o r m e d to

x(k + 1) = x(k) + hf(x(k), u(k)) where h is a fixed step of discretization. On the algebraic level, the differential algebra is t r a n s f o r m e d to the difference algebra. We prove t h a t this transf o r m a t i o n is a c t u a l l y a covariant functor from one category into the other. (For another m e t h o d of discretization see [1]) As an a p p l i c a t i o n we get the following:

If two continuous-time systems are dynamically equivalent then their Euler discretizations are dynamically equivalent. In particular, if a continuous-time system is dynamically linearizable then its Euler discretization is dynamically linearizable. 2

Equivalence

of Systems

We c o m p a r e here .dynamic equivalence for c o n t i n u o u s - t i m e and d i s c r e t e - t i m e systems. First we introduce some n o t a t i o n . By J(A) we will denote a disjoint sum of Jr (A) - the set of all sequences Zr = ( z ( r ) , z ( r + 1 ) , . . . ) , where z(i) E A a n d r E 25. I f A = JR', then the set j ( ~ s ) will be denoted by J(s) and J r ( ~ s) by Jr(s). Similarly, if A = ~" • x ~ " then J(]~" x ... x ]~') = J ( s l , . . . ,st). T h e shift operator Sk is the m a p sk : J(A)--+J(A) defined by: sk(Zr) := Yr+k, where y(i) = z ( i - k ) for i > r + k and k E 25. The restriction operator ci : J(A)---~J(A) is given by ei(Zr) := Zr+~ = ( z ( r + i ) , . . . ) , i > 0. We shall consider real m a p s defined on J(A) where A C lt~'.1 • . . . ] ~ m . We assume t h a t such m a p s are shift invariant (so we t r e a t t h e m as functions on Jo(A)) and d e p e n d on a finite n u m b e r of elements z(0), z ( 1 ) , . . . , z(q) of the sequence Z ~ J(A), b u t q depends on a given function ~. We then say t h a t is of finite order. A map r : J(A)--+I~ ~ is of finite order if all the c o m p o n e n t s of r have this property. Let A C ~ n l • . . . ~ n ' , B C ~,~a • . . . ~ k a n d 7 : J(_A)-+B. T h e n the extension of ~/ is the m a p F : J(A)--+J(B), F(Zr) = Z r , where 2 r ( i ) = 7(ci-r(Zr)) for i _> r.

2.1

Continuous-Time Systems

Let us consider an analytic or s m o o t h c o n t i n u o u s - t i m e control s y s t e m defined o n I~n:

Zc : ~(t) : f ( x ( t ) , u(t))

(1)

Euler's Discretization and Dynamic Equivalence

185

where u(t) E ]Rm and t E IR. By a trajectory of this system we m e a n any pair (x(.), u(.)) that fulfil (1) on some interval. The set of all trajectories of system Z'c forms a behavior of this system, denoted by B ( Z c ) (see [11] for the origin of the concept). di z By Z := Jz = (Trr)i_>0 we will denote the infinite jet extension of a s m o o t h function t ~-r z(t). Thus if z(t) E I~' then Z is a m a p with values in Jo(s).

Let us consider two continuous-time control systems:

2~c : x(t) = f(x(t),u(t))

L'c: ~(t) = ](~(t), ft(t)

and

where x(t) E ]Rn, ~:(t) E ]R'~, u(t), fi(t) E ]Rm, t E 1R. We say that systems G'c and ~Pc are dynamically feedback equivalent [6,7] if there exist transformations:

: r

r

=

=

0)

(2)

u)

(3)

where r r r r are maps of class C ~, s -- w or s -- o(), of finite order, such that the induced maps on pairs (x(.), u(.)) and (~(.), fi(.)) preserve behaviors of the systems and are mutually inverse on these behaviors. System ~wc is dynamically (feedback) linearizable if it is dynamically feedback equivalent with a linear controllable one.

2.2

Discrete-Time

Systems

Let us consider now a nonlinear analytic or s m o o t h discrete-time control system defined on ~n:

(4)

x(t + 1) = g(x(t),

where t 6 Z and u(t) 6 ]Rm. Let us consider the following sequences: Xr = (x(r),x(r+ 1 ) , . . . ) E Jr(n) and Ur = (u(r),u(r+ 1 ) , . . . ) E Jr(m). A trajectory of system Ea is any pair (Xr, Or) that satisfies (4) for k > r. T h e set of all trajectories of the system Sd forms the behavior of this system. We denote it by B(S,d). Moreover B(L'd) = U Br(Zd) where Br(,Ud) is the set rEg

of trajectories starting at instant r. Let us consider two nonlinear discrete-time systems defined, respectively, on ]Rn and ]R~:

,U,d: x(t + 1) = g(x(t),u(t))

and

~ d : ~(t + 1) = ~(~(t), fi(t))

where u(t), fi(t) E ]Rm. Let us consider also maps: r

J(fi)--~lR n, r

d(fi, m)--+~ m and

$ : g(n)--+lR r~, r : J(n, m)--->lR'~

186

Ewa Pawluszewicz and Zbigniew Bartosiewicz

T h e extensions 45, ~, ~3, ~ of r r r r define m a p s

and ~ : (Xr,Vr)

~ (~(Xr),~(X,,V~)).

T h e systems ~ and ~d are dynamically feedback equivalent if there exist m a p s of finite order (r r and (r r such that x(B,(~))

= Br(~),

~(B,(~))

= B~(~)

and X and ~ are mutually inverse on the behaviors of systems. In other words, r 0, r and ~ define transformations of states and controls of both systems of the form:

x(t) = r ~(t) = r

+ q)), ....

, ~(t + q), ~(t) . . . .

, ~(t + q))

and ~(t) = ~ ( x ( t ) , . . .

, x(t + q)),

~(t) = (~(x(t), . . . , x ( t + q), ~ ( t ) , . . . , ~(t + q) ).

Discrete-time system Z:d is dynamically (feedback) linearizable if is d y n a m i cally feedback equivalent to a controllable linear one.

3

Differential and Difference Algebras

A differential algebra is a c o m m u t a t i v e algebra A over ~ together with a differential operator D : A-+A, i.e. a linear m a p satisfying the Leibniz rule for product. A m a p 7-: A1--+A2 is a h o m o m o r p h i s m of differential algebras (A1, D1) and (A2, D2) if it is a h o m o m o r p h i s m of algebras and satisfies the condition D~ o r = r o D1.

(5)

A difference algebra is a a c o m m u t a t i v e algebra A over It~ together with a h o m o m o r p h i s m d : A--~A. If ( A l , d l ) , (A~, d~) are two difference algebras then a m a p 7- : A1--+Au is a h o m o m o r p h i s m of difference algebras if it is a h o m o m o r p h i s m of algebras and satisfies the condition d2ov=7"odl.

(6)

Euler's Discretization and Dynamic Equivalence

187

If r is a bijective map, then v is an isomorphism of differential (difference)

algebras. Let b/(n, m) denote the algebra of all real functions defined on ]Rn x J(m) that are of finite order and are shift invariant. Therefore we m a y treat them as functions on ]Rn • Jo(m). We will assume that functions ~o E / / ( n , m) are of the same class as the dynamics f of the given continuous- or discrete-time system. By the differential operator associated with the system L'c we will mean the m a p DEo : hi(n, m)-+N(n, m) [6]: D,vc : =

Z

l
s__0 qOxq"4-Z "

(j+l)

i,j

0 ) Ou}. '

(7)

where u~j) : IRn x J(m)--+l~, u~J)(x,U) = ui(j), i = 1,... ,m, j = O, 1,.... The second sum is treated as a formal sum. I f / > 2, then D~flv := DE~ The algebra U(n, m) together with the differential operator Ds forms a differential algebra, which we will call the differential algebra of the system ,Uc and denote by (L/(n, m), DEc) or shortly by H~o. By the difference operator associated with the system s dEd: U(n, m)--+U(n, m) defined by:

(dEs~p)(x, U) = ~(g(x, u(0)), clU)

we mean the m a p

(8)

where ~o E L/(n,m) and U = ( u ( 0 ) , u ( 1 ) , . . . ) . I f / _> 2, then d~fio := l-1 dEs(dEs ~o). The algebra 14(n,m) together with the operator des forms a difference algebra. We call it the difference algebra of the system •d and denote it by (U(n, m), d~d) or shortly by L/~e. Observe that each L'r or ~Ud system uniquely defines respective differential or difference operator and vice versa. In [6] it was proved that two analytic (or smooth) systems 57c and ~c are dynamically feedback equivalent if and only if their differential algebras are isomorphic. In [2] this result was carried over to the discrete-time case: two discrete-time systems ZTd and ~d are dynamically feedback equivalent if and only if their difference algebras are isomorphic. Both results hold under the following assumptions concerning the right-hand sides of (1) or (4): A1. For every x and y in ~'~ there is at most one u E ~ m such that y = f ( x , u) (y = g(x, u) for discrete time). A2. For every x E Xn and u e ]l;ttm:rankOS/Ou(x, u) = m (rankOg/Ou(x, u) = m for discrete-time). A3. The m a p (x,u) ~-~ (x, f ( z , u)) ((x, u) ~-+ (x,g(x, u ) ) f o r discrete time) is proper.

188

4

Ewa Pawluszewicz and Zbigniew Bartosiewicz

Categories of Differential and Difference Algebras

We define the category C of differential algebras of continuous-time systems as follows: 1. By the class of objects ObC of C we will mean the class of all differential algebras b/Eo corresponding to continuous-time systems ~'c satisfying assumptions A1, A2 and A3. 2. By the set of morphisms of C, denoted by Mor C, will be meant the set of all homomorphisms between differential algebras from Ob C. If (b/i, D1), (b/s, Ds) and (b/a, D3) e ObC and v : (b/1,D1)-+(b/~,Ds), r : (/gu, Ds)--+(b/a, D3) are morphisms from MorC, then r o v E MorC, r o v : (b/l, D~)--~(b/3, Da), is the composition of morphisms. It gives again a morphism in Mor C. The identity m a p b/--+b/is the unit morphism. Let us also define a category 1) of difference algebras of discrete-time systems. 1. The class of objects Ob D of this category is formed by all difference algebras b/Ed corresponding to discrete-time systems IYd satisfying assumptions A1, A2 and A3. 2. The class of morphisms, denoted by Mor D, consists of all homomorphisms between difference algebras from Ob D. Composition of morphisms and the unit morphism are defined as before. Let h > 0 be fixed. By ~" : C--~D we will denote a functor from category C into category D defined in the following way: .T := (~l,3rs) and .T1 : O b C - + O b D , .T'2 : MorC-+MorD. Moreover .Tl(b/(n,m)) := Ll(n, rn) and ~1 D.~o is a difference operator defined by

(~'tDso)~ ~ := (id + hDr, o)~ ~

(9)

and (~'lDEc)~o(x, U ) : = ~0((hrlDroxl,... ,J:IDzox")(x,U),DroU)

(10)

for any ~o E U(n, m) where id denotes the identity m a p on U(n, m), x ~ : x J(m)~]R, xi(x,U) = xi, DzcU = clU = (u(1), u ( 2 ) , . . . ) . Since D~~ U) = fi(x, u), where u = u(0), the difference operator d = .T1D.vo may be written as

d~o(z, U) -- ~o(x + hf(x, u), U1). It corresponds to a discrete-time system ,Ud with g(x, u) = z + h f ( x , u). It can be easily checked that ZTd also satisfies assumptions A1, A2 and A3 (because ,Ue does). This system is the Euler discretization of Z:e with the step h. Finally, if r : AI-+A2 is a homomorphism of differential algebras (A1, D1) and (Au, Du) from ObC then ~ s ( r ) - r as the homomorphism of algebras (recall that ~1(A1) = A1 and 3el(As) = As). We have to show that r is in fact a homomorphism of the difference algebras (AI,J:ID~) and (As,.~'IDu).

Euler's Discretization and Dynamic Equivalence

189

L e m m a 1 Let r E MorC, r : (lg(n, rn),Dz,)-4(H(fi, m),Ds Then r e M o r V , i.e. r : (H(n,m),~Dxr m ) , Y i D 2 o ) is also a homomor-

phism of difference algebras. Proof." From the assumption we have

rD:co = Ds r. We shall show that r(.T1Dzo) = (3:lD2~ Let ~1 and ~2 be the projections (s U) ~-4 s and (~, ~r) ~ ~. It was shown in [6] that the m a p v is a pullback, i.e. there is a m a p p :

~r~ x Jo(m)-+]R n x Jo(m) such that 7" = p*. This implies t h a t v c o m m u t e s with substitutions. Indeed, let F : ~k__+~. T h e n (~- o F ) ( ~ I . . . . , ~ )

= ~-(F(~,...,

~k))

= r F ( ~ l , . . . ,~k) = F(~oi o # , . . .

,~k o/1)

= F(r~l,...

,rpk).

We are using this property in the following calculation which proves the required equality.

(3~l D2~

hD2c)~rl , D 2o~r~) + hDs r(Ds hD~,, Dxo)(r(~rt, ~2)))

= (v~,)((id + = ~v(r((id = ~((id +

= ((Y-lDzo)~)(r(~t, ~ ) ) = r ( d z d ~ ) . [ ] L e m m a 1 gives the following Proposition

2 9r is a covariant functor from the category C into the cate-

gory I). E3

Let :DE denote the subcategory Y(C) of:D. It consists of all difference algebras that correspond to Euter discretizations (with fixed step h) of continuoustime systems satisfying assumptions A1, A2 and A3. It is a full subcategory of category :D, i.e. for any A, B E Ob:DE the set of morphisms from A into B is the same in category :DE as in :D. Thus there exists the inverse functor = ~--1 from :DE to C.

190 5

Ewa Pawtuszewicz and Zbigniew Bartosiewicz Discretization,

Equivalence

and

Linearization

Now we m a y apply the result of the previous section to the problems of dynamic equivalence and dynamic linearization. 1. If two continuous-time systems are dynamically equivalent then their Euler discretizations are dynamically equivalent.

Theorem

P r o o f . If two continuous-time systems are dynamically equivalent then their differential algebras are isomorphic. Functors ~" transfers the isomorphism of the differential algebras to an isomorphism of the difference algebras. This means that the Euler discretizations of the continuous-time systems are also dynamically equivalent. [] C o r o l l a r y 3 If a continuous-time system is dynamically linearizable then its Euler discretization is dynamically linearizable. P r o o f . This follows from Theorem 1 and the fact that if a linear system is controllable then its Euler discretization (also linear) is controllable. O

A differential algebra (Ll, D) (respectively difference algebra (U, d)) is free if ([7,9]) there exist w l , . . . , wr E/at such that 1. for any function u : ~k_._~ of class C" holds: u o W - 0 ==~ u - 0, where W = {DJwi}i=l ...... ;j=o,z..... (respectively W = {dJwi}i=l ...... d=o,1,...) 2. for ~ E U there exists k E 1~ and a function u : ~ k _ _ ~ of class C ~ such that ~ = u o W . Functions w l , . 9 9 , wr are called free generators of differential algebra (U, D) (respectively of difference algebra (L/, d)). In [7] it was proved that the continuous-time control system ,Uc is dynamically feedback linearizable if and only if its differential algebra is free. Moreover in [9] it was proved that discrete-time control system 5:a is dynamically feedback linearizable if and only if its difference algebra is free. E x a m p l e 4 Let us consider a linear controllable system with scalar input (control). We m a y assume that it is in a Brunovsky canonical form: ~1 = x 2 ~2 = X3

Then xl is a free generator of the differential algebra of the system. T h e succesive derivatives of xl using the differential operator of the system yields

Enler's Discretization and Dynamic Equivalence

191

the r e m a i n i n g coordinates a n d derivatives of u. Euler d i s c r e t i z a t i o n of the s y s t e m takes the form: x l ( k + 1) = x l ( k ) + hx2(k)

x , ( k + 1) = x,(k) + hu(k). Here again z l is a free generator of the difference algebra. F r o m Corollary 3 we o b t a i n the following C o r o l l a r y 5 Let consider the system Sc and its Euler discretization Za.

If (U(n,m),Dso) is a free differential algebra then (H(n, rn),d~d ) is a free difference algebra. References 1. Arapostatis, A., Jakubczyk, B. et al. (1989) The effect of sampling on linear equivalence and feedback linearization, Systems ~ Control Letters 13. 2. Bartosiewicz, Z., 3akubczyk, B., Pawluszewicz, P. (1994) Dynamic feedback equivalence of nonlinear discrete-time systems. Proc. First Internat. Symp. on Mathematical Models in Automation and Robotics, Sept. 1-3, 1994, Mi~dzyzdroje, Poland, Tech. Univ. of Szczecin Press, 37-40 3. Fliess, M. (1987) Esqnisses pour une theorie des systems non lineaires en temps discret, in: Rediconti del Seminario Matematico, Universithe Politecnico Torino, Fasciolo speciale. 4. Fliess M. (1990) Automatique en temps discret at alg~bre aux diff6rences, Forum Mat. 2. 5. Fliess, M. et al. (1995) Flatness and defect of nonlinear systems: introductory theory and examples, Internat. J. Control 61. 6. Jakubczyk, B. (1992) Dynamic feedback equivalence of nonlinear control systems. Preprint. 7. Jakubczyk, B. (1992) Remarks on equivalence and linearization of nonlinear systems. Proc. Nonlinear Control Systems Design Symposium IFAC, Bordeaux, Fraxice

8. Pawluszewicz, E., Bartosiewicz, Z. (1999) External Dynamic Feedback Equivalence of Observable Discrete-Time Control Systems. Proc. of Symposia in Pure Mathematics, vol.64, AMS, Providence, Rhode Island, USA, 73-89 9. Pawtuszewicz, E. (1998) External dynamic linearization of nonlinear discretetime systems. IV Int. Conf. on Difference Equations and Applications ICDEA'98, Poznafi, Poland 10. Pomet J.-B. (1995) A differential geometric setting for dynamic equivalence and dynamic linearization, in: Banach Center Publications, Vol. 32, pp. 319-339. 11. Willems, J. (1991) Paradigms and puzzles in the theory of dynamical systems, IEEE Trans. Automat. Control 36.

Singular S y s t e m s in D i m e n s i o n 3 : Cuspidal Case and Tangent Elliptic Flat C a s e Mich~le Pelletier Universit~ de Bourgogne Laboratoire de Topologie 9, avenue Alain Savary - BP 47870 21078 Dijon Cedex, France

mpellet ieu-bourgogne, fr A b s t r a c t . We study two singular systems in R 3. The first one is affine in control and we achieve weighted blowings-up to prove that singular trajectories exist and that they are not locally time optimal. The second one is finear in control. The characteristic vector field in sub-RiemannJan geometry, is generically singular at isolated points in dimension 3. We define a case with symmetries, which we call flat, and we parametrize the sub-Riemarmian sphere. This sphere is subana]ytic.

1 Stratification We consider a control system

q(t) = X(q(t)) + u(t)Y(q(t)).

(1)

where q lies in U, some open neighborhood of a fixed point q0 E R 3. T h e germs of vector fields X and Y are C~176The control u is bounded and measurable on some interval [0, 7]. We suppose throughout the paper that (GC1)

O"(qo) = det (X, r , [x, Y]) (q0) r 0.

We compute the singular control along normal extremals. It is denoted by u, and defined on some interval [0, Ts]. We apply the Pontryagin M a x i m u m Principle ([10]). Such a trajectory has an extremal lift (q(t),p(t),u,(t)) in T'U, such that

(ST)

Vt E [0,T,], p(t).Y(q(t)) - O, p(t).[X, Yl(q(t)) - O, p(t).X(q(t)) > O.

Here p(t) denotes the adjoint vector. We set

D(q) = det (Y, [X, Y], ad2Y.X) (q), D'(q) = det (Y, [X, Y], ad2X.Y) (q). Deriving twice the constraint p.Y = 0 and using p.X > 0, we get the singular control as a smooth feedback

p.ad2X.y u,(q) = p.ad2Y.X (q) r D'(q) - u,(q)D(q) = 0

(2)

194

Michble Pelletier

under the condition (GC2)

D(qo) ~k O.

Thus generic points are such that

D"(qo) • 0 and D(qo) # O. The singular vector flow is then defined by

Xs(q) = X(q) + us(q)Y(q) and it is smooth. Since the condition D = 0 cannot be true on any interval along singular trajectories (see [5]) the next stratum is at least of codimension 2, that is, defined by at least two conditions on the Lie brackets of the germs X and Y : D(qo) = D'(qo) = O, D"(qo) # O. T h e singular control is computed at q0by:

p.adaX.Y(qo) - 2us (qo)p.[X, ad2Y.X](qo) - u~s(qo) p.aday.X(qo) -- 0. (3) N o t a t i o n s We denote A(q) _-- det (Y, [X, r ] , adar.x)(q), A'(q) = det(Y, [X, Y], adaX.Y)(q), 3 " ( q ) = det(Y, IX, Y], IX, ad=Y.X])(q). The system has at q0 a singularity of codimension two if

D(qo) = D'(qo) = O, D"(qo) # O, A(qo) ~k O.

(4)

D e f i n i t i o n The system (1) has at q0 a cuspidal singularity if (o's)

D(q0) D'(qo) = O, D"(qo) • 0 A(q0) =7--0, A'(q0) ----0, A"(q0) ----0"

In fact, this is generically a codimension 3 singularity. We will investigate local time optimality. First we give a precise definition of strong time optimality of the singular trajectory (q, (.), us(0). Given two trajectories (q(.), u(.)), (qs(.),Us(.)), both defined on [0, T], we define the WI,1 norm : [Iq(.) - q, (-)111,1 = Sup[o,TI[q(t) -- q, (t)l +

I

lu(t) -- u,(t)ldt.

(5)

D e f i n i t i o n s The trajectory (qs(.), us(O) is strongly time optimal if there exists a Wl,l-neighborhood of (qs(.), us(.)) in which there exists no trajectory

Singular Systems in Dimension 3

195

(q(.), u(.)) defined on [0, T] with T < Ts and satisfying the given initial condition q(0) = q,(0) and the given final condition q(T) = qs(T,). A trajectory (qs (.), us (.)) is strongly locally time optimal if every sufficiently small piece of (qs(.), us(.)) is strongly time optimal. At generic points, it is known t h a t there exist one singular trajectory. It is smooth. It is strongly locally time optimal if D D " < 0. It is slow if D D " > O. (See [4].) In the codimension two case, we prove : T h e o r e m 1 ([9]) Suppose that D(qo) = D'(qo) = O, D"(qo) # O, A(qo) 0, A(q0).A'(q0) > 0. Then there exists a nonsmooth singular extremal trajectory through qo. It contains smooth strongly time optimal ares. They lose their strong time optimality at qo. (See point H on Fig. 1 where we draw the trajectories of X, .) One can define weak time optimality and local weak time optimality in the same way as strong time optimality, only replacing the Wi,i- norm by the following W i , ~ - n o r m :

IIq(.)

- q,(.)[[1,~o = SuPtO,T]lq(t) -- q,(t)l + EssSupt0,TllU(t) - u,(t)l.

(6)

In the cuspidal cas, we prove : T h e o r e m 2 If (CS) is satisfied, then generically there e~ist two singular arcs at qo. The smooth pieces of the singular extremals at qo are not locally weakly time optimal. We prove these results in Section 2. We also give a bifurcation model.

.... c

Fig. 1. Singular field at non generic points

2

Proofs

of the

Theorems

We recall briefly the proof of T h e o r e m 1 (see [9] for more details). Here we have :

(GC)

D"(qo) • O, D(qo) = O, D'(qo) -- O, A(qo) • 0

196

Mich~lePelletier

and we suppose that the surfaces (D = 0), (D ~ = 0) and the vector field Y are in general position. Using semi-covariants under the action of the feedback pseudo-group, we can normalize (1) :

{qo = (o,i,o); q=(x,u,z); Y(q)3o~ b~.z~ o+(x+c(~)u)oX(q) =

1 +~+j_>~,~r

,:( )

oy

0-7

(7)

We suppose c(0)bl,0(0) r 0 thus the system is controllable. After calculation, we get that A'(qo) r 0 and A"(qo) = O.

Definition The associated veetor lield is 2,(q) = D(q)X(q) + D'(q)Y(q). L e m m a 1 If the singularity of (1) is of codimension 2, then X,(qo) has an elementary singularity at qo. P r o o f Usin~ the preceding normal form, we compute the eigenvalues of the one-jet of X~ (q) at the origin : if A.A" :> 0, the eigenvalues are 0, A, - A with A real not equal 0 (point H of Fig. 1), whereas if A.Z~" < 0~ the eigenvalues are O,iA, - i A with A real not equal 0 (point E of Fig. 1) Actually, we will improve Theorem 1 :

T h e o r e m 1' Suppose that D(qo) = D'(qo) = O, D"(q) ~ O, A(q) O, A.z~' > O. Choose the smooth singular arc arriving at qo, and lying in the halfspace D D " < O. Suppose that admissible controls are uniformly bounded by some constant C : In(t) - u,(t)l < C. Then there exists a critical value Co such that if C < Co, then the singular arc is time optimal, if C > Co, then it is not. S k e t c h o f t h e p r o o f o f T h e o r e m 1' We rectify this arc because it is smooth. Then u,(t) = 0 f o r t in [0, Ts], and [u(t)[ < C. First we use the Baker-Campbell-Hausdorff formula and construct a variational vector, the variation being of order 4. The High Order Maximum Principle ([7]) allows us to conclude that if C is large enough, the arc is not time optimal. Next we compare the arc with other extremals, containing singular arcs or regular arcs. We bound the number of switchings if C is small enough. Then the singular arc is the only extremal trajectory steering its origin to q0, hence it is optimal.

Singular Systems in Dimension 3

197

P r o o f o f T h e o r e m 2 First we have to check that there exist singular arcs through q0. We use (7). F a c t Since A'(q0) = A"(qo) = 0, the one-jet of 2 , (q0) is nilpotent. Thus we have to blow it up (see [6] for the method). Straightforward computations give D(q) = 6x (1 + 2b4,o(z)x + ba,l(z)y + o(llx, y, zll)) DS(q) "- y (6z + 61y at- o([l(y, zll) + *" (6' + o(1))

D"(q) = 1 + o(llx, y, zll) with 6, d[1,6~ certain generically nonzero coefficients. Notice t h a t the curve O(q) = / Y ( q ) = 0 has a self-intersection at q0. L e m m a 2 One can suppose that 6t = 1, J > 0, and c(0) > 0. P r o o f Replacing t by ~'71t, y by ~'71y and z by ~'~Tz we get 61 = 1. Replacing if necessary (z, u, y, t) by ( - z , - u , - y , - t ) we get $ > 0. The cases c(O) > 0 and c(O) < 0 are symmetric. Hence x , = (u ~ + 6zu

0

0

+ 6'x ~ + . . . ) ~ + 6 (~ + u(bo,~(O) + b~,,(O)) + . . . ) 0-~ 0 + 6 ~ (c(O)y + ~ +...) ~ .

The coefficient b3,1(O) is not 0 because the point q0 is an isolated singularity on the trajectories of (1). P r o p o s i t i o n The associated vector field has at qo a cusp. One separatrix is repelling and the other one is attracting. P r o o f We make a blowing-up with weights 3,2,4 in the chart z -- 1. It turns out that the terms of quasi-degree no more than 3 are sufficient. These are the terms which are explicitly given in the preceding equation. T h u s we put x = ~5 3, y = ~)52, z = 5 4. After dividing by 5 and replacing bar - letters by x, y, z, we obtain the following blown-up germ of vector field :

+3zx~ \ - 3 + z ~ ( ( J y + J ' x ~) s

~ - Y~ + +6xy(b0,, +b3,~)~)

+ ...

where c ~= 0 stands for c(0), bi,j stands for bi,j(O). Since c > 0 the first singularity is y =

~f~, x = +~1 (2c \ - / ~3/4 , z =

0 and it is elementary. T h e other

198

Mich~le Pelletier

singularity is x = y = z = 0 and we have to blow-up again in the same chart, this time with weights 3, 2, 1. We get

(,v +

+

+ z" (. . .)

with a certain r > 3. Notice t h a t on the exceptional divisor, z - 0, the vector 9 field has 5 ~ [ + ~ - - 3x 2 as first integral. It is no longer true outside of this exceptional divisor. We also blow-up in the chart z = - 1 and then in the other charts, where there is no new singularity. We see that restricted to the exceptional divisor, the blown-up vector field is invariant under the s y m m e t r y x ~ ~ - x . It possesses 6 singularities on the exceptional divisor : two centers, two resonant saddles, a sink and a source (Fig. 2). From [11] we know that the trajectories at q0 are C ~176 We can verify t h a t the singular control is also smooth. We can deduce the phase portrait of X , . One has only to reverse the orientation of the trajectories in the half-space D < 0. (Fig. 3)

t,

i

po

Fig. 2. Phase portrait on the exceptional divisor

R e m a r k Since one arc is repelling and the other one is attracting, there is no nontrivial continuous first integral. Notice that the controllability is essential. We have :

Proposition The vector field X , has two singular smooth extremals arriving onto qo. Both are tangent to X(qo). This is the first part of T h e o r e m 2. For the second assertion of this T h e o r e m , we construct the same 4-order variation as for T h e o r e m 1'. The variational vector is +flad3y.x(o, 0, 0). Thus the High Order M a x i m u m Principle applies and we are done. P r o j e c t e d f a m i l y T h e determining jet of X, is

Singular Systems in Dimension 3

199

\ ~

Fig. 3. Germ X,

(Szy + y~)

0 + 6x0_~+

6c(O)xy~z"

We know that the trajectories are almost everywhere tangent to the distribution span (X(q), Y(q)). Thus it is very natural to project onto the plane z = 0 and to consider the family of plane vector fields with parameter z :

0 + 5zyff--~.

6x0-~ + Y ~xx

C o m p u t i n g further with our normal form we see that actually the germ is a cusp of order 3, whose versal unfolding is described in [8], Figure 3.1.7. This comes from the fact that a cusp of order 2 cannot have 2 singularities at which the divergence is 0. The normal form of a cusp of order 3 is

0

0

The special singularity we encounter in our problem is obtained for Ao=O, AI=-A~,

A~_>O.

This half line is described twice. We have a bifurcation model for our singu-

larity.

200

3

Mich~lePelletier

T a n g e n t e l l i p t i c fiat case

P r e s e n t a t i o n o f t h e p r o b l e m The local normal form of Sub-Riemannian geometries in dimension 3 is the following (see [2]) : the 2- distribution is generated by the germs of vector fields

0 El(x, y, z) = (1 + y~fl(x,y, z)~, a_ o* - xyfl(x, y, ~) ~a + ~7(x, y, z) ~7 F~(x, y, z) = - x y , 8 ( x , y, z) a

+

(1 + x2,8(x, y, z)) ~ - {"7(x, y, z)~-~ (8)

and these two vector fields are orthonormal for the sub-Riemannian metric. In the contact case, 7(0, 0, 0) ~ 0. In the generic Martinet case, 7 has order 1 in x, y, z. We suppose that 7 is quadratic in x, y and does not depend on z. Moreover we suppose that

(EN)

x2 7(x,y,z)= ~+

y2

b---s f l ( x , y , z ) = O .

We call this the elliptic case. It is already known that in this case the abnormal geodesics are not length minimizing and that the small sub-Riemannian spheres are subanalytic ([1]). Since 7 does not depend on z, this is an isoperimetric problem. We have

2

S~(x,y,z)

t,a:~

O L~ O a--d - ~x ( ~x ~ 4- /b,)5";

"

The Lie algebra generated by Ft and F2 is nilpotent and has rank 6. W e i g h t s The weights are 1, 1, 4 and the coordinates are adapted in the sense

ol [3]. P r o o f One has to compute and to apply the definition of [3]. We are going to choose the following model with one more symmetry. D e f i n i t i o n : The symmetric flat case is defined by the system

(SF)

Fl(x,y,z) = s

y2~ o + ~(x2 + , z ~ y21 a 9

The structure of the Lie algebra at the origin is given by

[F~, r2] = r3, [F~, r 4 = F, [F~, r~] = F~, [F~, F4 = [r~, F~] = F~, [F~, F4 = [F~, F,] = o.

Singular Systems in Dimension 3

201

T h e o r e m 3 The system (SF) is completely integrable. If t > 0 is small, the

sphere of radius t can be computed using hyperelliptie integrals. It is subanalyric. The basic facts are the quasi homogeneity of (SF) and the invariance of the sphere under rotation around the z - a x i s . In order to prove this theorem, we are going to compute the normal geodesics, and then the sphere. We denote (p, q, r) the adjoint vector and the initial condition is ((0, O, 0), (cos 8, sin 8, A)). The only non trivial geodesics are normal. We first compute the normal hamiltonian

H(x,y,z,p,q,r)-- -~(p+ r (x u + y2))2 + l ( q - r 2 ( x 2 + y2))2

(9)

and the equations of the geodesics

]c=u=p+r

: (~y_y~)

x~

~( x ~ +y2), ~ ) = v = q - r {( x2+y2), x~ y2 _ (py_qx)__V_+~(x+

y2

+2

r

,+yW.

Then we notice some symmetries : ( t , p , q, ~) ( t , - p , - q , ~) (t,p, q, - r )

, > , > ~ "

(~(t),y(t),z(t)) (-~(t),-y(t), z(t)) (-x(t),-y(t),-z(t))

This allows us to suppose that A _> 0 and 8 E [0, 7r]. F i r s t i n t e g r a l s The two first integrals H and r are clear from (9). We notice that py - qx is also a first integral and that they are in involution. G e o d e s i c s They are parametrized by

x(t) = p(t) cos r

p=(t) + ~p'(t) ~(t) = 88

y(t) = p(t) sin r = 89 $(t) = -}p=(t)

z(t) (10)

A such that ,(A) > Here

~(~) =

2/(~)'~'r ~o v~=:~p ~ dp

202

Mich~le Pelletier

P r o o f F r o m H = 2' we get p~(t) + p~(t)r Hence -p2(t)r

= 1/2. Moreover py - qx = O.

- 2P4(t) = 0. T h e n the computations are straightforward.

The subanalycity is already k n o w n because there is no non trivial abnormal geodesic ([1]). The new fact is t h a t we have an explicit c o m p u t a t i o n and we see that the initial condition is a compact cylinder.

Now using quasi homogeneity, we change the time and the coordinates.Setting 7" = Al/3t, ~ = 0, p = AI/3p, 2 = A4/3Z and denoting by ' the derivative with respect to r, we get the new parametrization

(~,2(7-)) + ~(,-) -7---

1

~(7-) --, ~,(7-) = 1 / ( 7 - ) , 7- e [0, 2,~] 2

2' ~ , ( T ) da

with a = 2 J0

2 " ~ - ~~6 "

S p h e r e There are neither conjugate points nor cut points if r 6 [0, 2a[. P r o o f As A. Agrachev pointed out to us, the s y m m e t r y around the z - a x i s reduces the problem to a plane problem. Moreover, two spheres of different radii cannot intersect. We have to consider the curve (/5(r), 2(r)) with v 6 [0, 2a[. From the above parametrization, the result is clear. This finishes the proof of T h e o r e m 3. R e m a r k There is another normal form due to B. J a k u b c z y k and M. Zhitomirskii ([12]). The distribution is dz + (xz + x3/3 + x y 2 + bx3y2)dy. We take here as a model the nilpotent approximation dz + (x3/3 + xy2)dy. Actually, this is the same sub-Riemannian geometry as above if we take the orthonormal frame 0

x, = ~,

0

x2 = ~

;q -- (z3/3 -}- xy2)~zz

The Lie algebra is the same as before. The computations can be done as follows : we put Pi -- p.Fi. T h e n

/'1 = P2P3, /,~ -- -P1P3, /'3 = - - PIP4+P2P~, /:'4 = P~P6, /'5 -- P2P6(11) a n d / ' 6 -- 0 because of the nilpotency. One can check that P6, P12 + P ] and P1Ps - P~P4 - -~- are first integrals. Thus computations can be done using elliptic functions. Actually, & =/:'1 -cos 0, y -- P~ -- sin 0. Thus 0 - / ' 3 and

Singular Systems in Dimension 3

0"2 ~6 ~

203

04

2 8 for trajectories starting from the origin. Again, the s y m m e t r y around the z - a x i s is clear.

References 1. Agrachev, A. A. (1999) Compactness for Sub-Riemannian Length-minimizers and Subanalycity. Preprint SISSA 2. Agrachev, A. A., Gauthier, J. P. (1998), Subriemannian metrics and isoperimettic problems in the contact case. Preprint Universitd de Bourgogne 3. Bellai'che,A., (1996) Tangent space in sub-Riemannian geometry. Birkh/iuser 4. Bonnard, B., Kupka, I. (1993) Thdorie des singnlaritds de l'appllcation entrdesortie et optimalit6 des trajectoires singnli~res clans le probl~me du temps minimal. Forum Mathematicum, 5, 111-159 5. Bonnard, B., Kupka, I. (1997) Generic properties of singular trajectories. Ann. Inst. Henri Poincard, 14, 167-186 6. Dumortier, F., Ftoussarie, Ft. (1999) Geometric singular perturbation theory beyond normal hyperbolicity. Preprint Universitd de Bourgogne 7. Hermes, H. (1988) Lie Algebra of Vector Fields and Local Approximation of Attainable Sets, S I A M J. Control and Opt.26 715-727 8. Mardesic, P. (1998) Chehyshev systems and the versal unfolding of the cusp of order n. Hermann 9. Pelletier,M. (1998) Time optimality of nonsmooth singular trajectories of afllne one input systems in R 8. Proceedings of the Pontryagin Conference 10. Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, Ft. V., Mishenko, E. F. (1964) The Mathematical Theory of Optimal Processes. Pergamon Press 11. Takens, F. (1974) Singularities of Vector Fields. Publ. Math. IHES 43 47-100 12. Zhitomirskii, M. (1995) Singularities and normal forms of smooth distributions. Geometry in nonlinear control and differentialinclusions, Banach Center publications 32 395-409.

Flatness of Nonlinear Control Systems and Exterior Differential Systems Paulo S6rgio Pereira da Silva Escola Polit6cnica da USP - P T C Sgo Paulo - SP - 05508-900, Brazil paulo@lac, usp. br

A b s t r a c t . Necessary and sufficient conditions for k-flatness are given. We construct an exterior differential system (27,f2) and show that (local) k-flatness is equivalent to the existence of (local) integral manifolds of (27, f/), which is in turn equivalent to the existence of a solution of a partial differential equation. As a consequence, the k-flatness of a nonlinear system can be checked with convenient applications of Cartan-Kghler and Cartan-Kurauishi theorems. Some academic examples are presented to illustrate the result.

1

Introduction

The notion of Differential Flatness of nonlinear control systems was introduced by Fliess et al [4,6] 1. In [8], it is shown t h a t checking flatness of a system is closely related to the problem of finding integral manifolds of exterior differential systems. In this paper we show how to c o m p u t e this exterior differential system using some symbolic operations. We m a y summarize our main result in the following way. It is well known t h a t a system is fiat if and only if there exist a set y of m differentially independent functions (where m is the eardinality of its input) such that the state and the input m a y be c o m p u t e d as a function of y and its derivatives. This last condition is shown to be equivalent to say that the jet of the fiat o u t p u t is an integral manifold of an exterior differential system (Z, ~ ) . Our main result is a systematic (and a rather economic way) for constructing (Z, s that can be implemented by using computer algebra. From an algorithmic point of view, we state a general m e t h o d for constructing a partial differential equation for which the existence of solutions is equivalent to the k-flatness of the system. After this construction, one could apply the theorems of C a r t a n - K ~ h l e r [1, Chap.3, 6 and 8] and C a r t a n - K u r a n i s h i prolongation theory [12] for checking flatness of a given nonlinear system. Recall that C a r t a n - K ~ h l e r theory states that if an exterior differential system is involutive then, for every integral element E~ at a point ~, there exists integrM manifolds whose tangent space 1 By reasons of space our list of references is rather incomplete. The reader may refer to [7] for a more complete bibliography.

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P . S . Pereira da Silva

at ~ coincides with E~. A l t h o u g h this is only a sufficient c o n d i t i o n for t h e existence of integral manifolds, C a r t a n - K u r a n i s h i p r o l o n g a t i o n t h e o r e m assures t h a t , if a s y s t e m a d m i t s integral manifolds, then after a finite n u m b e r of prolongations, one o b t a i n s an involutive s y s t e m . A l t h o u g h those results are "existence t h e o r e m s " , we believe t h a t the s y m b o l i c o p e r a t i o n s defined here are useful for finding the flat o u t p u t in practice, as illustrated in the e x a m p l e 1. T h e c o m p u t a t i o n s for even a s i m p l e e x a m p l e are r a t h e r c u m b e r s o m e 2, so the a p p l i c a t i o n of this m e t h o d in interesting e x a m p l e s m a y d e p e n d on t h e d e v e l o p m e n t of a s y m b o l i c c o m p u t a t i o n p r o g r a m . If a system is flat, by definition it is k-flat for k large enough. Hence, an i m p o r t a n t question r e m a i n s open, namely, if there exists a b o u n d on k d e p e n d i n g on the d i m e n s i o n of the state a n d of t h e i n p u t [8]. This p a p e r is organized as follows. In Section 2 we present n o t a t i o n s a n d some preliminaries. In Section 3 we recall well-known c h a r a c t e r i z a t i o n s of 0flatness a n d s t a t e an auxiliary result. In Section 4 we present the m a i n result. In Section 5 we a p p l y our results to some a c a d e m i c examples. In Section 6 we s t a t e some conclusions. Some a u x i l i a r y results are proved in A p p e n d i x A.

2

P r e l i m i n a r i e s and N o t a t i o n s

2.1

Notations

T h e field of real n u m b e r s will be d e n o t e d by ~ . T h e set of n a t u r a l n u m bers (including zero) will be d e n o t e d by N. T h e subset of n a t u r a l n u m b e r s {1 . . . . , k} will be d e n o t e d by [k]. Let Z be a set. T h e n card Z s t a n d s for its cardinality. A s y m m e t r i c m u l t i i n d e x K of class s and length k is a set o f elements of the form ( i l , - - - , i k ) , where ij E [s] for j E [k], and where all the p e r m u t a t i o n s of (il, 9. . , ik) are identified with each other. T h e set of all multiindices of class s is denoted by L'(s). T h e length of K E 27(s) is d e n o t e d by ItKtt. Note t h a t if K E ~7(s), IIKII = k is identified to a unique ( i l , . . . , ik) such t h a t il _< i2 < . . . _< ik < s. Given K E ~7(s), K = ( i l , . . . , i k ) , a n d i E [s], then (Ix'i) s t a n d s for ( i l , . . - , ik, i) E r ( s ) . We will use the s t a n d a r d n o t a t i o n s of differential g e o m e t r y a n d exterior algebra [20,1]. Let ~ be a s m o o t h m a n i f o l d of d i m e n s i o n p. Let j r be a set of s m o o t h functions defined on 9 . Let Z = {z E ~ ] f(z) = 0, V f E j r } be the set of c o m m o n zeroes of all f E j r . T h e n z E Z is an ordinary zero if: (i) T h e r e exists a subset F = { f l , . . . , f r } C j r such t h a t the set d f = ( d f l , . . . , d f r ) is independent on z; (ii) There exists an open n e i g h b o r h o o d U of z such t h a t the set of c o m m o n zeros of F t h a t are inside U coincides with Z U U. In particular, Z U U is a s u b m a n i f o l d of T'. As claimed in [8].

Flatness of Nonlinear Systems

207

Given a vector field f and a 1-form w on :P, we denote w(f) by (0~, f ) . T h e set of s m o o t h k-forms on :P will be denoted by Ak(7~) and A ( P ) = OkeNAk(P). Given two forms r / a n d ~ in A(~o), then 7/A ~ denotes their wedge product. The exterior derivative of ~/ 6 A ( P ) will be denoted by d7l. Note t h a t the graded algebra A(P), as well as its homogeneous elements Ak(7)) of degree k, have a structure of C ~176 ( P ) - m o d u l e (see [20] for details). Given a family = ( v l , . . . , vk) of a C~176 then span { v l , . . . , ~k} stands for the span over c ~ 1 7 6 ). An ideal Z is a C ~ (T))-submodule of A ( P ) such that, given two forms w and 0 in 27 then w A0 E 27. Given a subset S C A(:P) then {8} stands for the least ideal that contains 8.

A differential ideal Z is an ideal t h a t is closed under exterior differentiation, i.e., dZ C 27. A differential ideal is also called an exterior differential system. We present some definitions and results about exterior differential systems. The reader m a y refer to the treatise [1] for details. Let M be an analytic manifold of dimension m and let Z be a differential ideal defined on M . Let F2 = Wl A . . . A wn, I2 ~ 27 be an n-form on M . Then the pair (27, 12) is called an Exterior Differential System with independence condition. An integral element E of:/: on x 6 M is a subspace of T~M such that 0]E = 0 for all forms 0 of Z. An integral element of (27, I2) on x E M is a subspace E of dimension n of TxM such that 0le = 0 for all form 0 of Z and DIE r 0 (this last condition is called independence condition). An integral manifold of (27, 12) is an immersed manifold i : N --+ M of dimension n of T~M such that, for every point ~ E N, its tangent space E = i.(T~N) is an integral element of (27, 12), or equivalently, i*Z = 0 and i* I2 r 0. For every immersed manifold i : N --+ M (not necessarily an integral manifold) we define the restriction of (Z,D) to N by (27,12) = (i'27, i'I2). We denote by Gn (TM) the Grassmann bundle of all n-subspaces E C T~:M. The bundle of all integral elements E of 27 of dimension n is denoted by Vn (27) and is a subbundle of Gn (TM). Similarly, Gn(TM, 12) denotes the bundle of all n-subspaces E C T~M satisfying the independence condition Y2IE ~ O. The bundle of all integral elements of (27, I2) is denoted by V,, (27, I2) and is a subbundle of Gn (TM, 12). Let (27, 12) be an exterior differential system defined on a manifold M. Let N be an immersed submanifold of N and let t : N --+ M be the corresponding immersion. Let 45 = t*I2 and let s = t'27. Assume that r is a volume form on N. Then (s r is an exterior differential system with independence condition defined on N and called the restriction of (27,12) to N. We denote by fir (X, Y) the set of r-jets of all s m o o t h maps y : X -+ Y between s m o o t h manifolds X and Y. T h e n fir (X, Y) has a structure of s m o o t h manifold. For instance, consider the manifold fly(X, Y), where X has di-

208

P.S. Pereira da Silva

mension n and Y = ]Rm. If X has local coordinates x = ( x l , . . . , x fir (X, Y) has local coordinates Zi

a ,y a ,YK : i 9 tnl,a 9 tm],K 9 S(n),llKII < r),

n)

then

(1)

where ya E ~ represents the function evaluation ya (x) and y~ represents, for OkY~ . In a symmetric multiindex K = (il . .. ik), the partial derivative 0~'~...0~'k fir (X, Y) one m a y define the contact forms (see (6) with u = r - 1) and the exterior differential system (27, $2), where 27 is the exterior differential system generated by the contact forms, and where 12 = d x 1 A . . . A dx n. Note that partial differential equations are nothing more than relations between the coordinates of a jet-space that m a y be represented by an immersion N --~ f i r ( x , Y ) . The submanifold N is called restricted jet-space and the exterior differential system (/2, 4) corresponding to the restriction of (27, I2) to N is such that any solution of the partial differential equations corresponds to an integral manifold of (Z, ~i). 2.2

Diffleties, S y s t e m s a n d F l a t n e s s

We recall briefly the notions of diflieties, systems and flatness. For details the reader may refer to [5,17,7]. We consider analytic nonlinear control systems of the form 3

(2)

ki(t) = f i ( x ( t ) , u ( t ) ) , i = 1 , . . . , n ,

where x(t) evolves on an open subset X C ~ n , and the input u(t) is in the open subset U (~ C ~m. Associated to (2) one m a y define the diffiety S of (global) coordinates {t, r, (u (k), k 9 l~l)} and the Caftan-field d

0

d-7 =

~,

0

+ '=

m

k+l)

+ kel~l j = l

0

OUJk) "

A system is said to be (locally) flat if the diffiety S admits a (local) coordinate system {t, (yJk) : j 9 [m], k 9 N)} for which the Caftan-field is (locally) given

by

d

d-7 =

0-70 V ' ~ y ~ k + l ) +

kEl~j=l

0

O.,vj k

)"

The family of functions y = ( Y l , . . . , y m ) , where yj = y J ~ j 9 Ira] is called flat output. Note that the flat output m a y depend on the 3 There is strong reason to consider analytic systems. In fact the Cartan-Ks theorem does not hold in the smooth category because it is a generalization of Cauchy-Kowalevsky Theorem.

Flatness of Nonlinear Systems

209

state x and on the input and its derivatives u (s) for s = 0 , . . . , k - 1. When such k is minimal, the system is said to be k-flat. W h e n there exists a flat output that depends only on the state, the system is said to be 0-flat. We will assume that the state-representation (2) is well-formed (see [19]) i.e., Of/Ou has constant rank m for every (x, u) E X x U (~ This is equivalent to say that duj 9 span {dx, d~} , j 9 [m]. This a s s u m p t i o n is not restrictive since one m a y obtain a well-formed state representation by d y n a m i c extension, i.e., by adding one integrator in series with each input component. For instance, one m a y check 1-flatness of a state representation that is not well-formed by checking 0-flatness of the dynamically extended state representation.

3

A C h a r a c t e r i z a t i o n of O-Flatness

Consider a nonlinear system of the form (2) with o u t p u t y = ( Y l , . . . , y,n) given by

yj = hi(x), j E [ m ] .

(3)

The following proposition gives a well known characterization of 0-flatness 4 according to the definition of flatness given in [7]. For this consider system S given (2) as a diffiety (see w 2.2) and define on S the codistributions Ya -span { d y , . . . , d y (a)} and Yt, = span { d x , d y , . . . , d y ( k ) } , k 9 1~1. P r o p o s i t i o n 1. [13] Assume that the state representation (2) of S is wellformed (see w 2.2). Then S is locally O-flat around ~ 9 S with flat output y if and only if there exists k* 9 I~ such that { d y , . . . , d y (k')} is linearly independent in ~ and span {dx} Iv C Yk. I- for ~ in some open neighborhood

u oy~. Remark I. This result was originally stated with the additional a s s u m p t i o n that span {du} I,, C Y~. I-, v 9 U [13,8]. The extra a s s u m p t i o n is not necessary for well-formed state representations. A proof of this result is available in [16]. With some regularity assumptions, the characterization of 0-flatness above m a y be related to the algebraic structure at infinity [3]. P r o p o s i t i o n 2. Consider an analytic system S defined by (2) and assume that the state representation (2) is well formed (see w 2.2}. Consider the output (3). Assume that ~ 9 S is a regularpoint ofY~ and y k , k e { 0 , . . . , n } . Then S is (locally) O-flat around ~ with (local} flat output y if and only if there exists k* 9 In] such that one of the following equivalent conditions are satisfied: 4 In [5,17] it is shown that, if u is the input of a system and y is a local fiat output, then card y = card u (see also [16] for an alternative proof).

210

P.S. Pereira da Silva

(i) The algebraic structure at infinity {~1,---, ~,} obeys the conditions k* - 1

n+~'-~i=l ~ i = m k * , a k < m f o r k < k * and~rk=mfork>k*. (ii) span {dx} C Y k ' - i and span {dz} ~ Yk, for k < k* - 1.

(4)

Proof. See Appendix A. Remark 2. If~ 9 S is a regular point of Yk and Y k , k 9 { 0 , . . . , k * - 1}, and ~rk. = m, then Lemma part 6 of 2 in Appendix implies that ~ is a regular point of Yk and Yk, k 9 1~.

The following result is instrumental for the computations of Section 4. P r o p o s i t i o n 3. Assume that system S with state representation (2) is locally O-flat around ~ with flat output, y and that ~ is a regular point of Yk,~))k,k 9 {0,...,k*}. Let { ~ 1 , . . . , ~ , } be the algebraic structure at infinity of this system. Let k* be the integer considered in Prop. 2. Then, there exist a nested family of subsets of the input u D Ul D . . . D uk.-1 with card (f~k) = m -- qk, such that the family {dy(O),..., dy(k"- 1), d ~ k ' - 2 ) , . . . , d~(O) 1} is a basis o/span {dz, d u , . . . , d u ( ~'-2)} around ~.

Proof. See Appendix A. 4

Necessary

and

Sufficient

Conditions

for k-Flatness

In this section we give necessary and sufficient conditions for 0-flatness of a system (2). This section is organized in the following way: 1. In w we construct a jet-space 2t7/ and an exterior differential system (Z, 12) that is generated by some contact forms; 2. In w we define a symbolic formalism for computing the time derivatives of the outputs as a function defined on the jet-space 2iS/; 3. In w we show two different approaches for obtaining necessary and sufficient conditions of 0-flatness. 4. In w we present the generalization of our main results to considering the problem of k-flatness. 4.1

Construction of the Jet-Space

Consider the system (2), fix k* E In], set u = k* + 1 and let U (k) = ~rn, k E { 0 , . . . , v}. Consider the manifold Z = X • U (~ • ... • U (v), of dimension p = n + (v + 1)m, with global coordinates

(z': i 9 tpl) = (x', u~k): i 9 tnl,J 9 Lml,k 9 (0,...,,})

(5)

Flatness of Nonlinear Systems

211

Consider Y = 1~'n with global coordinates ( y l , . . . , y m ) . Consider the jetspace , I u+l (X, Y) and let I u+l be the codistribution generated by the corresponding contact-forms:

dyg.

(7

o"

j

- ?-..,j=l Ykj d x , o" E L m ] , k 9 Lnl,

(6)

Ilgll _< ..

d y ~ - Y'~.~=I Y~j d x j , ~ 9 L~], K 9 E(,,), Now let ~/=

U (~ x . . . x U (~) x J ~ + l ( X ,

Y)

with global coordinates

{~',.~),y~, y~ : i 9 t-l,J 9 Lml,k 9 {0, ...,~},

(7)

~ 9 Lm],g 9 E ( n ) , I I K I I _ < v + 1 } . Denote by ] the pull-back of I ~+1 from J " + I ( X , Y) to ,~/. Let :~ = {[, d]} and let = dx 1 A...Adx" ~

A du~ ~ A...Adu(m~ A . . . A du~v) A . . . A d u ~ ) .

(8)

~

Then (Z, 12) is an exterior differential system with independence condition defined on M. It is easy to verify that an integral element Ef = span {e,, i 9 t~l} of (Z, D) is given b y : 0

" a----i

~0

"1

o

0

.

E y: yj, Ty,,etnl,

(9)

a----I aE~(~)

llalI_<~+~ 0 ei = OuJk ) , i = n + j ( k + 1), k E { 0 , . . . , v}, j E [m],

(10)

where y~p = y~q if the symmetric multiindex (Jp) may be identified with (gq). The y~: are functions defined on M for IIKI] < v + 1, but this is not the case of Y~i for ]lJll = v + 1 that are in fact the coordinate functions added to the ones of M in order to construct coordinate functions of the prolongation .Q(I) of .~/. 4.2

A Symbolic Formalism

Consider an output y'~ = ha(x) of system (2). Let K - ( i l , . . . , i k ) E Z ( n ) be a symmetric multiindex. Denoting the partial derivatives O k y a / O ( X i l , . . . , x ih) by y~, one may compute symbolically the output derivatives of any order. From an algorithmic point of view, the equations (22) and (23) obtained

212

P.S. Pereira da Silva

in the end of this section can be used for the computations in the examples. However, it will be useful to consider this idea in a more abstract setting, at least for establishing the proofs of our results. Consider the vector field f defined on Z given by n

rn

i=l

J 0mi

v--1

0UJ k ) "

j=l k=0

Let y = h(z) be an output for system (2). Then it is easy to show that (see [14])

y(k) = L ~ h ( z ) , k E { 0 , . . . , v + 1}.

(11)

By Prop. 1 we conclude that 0-flatness is completely characterized by the vector field f on Z. From now on we will say that S is 0-flat on ~ E Z if there exists an analytic output y = h(x) and some k* E [hi such that the relationship span {dr} C span { d y , . . . , dy (k') } is satisfied locally around and the set { d y , . . . , dy(k')} is linearly independent on ~. Consider the tangent vectors ei defined by (9)-(10). Let Y~Ji= 0 for a E [m], i ~ [n~ and IlJll -- ~ in (9). Note that this assumption defines vector fields ri on M given by 0 m o0 ~-, r,=b-~zi + ~ l u / b - 7 + , . . . , ~ =

o 0 yji-ff~yj,ieLn],

(12)

o ' = 1 J'E.C'(~ )

liJIl_<,, O

Ti

Ou~k),i = n + j(k + l ) , k e { O , . . . , v } , j

e tm].

(13)

By definition, it follows t h a t / ~ = span {v/(~), i E [p]} is an integral element. IfE~ = span {ei(~), i E Lp]} is another integral element, then by (9)-(10) and (12)-(13) it follows that ri(~)-eiespanLoy

0,j j

~S(n),llJIl=J'+l

} ,~M,

ie[~].

(14)

Define the vector field f on M by

]

-, +

=

i=t

rn u--I

j=~ k=o

u?

o cgu~k) "

L e m m a 1. Consider the filtration 1"o C ... C ['L, C T*I~I, where: Fo = span {dxi,dy ~ : i e [n],a E [ m ] } ,

(15)

F1 = F0 + span {dy~.,duj : i E Lnl,,r,j E Lml},

(16)

Fk+l = Fk + s p a n {dyj,du~k+t): J E

~:(n), IlJll--k

+

x,j e [m]}.

(17)

Flatness of Nonlinear Systems

213

Then L~-ro c rk, k e L~] Proof. See Appendix A. The following definition establishes our symbolic formalism. We point out again that this formalism is essentially equivalent to considering symbols y~: standing for partial derivatives Oky~/Ox K. D e f i n i t i o n 1. The symbolic k-derivative of ya is defined by Lkl-y~ , k E {0,..., ~}. Note that the Lkl-yV is a function defined on /17/. The next result shows that, when specialized to an integral manifold of (:~, J~), the symbolic derivatives of ya coincide with the usual time derivatives of outputs. P r o p o s i t i o n 4. Let i : Z --~ l~I be a local integral manifold of (Z, [2). Then (Lkfy a) o i = L~^(y ~ o i ) , k e {0, 1 , . . . , v } ,

(18)

i* (dLkl-y~ = dLkl^ (yO o i), k e {0, 1 , . . . , u}.

(19)

Proof. Note that (19) is a consequence of (18) and from the fact that d(r = i ' d e = di*r for all functions r on M. We show (18) by induction on k. For k = 0, (18) is obvious. Assume that it holds for k. Taking local coordinates (5) for Z, note that i* o0~, = ei, where the ei, i C [p] are of the form (9)-(10) (this is a consequence of the fact that i : Z -+ M is an integral manifold of 27). By (14) and from the definition of ], it follow that ](i(z)) - i*f(z) E span {0--~ : HJII : v + l } .

Now note that

Lf[Lkf(yVoi)]----

Lf[(L~y~

:

(d[(L~y o) oi], f ) = (,9. d L fky a , f ) : (dL~yV i . f ) . By Lemma 1 and from the fact that F~ C span {o--~ : I [ J H - - u + 1}, it follows that we may replace i . f by f in the last equation, showing the desired result. Remark 3. If system (2) is affine, then it is easy to show by induction that Lfyk a, k E {0, ... , v} are polynomials in the variables (u (~ . . . , u (k-i)) with coefficients that are analytic functions of x, yO and the y~. Let i : Z --~ M be an integral manifold of (:Z, J)). Choose coordinates (5) for Z and (7) for .~/. Since i*:~ = 0 and i*J~ r 0, the ]~-linear operator i* has the following properties: i*dx i = dx i, i E [n],

i'du~ k) -- duJ k), J ~ tin1, k ~ {0,..., ,~}, i*dy a E~'--I(Y~ ~ dxi, (7 E I - d , i*dy~ = ~ = nl ( y J i a o i)d~ ~, ~ ~ t'.], J ~ x ( . ) , IlJll < ~-

(~o)

214

P.S. Pereira da Silva

Equation (20) shows that i* transforms every form in Ft, of L e m m a (1) into a form in T * Z contained in span { d x , (du~ a) : j E [ m l , k E {0, 1,...t~}}. T h e next definition is in fact a symbolic operation that is closely related to the effect of i*. D e f i n i t i o n 2. The C~176

operator

rr: F~ --+ span { d z ' : i E [ d

}c

T*~

is defined by: rrdx i = dx i, i E [n],

.d=5 k) = du?), j e Lml, k e {0 . . . . , . } , v.-.,fi

a-

i

rdY a 2-.,i=l Yi ax , er E [rn], rrdy~ = y']~in__lY~idx i, a E [m 1 , J E ~7(n), IljH < v.

(21)

Remark 2;. The following points will be important in the sequel: 1. Note that forw E F,~ then ~rw is the unique 1-form in span {dz i : i E [p] } such that (w - 7rw) mod ] = 0, where ] is the codistribution generated by the forms (6) (this fact is a simple consequence of the decomposition T ' , ~ / = span {dz} | I | span {dy.~ : ~ E t " l , J E ~ ( n ) , I I J l l = u + l}). 2. Let i : Z -+ M be an integral manifold of (M, S2). Note that :r is a symbolic projection with the property that i*rr = i*. Let O C F~, be a codistribution. Then dim rrO]i({) = dimi*Olr In other words, the operator i* preserves the dimension of ~rO. The following proposition shows that the computation of 7r(L~-dya) depends only on the computation of 7r(L~--ldy('). P r o p o s i t i o n 5. , ( L ~ d y a) = Lf{rr(L~--ldy")}, k E [~] Proof. See Appendix A. Note that the last proposition simplifies the computations since we only have to work with 1-forms in span {dz i : i E [p] }. It is easy to show by induction that

y: = L i f = ~

y~S~(:, ~),

" a = L fy~ = ~-~in__,Ysifi( a X , u), yy OL k a a a k+l a n f i ~ a-i m (j+l)~ Lf Y = ~i=1 + ~ i = o ~ i = 1 ui o,,,"' +

n

Y"~l_
(gL~Y

y,~

~

(22)

Flatness of Nonlinear Systems

215

By using the rule, rdy~j = z..,i=l X-~n YJi a ~,~xi , one obtains also

n y.ZAL_dxi. ,--~k-1,--,,n ~du!J)+OL,,h ~rdLk/ya = )-'~i=1 ax, "t- 2.,j=0 ~--,i=1 Ou~"

'

(23)

ElIJll
4.3

Necessary and Sufficient Conditions

of 0-Flatness

We present two different approaches. T h e first one is more effective and simple (this fact will be explained further in the presentation of the second approach). F i r s t a p p r o a c h We will restrict the jet-space ~ / constructed in w by adding the constraints 5 of the characterization of Prop. 2 part (ii). Notice t h a t a n y local fiat output Y induces a (local) integral manifold i : V C Z --> ~?/ of (Z, s Proposition 2 part (ii) and (11) implies that

span{dx}Cspan{dL~(y

a oi):ere[m],ke{O,1,...k*-l}}.

(24)

Define the codistribution y defined o n / ~ / by

f d L ]y ~ ,, : ~ E [m], k C {0, 1 , . . . k * - 1} } . y = span ~,

(25)

By Prop. 4, the condition (24) is equivalent to have span {dx} C i*Y. From Part 2 of Rem. 4, this is also equivalent to span {dx} C i*lry. From the fact that the image of zr is contained in span {dz i : i E [p] }, by the first two equalities of (20) it follows t h a t span {dx} C Try. Let ( a l , . . . , ak.) be a nondecreasing sequence of natural numbers satisfying condition (4), and let u D 31 D . 9 9 D u k . - t be a nested family of subsets of the input such that card uk = rn - ~ra. By arguments similar to the above, the condition of Prop. 3 is equivalent to saying that

{i'wdL~-y a :~r E [m],k E { 0 , 1 , . . . k * -

1},d~k'-~),...,d~(~

is a basis of span {dx, d u , . . . , d u (k'-2)} C T * Z . Since the operator i* preserves dimensions when applied to a codistribution zrF for F C F~,, then, from Prop. 3, around a point i(~) E M the set B given by B = BI U Y~,

Lm],k e {0,1,...k* - 1}},-

(26)

/~ = { d ~ [ k ' - 2 ) , -.. , d~(~ 1} . 5 We will assume that we are working around a regular point of the codistributions gk,Yk,k E {0 . . . . . k*}.

216

P.S. Pereira da Silva

must be a basis of span {dx, d u , . . . , du (k'-2) } C

T*M.

A l g o r i t h m 1 For a given sequence {~rl,... , ha. } satisfying (4), execute the following steps: 1. C o m p u t e , using (22) and (23)

zrdLkl'Ya : Z

a~dzi'

k E {O,...,k*-

1}

i=1

2.

3. 4.

5.

where z i are defined by (5). These c o m p u t a t i o n s can be done also by using proposition 5. Write the square matrix R((), ~ E M with p - 2m rows corresponding to the row vectors ( a ~ , . . . , a~_m) , describing the forms of (26) in the basis induced by coordinates (5). Inverting R(() formally, one obtains R - I ( ~ ) = cofR(~)T/det R(~). Referring to eq. (26), note now that card B1 = rnk* and card B2 -" n - r e . Note also that 131 is a basis of 3) = zry, where~ y is defined by (25). Assuming that det R(~) r 0 around some ~ E M , then the last n - m columns of cofR(~) T will generate 3) • Denote the matrix formed by these n - ra columns by R((), and let A(~) be formed by the first n rows o f / ~ . Note now that condition 3) • C span {dx} • is equivalent to the equalities {A(~)}ij = 0, i E In], j E [n - m ] . If system (2) is affine, by rem. 3, it is easy to verify that {A(~)}ij are polynomials in the variables ( u ( ~ u (k'-2)) with coefficients that are analytic functions of x, y~ and y~.. Denoting by {mr, r E N} the set of monomials in these variables, we can write {A}ij as a finite sum )-'~, r r m ," As the conditions {A(~)}~j = 0 must hold on some open subset of Z, then the conditions {A(~)}ij = 0 are equivalent to

r

: o, i E L-I,J E Ln - - , 1 , r E Ls~l.

(27)

We stress that the r are analytic functions of x, ya, and Y~2-If the system is not affine, a similar construction can be made. In this case the {A}ij will be polynomials in { u ( 1 ) , . . . , u ( k ' - l ) } with coefficients r t h a t are functions of x, u (~ yO, and y.~. We are ready to define our exterior differential system (Z, 12). Note that, since restrictions (27) depend only on the coordinates of y , + l (X, Y), to search the integral manifold in this jet-space it is not necessary to include the inputs and their derivatives 6. 6 If the system is not affine then a similar construction could be made on U (~ x

y~+~(x, v).

Flatness of NonlKtear Systems

217

D e f i n i t i o n 3. Assume that system (2) is affine. Consider the jet space MI = f f v + l ( X , Y ) , let Z1 be the exterior differential system generated by the contact forms (6) and let 121 = dx 1 A . . . A dx n. Assume t h a t the analytic restrictions (27) define a local analytic immersion 7 t : M ~ M1. Then we define 27 = t'271 and 12 = t'121. If system (2) is not affine, we consider M1 = U (~ x J L ' + ~ ( X , Y ) , 121 = dx 1A . . . A d x " A d u (~ and define (27,12) accordingly. The last construction allows us to state the main result. T h e o r e m 1. Let (2) be an affine well-formed system. Then system (2) is Oflat around some ~ E Z that is a regular point of the codistributions Y~, ya, k E I~I if and only if the following three conditions hold: (i) There exist a nondecreasing sequence of natural numbers ( a l , . . . , a k . ) satisfying condition (4), a nested family of subsets of the input u D ul D 9 .. D ~k~ such that card uk = m - ~k and ~ E M whose canonical projection on Z is ~ and furthermore the set of 1-forms

{TrdLkz-y~':~r E t m l , k e {0, 1 , . . . k " - 1 } , d ~ k ' - 2 ) , . . . ,d~(2)1} are independent in ~. (ii) (Z, 12) admits a local integral manifold i : V C X -+ M defined around x E X such that the immersion of i(z) in M1 is the canonical projection of ~ onto M1. (iii) The codistributions span {dx, (dL}-yC')oi:a E Lml,k E {0, 1 , . . . k ' } }

are nonsingular around ~. Proof. If (i) is satisfied then we m a y construct (Z, 12) as in Algorithm 1 If (ii) is satisfied, then the functions y~' o i : V C X --+ l~ are (local) o u t p u t s of the system. From (ii), (iii), from the identity i*rr = i* and from Prop. 4 it follows that the conditions of Prop. 2 are satisfied for these outputs. In particular the system is 0-fiat. If the system is 0-fiat and the assumptions of regularity of Yk and yk hold, from the construction of Algorithm 1 it follows t h a t (i), (ii) and (iii) are satisfied.

Remark 5. Condition (iii) holds generically on V C Z for a given integral manifold. Even if (iii) were not true in some point ~ (i.e., ( is not a regular 7 This is true for instance if we are working around an ordinary zero of the conditions {r = 0, r E [ v ] , i E [n], j E [n - m]} (see w 2.1). If this is not the case, one has to study the strata of the analytic submanifold defined by conditions (27).

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point of the Yk,J;k, k E { 0 , . . . , n}), (i) and (ii) assure t h a t span {dx} C Yk'-1 and the nonsingularity of the Yk, k E { 0 , . . . , k* - 1}. Hence, by Prop. 1, the conditions (i) and (ii) are still sufficient conditions for flatness of the system around ~. We stress t h a t a similar result holds for non-affine systems replacing X by X x U (~ and taking 12 -- $'21 -- dx 1 A . . . AdxnA du (~ S e c o n d a p p r o a c h This approach is based on the ideas of [8]. It is a direct consequence of the characterization of 0-flatness of Prop. 1 and the symbolic formalism of w 4.2. It m a y appears more simple t h a n the first approach, but it produces an exterior differential system that is not Pfaffian. Hence, to apply Cartan-K~ihler theory one m a y c o m p u t e the first prolongation. This c o m p u t a t i o n produces much more relations than the n - m restrictions found in the first approach. Furthermore, this second approach contains all the possible solutions of (4) for a given k*, whereas the first approach can check for flatness for a fixed structure at infinity obbeying (4).

k" 7rdLkfya T h e o r e m 2. Consider the notations of w Let w = A an= I Ak=o Let f_. be the exterior differential system generated by (6) and the forms wi = dx i A w , i E [,2]. Let I2 be the volume form on M given by (8). Then the system S defined by (2) is locally flat around ~ E Z if and only if there exists some k* E 1~ and a local integral manifold i : Z -+ M of (12, 12) such that ~li(r r 0. Proof. Note that the transversality condition a; ~ 0 means that { d ( y o i ) , . . . , d(y o i)( ~*)} is linearly independent in ~ and the conditions a;i - 0, i E In] mean that dxi E s p a n { d ( y o i ) , . . . , d ( y o i ) ( k ' ) } , i E In]. Hence the result follows easily from Prop. 1, equation (11) and Prop. 4. Remark 6. Consider the notation of L e m m a 2 in the appendix. Consider a partition h -- (]~k, hk), k ---- 0 , . . . , k* such that card (]~k) = ak, where the integers ah satisfies (4). A third approach that is very close to the first one m a y be obtained based on the relations : d~(~~) ^ d x ' ^

. . . ^ dxn ^ . . .

^ d~~

. . . ^ d~(~) = 0 , k = 0 , . . . , k *

- 1

d~ ^ A ~ ' 0 d~(~) ~ 0 h ~ ' 0 dh(k) ~ 0 4.4

Necessary and Sufficient Condition

of k-Flatness

Consider a system S defined by (2) and consider the d y n a m i c extension of system S obtained by putting k integrators in series with the input m

](~)+ ~]~j(~)vj I

=

j----1

Flatness of Nonlinear Systems where f = E n = l

u(~-I)T)T

fi(x,

u(~ ~r~,+ E ~ - - 1 E ~ - 0 ~ uJq+l) ~

, and ~j = ~ o

219

~ = ( x T , u(O) T, .. ",

, j E [m]. It is clear from the definition of flatness

of [7] that system (f, g) defined by (2) is k-flat if and only if system ( f , ~) is 0-fiat. In particular, the following result holds:

System (2) is k-flat if and only if the conditions of Thm. 1 (or the ones of Thm. 2) hold for system (f, ~). C o r o l l a r y 1.

5

Examples

We begin this section with a very simple example, for which it is well known that x 1 and x 2 are flat outputs. T h e flatness of this system could be verified based on known results (for instance results about systems with n states and n - 1 inputs or the results of [15]). However this example is useful for illustrating the method and for exhibiting the complexity of its computations.

Example 1.

Consider the system

~ i = Ul ' X2 = X3Ul, x3 :

(28)

u2.

Here we have n = 3 and m = 2. T h e only possible solution of (4) is al = 1,or2 = 2, a3 = 2, and k* = 2. Hence the basis of Prop. 3 is of the form {dy t , dy 2 , d~tl, dy 2, dul } where ul can be either ut or to u2. Consider ~1 = u2. Using (22) (23), one m a y compute the m a t r i x R of w obtaining

~rdy~

/ Y~ Y~

rcLfdy 1 ~rL f d,

Y]

0 0

/

= |Y~ il~ Y~ + y~ut y~ + y~x 3 y~ l ~ ~l~Oy~ + y~ U Y~ + Y~Xa Yl~

(29)

where r denotes Lfr Condition (27) is obtained by c o m p u t i n g the last n - m columns and the first n rows of c o f R T, resulting in: det ( ~ i ~ i

detF = 0

det(:i

det F = 0

det [' y~ yal

detF=O

yl

(30)

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P.S. Pereira da Silva

where F=

/y,'

If the three determinants multiplying det F were all equal to zero, then the first two rows of R would be linearly dependent. Hence det F = 0 corresponds to the constraints (27). To construct the exterior differential system (Z, 12), assume that (y~ +y~x 3) ~s 0 locally. Then the condition det F = 0 is equivalent to the partial differential equation: y32 - ~31(Y~+ Y~z3)

(31)

+

This partial differential equation is in turn equivalent to the exterior differential system defined on the manifold M of global coordinates (x 1, x 2, x 3, yl, y~, y~, y21,y~, yl2, yg) defined in the open set where y32 is well defined and generated by the forms 3

wa = d Y # - Z y ~ d x "

tre [2]

(32)

i----1

where y3~ is given by (31). Consider the independence condition [2 = dx 1 A d x 2 A d x 3. We will show that the system (Z, ~ ) is involutive. For this compute 3

dw~ = - Z

dye' A dx i, tr E [2].

(33)

i=l

Let

E =

span

{el,e2,e3} E Gn(TM) given by 0 +x2-'2 -~ ei :

~

o----1

0

3 j----1

0

2

0

j=l

In order to compute the codimension of Vn(Z, ~ ) it is easy to show that the conditions wa(ei) : O, dwa(ei,ej) = 0 for ~r E [2], and i,j E [ 2 ] , j < i produces 12 independent linear relations in the variables ~o, ~,, .~j with coefficients that are functions of~ E M. Let E0 = {0}, E1 = span {el}, and E2 = span {el, e2}. The codimensions ci of the polar spaces H(Ei) can be calculated via the following conditions:

H(Eo) = {v E T~,M I w~ H(E1) = {v 9 T~M [c0~ g(E2)={v 9176

= 0, a E [2]}, : O,dw~ el) = O,a 9 [21},

dw~

ej)=O,o' 9149

(34) (35) (36)

It easy to verify that (34), (35), (36) give respectively co = 2, c~ : 4, c~ : 6 independent restrictions (this will always occur when only one partial derivative is restricted). By Cartan's test (Thms. 1.11, p.74 and Cor. 2.3 [1]) it

Flatness of Nonlinear Systems

221

follows t h a t system ( I , / 2 ) is involutive. In particular there exists an integral element passing through every point of M . C o m p u t i n g the prolongation of (Z,/2) one constructs an involutive s system on a manifold M 0 ) with coor.1 .1 dinates (x 1 , x ~ , x 3 , y l , y 2 , y ~ , v ~1 , y ~1, v ~ , v ~ , v11, v12, 1 v h , .1 H22~ Y23J V 3 3 , Y11 Y~2,

Y~2, Y~3)- In particular one may choose the partial derivatives yljfreely.

Note now that y31 = Lfy~ = )-']~=1 y~jfj, 1 where f l = Ul,f2 : X3Ul and f3 = u2. Hence for the open set where (ul, u2) # 0, by choosing y~j one m a y impose any value of y31. By c o m p u t i n g det R symbolically, one m a y show that this free choice of y31 m a y assure that det R r 0. Hence theorem (1) implies that such integral manifolds produce flat outputs for the points such that (ul, u2) # (0, O) Now we will consider an e x a m p l e given in Fliess et al. [8]. Although they have given a flat output for this example we will show that the conjecture that this system is not 0-flat is true (the flat o u t p u t given depends on the input) .

Example 2. Consider the system [7] ;~1 ~

Ul ~

x~ = u2, X3 :

(37)

UlU2"

In this example we have n = 3 and m : 2. As in the first example the only possibility is having al = 1, or2 : 2, 0"3 : 2, and k* : 2. T h e basis of Prop. 3 is of the same form as in the last example. Consider first ~ = ul. Using (22) and (23), c o m p u t e the m a t r i x R of w obtaining

( ~dv1 \ ~dv2 I

[ vt vt y~ | y~ v~ ~]

0 0

0 0

7rLfdyl ] = | 9~ 91 9~ Y~ + y~u2 y~ + y~ul

J, [ o

+

\

J .

(38)

Condition (27) is obtained by c o m p u t i n g the first n - m columns and n rows of c o f R T. This will give: Y~

det _F _- 0

det ( Y ! Yi

detF = 0

det ( Y i Yi

detF = 0

(39)

a The prolongation of an involutive system is involutive (see [1, Thm. 2.1, p.248]).

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P.S. Pereira da Silva

where

+ y u2

+ y ul

+ y3u2 y~ + y~ul ) " If the three determinants that multiplying det F were all equal to zero, then the first two rows of R would be linearly dependent. Hence det F -- 0 corresponds to the constraints (27). Now note that det F = 0 is equivalent to

U2~---0.

det

(40)

If this polynomial equation holds in an open subset of Z, then the three determinants that are the polynomial coefficients must be identically equal to zero. This means that matrix R cannot be invertible. Choosing 01 = u~ a similar result is shown, thus this system cannot be 0-fiat around any point.

6

Conclusions

Given a control system, an exterior differential system (2:,/2) was constructed in a way that the existence of (local) integral manifolds (around points obeying certain dimensional properties) is equivalent to the existence of a 0-flat output (k-flat output if one extends the state of the system accordingly). This construction can be performed by computer algebra and it is equivalent to obtaining partial differential equations whose solution, if it exists, is a flat output of the given system. It is also important to say that Cartan-KKhler and Cartan-Kuranishi theorems may be also performed by computer algebra. Some related results may be found in [15].

Acknowledgements The author is indebted to B. Jakubczyk, E. Delaleau and F. M. Pait for several suggestions concerning this paper. This work was partially supported by Conselho Nacional de Desenvolvimento Cientffico e Tecnol6gico (CNPq) under grant 300492 / 95-2 and FAPESP under grant 97 / 04668-1.

A A.1

P r o o f of Auxiliary Results Proof of Prop. 2

Before giving the proof of the Proposition 2, we need to recall some facts about the structure of nonlinear systems. It is well known (see [3,2]) that the Dynamic Extension Algorithm (DEA) has an intrinsic interpretation. The approaches of [3,2] are algebraic, but it

Flatness of Nonlinear Systems

223

is possible to adapt these results to the differential geometric setting of [7] (see [16]). Let us recall the main aspects of this algorithm. Given system S defined by (2) with classic analytic state representation (x, u) and output y, the dynamic extension algorithm is a sequence of applications of regular static-state feedbacks and extensions of the state by integrators. Note that this algorithm can be regarded a sequence of choices of new local state representations 9 of system S. Let (x, u) = (x0, u0) be the original state representation of S with output y. Using the notation of [3], in step k of the algorithm one has constructed a classical state representation (xk, uk) with output y(k). Note that 9 ($1) xk+l = (xk, ~)(k~.+U),where ,Ok+l is chosen among the components o f y by completing {dxj,} into a basis {dxk, d~)k(~.+1) } for span {dxk, dy (k+l) } ; 9 ($2) uk+t = (~(k++1}, uk+l), where Uk+t is chosen among the components of uk by completing {dxk, d~)(k++1) } into a basis {dxk, d~(k++t) , d~k+l} for span {dxk, duk}. Then we have the following Lemma: L e m m a 2. Denote Yk = span {dx, d y , . . . , d y ( k)} and Yk : s p a n { d y , . . . , dy (k) } for all k E i~. Let Sa be the open and dense set of regular points of the codistributions Yi and Yi for i = O , . . . , k. In the kth step of the dynamic extension algorithm, one may construct, around ~ E Sk, a new local state representation (xk, Uk) of the system ,b" with state Xk = (x, f / o ) , . . . , f/(k)) and input uk : (y(k), Uk) such that

1. span {dxk} = span { d x , d y , . . . ,dy (k) }. 2. span { d x k , d u k } = span {dx, d y , . . . , d y ( ~ + U , d u } . ~(k+~) 4- ~k+~ C ~k5. Let :D(C) denote the generic dimension of a codistribution C generated by the differentials of a finite set of analytic functions. The sequence ak = V ( Yk ) -- V ( Y k - 1 ) is nondecreasing, the sequence Pk = D ( Yk ) - D ( Y~ _ t ) is nonincreasing, and both sequences converge to the same integer p, called the output rank, for some k* < n = dimx. 6. Sk : Sk+l for k >_ k*. 7. The sequence vk = l) (Y} N span {dx}) converges for k = k* - 1. Proof. The proof is only an adaptation of the algebraic results of [3,2] to the approach of [7]. For a complete proof, see [16].

9 We stress that the words state representation here may be considered in the sense of Iv].

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P.S. Pereira da Silva

Proof. ( O f P r o p . 2). We show first that 1 is equivalent to local O-flatness. If a system S is Ofiat with fiat output y, from the differential independence of the fiat output we have that d i m Y k . - 1 = re(k* - 1). Furthermore, given any function r defined on the diffiety S (see [7]), d e is contained in the codistribution Yk = span { d y , . . . , dy (k) } for k large enough. In particular Yk must contain span {dx} for k large enough. By L e m m a 2 part 7, we m a y take k = k* - 1. k*--I So, Yk.-1 = Y k . - 1 and d i m Y k . - 1 = n + ~']i=1 ~ri. From L e m m a 2 part 5 and from the fact that P0 -- m, it follows t h a t O'k. = m. We conclude t h a t (4) holds. Now assume that (4) holds for system (2). Around a regular point of the codistributions Yk and Yk for k ---- 1 , . . . , n , the D E A m a y be performed and will produce a state representation with state Xn of dimension n n + }"]~k=l o'k. Note that ~rn = m. By L e m m a 2, we know that Pk ~ O'n. It follows that pk = m, k = 0 , . . . , n. By (4) it follows easily that state representation (fn,gn) with input un = y(n+t) is linearized in Brunovsky form, since span { d ~ . } = span {dy,... ,dy(")} and { d y , . . . ,dy(")} is a basis for the last eodistribution. (ii) It suffices to show that (ii) is equivalent to (i). If (i) holds, from the proof of the first part it follows Y k ' - 1 = Yk'-~ and so (ii) also holds. Assume that (ii) holds. Since the system is well formed, by derivation it follows t h a t Yk. D span {dx, du}. By L e m m a 2 part (ii), note that span {dXk._ ~, d u k . _ 1} k*-I * = Yk'- Taking dimensions, we conclude that u.-J-~-~i=1 o'i+m = n+~]~/k;1 o"i. It follows that O'k. = m and from L e m m a 2 part 5 we have P0 = -.. = P k ' - t = rn. This show that the set { d y , . . . , d y (k-t)} is independent. Then, the equality Y k ' - I = Y k ' - I implies that (4) holds.

A.2

Proof of Prop. 3

Proof. Using the same notation as in L e m m a 2, note that (xk, Uk) is a state representation of system S for which span {dxk} = Yk and uk is a subset of Uk with m - Crk elements. It follows that the set .

(k'-k-~)~

{dxk, d u k , . . . , au k

(a'-k-~)~

, auk

1 is lin. indep, for k E [k* - 1]. (41)

Denote H0 = s p a n { d x } and Hk = s p a n { d x , d u , . . . , d u ( k ) } , k = 1 , 2 , . . . . From L e m m a 2 part 2 and from the fact t h a t span {dx} C { d x , d u } , one concludes by derivation of the equality Yk+l +/40 = span {dxk, duk} that Yk+r+t + H r = span { d x k , ( d u ( J ) : j ~ { 0 , . . . , r } ) } . In particular, for r = k* - k - 2 and k E [k* - 2] we have

Y k ' - I + / 4 k . - k - 2 = span { d x k , ( d u ( J ) : j ELk* - k-

2])}.

(42)

Flatness of Nonlinear Systems

225

From (41), and from the fact that uk C uk then

(~'-k-2).^(k'-k-1)~ {dxk, d u k , . . . , au k , au k j, is lin. indep, for k 9 [k* - 1]. (43) .

We will show the desired result by induction. Let j 9 { 0 , . . . , k* - 1}. Define

=

de

By construction it is clear that from the fact that uk C u that

Bj C Yk.-1 + / / j .

(44)

If the system is fiat with fiat o u t p u t y then { d y , . . . , d y (k)} is linearly independent for k 9 N. Let k = k* - j - 1, (i.e., j -- k* - k - 1). From (42) and (43) for k -- k* - 1, it follows t h a t /30 is independent. Assume that B j - 1 is independent for j < k* - 1. From (43), (42) and replacing j by 9 1 in (44), it follows that Bj = B j _ t O { d ^u(kJ ). _ j _ l } i"s also independen t . ~V; conclude that B k . - ~ is independent. To complete the proof it suffices to show that card B k . - 2 = card { d x , d u , . . . ,du (~'-2)} = n + re(k* - 1). Since .~ card uk ----m -- irk it follows that card Bk._~ = ink* + v-...~k ) . k =* l-- l tI m -- o'k]. Using (4), the desired result follows.

A.3

Proof of Lemma 1

Proof For k = 0 the statement is obvious. For k = 1 it suffices to show t h a t L ] d x i, L ] d y ~ are in F1. Note that dxi(vj) = Jij, dya(ri) = y~,i = 1 , . . . , n and dya(vi) = 0, i > n. Hence L f d x ~ = d L f d x ~ = d f i ( x , u (~ E F t and d L f y ~ = d[y~. (fi(x, u(~ 9 Assume that this statement holds for k. Since LkfFo C L f ( F k - l ) , it suffices to show that Lfdy~ C Fk and Lfdu~ k) C Fk. Since ri(yJ) -- Y~i for i __ n, ~ = O, i > n, 7 " n + i ( j + 1 ) ( U ' i( J )) ~: 1 ,rk(u} j)) = O , k • n + i ( j + l ) , i t i s e a s y i(Yj) to complete the proof the using the same ideas as above.

A.4

Proof of Prop. 5

Proof Let I be the codistribution defined on .~/generated by the forms (6). Let 12 C ] be generated by the same set with the exception of the forms w~ = d y ~ - ~"~i~1 Y~i with I I K I I = u. Consider the filtration Zk { 0 , . . . , u } and note that

= span {dx, d u ( ~

L f Z k C Zk+l C span {dz}, k E

L~'I.

C T*.~I, k E

(45)

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P . S . Pereira da Silva

We will show first t h a t L.ff2 C f. For this let I[KII < ~, a n d note t h a t Lr, dx j = ~fij, Lr, y~: = Y~i for i , j E Lnl, Lo/ou~)y ~ = 0 for k E [~] and j E L m ] , a n d (w~:, ri) = 0, i e L/l]. Using the fact t h a t Ly_,T,w~: = d f i ( w ~ , vi)+ finr, w~ it follows easily t h a t L ] w ~ = ~i~=t fiwgi C I. Let w e F u - 1 . By (45), it follows t h a t L]~rw C s p a n {dz}. Note t h a t , the form ~rw is the only form in span {dz} such t h a t (~rw - w) m o d / 2 _-- 0 (see p a r t 1 of Rem. 4 a n d (21)). In p a r t i c u l a r Lfrrw = ~rn]w + O, 0 e i. Since f A s p a n {dz} = 0 it follows t h a t 0 = 0. So, 7rL~-dy a = 7rLfLkf-tdy a =

L](~rL~f-t dya).

References 1. R. Briant, S. Chern, R. Gardner, H. Goldschmidt, and P. Griffths. (1991) Exterior Differential Systems. Springer Verlag, New York. 2. E. Delaleau and P. S. Pereira da Silva. (1998) Filtrations in feedback synthesis: Part l - - systems and feedbacks. Forum Math., 10(2):147-174. 3. M. D. Di Benedetto, J. W. Grizzle, and C. It. Moog. Rank invariants of nonlinear systems. (1989) SL4M J. Control Optim., 27:658-672. 4. M. Fliess, J. L~vine, P. Martin, and P. Rouchon. (1992) Sur les syst~mes non lin~aires diff~rentiellement plats. C. R. Acad. Sci. Paris Sdr. I Math., 315:619624. 5. M. Fliess, J. L~vine, P. Martin, and P. Rouchon. (1993) Lin~arisation par bouclage dynamique et transformations de Lie-Bhcklund. C. R. Acad. Sci. Paris Sdr. I Math., 317:981-986. 6. M. Fliess, J. L~vine, P. Martin, and P. Rouchon. (1995) Flatness and defect of non-linear systems: introductory theory and examples. Internat. J. Control, 61:1327-1361. 7. M. Fliess, J. L~vine, P. Martin, and P. Rouchon. (1999) A Lie Bs approach to equivalence and flatness of nonlinear systems. IEEE Trans. Automat. Control, 44(5):922-937. 8. M. Fliess, J. L~vine, P. Martin, and P. Rouchon. (1999) Some open questions related to flat nonlinear systems. In V.D. Blondel, E. Sontag, M. Vidyasagar, and J.C. Willems, editors, Open Problems in Mathematical Systems and Control Theory, pages 99-103, London. Springer Verlag. 9. L. R. Hunt, R. Su, and G. Meyer. (1983) Design for multi-input nonlinear systems. In R. Brocket, R. Millmann, and H. J. Sussmann, editors, Differential Geometric Methods in Nonlinear Control Theory, pages 268-298. 10. B. Jakubczyk and W. Respondek (1980). On linearization of control systems. Bull. Acad. Pol. Sc., Ser. Sci. Math., 28:517-522. 11. Kugi A., Schlacher K., Novak R., (1999) Software Package: Non-Linear Affine-Input System, In: Maple Application Center, Engineering, Control Systems, h t t p : / / w ~ . m a p l e s o f t . c o m / a p p s / c a t e g o r i e s / e n g i n e e r i n g / controlsystems/ accontrol.html. 12. M. Kuranishi. (1957) On E. Cartan's prolongations theorem of exterior differential systems. American J. Math., 79:1-47.

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227

13. P. Martin. (1992) Contribution d I'dtude des syst~mes diffdrentiellement plats. Th~se de doctorat, ]~cole Nationale Suprrieure des Mines de Paris. 14. P. S. Pereira da Silva. (1996) On the nonlinear dynamic disturbance decoupling problem. J. Math. Systems Estim. Control, 6:1-26. 15. P. S. Pereira da Silva. (2000) Flatness of nonlinear control systems : a C a r t a n K~hler approach. In CDROM Proe. Mathematical Theory of Networks and Systems - MTNS'2000, pages 1-11, Perpignan, Jun. 19-23. 16. P. S. Pereira da Silva. (2000) Geometric Properties of the Dynamic Extension Algorithm. Internal Report, Escola Politrcnica da USP - B T / P T C , available http://www.lac.nsp.br/,,~paulo/down.html. 17. J.-B. Pomet. (1995) A differential geometric setting for dynamic equivalence and dynamic linearization. In B. Jackubczyk, W. Respondek, and T. Rzezuehowski, editors, Geometry in Nonlinear Control and Differential Inclusions, pages 319-339, Warsaw. Banach Center Publications. 18. J.-B. Pomet. (1997) On dynamic feedback linearization of four-dimensional afline control systems with two inputs. ESAL~I Control Optim. Calc. Var., 2:151-230 (electronic), 1997. 19. J. Rudolph. (1995) Well-formed dynamics under quasi-static state feedback. In B. Jackubczyk, W. Respondek, and T. Rzezuchowski, editors, Geometry in Nonlinear Control and Differential Inclusions, pages 349-360, Warsaw. Banach Center Publications. 20. F. W. Warner. (1971) Foundations of differentiable manifolds and Lie Groups. Scott, Foresman and Company, Glenview, Illinois.

Motion Planning for Heavy Chain Systems Nicolas Petit a n d Pierre Rouchon Centre Automatique et Syst~mes t~cole Nationale Supdrieure des Mines de Paris 60, bd. Saint-Michel 75272 Paris Cedex 06, France {pet i t , rouchon}@cas, ensmp, f r

A b s t r a c t . In this paper we address the motion planning (steering from one state to another state) of heavy chain systems. We parameterize the system trajectories by the trajectories of its free end. It was shown in [7] that such systems are flat [2,3] when considered as a finite set of small pendulums. Our study is an extension to the infinite dimensional case. Under small angle approximations, heavy chains are described by a 1D partial differential wave equation (dynamics of the chain X(x, t) with speed depending on x, the space variable). Dealing with this infinite dimensional description, we show how to get the explicit parameterization of the chain trajectory t *-r X(x, t) by using (distributed and punctual) advances and delays of its free end t ~-+ y(t) = X(O, t).

Introduction It was shown in [7] t h a t heavy chain systems are flat [2,3] when considered as a set of finite small p e n d u l u m s : their t r a j e c t o r i e s can be explicitly p a r a m eterized by the trajectories of their free ends. Such flatness-based relations ( p a r a m e t e r i z a t i o n of the trajectories) involve n u m e r o u s derivatives (as m a n y as the double of the p e n d u l u m n u m b e r ) . W h e n this n u m b e r tends to infinite, the derivative order tends to infinite which m a k e s t h e m difficult to handle and to use in practice. In order to u n d e r s t a n d and overcome such difficulties, we consider here an infinite dimensional description. Under s m a l l angle a p p r o x i m a t i o n s a r o u n d the stable vertical s t e a d y - s t a t e , the d y n a m i c s are described by second order o r d i n a r y differential equations ( d y n a m i c s of the load at p o s i t i o n y(t)) coupled with 1D wave equations ( d y n a m i c s of the chain X ( x , t)) where waves speed depends on x, the space variable. Dealing with such infinite d i m e n s i o n a l description, we show how to get a useful explicit p a r a m e t e r i z a t i o n of the t r a j e c t o r i e s by using, instead of an infinite n u m b e r of derivatives, d i s t r i b u t e d a n d p u n c t u a l advances and delays. Such an explicit p a r a m e t e r i z a t i o n provides a simple answer to the m o t i o n planning p r o b l e m .

230

Nicolas Petit and Pierre Rouchon

For the heavy chain of figure 1 (see section 1 for details), our explicit parameterization says that the general solution of cO cOX ~(ax-Sg ~ ) is given by the following integral

X(x,t) = ~

cO2X

cOt----~- _ o

~ y(t +

2v/Ugsin 0) dO

where t ~ y(t) is any smooth enough time function: X(0,t) = y(t) corresponds then to the free end position; the control u(t) = X ( L , t) is the trolley position. Such parameterization is closely related to ~r-freeness introduced in [5,4,6] a notion derived from flatness (see [2,3]) and specifically addressing delay systems. Here, as in [6], we show that a "distributed delay" point of view provides a simple numerical algorithm for steering the system from a state to another one. Thus the resulting control method is open-loop. More generally speaking, given any mass distribution along the chain and any punctual mass at x = 0, there is a one to one correspondence between the trajectory of the load t ~-~ y(t) = X(O, t) and the trajectory of the whole system, namely the cable and the trolley: t ~+ X ( x , t) and t ~-~ u( t ) = X ( L, t ) . Such a correspondence yields an explicit parameterization of the trajectories: X ( x , .) = .A~y where {.Ax} is a set of operators including time derivations, advances and delays. In other words, ( x , t ) ~-+ (.A~y)t verifies the system equations for any smooth function t ~+ y(t). For each x, the operator .A~ admits a compact support. We detail the case of a simple homogeneous chain and briefly describe the case of an nonhomogeneous cable. 1

The

homogeneous

chain

without

any

load

A heavy chain in stable position, see figure 1, is ruled by the following dynamics (small angle approximation) cO cOX cO2X ~ x (gX-~x ) cOt2 -- 0 X ( L , t ) = u(t).

(1)

where X ( x , t) - u(t) is the deviation profile, g is the gravity acceleration. The control u is the trolley position. Thanks to the classical mapping y -= 2 1 g , we get cOaX cOX cOaX u-~u~ (u, t ) + -~u (u, t) - u - g i ~ ( u , t ) = o.

Motion Planning for Heavy Chain Systems

231

i I

u(t).x(c,t)

'~

1 I

X(x,t)

[I'

Fig. 1. The homogeneous chain without any load. Use Laplace transform of X with respect to the variable t (denoted by ~7 and with zero initial conditions i.e. X(., 0) = 0 and ~cOX ( . , 0) = 0) to get

a~2

Co2

~-~y~ (y,s) + -g-~y (y,s) - y s ~ 2 ( y , . )

: o.

Less classically the mapping z = ,sy gives

a2k of( z , s) + z X (z, s) o. ~ -5~S ( z , s ) + -~z ( =

(2)

This is a Bessel equation. Its solution writes in terms of J0 and Y0. Using the inverse mapping z : 2~s~//x, we get t - - - -

x ( ~ , s) = A J0(2,~Vq~) + B

Yo(~-~/~-~).

Since we are looking for a bounded solution at x --- 0 we have B = 0. Then X ( x , s) -- g 0 ( 2 , s v / ~ ) X ( 0 , s).

(3)

where one can recognize the Clifford function C~ (see [1, p 358]). Using Poisson's integral representation of J0 [1, formula 9.1.18] 1 [

Jo(z) = ~ J-] exp(~z sin O) dO,

232

Nicolas Petit and Pierre Rouchon

we have J~

= ~i f _ _ ~,r e x p ( 2 s v / ~ s i n

0) dO.

In terms of Laplace transforms, this last expression is a combination of delay operators. Turning (3) back into the time-domain we get

ii

X ( x , t) = - ~

y(t + 2 V / ~ s i n O) dO

(4)

?f

with y(t) = X(O,t). Relation (4) means that there is a one to one correspondence between the (smooth) solutions of (1) and the (smooth) functions t ~-~ y(t). For each solution of (1), set y(t) = X(O, t). For each function t ~-+ y(t), set X via (4) and u via u(t) = ~i f _ 'r y(t + 2 V / ~ s i n

O) dO

(5)

to obtain a solution of (1). Finding t ~-+ u(t) steering the system from a steady-state X = 0 to another one X _= D becomes obvious. It just consists in finding t ~+ y(t) that is equal to 0 for t < 0 and to D for t large enough (at least for t > 4 x / ~ ) and in computing u via (5). Figure 2 illustrates computations based on (4) with [0 i f t < 0 y(t)= {-~*(~)~ (3-2(-~)) (~ift>T

if0
where the chosen transfer time T equals 2A with ,4 = 2 V / ~ , the travelling time of a wave between x -- L and x = 0. For t < 0 the chain is vertical at position 0. For t > T the chain is vertical at position D = 3L/2. Plots of figure 3 show the control [0, 7] ~ t ~-) u(t) required for such motion. Notice that the support of ~ is [--A, T + A] while the support of y is [0, T].

2

The inhomogeneous (i.e. variable section) chain without any load

The case of a heavy chain with variable section (see figure 4) deserves a special treatment because of the singularity of the partial differential system at x = 0 .

Motion Planning for Heavy Chain Systems

233

Fig. 2. Steering from 0 to 3L/2 in finite time T, the successive position of the heavy chain and its trolley. 3L

/ I/ illII

2

Y

/I/!/ iI/u~ /// i

0

-A

0

T

I

T+A

Fig. 3. The steering control, trolley position u, and the "flat output", the free end y. This time the system is ruled by the following equations - -

- -

--0

(6)

X ( L , t) = u(t) where u is the control. The tension of the chain is r ( x ) with r(O) = 0 and r(x) = gx + O(x~), while r ' ( x ) / g > 0 is the mass distribution along the chain. Furthermore, we assume r ( x ) > ax >_ O. T h e o r e m I Consider (6) with [0, L] ~ x ~-+ r ( x ) a smooth increasing function with r(0) = 0 and r' > O. There is a one to one correspondence between

234

Nicolas Petit and Pierre Rouchon

u(t)-X(L,t)

X(x,t)

x=OI

Fig. 4. The inhomogeneous chain without any load.

the solutions [0, L] • ]~ 9 (z, t) ~-~ ( X ( x , t), u(t)) that are C 3 in t and the C z functions l~ ~ t ~-~ y(t) via the following formulae

x(~,t)

=

2~r3/~ (r(x))t/" +

y

t+

j_2C~./C(G(2v~),~)y

Ka(2v~)sin0 t+

dO

d~

u(t) = X ( L , t)

(7)

with

u(t) = x ( o , t)

Motion Planning for Heavy Chain Systems

235

where the constant K and the functions G and IC are defined by the function via the following equations

r

K = -71" l foL V ~

,z =

l ~o~ ~f~

h(z) = F"(z) F(z)

with F(z) =_ (r(x)) 1/4

- G(2v~)

(8)

(9)

D(z, s) = It'sl .I, [_vzsinh(Ks(z - t)) (h(z~) + 4 ~ ) (Lg)l/aV~JoOh.st)d t + ~s o-g'~

sinh(Ks(z

( , ~ ) exp (s~)d~ =

-

t))h(t~)D(t, slat.

D(z, s)/s.

(10)

(11)

where Jo is, as before, the zero-order Bessel function. The proof of this result includes symbolic computations where the time derivation is replaced by the Laplace variable s, followed by a Liouville transformation [9, p. 110]. Then one may use the Paley-Wiener theorem [8, page 375] to turn the resulting symbolic expression back into the time-domain (inverse Laplace transform) and one gets (7) at last.

Remark

In the case of an homogeneous chain, we can substitute

,-(~)

K=;VT,

= g~,

,-'(~,) = g,

z=G(2,/7)=.

,

}(~ --~ 0,

and equation (7) reads

X(x,t) = ~

r y(t - 2

sinO)dO

which is indeed identical to (4). 3

Conclusion

Around the stable vertical position, heavy chain systems with constant or variable section are "flat": the trajectories of these systems can be parameterized by the trajectories of their free ends. Relations (4) and (7) show that such parameterizations involve operators of compact supports.

236

Nicolas Petit and Pierre Rouchon

It is surprising t h a t such p a r a m e t e r i z a t i o n s can also be a p p l i e d a r o u n d t h e inverse and u n s t a b l e vertical p o s i t i o n . For t h e h o m o g e n o u s h e a v y chain, we j u s t have to replace g by - g to o b t a i n a f a m i l y o f s m o o t h s o l u t i o n s to t h e elliptic equation (singular at x = 0)

0

OX

02X

( gx-6-; ) + -b-P- = o by the following integral

1/:

X(x,t) = ~

y(t +

2,v/~sin0)

dO

where y is now an h o l o m o r p h i c function in lI~ x [ - 2 V / ~ , +2V/~] t h a t is real on the real axis. T h i s p a r a m e t e r i z a t i o n can still be used to solve the m o tion planning p r o b l e m in spite o f the fact t h a t the Cauchy p r o b l e m a s s o c i a t e d to this elliptic equation is not well-posed in the sense of H a d a m a r d .

Acknowledgments T h e a u t h o r s are i n d e b t e d to Michel Fliess a n d P h i l i p p e M a r t i n for fruitful discussions relative to P a l e y - W i e n e r t h e o r e m a n d Liouville transformations.

References 1. M. Abramowitz and I. A. Stegun, editors. Handbook of mathematical functions. Dover Publications inc., 1965. 2. M. Fliess, J. L~vine, Ph. Martin, and P. Rouchon. Flatness and defect of nonlinear systems: introductory theory and examples. Int. J. Control, 61(6):1327-1361, 1995. 3. M. Fliess, J. L$vine, Ph. Martin, and P. Rouchon. A Lie-Bs approach to equivalence and flatness of nonlinear systems. IEEE Trans. Automat. Control, 44:922-937, 1999. 4. M. Fliess and H. Mounier. Controllability and observability of linear delay systems: an algebraic approach. ESAIM: Control, Optimisation and Calculus of Variations, 3:301-314, 1998. 5. H. Mounier. Proprigtgs structurelles des syst~mes lingaires ~ retards: aspects thgoriques et pratiques. PhD thesis, Universit$ Paris Sud, Orsay, 1995. 6. H. Mounier, J. Rudolph, M. Fliess, and P. Rouchon. Tracking control of a vibrating string with an interior mass viewed as delay system. ESAIM: Control, Optimisation and Calculus of Variations, 3:315-321, 1998. 7. R. M. Murray. Trajectory generation for a towed cable flight control system. In Proc. IFA C World Congress, pages 395-400, San Francisco, 1996. 8. W. Rudin. Real and complex analysis. McGraw-Hill, New York, St Louis, Paris, 2nd edition, 1974. 9. K. Yosida. Lectures on differential and integral equations. Interscience Publishers, New York, 1960.

Control of an Industrial P o l y m e r i z a t i o n R e a c t o r U s i n g Flatness Nicolas Petit 1, Pierre Rouchon 1, Jean-Michel Boueilh 2, Fr&16ric Gu~rin 2, and Philippe Pinvidic 3 i Centre Automatique et Syst~mes l~cole des Mines de Paris 60, bd. Saint-Michel 75272 Paris Cedex 06, France {pet i t , rouchon}@cas, ensap, f r 2 Centre Technique Atochem Chemin de la L6ne BP 32 69492 Pierre B6nite Cedex, France s APPRYL, Usine PP2 BP 21 13117 Lav6ra, France

Abstract. We present our work on the APPRYL PP2 polypropylene plant. We give a physical nonlinear model of the system with a delay on one of the two inputs. This model is fiat. Using this flatness property, we show how to design a controller capable of fast and precise transients. Industrial results prove the relevance of our approach. Our controller is in full service since July 1999.

Introduction A P P R Y L is a joint venture with A T O F I N A and B P A M O C O . There exists three distinct A P P R Y L plants in Europe, n a m e d PP1 in Gonfreville (France), P P 2 in Lav6ra (France) and P P 3 in G r a n g e m o u t h (Scotland). PP1 is the first of these three plants. It produces 135kT/year. P P 2 is the largest polypropylene plant in the world [6]. It produces 250kT/year. P P 3 is very similar to P P 2 and will start soon, it is expected to produce 250kT/year. The applications of polypropylene are numerous (see [1] for more information). By thermoforming one can produce trays, dishes, . . . . By injection it is possible to produce cars large sized-parts such as bumpers, dashboards . . . [7]. W h e n producing polypropylene, two quantities are of particular interest: the a m o u n t of production and the melt-index of the polymer. The melt-index is an indicator of the mechanical properties of the polymer. It is very i m p o r t a n t for injection and thermoforming transformations. In order to control these two quantities, one has to specify the a m o u n t of catalyst and hydrogen that enters the reactor. From a m a t h e m a t i c a l point of view, this is a two inputs two outputs system.

238

N. Petit, P. Rouchon, J.-M. Boueilh, F. Gu~rin, Ph. Pinvidic

The production is planned with respect to economical considerations (the market of polymers). This induces frequent changes in the setpoints. We show here how we designed a controller capable of doing fast and precise transients for both the amount of production and the melt-index. In section 1 we give a model of the unit. In section 2 we show that this model is flat [2,3] and we use it to design open-loop control strategies. In section 3 we give industrial results of our controller. 1

Modeling

Fig. 1. APPRYL PP2 plant by night. The APPRYL PP2 plant is pictured on figure 1. The polymerization process is depicted on figure 2. The hydrogen enters directly the reactor while the catalyst enters the reactor after a delay due to activation processes. Roughly speaking, the catalyst acts upon the amount of production, while the hydrogen acts upon the melt-index of the polymer. To write the following model we use balance equations (heat and mass) which are coupled and

Control of an Industrial Polymerization Reactor Using Flatness

CataIst: u

Hydrogen:v

~

239

Cooling

Reactor

Fig. 2. The polymerization process: 2 inputs (u, v) , 2 outputs (quality and amount of production).

nonlinear and on the other hand we use nonlinear inferences arising either from the literature or from dedicated studies we carried out. Notations: Qa is the amount of catalyst in the reactor. X is the rate of solid (mass ratio between solid and liquid particles). P r o d is the instantaneous amount of produced polymer. CH~ is the hydrogen concentration. M I is the

melt-index of the polymer, u is the amount of catalyst coming in the reactor per unit of time. v is the amount of hydrogen coming in the reactor per unit of time. r is the residence time. d

Qa

- ~ ( Q a ) = u(t - 6 ) - - - r d d---~(X) = Q . ( c X + d) - e X + / 1

d

(1) X - X

(2)

X Yl = P r o d = g l - X

(3)

(CH~) = v -- g(CH2, Qa)

(4)

d(log MI)

alogCH2 + b - logMI 7-

Y2 = M I

(5) (6)

240

N. Petit, P. Rouchon, J.-M. Boueilh, F. Gudrin, Ph. Pinvidic

where a, b are c o n s t a n t coefficients, a n d c, d, e, f , g are c o m b i n a t i o n s of densities and other known p a r a m e t e r s . E q u a t i o n (1) is a d i l u t i o n e q u a t i o n w i t h a d e l a y on the input. E q u a t i o n (2) is a mass balance equation. E q u a t i o n (4) is a b a l a n c e e q u a t i o n a n d includes a nonlinear inference. E q u a t i o n (5) is a m i x i n g e q u a t i o n where t h e source t e r m arises from theoretical chemical studies of p o l y m e r growth.

2

Flatness

of the

model

One can write the previous e q u a t i o n s under this m o r e f o r m a l form

}1=u(t-~)

(7)

xl T

(8) (9)

x2 = x J ( x 2 ) - t ' h ( x 2 )

~a=v-g(xa,x~) x4 = a log(xa) + b - x4 T Yl = k(x2)

(10) (11)

y2=exp(x4).

(12)

It is easy to see t h a t this s y s t e m is flat1: all its variables are p a r a m e t e r i z e d by the flat o u t p u t s x2 -- X , x4 = M I . More precisely speaking: xa = exp

(

'rx4 "4- x4 -- b ) -a

5c2 - h(x2) xl -

yl

=

/(x2)

k(x2)

Y2 = exp(x4) and

u(t-

6) - x 2 - x~h'(x~) f(x2)

v ----exp

(&2 -

h(x2)) &2f'(x2_____~)"4- 5~2 - h(x2)

f2(x2)

(X4 7"- --b'~- x4 1 ;~'4v'3t"~4 -4- g(x4, x4, x2, x2). a

a

i More precisely speaking, it is (f-flat, see [4] for details.

7"f(x2)

(13)

(14)

Control of an Industrial Polymerization Reactor Using Flatness

241

Open-loop control strategy As usual with fiat systems (see again [2,3]), it suffices to control the flat outputs to control the whole system. The open loop controls are given by equations (13, 14). We detail here an example of open-loop control computation. Assume that the operator wishes to increase the setpoint for the amount of production: Prodinitiat ) Prodobjective. The controller expresses this wish in terms of the fiat outputs. X i n i t i a I ~ X o b j e e t i v e thanks to the relation

A transition x2(t) between Xinitlat and Xobjeetive is prescibed (for example a polynomial). Then the open-loop control is computed as

~,(t) _ ~ ( t + ~) ~(t + ~)h'(x~(t + ~)) -

f(xz(t + ~)) - (x~(t + ~) - h ( ~ ( t

+

+

~2(t + ~) - h(~2(t + ~f(~(t

+

~)))x~(t ~-t- ti)f'(x~(t + ~

+ $))

(15)

~))

~))

X_objective ~t /

~

tO-~ tO X_initial

/ Flatoutput (t) T_transition

Fig. 3. Open-loop control strategy. The operator's request is expressed in terms of a transition for the fiat output X and the open-loop control is computed via (13)

242

N. Petit, P. Rouchon, J.-M. Boueilh, F. Gu6rin, Ph. Pinvidic

A few words about the closed-loop In fact the open-loop strategy must be completed by a feedback. It is not possible to describe here the exact design of the control law for confidentiality reasons. Roughly speaking we use standard flatness-based closed-loop strategies, which are quite equivalent to a linear controller using appropriate reference trajectories (given by our open-loop design). On the other hand, some required variables are not measured. To overcome this we use estimators, predictors, and Luenberger-style observer.

i

I

I

:

I

'

~m

Production, septpoint

Catalyst: u jJ/

// f

i

i ,

:

;I

;

I,

i

Fig. 4. Industrial results over 2 days. Production (Prod) transient. The transients are fast and precise. Scales are omitted for confidentiality reasons.

3

Industrial results

Our controller is in full service since July 1999 and allows optimization of profit. The closed-loop response of the system and the open-loop computations are almost equal. As one can see it on figure 4, the controller allows very fast and precise transients. One can clearly see the effect of the delay compensation by an "advance" in the controller design (see equation (15)): before the system meets the setpoints, the controller stops changing the value of the input (catalyst). This prevents any overshoot. The controller is capable of simultaneous transitions for the amount of production Prod and for the melt-index MI. We give industrial results on meltindex transitions on figure 5. The flatness of the system allows us to take into account the nonlinearities and the delay of the system. This industrial realization proves the relevance of

Control of an Industrial Polymerization Reactor Using Flatness

\

k I f

If

f

/-I

"--e-..__

-

243

Hydrogen: v

9

MI,

sotl~nt

Fig. 5. Industrial results 10 hours. Melt-index ( M I ) transient. The transients are fast and precise. Scales are omitted for confidentiality reasons. (The off-limit values are only sensor failures).

our approach in c o n t i n u o u s process control. More details a b o u t this p a r t i c u l a r application and other i n d u s t r i a l control realizations in process control can be found in [5].

References 1. B. Elvers, S. Hawkins, and G. Schulz, editors. Ullmann's encyclopedia of industrial chemistry. VCH, 1993. 2. M. Fliess, J. Ldvine, Ph. Martin, and P. Rouchon. Flatness and defect of nonlinear systems: introductory theory and examples. Int. J. Control, 61(6):1327-1361, 1995. 3. M. Fliess, J. Ldvine, Ph. Martin, and P. Rouchon. A Lie-B/ickhmd approach to equivalence and flatness of nonlinear systems. [EEE Trans. Automat. Control, 44:922-937, 1999. 4. H. Mounier. Propridtgs structurelles des systbmes lindaires d retards: aspects th(oriques et pratiques. PhD thesis, Urtiversitd Paris Sud, Orsay, 1995. 5. N. Petit. Syst~mes ~t retards. Platitude en ggnie des procgd~s et contr61e de certaines gquations des ondes. PhD thesis, Ecole des Mines de Paris, 2000. 6. M. Roberson. APPRYL investments worth 1 billion french francs. Hydrocarbon Processing, 77(6), June 1998. 7. URL. ttttp://v~n~, appryl, f r

Controllability of Nonlinear Multidimensional Control Systems Jean-Franqois P o m m a r e t CERMICS Ecole Nationale des Ponts et Chauss~es 6/8 Av Blaise Pascal 77455 Marne-In-ValiSe Cedex 2, France polmaret@cermics, enpc. f r

A b s t r a c t . It is now known that the controllabifity of a linear multidimensional control system is a structural property equivalent to the lack of torsion of the corresponding differential module and can be tested, even in the case of nonconstant coefficients. It is therefore tempting to decide about the controllability of a nonlinear control system through the controllability of its generic linearization. The main purpose of this paper is to answer this question negatively, in general, by presenting for the first time the counterexample of a nonlinear system which is controllable while its generic linearization is not controllable. We also provide a test for searching autonomous observables, extending the one already existing in the 1-dimensional case.

K e y w o r d s : Control theory, controllability, nonlinear systems, differential modules, m u l t i d i m e n s i o n a l systems.

1

Introduction

T h o u g h the controllability of nonlinear o r d i n a r y differential (OD, 1dimensional) control systems is the subject of an extensive l i t e r a t u r e [1-3,5,8], almost no result can be found a b o u t the controllability of p a r t i a l differential (PD, n-differential, n _> 2) control systems. In two Notes to the Science A c a d e m y of Paris [11,12], we gave tbr the first time a definition of controllability t h a t could be valid for an a r b i t r a r y line a r / n o n l i n e a r P D system, with n independent variables in the a n a l y t i c case or n derivations in the differential algebraic case. Now, if one has a nonlinear system, its generic linearization (see below) is a linear s y s t e m w i t h variable coefficients a n d it is only in 1994 [13] t h a t we presented an effective way to test the previous definition, based on d u a l i t y techniques (adjoint o p e r a t o r , differential sequence). Such a result, o b t a i n e d i n d e p e n d e n t l y by U. O b e r s t in [10] for the constant coefficient case, was confirming the i m p o r t a n c e of

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homological algebra and algebraic analysis [15] in the study of control systems, bringing thus the controllability of a linear control system to the lack of torsion of the corresponding differential module. More precisely, studying (analytic or differential algebraic) systems, we m a y call observable any scalar (analytic) function of the control variables. Such an observable will be called autonomous if it satisfies at least to a PD equation for itself and free otherwise. Of course, if there exists an a u t o n o m o u s observable, the system is surely not observable, in any sense. It is however not evident at all that, for linear O D / P D systems, the lack of any a u t o n o m o u s linear observable, usually called torsion element in the corresponding module framework, is equivalent to controllability in any sense ( m a x i m u m rank of the controllability matrix or Hautus test for K a l m a n type systems, primeness condition for higher order multidimensional systems [16]). It is thus possible to state the following definition:

D e f i n i t i o n 1. A linear/nonlinear control system is controllable if and only if any observable is free or, equivalently, there does not exist any a u t o n o m o u s observable.

In the linear case, the above definition becomes [10,13]:

D e f i n i t i o n 2. A linear control system is controllable if and only if the corresponding differential module is torsion-free.

The corresponding test has five steps [13,15]: 1) Write the control system as a differential operator T~I by suppressing the distinction between inputs and outputs. 2) Introduce the (formal) adjoint operator "/)1 = a d ( ~ l ) . 3) Determine the generating compatibility conditions of T~I as an operator T~ = ad(:D) for a certain ~D. 4) Introduce the adjoint operator "D = ad(~). 5) Determine the generating compatibility conditions of ~ as an operator ~D11. In general, ~ o'/)1 -- 0 ~ T~I o ' D -- 0 but ~)1 m a y not generate all the compatibility conditions of "D, a fact that we shall denote symbolically by ~31 _< ~)1'. We have proved [13,15]:

T h e o r e m 1. :D1 is controllable if and only if ~)l = :D1I. In that case, :D1 is parametrizable by 9 .

Controllability of nonlinear multidimensional control systems

2

Main

247

results

Having these definitions and results in hand, we m a y reach the main idea of the paper. Indeed, starting from a nonlinear system, its generic linearization (see below) is a linear system with variable coefficients to which we can apply the preceding test. 9 If the linear system is controllable (no torsion element), then the nonlinear system is also controllable because, if it should admit an a u t o n o m o u s observable, then, by linearizing this observable and the O D / P D equation that it satisfies, we should obtain a torsion element, a result leading to a contradiction. 9 If the linear system is not controllable, then one can find at least one torsion element of the corresponding differential module. If certain generating torsion elements can be "integrated ", that is, can be considered as linear combinations (with function coefficients) of the linearizations of a finite number of " g e n e r a t i n g ' a u t o n o m o u s observables, then the nonlinear system is not controllable. If not, the nonlinear system does not admit any autonomous observable and is therefore controllable. For n = 1, a few authors have recently proved in [1-3] by other different techniques adapted from the study of the so-called "strong controllability distribution" [5,8] that the second situation cannot happen and that, roughly speaking, one can say that, when n -=- 1, a nonlinear OD system is controllable if and only if its generic linearization is controllable. For n > 2, the problem, as quoted in [9], is open and one can either prove the result or disprove it by means of a counterexample. The main purpose of this paper is to exhibit such a counterexample that we found after six months intensive work as, in most explicit examples, all the torsion elements can be integrated. A more detailed study and another more sophisticated example will be given in a forthcoming book [14] showing how to use the techniques of algebraic analysis even on nonlinear systems. In first place, we present the technique of "generic linearization" by recalling the concept of vertical bundles [13] in the analytic case and that of Ks differentials [6] in the differential algebraic case, using the standard notations in this framework. The following OD example will help understanding the techniques involved. E x a m p l e 1: Let us consider the nonlinear SISO system u ~ ) - u = a = cst. Setting U = 5u, Y = 5y for the (vertical) variations, the generic linearization becomes:

248

J.F. Pommaret

~? + yu - u : 0

In that case, it is known [13] that the corresponding operator 7)1 is controllable if and only if its adjoint 251 is injective. Multiplying by a test function )~ and integrating by part, we get for the kernel of 7)1:

X+y~=0

,

uX+u~=0~a~=0

Hence ~1 is uncontrollable if and only if a = 0. In this case, u W = u Y - U is a generating torsion element as W = 0 and W = (fw with w = y - logu, that is W is integrable. A direct nonlinear study has been given in [13]. When 7)1 is not surjective or n > 2, this simple study is not sufficient and one needs the full five steps of the previous test. Let E be a fibered manifold over the base manifold X with d i m ( X ) = n and T~q C Jq(E) be a nonlinear system of order q on E, considered as a subfibered manifold of the q-jet bundle of E. If (x i) with i = 1,...,n are local coordinates for X and (x i, yk) with k = 1, ..., m are local coordinates for ~:, then (z, yq) = (x i, yk [ 0 <--I # I<- q) are local coordinates for Jq(,~) when # -- (Pl,...,P,~) is a multi-index with I /~ I-- Pl -t-... 4-p,~. T h e vertical bundle E -- V(E) will have local coordinates (x, y, Y -- (fy) where the variational notation is the one used by engineers. Introducing the total/tbrmal derivatives:

k

a

di = ~i + Y.+l, ay~ where p + li : (/~1, ...,•i-l,pi + 1,pi+l .... ,Pn), then d~ = dg~...du~ has a meaning because didj = djdi = dij. The generic linearization of 7~q is the linear system Rq = V(Ttq) C Jq(E) obtained by using the identification isomorphism Jq(V(E)) = V(Jq(E)) with d~,Y k = d j y k = dy~ = Y~. Hence, if 7~q is defined by PD equations of the form qHr = 0, then Rq is defined by the additional linear PD equations:

(~5r --- .~.k (x, yq)d~Y k = 0

uy;

,

(x, yq) E T~q

T~q is formally integrable, that is to say when T~q+r = J~(T~q) N Jq+r(S) is a fibered manifold for any r > 0 and the canonical induced projection maps 7~q.{_q+~+l r : ~'~q-{-r-{-1 ~ r~q+r are epimorphisms for any r ~ 0, then Rq is formally integrable. If, moreover, the Accordingly, when

Controllability of nonlinear multidimensional control systems

249

c o m m o n s y m b o l is involutive [13], then b o t h s y s t e m s are involutive. E x a m p l e 2: W i t h n = 2, m = 1,q = 2, the s y s t e m 7 ~ defined by 1 3 r -- Yl~- 89 2 = 0, ~ ----Y22-~(Y11) = 0 is involutive and the linearized system defined by ~r - dl~Y - y l l d m Y = O,~q52 - d22Y - ( y l l ) 2 d l l Y = 0 is also involutive. In the differential algebraic case, in place of X and g, we shall have a differential field K and a family y = (yl,..., yn) of differential i n d e t e r m i n a t e s . As any perfect/radical differential ideal is the intersection of a finite n u m b e r of prime differential ideals [7,13], we m a y suppose t h a t the differential p o l y n o m i a l s ~ r generate a p r i m e differential ideal p C K { y } = lirnq__,oo K[yq] and we m a y introduce the differential fi'eld extensio,~ L / K where L = Q( K {y} /p ). In t h a t case, the previous linear system defining Rq is j u s t providing a p r e s e n t a t i o n of the m o d u l e ~L/K of Ks differentials which is a differential m o d u l e over L with filtration induced by the inclusions K C ... C K[yq] C ... C K { y } . It is therefore essential to be able to recognize whenever a differential ideal is p r i m e or at least perfect. We have a l r e a d y p r o d u c e d such a test in [13] by a d a p t i n g the J a n e t - S p e n c e r - G o l d s c h m i d t criterion for f o r m a l i n t e g r a b i l i t y [4,13]. E x a m p l e 3: The following e x a m p l e , given by J. Johnson, shows t h a t the s i t u a t i o n can be quite tricky. Indeed, if ~1 - Yl~ - (y)2,~2 - Y2~ - x2Y, it is not evident t h a t the root of the differential ideal generated by ( ~ 1 , ~ 2 ) is the prime ideal g e n e r a t t e d by y alone as one needs 4 p r o l o n g a t i o n s . In any case, we can introduce the ring DK = K[dl, ..., dn] of differential o p e r a t o r s over K , so t h a t 12L/g becomes a D L - m o d u l e . We now s t u d y for a m o m e n t the linear s i t u a t i o n where Cr __- ak y~ with a 6 K and set D = DK, in such a way t h a t we can introduce the residual differential m o d u l e (also called D - m o d u l e ) M , quotient of D y = D y l + . . . + D y m by the D - s u b m o d u l e generated by the given P D equations. In OD control theory, the search for torsion elements can be done by bringing the given system to a K a l m a n form y = A y + B u and then showing easily t h a t any torsion element must be of the form w = Ay where A is a c o n s t a n t row vector. As tb = AAy + A B u is again a torsion element [7] page 102, c i t e l 3 page 278, we must have AB = 0 and thus also A A B = 0,..., a result leading to the controllability m a t r i x in the linear case and to the strong controllability d i s t r i b u t i o n in the nonlinear case [1-3], [13] page 324. In P D control theory, no previous work has been done on the subject. T h e

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first basic idea, implicit in the Kalman form, is to consider Rq+l C Jl(Rq) as a first order system on Rq with no zero order equation (Spencer form). Hence, from now on we shall suppose that q = 1, R1 C J1 (E) is involutive and rr01 : R1 -----+ E is an epimorphism. As our study tbr determining the torsion submodule t(M) is of a local nature, we shall use a ®ular coordinate system tbr computing the characters of the involutive symbol [13]. Accordingly, we m a y present the system in "solved form"with/3~ = m - a~ PD equations of (leading) class ,~, ..., and finally/311 = m - a,~ P D equations of (leading) class 1, where a i _> ... _> a~' _> 0 are the characters of the involutive symbol, appearing in its Hilbert polynomial. For simplicity when needed, we shall denote by z = (z 1, ..., z ai') a differential transcendence basis (formal input) and call y = (yl,...,yZ'~) the remaining system variables. Setting D I = K[dl, ...,dn-1] and keeping only the parametric jets of order q among ( ~ I 0 _<1 /J l< q, pn = 0, Zq), we m a y filter M by considering the family of D~-submodules M(~) tbr which z~l is such that 0 < u,, < r and obtain the nested chain of inclusions:

M(0) _C M(t) C ... C_ M(~) = M Setting now M ( - r - ~ ) = {m 6 M(-r) [ complementary nested chain of inclusions:

dnm C M(_r)}, we obtain the

M(_o~) C_ ... _C M ( - I ) C M(0) Our first key result is:

Proposition

1.

t(M) = M(-oo)

P r o o f : By construction, /14(-oo) is a D-module like t(M). However, the same proof as in the Kalman case shows that t(M) C M(o) and thus

,(M) c Conversely, if w E M(-oo) is of order q, the number of formal derivatives of w of order r is a polynomial with leading term equal to ( 1 / n ! ) r n. Counting now the jets of order q + r in dl, ..., dn-1, we get a polynomial with leading term equal to (1/(n - 1)!)(q + r) n-1. Accordingly, when r is large enough, we may eliminate the parametric jets and obtain at least one P D equation for w alone, that is M(_~) C_ t(M). Q.E.D. Replacing in [2,III,A] the rank over K by the differential rank over D', then using the additivity property of the differential rank and noetherian arguments, one can prove similarly:

Controllability of nonlinear multidimensional control systems

251

C o r o l l a r y 1. M ( - r - 1 ) = M(-r) for r large enough. According to the previous proposition, t ( M ) is the largest D'-submodule of M(r) which is stable by dn for any r < oo.

Our second key result is to extend this method to nonlinear systems with L in place of K. In the OD case, if we have a system of the form y = f ( y , u, it) for (y,u) quasi-linear in it, we may replace it by the system y = f ( y , u, v), it = v for (y, u, v) where now v appears in a quasi-linear way but without derivatives. This is the reason for which people usually consider systems of the form it = f(Y) + g(y)u where f, g are vector fields. The PD case is similar as 7~q+1 is defined by equations quasi-linear in the jets of strict order q + 1 and we m a y thus consider, with no loss of generality, a system 7r quasi-linear in the jets of strict order one, with solved equations of the form: qhj = yj, -- ak',j[yly i --

b}(y)=O

l
i
in a d-regular coordinate system. If w is an autonomous element, let us consider the vertical 1-form r = 5w E t ( M ) obtained by identifying 5 with the vertical exterior derivative dy as in [13] page 405, and let us introduce the following two distributions of (infinite) vector fiels where i( ) is the interior product:

t ( M ) • = {U E V(T~r162I i(u) A = {U E V(7r

= o,w 9 t(M)}

I i(U)w = 0, i(U)dw = 0,Vw e t ( M ) }

both with A• = {w E M I i(U)w = 0, VU c A}. The proof of the following results is absolutely similar to the one presented in (1) and will not be repeated:

P r o p o s i t i o n 2. A is an involutive distribution. P r o p o s i t i o n 3. A is invariant under d,~. C o r o l l a r y 2. A • is invariant under dn.

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J.F. Pommaret

Theorem

2. t(M) = A •

We notice that the specific use of involutive systems in Spencer form has allowed one to introduce (infinite) vector fields with components only depending on a finite number of parametric jets with multi-indices p such that p,~ = 0, exactly like in the OD case, but now with dn in place of d/dt. The main difference with the OD case lies in the following corollary which generalizes the Frobenius theorem (see [14] for more details):

C o r o l l a r y 3. The vertical exterior derivative of each torsion 1-form of a basis {w} of t ( M ) ca,, be expressed as a finite sum of exterior products of certain arbitrary 1-forms {p} with 1-forms of t ( M ) obtained from the

I-forms of the basis by applying operators in D ~, that is to say, suppressing the indices for simplicity, we have symbolic relations of the form:

5w = E

p A D~w.

R e m a r k 1: When t(M) has finite type, for example when n = 1, we get equations of the fbrm 5~ = ~ p A w and the Frobenius theorem can be applied, that is the torsion elements can be integrated. This is the particular case considered in [1-3].

Counterexample

_

_

y3yl

1: n = 2, rn = 3, q = 1

=

0,

_=

-

y3y

=

0

The linearized system Rt is defined over L by the two P D equations:

d2y1 _ y3dt y1 _ y~ y3 = O, d2y~ _ ya dl y2 _ y~y3 = 0 Multiplying by test functions (A t, AS) and integrating by part, we get the adjoint o p e r a t o r / ) t A = p over L in the form:

_d2A1 + y3dtAt + yatAl = pt, -d~A 2 + y3dlA2 + y~A 2 = iJ~,

Controllability of nonlinear multidimensional control systems

253

We get the generating compatibility condition 7~tt = 0 over L in the form:

y~l-t 1 -4- y~l~2

d2/-t3 -4- y3 dlIl3 + 2y3tt 3 = 0.

-

Multiplying by a test function ~ and integrating by part, we get 2D~ = Y over L in the form:

Accordingly, apart from 7?1Y = 0, we get the new compatibility condition/torsion element:

satisfying:

d2w - yadlw - y3w = 0

In this case, w generates t ( M ) and we have:

5w = dl~ A w + ~ A dtw

As w A 5w 7s O, it follows that w cannot be integrated. Finally, setting y3 = z as before and decomposing d~ along the jet inputs z, zl, z11, ..., we m a y introduce the vertical infinite vector fields:

U(z) = ~I ~

O

s0

0

+ yl ~

O

20

+ yh o%-f + Y~-g~u~ +

'

0

U(z~) = ~I oy---?~+ y~ o~--7~+

'

and so on, while obtaining on the infinite level (care):

[U(z),U(zl)]

=

-U(z),

...

It is easy to check that U ( z ) , U ( Z l ) , U ( Z l l ) , . . . are orthogonal to w, dlw, dllw, ... and that their interior product with ~w can be expressed as linear combinations of w, dlw, dllw, ..., though they do not kill any function of the parametric jets (yl, y~, y~, y~,...), in agrement with the previous results.

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J.F. Pommaret

R e m a r k 2: The example presented in [9] with n = 2, m = 4, q = 2 could be treated similarly, evet~ t h o u g h the d e l a y does 1~ot act as a d e r i v a t i o ~ o~ a p r o d u c t , because only the two linear equations involved are used for exhibiting the generating torsion element. R e m a r k 3: The reader not familiar with infinite differential geometry may not forget that di is an infinite vector field (with polynomial components) that do not kill any function though it is orthogonal to all the contact forms dY k Y~+I, d x i " -

C o n c l u s i o n : We have presented new methods for better understanding the theory of multidimensional nonlinear control systems. In particular, we have shown that the formal theory of systems of PD equations (jet theory, formal integrability, involution) canTwt be avoided. The forthcoming book [14] will prove that the techniques of algebraic analysis [15] can also be extended to nonlinear systems and optimal PD control, that is PD variational problems with PD constraints.

Controllability of nonlinear multidimensional control systems

255

References 1. E. Aranda-Bricalre, C. H. Moog, J. B. Pomet (1995) Infinitesimal Brunovsky form for nonlinear systems with applications to dynamic linearization, Geometry in nonlinear control, Banach Center Publications, Institute of Mathematics, Polish Academy of Sciences, Warszawa, 32, 19-33. 2. E. Aranda-Bricaire, C. H. Moog, J. B. Pomet (1995) A linear algebraic framework for dynamic feedback linearization, IEEE Transactions on Automatic Control, 40, 1, 127-132. 3. G. Conte, C. H. Moog, A. M. Perdon (1999) Nonlinear control systems, Lecture Notes in Control and Information Sciences, 242, Springer. 4. Goldschmidt, H. (1969) Integrability criterion for systems of nonlinear partial differential equations, J. Diff. Geom., 1,269-307. 5. A. Isidori (1985, 1989) Nonlinear control systems: an introduction, Lecture Notes in Control and Information Sciences, 72, Springer. 6. J. Johnson (1969) KKhler differentials and differential algebra, Annals of Mathematics, 89, 1, 92-98. 7. E. Kolchin (1973) Differential algebra and algebraic groups, Academic Press. 8. R. Marino, P. Tomei (1995) Nonlinear control design, Prentice Hall. 9. L. A. Marquez-Martinez (1999) Note sur l'accessibilit~ des syst~mes nonlin~aires, C.R. Acad. Sc. Paris, 329, 545-550. 10. U. Oberst (1990) Multidimensional constant linear systems, Acta Appl. Math., 20, 1-175. 11. J. F. Pommaret (1986) G~om~trie diff~rentielle alg~brique et th~orie du contr61e, C.R. Acad. Sc. Paris, 302, I, 547-550. 12. J. F. Pommaret (1989) Probl~mes formels en th~orie du contr61e aux d~riv~es partielles, C. R. Acad. Sc. Paris, 308, I, 457-460. 13. J. F. Pommaret (1994) Partial differential equations and group theory: new perspectives for applications, Kluwer. 14. J.F. Pommaret (2000) Partial differential control theory, Kluwer, to appear. 15. J. F. Pommaret, A. Quadrat (1999) Algebraic analysis of linear multidimensional control systems, IMA Journal of Mathematical control and Information, 16, 275-297. 16. J. Wood, E. Rogers, D. Owens (1998) Formal theory of matrix primeness, Mathematics of Control, Signals and Systems, 11, 40-78.

Stabilization o f a Series D C M o t o r by Dynamic Output Feedback Richard P o t h i n 1, Claude H. Moog I, and X. Xia ~ 1 IRCCyN, UMR C.N.R.S. 6597 1 rue de la No~, BP 92101 44321 Nantes Cedex 3, France {pothin ,moog}@irccyn. ec-nantes, fr 2 Department of Electrical and Electronic Engineering University of Pretoria Pretoria 0002, South Africa xxia@post ino. up. ac. za

The goal of this paper is to design a stabilizing output feedback controller for a series DC motor, without using observers. The control scheme is based on exact output feedback linearization and on output feedback stabilization. It avoids any approximation in the control of the model of the DC motor. Abstract.

1 Introduction A DC m o t o r where the field circuit is connected in series with the a r m a t u r e circuit is called a series DC motor. Since this connection allows to produce a very large torque with respect to the current, a series DC m o t o r is essentially used for electrical traction applications [1,5].

Rf

Lf E

V

_

Ra

La

Fig.1. Schematic circuit for a series-connected DC motor

258

Richard Pothin et al.

The armature inductance is denoted by La and its resistance Ra. R ! denotes the resistance of the field winding and L] its inductance. V denotes the voltage applied to the motor and E the electro-motrice-force. Following the notations and the assumptions as in [3], a state space representation of the series DC motor, at low speed, when there is no field weakening, can be written as :

it1

=

-klxlz:~

--

Ra + Ry rld-U K ki x~

x3 = 0

(1)

y = Xl~ where xl = r denotes the flux, x2 = w denotes the speed, and x3 = 7]/J denotes the load torque divided by the inertia of the rotor, kl -- Km is the constant ratio torque/back-emf and B is the friction coefficient. The measurement is then the flux xl = Cf(i) = Ki which is directly deduced from the physical measurement of the field current. This follows from the assumption made in [3] where the flux r is supposed to be linear when there is no magnetic saturation. Since all state variables are not available for measurement, the main idea, in this paper, is to seek directly a (reduced order) stabilizing dynamical feedback, without using observers. This solution has the advantage to circumvent the problem of superposition of the observer with the feedback. Indeed, when a nonlinear observer is combined with a stabilizing controller, this does not necessarily lead a stable system in closed loop [4]. It is also possible to consider a non constant reference, as in the trajectory tracking problem, since no use is made of any (stable) state estimation. The control purpose is to stabilize the flux. In practise, the flux and the torque are related by a static relation. Thus, the stabilization of the flux is equivalent to the stabilization of the torque. This paper is organized as follows: Section 2 is devoted to introduce the stabilization approach which is applied in Section 3 to a series DC motor at low speed (below 200 rd.s-l). In Section 4, some simulation results are given. Finally, some concluding remarks are done in the last section.

2

Stabilization

approach

The given local output stabilization approach can be decomposed into two steps. The first step consists in cancelling nonlinearities by (partial) feedback linearization or/and approximation. The second step consists in constructing the minimal linear dynamical output feedback which stabilizes the considered system, thanks to Algorithm 1 below.

Stabilization of a Series DC Motor by Dynamic Output Feedback

259

The algorithm, for constructing the minimal order dynamic stabilizing output feedback, is effective for the following class of SISO system:

~1 = 6 6=6

~-1 =r b = E_~,=l a,r + u ~+1 = fr+l (;, u)

(2)

& = s 1 6 2 u) Y=r whereaiER, i= 1,...,n. System (2) is supposed to be (weakly) minimum phase. This algorithm reduces to [2] in the special case where the linear system has no zero. A l g o r i t h m 1: Step 0: * put the linear subsystem in the controller canonical form (2). * Apply the static output feedback u = c01y + v0. 9 Check if the system can be stabilized by a suitable parametrization of

COl. 9 If yes, put v0 = 0 and Stop! Otherwise go to step 1. Step 1: 9 Apply the dynamic output feedback u =

clly+

ih = ~llY +

711

~12~I + vx.

9 Check if the system can be stabilized by suitable parametrization of e l l , Cll~ C12"

9 If yes, put vl = 0 and Stop! Otherwise go to step 2. Step i(i = 2 , . . . , r -

1):

9 put the linear subsystem in the controller canonical form (2) (of dimension r+i-1). 9 Apply the dynamic output feedback Vi_ 1 = c i l y + ~ i

//i = 5 i l y + ~i2~i + vi.

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Richard Pothin et al.

9 Check if the system can be s t a b i l i z e d by a suitable p a r a m e t r i z a t i o n o f ckl, ~1,5k2, k = 1 , . . . , i . 9 If yes, p u t vl = 0 and Stop! Otherwise go to s t e p i + 1. End of A l g o r i t h m . T h e d y n a m i c extension has a d i m e n s i o n which is at least 0 and at m o s t r - 1. In the case where this extension d i m e n s i o n equals r - 1, this corresponds t o a p p l y a reduced order observer, since t h e d i m e n s i o n of the linear observable s u b s y s t e m is r as shown in [2]. In t h e o t h e r cases (when the extension is between 0 and r - 2.), the d y n a m i c o u t p u t feedback is by no m e a n s an observer.

Now, this a p p r o a c h is applied on a series DC m o t o r described by s y s t e m (1) and which can be p u t into the form (2). Note t h a t the d y n a m i c o u t p u t feedback used to stabilize the linearized series D C m o t o r is not an observer since its d i m e n s i o n equals 1.

3

Application

to a series

DC

motor

Since s y s t e m (1) is i n p u t - o u t p u t (i/o) linearizable by i / o injection, a n d its relative degree equals one, there exists a d y n a m i c o u t p u t feedback,

u=yr/+k2ylny+ i] = v -

R,, K + R! xl

(3)

ks k2~1 - k ~ l n y + ~ -1~ y 2 ,

which linearizes the o u t p u t ~ = In y [7]. T h e equivalent closed loop s y s t e m can be w r i t t e n as

~2 ----f3 q- k27/+ k~ In y -

k21

~=o r

k=1 2 v - k2~ - k~ In y + ~-~-y = r

in the new s y s t e m of coordinates : ~1 : In x l ~ = -klx2 + k~lnxl ~3 = k l z 3 .

2

(4)

Stabilization of a Series DC Motor by Dynamic Output Feedback

261

Consequently, the resulting linear i n p u t - o u t p u t relation is ,~(3) __ t). Since the load torque is an unknown constant, the series DC m o t o r is weakly m i n i m u m phase. Now turn to Algorithm 1 to stabilize the system. Consider the DC m o t o r with zero load (~3 -- 0) and apply Algorithm 1 on the relation: = r

+ v.

(5)

A stabilizing output feedback is c o m p u t e d from step 1 of Algorithm 1,

(6)

v - -12(Z) - Yd) + ~71 //1 ----64(3 -- .~d) -- 6yl,

in which Ya = In ya, Yd is the desired reference o u t p u t (flux). Since the flux r = K i is controlled, the torque C is controlled thanks to the static relation C = k l r = k l K i ~. Simulation results are displayed in the next section.

4

Simulation

results

The following parameters are borrowed from [3].

R ! = 0.0148552 B = O.1N/rad.s -1

Ra = 0.0098952 K r n = 0.04329

Rp = 0.0169652 J = 30.1N/rad.s -1

K = 0.06.

The voltage V of the m o t o r is restricted to 0 < V < 100OV. The current limit is i = 1000 Amperes. The following simulation results show the stabilization of the flux and the torque at low speed (speed below 200 r d . s . - 1 ) .

These simulation results has been obtained with a non zero load-torque equal to 100 N m . The c o m m a n d velocity is adapted so t h a t the steady state reaches the desired value.

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Richard Pothin et al.

w

t ,,

f

9

Fig.2. Flux(Wb) versus time(s)

~

d

0

i

t#

12

~

Fig.3. Current(A) versus

ii

~

u

time(s)

," 0

z

4

0

i

II t~

it

,4

ii

Im qq

Fig.4. Torque(Nm) versus time(s)

Fig.5. Velocity(rad/s) versus time(s)

/ Fig.6. Control

voltage(V) versus time(s)

Stabilization of a Series DC Motor by Dynamic Output Feedback

5

263

Conclusion

In this paper, a dynamic output feedback for stabilizing the current and the torque of a series DC motor is given. Moreover, the way to design this solution is simple and constructive. Indeed the compensator is composed of two distinct dynamics whose designs are completely decoupled. The first one linearizes the series DC motor and the second one controls the current and implicitly the torque. No explicit use of observer is made. In the case where the speed is larger than 200 rd.s. -1, the model of the motor is slightly different but keeps the same structure. Then, a similar reasoning can be applied in the field weakening region, as well [8].

References 1. R. D. Begamudre, Electro-Mechanical Energy Conversion with Dynamics of Machines, Wiley, 1988. 2. F.M. Brasch and J.B Pearson, Pole placement using dynamic compensators, IEEE Trans. Aut. Contr., 15, 1970, pp. 34-43. 3. J.N. Chiasson, Nonlinear differential-geometric techniques for control of a series DC motor, IEEE Trans. Contr. Syst. Technology, 2, 1994, pp. 35-42. 4. P.V. Kokotovic, The joy of feedback: nonlinear and adaptive, IEEE Contr. Syst. Magazine, pp 7-17,1992. 5. W. Leonhard, Control o] Electrical Drives, Springer-Verlag, 1985. 6. R. Pothin, S. Celikovsk3~ and C.H. Moog, Simulation results of stabilization of an inverted pendulum by dynamic output feedback, in Proceedings of IFAC Rocond, Prague, Czech Republic, 2000. 7. R. Pothin, C.H. Moog, and X. Xia, Input-output linearization of nonlinear SISO systems by output feedback, submitted for publication. 8. R. Pothin, C.H. Moog, and X. Xia, Nonlinear stabilizing dynamical output feedback for a series DC motor, In Poceedings of IFAC Decom, Pretoria, South Africa, 2000.

Stabilization of Nonlinear Systems via F o r w a r d i n g mod{LoV } Laurent P r a l y l, R o m e o O r t e g a 2, and G e o r g i a K a l i o r a a* i Centre Automatique des Systmes Ecole Sul~rieure des Mines de Paris 35 Rue Saint Honor6 77305 Fontainebleau, France praly@cas, ensmp, fr Laboratoire des Signaux et Syst~mes CNRS-SUPELEC, Plateau de Moulon 91192 Gif-sur-Yvette, France rort ega@iss, supelec, fr

3 Dept. of Electrical and Electronic Engineering Imperial College London, SW7 2BY, Great Britain g . k a l i o r a @ i c . ac .uk A b s t r a c t . Forwarding is a powerful tool for constructing stabilizers for nonlinear systems which is applicable to various practically interesting examples. A key step in the design technique involves the solution of a partial differential equation (PDE), which may be hard to find -actually, the PDE may even not be solvable at all. In this brief note we show that it is possible to provide an additional degree of freedom for the solution of the aforementioned PDE, hence effectively extending the realm of application of the forwarding methodology. Our contribution is illustrated with the example of an inverted pendulum with a disk inertia.

1

Background

In this section we will briefly review the basic f o r w a r d i n g technique for stabilization of n o n l i n e a r systems from a g e o m e t r i c perspective. For further details on this technique the reader is referred to [5,4]. To m o t i v a t e the d e v e l o p m e n t s let us consider first a cascade o f two scalar s y s t e m s of the form

= h(x) k ----f ( x ) where the origin of the x - s u b s y s t e m is a s y m p t o t i c a l l y stable, n a m e l y f ( 0 ) = 0 and there exists a positive definite L y a p u n o v function V(x) such t h a t 1 The work of Ms. Kaliora is supported by the State Scholarships Foundation of Greece and by the TMR Network, NACO 2

,gv I We use the standard Lie derivative notation LIV ~=--~.f.

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Laurent Praly et al.

LI V < 0 for all x r 0. To study the stability of the cascade we look for the existence of a stable manifold of the origin described by the graph z = M(x). That is, we want to find a C 1 function M(x), with M(0) = 0, such that the following implication is true (z(0), x(0)) E / 2 ~ {(z, x)lz = i ( x ) }

:0

(z(t), x(t)) E/2, Vt >_ 0

It is easy to see (e.g., see the proof of Lemma A in the Appendix) that the existence of/2 is equivalent to the solvability of the PDE

LfM = h

(1)

with boundary condition M(0) = 0. If we can solve the PDE (1) a Lyapunov function for the overall system is given by

w ( x , z)

v(x) + 1

- M(x)]~

(2)

whose derivative is simply L 1 V. Forwarding builds upon this basic idea to stabilize cascaded systems of the form

,V,:

f ( x ) + g(x)u.

(3)

In this case the derivative of W yields

W = L]V + [ L g V - (z - M)LgM]u which clearly suggests the control law

u = - [ L g V - (z - M)LgM].

(4)

C a v e a t For the sake of clarity we restrict our attention to the case of scalar subsystems. As will become clear in the example our developments extend verbatim to the vector case. We also avoid the technical details concerning the domains of validity of the various statements, this in the understanding that if the assumptions hold globally then the results are also global. 2

Forwarding

mod{LgV}

The main stumbling block of the forwarding procedure is, of course, the solvability of the PDE (1), a question which is difficult to answer in general. The main objective of this paper is to provide an additional degree of freedom for the solution of the PDE, consequently enlarging the class of systems that can be stabilized with the forwarding procedure. Towards this end we show that we can add to the right hand side of the PDE a "free" term, and still be able to synthesize a stabilizing controller.

Forwarding mod{LgV}

267'

P r o p o s i t i o n 1 Consider the system (3) with the following assumptions: A.1 (Stability of the x-subsystem) f(0) - 0 and there exists a positive definite Lyapunov function V(x) such that LI V < 0 for all x r 0. A.2 We know a s function k(x) and a C 1 function M ( x ) , with M(0) = 0, such that (i) (New PDE) (compare (5) with (1).) LIM = h + kLgV (5) (ii) The function rain { k ( x ) ( i g U ( x ) ) -z, 0} i a V ( x ) is continuous and

i a i ( O ) ~ O. (iii) The following inequality holds L f V <_ k ( L a M ) - I ( L g V ) 2 (6) Under these conditions, the equilibrium of the system (3) is globally asymptotically stabilized by the control u = k ( z - M) - 7 [ n g g -

(z - M ) L a M ]

- min{0, k L g M } ( L g M ) - ~ [ L g V + (z - M ) L g M ]

(7)

where "y is any continuous positive definite function of (x, z). Proof Evaluating the derivative of the Lyapunov function candidate (2) along the trajectories of (3), and using (5), yields ~V = L I V + LgVu - (z - M ) [ L g M u + kLgV] We consider two cases: 1. For points x where k ( x ) L g M ( x ) >_ O, by adding and subtracting k L g M ( z - M ) ~ we can write W in the form W -- L f V + [LgY - (z - M)LgM][u - k(z - M)] - k L g M ( z - M ) ~ So, the control u = k ( z - M) - 7 [ L g V - ( z - U ) L g i ]

(8)

with 7 as given in the statement, yields I/V = LI V - ~/[LgY - (z - M ) L g M ] 2 - k L g M ( z - M) 2

(9)

Here, we notice the following chain of implications W=O=> LIV=O ::~ x -= 0 (r (A.1)) => ~o xv (o) : 0 (r

V is positive definite)

:::v LgV(O) = 0 We also have that W = 0 ~ ( z - M ) L g M - LgV = 0. Given that i ( O ) = O, LgM(O) ~ 0 and LgV(O) ---- 0 the latter implies that z -= 0. So when restricted to the set of x where k ( x ) L g M ( x ) >_ O, I~ is negative definite with the control (8).

268

Laurent Praly et hi. LV

2. For points x where k ( x ) L g M ( x ) < 0, by adding and substracting k ( L sg M/

~

we can write I)d in the form

(LgV) 2 LgV IJV = L I V - k ~ + [LgV - (z - M)LgM][u + k L - ~ ] So, the control

u = - k ( L g M ) - I L g V - 7[LgV - (z - M ) L a M ]

(10)

yields

PV = [LIV - k ( L g M ) - I ( L g V ) ~] - 7 [ L g V - (z - M)LgM] ~ Again we notice

I)V = 0 ~ [LyV - k ( L g M ) - l ( n g v ) 2] = 0 LIV = 0

( ~ d.2(iii))

( ~ k(LgM) -1 < 0 and (A.1))

So exactly as above we conclude that, when restricted to the set of x where k(x)LgM(x) < O, W is negative definite with the control (10). To conclude we observe that the expressions (8) and (10) are given by (7) which is continuous by assumption. This control provides a negative definite function i)d. R e m a r k 1 To get some further insight into the assumptions of the proposition it is interesting to consider the linear case = cTx

= Ax + bu We then have V = 89 with P = pT > 0 the solution of the algebraic Lyapunov equation P A + A T p = - Q < 0, and we can take M ( x ) = M T x , with M defined as M = A - l c + kbTp. Now, as LgM = MTb, the condition A.2 (ii) reduces to MTb :fi O, which is a necessary and sufficient condition for controllability of the system. Also, assumption A.2 (iii) reduces to 1

- ~ Q <_ k ( M r b ) - l PbbV p which is a measure of stability (actually, of excess of o u t p u t feedback passivity) of the x subsystem. R e m a r k 2 Using forwarding mod{LgV} allows us, in some cases, to relax the assumption A.1 of stability of the z - s u b s y s t e m to L ! V ~ 0. This feature is illustrated in the example below.

Forwarding rnod{LgV}

3

269

Stabilization of the disk inertia pendulum

In this section we apply Proposition 1 to stabilize the upward position of the pendulum device shown in Fig. 1, which consists of a physical pendulum with a rotating mass at the end. The motor torque produces an angular acceleration of the end-mass which generates a coupling torque at the pendulum axis. It is worth mentioning that the present study was precisely motivated by this example, where the input constraint puts a hard bound on our ability to stabilize the upward position. First, we show that the problem is not solvable with "standard" forwarding. In particular we prove that the PDE (1) is not globally solvable. To overcome the problem we apply the forwarding m o d { L g V } technique and show that it is possible to design a saturated controller that "almost" globally stabilizes the upward position. ~ In [6] the system is stabilized using passivity-based control, we refer the reader to this reference for further details about this device.

]

Fig. 1. Experimental apparatus balancing at the unstable equilibrium point A. M o d e l The dynamic equations of the device can be written in standard Lagrangian form as

In 1

The change of coordinates xl = ql, xa = 01, zl = 2ql + q~, z2 = 201 + 02 and v = sin(x1) - u leads to a system in the {block) forwarding form (3) as 2 The qualifier "almost" is needed because there is a set of initial conditions which do not converge to the upright position, but it has zero measure.

270

Laurent Praly et al. 7,1 ~ Z2

~2 = sin(xx)

(12) 9~ 1

=

x3

dr3:u where, to avoid cluttering the n o t a t i o n , we have t a k e n mgl = ml = m2 = 1. 3 T h e final control objective is to find a stabilizer o f the origin of (12) which is bounded. T h i s l a t t e r constraint m a k e s the whole difficulty of the p r o b l e m . W i t h o u t it, a b a c k s t e p p i n g technique or s t a b i l i z i n g a p a r t i c u l a r p l a n n e d traj e c t o r y would provide a global a s y m p t o t i c solution. Here, before presenting a solution to the constrained p r o b l e m , we solve the n o n - c o n s t r a i n e d p r o b l e m with the new forwarding mod{LgV}.

B. S t a n d a r d f o r w a r d i n g

(Step I) We consider the s u b s y s t e m ~1 = X3

5r3:u and design a controller invoking passivity, i.e. with the L y a p u n o v function v~(x~,x~)

= (1 -

Ot 2 cos(x,)) + ~x~

where a > 0 is a tuning p a r a m e t e r . T h i s yields 1)'1 = x3 [sin(x1) + au] Hence the first control loop is

1 u = - - - sin(x1) - / 3 x 3 + vl O~

where fl > 0 and vl will be defined in the next step.

(Step 2) We a d d an integration to the s u b s y s t e m as ;?2 = sin(z1) xl = x3 9 ab3 = - 4

sin(x1) -

(13) /3z3 + vl

3 In Subsection 3.E, where we present some simulation results, we give the expression of the control law for the general case.

Forwarding

mod{ Lg V }

271

and, following [5], look for a function M~(x~, x3) such that, with v~ = 0, we have M~ = sin(x~). This is t a n t a m o u n t to solving the P D E (1) which, in this case, takes the form

c9M1

c3M1 [ 1 sin(x1) +/~x3] = sin(xx)

(14)

A solution is given as M1 = - a ( x 3 + ~xl). The second controller is defined by the formula (4) and takes the form vl = z 2 5 r

+ .(~

LgV1

v

+ z~)] (-0) +.~

z~-M1

s

~

LgM1

(Step 3) We add the last integration to the subsystem as Z1 =

Z2

~u = sin(x1) ;~1 :

X3

x3 = - ~

sin(~l) - (fl + ~ + ~ ) ~ 3 - ~z2 - Z~2~1 + v~

Now, we look for a function M2(z2,xl,x3) such that, with v~ = 0, we have M~ = z2. Unfortunately, the associated P D E does not have a global solution. Indeed, in L e m m a A of the Appendix we show that a necessary condition for the existence of a global solution of the P D E (1) is that the function h(x) is equal to zero at the equilibria of the subsystem x = f(x). In our ease, the equilibria of the " f subsystem" are given by (5~, ~1, ~3) = (afljTr, j r , 0), with j E Z, hence the "h function", which is equal to z2, is nonzero at some of the equilibria and we cannot complete our design. We should underscore that the procedure was stymied by the presence of the term fla2xl in the " f subsystem". We will show below that this term can be removed with the new forwarding technique. C. F o r w a r d i n g

rnod{LgV}

(Step 1) Is the same as above. (Step 2) Proceeding from the subsystem (13) we look now for a solution of the new P D E (5), which in this case takes the form OM1 OM1 [ 1 cOx----~x3 - ~ -d sin(x1) + fix3] = sin(x1) + k(xl, x 3 ) , ~ LgVx

272

Laurent Praly et al.

Compare with (14). To remove the term dependent on xl we propose a solution M1 = - a x 3 and k = ft. Noting that kLaM1 - -]~(~ < 0 we evaluate the control law (10) to get the second loop

v~ = _ l [ z ~ + (1 - ~ # + ~ ) ~ ] + v; where, for ease of presentation, we have taken 7 = aA~- R e m a r k t h a t we have succeeded in eliminating the troublesome term. The Lyapunov function corresponding to this step is V2(zl, z3, z2) = (1

2

-

1

cos(x1)) + ~ z 3 + ~(z2 + ax3) ~

(15)

(Step 3) The last step is classical forwarding similar to Step 3 above, with the fundamental difference that the P D E is now OM2 sin(x1) + OM~ OM2 1 Oz2 - ~ x l z3 - c9x3 [sin(xx) + z2 + (1 + a)z3] = z2 for which we have a solution M2 = - z 2 - ax3 - (1 + a ) x l . We then c o m p u t e the final outer loop control v~ = - a [ z l + 2z2 + (1 + a ) x l

+ (1 + 2~)x3]

By invoking LaSalle's invariance principle, we get the following stabilization result. P r o p o s i t i o n 2 The system (12) in closed-loop with the static state feedback control 1

1

u = ---a sin(x1) - aZl - ( ~ + 2a)z~ - a(1 + c~)xl -(~

1

+ ~ + 1 + 2c~)x3

with a > 0, has an asymptotically stable equilibrium at zero with Lyapunov function a

w ( x l , x3, zl, z2) = (1 - cos(~,)) + ~ 3

+~(zl + z~ + ~

2

1

+ ~(z2 + ~ 3 ) ~

+ (1 + ~)x~)~

L

Its d o m a i n of attraction is the whole space minus a set of measure zero.

Forwarding m o d { L g V }

273

R e m a r k 3 It is interesting to note that the controller derived in [6] using passivity considerations is of the form of the controller above, that is u = - c l s i n ( z 0 - c~zl - c3z2 - c 4 x l - c s z s with c;, i = 1 . . . , 5 some suitably defined positive constants. Also, the Lyapunov function in that paper is the sum of a quadratic function of the state and the potential energy term as W above. R e m a r k 4 As it is well known, the commissioning of nonlinear controllers on actual physical devices is far from obvious. Hence it is interesting to know what are the available degrees of freedom in the tuning parameters. In Proposition 2 above we have restricted, for the sake of clarity of the presentation, to the single p a r a m e t e r a. A natural question is then what is the largest range of the constants ci, i = 1 - - - , 5, so as to globally stabilize the p e n d u l u m w i t h a Lyapunov function consisting of the s u m of a quadratic function of the state and the potential energy term. T h e answer is provided by the following choices c

ae

b

cl = ~, c2 = --~-, c3 = [ ( d + c ) a + e

ce

b

ab

e

b---~], c4 = [ ( d + c)(C + -~-) + ~ 1 ,

ab

c~ = ~ ( c + c ) where a, b, c,d, e > O. One final remark is that it is possible to prove that there does not exist gains ci, i = 1 . . . , 5 , such that W is a s t r i c t L y a p u n o v function. D, S a t u r a t e d c o n t r o l We will consider in this subsection the practically i m p o r t a n t case when the control signal must satisfy a bound lu] < U M . To obtain a saturated control we take-off from the second step above and evaluate the derivative of V~ (15) (along the trajectories of (12)) as V2 = [z2 + (1 + a)z3][otu -~ s i n ( z 0 ] Let us define /x

Yl = zl - Ms = zl + z~ + a z 3 + (1 + t~)xl

(16)

whose derivative yields ~)1 = [z~ + (1 + a)x3l + [au + s i n ( x 0 ] Instead of proceeding with the third forwarding step we propose a L y a p u n o v function candidate which is suggested by the c o m p u t a t i o n s above V3(zl, z 3 , z l , z2) = V ~ ( z l , x 3 , z2) +

a(s)ds

274

Laurent Praly et al.

where a is any continuous b o u n d e d o d d function. Indeed, this yields

93 = a(yx) [(1 + a)x3 + z2] + [(1 + a)x3 + z~ + a(yl)] [sin(xl) + au] which leads to the control u =

1

-a

sin(x1) - b(yo)

(17)

where w e have defined /x

Y0 = (1 + a)x3 + z~ + a(yl)

(18)

where b is, again, any continuous bounded odd function. Notice that the control is bounded by any prescribed bound UM by choosing appropriately ot > 1/UM, and the function b to be bounded in norm by UM 1. The derivative of the Lyapunov function can be expressed in the form

93 = -a(ul )~ + Uoa(Ul) - ,~Uob(Uo) which is a quadratic form in a. Letting aM > 0 be a bound for ]a I and maximizing the polynomial we see that, to ensure negativity of i~'3, we need the implications

lu01 aM < T ~ aMlYol -- a]4 < ,~yob(yo) Y~ < OtyO b(y0) "ly01 ~ < a M :::r --~ The latter imposes some restrictions on the function b as 2aM_< l Y 0 I : : > - Ot

1--

_< Ib(y0)l

lu0i lyol < 2aM ~ ~ < Ib(yo)l which are easily satisfied. In this case, we get ~'3 < 0 and consequently 93 = 0

::~

{Y0 = O, Yl = O}

Finally, an invariant set analysis for V3 = 0 leads successively to: 0=(l+a)xt+zl+z~+~x3 fromyl =0 (19) 0=(1 + a)x3 + z~ from Y0 ---- 0 and a(0) --- 0 (20) O=sin(xl) + a u from (17) and b(0) = 0 (21) 0=(1 + a)u + sin(xz) from differentiation of (20) (22) O=sin(xl) = u =r xl = j ~r from solving the linear system ((21),(22)) (23) from differentiation of (23) (24) O=x3 from (20) (25) O~g 2 O = [ l + a ] j ~ " + zl from (19), (24), (25) (26)

Forwarding mod{ Lg V }

275

The main result of this section is contained in the proposition below. P r o p o s i t i o n 3 Consider the system (12) with bounded input lu[ 0 such that 1/UM >> a, and choose a, b such t h a t lal g a M and Ibl < u M - - 1. Further, select the function b such that

a--~M ( 1 - - a7-~-7~ < ,b(yo), if 2aM <_ lYo, Ct

\

lY01/

Iv01 < Ib(y0)l if lY01 < 2aM 4a Under these conditions, zero is an asymptotically stable equilibrium of the closed-loop, with domain of attraction the whole space minus a set of measure zero. E. S i m u l a t i o n s Repeating the derivations leading to Proposition 3 for the general model (11) we obtain the controller u = -

1 ~tTl 2

sin( xl ) - b(y0), 77/2

y0 ~ (1 + aflrn~(mgl)~)xa + flm~mglz~ + "/a(yl) ZX

(1----L--z2 + ~ 3

yl = zl + "r \ m ~ m g l

+

1 +,~,.~(mgl)~ 3m~(mgl) 2

~1

where, to provide more tuning flexibility, we have included some additional gains that were set to one in the procedure described above. Note also, that the change of coordinates is now Xl = m2ql, x3 = m2(ll, zl = (ml + m2)ql + mlq~,z2 = (ml + m~)ql + m1(12 and v = m g l s i n ( x l / m 2 ) - u. We simulated the response of the pendulum using the system parameters of the hardware setup reported in [7], namely: ml = 0.3x 10 -6, ms = 0.0048, ml = 38.7• 10 -3 [Kg]. The controller gains were set at a = 700, fl = 0.0025, 7 = 10. Figure 2 shows the swingup response of the pendulum starting at nearly the vertically downward position, with the remaining initial conditions zero. Notice that the response is very fast without any initial swinging of the pendulum, this should be contrasted with the simulations shown in [7] (e.g., Fig. 5). The following remarks are in order: 9 It is clear from the simulations that the stabilization mechanism of our controller consists of spinning-up the disk inertia to lift the pendulum, which might impose some unrealistic values to the disk speed. This should be contrasted with the alternative m e t h o d of [8] - a l s o studied in [1,3]- where the

276

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Laurent Praly et al.

*s

t

IJ

t

Ii

9

*t

9

~

9

2. Swingup response of the Pendulum

energy is first pumped-up through a balancing motion before lifting the pendulum. Two drawbacks of the latter approach are the slow convergence and the need to switch the controller close to the upward position. From the theoretical viewpoint both methods also differ, our controller (as well as the one reported in [6]) stabilizes the equilibrium point, while the energy-pumping methods stabilizes the homoclinic orbit, hence the need for the switching. 9 Although we have solved the stabilization problem of the system (12) with any prescribed saturation of the control, when we come back to the original disk inertia pendulum (11), we have to add sin(x1) to the above control. So the above procedure does not give an answer to the problem where the maximal torque that the motor can deliver is smaller than the maximal gravity torque. Our simulations have shown that stability cannot be guaranteed if we impose this saturation limit.

References 1. Astrom, K., Furuta, K. Swinging up a pendulum by energy control (1996) Proc. 13th IFAC World Congress, San Fransinco, USA, E, 37-42. 2. Isidori, A. (1995)Nonllnear Control Systems. Springer-Verlag, 3rd ed. 3. Fradkov, A.(1996) Swinging control of nonlinear oscillations. Int. J. Control, 64, 6, pp. 1189-1202. 4. Jankovi~, M., Sepulchre, R., Kokotovi~, P. (1996) Constructive Lyapunov stabilization of nonlinear cascade systems. IEEE Transactions on Automatic Control, 41, 12. 5. Mazenc, F., Praly, L. (1996) Adding integrations, saturated controls and global asymptotic stabilization for feedforward systems. IEEE Transactions on Automatic Control, 41, 11. 6. Ortega, R., Spong, M. (2000) Stabilization of underactuated mechanical systems via intercormection and damping assignment. IFAC Workshop on Lagrangian and Hamiltonian methods in nonlinear systems, Princeton, NJ, March 16-18, 2000.

7. Spong, M. W., Corke, P., Lozano, R. (1999) Nonlinear Control of the Gyroscopic Pendulum. Automatica, (submitted). 8. Spong, M., Praly, L.(1997) Control of underactuated mechanical systems using switching and saturations. Lecture Notes in Control and In]ormation Sciences, 222, Springer.

Forwarding m o d ( L g V }

A

277

Appendix

L e m m a A A necessary condition for the global solvability of the PDE (1) is that h vanishes at all the zeros of f , that is, f(~-) = 0 ::~ h(~') = 0. P r o o f Let us assume that the PDE (1) is globally solvable. This clearly implies that the function C(x, z) ~ z - M(z) is constant along the trajectories of the system = =

Consequently we have that, for all initial conditions, z(t) - M(x(t))

= z(O) - M ( x ( O ) )

(27)

On the other hand, the trajectory starting at the initial conditions (z(0), x(0)) = (z0, ~) is given as z ( t ) = zo + h ( ~ ) t ,

x(t) =

Hence, for this trajectory we have z ( t ) - M ( ~ ) = zo - M ( ~ ) + h ( ~ ) t

which together with (27) implies that h(~) -- O.

A Robust Globally Asymptotically Stabilizing Feedback: The Example of the Artstein's Circles C h r i s t o p h e Prieur Universit~ Paris Sud, Laboratoire d'analyse num~rique et EDP b&timent 425 91405 Orsay, France Chr ist ophe. Prieur@math. u-psud, fr

phone : 33-1-69-15-57-81, fax : 33-1-69-15-67-18.

A b s t r a c t . We study a two dimensional system which is globally asymptotically stabilizable with a discontinuous feedback but for which there exists no smooth stabilizing feedback. However this asymptotic stability is not robust to measurement, actuator or external noise. We show that such a robustness property can be achieved with an hybrid controller. In doing so we need to introduce an appropriate notion of solutions for hybrid systems.

1

Problem statement and related results

1.1

Introduction

T h e general p r o b l e m under consideration in this p a p e r is the a s y m p t o t i c s t a b i l i z a t i o n via h y b r i d state feedback. Let us recall t h a t asymptotic stabilization means the satisfaction of two properties: s t a b i l i t y of the origin of the closed-loop system and convergence to this point of all the solutions of the closed-loop system. We focus our s t u d y on the e x a m p l e of the A r t s t e i n ' s circles, i.e. the following two d i m e n s i o n a l system, see [1]: z'l = u ( - x ~ + z ~ )

x~

-2uxlx~

with u in ~ . Let g be defined for all (xl, z2) in ~2:

,q(X) --~ (--X~ -~- X 2,-2xlx2)'. In (1), all m o t i o n s are allowed along the integral curves of g i.e.: * the origin 9 all circles centered on the x2-axis and t a n g e n t to the x l - a x i s

(1)

280

Christophe Prieur

9 the zl-axis With u > 0 the circle is followed clockwise if x2 > 0 and anticlockwise if x2<0. In the complex plane (z = xl § ix~) we can rewrite the system (1) as: ~ = - z ~ u . The restriction of any neighborhood of z = u = 0 of the m a p (z, u) - z 2 u takes value into a neighborhood of the z = 0. So the necessary condition [2, Theorem 1, (iii)] for the existence of a continuous control law which makes the origin globally asymptotically stable is satisfied. However it is proved in [12] that it can not exist a continuous stabilizing feedback. Nevertheless there are m a n y obvious discontinuous stabilizing feedbacks e.g. the following: u(x) =

(-lifxl <0 1 if xl > 0

(2)

which makes the origin of the closed-loop system a globally asymptotically stable equilibrium when we restrict our attention only to Carath~odory solutions. The meaning of the solution of the discontinuous right-hand side differential equation (1) with u given by (2) is an i m p o r t a n t issue. Indeed the origin of this system is not locally attractive for Filippov solutions (in particular not locally attractive for Krasovskii solutions i.e. the limit of the perturbed Carath~odory solutions as the perturbations tend to 0, see [4].). The reason is that every point of the x2-axis is an equilibrium for Filippov solutions. So the system (1) in closed loop with (2) is very sensitive to measurement noises. However in [6] when we consider the 7r-solutions (i.e. (1) with the feedback (2) computed with an arbitrary small sampling schedule) the origin of the closed-loop system is a globally asymptotically stable equilibrium. Moreover this controller is robust with respect to external disturbances but not with respect to measurement noise. We study in this paper the robust stabilization i.e. the insensitivity of the feedback's performance with respect to measurement errors, actuator errors and external errors. We know by [1] that there exists no s m o o t h control Lyapunov function for (1). Then, due to [6, Theorem 1], there exists no robust stabilizing feedback u = u(z). So we must enlarge the class of controller if we want to robustly stabilize the system (1). In [7] the authors introduce the notion of "dynamic hybrid controller" which is c o m p u t e d with an "external model". This controller has the following form: u =

k ( ~ ' , z)

where z has the same size as z and denotes the state of the external model, and x ~is the measured estimate of state vector x. The origin is a robustly globally asymptotically stable equilibrium for the ~'-solutions. But as remarked

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281

in [12] it requires a resetting of the controller which m a y be difficult to construct. Moreover we prove in this paper the same result for a larger class of solutions. In [12], E.D. Sontag transforms the controller of [3] in a controller which is robust to measurement noise and makes the origin of the system (1) a semiglobal practical stable equilibrium (i.e. driving all states in a given compact set of initial conditions into a specified neighborhood of the origin). We exhibit here a robust global asymptotic stabilizing controller. In this paper we propose a robust stabilizing (dynamic hybrid) controller u such that: 1. The controller is easy to c o m p u t e and does not differ substantially from

(2). 2. The solutions are a generalization of Carath@odory solutions. A very natural way to overcome the nonrobustness to noise encountered with static time-invariant discontinuous controllers is to enlarge the surface of discontinuities and to introduce hysteresis. See [10] and [9], where the authors introduce hysteresis between two different controllers: one local and one global. Here we introduce hysteresis between two controllers, but we do not consider one local and one global controller but rather one controller which is defined on the right-hand side of the plane II~~ and another one which is defined on the left-hand side. In [10], the hysteresis technique allow us to join the local and the global controller with robustness to noise. Here we join the right-hand and the left-hand side of the plane with robustness to noise.

1.2

O r g a n i z a t i o n o f this p a p e r

In Section 1.3 we define the class of admissible controllers and the notions of solutions. In Section 2, we introduce two controllers which are basic components of the robust stabilizing hybrid controller presented in Section 3.1. In Sections 3.2 and 3.3 we study solutions of the closed loop-system and we prove the main Theorem 1 in Section 3.4. In Appendix A, we prove the Propositions 1, 2 and 3. 1.3

Class of controllers and notions of solutions

In this section we make more precise the notions of controller and solutions under consideration. We dot not restrict the system under consideration to be (1) but, in this section only, we allow us to study a larger class of systems. Let D be a subset of ~ n containing the origin. Let f : D x ~ ' ~ --+ ~ n be a locally Lipschitz function such that f(0, 0) = 0. We consider the system

=

u).

(3)

282

Christophe Prieur

T h e controllers under consideration in this p a p e r a d m i t the following description (see [14])

u = k(x,s~)

,

Sd = k d ( x , s ; )

(4)

where Sd evolves in some finite set .T, k : ~ n x j r ~ ~rn is continuous in x for each fixed sd, kd : ]1~n • .T" ---+ .T" is a function and s d is defined, at this stage only formally, as

s d(t) = limsd(s) . s
For this to m a k e sense, we equip .T with the discrete topology, i.e. every set is an open set. T h e above controller is h y b r i d due to the presence of the discrete d y n a m i c s of Sd. It gives rise to a non classical o r d i n a r y differential e q u a t i o n describing the d y n a m i c s of the closed loop system. In p a r t i c u l a r this s y s t e m is infinite dimensional since to evaluate s~ (t) at t i m e t, we need to know the past values of Sd(t). In this p a p e r we are interested in a notion of robustness to s m a l l noise. Let three functions 9 e and d in Ltoooc(D • [0, + ~ ) ; ~ n ) , which are continuous in x for each t, 9 a in L,o~(D • [O,+cx~);~'n), which is continuous in z for each t. We introduce these functions as a m e a s u r e m e n t noise e, an a c t u a t o r noise a and an external noise d of (3) and s t u d y the following p e r t u r b e d system:

{

x(t) = f(~(t), k(x(t) + e(~, t), sd(t)) + a(~, t)) + d(~, t) s~(t)

k~(x(t) + e ( x , t ) , s ~ ( t ) )

(5)

As noted in [7, R e m a r k 1.4], we can o m i t any explicit reference to a c t u a t o r errors because f is supposed to be locally Lipschitz. So in the following we suppose that, for all x in D and for all t > 0, we have:

a ( x , t ) = O. We have to make precise w h a t we m e a n by solution of the c o r r e s p o n d i n g differential equation. We want to s t u d y the i m p l e m e n t a t i o n of the controller (4). A n a t u r a l framework is the n-solutions. These r - s o l u t i o n s are s t u d i e d in [12,6,13] in the case of a o r d i n a r y differential equation. In our context a n a t u r a l definition is D e f i n i t i o n 1 Let rr be a sampling schedule of [O, T) with T > 0 (i.e. a partition ~r = {to = 0 < tl < " ' " < t o o } ) . Given ( x o , s - 1 ) E ]Rn x ~ .

We say that (X, Sa) is a s a m p l e d solution, startin 9 from (xo, s_ 1), of (3) on [0,T) if

The Artstein's Circles

283

i. X is absolutely continuous on [0, T) 2. For all i in N and for almost all t in [ti,ti+l), we have 1

x ( t ) = f (x(t), u(x(t~), sd(t~))) , 3. For all i in I~ and for all t in [ti,ti+l), we have Sd(t) = k d ( X ( t i ) , S d ( t i _ l ) ) .

(6)

~. We have: x(0)

= ~0.

(7)

We say that X is a ~r-solution, starting from xo, of (3) on [0, T) if there exists a sequence ( X , , Sa,n) of sampled solutions of (3) defined on [0, T) such that suPn_~o o IXn - X I = 0 and such that we have (7).

R e m a r k 2 By invoking Zorn's L e m m a exactly as in the proof of [11, Proposition 1], we can prove that every sampled solution (resp. r-solution) can be extended to a maximal sampled solution (resp. r-solution) (X, Sd) defined on an interval [0,T) with T < + o o (i.e. for which there exists no solution defined on an interval [0, T') with T ' > T and whose restriction is (X, Sd) on

[0, T)).

o

In this context our definition of global a s y m p t o t i c stability is D e f i n i t i o n 3 Let e, d be two functions with our standing regularity assumption. The origin is said to be a globally asymptotically stable equilibrium of the system (5) on D if the following three properties hold I. For every xo in D, there exists so in ~ and a sampled solution starting from (xo, so). 2. All the maximal sampled solutions are defined on [0, +oo). 3. There exists a function fl of class I(.s such that each maximal sampled solution ( X ( t ) , Sd(t)) satisfies for all t > O:

IX(t)l < D(Ig(0)l,t).

(8)

Note that (8) holds for all m a x i m a l 7r-solutions if the origin is a globally asymptotically stable equilibrium. R e m a r k 4 We observe, with [8, R e m a r k 2.4 and Proposition 2.5], that (8) is equivalent to the set of following two properties: 1. There exists a class-K;oo function a such that we have

IX(t)l _< ~(IX(O)l) ,vt _> o. 1 we denote Sa(t-1) = s-1.

(9)

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Christophe Prieur

2. For any r > 0 and e > 0, there exists T > 0 such that IX(0)l _< r

~

(10)

IX(t)l _< e ,Vt _> T .

Actually, we are interested in a notion of the robustness with respect to small noise. For this reason, we introduce the notion of robust stabilizing controller: D e f i n i t i o n 5 We say that the controller (k, kd) is a robust global asymptotically stabilizing controller if there exists two continuous functions Pc, Pd defined on O such that pj (x) > 0 for any x # 0 and for any j in {e, d} and such that for any perturbed system (5) with, for all x in D sup~_>0le(x, .)1 < p,(x)

,

sup~_>0ld(x, .)1 < pa(x) ,

(11)

where e,d are two functions with our standing regularity assumption, the origin is a globally asymptotically stable equilibrium on D as characterized in Definition 3.

2

A local c o n t i n u o u s controller

In this section we define two controllers of the control system (1) which are defined respectively on the right-hand and the left-hand side of the plane ~ . We overlap the domain of definition of these controllers to define the robust global asymptotic stabilizing controller in Section 3.3. For any a in (~, ~-), let us define the following set D1 = { x e ~ 2 : - a

< 0 < a } U {(0,0)} ,

(resp. D - 1 = { x : r r - a < 0 < T r o r - r r < 0 _ < - T r + a }

U {(0,0)}),

where 0 in (-rr, 7r] denotes the polar angle of the point z ~ 0. On this set, we consider the controllers : u = 1

(resp. u = - 1 ) .

P r o p o s i t i o n 1 There exist two continuous functions Pe and Pd defined on ll~~ such that pc(x) > 0 and pd(x) > 0 for any x ~ 0 and c~ in (~, ~-) such that the origin of the system

= g(~) + d

(resp. ~ = - g ( = ) + d)

(12)

is globally asymptotically stable on D1 (resp. D - I ) , for all noises satisfying (13), for all x in ~2.

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285

R e m a r k 6 In this proposition the notion of solutions is the classical one i.e. the Carath~odory solutions. Moreover we must take care of the fact that the origin is on the boundary of the domains D1 and D_z. T h u s to prove this proposition, we cannot use 9 every continuous stabilizing feedback is a robust stabilizing feedback ([5, Theorem 19.1]). 9 a Lyapunov function and the characterization of the robust stabilization given by [8, Theorem 2.9]. O

To explicit the robust controller, we need to define the angles in (~a) 2~ j3 ----arctan (tan(a) - 1)

,

3; ----arctan (3 (tan(a) -- 1)) .

(13)

The following proposition collects all technical properties t h a t we use in Section 3.1 to study the robust stabilizing controller. We denote I.I the norm

Ixl = I ll+ Ix21. P r o p o s i t i o n 2 We find two functions Pd and Pe such that the statements of Proposition 1 hold and we have the property: For all x2 in ~ and for all e = (el,e2) satisfying lel <_ 2pe(x), we have

p (0,

< --

(14) 2

'

for x # O, we have x+e

7~ O,

(15)

and the following implications hold

1=21 ___ tan(a)

zz,

:=>

Ix2 + e21 < tan(/3)(xl + el) ,

(16)

1~21 > tan(7) zz,

~

Ix2 + e21 > tan(/3)(zl + el) ,

(17)

I~1 < tan(z,)x~,

~

zz + e~ < 0 .

(18)

The proofs of Propositions 1 and 2 are technical. We postpone them to Appendix A.

3

Dynamic

time-invariant

controller

with

hysteresis

In this section we use the stabilizing controllers expressed in the above section and we join the domain of definition of these functions by making an hysteresis.

286 3.1

Christophe Prieur A robust stabilizing controller

Our main result is: T h e o r e m 1 With/~ given by (13), the controller below makes the origin of the system (1) a robust globally asymptotically stable equilibrium on 1R2:

u = Sd

,

Sd = k d ( x , s ~ )

(19)

where Sd is in {--1, 1}, the function kd satisfies:

kd(x, S d ) =

--1 i f - z r < 0 <_ - f l or ]3 <_ 0 <_ ~r s d i f - - f l < 0 < - - r r + f l or r - - ~ < 0 < fl 1 i f - - r r + f l <_ 0 <_ rr--fl

,

(20)

for x ~ 0 and =

(21)

x_l

Sigma_{-1 }

Sigma_l

Fig. 1. Definitions of the angles a, fl, 7 and the sets D,o, 2Y,o for so in { - 1 , 1}

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287

R e m a r k 7 T h e function k d makes an hysteresis between the two functions 1 and - 1 on a p p r o p r i a t e subsets of ~ . For any sa in { - 1 , 1}, the function kd(., sa) is continuous except on the b o u n d a r y of the sets defining the hysteresis. This r e m a r k is i m p o r t a n t to establish P r o p o s i t i o n 4. To prove T h e o r e m 1 we need to introduce a class of solutions which c o n t a i n the s a m p l e d solutions with a sufficiently fast schedule (see Section 3.2) then we prove p r e l i m i n a r y results on the solutions of the closed-loop in Section 3.3. F i n a l l y we prove T h e o r e m 1 in Section 3.4.

3.2

D e f i n i t i o n o f t h e "R.C-solutions

T h e p e r t u r b e d system under consideration is

]r

,

sd = k d ( x + e ( x , t ) , s d ) ,

(22)

In this section we define a class of solutions which c o n t a i n the solution defined in Definition 1. For the s t a t e m e n t of this definition we need the following closed sets (see Figure 1): L'-I = {x: -~ < 0 < -7or7 Z'I = i x :

-~'+7

< 0 < a } U {(0,0)},

< 0 < -~r+c~or~r-~

< 0 < ~r-7}

(23) tO { ( 0 , 0 ) } , ( 2 4 )

and the open set R C = It~2 x { - 1 , 1} \ (~U1 • i - l }

U ,U-1 x i l } ) .

(25)

D e f i n i t i o n 8 ( X ( t ) , Sd(t)) defined on [0, T) is a T~C-solution of (22) if

1. X is absolutely continuous on [0, T) and takes values in D. 2. For each t in [0,T) such that ( X ( t ) , S d ( t ) ) is in T~C, Sd is right continuous. 3. For almost all t in [0, T), we have J((t) -- sdg(X(t)) + d ( X ( t ) , t ) ,

(26)

and, for all t in (O,T) where S . ( s ) has a limit as s tends to t from the left, we have 2 :

s~(t) = k~(X(t) + e(X(t), t), S~ (t)) .

(27)

Note t h a t we can make R e m a r k 2 for the 7~C-solutions. Now we define the notion of s a m p l e d solutions with a sufficiently fast schedule. Let p: 11~n --+ l~ be a continuous function such t h a t , for all x :/: O, p(x) > 0. We say t h a t a 2 Note that we do not ask for (27) to hold at t = 0.

288

Christophe Prieur

sampled solution (X, Sa) has a sampling rate less than p if for all i in l~l and for all t in [ t i , t i + l ) , w e have

t~+~ - t~ <_ p ( x ( t ) + e(X(t), t ) . Every sampled solution is a ~C-solution. More precisely we have the following P r o p o s i t i o n 3 There exists Pc, Pa and p three continuous functions such that pc(x) > O, pd(x) > 0 and p(x) > 0 for all x # 0 and such that, for any e, d: II(2 x IR+ -+ ]R2 with our regularity assumption and satisfying (11) for all x in ]R~, every sampled solution (X, Sd) starting from (xo, s - l ) of (22) with a sampling rate less than p is a 7~C-solution of (22) on [to, T) starting from (xo, so) with so = k~(xo + ~(xo, to), s_l). The proof of Proposition 3 requires technical properties of'Re-solutions and of sampled solutions. Therefore we postpone it in Appendix A. 3.3

B a s i c s p r o p e r t i e s o f t h e "R~C-solutions o f (1) w i t h u g i v e n by (19)

In this section we study properties of 7~C-solutions of the closed-loop system (1) with the controller stated in Theorem i. The unperturbed system under consideration is: = sag(x)

,

s,~ = k~(x, s ~ ) ,

(28)

where kd is defined by (20)-(21). Given two function e and d with our standing regularity assumption, the perturbed system under consideration is (22). D e f i n i t i o n 9 A function (X, Sa) defined on [0,T) is said to have a switch at time t 6 [0, T) if Sa is not continuous at t. We start by locating the points where a 7~C-solution may have a switch: P r o p o s i t i o n 4 Let (X, Se) be a T~C-solution of (22) defined on [0, T) with a switch at time t 6 [0, T). Consider the following sets. S 2 ( t ) = {s: 3tn 6 [0,t], t,, ~ SPd(t) = {s: 3t,~ 6 [t,T), t,, ~

t, Sd(tn) ~ t, Sd(t,) ~

s} ,

(29)

s} .

(30)

P * t i t h e switch is such that - 1 6 S~m (t) and 1 6 Sd(t ), then X ( t ) is in ~1, . If the switch is such that 1 6 ST(t) and - 1 6 S~d(t), the,, X ( t ) is in ~ _ , .

Note that S~ = {Se(t)}, for a n t such that (X(t),Se(t)) is in 7~C. Proof: Suppose that, for t 6 [0, T ) , w e h a v e - i 6 S~n(t) and 1 6 S dp(t). From (29), there exists a sequence t,~ _< t converging to t such that Sd(t,~) = --1. We have to consider two cases.

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289

1. Suppose that there exists a sequence tn <_ t converging to t such t h a t Sa(t~) = - 1 and such that Sd is not right-continuous at each tn. Therefore (X(t,),Sd(t,)) is not in T4C. Thus due to (25), X ( t , ) is in 271. Therefore by continuity and closedness of •l, X(t) is in L't. 2. Suppose that there does not exist a sequence tn _< t converging to t such that Sd(t~) = --1 and such that Sd is not right-continuous at each tn. Then there exists a sequence tn converging to t such that Sd is rightcontinuous at each tn and such t h a t Sd(tn) = --1. Thus there exists a maximal en > 0 such that Sd(S) = --1 for s in [tn,t, + en). This implies t~ + r _< t, since if not we should have Sa(t) = - 1 and Sd right-continuous at t which contradicts 1 E S~ (t). (tn + en) is a sequence converging to t, so we have three (non exclusive) cases (a) there exists a subsequence such that Sa(t, + ~,) = 1 (b) or there exists a subsequenee such that Sd(tn + en) = --1 and Sd is right-continuous at each tn + r (c) or there exists a subsequence such that Sa(t, + en) = - 1 and Sd is not right-continuous at each tn + en. The case 2b is not possible because en is supposed to be maximally defined. The case 2c is not possible because we have supposed that there does not exist a sequence t , _< t converging to t such that Sd(tn) = - 1 and such that Sd is not right-continuous at each tn. Thus Sa(t~ + en) = 1. This implies with (27) that

I = &(t,+e,)

= ka(X(t~+en)+e(X(t,+zn),tn+z,),-1).(31)

So, from the definition of kd, the polar angle of X(t,~ + en) + e(X(t,~ + E,), tn + e,~)is [-Tr +/3, ~r-/3]. From (16)-(18), this implies that the polar angle of X (tn + en) is in [ - n + 7, ~r - 7]. Similarly, for all s in [t,~, tn + r the polar angle of X(s) + e(X(s), s) is in (~r - / 3 , 7r] U (-rr, -Tr +/3), and therefore X(s) is in D _ I . By continuity this implies that X(t~ + r is in the closed set Z1 and X(t) must also be in ~1. The case 1 E S~(t) and - 1 E SPa(t) is established in the same way.

[]

We are now in order to study the regularity of the function Sd. P r o p o s i t i o n 5 Let (X, S~) be a Tee-solution of (22) defined on [0, T). Then, for au t i,, (O,T) such that X(t) # O, X(t) is right-co,~tin,,ous ,,t ~ or l r continuous at t. P r o o f : Let t in (0, T) such that Sd is not right-continuous at t and such that Sd is not left-continuous at t. Therefore there exists two sequences s,~ < t and tn :> t converging to t such that Sd(sn) = Sd(tn) 7s Sd(t). Therefore S~(t) = { - 1 , 1} and S~(t) = { - t , 1}. Thus due to Proposition 4, X(t) is in r l A 27-1 which implies with (23) and (24) that X(t) = O. []

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Consider the sets: Let M be the set of the origin and the pairs (x0, so) such that 1. so is 1 or - 1 and 80, the polar angle of x0, is in ( - f l , - r e + f l )

(~

-

or in

fl, ~),

2. so is - 1 and 80 is in (-Tr,-/3] or in [fl, re], 3. so is 1 and 80 is in I-re + fl, re - fl], and M ~ be the set of the origin and the pairs (x0, So) such that 1. so is 1 or - 1 and 80 is in ( - 7 , - r e + 7) or in (Tr- 7,7), 2. so is - 1 and 80 is in ( - T r , - 7 ] or in [7, 7r], 3. so is 1 and 8o is in [-Tr + 7, re - 7], Note that M ' C M and that, for every xo is in ll~2, there exists so in { - 1 , 1} such that (xo, so) in M q We prove the following result of existence of 7~Csolutions. P r o p o s i t i o n 6 For every (xo, so) in M , there exists a T~C-solution of (28) starting from (xo,so). Similarly, for every (x0,s0) in M ~ and every noise (e, d) satisfying (11) for all x in ]R2, there exists a T~C-solution of (22) starting

fro,, (xo, so). P r o o f : We prove the result for M ' . The case M is analogous. For x0 = 0 and any so, we get from (21) that X ( t ) - O, Sd -- so is a 7~C-solution of (22). Let (x0, so) in M ' , x0 :/= 0. From our standing regularity assumption on f , e, and d, the Carath~odory conditions are met for the system = sog(Y) + d

,

Y(O) = xo .

So, with the specific definition of M ' , there exists T > 0 and an absolutely continuous function Y defined on [0, T) and such that (Y(t), so) is in M ' . Due to (16)-(18), we get kd(Y(t) + e(Y(t), t), so) = so, for all t in [0, T). This implies that, by letting: X(t)

= Y(t)

,

&(t) = so,

for t E [0, T), we get a nC-solution of (22) starting from (x0, so).

[:3

R e m a r k 10 If (X(t), Sd(t)) is a T~e-solution defined on [0,T), then for any s i n [0,T), ( X ( t + s ) , S d ( t + s ) ) is a 7~C-solution defined on [ 0 , T - s).

P r o p o s i t i o n 7 For every noise satisfying (i1) for all x in ~ , every T~Csolution of (22) can be extended to a maximal RC.solution (X, Sd) defined on an interval [0, T) with T < +oe. Moreover i f T < +oe then lim IX(t)[ = +oo. --

t--* T

The Artstein's Circles

291

Proof :

Part 1: Maximal extension: See R e m a r k 2. Part 2: Explosion in finite time: Let (e,d) satisfy (11), for all x in 11~~, and (X(t),Sa(t)) be a T~C-solution maximally defined on [0, T). Suppose the conclusion of Proposition 7 does not hold, i.e. there exists K a compact set of ]R'~ and times t , in [0, T) tending monotonically to T such that (X(tn), Sd(tn)) is in K • { - 1 , 1} for all n. We first establish C l a i m 1 For some n sufficiently large, for all t E [t., T), X ( t ) is in the bounded open set K + B with B = {x E ]R2, Ixl < 1}. P r o o f o f C l a i m 1: If the conclusion of this claim is not true, the continuity of X implies the existence of sn E (tn, T) such that ] X ( t n ) - X(sn)] = 1

and

]X(t,) - X(t)] < 1, Vt E [ t , , s , ) .

(32)

It follows that X(t) is in the c o m p a c t set K + c l o s ( B ) , for all t in [tn, Sn]. Let P = reKl~c~Xs(.B)

(IPe (r)l,

Ipd(X)I), ~ :

xeg+clos(m)a,Xdepclos(B) Ig(x)

Then we have for all (s,t) in [tn,sn], IX(t) - X ( s ) ]

1 = [X(tn)--X(sn)l < gls.-t,,I

-4- d] .

< ~l t - s I. Therefore:

<_ ~ l T - t . I .

This cannot hold for n sufficiently large and proves Claim 1.

[3

The Claim 1 implies that there exists ~ such that for all (s, t) in [t,, T), we have ] X ( s ) - X(t)[ < ~ Is-t[. It follows (by invoking Cauchy criterion) t h a t X(t) has a limit x0 when t tends to T.

Part 1: If S ~ ( T ) is not a singleton Then there exists a sequence of times of switch tn < T tending to T from the left such that: 1 E S~(t~n) , - 1 E 5~(t2~),

- 1 E S'~(t~n+l) , 1 E S~(t2,~+1) .

Then due to Proposition 4, X(t2n) is in 57-1 and X(t2n+~) is in 571, for all n in iN. Then by continuity and closedness, z0 = lim,~+o~ X ( t , ) in 5Y_1 n 271. So x0 = (0, 0). We define a function (X', S~) on [0, +cr by letting Vt E

[0,T),X'(t)

Vt ~ IT, + o o ) ,

= X(t)

X'(t)

= 0

,

S'a(t) = ,

S'~(t) =

Sd(t) , 1,

We have no limit of Sd at t = T, so we do not need to have (27) to hold at time t = T. So the function (X', S~) is a 7~C-solution of (22) defined on [0, +oo) whose restriction on [0, T) is (X, Sd). Part 2: If Sr~(T) is a singleton {so} We have to consider two cases:

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Christophe Prieur

1. if x0 = (0, 0). We define a function (X', S'd) on [0, +oo), by letting Vt E [0, T ) , X ' ( t )

vt

[T,+oo),

:

X'(t)

X(t) :

0

, ,

S'd(t) : Sa(t) , S'At)

:

so,

It follows from (21) that the function (X', S~) is a 7~C-solution of (22) defined on [0, +oo) whose restriction on [0, T ) i s (X, Sd). 2. if x0 # (0,0). Let

sl = k (x0 + e(x0, T), s0).

(33)

There exists Y defined on [0, Ti) a solution of

Y(t) = sig(Y(t)) + d(Y(t), t + T) starting from x0. We have to consider two cases: (a) If there exists 0 < To __ Ti such that, for all t in [0, To), we have kd(Y(t) + e(Y(t),t + T), si) = si . (34) Then we define a function ( X ' , S~) on [0, T + To), by letting Vt E [0, T), X'(t) = X(t) , S'd(t ) = Sd(t) ,

VtE[T,T+To),X'(t)

= Y(t-T)

,

S'd(t ) = s i .

(34) implies that kd(X'(T) + e(X'(T),T),limt 0 of Z(t) = - s i g ( Z ( t ) ) + d(Z(t),t + T) . (36)

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293

Then we define a function ( X ' , S~) on [0, T + T2), by letting Vt e [0,T), X ' ( t ) = X ( t ) , Sh(t ) = Sd(t) , X ' ( T ) = xo , S'd(T) = s l , VtE(T,T+T2),X'(t) = Z(t - T) , S'a(t ) = - s l (33) implies that k d ( X ' ( T ) + e ( X ' ( T ) , T ) , l i m t < T S'd(T)) = "S'a(T ), whereas Claim 2 implies that we do not need Sd to be right-continuous at t = T. So (X', S~) is a TiC-solution of (22) defined on [0, T + 712) whose restriction on [0, T) is (X, Sa). In the above various cases, we have obtained a contradiction with the fact that (X, Sd) is a m a x i m a l TdC-solution. rn Now we exhibit points of JR2 x { - 1 , 1} for which there exist no TiC-solution of the initial condition problem (22) in positive time. P r o p o s i t i o n 8 For all so in { - 1 , 1}, f o r all xo in ~ \ Dso and for all functions e, d satisfying ( I i ) for all x in ~ , there is no TiC-solution of (22). P r o o f : Let x0 be in IR2 \ D_ 1 and so = 1. There is no TiC-solution starting from (x0, 1). Indeed for such a , we have (x0, 1) is in TiC and then, due to Definition 8, there exists s > 0 such that Sd is constant on [0, s), and X being continuous, X is in the open set ]R 2 \ D - 1 (neighborhood of x0). But, for all x in a neighborhood of x0 and all e satisfying (11), (16)-(18) yield Sd = k d ( x + e , s ~ ) = --1. So Sd must be - 1 . So there is no TiC-solution. T h e other case is established in a same way. rn As a conclusion of Proposition 8, and due to R e m a r k 10, we can claim that for all TiC-solution (X, Sd) of (22) defined on [0, T0) and, for all t in [0, T), by denoting So = Sd(t), we have: X ( t ) is in D,0. We end this section by counting the number of switches: P r o p o s i t i o n 9 Let so in { - 1 , 1}, let (e, d) be noises satisfying (11) for all x in ]I(2. Let (x0, so) be an initial condition such that there exists a TiC-solution (X, St) of (22). Only two cases can occur: 1. There exists no switch i.e. Sd -: so and X is a solution of = sog(x) + d

(37)

and is contained in D,o. 2. There exists a switch at the time a > O. Then: (a) X ( a ) is in ,U_,o, (b) For all t in [0, or), X is a of (37) and, for all t in [0,+oo), X is a solution of: = -sog(x) + d. (38)

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Christophe Prieur

(c) If there exist two switches at times 0 < a < ~r' then for all t >_ ~r', X ( t ) = O.

P r o o f : Let us start with a remark: If there exists to such t h a t X(to) = 0 then, for all t > to, X(t) = O. We prove now the following:

C l a i m 3 If X(t) ~ O, for all t in an open interval I, then there is at most one switch of Sa in I. P r o o f o f C l a i m 3: Suppose t h a t so = 1 and there exist two switches at times ~' > a in I. In view of Proposition 4, we can suppose t h a t X ( a ) is in ,U_l and X(vr') is in L:l. Then by continuity there exists a time t in (a, or') such that:

xl(t) = O ,

z2(t) r O ,

Sa(t) = - I

,

zl(t)>0.

(39)

But (14) implies that if Sd(t) = - 1 then zl (t) < - x](t) Therefore if Sa(t) = -i then zi(t) < 0 which is a contradiction with (39). n The Proposition 9 is a consequence of Claim 3 and Proposition 8.

I:]

Now we are in order to prove Theorem 1. 3.4

Proof of Theorem

1

Maximality of trajectories." Let (x, Sd) E ]Ks x .T be any point so t h a t there exists a 7~C-solution maximally defined on [0, T) (see Propositions 6 and 7). Let us show that, for each such T~C-solution, we have T = + o c . Suppose not. Then with Proposition 9 there exists to such that, for some so in { - 1 , 1}, with X(to) E D,o, X is a on [t0,T) of (37) which tends to +oo. This is a contradiction with Proposition 1. So we must have T = +oo. With the help of Proposition 9 and Remark 4, we can now prove t h a t the controller (19) is a robust global asymptotic stabilizing controller: From Proposition 6, for any x0 there exists so such that, for any noise (e, d) satisfying (11), there exists a 7~C-solution of (22) starting from (x0, so). Also, from the previous paragraph, all the s of (22) are defined on [0, + o o ) .

Establishing (9}: Let (e, d) be a noise satisfying (11). Due to Proposition 1 there exists for so in { - 1 , 1}, a class-/Coo function aso such that, for all x0 in D,o , we have I X , 0 ( t ) l _ a,o(Ix01) ,

Vt ___0 ,

(40)

where X, o is any RC-solution of (37) starting from x0. Let us show that, given a noise (e,d) and an initial condition (x0, so), a 7~C-solution (X, Sd)

satisfies: IX(t)[ _< m a x ( a _ l ( a l ( ] x o [ ) ) , a l ( a - l ( ] x o [ ) ) )

,

Vt _> 0 .

(41)

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295

Indeed 9 if X has no switch then, due to Proposition 9, X is a solution of (37). So, from (40) we have (41). 9 i f X has a first switch at time tr > 0, then for all t in [0, tr):

Ix(t)l < ~,o(Ix01) _< c,,o(c~-,o(Ix01)) and, from Proposition 9, for all t in [or, +~x)): IX(t)l _< ~ - , o ( I X ( ~ ) l ) _< a-,o(a,o(l~01) , and therefore we have (41).

Establishing (10): Let r > e > 0. Let R _> r and 0 < e ~ < ~ be defined by R = max(u_l(r),al(r))

,

e' = min(a_-~(e),a~'l(~)) .

For all so in { - 1 , 1}, for all x in D, o and for all X , 0 (t) of (37), we obtain:

IX, o(0)l _< r IX, o(0)l < e'

~ ~

IX,0(t)l < R ,Vt ___0 , IX, o(t)l _< E ,vt > 0,

(42) (43)

Due to the global asymptotic stability of the systems (37), for so in { - 1 , 1}, there exists Tso < + ~ such that: x E O,o, IX, o(0)l < R

~

IX, o(t)l _< e', Vt > T, o .

(44)

Let T = T-1 + 711. Let us show that: IX(0)l < ,"

:*

IX(t)l < e ,Vt > T ,

(45)

where X(t) is any solution of (22) with initial condition (z, so). Indeed 9 i f X has no switch then, due to Proposition 9, X is a solution of (37) and is contained in D, o. So (45) is a consequence of (44). 9 i f X has one switch at time a, due to Proposition 9, then X(tr) is in S _ , o , X is a solution of (37) on [0, a) and a solution of (38) on [or,+c~). Two cases m a y occur 1. suppose a < Tso. Due to (42), since X(0) < r, we have IX(~)l ___ R and X is a of (38) on [a, +oo). So due to (44), we have, for all t _> ~ + T-,o, IX(t)l < g < e. In particular we have (45). 2. ifTso < a then due to (44) we have, for all t in [T,o, or], IX(t)l < s' <_ e. Therefore due to (43), for all t in [tr, +cx~), we have IX(t)[ < z. So in particular we have (45). This achieves the proof of Theorem 1.

rn

296

A A.1

Christophe Prieur

Technical

proofs

Proof of Proposition 1

Let a > 0, b > 89 be such that 3a ~ < 2 b - 1 .

(46)

P a r t 1: Choice o f a

C l a i m 4 Let P ( w ) = w 3 - 3aT 2 + (2b - 1)w + a. There exists wo in ( - 1 , 0) such that f o r any w >_ To, we have P ( w ) > O. The proof of the Claim 4 results from the following three remarks: 1. Under the condition (46), P is strictly nondecreasing. 2. P is continuous. 3. P ( - 1 ) < 0 and P(0) > 0. Let a in ~2, t~- 3~4j be defined by t a n ( a ) = W o1.- A consequence of the Claim 4 is V0 E (0, a], P ( ( t a n 0 ) -1) > 0

(47)

Part 2: Computation of a Lyapunov function f o r (1) with u = 1

Let V1 be defined for all x in D1 by b

vx(xx, x~) = ~x~ - axxlx21 + ~ x 2 . Since (46) implies a ~ < b, we have, for all x in D1, V1 is non negative and Vl(x) = 0 ~ x = 0. Along a solution of (1) with u = 1, we have

91(xl, x~) = - x ? + (-2b + 1)~1x~ + 3alx~lx~ - alx213 . If x2 r 0 then it follows

91(Xl, Ix21) --- -P(i-~l~l)lX213

,

and if x2 = 0 then we have 91 (xl, Ix21) = - x ~ . Therefore with (47)

we

deduce

that for all x in D1 \ {(0,0)}, we have Vl(X) < 0. Let K be the compact set g = {x e 01, Vl(x) = 1}. Let el = max~eK - V l ( x ) . Thus, for all x in K, 91(x) <_ - e l ( V ~ ( x ) ) ]

(48)

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297

But by homogeneity this inequality holds for all x in D1. We use now this Lyapunov function to study the perturbed system k = g(x) + d(x, t). Let

c 2 = m~eK a x / lt _ ~ +

~ OV1I + 12~1~11b-~1' OV~ IOVII ~llb-g-~ 0~1 + iOV~ ~ ~) "

By denoting V1 the derived function of 89 along a solution of this system we deduce from (48) that for all z in D1 \ {(0, 0)}

fzl <_-c~(V~(x))~ + c~(V~(x))89 Therefore by letting c3 = ~2c~ we get that for any (e, d) satisfying Vz E D1, sup~>ole(x,-)l < caVl(z) ,

sup~>old(x,-)l _< r

every solution satisfies VI(x) < - ~ ( V l ( x ) ) } . Then u = 1 is a robust global asymptotically stabilizing controller for the restriction of the system (1) on Dl. Part 3: solution of (1} with u = - 1

We remark that for every solution (Xl(t), x2(t)) of a perturbed system of (1) with u = -1, the pair ( - x l ( t ) , x ~ ( t ) ) is a of a perturbed system of (1) with u = 1. We deduce that u = - 1 is a robust global asymptotically stabilizing controller for the restriction of the system (1) on D-1. Correspondingly, the Lyapunov functions is b 2 V-1 (,~1, x2) = lx12 + a.~l Ix21-Ji- ~ x 2 .

Part 4." P~ and Pd can be defined on IR ~

Let 1

pe(x) : 5 min

{

1 ((~b) (l_tan(a))2,ea 1--

pd(x) = min ( 1~, ca ( ( 1 a-- )--~

min{1,b}) } } 4 ([x[),(49)

man{1, b} ) ] }(ixl)~ "

(50)

On D1 and on D - l , we have p~(z) < caVi(z)89 and pd(z) <_ caVl(x). Finally, from the above, for so in {-1, 1}, the origin of the system ~: = sog(x) + d is globally asymptotically stable on D, 0 for every noise (e, d) satisfying (11). t:]

298 A.2

Christophe Prieur Proof of Proposition

2

Note that 1 - t a n ( a ) >-- 2 and (49) imply that Pe <-- I~1 8 " Also for all e such that ]e] < 2pe(X) we have

Thus (15) holds. Moreover (14) holds since we have, for xl = 0, s pd(0, z2). The implications (16)-(18) result from (49) and the inequality: 2p~(~1,~2) ~

_>

1

(1 -- tan(a)) 2 (1~11+1~1) 1:3

A.3

Proof of Proposition

3

Let p: ~ n __~ ]~ be a continuous function such that 1. for all z r 0, we have p(z) > O. 2. for all e and d satisfying (11) for all x in ~2 and for all x such that the polar angle of x + e(z, 0) is in (-rr, -fl] U [fl, 7r] (resp. [-fl, fl], resp. ( - T r , - f l ] U [rr - #, 7r], resp. [-rr + #, 7r - fl]) we have 1

min p ( x 31- e ( x , 0)) < ~ t~e{-1,1}

{T, X(0) = x,

V t e [0,T], the polar angle o f X ( t ) e ( - l r , - 7 ] t.J [7, 7r] (resp. [ - a , a], resp. (-rr, - a ] U [rr - a, rr], resp. [-rr + 7, rr - 7]) X is a solution of ~/" = u g ( X ) + d ( X , t ) } The existence of such function p results from (11) and (16)-(18). Let us prove Proposition 4 in the case of the sampled solutions. We can remark that for every (X, Sd) sampled solution of (22) on [0, T), due to (6), if there is a switch at time t in [0,T), then there exists i in N>0 such that t = tl, S'~(ti) = {Sd(ti-1)} and SdP(ti) = {Sd(ti)}. L e m m a 1 Let (X, Sd) be a sampled solution of (22) on [0, T) such that its sampling rate is less than p and such that Sd has a switch at time ti E (0, T). 9 If the switch is such that S d ( t i - t ) = --1 and Sd(ti) = 1, then, for hilt in

[t.ti+O,

x(t) is in z , .

9 If the switch is such that S d ( t i - t ) = 1 and Sa(ti) = - 1 , then, for hilt in

[t.t~+0,

x(t) is in s_1.

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299

P r o o f : Let i in N>0 such that Sa(t~_l) = 1 and Sa(tl) -- - 1 . T h e n due to (20), the polar angle of X(tl) + e(X(ti), tl) is in (-Tr, -fl] U [fl, 7r]. From the Assumption 2 this implies that for all t in [t#,ti+2), the polar angle of X(t) is in ( - ~ r , - 7 ] U [7, 7r]. Similarly the polar angle of X(ti-1)+ e(X(ti-1),ti-1)is in ( - f l , fl). From the Assumption 2 this implies that for all t in [ / i - l , t i + l ) , the polar angle of X(t) is in [ - a , a]. Therefore for all t in [ti,ti+l), the polar angle of X(t) is in { ( - ~ r , - 7 ] U [7,7r]} N {[-a,c~]}, and thus, for all t in [ti,ti+l), X(t) is i n / ? - 1 . T h e case Sd(ti_l) : - 1 and Sd(ti) = 1 is established in the s a m e way. Q Now we are in order to prove Proposition 2. P r o o f o f P r o p o s i t i o n 2.' Let (X, Sd) be a sampled solution of (22) with a sampling rate less than p. Let i in l~l. Note that if i > 0, then we have S~(ti) = Sd(ti-1). Therefore we have (27) at time t = ti. If there is no switch at time ti, then for all t in (ti,ti+l), Sd(t ) = Sd(tl) =S~(ti-1). And therefore we have (27) for all t in (ti,ti+l). Suppose that there is a switch at time ti such that Sd(ti-1) = --1 and Sd(ti) = 1. Then, for all t in (ti, ti+l), S d (t) = Sd(tl) = 1 and due to L e m m a 1, X(t) is in 571 and due to (16)-(18), the polar angle of X(t)+e(X(t), t) is in [ - r + 7 , 7]Thus (20) implies that we have (27), for all t in (ti,ti+l). The case S~(ti-1) = 1 and Sd(ti) = --1 is established in the same way. Therefore we have (27) for all t. Moreover for all i in 1~ and for all t in (ti,ti+l), we have Sd(t) = Sd(ti). Therefore we have (26) for almost all t. D

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Christophe Prieur

8. Lin Y., Sontag E.D., Wang Y. (1996) A smooth converse Lyapunov theorem for robust stability. SIAM J. Control and Optimization 34 (1): 124-160 9. Prieur C. (1999) Uniting Local and Global Controllers. In Proc. of the first NCN Pedagogical School, Athens, Greece, 457-461 10. Prieur C., Praly L. (1999) Uniting Local and Global Controllers with Robustness to Vanishing Noise. Submitted for publication in MCSS 11. Ryan E.P. (1990) Discontinuous feedback and universal adaptive stabilization. In Control of Uncertain Systems, Hinrichsen D., Ms B. (Eds.), Birkhs 245-258 12. Sontag E.D. (1999) Clocks and insensitivity to small measurement errors. ESIAM: COCV, www.emath.fr/cocv/4:537-557 13. Sontag E.D. (1999) Stability and stabilization: Discontinuities and the eitect of disturbances. In Nonlinear Analysis, Differential Equations, and Control (Proc. NATO Advanced Study Institute, Montreal, Jul/Aug 1998; Clarke F.H., Stern R.J., Eds.), Kluwer, 551-598 14. Tavernini L. (1997) Differential automata and their discrete simulators. Nonlin. An., Th., Meth., App. 11 (6): 665-683

Robust Stabilization for the Nonlinear Benchmark Problem ( T O R A ) Using Neural Nets and Evolution Strategies Cesfireo Raimdndez Universidad de Vigo, Vigo (Pontevedra) CEP 08544, Spain c e s a r e o 0 u v i g o , es A b s t r a c t . Evolution Strategies (ES) are stochastic optimization techniques obeying an evolutionist paradigm, that can be used to find global optima over a response hypersurface. The current investigation focuses on robust controller synthesis, using the unsupervised learning capabilities of ES's issued from their evolutionist paradigm. The training process intents to construct a Lyapunov Function which guarantees internal stability, performance and disturbance rejection.

1

Introduction

The problem of design and syntheses of nonlinear control systems have aroused wide research interest in recent years. Obtaining robust nonlinear controllers by a regular methodology has been a real challenge. Many design methodologies have been developed mainly with the use of algebraic tools [9], [13]. In this method, the nonlinear system is assumed to be a perturbation of a linear system. The optimal cost and control are assumed to be analytic and are expanded in a Taylor series. Various techniques are then employed to find the first few terms in the series. The first term corresponds to the solution of the matrix Ricatti equation obtained by linearizing the system about the origin. The difficulty with perturbation methods is that they are limited to a small class of systems, i.e. systems that are small perturbations of a linear system and that have analytic functionals describing their dynamics. These methods depend on the convergence of a power series for which is difficult to estimate the region of convergence as well as the attraction basin for the control. For bilinear systems however, it appears that the region of attraction can be estimated [4]. A methodology for approximation to the generalized Hamilton-Jacobi-Bellman equation is presented in [2] using basis functions and the Galerkin method. This method seems powerful but suffers from truncation considerations. Also the controller order for a expected performance can be fairy high. It is not clear how to introduce state-space restrictions as well as limited actuation. In [6] a method is presented to satisfy the nonlinear Hamilton-Jacobi-Bellman partial differential equation in a restricted domain of the state space for a

302

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given nonlinear feedback controller, training for this purpose, a (squared) feedforward neural net which materializes the Lyapunov function. This method uses back-propagation as the main algorithmic tool, needing so a previous reduction (adjusting) to a neural net of all the objects involved. The controllers obtained through all those methods require a posteviori performance checking because they accomplish necessary conditions. Systematic simulations must be done to assure the needed characteristics. The evolutive approach focused in this paper also intents to produce controllers which obey the necessary conditions, but knowing in advance that some desirable characteristics are already incorporated like limited actuation, state-space restrictions, settling time, etc. The evolutive approach must be used advantageously when other techniques fail to succeed because the method has no restrictions concerning regularity. Controlability of course is mandatory. The calculation process only involves point wise function evaluations. As in the other methods, the quest is for Lyapunov functions but in the present case, the Lyapunov function structure as well as the controller complexity, can be fixed. Under the evolutive paradigm, the problem of control synthesis is reduced to find the minimum of a function (measure) over a feasible set of values. Evolution Strategies are used to find out the solution to that minimization problem. The content of the paper is as follows. In Section 2 are briefly explained the main results in nonlinear robust control, needed to work out the results. In Section 3 is presented the Liapunov function modeling using a perturbed gradient field generated by a neural net. In Section 4 Evolutionary MetaHeuristics and Evolution Strategies are described. In Section 5 the fitness (measure) of the system (plant + controller + perturbation) dynamic performance is proposed. In Section 6 is presented a case study which illustrates the line of research. Section 7 is dedicated to conclusions and Section 8 to the acknowledgements. The main notation signs used are: 0 T for transpose, oc for proportional to, < , , . > for scalar product, II * II for Euclidean norm [ 9 [ for cardinality, = for equivalent and C 1 as the set of derivable real functions. 2

Hamilton-Jacobi-Isaacs

The formulation of nonlinear Hor is introduced briefly. Let be an affine nonlinear control system given by:

(1) where f(0) = 0, h(0) = 0. Here w includes all external disturbances, u is the control and z the state. In addition k T ( x ) k ( z ) is of full rank assuring

Robust Stabilization for the Nonlinear Benchmark Problem (TORA)...

303

the existence of a unique (local) saddle solution of the Hoo control problem [7], [8]. Also we will assume that hT(z)h(z) = O. Nonlinear Hoo control via state feedback m a y be stated as: For w E L~[0, t+), t + > 0, find a positive constant (as small as posible) 7", and for 7 > 7", a state feedback law u = u ( x ) such that: 1. for the initial condition *(0) = 0 occurs that ~ + zTzdr < 72 f~+ w T w d r 2. if w ----0 the closed loop is asymptotically stable.

To solve this problem, define

o v ..f + H(V,x,u, to) = -~z ( glU-["g2tO)-{" zT z--72toTw

(2)

and its saddle solution

--~(k k) m \ o ~ ] i

=

TfovhT k ]

(3)

Then inequality fo + zTzd'r ~ "72fo + w T w d r can be satisfied if there is a nonnegative function V ( z ) s u c h that H(V,:r,u*,w*)<_ 0 (HIJ inequality) holds. Given a system as in (1) and a constant 7 the problem of control synthesis consists in the determination of a Lyapunov function V(~) such that with u*,w* as given by (3), H(V,;~,u*,w*) <_0 holds all over the region of interest.

3

Realization of L y a p u n o v functions

The choice of the Lyapunov function is most important because the controller inherits all desired properties from it: closed loop asymptotic stability under initial conditions as well as disturbance rejection, to cite some of them. Shaping a positive function as a Lyapunov candidate can be done as in [6] imposing a squared neural net. The gradient field could be obtained using the partial derivatives formally. In this paper, the proposed methodology tries to find a perturbed gradient field which approaches a solution through the evolutive learning process. This approach is theoretically possible because the Hamilton-Jacobi-Isaacs

304

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Raimtlndez

inequality allows the existence of an open perturbation field added to the gradient solution. Calling ~b(~r, z) the perturbed field, ~b(~r, z) will be modeled using a feedforward net with an inner layer, and as many outputs as inputs. Being m the number of neuronas in the inner layer, and n the state dimension, the total number of weights needed is given by [ ~" [= 2- n 9rn + m + n. An additional condition is needed. Null output with null input (A/'(~r, O) = O) which reduces the dimension of parameter search space by n. For a neural net with an inner layer .Af(Wl, hl, W2, h2, z) -- o'(h2 -[- W 2 o ' ( W l x -[- hi)) that can be abridged as = vmo

v(.,

with W denoting weights and h offsets, where lr -- W1 U hi U W~ tAh~ and Umaz is a diagonal matrix with the maximum field strength. The condition A/(n', 0) : 0 reduces to h~ : - W 2 o ' ( h l ) and the set of operating equations DV with the substitution of ~-5 by ~b(Ir, z) H(~b, lr, z, u, w) = O(lr, z ) ( f + g l u + g~w) + z T z -- 7~wTw

U*(4, 7t'~Z) = -,

=

-- ~

(kT~) - IgT~T (TT,~) g

(4)

The learning process about ~5(7r, ~:) will be guided by two additional conditions: 1)< ~b(Tr,z), z > > 0 (divergence in the origin) and 2) < ~b(~r, z), ~ > < 0 (energy losses along the path). The integral storage function is obtainable numerically by solving the partial differential equation with boundary conditions dV = dp(~r, x) dz v(0) = 0

4

Evolutionary

(5)

strategies

Simulated Evolution is based on the collective learning processes within a population of individuals, in the quest for survival [11]. Each individual represents a search point in the space of potential solutions to a given problem. There are currently three main lines of research strongly related but independently developed in simulated evolution : Genetic Algorithms (GA), Evolution Strategies (ES), and Evolutionary Programming (EP). In each of these

Robust Stabilization for the Nonlinear Benchmark Problem (TORA)...

305

methods, the population of individuals is arbitrarily initialized and evolves towards better regions of the search space by means of a stochastic process of selection, mutation, and recombination if appropriate. These methods differ in the specific representation, mutation operators and selection procedures. While genetic algorithms emphasize chromosomal Ol>erators based on observed genetic mechanisms (e.g., cross-over and bit mutation), evolution strategies and evolutionary programming emphasize the adaptation and diversity of behavior from parent to offspring over successive generations. Evolution is the result of interplay between the creation of new genetic information and its evaluation and selection. A single individual of a population is affected by other individuals of the population as well as by the environment. The better an individual performs under these conditions the greater is the chance for the individual to survive for a longer while and generate offsprings, which inherit the parental genetic information. The main contributions in the evolutionary computation approach are: 9 9 9 9 9

Model regularity independence. Parallelization to cope with intensive cost fitness computation. Population search • individual search (classical). General meta-heuristics. Good convergence properties.

Evolutionary algorithms mimic the process of neo-Darwinian organic evolution and involves concepts such as: t Time or epoch. ~(a', er) Individual. n" Exogenous parameters. (Search Space). ~r Endogenous parameters. (Adaptation). P(t) Population. P(t) = U~i(~ri, o'i) ~(f(Tr)) Fitness. ~ ( f Q r ) ) : ~1~1 ._+ 3 + operators (Mutation, Selection, Variation, etc.) A simple evolutionary algorithm follows: t+--0 initialize P(t) evaluate

4~(P(t))

while not

P'(t) +-

terminate variation P(t)

evaluate ~(pt(t)) P(t + 1) ~ s e l e c : P'(t) U Q

t~-t+l end

306

CesS.reo Raimtlndez

Q is a special pot of individuals that might be considered for selection purposes, e.g. Q = {1~,P(t),..-}. An offspring population P'(t) of size )~ is generated by means of variation operators such as recombination a n d / o r m u t a t i o n from the population P(t). T h e offspring individuals ~i(~r~,a~) E P(t) are evaluated by calculating their fitness represented by ~ ( f ) . Selection of the fittest is performed to drive the process toward better individuals. In evolution strategies the individual consist on two types of parameters: exogenous lr which are points in the search space, and endogenous et which are known too as strategic parameters. Then ~ = 1~(~',a ) . Variation is composed of mutation and self-adaptation performed independently on each individual. Thus

,,')

.)) u adapt( (-, ,,))

(6)

where m u t a t i o n is accomplished by zr~ = rri + a l . N(0, 1)

(7)

and adaptation is accomplished by a~ = c i . e x p { r ' - N ( 0 , 1 ) + r . N ( 0 , 1 ) }

(S)

where r ' o((V/2"n) -1 and 7" oc (X/2X/2X/2X/2X/2X/2X -1~) . N(0, 1) indicates a normal density function with expectation zero and standard deviation 1, and n the dimension of the search space (n =1 ~" I)Selection is based only on the response surface value of each individual. A m o n g m a n y others are specially suited: * Proportional. Selection is done according to the individual relative fitness . Rank-based. Selection is done according to indices which correspond to probability classes, associated with fitness classes. 9 Tournament. Works by taking a r a n d o m uniform sample of size q > 1 from the population, and then selecting the best as a survival, and repeating the process until the new population is filled. 9 (X,p). Uses a deterministic selection scheme, p parents create ~ > p offsprings and the best p are selected as the next population [Q = 0]. 9 (~ + p). Selects the p survivors from the union of parents and offsprings, such that a monotonic course of evolution is guaranteed [Q = P(t)]

5

Fitness evaluation

Each individual is characterized by a set of exogenous and endogenous parameter values a- and ~r respectively. T h e exogenous parameters are inherited

Robust Stabilizationfor the Nonlinear Benchmark Problem (TORA)...

307

from the Lyapunov function. The endogenous parameters does not influence the fitness measure. Each individual represents a set of independent paths over the phase space issuing from different initial conditions, under the influence of the same controller. This set of orbits should cover conveniently the phase space and can be randomly generated at the very beginning, being common to all the individuals of the population. Care must be taken in the process of initial conditions generation. The set of initial conditions must spread over the expected attraction basin and to avoid overfitting [12] must be considered a minimum and to avoid prohibitive computational costs must be a maximum. Under the controller action, the resultant set of orbits should approach the origin considered the rest point. The fitness must measure this performance to serve as learning factor. Thus being z k (0), {k = 1,---, ns} initial conditions, each orbit can be represented as: ~k (lr, t) - =k (lr, z ~ (0), t)

(9)

Settling time performance is measured through p ( x , t ) > 0 which normally has the structure:

p(~, t) = I1=11. t~

(10)

with a > 1. A typical fitness measure can be obtained as:

k

k

kz, k~ and ka being scale factors and g(lr) is a measure of closeness from the ~" parameters to the origin, as a means to guarantee regularity [12] in the approximator. Usually #(lr) = [[lrl[ and b(~ k (lr, t), t) is a barrier function [5] which penalizes unwanted states or control efforts. The evolutionist approach here adopted comprehends the following steps. * An indirect positive function definition, using a perturbed gradient field ~b(n', a~) with two additional conditions: 1)< ~b(~r, z), x > > 0 (divergence in the origin) and 2) < ~b(n', a~), ~ > < 0 (Energy losses along the path). * Definition of p(~, t) = zTzt a. * The control action and the worst perturbation definitions, derived from ~b(~r, ;r) as shown in (3). 9 With the expressions in (3) for u*(~-, a~) and w*(Tr, z) integrate the dynamical paths for the system (1) under initial conditions contained in a previously defined ball. Follows a piece of C-like pseudo language explaining how to calculate the individual fitness.

308

Ces~reo

Raimdnclez

float Individual :: fitness( void ) { int j,k; float t,f.err; f l o a t k . 3 8 0.1; f l o a t s t o t z O; f l o a t e r r , e.p; f l o a t HJI, ueight.HJ1 s lO0; f l o a t norn.parms s f i e l d ( p i , x ) - > n o r m ( ) ; VECTOR x ( d i m . s t a t e ) ; VECTOR u ( d i m _ c o n t r o l ) ; VECTOR g ( d i m . p e r t u r b a t i o n ) ; f o r ( j z O; j ( n u a . s a m p l e s ; ~§247 { // z = init[j] ; I/ err = O; f o r (k = O; k < n u n . s t e p s ; k++) { t = k*dt; // i f ( f i e l d ( p i , x ) * x < O) // e.p = f.err; break; } u = -O.5*Ol(x)'*field(pi,x); // u = O.5*G2(x)'*field(pi,x)/SQ(laJbda); // @.p = m e a o u r e ( x , u , e , t ) ; // i f (e_p == VERY_LARGE) { o.p 9 f _ e r r ; goto c o n t ; ) if (field(pi,x)e(F(x) § G l ( x ) * u + G2(x)*g)) e.p 9 f_err; HJI = f i e l d ( p i , x ) * F ( x ) if

+ H(x)'oH(x)

) x = x + (F(x) + G l ( x ) * u + G2(x)*e) e r r += e . p ;

* dr;

+ k.3*norm_parmo;

float Environment : : measure( VECTOR x, u, w, float t ) { f l o a t norm = x . n o r a ( n o r n . t y p e ) ; i f (nom > max.error) retua-n INFINITE; norm 4= B A R R I E R ( x , u , t ) ; roturu norn*goigth(t); }

Assuring field at the o r i g i n .

divergence

Control. Perturbation. performance.

> 0){ / / A s s u r i n g d i s s i p a t i o n .

+ SQ(lambda)u~eu - e ' e r ;

(HJX > O) e . p 48 e e i g h t . H J l * m a x ( O , H J l ) ;

) c o n t : s t o t 4= e r r ; } value = k.l*stot return value; )

same s o t o f i n i t i a l c o n d i t i o n s i n e v e r y c a l c u l a t i o n round

//

Robustness

// I/

Integration step. A c c u a u l a t i n g f o r each p a t h .

I/

Accumulating f o r a l l

// //

k.2 is implicit in barrier. Calculated fitness.

//

avoiding state

//

state

explosion

restrictions

samples.

Robust Stabilization for the Nonlinear Benchmark Problem (TORA)...

309

Follows the meaning of the main names used: field Represents ~(n-, x) F,G1,G2,H As in (la) and (15) dt Integration step measure Builds the integral along the closed loop path for a set of orbits (num_samples) for an integration time of

num_steps*dt numindividuals The number of individuals [ ~ I in a population num_epochs The number of real epochs until the process finish num_samples Number of independent paths whose behaviour characterizes an individual The integration is performed according to an Euler rule. Care must be take when choosing the integration step to avoid observability deterioration.

6

Case study

///////I//////I

()

()

I I Fig. 1. Rotational Actuator to Control a Translational Oscillator A cart of mass M which is constrained to translate along a straight horizontal line. The cart is connected to an inertially fixed point trough a linear spring of constant k (see [1]). Attached to the cart there is an actuator of mass m and moment of inertia I. The actuator is responsible for the control torque N and F is the external disturbance on the cart. The motion occurs in the horizontal plane, so that no gravitational forces need to be considered. Let q and q denote the translational position and velocity of the cart and let 0 and ~ denote the angular position and velocity of the rotational actuator. The equations of motion are given by

(M + re)q+ kq. = -me(~cosO - ~i2 sin 0) + F (I + rne2)~ = - m e , cos 9 + N

(12)

310

Cess

Raimfindez

W i t h normalizations and arrangements [3] the system is reduced to:

=

f(z)+gl(~)u+g2(~)w

where

--xl

I(~,) =

+ ea:~ s i n x s

1 -- s 2 cos:2 ~'3 ~r 4

9 c o s ~'s(a:l - cx'~ sin z 3 ) 1 ~ s 2 C 0 8 2 ~'3

I i l los.3 -__Seco, ~____L

g l ( Z ) ~---

,

g~l(Z) =

r

~

x'3

(13)

with the constraints Ix1 I -< 1.282 l u l < 1.411

(14)

The linearized disturbance-attenuation problem has solution (saddle point) for A > A* ~ 5.5 [10]. In this application is adopted A = 6 and according to [13]

h(~)=[~l k(z)=[0

cz2 cx3 c~4 0 0 0 1]'

0]' (15)

with c = 0v/-~-~.l.Supposing full state knowledge, it is proposed a state feedback controller, being submitted to the evolutionist selection under the assumption of a perturbed gradient field. In the fitness formulation were considered state and control effort restrictions, according to (14). Our purpose is c o m p a r e the results here presented with the results obtained in [13]. The results obtained are presented as follows.

Robust Stabilization for the Nonlinear Benchmark Problem (TORA)...

U . ~ x = [ 5.0 5.0 5.0 V1 = [ 1.18162 -1.30304 2,35275 0.50982 1.63484 1.97930 0.74528 -1.69658 bl = z e r o s ( 5 , 1 ) ; V2 = [ 0.01824 2.07790 -4,06098 1.36253 1,53140 0.87103 -3.31062 -0.79987 1.03472 1.92860 b2 : z e r o s ( 4 . 1 ) ; c1 : ' r C2 ffi ' t ~ J l l i g ' ;

311

5.0 ] ' ; 1.57378 0.93887 0.93666 -1.56234 -1.20159 -0.36271 -0.16426 -3.88394 -0.49461 -0.17435 0.18411 -1.08677 ] ' ; -3.24036 1.56093 1.57604 -3.53676 0.40685

-0.02459 -0.60562 -1.14073 -1.81314 -2.97264 ] ' ;

The perturbation rejection in this controller maintains limited behaviour even with w(t) = 0.24sin(t) as in [13].

2

~ ~

[

!~

.

,

,

1,6

2 ...

'

1.tS 1

1

0,50

i

0

I

--0~ -0.5

-1

-1

-1,5

-~ 5

-1

3

-0.5 :

0

O.G

1

1.5

Actuacion

-0.5

0

O.G

1

1.5

HJIE

-11

,L

i ...........................................

-2~

-2 . . . . . . . . . ..................

-~

;

1'o

is

2o

....

"11

-1

%

-1

(

1

0

~-~.5

~

"

~. . . . . .

i ......

.........................................

~s 4 o

Fig. 2. Phase space projections x, (t) xx:(t), ~3(t) • along the path for ~c(0) = [-1.2 1.2 -1.2 1.2]'

;

lO

,5

~

~5

actuation u(t) and HJ[(t)

312

7

Cess

Raimfindez

Conclusions

The Evolutive approach has as main advantages: 1. The controller structure and complexity are established a priori. 2. The method is applicable even with nonregular models (no derivative needs). 3. An attraction basin is assured by simulation. The computational effort in each epoch considered mainly as computing time, is roughly proportional to

t~,,,.,,p o< ,,,,t,,, I ~ I (t,/
A ----o f f s p r i n g s ,

ns = num_samples,

[ x I ---

Space dimension, ts = num_~geps*dt expected settling time for the closed loop system and 6t = dr. The total time can be reduced to a fraction, depending on parallelization. Phase

The tuning process depends on few parameters associated to the open-loop x closed-loop system dynamic behaviour: num_steps,dt and training basin (where initial conditions are generated).

8

Acknowledgements

This work is supported by CICYT, under project TAP99-0926-C04-03.

References 1. T. B~ck, U. Hammel and H.-P. Schwefwel, Evolutionary computation: comments on the history and current state, IEEE Trans. on Evolutionary Computation, vol. 1 n ~ 1, April 1997. 2. R.W. Beard, G. N. Saridis and J. T. Wen, Galerkin approximations of the generalized Hamilton-Jacobi-Bellman equation, Automatica, vol. 33,n'12, pp 2159-2177, 1997. 3. R. T. Bupp, D. S. Bernstein and V. T. Coppola, A Benchmark problem for nonlinear control design, International Journal o] Robust and Nonlinear Control, Vol. 8, pp. 307-310, 1998. 4. W.A. Cebuhar and V. Costanza, Approximation procedures for the optimal control of bilinear and nonlinear systems, J. Optim. Theory Appl., 43(4),pp 615-627. 5. A. V. Fiacco and G. P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley and Sons, Inc. 1968.

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6. C. J. Goh, On the nonlinear optimal regulator problem, Automatica, vol.29 n ~ 3 pp 751-756, 1993. 7. A. lsidori, Feedback control of nonlinear systems, International Journal of Robust and Nonlinear Control, Vol. 2, pp. 291-311, 1992. 8. A. Isidori and A. Astolfi, Disturbance attenuation and Hoo control via measurement feedback in nonlinear systems, IEEE Transactions on Automatic Control, V o l . 3 7 , n ~ 9 pp. 1283-1293, 1992. 9. A. Isidori and W. Kang, Hoo Control via measurement feedback for general nonlinear systems, IEEE Transactions on Automatic Control, Vol. 40 n ~ 3, pp. 466-472, 1995. 10. A., J. van der Schaft,/~ Gain and Passivity Techniques in Nonlinear Control, Lecture Notes in Control and Information Science, VoI. 218, Springer Verlag, London, 1996. 11. H.-P. Schwefwel and G. Rudolph, Contemporary Evolution Strategies, Advances in Artificial Life. Third International Conference on Artificial Life, vol. 929 of Lecture Notes in Artificial Intelligence, pp. 893-907. Springer, Berlin. 12. J. SjSberg and L. Ljung, Overtraining, regularization and searching for minimum in neural networks,IEEE 4th IFAC Symposium on Adaptive Systems in Control and Signal Processing, pp. 669-674, Grenoble, France. 13. P. Tsiotras, M. Corless and A. Rotea, An /22 disturbance attenuation solution to the nonlinear benchmark problem, International Journal of Robust and Nonlinear Control, Vol. 8, pp. 311-330, 1998.

On Convexity in Stabilization of Nonlinear Systems A n d e r s Rantzer Department of Automatic Control Lund Institute of Technology Box 118 S-221 00 Lurid, Sweden Phone: +46 46 222 03 62 - r a n t z e r @ c o n t r o l , l t h .

se

A b s t r a c t . We recently derived a stability criterion for nonlinear systems, which can be viewed as a dual to Lyapunov's second theorem. The criterion has a physical interpretation in terms of the stationary density of a substance that is generated in all points of the state space and flows along the system trajectories. If the stationary density is finite everywhere except at a singularity in the origin, then almost all trajectories tend towards the origin. Here we consider consider state feedback for nonlinear systems and show that the search for a control law and density function that satisfy the convergence criterion can be stated in terms of convex optimization. The method is also applied to the problem of smooth blending of two given control laws.

Keywords

1

Stabilization, nonlinear systems, linear p r o g r a m m i n g , convexity

Introduction

L y a p u n o v functions have long been recognized as one of the most f u n d a m e n t a l tools for analysis and synthesis of nonlinear control systems. See for e x a m p l e [1,3,5-8]. T h e i m p o r t a n c e of the criterion s t e m s from the fact t h a t it allows s t a b i l i t y of a system to be verified w i t h o u t solving the differential e q u a t i o n explicitly. L y a p u n o v functions play a role similar to p o t e n t i a l functions and energy functions. Moreover, when a s y m p t o t i c s t a b i l i t y of an e q u i l i b r i u m has been proved using L y a p u n o v ' s theorem, i n p u t - o u t p u t s t a b i l i t y can often be proved using the Lyapunov function as a "storage function" [14]. Surprisingly, it was recently found t h a t L y a p u n o v ' s second t h e o r e m has a n a t u r a l dual t h a t has been neglected until present d a t e [11]. T h e result is a criterion t h a t for a given nonlinear s y s t e m guarantees convergence to the equilibrium from almost all initial states. In this paper, we consider consider s t a t e feedback for nonlinear s y s t e m s and show t h a t the search for a control law a n d density function t h a t satisfy the

316

Anders Rantzer

convergence criterion can be stated in terms of convex optimization. The method is applied to the problem of smooth blending of two given control laws.

The notation V V = [ aP-~( . "" oz.] ov

V :Rn -+R

011 Ofn ~7. f = ~xl _k... + Cgx----~

f :Rn "-+Rn

will be used throughout the paper.

2

A dual to Lyapunov's

second

theorem

One way to interpret a Lyapunov function V for the globally stable dynamical system k = f ( x ) is to view V(xo) as the "cost to go" from the initial state x0 to the equilibrium. In fact, existence of a Lyapunov function is often proved by introducing a penalty function l(x) > 0 and defining V by the formula y(~,o) =

l(x(t))ex

where k(t) = f ( x ( t ) ) , x(O) = xo. The definition immediately implies that ~7v(x) . f(x) = -t(x)

< o

Inspired by the duality theory for discrete transportation and network flow problems [4] and convexity arguments in optimal control [12,15,13], we consider the "flow" of trajectories generated by a system. Given the rate of generation r > 0, the integral xV(X)r can be viewed as the t o t a l s t a t i o n a r y cost per t i m e unit, when the s u b s t a n c e

flows along the system trajectories towards the equilibrium. If the flow gives rise to the stationary density p(x) in each point, another expression for the s t a t i o n a r y cost per t i m e unit is

x P ( X ) l ( z ) dt

Equality between the two expressions follows from Gauss' theorem:

Proposition 1 ([11]) Given f E C I ( R n , R n ) , let ~b,l E C ( R n) and p, V E C 1( R n) satisfy ~TV. f + l = 0

~7. (fp) = r

(1)

On Convexity in Stabilization of Nonlinear Systems

317

in X C R n while V ( x ) = 0 f o r x on the boundary of X . Then

fx V(X)r

= fx P(X)l(x)dz

Proof. /x[Vr

pl]dX = / x [ V ( ~ Y " (fP)) + V V " f p ] d x = / x ~Y" (V f p ) d x = O

where the last equality is due to Gauss' theorem and the fact t h a t V ( x ) - 0 on the boundary of X . The duality between the V and p is apparent in Proposition 1. Since the first equality in (1) with l(x) >_ 0 is the basis for Lyapunov's second theorem, it should come as no surprise t h a t a dual criterion can be stated based on the second equality in (1) with r _> 0. This is even more clear from the interpretation of p as stationary density of a substance generated with rate r For a stable system, a finite stationary density can be achieved everywhere except at the equilibrium, where the substance will accumulate and the density will be infinite. Conversely, the existence of a stationary density indicates that almost all substance must accumulate at x = 0. This intuitive a r g u m e n t is formalized in the following theorem. T h e o r e m 1 ( [ 1 1 ] ) G i v e n the equation Jc(t) = f ( x ( t ) ) , where f E C I ( R n , R n ) , suppose there exists a non-negative p E C I ( R n \ {O},R) such that l(~)p(x)/l~l is integrable on {x e R " : I~1 _> 1} and

[~7 . (fp)](x) > 0

for almost all x # 0

(2)

Then, for almost all initial states x(O) the trajectory x(t) tends to zero as t ---r oo. Moreover, if x = 0 is a stable equilibrium, then the conclusion remains valid even if p takes negative values. E x a m p l e 1 For scalar x, define f(~)

=

p(~)

= _ _

I

X4

Then [ V . (fp)](x) = 3 / x 4 > 0, so all conditions of Theorem 1 hold except for non-negativity of p and stability of x = 0. Example 2 With

f ( x ) = (x 2 - 1)x

1

p(x) ---- ~-~

we have [~7. (pf)](x) = 1 -I- x -2 > 0, so all conditions of T h e o r e m 1 hold except for the integrability of pf/[x[. In this case, all trajectories starting outside the interval [ - 1 , 1] have finite escape time.

318

Anders Rantzer

-3

-2

-1

~

1

2

3

Fig. 1. Phase plane plot for Example 3 Example

[/;1]

3 T h e system

1

/:2 = [ - 2 z ~ + 2 x l x 2 J

has two equilibria (0, 0) and (2, 0). See Figure 1. Let f ( z ) be the right hand side and let p(x) = Ixl - ~ . Then

[v. (pf)](~) = v p . f + p ( v . f) = --alxl-~'-2zTf + Ixl-~(4zl

-- 4)

= --,~1:~1-'~-2(~ -- 2)1~1 ~ + Ixl-~(4x~ -- 4)

= Izl-~[(4-a)zl

+ 2a-4]

With a = 4 all conditions of T h e o r e m 1 hold, so almost all trajectories tend to (0, 0) as t --4 cx~. The exceptional trajectories turn out to be those t h a t start with xl > 2, x2 = 0. Example

[/;1] /;2

4 The system

= l

(3)

-6x2 "4-2zlx2 J

has four equilibria (0, 0), (2, 0) and (3, :t:v~). See Figure 2. In this case,

p(~) = I~1-' gives [V. (pf)](x) = -41xl-%Tf + Izl-a(4z~ - 8) = -4lxl-6[(xi = 16z~lx1-6

-

2)1~12 4~] + -

Izl-a(4xl

-

8)

On Convexity in Stabilization of Nonlinear Systems

319

,

i -4

i 2

i o

i 2

i 4

i e

Fig. 2. Phase plane plot for Example 4 so again Theorem 1 shows that almost all trajectories tend to (0, 0) as t --~ cx~. The exceptional trajectories are the three unstable equilibria, the axis x2 : 0, xl _ 2 and the stable manifold of the equilibrium (2, 0), t h a t spirals out from the equilibria (3, -l-x/~).

3

Convexity in state feedback synthesis

An important application area for Lyapunov functions is the synthesis of stabilizing feedback controllers. For a given system, the set of L y a p u n o v functions is convex. This fact is the basis for m a n y numerical approaches, most notably computation of quadratic Lyapunov functions using linear matrix inequalities [2]. However, when the control law and L y a p u n o v function are to be found simultaneously, no such convexity property is at hand. In fact, the following variation of an example by [9,10] shows that the set of "control Lyapunov functions" (functions that can be used as L y a p u n o v functions for some stabilizing control law) m a y not even be connected. E x a m p l e 5 Every continuous stabilizing control law u(x) for the system 5:2 -- f ( x , u ) -

u(x)

J

(4)

must have the property that u(x) has constant sign along the half line xl > 0, x2 = 0. The reason is that a zero crossing would create a second equilibrium. A strictly decreasing Lyapunov function satisfies

OV 0 > "~V. f ( x , u) = .-~--u(x) ox~

for xl > 0, x2 = 0

320

Anders Rantzer

so also OV/Ox~ must have constant non-zero sign along the same half line. The control law ut(x) = - x l - 2x~ is stabilizing with strictly decreasing Lyapunov function Vl(x) = x~ + x~ + xlx~. Apparently OVt/Ox~ is positive along the half line. Similarly, the control law ua(x ) = xl - 2x~ is stabilizing with L y a p u n o v function Vg(x) = x~ + x~ - xlx2, with OVg/Ox2 negative along the half line. In particular, we see t h a t the two control L y a p u n o v functions Vt and Vg can not be connected by a continuous path without violating the sign constraint on

OV/Oz~.

Given this negative example, it is most striking to find that a convexity result is easily available when instead of Lyapunov's theorem we consider the new convergence criterion. Given a system of the form

the problem is to find a contral law u(x) and a density p such that [ V - ( ( f + gu)p)] (x) > 0

for almost all x r 0

while (f + .qu)p satisfies the integrability condition. Let pg(x) = p(x)u(x). Then the divergence inequality becomes

x7. (fp + gpg) > o which is a convex constraint on the pair (p, pg). Once a feasible pair has been found, the control law is obtained as u(x) = p(x)/pg(x). The convexity is also useful when a s m o o t h transition between two given controllers is to be designed. This is seen in the following continuation of Example 5. E x a m p l e 6 Theorem 1 applies to the system (4) with the controller ut(x) = - x l - 2x~ and the density function pt (x) = (x~ + x~ + xl x ~ ) - a ' provided t h a t ~t is sufficiently large. The same is true with the controller ug(x) = xl - 2x~ and the density function pg(x) = (x~ + x~ - x l x ~ ) - ~ , for sufficiently large Otg. Suppose that we are looking for a s m o o t h control law u(x) that acts as ut(x) for small x and as ug(x) for large x. This can obtained by putting c~t > ~g sufficiently large and

It it is easy to see that the conditions of T h e o r e m 1 hold and that u has the desired properties. Note that by the earlier argument such a s m o o t h blending of ut and ug can not be stabilizing in the sense of Lyapunov. However, with the method proposed here, we do get convergence in the sense of T h e o r e m 1.

On Convexity in Stabilization of Nonlinear Systems

4

321

Conclusions

A new convergence result for nonlinear s y s t e m s has been shown to have significant convexity p r o p e r t i e s in synthesis of control laws. In p a r t i c u l a r , this allows s m o o t h blending of a given set of stabilizing conrollers.

5

Acknowledgement

S u p p o r t by the Swedish Research Council for Engineering Sciences is gratefully acknowledged.

References 1. Z. Artstein. Stabilization with relaxed controls. Nonlinear Analysis TMA, 7:1163-1173, 1983. 2. S. Boyd, L. E1 Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory, volume 15 of Studies in Applied Mathematics. SIAM, Philadelphia, 1994. 3. R.W. Brockett. Asymptotic stabifity and feedback stabilization. In R.W. Brockett, R.S.Millman, and H.J. Sussmann, editors, Differential Geometric Control Theory, volume 27 of Progress in Mathematics. Birkhauser, Boston, 1983. 4. L. K. Ford and D. K. Fulkerson. Flows in Networks. Princeton University Press, Princeton, New Jersey, 1962. 5. W. Hahn. Theory and Applications of Lyapunov's Direct Method. Prentice-Hall, Englewood Cliffs, New Jersey, 1963. 6. A. lsidori. Nonlinear Control Systems. Springer-Verlag, London, 1995. 7. M. Krstic, I. Kanellakopoulos, and P. Kokotovich. Nonlinear and Adaptive Control Design. John Wiley & Sons, New York, 1962. 8. Yuri S. Ledyaev and Eduardo D. Sontag. A Lyapunov characterization of robust stabilization. Nonlinear Analysis, 37:813-840, 1999. 9. Laurent Praly. Personal communication. 10. Christophe Prieur and Laurent Praly. Uniting local and global controllers. In Proceedings of IEEE Conference on Decision and Control, pages 1214-1219, Arizona, December 1999. 11. A. Rantzer. A dual to Lyapunov's second theorem. Submitted for journal publication, March 2000. 12. A. Rantzer and M. Johansson. Piecewise linear quadratic optimal control. IEEE Trans. on Automatic Control, April 2000. 13. R. Vinter. Convex duality and nonlinear optimal control. SL4M J. Control and Optimization, 31(2):518-538, March 1993. 14. J.C. Willems. Dissipative dynamical systems, part h General theory; part II: Linear systems with quadratic supply rates. Arch. Rational Mechanics and Analysis, 45(5):321-393, 1972. 15. L. C. Young. Lectures on the Calculus of Variations and Optimal Control Theory. W. B. Saunders Company, Philadelphia, Pa, 1969.

Extended Goursat Normal Form: a Geometric Characterization W i t o l d Respondek 1 and W i l l i a m Pasillas-L~pine~,* 1 Laboratoire de math~matiques INSA de Rouen 76 131 Mont Saint Aignan, France wresp@imi, irma-rouen, fr

2 Laboratoire des signaux et syst~mes CNRS-Sup~lec 91 192 Gif-sur-Yvette, France pas illas@iss, supelec, fr

A b s t r a c t . We give necessary and sufficient geometric conditions for a nonlinear control system to be feedback equivalent to an extended Goursat normal form. We study the geometry of distributions, equivalently, the geometry of Pfaffian systems, which describe that class of control systems, and, in particular, the existence of corank one involutive subdistributions and the geometry of their incidence with characteristic distributions. We illustrate our results by applying them to two examples of nonholonomic multi-trailer systems.

1

Introduction

Consider a n o n h o l o n o m i c control system, with m + 1 controls, of the ~ l l o w i n g ~rm: m

(1)

~=~-~9,(x) ui, i--0

where x(.) belongs to ]1~N and g o , . . - , g,n are s m o o t h vector fields o n ]~N. We will say t h a t the control system (1) is in extended Goursat normal form [19] if we have 0

0 m n j--1

O

j

b-~'

j = l i=0 Research supported in part by the French company PSA Peugeot Citroen.

(2)

324

Witold Respondek and William Pasillas-L6pine

w h e r e x = ( x o ~ 1 7 6 9 , x ? ' , . .. , x mo, . . . other words, the system (1) reads as X~ =

U0

xo 9nz--1 X 1

iuo nz - ~ X 1 UO

x ' ~ = ul

o

m nj. In , X m, . ) and N = m + 1 + }-~j=l

kuo

"nm--I xnnmuO Xrn .-~ 9n m

xm

=urn.

For nonholonomic control systems in extended Goursat normal form, which is a particular case of multi-chained form [20], m a n y problems like motion planning, trajectory tracking, and point stabilization have been succesfully solved (see e.g. [9], [12], [16], [20], [24], [25], [26], [27], [28], and the references given there). A particular case of extended Goursat normal form (2) is when all chains of integrators have the same length, t h a t is, when nl = n2 = .-. -- n m = n. In this case, the extended Goursat normal form gives the canonical contact system on J n ( I [ ~ , ~ m ) , the space of n-jets of maps from ~ into ~m, that is, the space of n-jets of curves in ~m. The n a m e follows from the fact that the canonical contact system (2), with nl -- n2 -- --- -- nm = n, appears naturally (see e.g. [3] and [30]) when describing those curves in j n ( ~ , ~ m ) that are n-graphs of functions (note t h a t the canonical contact system is also called the C a r t a n distribution, see e.g. [30]). Nonlinear control systems equivalent to canonical contact systems for curves have been studied by the authors in [22] and [23]. If rn = 1, the extended Goursat normal form (2) gives Goursat normal form. Nonlinear control systems equivalent to Goursat normal form, and the form itself, have been extensively studied during last years (see [13], [16], and [19]; see also [6], [17], [18], and [21]). Consider two control systems m

Z:

x = Zfi(x)

ui = f ( x ) u

(3)

i=0

and m

(4)

=

i=0

evolving respectively on two open sets X and )( of ~ N , where u = ( u o , . . . , U r n ) t , fi : ( r i O , . . . , s t, f = ( f o , . . . , f m ) t, and f = ( ] 0 , . - - , ] m ) t. We say that the systems E and Z" are f e e d b a c k equivalent if there exits a smooth diffeomorphism ~ : X --+ A" and a feedback u =/3fi, where the matrix/3, whose entries /3j are s m o o t h functions, is invertible at any x E X, such that ~. (ffl) = ] ,

Extended Goursat Normal Form

325

where any vector field V on X is transformed by a diffeomorphism p into a vector field p . V on ~: according to

(~.y)(~) = -~(~-~(~)). y(~-~(~)). We say that the systems L' and ~' are locally feedback equivalent if the above p is a local s m o o t h diffeomorphism and fl is defined locally. To the system 2Y we associate the distribution :P spanned by the vector fields f 0 , . . . , fro, which will be denoted by = ( f o , . . - , fro). Since the distribution ~ remains invariant under any invertible feedback transformation u = tiff, all objects that we construct with the help of :P can be considered as feedback invariant objects attached to X'. This paper is devoted to the problem of when a given system, linear with respect to controls (in other words, a distribution), is locally equivalent to extended Goursat normal form. In the case m = 1 this is the problem of equivalence to Goursat normal form, which has been studied and solved for N = 3 by D a r b o u x [7], for N = 4 by Engel [8], and in the general case by yon Weber [31], C a r t a n [5], Goursat [11] (at generic points) and by Libermann [15], K u m p e r a and Ruiz [13], and Murray [19] (at an arbitrary point). In the general case m > 1, the problem of equivalence to extended Goursat normal form has been studied by Gardner and Shadwick [10], Murray [19], and Tilbury and Sastry [28]. Their solutions are based on a result of [10] that assures the equivalence provided that a certain differential form satisfies precise congruence relations. The problem of how to verify the existence of such a form had apparently remained open. This difficulty was solved by Aranda-Bricaire and Pomet [1], who proposed an algorithm which determines the existence of such a form. Their solution, although being elegant and checkable, uses the formalism of infinite dimensional manifolds and thus goes away from classical results characterizing particular extended Goursat structures and general contact systems. In this paper we give, in terms of classical invariants of (co)distributions like the Engel rank and characteristic subdistributions, geometric necessary and sufficient conditions for a nonlinear control system to be feedback equivalent to an extended Goursat normal form. Our study is a continuation of our recent work [22] and [23], where we characterized distributions and nonlinear control systems equivalent to a canonical contact system for curves. In Section 2 we introduce geometric tools which allow us to give our necessary and sufficient conditions: derived flags, Lie flags, and characterisitic distibutions. In Section 3 we formulate the main result of the paper: T h e o r e m 1, which gives a geometric characterization of extended Goursat normal form and thus

326

Witold Respondek and William Pasillas-L~pine

of control systems feedback equivalent to that form. An essential condition states that each distribution :D(1) of the derived flag of the distribution :D, defined by the system, contains an involutive subdistribution/:i C :D(i) that has constant corank 1 in :D(i). A natural problem which thus arises is whether a given distribution contains an involutive codistribution of corank 1. In Setion 4 we recall an answer to this problem given in [22], which follows directly from Bryant's algebraic lemma [2]. Finally, in Section 5 we illustrate our results by applying them to two examples of nonholonomic multi-trailer systems.

2

Notations

and Definitions

A rank k distribution l) on a smooth manifold M is a m a p that assigns smoothly to each point p in M a linear subspace :D(p) C TpM of dimension k. Such a field of tangent k-planes is spanned locally by k pointwise linearly independent smooth vector fields f l , . - - ,.fk on M, which will be denoted by l) = ~ f t , . . . , fk). Two distributions :D and /) defined on two manifolds M and M, respectively, are equivalent if there exists a smooth diffeomorphism between M and M such that

(~.:D)(~) =

~(~),

for each point i5 in 21~/.They are called locally equivalent if ~ is a local diffeomorphism. Clearly, two distributions :D and :D are locally equivalent if and only if the corresponding control systems L' and ~:, given respectively by (3) and (4), are locally feedback equivalent The derived flag of a distribution :D is the sequence of modules of vector fields :D(0) C :D(1) C "-" defined inductively by D (~ = D

and

D(i+I) : ~)(i) _]_ [~)(i),~)(/)],

for i > 0.

(5)

The Lie flag is the sequence of modules of vector fields :Do C :Dr C --- defined inductively by :D0 = :D

and

:Di+l = :Di + [:D0, :Di],

for i _> 0.

(6)

In general, the derived and Lie flags are different; though for any point p in the underlying manifold the inclusion :Di (P) C :D(i) (p) clearly holds, for i > 0. Obviously, a distribution :D is involutive if its first derived system satisfies D(l) = D (~

A characteristic vector field of a distribution :D is a vector field f that belongs to :D and satisfies If, :D] C D. The characteristic distribution of :D, which will be denoted by C, is the module spanned by all its characteristic vector fields.

Extended Goursat Normal Form

327

It follows directly from the Jacobi identity that the characteristic distribution is always involutive. We will denote by Ci the characteristic distribution of D(i). A distribution 29 (or the corresponding control system), defined on a manifold M of dimension N, is said to be completely nonholonomie if, at any point p in M, we have 29i(p)(p) = T v M for a sufficiently large integer i(p). The smallest integer with this property will be called the nonholonomy degree at p and will be denoted by n(p). Let nonnegative integers nl > ." 9 > nm form a partition of N - (m + 1), that is nl + " " + nm = N - (m + 1). Denote the cartesian coordinates of ] ~ N b y 0 0 (X0, X 1 .

nl 0 . , X. 1 ., ; r 2. , . - -.

n2 ,.X 2 . , .

0 ., X r n ,

, X~n-n'n).

An extended Goursat normal form, with indices ( n ~ , . . . , nm), is the (m + 1)tuple of vector fields: g l = ~ , . .0. ,

g,~=

0

rn h i - - 1 0

(7) j=l i=0

A nonlinear control system ~ = E i =,nO g i u i , where the vector fields gi are given by (7), is said to be in extended Goursat normal form.

3

The

main

result

Consider a distribution 29 and its derived flag (5). For any i ~ 0, put d~(p) = dim29(i)(p) and r~(p) = di+l(p) - d~(p). Recall that for a completely nonholonomic distribution we denote by n(p) the nonholonomy degree at a point p and if it is constant we will denote it by n. The following result gives a geometric characterization of extended Goursat normal form. T h e o r e m 1 ( e x t e n d e d G o u r s a t n o r m a l f o r m ) . A nonlinear control system of the f o r m (3), defined on an open subset X o f ~ N, is feedback equivalent, in a small enough neighborhood of any point p in X , to an extended Goursat normal f o r m if and only if the distribution 29 = (fo, . . ., fro) satisfies the following conditions. (i) The distribution 29 is completely nonholonomic, with a constant nonholonomy degree n. (ii) Each element 29(Q of the derived flag has constant rank di, f o r 0 < i < n. (iii) Each element 29(i) of the derived flag contains an involutive subdistribution s C 29(0 that has constant corank one in 29(i), f o r 0 2 and there exists 0 2 and r~+l = 1, then, additionally, 1:i C Ci+l. This result yields a verifiable test for the local feedback equivalence to an extended G o u r s a t n o r m a l form, provided t h a t we know how to check whether or not a given d i s t r i b u t i o n a d m i t s a corank one involutive s u b d i s t r i b u t i o n . T h e second s t e p o f our geometric c h a r a c t e r i z a t i o n of e x t e n d e d G o u r s a t normal forms is to give, in the next Section, a checkable necessary and sufficient condition for the existence of such an involutive s u b d i s t r i b u t i o n . We will also provide an explicit f o r m u l a to calculate such an involutive d i s t r i b u t i o n , which is needed to check the condition (v). It is s t r a i g h t f o r w a r d to observe t h a t if a control s y s t e m is equivalent to extended G o u r s a t n o r m a l form, then we have

1:i C Ci+l for any O 2 and r i + l = 1. This is so, because in all other cases we have the following result proved in [22]. 1. Let l) be a distribution such that :D(~ :DO), and 79(2) have constant ranks do, dl >_ do + 2, and d2 > dt + 2, respectively. Assume that each distribution l) (i), for i = 0 and 1, contains an involutive subdistribution l:i C l) (i) that has constant corank one in l) (i). Then 1:o C C1. Lemma

The following e x a m p l e shows t h a t the above l e m m a does not hold in the case r0 3> 2 and r l = 1, which explains the presence of the condition (v) in the s t a t e m e n t of T h e o r e m 1. E x a m p l e . Consider the d i s t r i b u t i o n 0

0

V (~ = ( f o , f l , f 2 ) = (-~xl + X 2 0 y 2 '

0

0

0

0

Ox2' Ox3 q-xl-~yl d - y l ~ y 3 ) .

We have [fl, f2] = 0, [fl , f0] = --oy~a,and [f0, f2] = b-~y~ " ~ Hence 1:o ----(0-~2, f2)We calculate /)(1) = (00xl,

0

0

0

Ox2' Oyl' by2' f2), and thus one possible involutive c o d i s t r i b u t i o n of corank 1 in :D1 is 1:1=(

a o a 0 Oxi'-- Ox2' a y t ' Oy2 ')"

Extended Goursat Normal Form

329

We have

C1 =

b

,

b

(0-1 0*2'

b

ay=).

Clearly, the conditions (i)-(iv) of T h e o r e m 1 are satisfied but /:0 is not contained in C1 and thus the distribution/)(0) is not equivalent to any extended Goursat normal form, which proves the necessity of the condition (v) in that case.

4

Corank One Involutive Subdistributions

The aim of this Section is to give an answer to the following question: "When does a given constant rank distribution 7) contain an involutive subdistribution s C 7) that has constant corank one in 7)?". In fact, the answer to this question, together with a construction of s if it exists, is an immediate consequence of a result contained in B r y a n t ' s Ph.D. thesis [2]. Links between Bryant's result and the characterization of the canonical contact system for curves have also been observed by Aranda-Bricaire and P o m e t [1]. Recal that by C we denote the characteristic distribution of ~D. We will also denote c0(p) = dim C(p). The Engel rank [3] of a distribution 7), at a point p, is the largest integer p such that there exists a 1-form w in 7)• for which we have (d~)"(p) ~ 0 m o d : D • where 7) • denotes the Pfaflian system that annihilates the distribution 7). Obviously, the Engel rank p equals zero at each point if and only if the distribution is involutive. We send the reader to [22] for an equivalent definition of the Engel rank, in the language of vector fields, in the particular case when p = 1 that will be i m p o r t a n t in the sequel. Recall that for a distribution 7) such that 7)(0) and 7)(1) have constant ranks do and dl, respectively, we denote r0 = dl - do. The following result is a direct consequence of B r y a n t ' s algebraic l e m m a [2] (see also [22]). In order to avoid the trivial case r0 = 0, for which the existence of a corank one involutive subdistribution is obvious, we will assume that r0 > 1. P r o p o s i t i o n 1 ( c o r a n k o n e i n v o l u t i v e s u b d l s t r i b u t l o n s ) . Let 7) be a distribution such that D (~ and 7)(1) have constant ranks do and dl, respectively. A s s u m e that r0 > 1. Then the distribution 7) contains an involutive subdistribution f_. C 7) that has constant corank one in :D if and only if the three following conditions hold: (i) The characteristic distribution C of 79 has constant rank co = do - ro - 1; (ii) The Engel rank p of :D is constant and equals 1;

330

Witold Respondek and William Pasillas-Ldpine

(iii) If ro = 2 then, additionally, the unique corank one subdistribution B C 9 such that [13,B] C D must be involutive.

Moreover, if an involutive subdistribution s C D of corank one in D exists and r0 > 2 then it is unique. We would like to to emphasize that the above conditions are easy to verify. Indeed, for any distribution, or the corresponding Pfaffian system, we can c o m p u t e the characteristic distribution C and check whether or not the Engel rank equals 1 using, respectively, the formula (8) and the condition (9) below. This gives the solution if r0 ~: 2. If r0 = 2 we have additionally to check the involutivness of the unique distribution B satisfying [B, B] C D, whose explicit construction is also given below. Consider a distribution D of constant rank do on a manifold of dimension N. Let w l , . . . , w, o, where so = N - do, be differential 1-forms locally spanning D • the annihilator of 9 , which we denote by

/)i = ( ~ i , . . - , ~ , 0 ) We will denote by 2: the Pffafian system generated by w i , . . . , w, o. For any form w E D • we put

W(w) = { f el) : f a d w E V • Clearly, the caracteristic distribution C of D is given by $o

c = N w(~,).

(8)

i=1

Now assume that D (1) is of constant rank dl > do, that is r0 > 1, or, equivalently, that the first derived system Z (1) is of constant rank smaller than so. By a direct calculation we can check (see e.g. [3]) that the Engel rank of the distribution D, or of the corresponding Pfaffian system 27, equals 1 at p if and only if

(dwi A dwj)(p) = 0

mod I,

(9)

for a n y l < i < j < s 0 . Now let us choose a family of differential 1-forms

@ l , . . . , ~r0, ~r0+l,..., ~,0 ) such that (V(~ • = ( w l , . . . , w , o ) and (D(I)) • = (wr0+l,...,w,0). Independently of the value of r0 > 2, the unique distribution B C D satisfying [B, B] C D is given (if it exists, that is if D satisfies the conditions (i) and (ii) of Proposition 1), as shown by Bryant [2], by 1"0

B = ~ i=1

W(wi).

(10)

Extended Goursat Normal Form

331

In fact, Bryant has also proved that it is enough to take in the above sum only two terms corresponding to any 1 < i < j < r0. In order to verify, in the case r0 = 2, the conditions of Proposition 1 we have additionally to cheek the involutivity of this explicitely calculable distribution B. Moreover, as shown by Bryant [2] (see also [22]), the unique distribution B satisfying [B,B] C /), and thus given by (10), coincides with s an involurive subdistribution of corank 1 i n / ) , provided t h a t either r0 = 2 and B is involutive or r0 > 3. In those two cases s is also unique and the formula 7"0

s -- Z

W(wl).

(11)

i=1

gives a constructive way to find it.

5

Examples:

multi-steered

trailer

systems

Consider the following nonholonomic model of multi-steered trailer systems introduced by Bushnell, SCrdalen, Tilbury, and Sastry (see [4], [28], and [29]). The system is a "train" consisting of m consecutive chains of hi-trailer systems. Each hi-trailer system, for 1 < i < m, is a mobile robot towing ni - 1 passive trailers. For each 1 < i < m and 1 < j < m , the point (x~, y~) gives the position, in the (x, y) plane, of the center of the axle of the j - t h trailer located in the i-th chain of trailers. T h e angles 03, either for 1 < i < m and 1 < j < h i - 1, or for 1 < i < m - 1 and j = h i + l , represent angles between consecutive trailers while the angles 0~,, for 1 < i < m - 1, are freely steerable, as they correspond to mobile robots, that is steerable trailers, situated at the front of each train. All trailers are supposed to roll without slipping, which imposes the following nonholonomic constraints on the system: the velocity of the system along any trajectory is anihilated by the following differential I-forms /~j = sin" O~idzj~ - cos O~dyj,i i

for 1 < i < m and 1 ~ j < m.

A detailed description of this class of systems can be found in [28]. Any configuration of a multi-steered trailer system is described by positions (x~., y~), for 1 < i < m and 1 < j < hi, of all trailers and by all angles 0~., for 1 < i < m+l and 1 <- - j <- - h i + 1 (except for 0nmm which has not been introduced). Therefore any configuration is given by a point in Ii~N x ($1) L, where N = 2 ~-~=1 m ni and L = m - 1 + ~-~i=l m hi. Notice, however, that the positions on the (x, y)-plane of any two consecutive trailers are related by the following holonomie constraints (static constraints) i 1 = xji + cos 03 xj+ Y~'+i =Y~" + sin 0~.,

332

Witold Respondek and William Pasillas-L~pine

forl
l<j
and

Xi ni + COS 0 n i + l 9 i

9 yl+l = y~, + sin0.,+1,

for l < i < m - 1 . Denote by i the i m m e r s i o n i : ]t~ 2 • ( S 1 ) L ~

]1~N • ( S 1 ) L

defined by the above holonomic constraints. T h e s t a t e space o f the m u l t i steered trailer s y s t e m is thus 1 ~ • ( S 1)L a n d its evolution on t h a t s t a t e space is subject to the n o n h o l o n o m i c c o n s t r a i n t s given by w ji = i* (12j),

for l < i < m and l < j < ni.

i for 1 < i < m a n d A r o u n d any point a t which the differential 1-forms wj, 1 < j < hi, are independent, the m u l t i - s t e e r e d trailer s y s t e m is thus a controllinear s y s t e m , with m + 1 controls, of t h e form

= 5 f~(x)ui, i=0

where x(.) E I~ 2 • ($1) L and fi, for 0 < i < m, are pointwise l i n e a r l y independent vector fields s p a n n i n g the d i s t r i b u t i o n 9=

(f0,fl,...,fm)

a n n i h i l a t e d by all w~., for 1 < i < m and 1 _< j _< hi. T h e nonholonomic c o n s t r a i n t s of any m u l t i - s t e e r e d trailer s y s t e m satisfy the two following f u n d a m e n t a l identities. F i r s t l y dw~ A w~

A~i+l '

=

O,

(12)

for 1 < i < m a n d 1 < j < ni - 1. Secondly

dw) Aw~ Awk+ A for 1 < i < m a n d 1 < j

9. . A w ~ . _ , A w ) = - d x ~ A d y i k A d O ~ : A

''.AdO#i, (13)

< ni and 1 < k < j .

In order to i l l u s t r a t e the i m p o r t a n c e of those relations, let us consider the case of a single chain of trailers, t h a t is the c e l e b r a t e d n - t r a i l e r s y s t e m . If we drop the double i n d e x a t i o n we o b t a i n n n o n h o l o n o m i c c o n s t r a i n t s #2i = sin Oi dxi - cos Oidyi,

for 1 < i < n,

Extended Goursat Normal Form

333

each relating the velocity (&i, Yi) of the i-th trailer with the angle 0i. To these nonholonomic constrains, the following n - 1 holonomic constraints are added x i + i = Xi + cOS Oi Yi+l = Yl + sin0i,

for 1 < i < n - 1, By (12) and (13), the nonholonomic constraints of the n-trailer system satisfy dtoiAtoiAwi+l

=0,

forl
and dton A tol A to2 A 9 9 9 A t o n - 1 A ton = - d z l

A d y l A dO1 A . 9 9 A d o n .

If we represent these kinematical contraints as the Pfaffian system Z =

@1,,~2,... ,to,-1,~,),

then the two last relations clearly imply that Z (k) = ( t o l , w ~ , . . .

,wn-k-l,w,~-k),

for 0 < k < n,

(14)

which gives an alternative proof of a now classical result: the n-trailer system is controllable [14]. In fact, the relations (14) imply that the n-trailer system is a Goursat structure and can thus be converted into Goursat normal form on an open and dense subset of its configuration space. We will now apply our characterization of the extended Goursat normal form obtained in the paper to two concrete examples of multi-steered trailer systems: the fire truck, that is a car to which a steerable trailer has been attached, and a system of two cars attached together, one behind the other. The equivalence of both systems to extended Goursat normal forms has been already observed in [4] and [28]. We show how the same conclusion can be obtained from our characterization, in particular, we describe singular configurations at which the equivalence does not take place. E x a m p l e ( t h e fire t r u c k ) . T h e fire truck system [4] is the multi-steered trailer system defined by the three following nonholonomic constraints Dtt = sin O~dx~ - cos 0~ dy~ J2~ = sin o~tax~ - cos O~dy2t D~ = sin e ~ a x ~ - cos o~ay~ to which the following holonomic constraints =

+ cos 021

yl2 = y~ + sin O~ =

+ cOSOx

y22 = y~ + sin 0~.

334

Witold Respondek and William Pasillas-L~pine

are added. It is thus a system evolving on ~ • (S 1)4 subject to the three above nonholonomic constraints. By relations (12) and (13) we have

d~ ^ ~ ^ ~ = 0

On the one hand, since 1 21 + w~ = sin O~dz~ - cos Otdy

cos(0~

t 1 - 01)d02,

we have

On the other hand, since ~ = sin O~dx~ - cos O~dy~ - cos(012 - 0~)d0~

~. 1i . cos(0~. w~ . sin0~dx~ . . cos0~dy

0~)~0~

cos(0~

01~ )d01,

we have

dw~ A w~ A w~ A w~ = - cos(0~ - 02I) cos(0~ - 02)dx~ A dy~ A dO~ A dot9 A dO~. We can thus conclude that all elements of the derived flag of the fire truck are of constant rank if and only if we have 0t - 0 ~

7[

# ~mod~

and

7r

012-0~ # ~modTr

and

7~

0 ~ - 0 2 # ~mod~r.

In what follows we will consider the fire truck on the open submanifold M of 1~2 x (S 1)4 defined by the three above conditions. Denote by Z the Pfaffian system spanned by the fire truck on M. We have z ~~ = ( ~ , ~ 2 , ~ ) z (~) : (~2) z (2) : (0).

Now, denote I)(k) = (J[(k))j-. The ranks of/) (~ I) (I), and 2)(2) are constant

and equal to 3, 5, and 6, respectively. The distribution :D(~ clearly contains an involutive subdistribution s = (0/00~, c3/00~) that has constant corank one in :D(~ Since the characteristic distribution C1 of T~(1) is given by 61 = (c3/00~, a/00~), we obviously have s C C1 (compare the condition (v) of Theorem 1 and Example following it). Since, moreover, in the (3, 5, 6) case the regularity condition on the growth vector is always satisfied, the fire truck system satisfies all the conditions of Theorem 1, and thus can be converted

Extended Goursat Normal Form

335

locally into extended Goursat normal form in a small enough neighborhood of each point of M , as observed by Bushnell, Tilbury, and Sastry [4]. E x a m p l e ( t h e t w o - e a r ) . The two-car system is the multi-steered trailer system defined by the four following nonholonomic constraints ~2~ = sin O~dx~ - cos O~dy~ r

= sin 02dx2 1 i - cos O~dv~1

1212 = sin 01~ dx 2i - cos 8~dy~ ~

= sin O==dx~ - cos O~dy~,

to which the following holonomic constraints

x,' = ~I + cos el yl = y ~ + s i n O ~

9 ~ = 4 + cos 01 y12 = y21 + sin 03i y== = y12 + sin 912 are added. It is thus a system evolving on ]R2 x ($1) 5 subject to the four above nonholonomic constraints. By relations (12) and (13), we have

6~I ^ ~I ^ ,,4 = 0 6~ ^ ~

^ 4

= - 6 ~ I ^ 6y~ ^ 601 ^ 60~

- cos(e== - e12)aet we have

= - cos(el2 - 03') cos(e~ - e12)6~I ^

6v~ ^ 6el ^ 6e~ ^ 6e~ ^ 6o12.

On the other hand, since w~ = sin O~dx12 - c o s O~dy12 + cos(O~ - OI)6Ol + 6O~ w~ = sin O~dz12 - cos O~dy~ + cos(Oai - e~)6e 1,

336

Witold Respondek and William Pasillas-L~pine

we have

We can thus conclude that all elements of the derived flag of the two-car system are of constant rank if and only if we have 0 3 1 - 0 ~ r ~modTr

and

0~-031 r

and

0~-8~r

~modTr.

In what follows we will consider the two-car on the open submanifold M of R 2 x (S 1)5 defined by the three above conditions. Denote by Z the Pfaffian system spanned by the two-car system on M. We have i;(o)

i i 2 = (~:,~2,~:,~2)

-Z(2) = (0).

Denote ?)(k) = (i(k))• The ranks of :D(~ :D(U, and :D(2) are constant and equal to 3, 5, and 7, respectively. The distribution :D(~ clearly contains an involutive subdistribution s = (0/00~, c3/c3~) that has constant corank one in :D(~ Observe that the distribution :D(U contains the characteristic distribution Ci = (c3/c38~,c3/00~) and thus decomposes as :D(i) = Ci ~.T'. Therefore instead of checking whether the distribution 1)(i) of type (5,7) containes an involutive subdistribution of corank 1 we can check whether the distribution .T of type (3, 5) contains an involutive distribution of corank 1. The distribution .~ is given by ~" = (w~,w~) • Using (10), we calculate the unique rank 2 subdistribution B C ~ such that [B, B] C 9c. A direct calculation gives B = (0-~1~ , sin031~z~-c~

10

-cos(O~--03)~0T+0-~31 ) -

Clearly, B is involutive and hence s the involutive distribution of corank 1 contained in 1) (1), is given by s = B. Therefore the two-car system is equivalent on M to an extended Goursat normal form, as already observed by Tilbury and Sastry [4].

6

Acknowledgements

The second author would like to thank Franqoise Lamnabhi-Lagarrigue for the Postdoctoral position she proposed him at LSS, where this paper was finished.

Extended Goursat Normal Form

337

References 1. E. Aranda-Bricalre and J.-B. Pomet. Some explicit conditions for a control system to be feedback equivalent to extended Goursat normal form. In Proceedings o] the IFA C Nonlinear Control Systems Design Symposium, Tahoe City (California), 1995. 2. R. Bryant. Some aspects o] the local and global theory of P]aj~:ian systems. Ph.D. thesis, University of North Carolina, Chapel Hill, 1979. 3. R. Bryant, S-S. Chern, R. Gardner, H. Goldschmidt, and P. Griffiths. Exterior Digerential Systems. Mathematical Sciences Research Institute Publications. Springer-Verlag, New York, 1991. 4. L. Bushnell, D. Tilbury, and S. Sastry. Steering three-input nonholonomic systems: The fire truck example. International Journal o] Robotics Research, 14(4):366--381, 1995. 5. E. Cartan. Sur l'~quivalence absolue de certains systtmes d'~quations diff~rentielles et sur certaines familles de courbes. Bulletin de la Socigtg Mathgmatique de France, 42:12-48, 1914. ~T~uvrescompletes, Part. II, Vol. 2, Gauthiers-Villars, Paris. 6. M. Cheaito and P. Mormul. Rank-2 distributions satisfying the Goursat condition: All their local models in dimension 7 and 8. ESAIM Control, Optimisation, and Calculus of Variations, 4:137-158, 1999. 7. G. Darboux. Sur le probl~me de Pfaff. Bulletin des Sciences mathdmatiques, 2(6):14-36,49-68, 1882. 8. F. Engel. Zur Invariantentheorie der Systeme Pfaff'scher Gleichungen.

Berichte Verhandlungen der Koniglich Sachsischen Gesellshaft der Wissenshaften Mathematisch-Physikalische Klasse, Leipzig, 41,42:157-176;192-207, 1889,1890. 9. M. Fliess, J. L~vine, P. Martin, and P. Rouchon. Flatness and defect of nonlinear systems: Introductory theory and examples. International Journal of Control, 61(6):1327-1361, 1995. 10. R. Gardner and W. Shadwick. The GS algorithm for exact linearization to Brunovsk~ normal form. IEEE Transactions on Automatic Control, 37(2):224230, 1992. 11. E. Goursat. Lemons sur le probl~me de P]aff. Hermann, Paris, 1923. 12. Z-P. Jiang and H. Nijmeijer. A recursive technique for tracking control of nonholonomic systems in chained form. IEEE Transactions on Automatic Control, 44(2):265-279, 1999. 13. A. Kumpera and C. Ruiz. Sur l'~quivalence locale des syst~mes de Pfaff en drapeau. In F. Gherardelli, editor, Monge-Amp~re equations and related topics, pages 201-247. Instituto Nazionale di Alta Matematica Francesco Severi, Rome, 1982. 14. J.-P. Laumond. Controllability of a multibody mobile robot. IEEE Transactions on Robotics and Automation, 9(6):755-763, 1991. 15. P. Libermann. Sur le probl~me d'~quivalence des syst~mes de Pfaff non compl~tement int~grables. Publications Paris VII, 3:73-110, 1977. 16. P. Martin and P. Rouchon. Feedback linearization and driftless systems. Mathematics of Control, Signals, and Systems, 7:235-254, 1994. 17. R. Montgomery and M. ZhitomirskiY. Geometric approach to Goursat flags. Preprint, University of California Santa Cruz, 1999.

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18. P. Mormul. Goursat flags: classification of codimension-one singularities. Preprint, Warsaw University, 1999. 19. R. Murray. Nilpotent bases for a class of nonintegrable distributions with applications to trajectory generation for nonholonomic systems. Mathematics o] Control, Signals, and Systems, 7:58-75, 1994. 20. R. Murray and S. Sastry. Nonholonomic motion plarLrfing: Steering using sinusoids. IEEE Transactions on Automatic Control, 38(5):700-716, 1993. 21. W. Pasilias-L6pine and W. Respondek. On the geometry of Goursat structures. Submitted to ESAIM Control, Optimisation, and Calculus of Variations, 1999. 22. W. Pasillas-L6pine and W. Respondek. Contact systems and corank one involutive subdistributions. Submitted to Acta Applicandae Mathematicae, 2000. 23. W. Pasillas-L6pine and W. Respondek. A new intrinsic characterization of the contact system for curves by corank one involutive subdistributions. In

Proceedings of the International Symposium of Mathematical Theory of Networks and Systems, Perpignan (France), 2000. 24. J.-B. Pomet. Explicit design of time-varying stabilizing control laws for a class of controllable systems without drift. Systems and Control Letters, 18:147-158, 1992. 25. C. Samson. Control of chained systems: Application to path following and timevarying point-stabilization of mobile robots. IEEE Transactions on Automatic Control, 40(1):64-77, 1995. 26. O. Scrdalen and O. Egeland. Exponential stabilization of nonholonomic chained systems. IEEE Transactions on Automatic Control, 40(1):35-49, 1995. 27. A. Teel, R. Murray, and G. Walsh. Nonholonomic control systems: From steering to stabilization with sinusoids. International Journal of Control, 62(4):849870, 1995. 28. D. Tilbury and S. Sastry. The multi-steering n-trailer system: A case study of Goursat normal forms and prolongations. International Journal of Robust and Nonlinear Control, 5(4):343-364, 1995. 29. D. Tilbury, O. S0rdalen, L. Bushnell, and S. Sastry. A multi-steering trailer system: Conversion into chained form using dynamic feedback. IEEE Transactions on Robotics and Automation, 11(6), 1995. 30. A. Vinogradov, I. Krasil'shchik, and V. Lychagin. Geometry of Jet Spaces and Nonlinear Partial Differential Equations. Gordon and Breach, New York, 1986. 31. E. yon Weber. Zur Invariantentheorie der Systeme Pfatt'scher Gleichungen. Berichte Verhandlungen der Koniglich Sachsischen Gesellshafl der Wissenshaflen Mathematisch-Physikalische Klasse, Leipzig, 50:207-229, 1898.

Trajectory Tracking for r-flat Nonlinear Delay Systems with a Motor ExamPle Joachim Rudolph 1 and Hugues Mounier ~ i Institut f/it Regelungs- und Steuerungstheorie TU Dresden, Mommsenstr. 13 01062 Dresden, Germany rudolph@erss t 1. et. t u-dresden, de

2 lnstitut d'61ectronique fondamentale Universit6 Paris-Sud, BAtiment 220 91405 Orsay, France [email protected], fr

A b s t r a c t . Stable trajectory tracking is discussed for the class of r-flat nonlinear systems. The main definitions introduced are: r-flatness, a generalized controller form, Brunovsk#-states and a corresponding notion of equivalence under predictive quasi-static feedback of an extended Brunovsk3~-state. These concepts lead to stable tracking with so-called weak state representations, which involve delayed time derivatives of the weak state variables, through "feedback lineaxization", i.e., through the equivalence of the 7r-flat systems to a Brunovsk3Llike form. Formal definitions and results axe given in a difference-differential algebraic frame. A motor with a flexible shaft serves for illustration.

1

Introduction

Simple and efficient methods for motion planning and trajectory tracking are available for nonlinear systems that are differentially flat [12-14,19]. For linear systems with time delays a counterpart of the flatness is the freeness, and more generally the d-freeness and the lr-freeness [15,20]. These flatness and freeness properties have been shown to be quite currently encountered in applications, see e.g. [13,18,19,21,26]. It is therefore most natural to ask whether there is a useful generalization of these concepts to the case of nonlinear delay systems. In this spirit, the J-flatness of nonlinear systems with time delays has been introduced [22,23]. A type of feedback which turns out to be particularly useful for the control of flat systems is the so-called quasi-static state feedback. This notion has been introduced in [3] for the i n p u t - o u t p u t decoupling of nonlinear multiinput multi-output systems x = f ( x , u), y = h(x). In the sequel this class of feedback was shown to be useful also for disturbance rejection and feedback linearization, see [4,5,7,28] for example. The main result of the present paper is a generalization to 7r-flat nonlinear delay systems with a single delay of the feedback linearizability result of [6,7]:

340

Joachim Rudolph and Hugues Mounier

The ~r-flat dynamics with delays are equivalent by predictive quasi-static feedback of an extended Brunovsklj-state to a system admitting a Brunovsk~-like form. We thus add an item to the long list of results on feedback linearization. As a particular feature, in this kind of feedback, besides states, delayed time derivatives of state variables are fed back. The mathematical framework used in the present paper is differencedifferential algebra [2], an extension of differential algebra [17,25]. This tool has been proposed for the study of nonlinear delay systems by M. Fliess [9]. It deals with algebraic structures, namely difference-differential field extensions, which may be seen as being obtained from the differential algebraic ones - - as used for systems without delays - - by adding so-called transformations. These latter correspond to the delays, which have finite amplitudes. As in the delay-free case, this algebraic framework is most appropriate for the definition and investigation of both flatness and quasi-static state feedback. The basic ingredients are the generalizations of purely transcendental field extensions (for flatness) and filtrations (for the feedback). The paper is structured as follows. The systems under consideration, the rrflatness, and the basic concepts of inputs and states are given in section 2, where also extended Brunovsk3~-states and a generalized controller form for re-flat systems are introduced. The equivalence by predictive quasi-static state feedback of an extended Brunovsk~-state is then defined in section 3, and the main result on the equivalence of 7r-flat systems and a Brunovsk~-like form is stated. In section 4 the mathematical model of a motor with a flexible shaft borrowed from [24] is considered. While only open-loop steering has been considered in [24], here we indicate how the feedback equivalence result can be exploited in the design of a feedback law for trajectory tracking. This example serves for illustration in the former sections, too. A sketch of the mathematical background is postponed to section 5. In order to simplify the notations, we restrict our attention to the case of a single delay operator. (The generalization to the case of multiple delays is possible in a direct manner.) 2

2.1

Systems,

n'-flatness,

inputs

and

generalized

states

Systems with delays

We consider nonlinear systems with a single delay amplitude r that can be described by a (finite) set of N differential-difference equations of the form

Fj(z,... , z ( O ( t - s t ) , . . . ,z(~J)(t - a t ) ) = O,

j = 1,... ,N,

with the system variables z = ( z l , . . . , zq). Denoting by J the delay operator of finite amplitude v, which maps f(t) on J r ( t ) = f ( t - r), the system equations can be written more concisely as

F(z,

= 0.

(1)

Tracking for w-flat Nonlinear Delay Systems

341

Notice that 5"z (i) indicates the typical form of the arguments of F , which means that each of the time derivatives of z m a y occur with several different delays. In the algebraic framework detailed in section 5 the following definition can be given. D e f i n i t i o n 1 ([9]). A (nonlinear) delay system (with a single delay amplitude) is a finitely generated &differential field extension S / k . Example 1. A PM stepper m o t o r with a flexible shaft has been considered in [24] with a model of the form d L - ~ i d = vd - R i d + N . L O i q d. L - ~ q = Vq - R i q - N , LOid - Km JO=Kmiq-BO-p 0 1

aGI 0

If the constant parameters are considered to be elements of k, and 1 M = k(8, p, id, iq, vd, Vq, r/) a the corresponding system can be defined as M / k .

2.2

rr-flatness

Roughly speaking, the notion of flatness [13] is extended to delay systems of the form (1) in the following way: A system with a delay 5 is called w-fiat if there exists a polynomial rr E k[5] and a collection y = (Yl,-.. ,Ym) of variables, called a w-fiat output, with the following three properties: 1. The components of y can be expressed in terms of the system variables z using only differentiations, delays, and rr-advances (i.e., the inverse 7r- I of 7r) - - via difference-differential relations of the type Yi = Pi(z,

...

, 5sT:-Sz(J),

...

, 50~-~

i = 1,...

, m.

2. The components of y are difference-differentially independent, i.e., they are not related by any (non-trivial) differential-difference equation Q ( y . . . . ,~Sy(j),... ,8Zy(a)) : o. 1 With a slight abuse of notation, we use (f = {5} for simplicity.

342

Joachim Rudolph and Hugues Mounier

3. Every variable zl used to describe the system, for instance states or inputs - - and with these all their derivatives, the delayed variables, and all functions of those variables - - can be calculated from y using only differentiations, delays, and ~r-advances. In other words, any such zi satisfies a relation of the type zi = R i ( y , . . . , t~" Tr-$ y(J) , . . . , t~eTr-~y(Y)).

In the framework of difference-differential algebra - - see section 5 and [23] - - a formal definition can be given: D e f i n i t i o n 2. A delay system E / k is 7r-flat if 1. there does not exist any z E ~ , z ~ k which is &differentially k-algebraic; -= k(y)6u,r-~, with k ( y ) 6 u ~ - i / k a $ U 7r-~-differentially purely transcendental extension.

2. E 6 u ~ - i

Then any such y is called a 7r-ftat output. In case lr -- $, one speaks about &flatness, and in case lr -- 0 one simply speaks about flatness. Of course, flat systems are if-flat and &flat systems are 1r-flat. R e m a r k 1. The first of the two conditions for 1r-flatness does not have a counterpart in the delay-free case [7,13]. It corresponds to the torsion-freeness in linear delay systems [15] and excludes the existence of "autonomous" equations

, ~ % ( i ) , . . . ,~bz(~)) = 0

R(z,...

with R a (non-trivial) polynomial with coefficients in k. E x a m p l e 2. The m o t o r model M / k of example 1 is &flat and y = (rl, id) is a &flat output. One has

1

0 = ~ [~-1~ + ~,~] + ~ 1

iq = I ~m

O

[ j O + B t ~ + 7 [ ,0~ _

[~-1~

_

~]

1

crGI /~+~/~]+......~_[3_1/1_~//] ]

d. Vd = L - ' ~ d + R i a - N ~ L O - iq d. vq = L --~Zq -~- R iq + Nr L O iq + Krn O.

2.3

Inputs and dynamics with delays

D e f i n i t i o n 3 ([9]). An ( i n d e p e n d e n t ) input u of a delay system E / k J-differential transcendence basis of 22/k.

is a

Tracking for 1r-flat Nonlinear Delay Systems

343

This definition m e a n s t h a t for any s y s t e m v a r i a b l e w E Z' there exists a relation of the t y p e

~ ( w , . . . , 5~w (~), u, . . . , 5~u (~)) = 0 with ~ a p o l y n o m i a l with coefficients in k. Thus, the t r a j e c t o r y of any s y s t e m variable is d e t e r m i n e d via a p o l y n o m i a l differential-difference e q u a t i o n with coefficients d e p e n d i n g on the p a s t t r a j e c t o r i e s of the i n p u t s on a finite t i m e interval. In other words, once those i n p u t t r a j e c t o r i e s have been fixed, the t r a j e c t o r i e s of all o t h e r system variables d e p e n d only on their p r o p e r initial conditions on an a p p r o p r i a t e finite interval. As in the delay-free case [13], the following result is now a simple o b s e r v a t i o n on the definitions of an i n d e p e n d e n t i n p u t a n d a 7r-flat o u t p u t respectively.

Proposition 1. A r-flat output of a re-flat delay system has the same cardinality (i.e., number of components) as an independent input. We can also directly see: 2. Given a :r-flat delay system ,V,/k with rr-flat output y, for any set {~1, . . . , ~,~ } of non-negative integers, the collection v = (vl, 9 vm) defined by vi = y~X,), i = 1 , . . . ,m, is an independent input o f - ~ / k . Proposition

D e f i n i t i o n 4 ([9]). A 5-differential field extension S / k ( u ) 6 where u is an i n d e p e n d e n t i n p u t of the delay s y s t e m S / k is called a (nonlinear) dynamics with delays. It is called :r-flat if S / k is 1r-flat.

2.4

Generalized

state representations

D e f i n i t i o n 5 ([9]). A weak (generalized) state of a (nonlinear) d y n a m i c s with delays •/k(u)6 is a (non-diff.) 5-transcendence basis x = ( x l , . . . , xn) of S / k ( u ) ~ . T h e state dimension is n = 5-tr d~ As a consequence of this definition, for every xi, i = 1 , . . . , n, there exists a difference e q u a t i o n of the form

Pi(~:i,... , ~ a x i , . . . ,~b~ri, x , . . . ,~Cx,... ,~dx) -~ 0, where the Pi, i = 1 , . . . , n, are p o l y n o m i a l s with coefficients in k(u)~. Explicitly writing down the dependence on u, one has the (generalized) weak state

representation F,(x,,...,Sbxi,x,...,Sdx,

u,...,Seu((~)) =0,

i=

i,...,n,

where the Fi, i = 1 , . . . ,n, are p o l y n o m i a l s w i t h coefficients in k and the p a r t i a l derivative OFi/O(Ss&i) ~ 0 for at least one s E N.

344 2.5

Joachim Rudolph and Hugues Mounier Brunovsk~-state

and generalized controller form

For 7r-flat systems a particular type of states can be introduced. L e m m a 1. Let E / k ( u ) 6 be a 7r-flat dynamics with delays, y = ( Y l , . . . , ym)

a 7r-flat output. Then there exist t q , . . . ,Kin E 1~+ in such a way that (with the convention y~-l) = 0) "~ = (Yx, Y l , ~

, y ~ l - - ] ' ) , y0, . . 9 , y~'rt*--]-))

(2)

is a weak (gener.) state of Z / k ( u ) 6 . Such an x is called a weak Brunovsk~state of Z / k { u ) 6 . The proof based on considerations of filtrations quite closely follows the lines of the one of the delay-free ease [7].

Remark 2. The xi s u m up to the state dimension:

m Ki Ei=I

= n.

R e n u m b e r i n g the c o m p o n e n t s of y allows a simplification of n o t a t i o n s : Unless all xi = 0, i = 1 , . . . , m , define rh in such a way t h a t ~i > 0, for i = 1 , . . . ,fn _< m,

~ = 0, for i = rh + 1 , . . . , m.

If all tq -- 0, let rh -- m: in this case x is void. We assume such a r e n u m b e r i n g when using rh in the sequel. W i t h this, the Brunovsk~-state is

For n o t a t i o n a l ease we also define I = { 1 , . . . , m} and lm = { 1 , . . . , m } . D e f i n i t i o n 6. T h e weak state representation with a B r u n o v s k ~ - s t a t e x of E / k ( u ) ~ is called a (generalized) controller form. It is of the form

x, =

{1,...

,,,.,},

O= Fj(SaJ'lx~,,... ,5~J"JJ:~j,x,...,SbJx, u , . . . ,5~Ju(~i)), 0 = F j ( 5 a J " y j , . . . , 5 " J " J y j , x , . . . ,SbJx, u , . . . ,SCJu(aJ)),

j E Im, j E I\Im.

D e f i n i t i o n 7. Let ~ / k ( u ) ~ be 7r-flat, x a (weak) Brunovsk~-state, a n d 5 u j, pj E 1~ , j = 1 , . . . ,m, the operator 5aj,, of the lowest degree actually occurring in the (generalized) controller form in front of k~j resp. yj. T h e n the pair (x, B u ) w i t h Bu = k ( S U ~ + l y ~ ) , . . . , 5u,,+ly~,~))~ is called the extended Brunovsk#-state corresponding to x. W i t h this, the controller form can be rewritten as :/', = xi+x,

i E { 1 . . . . . n } \ { ~ l , . . . ,~m},

O= Fs(5"Jk~s,Sa~'~k~j,... ,SaJ"s-lk~j,x,... ,SbSx, u , . . . ,5~iu(~J)), j E Im 0 = F3(5"iys,(faj'*yo . . . . . 5aJ"i-~ys,z . . . . . 8~ix, u . . . . . 5~iu("S)),

j ~ I\le,,

Tracking for r-flat Nonlinear Delay Systems

345

where for fixed j the degrees of the operators 6 d~,~ are higher than those of ~u,. To each extended Brunovsk~,-state (x, Bu) corresponds an operator /5 = lcm(~U~,... , ~ u ' ) , where lcm denotes the least common multiple.

Example 3. With the input u = (vd, Vq) one gets the dynamics with delays of the motor example as M/k(u)~. Using the d[-flat output y = (7/, ia) a weak Brunovsk:~-state x may be defined with x~ = r/, x2 = //, x3 = /?, x4 = 0 (3) and xs = ia. The corresponding (generalized) controller form is of the form xi=xi+a,

iE{1,2,3}

where al and a2 are constants. The extended Brunovsk~,-state is (x, Bu) , with B u = k(Sr/(4), j d i a ) 6 .

3

Quasi-static

feedback

of extended

Brunovsk2~-states

D e f i n i t i o n 8. Let 27/k(u)a be a w-fiat dynamics with delays and (x, Bu) an extended Brunovsk2?-state, p = lcm(6~'~,... ,Su.,) the corresponding operator. Then the extended input-state filtation H ~ = (H~)~e~. of 57/kk(u)a associated with (z, Bu) is the filtration of L:---6ur-,u~-, with the ~ U r -~ U/5-~-fieids Urx : k //~1 = (Bu(x)),su,r-'u~-~

//~ -- (Bu(x,u,/L,... ,u(~)))6u,r_~up_ ~

for r ~ - 2 , for r = - 1 , for r > O.

D e f i n i t i o n 9. Two dynamics with delays, Zlk(u)~ and ~lk(fi)6 , are called

equivalent by predictive quasi-static feedback of an extended BrunovskO-state (z, Bu) if (x, Bu) is an extended Brunovsk~,-state of both these dynamics and the corresponding extended input-state filtrations have bounded difference. If Z/k(u)6 and ~/k(fi),~ are equivalent by predictive quasi-static feedback of an extended Brunovsk3~-state, then there exist relations of the type o

=

O = ~i~'~(j) tu ,

x,...,

. ,,~ai~-bTr-~x,u, . . . .

. . . ,

,,SJ#-~Tr-fu (r~A) ,

where ~ / , ~ ' , i = 1 , . . . , m , j > O, are polynomials with coefficients in (Bu)6u,~-lu~-l. Notice that higher order derivatives of ~ m + l y ~ . ) m a y occur, which, in general, cannot be eliminated!

346

Joachim Rudolph and Hugues Mounier

L e m m a 2. Let S/k(u)~ and ~/k(fi)6 be two r-flat dynamics with delays

such that there exists an extended Brunovsk~-state of ,U/k(u)~ which is an extended BrunovskO-state of ~/k(fi)6 , too. The extended input-state filtrations of X',~u~r-aua-~ corresponding to this extended Brunovsk~-state have bounded difference. Sketch of the proof: The filtrations are good, because, for a large enough, x (i) E H i . As for the exhaustivity, observe that for any z E r there exists an r such that z e H~, because 2?~u~-~u#-, = k(y)6ur~up_~. P r o p o s i t i o n 3. A n-flat dynamics with delays ~/k(u)6 and n-flat output y is equivalent by predictive quasi-static feedback of an extended Brunovsk!jstate to the dynamics with delays ~?/k(v)6 with

gUsy~';i) = ~ " v j , j = 1 , . . . ,m. Sketch of the proof: The extended Brunovsk3)-state x of iY/k(u)6 is an extended Brunovsk~-state of ~/k(v)6. Furthermore, the extended input-state filtration of iY/k(v)6 defined with x (and v) and the one of S / k ( u ) z defined with x (and u) have bounded difference.

Example 4. The dynamics M/k(u)6 of the motor is equivalent by predictive quasi-static feedback of the extended Brunovsk~-state (x, By) to M/k(v)6 with the representation d

71(4) = vi,

-~ia = v2.

The inputs vi and v2 are related to u = (Vd, Vq) by

Vl : ~2X4 "4- ~l(X, ~X, t~2X) "4- al ~Vq

(3)

v2 = ~2(x,~x,~2x)+ a~va. 4

Tracking

control

for the

motor

with

flexible shaft

The equivalence of M/k(u)~ to M/k(v)~ can now be used for designing a feedback law for stable trajectory tracking. The relations (3) can easily be solved for u = (va, Vq): 1 [~-]vl

al 1

a2 The need for an inversion of 6, i.e., the prediction by rl, is obvious. Studying the relations in detail one observes that only g-aq(j) j = 0 , . . . , 3, occur.

Tracking for r-fiat Nonlinear Delay Systems

347

The stabilization is now simple. For, the injection of 3 j=0

d

V~ = "~id,r -- 3~2,0(id -- id,r) with reference trajectories r/r and id,r (see [24]) and with appropriate real positive gains )q,j, j = 0 , . . . , 3 , and A2,0 ensures an exponentially stable dynamics of the tracking error. A discrete time prediction of 6-1r/is simple here (cf. [23]): With Sk = to + ( k - 1)At, At = v l / N , k = 1 , . . . , N , and ~(~k+~)

= e

--

GI~

-~-

Z~t

(e - ~ '

^

~(sk) + - 1)(~(sk

- 2 ~ ) - 2 0 ( s k - ~) -

0

~ - T ~ ~ ( s k -- 2 T ) )

an estimate ~(t0 + r) of r/(t0 + v) is obtained in N iterations./,From this, the derivatives can be predicted recursively:

= 671(i) + ~

a(j_l)

_

G[~O ( j - l ~ ( j - 1 ) + 60(i-i)),

j = 1,2, 3.

All these prediction equations are derived from the last equation of the model given in Example 1.

Remark 3. One might believe a stabilizing feedback could be obtained by using an equivalence to (1 - 62)r/(4) = (1 - 6~)vl, d i d = v2 and injection of stabilizing feedback as above. However, this would not lead to an asymptotically stable behavior, because (1 - 62)( = 0 admits periodic solutions.

5

Mathematical

background

and

notations

The differential algebraic approach to nonlinear control systems has been introduced by M. Fliess more than ten years ago, and many introductory references are now available [7,8,10,11,27]. As indicated in [9], nonlinear systems with time delays may be treated along analogous lines by using differencedifferential algebra [2] (see also [1] for aspects concerning difference algebra). 5.1

D i f f e r e n c e - d i f f e r e n t i a l fields

A difference-differential ring R, ordinary w.r.t, differentiation, is a ring which is equipped with:

348

Joachim Rudolph and Hugues Mourtier

1. a single derivation, here d e n o t e d by ~ --~ at -. . . . . . , i.e., such t h a t for all a,b E R:

d di(ab) = ab + ab'

hER,

did ( a + b ) = h +

b

2. a finite set of transformations A = { ( f l , . . . , g , } , i.e., o f m o n o m o r p h i s m s 3 on R such t h a t for all a , b E R , i E { 1 , . . . , r } :

6i(ab)

=

6 (a + b) =

+ 6ib

These t r a n s f o r m a t i o n s c o m m u t e with one a n o t h e r a n d also with ~7" d An (ordinary) difference-differential field F is an ( o r d i n a r y ) differencedifferential ring which is a field. We also use the t e r m A-differential field for short. T h e set O* of formal power p r o d u c t s (including the identity) o f t h e transform a t i o n s (fi, i E A is called the set of difference o p e r a t o r s .

A A-differential field extension L / K consists in two A-differential fields K and L such t h a t K C L. All A-differential field extensions considered here are (A-differentially) finitely generated up to algebraic closure, a n d o f characteristic zero. T h e A-differential field generated over K by a finite set z = { z l , . . . , Zq} of elements of a A-differential field c o n t a i n i n g K is d e n o t e d as K(z)za, the difference field generated by z as K(z)z~. T h e algebraic closure of a field F is d e n o t e d as F . 5.2

Algebraic dependence

and transcendence

Let L / K be a A-differential field extension. A f a m i l y z = ( z i , . . . , Zq) o f elements of L is called A-differentially algebraically independent (resp. A algebraically independent) over K if the set {Ozi, 0 E O*, zi E z} has the following two properties: firstly, it consists o f distinct elements, in the sense t h a t Otzl = O~z2 implies both 01 = 0~ and zl = z2; and, secondly, it is differentially algebraically i n d e p e n d e n t (resp. algebraically i n d e p e n d e n t ) over K . T h e t e r m A-differentially algebraically dependent over K (resp. A-algebraically dependent over K ) is used in the o p p o s i t e case. A set of elements of L which is m a x i m a l with respect to inclusion a n d Adifferentially algebraically i n d e p e n d e n t (resp. A - a l g e b r a i c a l l y i n d e p e n d e n t ) over K is called a A-differential transcendence basis o f L / I f (resp. a A transcendence basis 4 o f L/K). T w o such bases have the s a m e c a r d i n a l i t y , As usual, the derivatives ( ~a) , a of any order are denoted as a (0 . 3 Following [9], we use monomorphisms, whereas R.M. Cohn [2] uses isomorphisms. 4 The fields are then difference fields in the sense of difference algebra. In this context, the terms transformal transcendence basis, transformal transcendence degree, etc. are commonly employed [1,2].

Tracking for rr-flat Nonlinear Delay Systems

349

which is called the At-differential transcendence degree (resp. A-transcendence degree) of L / K , denoted as At-diff tr d ~ (resp. as A - t r d~ If L / K is A-differentially algebraic its A-transcendence degree is finite. If At-diff tr d~ = q, the cardinality of z, then the extension K(z)z~/K is called

A-differentially purely transcendental. 5.3

Filtrations

Filtrations with respect to the differential structure of A-differential field extensions are defined as in the case without transformations, see [7,16,29]. This will be done now. The (non-differential) difference fields considered in the following are fields not equipped with a derivation but with the transformations in A; we thus also speak of A-fields for short. For a given subset A of a A-differential field d K, we denote the set {b 9 K ] qa 9 A, b = d a } as aTA. A (differential) filtration of a A-differential field extension L / K is a nondecreasing sequence s : = ( / : r ) , e z of (non-differential) At-fields L:~, algebraically closed, such that K C / : r C T a n d / : r C s for all r 9 2~. Such a filtration/: of L / K is exhaustive (in T) if U~ez/~r = T. It is discrete if/~, = K for every r 9 Z small enough; finite if up to algebraic closure all L:, are finitely generated A-field extensions of K ; good if for some r ' 9 E, s > r ' implies/:s+l = /~s(~/::~); d and excellent if it is both finite and good - cf. [16]. Two filtrations s and ~. have bounded (or finite) difference if there exists an integer r0, called the difference (of the filtrations), such that s C_ 1:~+~o and E~ C_ s for all r 9 Z. Having bounded difference s is an equivalence relation on the set of filtrations of L / K . The s y m m e t r y and the reflexivity are obvious. In order to show the transitivity, let be given three filtrations s s and L: of L / K such that for some r0, rl 9 Z and for all r 9 E,/2~ C s 1:~ C s and, moreover, s C s L:~ C s Then, for all r 9 Z, /:r C s and s C/:r+~o+~,-

Two filtrations of a A-differential field extension L / K which are discrete, excellent, and exhaustive have bounded difference. L e m m a 3 (cf. [7,16]).

P r o o f : Let be given two filtrations, L: and L:, satisfying the conditions of the lemma. The filtrations being discrete, s = s = K for r small. Moreover, from their exhaustivity in L together with the finiteness it follows that, for all r E Z, /::r C ~:sr+~ and / ~ C /Z~r+~ for some s~,~r E Z large enough, depending on r. For large r, because the filtrations are good, d~ L:~+l = L~(~ts ) C_ ~:~+'r(~7 r+,r) = ~ + , r + i

and ~:r+l

~

-~-~(~/Zr)d- C

5 Of course, the term "difference" here, is not related to the transformations defined on the field, i.e., to its difference-differential character.

350

Joachim Rudolph and Hugues Mounier

/ : r + ~ r ( ~as r + i r ) = s Therefore, for r large enough, the integers sr and sr do not depend on r. The b o u n d of the difference follows now by choosing r0 = s u p ( s t , st) for r large enough.

References 1. R. M. Cohn. Difference Algebra. Interscience, New York, 1965. 2. R. M. Cohn. A difference-differential basis theorem. Can. J. Math., XXI1:12241237, 1970. 3. E. Delaleau and M. Fliess. Algorithme de structure, filtrations et d4couplage. C. R. Acad. Sci. Paris Sgr. I Math., 315:101-106, 1992. 4. E. Delaleau and P. S. Pereira da Silva. Filtrations in feedback synthesis: Part I - Systems and feedbacks. Forum Math., 10:147-174, 1998. 5. E. Delaleau and P. S. Pereira da Silva. Filtrations in feedback synthesis: Part lI - Input-output decoupling and disturbance decoupling. Forum Math., 10:259275, 1998. 6. E. Delaleau and J. Rudolph. Decoupling and finearization by quasi-static feedback of generalized states. In Proc. 3rd European Control Conference, pages 1069--1074, 1995. 7. E. Delaleau and J. Rudolph. Control of flat systems by quasi-static feedback of generalized states. Internat. J. Control, 71:745-765, 1998. 8. M. Fliess. Automatique et corps diff~rentiels. Forum Math., 1:227-238, 1989. 9. M. Fliess. Some remarks on nonlinear input-output systems with delays, volume 122 of Lecture Notes in Control and Inform. Sci., pages 172-181. SpringerVerlag, Berlin, 1989. 10. M. Fliess. Generalized controller canonical forms for linear and nonlinear dynamics. IEEE Trans. Automat. Control, AC-35:994-1001, 1990. 11. M. Fliess and S. T. Glad. An algebraic approach to linear and nonlinear control, volume 14 of Progr. Systems Control Theory, pages 223-267. Birkhs Boston, 1993. 12. M. Fliess, J. LSvine, P. Martin, and P. Rouchon. Sur les systSmes non linSaires diff4rentiellement plats. C. R. Acad. Sci. Paris Sgr. I Math., 315:619--624, 1992. 13. M. Fliess, J. L~vine, P. Martin, and P. Rouchon. Flatness and defect of non-linear systems: Introductory theory and examples. Internat. J. Control, 61:1327-1361, 1995. 14. M. Fliess, J. LSvine, P. Martin, and P. Rouchon. A Lie-B/icklund approach to equivalence and flatness of nonlinear systems. IEEE Trans. Automat. Control, 44, 1999. 15. M. Fliess and H. Mounier. Controllability and observability of linear delay systems: an algebraic approach. COCV (Control, Optimization and Calculus of Variations), 3, 1998. (URL: http://www.emath.fr/COCV/). 16. J. Johnson. Differential dimension polynomials and a fundamental theorem on differential modules. Amer. J. Math., 91:239-248, 1969. 17. E.R. Kolchin. Differential Algebra and Algebraic Groups. Academic Press, New York, 1973. 18. J. L6vine. Are there new industrial perspectives in the control of mechanical systems? In P. M. Frank, editor, Advances in Control, pages 197-226, SpringerVerlag, 1999.

Tracking for r-flat Nonlinear Delay Systems

351

19. P. Martin, R. M. Murray, and P. Rouchon. Flat systems, In G. Bastin and M. Gevers, editors, Plenary Lectures and Mini-Courses, 4th European Control Conference, Brussels, Belgium, pages 211-264, 1997. 20. H. Mounier. Propridtds structurelles des systhmes lindaires d retards : aspects thdoriques et pratiques. Th~se de Doctorat, Universit~ Paris-Sud, Orsay, 1995. 21. H. Mounier, P. Rouchon, and J. Rudolph. Some examples of linear systems with delays. RAIRO--APII--JESA, 31:911-925, 1997. 22. H. Mounier and J. Rudolph. First steps towards flatness based control of a class of nonlinear chemical reactors with time delays. In Proc. 4th European Control Conference, Brussels, Belgium, paper 508, 1997. 23. H. Mounier and J. Rudolph. Flatness based control of nonlinear delay systems: A chemical reactor example. Internat. J. Control, 71:871-890, 1998. 24. H. Mounier and J. Rudolph. Load position tracking using a PM stepper motor with a flexible shaft: a ~-flat nonlinear delay system. In Proc. 5th European Control Conference, Karlsruhe, Germany, paper F1010-5, 1999. 25. 3. F. Ritt. Differential Algebra. American Mathematical Society, New York, 1950. 26. P. Rouchon and J. Rudolph. R6acteurs chimiques diff~rentiellement plats : planification et suivi de trajectoires, in Automatique et procddds chimiques, J.P. Corriou, editor, Herm6s, Paris, 2000. 27. J. Rudolph. Viewing input-output system equivalence from differential algebra. J. Math. Systems Estim. Control, 4:353-383, 1994. 28. J. Rudolph. Well-Formed Dynamics under Quasi-Static State Feedback, volume 32 of Banach Center Publ., pages 349-360. Banach Center, Warszawa, 1995. 29. J. Rudolph and S. El Asmi. Filtrations and Hilbert polynomials in control theory, In U. Helmke, R. Mennicken, and J. Saurer, editors, Systems and Net-

works: Mathematical Theory and Applications, Vol. II, Proc. Int. Symposium MTNS'93, Regensburg, Germany, August 2 - 6, 1993, pages 449-452, Akademie Verlag, 1994.

Neuro-genetic Robust Regulation D e s i g n for Nonlinear Parameter D e p e n d e n t S y s t e m s Giovanni L. Santosuosso Dipartimento di Ingegneria Elettronica Universit& di Roma Tor Vergata via di Tor Vergata 110 00133 Rome, Italy sant osuosso@ ing. uniroma2, it

A b s t r a c t . In this paper we consider nonlinear continuous time systems perturbed by unmodelled dynamics, and we address the problem of the robust control law synthesis via a neuro-genetic strategy. To this purpose, we introduce a class of global diffeomorphism in the state space, (DIMENS) to obtain a class of positive definite and proper parameter dependent Lyapunov functions. We implement DIMENS with LISP hierarchies of automatically reusable sub-programs, in the context of Genetic Programming, which avoid the "curse of dirnensionality", and are closed with respect to the operation of "genetic crossover".

1

Introduction

and

motivation

Consider a nonlinear p a r a m e t e r d e p e n d e n t system Z

= f(x,p) +#,(x,p)d(t)+g~(x,p)u

(1)

with s t a t e x G ~'~, control i n p u t u 6 ~'~, exogenous i n p u t d 6 ~ u , t h a t can be both time varying disturbances and terms due to imperfect s y s t e m modelling, and p G ~ ' , is a vector of constant known p a r a m e t e r s , which m a y represent references to be tracked. We assume f ( . , - ) , 91(', "), 9 2 ( ' , ' ) to be s m o o t h functions, and t h a t f ( 0 , p ) = 0, for all p E ~ . We will concentrate in the a u t o m a t i c synthesis of a "good" r o b u s t s t a t i c controller,

= l(.,p)

(2)

with l(-, .) s m o o t h function, when there is direct access to the s t a t e of the system. A classic approach to the p r o b l e m of the global control for a n o n l i n e a r system can be solved in the context of state feedback linearization techniques (see [5] for a review of the subject). This control s t r a t e g y consists in finding a static state feedback and a global c o o r d i n a t e change such t h a t the closed loop system in the new coordinates results either p a r t i a l l y or g l o b a l l y linear, and

354

G.L. Santosuosso

possibly controllable via a second state feedback. However, for m a n y systems, there may be a subset 8 P of the state space composed of singular points i.e.

II~(~,p)ll

$'P = {x, E ~n : lim g

.--+;r

= o~,vt}.

m

This static feedback law is implementable only on a subset of the state space. If in the system enter also exogenous inputs d(t), the system m a y me easily driven towards the "forbidden" region 8 7 ) rendering this control strategy hardly implementable in many practical situations. Motivated by this argument, we define a suitable cost function

z ( x , u , p ) = /\k(x,p)u/ h(x,p)

(3)

for the closed loop system, where h(O,p) = 0 for all d(t) E ~P, which is a measure of the distance of the system state and input from equilibrium. It is also reasonable to assume ibr all exogenous inputs to be norm bounded, and without loss of generality, we can also assume that IId(t)U < 1 for all t > 0. As a matter of fact, if Nd(t)ll < ca, then by setting d = c~d and

~l(x,t) = gl(x,t)cd, system & = f ( x , p ) + g l ( x , p ) d + the hypothesis that

el(t)

g2(x,p)u complies

with

< 1. We will also assume in the same way without

loss of generality that IlPll < 1. Therelbre, it is desirable to introduce this intbrmation in the control design strategy with a criterion to quantify the performance of a control law, measuring the m a x i m u m m a g n i t u d e of the cost function with respect to the time. To be more specific, we tbrmalize the attenuation performance index that will be considered.

Problem 1. Consider system (1) together with the cost function (3). Let c, x E /R + be positive scalars. Search, if existing, a function l(x,p) such that by setting u = l(x,p), 9 the corresponding closed loop system is I n p u t to State Stable,(see [11]) ISS) i.e. there exist suitable functions r .) E /eL; and q(.) E K: such that

+,,

(4)

tbr all t _> 0, and all Hx(0)]] _< c~, all HpOI_< 1. 9 there exist two positive reals 7 E /R +, 5 E ~ + 3' § c5 = ~, such that I[z(t)[I < 5 , tbr a l l t > 0 ; a n d lim IIz(t)[ I < % --

--

t--4

O0

--

In recent years there has been.considerable interest, in the control community, for the above performance index (see [10]). In this context it is straightforward to show that I S S performance can be detected and verified introducing Lyapunov like positive definite functions.

Neuro-Genetic Robust Regulation Design

355

P r o p o s i t i o n 1. Consider system S , and the cost function (3). Let ~ E ~+, 7 E ~+, and W : ~n x ~ -+ ~ , W(O,p) = O, be a smooth positive definite parameter dependent and proper function. For any positive real c E ~+, consider the level set

c s ( w , c , p ) = { x : w(x,p) < c}. Let S~(p) = s

c*(p)) where

t~(c~) -- {~ : I1~11< e~} c*(p) = max {1, ce[0,~)min {B(c~)C s

aS~(p) =

and Sl(p) = f-.S(W, 1,p);

{x:

w(z,p)

= 1}. Define

K(W, u, ~,p) = ~O W (y( X ,p) + g,(x,p)u) 1 ow ~ ~ w(~,p) +~ll-z~-g~( ,p)ll +

(5)

Assume there exists a static feedback u = l(x,p) such that K ( W , l ( x , d ) , x , d ) < 0 V x E S~(d), V p, [Ipll < 1.

(6)

Then the dynamics of the closed loop system (1) - (2), for all initial conditions x(O) E B(cz) evolve in the compact set S2T = U $~(p). Besides, when vp, Ilpll<1 t --+ oo, is asymptotically stable for d = 0 in the compact set S2T. Note also that if

g =

=

sup [[z(x,l(x,p),p)ll e & (p) Ilpll < 1 sup [Iz(x,l(x,p),p)[I

(7)

z e os~ (p)

Ilpll < 1

then IIz(t)ll < ~, for aUt > O; and lim IIz(t)N < ~. - -

- -

t

~

c

r

- -

Proof. The arguments above can be proved adapting to parameter's dependent systems the sufficient conditions implying the property of input to state stability (ISS) in [10]

Proposition above provides the tools for robust control design, in the presence of norm bounded disturbances. First, it is necessary to find a suitable functional optimization algorithm yielding as a result the feasible parameter dependent Lyapunov function and control input that minimizes the cost function. There is also a second task to comply with, involving the complexity of representation of the solution proposed. In fact, there are families

356

G.L. Santosuosso

of approximating functions in which the number of terms -for an acceptable approximation-, grow exponentially with the dimension of vector x. This happens for instance tbr polynomial approximations, gaussian neural networks, wavelets et cetera. This is the well known "curse of dimer, siolmlity" problem, as introduced by [2]. A possible strategy to overcome this undesired feature is to make use of reusable parts in function representation. A popular approach in the computation of positive definite Lyapunov functions, as solutions of solutions of Hamilton Jacobi Bellman equalities for fixed 7, is represented by [1]. This approach, however, has the drawbacks of considering a Galerkin polynomial approximation of V(x), incurring in the "curse of dimer, sionality", and the positive definitiveness of V(x) is not guaranteed. On a different perspective, [9] has shown the feasibility of computing 7/0o norms via genetic algorithms, without anyway specific reference to nonlinear inequalities above, following the research path outlined in [3]. Motivated by the previous arguments, in this note will locus our attention on the tbllowing issues. 9 We introduce a class of global diffeomorphism in the state space, which depend on the vector of parameters p, that we call "Differentiable Invertible Map Encoding Neural Scheme", (DIMENS) extending the research line traced in [12] and [13], where DIMENS where introduced in the context of 7/0o/s optimization, and without dependence on the vector p,and use DIMENS to obtain a class of positive definite and proper parameter dependent Lyapunov functions. 9 We construct a LISP package, in the context of Genetic Programming (see [7]), implementing DIMENS in LISP automatically defined programs, possessing two important features: first, they are composed of hierarchies of automatically reusable sub-programs, thus avoiding the "curse of dimel,sionality". Second, in the genetic algorithm, the elements of the population, i.e. couples of programs consisting of control input functions and Lyapunov functions are closed with respect to the operation of "genetic crossover". This means that in the genetic algorithm, the offsprings generated by two parent positive definite and proper parameter dependent Lyapunov functions are themselves positive definite and proper parameter dependent Lyapunov functions. Note this is crucial for the genetic algorithm to proceed. We remark that the algorithm presented can be potentially applied to any continuous time nonlinear system affected by constant known references and unmodelled disturbances, by training off-line a state feedback control law, and therefore has a very broad range of potential applications.

2

A Differentiable Scheme

Invertible

Map

Encoding

Neural

We will address the problem of the approximation of the Lyapunov function in Proposition 1 with feedforward neural networks. The choice of a neural net,

Neuro-Genetic Robust Regulation Design

357

in our case, is m o t i v a t e d with the well known p r o p e r t y of these m a p p i n g s of a p p r o x i m a t i n g functions with relatively few elements. We remark, however, t h a t in our case we deal with a p r o b l e m of constrained approximation, i.e. we a p p r o x i m a t e a function which has to be positive definite, this implies a special care in defining classes of neural networks t h a t a u t o m a t i c a l l y satisfy the constraints for any real p a r a m e t e r value. We will briefly recall now the essential features of the neural networks we will use hereafter. D e f i n i t i o n 1. A sigmoidal function of order C k, ~ : ~ -+ ~ , is a non constant, limited, m o n o t o n e increasing function whose derivatives are continuous up to order k-Th. T h e e l e m e n t a r y perceptron is described by the relation: f(xl,...,xn,a,

wl,...,wn,

O) ---- vw"

wixi - 0

,)

.

A layer of q e l e m e n t a r y perceptrons can be described with a c o m p a c t n o t a t i o n as

F(x,w)

= [fl(xl,...,xn,al,wl,1,...,wn,l,01),..,

f q ( x l , . . . , z,~, ap,

W l , q , . . . , W n , q , Oq)] T

where x : [ X l , . . . , xn], w : [al, w 1 , 1 , . . . , Wn,l, 01 . . . . aq, W l , q , . . . , W n , q , Oq], and finally, a multilayer feedforward neural net is a o b t a i n e d t h r o u g h the c o m b i n a t i o n of a number, say r, of layers of perceptrons, R ( x , (v) = Fr ( F r - ~( 9 . . F , ( x , wl) . . . ) , w , _ l ) w , ) . In order to verify the capabilities of neural networks, it is necessary to recall some notions of a p p r o x i m a t i o n theory. T h e o r e m i. (see [4]). Let ~-~D C ~n, SC C ~r~be compact sets containing the origin, and let C ( S D ) be the set of f u n c t i o n s F : SD ~ $ c continuous on S o . Let

{

.

: wi,k, ai, Oi E ~ , } 03 m_ U N = I O J N "

Set x = I x 1 , . . . , x,~], and let f2 be a subset o f O ( S D ) , defined in the f o l l o w i n g way:

Y2 = {F(x): [f~(x),...,fm(x)] :f1(x),...,fm(x)

E~).

358

G.L. Santosuosso

Then 12 is dense on C(SD), i.e. for all e E ~+ and all g(x) E C(SD) there exists a function f(x) E i2 such that [ a ( x ) - f ( ~ ) l < e for all x E SD We will introduce hereafter, classes of neural networks representing global time varying diffeomorphisms, for any real parameter value, which represent "elementary" mappings in the role of "bricks" to build a more general diffeomorphism. We can set the three classes of mappings: 9 Pseudorotations: Let O i ( x l , . . - X i - 1 ) , i = 2 , . . . n , be n - 1 s m o o t h scalar functions, such that 01r ( 0 , . . . 0) = 0. Then the m a p p i n g ~ = F(x), ~ E /R n, x E ~ n , is a pseudorotation if ~1 = xl, r =xi+~(Xl,...xi-x).

~=F(x):

We approximate pseudorotations with neural networks defined as

rn -

i--1

E;:I a,,ja (-Oi,i)).

where

W = [Wl,l,l(p),...,Wi,m,i-l(p),Oti,l(p)...Oi,m(p),Oi,l(p)...Oi,m(p)] is a vector whose entries are functions of the parameter p in (1). 9 Scaling: Consider the set of the functions r : ~1 __~ ~1 that are invertible, and such that r = 0. The m a p p i n g ~ = Z ( x , w), x E ~'~ is a scaling in the space ~n, if~i = r wi(p)), i = 1 , . . . n , where ~i ( Xi, di, oq,1. . . Oti,rn, Oi,l . . . Oi,rn) :

E?=I ~2,jO" ( J'O',3 ) ' Wl,,l, 0l" O' e ~}~, and

w~ = [d~(p), ~,l(p)... ~,,,(p), o~,: (p)... 0~,~ (p)] is a vector whose entries are functions of p, for i = 1 , . . . n . 9 Coordinate exchanges: Let I : A[[1,n] --~ AY[1,n] an invertible function. Then a coordinate exchange is defined as the invertible m a p p i n g that rearranges the order of the vector entries.

f ~r = xi(i),

I. ~i

= xi(i)

We introduce now a class of neural networks, in the framework of the definitions above, whose peculiar feature is that of representing global diffeomorphisms for any real parameter value.

Neuro-Genetic Robust Regulation Design

359

Definition 2. Differentiable Invertible Map Encoding Neural S c h e m e . Let II(p) be the set of all neural networks defined either by parameter dependent pseudorotations, scaling, or coordinate exchanges. Let ~r (P) be a subset of all parameter dependent diffeomorphisms on ~ n obtained through a number of r compositions of pseudorotations and scaling neural networks. e r ( p ) = F r ( F r _ I ( . . . FI(~, w~(p))..-), w~_~(p))~,(p)), Fj E H(p), j = 1,...r, w(p) = ( w l ( p ) . . . w r ( p ) ) .

Then a D I M E N S is defined as ~(p) = U~=lCr(p ). It is straightforward to verify the following. C o r o l l a r y 1. The elements of the class of functions ~(p), for any real value

of their parameters are global time-varying diffeomorphisms. 3

Neuro-genetic computation

suboptimal

robust

control

solution

Let qh(x,w(p)) = [r162 E ~n and w : ~ " --+ ~m, be a mapping belonging to the class DIMENS as outlined above. We decide to approximate the function W(x,p) in Proposition 1 with the neural network g g ( x , w(p)) = r w(p)) + . . . + r w(p)), following the research line in [12], [13]. This mapping will define a positive definite function, for all w(.). The control input l(x,p) will be also be approximated with a neural network, NL(x, p). A natural way to obtain a couple (NN(x, w(p)), NL(x, p)) representing a good solution to the problem addressed, is to adapt genetic programming (GP) tools as developed in [6]. Genetic programming starts with primordial ooze of randomly generated computer programs composed of the available programmatic ingredients, and breeds the population using the Darwinian principle of the survival of the fittest and an analog of the genetic principle of sexual recombination. An interesting GP feature described in [7] is the capability of evolving automatically defined functions (ADF),consisting in subroutines, or modules that are dynamically evolved during a run of a genetic program which may be called by a calling program -(e.g. main program), that is simultaneously being evolved. ADF functions are basically reusable parts of the whole program, that are continuously modified by the genetic algorithm. The essential features of the LISP program are: 9 A set P = {Pl,...P~}, of couples Pl -- (NN(x,wi(p)),NL(x,p)), i = 1 , . . . p , Pi E /,where P is the population of programs that undergo the Darwinian evolution. Figure 1 shows the structure of the LISP expressions representing Pl.

360

G.L. Santosuosso

* A probabilistic operator, .T : I ~-~ ~, representing the function generating the performance index of the neural net. This is a fitness function associated with ( N N ( x , wi(p)), N L ( x , p ) ) . Practically, a training set of points(xj,pj) E ~n • ~v, j = 0, 1 , . . . k is chosen, and for each fixed i it is computed expression (5), for all j = 0, 1,.../~. If the neural candidate satisfies the constraint represented by inequality (6), then .T turns out to be a an estimate of -~+--~-$~ 1 in (7), and if not, it is a scalar negative value, performance measure of the "positiveness" of HJ inequality. In the simplest case, if 7' denotes the training subset such that K ( N Y ( x , wi(p)), YL(~:,p)) > 0, then

( N N ( x , w,(p)), Y L ( x , p)) = - E

(8)

It" ( Y Y ( z j , w,(p)), YL(~:,p)).

xjE~

Note the size of the training set is the actual critical issue of all the procedure, because if it were too accurate, then ~- it would turn out to be an excessively time consuming operator, while if .7" were c o m p u t e d on a small training set, it would become unreliable. 9 A stochastic operator (..9 : { } ~-~ I, yielding new elements to be added at the population, in order to span all possible neural networks. 9 A deterministic operator ~ : I ~ --~ I, of crossover that combining two parent elements of the population, generates two new offsprings, with intermediate topolog.$_.__and weights. If, for instance, the two parents are (N N(x, (v), :7) and (N N(x, ~), ~) associated respectively with the global mappings F~(F~-I('"/#1 (x, @1)'" '), tb:~-l)@~) and then choosing the integers i and j, such that 1 < i < /5 and 1 _< j < i6, then the offsl~ringswill be ~'~ ( ' " Fi-I ( Fj " " F1 (x, wl) " " "), ~vj)@i-i) " "~bp- l) @p) and

F/~("" Fj--l(Pi''' Fl(ff', I1)1)'" "), ~)/)Wj--I)''" ~)~6--1)W~)9 9 A stochastic operator, M : I ~-+ I, whose task is that of r a n d o m l y m u t a t ing - in our case with a normal gaussian probability density distribution,the scalar parameters of the neural structures of the population. 9 A selection operator, S : I" --+ I x, that chooses A elements with higher performance on ~, elements of each of the populations. 9 A termination operator t on P, yielding a boolean value according to the fitness of the population in the evolution process. The structure of the algorithm is as follows:

9 Initialize P with the operator O. 9 White (not ,(P)) DO:( Generate new elements of P with the operators 0 and M .

Neuro-Genetic Robust Regulation Design

361

e, Combine couples of elements of P with the operator Tr generating offsprings. ~, Apply the 'fitness function" qS, to all the elements of P. t, Select the best elements of P with operator 8, for P to remain eonstant.) In the Picture below is shown the structure of an elememt of the population of individulas that undergo the genetic algorhitm.

Level 2. :oordinate exchanges.

ients

t r gu ms72~pS~'i~+m:(~:~i.: : : ' '" %" ' Figure 1 At level 0, the program representing the Lyapunov function is structured as a DIMENS global diffeomorphism, together with a program implementing the control input. The DIMENS network at level 1 is composed of automatically defined and reusable programs of level 2, representing pseudorotation and scaling operators. These programs in turn, use as arguments the variables of the state space, together with the subprograms at level 3, i.e. the coefficients of pseudorotation and scaling which are nonlinear functions of the parameter vector p. The automatically defined and reusable functions in level 3 are simple LISP function using the four usual arithmetic operators (with division

362

G.L. Santosuosso

substituted with a nonlinear smooth pseudo-division "%", to avoid division by zero.), together with a standard sigmoidal function. The LISP program implementing NL(x,p) at level 1 is composed with the same operators of the functions in level 3.

4

A simple

case study

The proposed algorithm has been tested on the simple system

= F(x,p) + Gld(t) + G2(p)u

(9)

where x E ~2, p is a constant parameter, d(t) E ~l is an exogneous input, u E ~1 is a control input,

r (o.s + and G1 = (0.10, 0.10) T. The genetic algorithm, with a population of 6000 individuals, a training set of 300 points to test each element of the population, has run for 100 generations. The Lisp code "chromosome" of the best element founded at the end of the program run, was coded as the list:

(R2(EXI(b'2(RI(S2(R1ARG))))))

(- (Cll (c13 x2))(+ (, 3.0 (c14 Xl)) ( - (. 0.51 X 1 ) ( C l l (+ (, P (ES X2)) X2))))) (- (* (ES (ES (ES (% P 50.0))))(- - 0.2 52.0)) P) (+ (, ( + - 9 0 . 0 ( E S ( (- - 0 . 3 7 P ) ( + 8 9 . 0 P ) ) ) ) - 0.85) (ES (+ (+ ( - 0 . 5 4 - 0.82)(% (ES P ) ( E S 0.22))) P))) (+ P - 1.2)(, - 0.18 P ) ( + 0.6 P ) ( - - 33.0 7.4) (% P 4 . 9 ) ( - P - 14.0)(- P - 0.81)(% P - 0.1) (+ P - 1.2)(, - 0.18 P ) ( + 0.6 P ) ( - - 33.0 7.4) ( - 5.6 P)(% P P)(* P P)(* P (* P P))(ES P) (6962522939688578914109890597588426383 903575518008848752539448078979035)).

(10)

Recall that in Lisp a list is represented as a sequence of symbols closed in parentheses, and that in the execution of a list representing mathematical operations the first element of the list represents the operation that has to performed. For instance, the operation :3 + 2 * 5 is represented in Lisp as the list (+ 3 (* 2 5)). Expression (10) is a list of 18 Lisp objects. The first one is the code of the global diffeomorphism associated with the Lyapunov functions, as introduced in previous section. The second Lisp object is the code of the control input to the system. The lists 3, 4 , . . . 17 represent Lisp code of scalar functions, more precisely automatically defined functions (see [7]), CI(P),...Cls(P), Q(P) : ~ ~ ~, 1 < j < 15. The first 10 of these

Neuro-Genetic Robust Regulation Design

363

functions, C l ( P ) , . . . C10(P) are involved in the determination of the parameters of the neural network representing the Lyapunov function. Since these coefficients are more than 10, each scalar coefficient is mapped into one of the 10 functions available, via the correspondence expressed by the last list in (10), i.e. the Lyapunov function parameters are Cg(P), C~(P), C6(P), C~(P) ..., et cetera. Each of the first 10 coefficients is in turn a function of the parameter p that appears in (9). The Last 5 automatically defined functions, Cll ( P ) , . . . Cls(P), are used in the construction of the mathematical expression of the control law. This time the function argument P is restricted to be allowed to be the parameter p in (9) but can be one of the system states xl, x~. By (10) setting Cl~(P) = 5 . 6 - P , C13(P) = P . P , C14(P) = P*P*P, and II('T1,;g2,P)

:

C11(C13(;g2))

+(0.51,

--

(3'

C14(.T1)

+ p 9

-

the state feedback control deduced by (10) is u(xl,x2,p) = f i ( x l , x 2 , p ) fi(0, 0, p). We illustrate the performance of system (9) with the state feedback control u(xl, x~,p) by considering a disturbance d(t) = 0.9 * [sin(t) + 0.2 * cos(10 * t)] acting on the system. We simulate the system trajectories, as shown in figure (2), for 0 < t < 50, initial conditions ;gl(0) : 1, x2(0) = 1, and p = 0.3. Notice that after a transient period of few seconds, the system trajectories are attracted in a neighborhood of the origin.

0

1 ~.

8

.......

o.~,o

10

i .........

......

15

20

!....................

.........

:..........

.........

25

30

35

40

45

50

i ..........

i ..........

.........

! ..........

i .........

! ........

: .........

i ..........

..........

i ..........

i ..........

i ........

-? 0

1 .... o.,,s

5

10

15

,

i

+

!. .........

!..........

;

........

o

i

i

.

-o.5

.......

-.t

Fig. xl(t),

.......

2.

.

i ......

: .........

0

5

Plot x2(t),

25

30

35

40

45

,

,

,

,

i

! ..........

! ..........

! ..........

i .........

i ........

i

i

.... i ...... i ...........

i .........

x~

20

i

. i ......

i .........

10

respectively

. i

i .......

.

......

i .......

i .........

i ..........

15

20

of the

for 0 < t < 50.

[

i. . . .

: ..........

25

.

i .......... !

. i. . . . . . .

i ..........

30

disturbance

i ......... i .........

. . . . . i. . . . . . .

i

.

50

i

:

. i.

i .........

35

d(t)

--

i.

-i ........

i .........

40

and

i ........

45

the

50

vector

state

components

364

5

G.L. Santosuosso

Conclusion

In this paper we aimed at combining the traditional approach control theory to obtain robust stabilizing laws via Lyapunov functions with symbolic regression algorithms developed in a different context, i.e. LISP genetic programming (see [7]). The advantages of this approach are in the capability to express in a compact formulation (via reusable parts of programs) the solutions of the problem addressed, and as it is intrinsic in genetic algorithms, to yield a "feasible" solutions, even for almost degenerate situations, in relatively few algorithm iterations.

References 1. Beard, R. Saridis, G. and G. Wen "Galerkin Approximation of the Generalized Hamilton Jacobi Equation" Automatica, Vol 33, N. 12, pp 2159-2177, 1997. 2. Duda, R. and P. Hart, "Pattern Classification and Sc. Analysis" New York, Wiley. 1973. 3. Dasgupta,D and D. R. Mc Gregor, "Nonstationary Function Optimization Using the Structured Genetic Algorithm" Parallel Problem Solving from Nature,Vol.2, pp. 145-154.1992 4. Hornik, K. M. Stinchcombe and H. White" Multilayer Feedforward Networks are Universal Approximators',Neural Networks, Vol.2, pp.359-366. 1989. 5. lsidori, A. Nonlinear Control Systems. Springer-Verlag, New York, 3-rd edition, 1995. 6. Koza J. R. Genetic programming: On the Programming of Computers by Means of Natural Selection. MIT Press, Cambridge, Massachussets, fourth edition, 1994. 7. Koza J. R. Genetic programming lI: Automatic Discovery of Reusable Programs. MIT Press, Cambridge, Massachussets, first edition, 1994. 8. Lu, Wei-Min. "Attenuation of Persistent s -Bounded Disturbances for Nonlinear Systems" . In Proceedings of the 34-th CDC Conf. New Orleans, Lousiana, pp. 829-834, December 1995. 9. Man, K., K. Tang, S. Kwong and W. A. Halang "Genetic Algorithms for Control and sygnal Processing".Advances in Industrial Control, Springer-Verlag, New York, 1997. 10. Khalil, H.K. Nonlinear Systems. Prentice Hall, Upper Saddle River, (N J), 2-nd edition, 1996. 11. Sontag, E. D. "On the Input to State Stability Property". European J. Control, N.1, 1995. 12. Santosuosso, G. L. "On the Structure and the Evolutionary Computation of 7too/s 1 Sub-Optimal Robust Controllers for Nonlinear Systems", in the Proceedings of the American Control Conference, Albuquerque, (NM), USA, 4-6 June, 1997. 13. Santosuosso, G. L. "Remarks on the Computation and Structure of the suboptimal Solutions of Generalized Hamilton Jacobi Inequalities." in the Proceedings of the European Control Conference, Bruxelles, Belgium, 1-4- July 1997.

Stability Criteria for Time-periodic Systems via High-order Averaging Techniques Andrey Sarychev Department of Mathematics Universidade de Aveiro 3810 Aveiro, Portugal ansar@mat, ua. pt

A b s t r a c t . We study stability and asymptotic stability for time-periodic systems described by ODE's, in particular for systems with fast oscillating parameters. Since the discovery of stabilizing effect of vibration in the reverse pendulum example, there was much study regarding stability of such systems and design of fast-oscillating stabilizing feedback laws. The approach we suggest is kind of high-order averaging procedure. It is based on a formalism of chronological calculus - a method of asymptotic analysis for flows generated by time-variant ODE. We apply this approach to study stability properties of linear and nonlinear systems. In particular we derive conditions of stability for reduced second-order linear ODE with periodic fast-oscillating coefficients, study the stability of reversed pendulum with fast oscillating suspension point, consider high-order averaging procedure for nonlinear systems under homogeneity assumptions.

1

Introduction and problem setting

We study averaging techniques and stability and stability properties for periodically time-varying systems x = f ( t , x). In particular we deal with fast oscillating systems of the form

=

(1)

where ~ > 0 is a small parameter and [(t, x, ~) is 1-periodic in t, f(t, 0, ~) = 0, Vt, e. For our objectives it suffices that f ( r , x, e) is 'integrable' with respect to t and Ck-smooth (with k sufficiently large) with respect to x. To simplify the presentation we will consider f ( . , . , .) to be continuous and f(n,., .) to be real-analytic. This will allow us to talk about converging series instead of asymptotic ones. Differential equations with fast oscillating terms have been studied and averaging techniques have been developed since 18th century. What for the stability issues for periodic differential equations then substantial contributions have been already made by A.M.Lyapunov; further contributions have been

3~

An~eyS~y~ev

made by O.Perron, N.Chetaev, I.Malkin, N.Erugin, E.Barbashin, R.Bellman, N.Krasovsky, J.LaSalle, J.Massera, L.Cesari among others. In control theory and mechanics the discovery of stabilizing effect of vibration in the reverse pendulum example inspired study of time varying feedback laws. This area became much more active after discovery of obstructions (e.g. R.Brockett's criterion) to time-invariant stabilizability and especially after the J.-M.Coron's ([6]) established general result on time-variant stabilizability for controllable nonlinear control systems. In this paper we introduce high-order averaging method for time-varying systems with the objective of studying their stability properties. This method is useful for treating critical and nearly critical cases and is based on some results of chronological calculus - technique of asymptotic expansions for the flows generated by time-varying nonlinear vector fields. This calculus has been developed by A.A.Agrachev and R.V.Gamkrelidze in 70's ([1]). We introduce the notion of complete averaging of a time-periodic system as a logarithm of the corresponding monodromy map (diffeomorphism). The notion of the logarithm of a flow appeared in [1]. In linear case it can be computed via ordinary matrix logarithm. In nonlinear case the logarithm of a diffeomorphism P is a time-invariant vector field A such that P is included into the flow o f A : e at It=l = PThe complete averaging can always be constructed as a formal Lie series; its terms are computed via Lie brackets of the vector fields f ( t , .) for distinct t's. Its truncations correspond to high-order averagings of the system. The computation of these high-order averagings for an arbitrary time-varying vector field is technically rather involved. In Section 2.2 we collect the main constructions of chronological algebras (see [1-3] for details) utilized for this computation and provide formulae for the first three terms of the expansion of the logarithm. We also derive asymptotic stability of equilibrium for time-periodic system is implied by its asymptotic stability for the complete averaging. The high-order averaging technique allows obtaining new (up to our knowledge) stability results for time-varying linear systems x : A ( ~ - l t ) x with fast oscillating coefficients. In particular we study stability issues for the reduced second-order linear differential equations with periodic fast-oscillating coefficients. We also study stability of reverse pendulum with oscillating point of suspension. Nonlinear case is usually much more complex for treatment by most part of the techniques. In the scope of our approach we manage to handle the nonlinear case similarly to the linear one. We illustrate our method by establishing a local asymptotic stability condition for an equilibrium of time-periodic nonlinear vector field. An obvious alternative to the introduced approach is the one which makes use of time-varying Lyapunov functions. One can find many interesting results of

Stability Criteria via High-Order Averaging Techniques

367

this kind in a series of publications by D.Aeyels, J.Peuteman, L.Moreau (see [12,10] and references therein). We shall study the problem of asymptotic stability a n d / o r of stability for the zero solution of such a system for small values of e > 0. We study three types of stability properties, which can be formulated respectively as: there exists ~0 > 0 such that for all positive e < e0 the zero solution is locally asymptotically stable (property Sa,); respectively is stable (property S0) ; respectively is unstable (property Su,).

2

2.1

Method: high-order averaging for time-periodic s y s t e m s - definition and computational tools. H i g h - o r d e r a v e r a g i n g s f o r systems with f a s t - o s c i l l a t i n g parameters

Although the problem has been set for nonlinear systems, we will first outline the method for linear systems with periodic coefficients. Obviously the notions of global and local asymptotic stability coincide for linear systems. Consider the system

(dx/dt =) k(t) = A(t/e)x(t),

(2)

where x E R n or C n, A(v) is R n• or Cn• function continuous and 1-periodic with respect to r, e > 0 is a small parameter. Below C - , C + are open left and right complex half-planes respectively. What one can say about asymptotic stability of such a system? An explicit but partial answer to this question is provided by the standard averaging result: if all the eigenvalues of the corresponding averaged matrix

f~ A(r)dr are located in C - ,

then the system (2) is asymptotically stable for all sufficiently small e > O. The question which persists is what happens in

the critical cases where some of the eigenvalues of this averaged matrix are located on the imaginary axis (for example vanish)? A complete but hardly verifiable stability criterion can be formulated in terms of monodromy matrix of the system. Introducing fast time variable 7- = t/e we arrive to the system dx/dT- = e A ( r ) x ( r ) with 1-periodic coefficients. Consider the corresponding matrix differential equation

d X / d r = e A ( r ) X ( r ) , X E R nx" or, X E C n•

(3)

and the monodromy matrix M~ -- X(1).

If all the eigenvalues of .~ul~ (multiplicators) are located in the interior of the unit circle, then the system is asymptotically stable. If at least one of these eigenvalues lies outside the unit circle, then the system is unstable.

368

Andrey Sarychev

The principal difficulty in utilizing this result is impossibility to c o m p u t e the spectra of the matrix Me. The only feasible idea is dealing with expansion of Me in a power series with respect to e. Instead of Mt one can deal with its logarithm 1 At. The corresponding stability criterion becomes:

the system (2) (with fixed e > O) is asymptotically stable if all the eigenvalues of the logarithm At ave located in C - and unstable if any of them is in C + . The matrix Ae admits a power series representation:

At = cA (1) + e~A (~) + " ' .

(4)

(It is easy to see that A(U = A 1 = f~ A(r)dv.) Obviously the logarithm Ae of the m o n o d r o m y matrix Mt provides a complete information about turnpike or averaged behaviour of the solution of the system (2). According to Floquet theorem this solution 'oscillates a r o u n d ' the trajectory e ta, of the system X = AeX, X(O) = I. Therefore At can be seen as 'complete averaging' of the system (3), while the truncations of its expansion can be seen as partial averagings of the system. A m a j o r problem remains c o m p u t a t i o n of the expansion for At. In the Section 2.2 we will proceed with this c o m p u t a t i o n in nonlinear setting. Let us consider now the nonlinear system

= f ( c - l t , x), where f ( r , x) is 1-periodic with respect to r. We assume f(t, 0) = 0, Vt. Putting 7" = t / s we come to the equation d x / d v = ~f(r, x). Let P [ be the corresponding flow - the family of diffeomorphisms satisfying the o p e r a t o r differential equation

aPe~dr = P / o ~f(v, .), pO = I. Sometimes we will denote such a flow by P [ =e--~p ./o f(O, .)dO and call it, following [1], right chronological exponential. The diffeomorphism Mt = P~ is a time-l-map or a monodromy map. Obviously p [ + t = Me o P [ and the origin is a fixed point of the m a p Me. The fact of time-invariant vector field AE being a logarithm of the diffeomorphism Me means t h a t Me is included in the flow e a't of this vector field: Me = e A't [t=l = cA'. Again we introduce the expansion (4) of At into the power series (where A (i) are now time-invariant vector fields) and try to proceed in a similar way as in the linear case. 1 The logarithm is a more suitable object. As it will be seen the notion can be generalized onto nonlinear case, where the monodromy map is a nonlinear diffeomorphism. In this latter case the 'logarithm' of this map, (if exists!) is a vector field - an element of a Lie algebra with natural linear structure suitable for expansions into series.

Stability Criteria via High-Order Averaging Techniques

369

There are additional difficulties on this way. In contrast to the linear case the logarithm Ac may fail to exist. Indeed as it is known the exponential map of the infinite-dimensional Lie algebra VectR" of real analytic vector fields into the infinite-dimensional Lie group of real analytic diffeomorphisms Dill'R" is not locally onto. This obstacle can be overcome. This exponential map turns out to be 'locally onto in a given direction':

for any 1-periodic real analytic vector field f(r, x), 3r > 0 such that for any positive ~ < ~o logarithm A, of the diffeomorphism ex-~pf~ el(O, .)dO exists. In any case one is able to construct a formal power series (4) for At, and even if this series diverges, still the exponentials of its truncations provide nice asymptotics for the diffeomorphism Mt (see [1]). Along this contribution we assume the logarithm At to exist; obviously the origin is an equilibrium point for the vector field At. Coming back to the stability issue we formulate the following proposition. P r o p o s i t i o n I The origin 0 is asymptotically stable for the system if it is

an asymptotically stable fixed point for the map Mr, i.e. VZI > 0 qJ > 0 such that p(x,O) < 6 ~ p(Mn(x),O) < ZI and p ( M n ( x ) , O ) --r O. Thus the question is reduced to the one of stability of the origin for the diffeomorphism Mr. Passing to the logarithm Ae of the diffeomorphism Mt or, just the same, to the complete averaging of the system we conclude. T h e o r e m 2 The equilibrium point 0 of the system Jc = f ( ~ - l t , x) is asymp-

totically stable if it is asymptotically stable equilibrium point for A~. As in linear case one can not conclude automatically property Sas for a nonlinear system basing on the stability properties of a given truncated series ~'~N=Ir We illustrate the phenomenon by the following example. Example. Consider a system

xl = - x l + x2sint/e, de2 = xl cost/e + x~. The first-order averaging of this system - the first term of the expansion of the corresponding logarithm - is the vector field A0) = -xlcg/Ox~ +x30/Ox2. The origin is an unstable equilibrium for A (1). Still in contrast to this fact the system is asymptotically stable, for sufficiently small e > 0. To prove the asymptotic stability we may consider the system as a perturbation of its linear part by the nonlinearity x~O/cgx2. We will establish in the next section (see Example 2) that for a sufficiently small e > 0 the linear part is asymptotically stable, i.e. spectrum of the corresponding averaging At is in the open left half-plane. Since the linear system is time-periodic, it is reducible and hence Lyapunov regular, i.e. the sum of the characteristic numbers equals to the average of the trace of its coefficient matrix. Then by a standard result of Lyapunov the nonlinear system has an asymptotically stable equilibrium at the origin.

370

2.2

Andrey Saxychev

C o m p u t a t i o n of the high-order averagings

The complete averaging can be computed via 'formal Lie series'. The computation of the logarithm of a flow has been accomplished in [1]); in [3] another method of its computation has been provided. This latter is based on so called chronological product of time-variant vector fields (see [2,3]).

Definition 3 For two time-varyin9 vector fields Xt(x), Yt(x), which are absolutely continuous with respect to t, the chronological product (X. *Y. ). equals: (X. 9 Y.)t =

/:[ 2:] X,,

Y, dr,

(5)

where [., .] stays for the Lie bracket of two vector fields. This is a nonassociative product, satisfying the chronological identity: X 9

(r*z)-Y

* ( X *Z) = (X , r - Y

, X)*Z.

(In their recent paper [8] M.Kawski and H.J.Sussmann defined (right) chronological algebra by means of identity u . (v. w) = (u- v). w + (v. u). w.) We will need another operation in the space of integrable functions:

(u,v)(t) = ( u . v ) ( t ) - ( v . u ) ( t ) = v ( t )

I'

u(~-)d,--u(t)

/0'

v(,-)d,-.

(6)

Using the chronological product (5) and the product (6) one is able to compute the expansion )-'~i~=:eiA (i) of the logarithm At of the diffeomorphism M, =~xp f2 eXodO. The first terms of this series (expressed in terms of Lie brackets) are

A:(X.) =

i

Xtdt,

,j0 [J0, Xt~dt2,Xtl ] dtl,

A2(X.) = -~

'Jo (I"

A a ( x ) = - 2 1 [A:(X.),Az(X.) ] +-3

ad s

Xt,dt2

)

(7)

Xtldtz.

Here adX Y = IX, Y], adiX Y = [X, adi-:X Y], i >_ 2. For a particular case of linear vector fields A l x and Asx the Lie bracket is a linear vector field equal to [A:x, Asx] = -[a:, As]x, where [A:,As] is the commutator of matrices A: and As. Hence in the linear case the terms A(i)(A.) in the expansion (4) are (if we suppress x) matrices; to compute them one has to substitute X. by A. in (7) and multiply Lie monomials of even order by ( - 1 ) . We will now outline the method of computation of A (i). Consider the right-chronological exponent ~ [ =ex--~ fo eXtdt. Obviously mc = ~ . Differentiating ~ with respect to e we obtain (see [3]): (ala:)~: = z[ o

~;,

Stability Criteria via High-Order Averaging Techniques

371

where the Z~" is time-varying vector field depending absolutely continuously on time r; for ~ = 0, Z D "" f o Xtdt. The vector field Z r has been called in [3] 'angular velocity' of the flow; Z [ satisfies the equation

(alOe)z: = ( z ; 9 z ; ) " =

,

dO,

(s)

where , is the chronological product introduced in the Definition 3. The equation (8) allows us to compute recursively the terms of the expansion

z: = z0+~=~

z ( k~ ) ( ~ / k . ) . ,

Various methods for computation of Z(t,) were introduced in [3]. For example considering the free chronological algebra with a generator A and a derivation D defined by: DA = A 9 A one can construct a sequence of polynomials with respect to the product .: pi(A) = A, pi+i(A) = Dpi(A), i > 1. Then one can compute Z(~) as

Z~k) : Pk+l

(/o

(9)

We yet need another construction to compute the terms of the expansion - for ---k the logarithm A~. To this end we differentiate the equality e A" = M, =exp f: r with respect to ~, obtaining (see [1] for the details):

f0

1 e r a d A . ~. OA o ea" = Z:o ~ p fo i ~x, at.

and consequently f0 i e ~ a d a , a- ~ OA = Z :

Consider the function f : er~dr = (e~ - 1)/~ and denote by r Then the latter equality is equivalent to the formal equation

OA

0----~-= r (ad (A~)) Z~.

= ~/(ef - 1).

(10)

To obtain the expansion for A~ from this latter equation we involve Taylor expansion r = :~-]i"=i Bi(~n/n!), where Bi are known to be Bernoulli numbers. The first Bernoulli numbers are: B0 = 1, Bi - - 1 / 2 , B~ = 1/6, B~k+i = 0, k > 0, /34 = - 1 / 3 0 , B6 = 1/42, Bs = - 1 / 3 0 . From (9), (10) we derive the expressions for A (i) and in particular the expressions (7).

372 3

3.1

Andrey Sarychev Applications: properties

high-order

averaging

and

stability

Asymptotic stability for linear systems- two examples

Here we provide two simple examples of linear systems, whose stability properties are not determined by the standard averaging but can be established on the basis of high-order averaging(s).

Example 1.Consider (a pair of) linear system(s) ~1 = - x l + xz sin(2~r/e)t, x2 = xl cos(2~r/e)t.

(11)

Its first-order averaging A(U = diag{-1, 0} is 'critical' - one of its eigenvalues vanishes. Computing A (2) we obtain A~ = r (1) + EZA(z) + o(~ z) = ( - ~ + ~2/47r ~ z / 2 ~

0

q:e2/4zr] + ~

Its trace equals - r while its determinant equals -4-g3/47rA-o(c3). The system (11) possesses property Sas provided that the sign ' + ' is chosen in the first equation and possesses property S,s otherwise. [] Example 2. The averaging of the system

Zl = - z a + ax2sin(27r/~)t, i~ = xl cos(2rr/e)t + Ex2/3

(12)

is unstable for each a : A (1) = diag{-1, e/3}. Computing the terms of orders and ~ in the corresponding logarithm we obtain cA(I) +~2A (2) = ( - e + O

b ~ / 4 7 r ~2/3 - a ~ / 4 r c )

conclude that the original system (12) possesses property Sas, provided that a > 4~r/3, and possesses property Sus, if a < 4~r/3. [] 3.2

Reduced second-order differential equation

Let us consider the reduced second-order equation + p(t/E)x = 0.

(13)

Here q(t) -- 0 and according to Liouville theorem zero solution of this equation is not asymptotically stable. From a standard averaging result we deduce that the stability property So holds if t5 > 0 and the property S,s holds if P 0. We deal with p(-) of small period but with no sign assumption. For nonnegative p(.) of period e the Lyapunov's criterion requires 0 < 6P < 4 for stability. It holds trivially for sufficiently small 9 > 0.

Stability Criteria via High-Order Averaging Techniques

373

Much more interesting is the case where P = O. Here three terms of the expansion for A~ are needed for coming to a conclusion about stability. Indeed _r

,

where p(U = f~ tp(t)dt. This matrix is trace-free while its determinant is negative. Still one can not conclude the instability property Sus, because the determinant has order of smallness O(r 4) and the term of order ~3 in the lower-left corner may affect the sign of the determinant. It does affect it in a radical way; as we will see in a minute the determinant of the matrix eA(U + r (~) + ~3A(3) is always positive provided that p(r) ~ 0. Indeed direct computation involving the formulae (7) gives us ~2p(1)

r

+s~A(2) +s3A(3) =

r + O(r

"~

_ea fol(fop(r)dr)2dt _ ~ p ( 1 ) ] "

(14)

Hence the determinant of Ae equals E4 (_(p(1))~ A- fo1( f t p(T)dT_)2dt ) q_o(g4). J

Integration by parts transforms pO) = f~ tp(t)dt into - f01 (f0 p(T)dr) dt (recall that P = fo1 p(t)dt = 0). By virtue of Cauchy-Schwarz inequality (p(U)~ =

p(r)dr)dt

<

p(v)dv)~dt

dr.

Hence the determinant of A~ is positive provided that p(r) ~ 0 and e > 0 is sufficiently small. We have established the following property. T h e o r e m 4 If P : O, then the reduced second order equation (13) with fast-

oscillating coefficient p(t/e) is not asymptotically stable. If p(v) ~ O, then the property So holds for it, i.e. the zero solution is stable if the rate of oscillation ~-1 is su.~iciently large. 3.3

Stabilization of equilibrium of reverse pendulum

As it is well known the upper position of a pendulum - the reverse pendulum - can be made stable if the suspension of the pendulum is subject to (sufficiently) fast harmonic oscillation (see [5]). Here we shall treat the case where the suspension is subject to fast oscillation of arbitrary form ~s(kt) and will derive conditions for it to stabilize the upper position of the pendulum. We assume (f > 0 to be a small and k > 0 to be a large parameter respectively, and s(r) to be 1-periodic C2-function. Small oscillations of the pendulum in a neighborhood of upper equilibrium point are described by the equation

= (w 2 + tik2~(kt)) x

(15)

374

Andrey Sarychev

where w is proper frequency of the pendulum. Without loss of generality we may assume '(0) = 0. Putting & = y and proceeding with time substitution r = kt we can rewrite the equation (15) as a system

dz/dT = (k-tA + 6kBr) z,

(16)

where

z=(x,y)T, A =

0 ' By =

~('r)

"

Invoking the variational formula of the chronological calculus (see [1] or [4]) we derive the formula (see the subsection 2.2 for the notation): exp

(k-IA + 6kBr) dr = e--+ xp

j0

C a d e o e6k f~ Brd'r ,

where Ca is defined by the first equality in (17). The second factor of the composition in the right-hand side is the identity. Indeed fot g(r)dr = ,(1) - ,(0) = 0 and hence f2 Brd~" = 0. What for the first factor then direct computation gives us

C:,: e"adfoB~d*k-lA: k-lA+6ad
(

Ca

:

-6,(,) k-liM 2 -- 62/r

k -1 ) .) 6,((9")

"

The matrix C. is 1-periodic in e; taking the corresponding monodromy ma9 ----+ 1 trix exp fo (Jade we obtain for its logarithm the expansion, which starts with the (first-order) averaging A (1) =

/0 C~,de =

_62k

(18)

What for the (first-order) rest term of the expansion of In ~xp f2 Coda, then it admits an estimate 0(6 2 + k-2), as 6 -+ O, k ---+oo. As far as the logarithm is a trace-free matrix, then the stability of the system is defined by (the sign of) its determinant 9 According to the above estimates the determinant equals -k-2aj 2 + 62

Jo

,2(~y)de 2c

0 ( 6 4 --~ k - 4 ) .

This leads us to stability condition for the reverse pendulum.

Stability Criteria via High-Order Averaging Techniques

375

T h e o r e m 5 For each ~ > 0 there exist Jo > 0, ko > 0 such that the equilibrium of the reverse pendulum is stable provided that 0 < J < Jo, k > ko and j2 fo1 ~2(r)d r > kw-rg + e and is unstable provided that 0 < J < Jo, k > ko and 0) 2

3.4

High-order averaging and stability for time-varying nonlinear

systems Let us show how the technique works in periodic nonlinear non-fast-oscillating case.

Assume X ( t , x ) to be a nonlinear 1-periodic time-varying vector field in R n, continuous together with all its partial derivatives with respect to xi's;

X(t, O) = O, Vt. We will need to introduce some homogeneity. To this end let us consider an n-tuple r = ( r i , . . . , rn) >_ 0 and the dilation

~'~

: R n -~ Rn : J,(xl,...,~,)

= (~r,~,...,~r.~,),

E > o.

We define the weight of a monomial vector field x~ 1 ... x(~"O/Oxa as - r E + ~]k=i akrk. For the sake of brevity let us say that vector field is of weight _> s if all the monomials in its Taylor expansion with respect to x are of weights _> s (see [3] for 'more invariant' definition). We assume X ( t , x) to be of weight _> s for some positive s. Let us take Fourier expansion X ~(x) + )-]~nr ( X n (x) cos 21rnt + y n (x) sin 21rnt) of X (t, x). Obviously X ~ = f~ X ( t , x)dt coincides with the first-order averaging of X and all the vector fields X n, y n are of weights _> s. In addition we assume X ~ to be of weight >_ 2s. This includes the case X ~ = 0 (arbitrary weight can be assigned to vanishing vector field). Let us represent X i, i >_ O, y i , i > 1, as X i = ~ i + . . . , y i = ~ i + . . ", where )~0 is homogeneous of weight 2s, ~ i , I?i are homogeneous of weight s and the rest terms are of bigger weights. 6 If the equilibrium is locally asymptotically stable for the timeinvariant homogeneous vector field

Theorem

oo

X~

+ Z(47rn)-i[xn,r']

(19)

n----1

then it is asymptotically stable for the system Jc = X (t, x). R e m a r k . This result generalizes the one P.Morin in [9]. T h e y proved that if X ~ is and X i , y i are homogeneous of weight s totically stable for the system x = X ~

obtained by R.M'Closkey and homogeneous of weight 2s > 0 > 0, then the origin is asymp+ X i (x) cos 2rrt + y i(x) sin 2~rt,

376

Andrey Sarychev

provided that it is asymptotically stable for the time invariant system = X~ + (4~r)-I[X1,Y1]. We outline the proof of the theorem 6. Take the time-1 m a p for X(t,x). Calculating the first two terms of the expansion of its logarithm (the logarithm m a y not exist, but the expansion exists as a formal series!) we obtain A (1) = X~ A (2) = ~-~=l(4~rn)-l[Xn,Yn]. Denote by ,~(2) = )~0 + ~-~=1 ( 4 : r n ) - l [ ~ n , ~n]. Then A = A (1) + 2 (2) + - . . , where the omitted terms are of weights > 2s. Therefore the homogeneous vector field A (1) +.~(2), appearing in (19), is principal part of A. If the origin is asymptotically stable for (19) and A exists (i.e. the respective formal series converges) then the origin is asymptotically stable for A by virtue of Massera-Hermes theorem ([7]) and hence is asymptotically stable for X(t, x). If A only exists as a formal series still we are able to prove that the vector field (19) determines the asymptotics and stability of the trajectories of the system ~ : X(t, x).

References 1. Agrachev A.A., Gamkrelidze R.V[1978] Exponential Representation of Flows and Chronological Calculus. Matematich. Sbornik, 107:467-532. English transl, in: Math. USSR Sbornik, 35:727-785 2. Agrachev A.A., Gamkrelidze R.V.[1979] Chronological Algebras and Nonstationary Vector Fields. Journal Soviet Mathematics, 17:1650-1675 3. Agrachev A.A., Gamkrelidze R.V., Sarychev A.V.[1989] Local Invariants of Smooth Control Systems. Acta Applicandae Mathematicae, 14:191-237 4. Agrachev A.A., Sarychev A.V.[1986] On reduction of smooth system linear in control, Matematich. Sbornik,130:lS-34. English transl, in: Math. USSR Sbornik, 58:15-30 5. Arnold V.I.[1978] Mathematical Methods of Classical Mechanics, SpringerVerlag, Heidelberg 6. Coron J.-M.[1992] Global asymptotic stabilization for controllable systems without drift. Math.Control Signals Systems, 5:292-315 7. Coron J.-M.[1999] On the stabilization of some nonlinear control systems: results, tools, and applications, in: Clarke F.H., Stern R.J. [Eds.], Nonlinear Analysis, Differential Equations and Control, Kluwer Academic Publishers, 307-367 8. Kawski M., Sussmann H.J.[1997] Noncommutative power series and formal Lie-algebraic techniques in nonlinear control theory, in: U.Helmke, D.Prs Wolters, E.Zerz [Eds.] Systems and Linear Algebra, Teubner, Stuttgart, 111128 9. M'Closkey R.,Morin P.[1998] Time-varying homogeneous feedback: design tools for the exponential stabilization of systems with drift, Internat. J. of Control, 71:837-869 10. Moreau L., Ayels D.[1999] Stabilization by means of periodic output feedback, in: Proc. 38th IEEE Conf. Decision Control, Phoenix, USA, Dec. 1999,108-109 11. Morin P., Pomet J.B., Samson C.[1999] Design of homogeneous time-varying stabilizing control laws for driftless controllable systems via oscillatory approx-

Stability Criteria via High-Order Averaging Techniques

377

imations of Lie brackets in closed-loop, SIAM J.Control and Optimization, 38:22-49 12. Peuteman J., Aeyels D.[1999] Averaging results and the study of uniform asymptotic stability of homogeneous differential equations that are not fast time-varying, SIAM J. on Control and Optimization, 37:997-1010. 13. Sarychev A.V.[2000] High-order Averaging and Stability of Time-Varying Systems, Preprint 7/2000/M, International School of Advanced Studies, Trieste, ltaly,23pp.

Control of Nonlinear Descriptor Systems, A Computer Algebra Based Approach K u r t Schlacher 1 '~ a n d A n d r e a s K u g i 1 1 Johannes Kepler University of Linz Linz A-4040, Austria kurt @regpro. mechat ronik, uni-i inz. ac. a t Christian Doppler Laboratory for Automatic Control of Mechatronic Systems in Steel Industries Linz A-4040, Austria A b s t r a c t . Many problems in mathematical modeling of lumped parameter systems lead to sets of mixed ordinary differential and algebraic equations. A natural generalization are so called descriptor systems or sets of implicit ordinary differential equations, which are linear in the derivatives. This contribution deals with the geometric control of descriptor systems. Based on the presented geometric framework using the mathematical language of Pfaffian systems, we derive a canonical form of a descriptor system under some mild rank conditions. This form is equivalent to an explicit system, whenever some integrability conditions are met. This approach allows us to extend the well known concepts of accessibility, observability, equivalence by static feedback, etc., to the class of descriptor systems. The Euler-Lagrange and Hamilton-Jacobi equations for optimal control problems with descriptor systems are also derivable from this canonical form similar to the case of explicit control systems. In addition, this approach offers computer algebra based algorithms, which permit to apply the presented methods efficiently to real world problems.

1

Introduction

In the m a t h e m a t i c a l m o d e l i n g of l u m p e d p a r a m e t e r p a r a m e t e r s y s t e m s it has turned out t h a t the D A E - s y s t e m (differential algebraic equation) a p p r o a c h [1] is a very n a t u r a l one. A D A E - s y s t e m , also called descriptor s y s t e m , is a set of implicit o r d i n a r y differential equations, which are linear in t h e derivatives, such t h a t the relations e~(w)~b i = f ~ ( w , v )

,

tb i = ~-~w d i ,

i,a=

1,...,n

(1)

are met with a singular m a t r i x [e~]. w E R n denotes the d e s c r i p t o r s t a t e , v E R m is the descriptor i n p u t of the s y s t e m a n d t E R denotes the i n d e p e n d e n t variable. A special t y p e of a d e s c r i p t o r system is given by zba ' - - f a " (w,v) , 0 - - I t " (w,v) ae = 1 , . . . , n ~ , a, = nw + 1,... ,n,o + ns 9

(2)

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Kurt Schlacher and Andreas Kugi

T h i s s y s t e m is also called a descriptor s y s t e m in s e m i - i m p l i c i t form. T h i s t y p e arises e.g., if one can split an explicit control s y s t e m ~:a. _ f a . (Xae,Xa.,U) ,

lim ex a ' = f a . ( x a . , x~,., u)

~'--,4 0 a e ~

1,...

, n e ,

a s = ne + 1 , . . .

,n~:

into the so called slow and fast d y n a m i c s . It is worth m e n t i o n i n g t h a t nearly all s i m u l a t i o n p r o g r a m s for electric circuits, electric power systems or electronics are based on DAEs. Also m a n y p r o g r a m s for m u l t i - b o d y systems use this a p p r o a c h . R o u g h l y speaking, whenever syst e m s with ports, like p o r t controlled H a m i l t o n i a n s y s t e m s with d i s s i p a t i o n [13] are connected, then the result is a D A E - s y s t e m . Also o b j e c t o r i e n t e d m o d e l i n g leads in a n a t u r a l way to this t y p e of systems. In c o n t r a s t to an explicit control s y s t e m ~

= fa (x,u) ,

xa = dxa

,

a = 1,... ,n

(3)

with the s t a t e x E R n and the i n p u t v E R m, s i m p l e c o u n t e r - e x a m p l e s prove t h a t neither w of (1) is the usual s t a t e nor v of (1) the i n p u t of (3). In [9], [10] it was d e m o n s t r a t e d t h a t the d e s c r i p t o r a p p r o a c h becomes easier, if we give up the s e p a r a t i o n into w and v and m e r g e t h e m into z = (w, v) such t h a t w and v are considered on the s a m e level. In the t h i r d sectiont of this c o n t r i b u tion we show t h a t one can t r a n s f o r m a s y s t e m of the t y p e (1) to a s y s t e m of the t y p e (3) in principle, if some m i l d rank c o n d i t i o n s and some i n t e g r a b i l i t y conditions involving the functions e~ are fulfilled. T h e m a i n p o i n t is t h a t only the existence of this transform is required, b u t we need not to p e r f o r m the t r a n s f o r m itself. T h e fourth section is devoted to the a p p l i c a t i o n in g e o m e t r i c control, where we will show, how one can e x t e n d the well known concepts of accessibility, observability, s t a t i c feedback-equivalence, etc. from explicit control s y s t e m s to descriptor s y s t e m s in a s t r a i g h t f o r w a r d m a n n e r . T h e fifth section deals with o p t i m a l control p r o b l e m s , where the E u l e r - L a g r a n g e and the H a m i l t o n - J a c o b i equations for a general v a r i a t i o n a l p r o b l e m w i t h descriptor systems are derived. T h e sixth section presents the a p p l i c a t i o n of the p r o p o s e d approach to a simple e x a m p l e , a chemical reactor. T h r o u g h o u t this contribution, we consider only generic problems, because we are interested in m e t h o d s based on c o m p u t e r a l g e b r a to derive a l g o r i t h m s for the tests concerning accessibility, observability, s t a t i c feedback-equivalence, etc., but we do not investigate degenerate p r o b l e m s . Therefore, we a s s u m e t h a t all functions have sufficiently enough continuous derivatives t h a t t h e d i s t r i b u t i o n s or c o d i s t r i b u t i o n are regular a n d t h a t the o p t i m i z a t i o n p r o b l e m s are s t r o n g l y non-degenerate. We use the t e n s o r - n o t a t i o n for vector-fields a n d forms (e.g., [3]) and a p p l y Einstein's convention for the s u m s (see e.g. [2]) to keep the formulas short and readable. F u r t h e r m o r e , the range of an index will always be suppressed, whenever it is clear from the context.

Control of Nonlinear Descriptor Systems

2

381

R e m a r k s on C o m p u t e r A l g e b r a S y s t e m s

Low order descriptor systems look simple, it seems that all the calculations can be done by hand. This picture changes dramatically, if "higher" order systems are considered. Also an exclusively numerical approach is not straight forward because of the index problem. It is well known that higher index problems cannot be solved reliably by numerical methods alone [1]. Computer algebra systems offer a way out, because one can let them do all the laborious algebraic manipulations and they can work reliably on high index problems at least in the neighborhood of generic points. Computer algebra systems have historically evolved in several stages [14]. MACSYMA, SCRATCHPAD, REDUCE and MUMATH, available since the late 1960s, belong to the first generation. The second generation with MAPLE and MATHEMATICA made computer algebra applications popular in applied mathematics and engineering. Currently, the third generation with AXIOM, MAGMA or MUPAD is on the market. Of course, this enumeration cannot be complete, since many specialized products are available in addition. Often, problems in symbolic computing are NP-complete [14]. According to our knowledge, user assistance is the best help to make these difficult problems manageable. E.g., programs should be written in a way that rules for simplifiers can be added or deleted by the user, because the decision tree must be cut as early as possible. Factors of 10 to 100 for the execution time can easily be gained or lost. Nevertheless, the solver for nonlinear equations, ODEs and PDEs have a high standard, which is often beyond the capabilities of the majority of the even well-trained engineers. At present, there are only a few packages for problems in nonlinear control available [5], but the situation improves fast. E.g., the package [6] can deal with many problems in geometric control for explicit systems.

3

A Canonical Form

Although the generalized state w and the generalized input v seem to be similar to the state x and the input u of an explicit control system, there exist important differences. Therefore, we do not distinguish between w and v and combine them to z = (w, v) and investigate the system n 7"(z) ki = r n " ' ( z )

,

( ~ = 1 . . . . ,n~,

i= 1,...,nz

(4)

(see, e.g., [11]) for the linear case. The relations n ~ " = e~~e ,

n~,Ole = O ,

r n ~e = f - e ( z )

for i,c~e = 1 , . . . ,n, i ~ ----n + l , . . . , n + m ---- n z show that (1) is a special case of (4). From now on, t E R denotes the independent variable, z E R n" = .A4

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Kurt Schlacher and Andreas Kugi

are the dependent variables and ~ = R x.h4 denotes the total space, where the coordinates (t, z) are used. The first jet bundle of,~ with coordinates (t, z, k) is specified by J,~. Furthermore, we assume that (4) defines a submanifold of J g , at least locally. The prolongation of a section (t, z) = a (t) of ~: to a section of J s is denoted by (t, z, k) = a (1) (t) with ~i = O t ~ i . To avoid superfluous complications, we assume from now on that there exists at least one section a of s whose prolongation ~r(D is a solution of (4). Now, two observations are critical for the following. If there exists a nontrivial solution of the equations :~,.nT" = O,

(5)

i = 1 , . . . , nz

then (4) contains at least one additional constraint )~a. m~" = O. Introducing the total time derivative dr, dt -- ~iai , we see that the equation dt ()~a.m a ' ) ---- 0 is a consequence of (5). We call such a system ill-posed, because it contains algebraic constraints for the dependent variables. A system of the type (4) is called well-posed, if no hidden algebraic constraints for the dependent variables exist. Before we present an algorithm, which transforms an ill-posed system to a well-posed one, we look at the second observation. Let us assume that (4) is well-posed and has the special form nTzk ' = m

~x ,

dt~o a" = 0 ,

a~=l,...,n~

as = nx + 1,...

(6) ,ne ,

where the equations dt~o'~" = 0 follow from the pure algebraic constraints ~ ~ = 0. If there exist functions ~o"-, ~o~', a~=, azx , Z~ ~7. (f~, with the Kronecker-symbol J ~ : and/3~, 7x = 1 , . . . , n~ defined on ~: such that a'~: dt~o z " + a g : d t ! o " " = n (~" i: i

(7)

is met, with functionally independent functions ~0i, then there exists an invertible m a p ~0, (x, s, u) = ~0 (z),

u a"-n'=~o

au ,

C~u=ne+l,...,nz

with ne = n , + n , such that (6) together with (8) can be rewritten as

~'-~

= 0.

(9)

Control of Nonlinear Descriptor Systems with the simple constraints s a ' - n ~

383

= O.

For the derivation of an algorithm, which transforms an ill-posed system to a well-posed one, it is advantageous to rewrite the system (4) as a Pfaffian system P = ( { n ~ . ~ d z i - m ~ d t , d~o~' }, {~o~" }), , a~ = 1 , . . . ,n~ d~oa' , as = na~ 4- 1 , . . . ,he 9

n]~dz ' - ma~dt

(10)

Here, d denotes the exterior derivative operator. The solutions of (10) are constraint to the submanifold N = {(t,z) E ~: [ ~ a ' ( z ) = 0}. A section a (t) = (t, z (t)) of ~:, which lies in A/', meets the relations ~oa" (a) = 0, a* (d~o~') = 0, where a* denotes the pullback. Roughly speaking, we use the fact that the variables ~i enter the equations (4) in a linear way. T h e Pfaffian system P defines a submodule of T*E. In contrast to the forms, which span the module, the module itself is unambiguously defined. From now on, P is used synonymously for this module. Now, we restrict our considerations to generic points and we assume that the relation A d ~ ,c'~ r 0

(11)

as well as

(A

,, A (,r

- ,,,~

#0

(,2)

are met in addition, where A denotes the exterior product of forms. We call the system adjusted, if it fulfills these requirements for a generic point. Adjusting the system needs two steps. In the first step, we eliminate a minimal number of functions q0~' such that (11) is met. In the second step, we eliminate a minimal number of further forms to meet (12). It is easy to see that any system of the type (10) can be adjusted under some mild rank conditions. Now, the following algorithm will perform the required transformation. 1) Start with p0 = ( { n ~ x d z i - m ~ d t } , { }). 2) Adjust P~ to derive p k + l and determine n , , n~ for P~. 3) If ( A n ~ z d z i) A ( A d ~ ~') :~ 0 then stop. 4) Otherwise the system contains new constraints of the type (5). Add the constraints to the system and goto 2. This algorithm generates a sequence of Pfafiian systems p k , k = 1 , . . . ,1 constrained to manifolds N k such t h a t the n u m b e r n * is strictly decreasing and

N O ~ N I . . . ~ 77 l is fulfilled. Obviously, the last system p t is well-posed, since the conditions of (6) are met by the associated descriptor system. It is worth mentioning that

384

Kurt Schlacher and Andreas Kugi

the condition (7) can be checked easily, if we consider the Pfaffian system (10), because the Theorem of Frobenius (e.g. [3]) states, t h a t the relations d ( n T x d z i) = 0 m o d

{ n T ~ d z i , d ~ o a~ }

(13)

must be met. Furthermore, the 1-forms w a, w a" = n ~ . ' d z '

,

w a" = d ~ a" ,

w"" = d ~ o " "

(14)

together with dt form a basis B* = {dr, w ~ } of T*E. T h e canonical dual basis B of 7-$ will be denoted from now on by S = {Ot,Ox,Oa,,Oa,}

(15)

.

Of further interest are special descriptor systems like (2) in semi-implicit form.

k s' = f " ' (z) , 0 = f ~ " (z) ,

a' = l,...,n' c~" = n' + 1 , . . . , n , .

(16)

This type of systems meets the integrability conditions (7). Since the proposed algorithm preserves the integrability conditions, the final system (6) fulfills these conditions in a trivial manner and their fulfillment need not to be checked.

4

Geometric

Control

We consider a descriptor system of the type (4), which is transformable by (8) to the state space form (9) or equivalently (5) permits only the trivial solution and (7) is met. This property allows us to call the system (4) accessible, observable, linearizeable by static feedback, etc., iff this is true for the associated state space model. See, e.g., [4], [7] or [8] for the definitions and theorems in the case of explicit control systems. Now, the direct approach is to transform (4) into (9) and to perform the tests with (9). The disadvantage of this approach is that one has to determine the transform, which involves the solution of several PDEs. A nicer way is to do all the calculations directly with the original system in canonical form. Therefore, we consider the Pfaffian system P , P = ( { n ~ x d z i - maxdt,d~o ~. }, { ~ ' } ) a~: = 1 , . . . , n x , a s = n x + l , . . . , n e ,

(17)

see (10), which meets (13) and (14), in the following subsections. The solutions of (17) are confined to the manifold ,N', Af = {(t,z) E $ I ~ ' ( z )

= O} .

(18)

Control of Nonlinear Descriptor Systems

385

To complete the system, we add the o u t p u t y a , , y~y = c a ' ( z )

,

a~=l,...,n~,

(19)

and assume without loss of generality that

Ady ", #0 ~y

is met. The following subsections will show that the form (17) allows a straightforward implementation of the tests concerning accessibility, observability, etc. in a computer algebra system.

4.1

Input-to-Output

Linearization

Roughly speaking, the well known input-to-output linearization [4], [7] requires the derivation of several time derivatives of the o u t p u t functions yav as well as the test, if the functions y~y depend on u au. Because of (14) we get a,

~x

= haxm

a,

a,

dt+hauw

hay

a~

mod

and we see that (20)

O,~.c '~" = O~uJdc ~"

with the field 0~ u from (15) is met. The symbol J denotes the interior product of a vector-field and a form. If furthermore O~,uc ~', -- 0 is fulfilled, then we get dtca'=h~rn

(21)

~x

in addition. Obviously, (20) and (21) provide all necessary operations to determine the relative (vector) degree of an output (19). Nevertheless, the implementation of the control law requires the fixing of the input u ~".

4.2

Accessibility

The accessibility test uses a slightly modified version of the well known test for explicit systems based on the derived flags of a Pfaflian system (see, e.g., [8]). Let P be a Pfaffian system generated by the set of one-forms {~/}, then the first derived flag is given by pl__{O~Ep,

d~i=OmodZ(P)}

,

386

Kurt Schlacher and Andreas Kugi

where Z (P) denotes the exterior ideal generated by P . The ith-derived system is given by P i = ( P i - 1 ) 1 with p 0 = p . T h e system p N with p g : p g + t is called the b o t t o m system, it is the largest integrable subsystem of P . Let us denote the b o t t o m system of (17) by PA. Now, it is well known, t h a t an explicit control system ka* = fa= (x, u) is accessible, iff the b o t t o m system of P = {dx ~* - f ~ * d t } is trivial or p g = {0} is met (see, e.g., [8]). These facts lead immediately to the following theorem [10], [9]. T h e o r e m 1. The system (17) or equivalently (9) is locally (strong) accessible in a neighborhood U (z) r Af (see (18)) of a generic point z, iff the relation PA = ( { d ~ a ' } , {~oa'}) is met there. It is worth mentioning that this test requires only the derivation of the canonical form. Furthermore, integrability needs to be fulfilled only on H (z) N Af.

4.3

Observabillty

We consider the system (17) with the output (19) and regard u a- as (arbitrary) functions of the time t. Since we have to face the problem t h a t the input is not known, we add the 1-forms d~aa" - ~i~" (t) dt to (10) to overcome this problem and denote the extended system by Re = ( { n f ~ d z ' - m ~ d t , d ~ ' , d ~

~" - u ~ " d t } , ( ~ ' } )

.

(22)

Let us introduce the Pfaffian system O :- {dc% - ha~dt} with dc ~" - h % d t = O m o d P e . We define Po as the smallest integrable Pfaffian system, which contains O, or O C_ Po is met [9], [10]. It is worth mentioning that Po is well defined, furthermore, one can construct Po with the following algorithm. 1) Start w i t h P t = { d c o ' - C ~ ~dt} = O, c o = c ~,, c 1 = h ~,, 3~, = 1. 2) If Pk = {dci~l - c i~" d t } , i = l , . . , j a r isintegrable, t h e n P k = Po. 3) Otherwise find a minimal number of functions c k" such t h a t dPa = 0 m o d P k + l ,

Or,

Pk+l = Pk U { d e ; ' - Ck+idt }

Cry

with dc k" - %+1dr = 0 m o d P e is met.

4) S e t k = k + l a n d j % = j % + l9 ,

. a, lfdCk_l-c ka y d . r E Pk, then goto 2.

Roughly speaking, this algorithm calculates several time derivatives of the output y % . In addition this observation leads to the following result [9], [10].

Control of Nonlinear Descriptor Systems

387

T h e o r e m 2. The system (17) with the output (I9) or equivalently (9) is locally observable in a neighborhood 11 (z) N Af (see (18}) of a generic point z, iff the relation Po = P is met there. Again, this test requires only the derivation of the canonical form and integrability needs to be fulfilled only on L/(z) 0 Af. 4.4

Transformation to AI-form

Before we consider the descriptor case, it is worth mentioning that any explicit control system ~o~ = y ~ (~, u)

(23)

can be transformed to Al-form by

see [12]. Therefore, we restrict ourselves to transformations, which do not increase the number of state variables, and ask, when does a static feedback

u a u - : h a"

(x,v)

with the new input v exist such that f (x, h) is affine in v ? The following theorem gives the answer [9]. T h e o r e m 3. The system (23) is static feedback equivalent to an AI-system, iff the condition a a , a # , E span { a a , f }

is met. The proof of this theorem follows from the fact, that a m-dimensional manifold B C R", m < n is affine, iff (Vvi) (vj) E span{vi} is met with the Euclidean connection (see e.g. [2])

~vi~i = cOkvidxk | ai for all vi E T B . Since the system (17) is transformable into an explicit control system, we h ave d

=

^

for suitable forms r #~ r

- mY'd,) +

+

^ dt

and get

Oa.Jd (n/~dz i) = (O~.]r

7~) dt m o d {(cJ~ - mW~dt),co ~" } ,

388

Kurt Schlacher and Andreas Kugi

as well as

Oa.Jd (n~=dz i - rna=dt) = - g a ~ d t m o d {(w "yx - m~=dt) , w ~" } for some functions ga,. From (6), (7), (8) and (9) follows

aOtX therefore we are able to determine the first order derivatives for the Theorem 3 up to a transformation by the regular matrix ]a~:~. If we apply the same procedure like above for the virtual system

n ~ ' d z ' - ga=dt , then we are able to determine the second order derivatives for the Theorem 3 up to a transformation by ] a $ : [ . This allows us to perform the required test. r

L

4.5

Input-to-State

--

J

Linearization

The subsequent investigations are based on the following well known fact for explicit control systems. Let us consider the associated Pfaflian system P = {dx ax - f a x d t } , then P is static feedback equivalent to a linear system, iff the following conditions for the derived flags are satisfied (see, e.g., [8]). The b o t t o m system is trivial or p N __ {0} is met and the extended systems { p k , dt}, k = 0 , . . . , N - 1 are integrable. The extension to systems of the type (17) is straightforward and we get the following theorem [10], [9]. T h e o r e m 4. The system (17) or equivalently (9) is static feedback equiva-

lent to a linear system, i]f the derived flags meet the following conditions. The bottom system p N is given by p g = ( { d ~ a , } , {So(~,}) and the systems { P k , d t } , k = 0 , . . . , Y - 1 are integrable on 14 (z) N.Af (see (18)). Again, this test requires only the derivation of the canonical form.

5

Optimal

Control

To start with we consider the system (4) and use the shortcuts ga,, g"~ = n~"~ ~ - m"~ .

(25)

Let a = (t, z) denote a section of E and r (or) the functional r (~) =

l (~, (t)) d t . 1

(26)

Control of Nonlinear Descriptor Systems

389

We look for the determining equations of a section q, whose prolongation ~(1) meets (25) and minimizes (26). To complete the problem, we have to add suitable boundary conditions. For the sake of simplicity we consider fixed terminal points ~ (tl), a (t2), since we are interested only in the derivation of the determining equations. The standard approach to this variational problem starts with the set of all I-dimensional point transforms T : C -+ s such that the independent coordinate t as well as the terminal points remain unchanged. Roughly speaking, such a m a p Tr transforms a solution of (25) into another one. In addition, the minimizing section a must fulfill the inequality ~b (a) _~ ~b (~0r (~)). Now, it is well known that the determining equations for the extremal solution cr can be derived from the conditions given by v (1) (g"o) = 0,

v (tdt) = 0

(27)

where v denotes all vector-fields v E 1)s C TC,

= v%,

,

i v i = 0~T[~__0

,

(28)

which are induced by the point transforms ~, and v (1) their prolongations to the 1st-jet space given by

~(1) =

o_v(,)

07"

r=0

= v%,

+ d,V%,

(29)

We will not follow this approach, but we will study equivalently the Pfaifian system (17) assuming that the system (25) is well-posed. Now, one can show [3] that the equations (27) are equivalent to the set wc=ldt+A~O ~ ,

vJdwc = 0

modcr,

(30)

which depends only on the vector-fields v of (28), but does not depend on their prolongations vO) (see (29)) any more. The 1-form wc is defined on Z = g x R n. with local coordinates (t, z, A). The notation 0 m o d a means that the relations must be fulfilled only on the submanifold a ([t~,t2]) C Z. Still, there remains the problem to find the equations, which determine the optimal solution. From now on, we call these equations the Euler-Lagrange equations of the variational problem. Before we present the solution for the class of descriptor systems we look back to the case of explicit control systems.

5.1

Explicit Control Systems

Let us discuss the variational problem for the explicit control system x~'~ = f ' ~ ( x , u )

,

P= {f~

(31)

390

Kurt Schlacher and Andreas Kugi

with a~ = 1 , . . . ,ne, x E R n" = 2d, u E R m = U and a given 1-form 1 (z) dt. Setting A4 = X x / / with coordinates z = (x, u), we see t h a t w c of (30) together with dwc is given by wc = (1 + A ~ . f t ` ' ) dt - A,~.dx '~* dwc = (dl + At`.dft`-) A dt - dAt`. A (dx a" - l t ` ' d t )

(a2)

in this special case. A short calculation shows t h a t B* = { d t , w a } , wt`'=dx ~,

w ~" = d u a " - " " ,

w ~ =dAt`~_n.

with a~x = nz + 1 , . . . ,n~ + n~ form a basis of T ' Z , whose canonical dual basis of T Z is denoted by B = { at, a ~ . , aau, 8t`~ }. It is well known t h a t the Euler-Lagrange-equations follow directly from the pullback of the forms a a . J d w c -- at,. (l + A # . I #-) dt + dAa. Ot`~ ]dwc at`. (l + A # . f # ' ) at 0~, ]dwc = f t ` - d t - dx ~-

(33)

by the section a : t --+ (t, x (t), u (t), A (t)) of Z. It is worth mentioning t h a t (33) is a descriptor system. If the Hessian [cqt` cq#, (l + A#~J#-)] is regular, then (33) is well-posed, and one can determine fi -- fi (t, x, A). Otherwise one has to apply the algorithm of the previous section to transform (33) into a well-posed system. Let Y denote a submanifold of Z such t h a t the relations dtAAwa~50,

c9~u(l+Aa.f ~')=0

(34)

t`x

are m e t and the restriction of ~ c to y , denoted by w Cy ' fulfills the relation dwcy = O. Let ~ : Z --+ y denote the projection and s be a section (t, x, A (t, x)) with fro s = i, then the relations s*dwcy = ds*wcy = 0 are met. Therefore, there exists a function V = V (t, x) at least locally such that the equation -dV=s*wc

(35)

is fulfilled for Ad~ = Aa.x (t,x). Equation (35) together with (34) is nothing else than the Hamilton-Jacobi equation OtV+(l+Aa.fc'z)=O,

Oa.V=Ac,~

of this special variational problem.

Control of Nonlinear Descriptor Systems 5.2

391

D e s c r i p t o r Systems

In the case of descriptor systems (17), we have to face the problem that neither the input u is explicitly given nor a suitable basis B*of T * Z for the derivation of the variational fields v is known. On the other hand, if we know the basis B* together with the canonical dual basis B, then the derivation of the Euler-Lagrange-equations of the variational problem (25), (26) is a straightforward problem. From now on we assume that the descriptor system under consideration is well-posed. Analogously to (32) we introduce the form ~oc together with d w c , w e = (1 + A~m~x) dt - A , ~ n ~ z d z i - A~.d~ ~' d w c = (dl + Aaxdrn a* + dAazm a*) A dt ^

-

.zd ( n T " d z ' )

- dA

(36) . ^ de

.

LFrom (14) we see that the required basis of T * Z is given by B* = {dt,w'~}, ta'~ = n~. =dz ' ,

w a" = d ~ '~. ,

A

waz = dA~ . . . .

w '~" = d~oa"

A

wa. = d A a x _ n . _ n ~

x = n~ + n~ + 1 , . . . , n~ + n e . The canonical w i t h a ~x = n z + l , . . . , n , + n ~ , % dual basis of T Z is denoted by B = {Or, 0a~, 0,~,, 0au, 0a~, 0ax }. Repeating the same procedure like for the system (31), we get the system Oa~ldwc = 0,~] (dl + ) ~ d r n z~) dt + dad. - ) ~ O ~ ] d ( n ~ i ' d z ' ) O a . J d w c = O~.J (dl + A , ~ d m " ) d t + dA~,. - A,~ Oa.Jd (nZi'dz i) O..Jdwc

O..j(dl+)~a~dmZ~)dt_)~Z~O~.j(dn~dzi)\

"

(37)

Oa~Jdwc = m ' ~ d t _ n~.~dz '

0,~]dwc = - d ~ ~' . The Euler-Lagrange equations of this problem follow from the pullback by the section 0" = (t, x (t) ,u (t), A (t)). Of course, the relations (37) must hold only on the submanifold 2r (see (17)). In general, (37) is a complicated descriptor system. If (7) or equivalently (8) is met in addition, then we get

/

0a.Jd {n~i'dz i

= -r

O,.Jd (niZ~dz i

= O..]m'Y*r

+ O~.]m~r

(38)

mod { (w"~z - - r n ~ dt), w~' } from the relations (24) together with the last two relations of (37). The combination of (37) and (38) gives the final set of the Euler-Lagrange-equations

392

Kurt Schlacher and Andreas Kugi

for this variational problem m a ' d t - nT~dz i , 0a.J ( d / + A#=w#x) dt + dA~. + A#xr d~ a" , 0a.J ( d / + Aa~w z ' ) dt + dAa. + AZ~r " 0a.] (dl + AZ wZ* ) d r , w as = dma* - m~-r .

(39)

Furthermore, from (38) it follows that

0o.j0o.j

0,

0o.j0o ja

:0

or

r

= r

= 0 mod {(w 7" - m T ' d t ) , w ~" , at}

is met. Therefore, there exist functions h~:, h~: such that rr

= h~: dt t. m~ {(w'Y" - rn~*dt) ' w ~ " dt} h~:dt

is fulfilled or (39) corresponds to a system of explicit ODEs in A. Furthermore, if the matrix [0a. (0Z. j (dl + A~ ~ ' ) ) ] is regular, then the system (39) is well-posed. Again, one can project from Z to the submanifold y , where the relations d t A A w ~ ' A A w~"

r

Oo,.J(d/+AZ ~ z ' ) =0

(40)

are met. Like above, the restriction dwcy of d w c to Y vanishes there. Analogously, we get from (40) the equations OtV+l+Aa

m a" = 0 , Aa~ = O a ~ V , O,~.V = O , ~o~'" = 0

As, = O a . V

(41)

because of d V = OtVdt + Oa~ Vw a" + O~. Vw ~" + Oa. Vw a"

and (36). Of course, (41) is nothing else than the Hamilton-Jacobi equation of the variational problem. It is worth mentioning that the Euler-Lagrange equations and the Hamilton-Jacobi equations for a non-linear H2- or Hooproblem can be derived in a straightforward manner, if the function l is replaced by a suitable objective function and additional information for the descriptor state w and descriptor input v (see (1)) is available. 6

A Chemical

Reactor

We consider the simple example of a chemical reactor [1] O=ts = IQ (To - T ) + K ~ R - K3 (T - T c ) 0 = R-- KaCe -K4/T

(42)

Control of Nonlinear Descriptor Systems

393

which describes a first-order isomerization reaction. The system (42) is of the type of (16). The symbols Co and To are the known feed reactant concentration and feed temperature. C and T are the corresponding quantities in the product. R is the reaction rate per unit volume, the actuator signal T c is the temperature of the cooling medium and K i , i = 1, 2, 3, 4 are constants. The associated Pfaffian system P in well-posed form is given by P = ({e~,O1,ds},{s}) 01 = d C - ( K t ( C 0 - C ) - R ) d t

(43)

02 = d T - ( g l (To - T) + K 2 R - Ka ( T - T c ) ) dt s = R - KaCe - K ' / T .

A basis B* of (43) is given by (dr, dC, dT, as, d T c ) and its canonical dual basis is denoted by B = {cgt, c9c, OT, c9,, OTc }. To check local accessibility we determine the sequence of derived flags and get pO = p ,

pt

=

({e,,ds},{s})

,

Pa = ({ds},{s})

.

(44)

According to Theorem 1, the system (43) is locally strong accessible. Let us introduce the system Pe, Pe = ({81,02,ds, d T c - fi ( t ) d t } , {s})

to check, if (43) is locally observable. Now, we construct P o by the help of the proposed algorithm and get the sequence Pi= {dci-l-cidt}

,

co=C,

cl = K t ( C o - C ) - R

for i = 1 , . . . , 4. Unfortunately, the expressions for the forms are too big to be presented here. Since Po and P span the same submodule, the system (43) is observable because of Theorem 2. Applying Theorem 3, we are able to show that the system (42) is transformable to an AI-system. Finally, the combination of the sequence (44) with Theorem 4 tells us that the system is static feedback equivalent to a linear one. Now, we present the Euler-Lagrange- and Hamilton-Jacobi-equations for this chemical reactor. Let us choose a function I for the objective function (26). From w c (see (36)), w e = ldt + A~I ((K1 (Co - C) - R) dt - dC) +Aa} ((K1 (To - T ) + K 2 R - K3 ( T - T c ) ) dt - dT) +Aald ( R - K a C e - K 4 / T )

(45)

394

Kurt Schlacher and Andreas Kugi

we obtain the equations

OcJdwc = d ) ~ + (o-P-cl- KlAn, + (h'~A~ - A~, - o~l) K3e - K ' / T ) dt

arJd~c

d ~ + ( ( ~ t - ~,_ = K~X~) k~K~-2c~-K'/T

O,Jd~c = ( ~ l :

Xo; -/~'2,X,,:) dt +

(46) The combination of (43) and (46) form the sets of Euler-Lagrange equations for this variational problem. Since dwc y vanishes on the submanifold y , where the relations

dt A d T A dC A ds r O , OTcI + K3)~a$ = 0 are met, the Hamilton-Jacobi equation follows directly from (41) and we get 0t V + l + A~I (K1 (Co - C) - R) +Aa~ (K1 (To - T) + K2R - Ks ( T - Tc)) = 0 Aa~ = (~c V ,

Aa~ = OT V ,

OToV=O,

Aa, = O, V

s=O.

Of course, this set must be fulfilled only on the submanifold s -- 0.

7

Conclusion

This contribution has shown that there is no essential difference in geometric control and the calculus of variations for explicit control systems and descriptor systems, which are transformable to explicit systems in principle. Based on the presented geometric framework using the mathematical language of Pfaffian systems, we are able to derive a canonical form of a descriptor system under some mild rank conditions. This form is equivalent to an explicit system, whenever some integrability conditions are met. This approach allows us to extend the well known concepts of accessibility, observability, equivalence by static feedback, etc., from the class of explicit systems to the class of implicit systems in a straightforward manner. In addition, we are able to derive the Euler-Lagrange and Hamilton-Jacobi equations of a general optimal control problem from this canonical form. It is worth mentioning that only algebraic manipulations are needed to perform all required manipulations and tests. Of course, they can be done by any good computer algebra system. Finally, the example of a chemical reactor has shown the feasibility of the proposed approach.

Control of Nonlinear Descriptor Systems

395

References 1. Brenan K.E., Campbell S.L. and Petzold L.R. (1996) Numerical Solution of Initial-Value Problems in Ditterential Algebraic Equations, SIAM, New York 2. Choquet-Bruhat Y., D~Witt-Morette C. (1991) Analysis, Manifolds and Physics. North Holland, Amsterdam 3. GriIfiths P.A. (1983) Exterior Differential Systems and the Calculus of Variations. Birkh~nser Verlag, Boston, Basel, Stuttgart 4. Isidori A. (1995) Nonlinear Control Systems. Springer Verlag, N e w York 5. Kugi A., Schlacher K., Novak R. (1999) Symbolic Computation for the Analysis and Synthesis of Nonlinear Control Systems, In: Konrad A., Brebbia C.A. (Eds.), Software for Electrical Engineering, Analysis and Design IV, 255-264 6. Kugi A., Schlacher K., Novak R. (1999) Software Package: Nonlinear AiIineInput Systems, In Maple Application Center, Control Systems 7. Nijmeijer H., van der Schaft A.J. (1996) Nonlinear Dynamical Control Systems. Springer Verlag, N e w York 8. Sastry S. (1999) Nonlinear Systems. Springer Verlag, New York 9. Schlacher K., Haas W. and Kugi. A. (1999) Ein Vorschlag ffireine Normalform von Deskriptorsystemen. Z A M M , Angew. Math. Mech. 79, 21-24 10. Schlacher K., Kugi A. and Haas W. (1998) Geometric control of a class of nonlinear descriptor systems. N O L C O S , Enschede, 387-392 11. Willems J.C. (1997) O n interconnections, control, and feedback. IEEE Trans. on Automatic Control 42, 326-339 12. van der Schaft A.J. (1984) Linearization and input-output decoupling for general nonlinear systems. System & Control Letters 5 13. van der Schaft A.J. (2000) L~-Gain and Passivity Techniques in Nonlinear Control. Springer Verlag, London, Berlin, Heidelberg 14. yon zur Gathen J. and Gerhard J. (1999) Modern Computer Algebra. Cambridge University Press, Cambridge

Vibrational Control of Singularly Perturbed Systems Klaus R. Schneider Weierstrass Institute for Applied Analysis and Stochastics MohrenstraBe 39 D-10117 Berlin, Germany schneider@wias-berlin, de

Abstract. We extend the theory of vibrational stabilizability to systems with fast and slow variables. The mathematical tools for establishing corresponding results are the persistence theory of normally hyperbolic invariant manifolds, the averaging theory and appropriate transformations. At the same time we introduce modified concepts of vibrational stabilizahility compared with the 'classical' definitions.

1

Introduction

Vibrational control is an open-loop control strategy to modify the dynamical properties of a system by introducing fast oscillations with small amplitude into the system under consideration [10]. Compared with feedback or feedforward control, this method is in some sense unconventional since it does not need online-measurements of states, outputs and disturbances. A wellknown example for vibrational control is the inverted pendulum that can be stabilized by vertically oscillating the pendulum pin at a sufficiently high frequency and small amplitude. The corresponding mathematical model reads dxl dt

~ x2~

dx~ dt = [cl - aw2c3 sin r

(1.1) sin xl - c~x2,

where xl is the angular displacement measured from the inverted equilibrium position, x~ is the angular velocity, Cl, c~, c3 are positive physical constants, a is the amplitude and w the frequency of the applied vibration. From the representation (1.1) it follows that the applied control can be viewed as a variation of the parameter cl. If we horizontally oscillate the pendulum pin of the inverted pendulum, then we get the system

398

Klaus R. Schneider

dXl

dt dx2 dt

= X2~

(1.2) = Cl s i n x l - - a w 2 c 3 s i n w t c o s x 1 -- c 2 x 2.

Here, the applied control cannot be viewed as a parameter oscillation, and the origin is not more an equilibrium point. If we introduce the notation E := l/w,

a = ae,

(1.3)

then system (1.1) can be written in the form dxl dt

= X2,

dx2 dt

=

cl sin

xl

-

cux~

c3a

+

sin

(~ )

(1.4) sin xl.

It is well-known [1,10,14] that the coordinates Xl and x2 of (1.4) can be stabilized near xl = x2 = 0 for sufficiently small ~ and a s > 2cl/c~ (that is the frequency w and the amplitude a are sufficiently small). Concerning system (1.2) we can prove t h a t only the coordinate x~ can be stabilized near xl = 0 (partial stabilization) under the same conditions. Using (1.3), systems (1.1) and (1.2) can be represented in the form

d--i =

+

),

(1.5)

where U is T-periodic in the last argument. By introducing the fast time r by t = er we get from (1.5) dx

d--~ = ef(x) + U(x, a, v),

(1.6)

where U is T-periodic in r. First contributions towards a theory of vibrational control are due to S.M. Meerkov (see [10] for linear systems) and R.E. Bellmann, J. Bentsman and S.M. Meerkov (see [2,3] for systems affine linear in the applied control). I m p o r t a n t applications of the m e t h o d of vibrational control are the stabilization of plasmas [12], lasers [11], chemical reactors [2,6]. In what follows we extend the theory of vibrational control to systems with slow and fast state variables where we apply the control to the slow components. In section 2 we describe the class of control systems under consideration and introduce modified definitions of vibrational stabilizability compared

Vibrational Control of Singularly Perturbed Systems

399

with the 'classical' definitions. Section 3 contains the reduction of our control problem to some normal form by means of normally hyperbolic invariant manifolds and appropriate transformations. In section 4 we derive conditions for strongly vibrational stabilizability and illustrate our result analytically by means of a linear singularly perturbed system. In the last section we treat the case of partial vibrational stabilizability and d e m o n s t r a t e it by considering the singularly perturbed van der Pol system.

2

Formulation of the problem.

We are given a process containing slow and fast variables and which can be described by the singularly perturbed differential system dz

d--i = x ( z , y),

(2.1)

dy = Y ( z , y),

C-d7

where e is a small positive parameter. Concerning the functions X and Y we suppose (A1). X : G --+ R n, Y : G --+ R m are twice continuously differentiable where G is a neighborhood o f the origin in R n • R m. (A:). (x = O, y = O) is an equilibrium point o f (2.1) that is possibly unstable.

Our goal is to apply a vibrational control to (2.1) such t h a t the controlled system has an attracting invariant manifold whose projection into the z, yphase space is a compact set near the origin. Let G n be a neighborhood of the origin in R n. We denote b y / 4 the set of all functions U : G n • R --.+ R n which are twice continuously differentiable with respect to all arguments and T-periodic in the second argument. In the sequel we consider control systems of the type

- - = X ( z , u) + dt

U(z,

(2.2) = V ( z , y),

where r is a small parameter and U belongs to the set/4. (The case that U is almost periodic in the second argument can be treated in the same way.) It is clear that (z = 0, y -- 0) is not necessarily a stationary solution of (2.2). By means of the fast time 7- we m a y rewrite (2.2) as

400

Klaus R. Schneider dz

d--; = ~X(~, y) + U(z, r), (2.3)

d..2 =

dr

Y(z,y).

D e f i n i t i o n 1. We call the equilibrium point (z = O, y = O) of system (2.1) strongly vibrationally stabilizable if to any J > 0 there are a suiJiciently small positive number eo and a function U E 11 such that for 0 < e < eo system (2.3) has an exponentially attracting T-periodic solution (z v (r, r Yv (r, e)) satisfying Izp(r, e)l _< J, Ivp(r,e)l ___J f o r all r. R e m a r k . This definition of vibrational stabilizability differs from the definition introduced by Meerkov and others [3] as follows: In [3] it is required that only the average of the periodic solution (zp(r, e), yp(v, e)) is located in a J-neighborhood of the origin, and it is assumed that the time-average of the control is zero. D e f i n i t i o n 2. We call the equilibrium point (z = O,y = O) of system (2.1) weakly vibrationally stabilizable if to any J > 0 there are sufficiently small positive numbers r Jo and a function U E 11 such that for 0 < r < eo the solution of (2.3) starting for r = 0 at any point in a go-neighborhood of the origin exists for all r > 0 and stays for all v in a J-neighborhood of the origin.

In singularly perturbed systems the slow variables usually play a special role. Therefore, we introduce the concept of vibrational stabilizability with respect to the vector z of slow variables. D e f i n i t i o n 3. We call the equilibrium point (z = O, y = O) of system (2.1) strongly vibrationally stabilizable with respect to the slow variable z if to any J > 0 there are a suJJiciently small positive number eo and a function U E 11 such that system (2.3) has for 0 < ~ < So an exponentially attracting T periodic solution (zp (v, e), Yv (r, ~) ) with the property Izp (r, e)[ < J for all r. D e f i n i t i o n 4. We call the equilibrium point (z = O, y = O) of system (2.1) weakly vibrationally stabilizable with respect to the slow variable z if to any J > 0 there are sulficiently smallpositive numbers eo, Jo and a function U E 11 such that for 0 < e < eo the following properties hold: (i) any solution (~(r, zo, Yo), 9(r, zo, Yo)) of (2.3) starting for 7" = 0 at a point (zo, Yo) in a Jo-neighborhood of the origin exists for all 7- >_ O. (ii) The inequality le(r, z0, u0)l< J holds f o r all r > O.

Vibrational Control of Singularly Perturbed Systems

401

Our aim is to find a vibrational control U(z, r) stabilizing the equilibrium point (z = 0, y --- 0) of (2.1). To this purpose we first derive conditions on U and Y implying that we can reduce system (2.3) to a system in some normal form to which the m e t h o d of averaging can be applied in order to prove the existence of an attracting periodic solution near the origin.

3

Reduction

to some

normal

form

The first step in our reduction process consists in eliminating the term U(z, r) in the first equation of (2.3) by means of an appropriate coordinate transformation. To this end we assume: (Aa). To any 6 > 0 there is a J1 > 0 and a function U E bi such that the

differential system

de d-V = u(r r)

(3.1)

has the flrst integral ~ = h(r,c) where h is periodic in r, and Ih(r,c)l < 6 for

Icl _< al. The assumption that the image of h is in a small neighborhood of the origin is important for establishing the stabilizability property. As examples for (3.1) we consider the simple cases U(~, r) -_- a c o s r where we have Ih(%c)l := [ a s i n r + c[ _< [hi + Ic[, such that for [el < 6/2 and ]cl <_ J1 = 6/2 it holds Ih(r,e)l < 6; and U ( ~ , r ) -- c o s t z where Ih(r,c)l =

lee'"~l < eM. The solution h(r, .) of (3.1) represents for all r a diffeomorphism and can be used to introduce a new variable x by z = h(r,x).

(3.2)

By hypothesis (Aa) we get from (3.2), (3.1) and (2.3)

dz

Oh dx

4-7 = U ( z , r ) + &~ d,- - ~ X ( h ( r , ~ : ) , u ) + U(z,~-).

Thus, we have

d~

(ah~-i

d-7 = e ~ & /

(r, ~)X(h0-, x), y), (3.3)

dy = Y ( h ( r , x), y). dr

402

Klaus R. Schneider

T h e right hand side of (3.3) is periodic in % hence we consider system (3.3) in the extended phase space R n x R m x S 1. For e = 0 we get the system dx ~

~

O~

dr dY=y(h(r,x),y). dr

(3.4)

In the next step we will reduce system (3.3) to a system containing only slow variables by means of a compact exponentially attracting invariant manifold. To this purpose we assume

(A4). To any J > 0 there is a neighborhood G~ of the origin in R n such that for x E G n the differential system dy d--~ - Y ( h ( r , x), y)

(3.5)

has an exponentially attracting T-periodic solution y = Po (r, x) with the properties

(i) Po is differentiable with respect to x. (ii) Po satisfies [p0(r, x)l < ~/2. R e m a r k 1. Assumption (A4) implies that Fo := {(x, y, 7-) E G~ x R m x S 1 : y = p0(r, x)} is a compact normally hyperbolic invariant manifold of system (3.4) [7,15]. R e m a r k 2. Assumption (A1) does not imply that (3.4) has only one exponentially attracting invariant manifold. But it is clear that exponentially attracting invariant manifolds cannot intersect each other. R e m a r k 3. Under the hypotheses (A1) - (A4) the equilibrium point (z = O, y = O) of system (3.4) is weakly vibrationally stabilizable. This follows immediately from the property that (x = c E G n, y = p0(r, x)) is a solution of

(3.4). According to the theory of normally hyperbolic invariant manifolds they persists under small perturbations [7,15]. Thus we have T h e o r e m 1. Under the assumptions (A1) (A4) there exists a sufficiently small positive eo such that for 0 < e < eo system (3.3) has a compact exponentially attracting invariant manifold F, := {(x, y, r) E G n x R m x S 1 : -

Vibrational Control of Singularly Perturbed Systems

403

y = p(r, x, e) = p0(r, x) + O(e)} where p is T-periodic in r and has the same smoothness as X .

Our aim is to prove the existence of an asymptotically stable T-periodic solution of system (3.3). Since /~c is an exponentially attracting invariant manifold of (3.3) it is sufficient to consider system (3.3) on the manifold Fc that is, we study the system

dx

(Oh'~-l(r,x)X(h(r,x),p(v,x,e)).

(3.6)

dr = e ~Ox/

4

Existence of a small periodic solution

asymptotically

stable

T-

Equation (3.6) can be written in the form d---r = ~

-~x

(v'x)X(h(7-'x)'P~

(4.1)

Since the right hand side of (4.1) is T-periodic in 7- we use the averaging theory to prove the existence of a T-periodic solution of (4.1). To this end we have to introduce the following assumption. Let

1//

(As). X ~ -- 0 has a solution x = xo with Ix01 _ 51/2. The spectrum of the Jacobian A := X~ is located in the left half plane.

Then, applying the fundamental theorem of the theory of averaging [5,13] we have the following result. T h e o r e m 2. A s s u m e the assumptions (A1) - (As) are valid. Then, there exists a sujfficiently small positive el such that f o r 0 < e < el system (4.1) has an exponentially attracting periodic solution x = q(r,e) located in an 51-neighborhood o f the origin.

Under the assumptions of Theorem 2 it follows that for 0 < e _< el (z = h(r, q(r, e)), y = p(r, q(r, e), e)) is an exponentially attracting periodic solution of system (2.3) satisfying [h(7-, q(7-, e))[ + Ip(v, q(r, e), ~))] < 25. Thus, we have C o r o l l a r y 1. Under the assumptions of Theorem 2 the equilibrium point (z = O, y = O) o f system (2.1) is strongly vibrationally stabilizable.

404

Klaus R. Schneider

We illustrate our result by considering the following singularly perturbed linear system dz

d"'[ = az + by, dy ~--~ : z - y

(4.2)

with b > 0, 0 < e << 1. Hence, the equilibrium point (z -- 0, y---- 0) is a saddle that is, an unstable equilibrium. We want to apply a high frequency control to the slow variable z in the first equation in order to stabilize the system near the origin. The corresponding control system has the form .z

d'--t = az + by + dy e~-~=z--y.

\~/

z, (4.3)

Introducing the fast time v we get from (4.3) dz

d--'r = e(az + by) + cos r z, dy dr - z - y .

(4.4)

Using the coordinate transformation Z ---- e s i n r X

we get from (4.4) d_x_x= e(ax + bye- sin r), dr

dy e s i n d---~=

r

(4.5)

x-y.

If we consider x as a parameter in the second equation in (4.5), then to given x this equation has a unique 2~r-period solution yo(r,x) := po(r)x where p0(r) is defined by

po(v) .-- e2Y-e-" 1 [ ~o2'~ea+'i'*ada + (e~'~ - l) ~orea+'ina&y]

(4.6)

It is easy to check that y0(r, x) is exponentially stable and t h a t r o := {(~, u, ~') c G~ x n "

• 5"~ : u = po(~-) ~}

(4.7)

Vibrational Control of Singularly Perturbed Systems

405

represents a compact exponentially attracting invariant manifold of system (4.5) for c -- 0. Therefore, hypotheses (A1) - (A4) are satisfied and for sufficiently small e we get by Theorem 1 that (4.5) has a compact normally hyperbolic invariant manifold Fe r , := { ( x , y , r )

~ G~" • n "

• s 1 : y = p ( ~ , ~ ) x = (p0(~) + p l ( r ) ~

+

...) ~},

where p is 27r-periodic in r. On F~ (4.5) reads dx = ~(a A- bp(r, e ) e - sin r)x : e(a 4- bpo(r)e- sin ~')x + 0(~2).

dr

(4.8)

Using m :---- ~1 o~o~ p o ( r ) e - , i n " d r ~ 1.29 :~ 0 the averaged equation to (4.8) has the form dx - - = ~(a + bm)x. dr For a + b m < 0 hypothesis (As) is satisfied. Consequently, by Theorem 2 system (4.2) is vibrationally stabilizable. 5

Partial

vibrational

stabilizability

In the sequel we replace hypothesis (A4) by the following assumption. (A~). To any 5 > 0 there is a neighborhood G~ of the origin in R n such that for x E G'~ system (3.5) has an exponentially attractin 9 T-periodic solution y = po(v, z) that is differentiable with respect to x.

Compared with assumption (A4) we do not assume that the periodic solution P0 is located in a small neighborhood of the origin. A consequence of this hypothesis is that we are not able to guarantee that the y-component of system (2.1) can be vibrationally stabilized near y -- 0. The following observation is obvious. L e m m a 1. A s s u m e the hypotheses (AI) - (A3) and (A~) are valid. Then system (2.1) is weakly vibrationally stabilizable with respect to the slow component z.

The following theorem can be proved in the same way as Theorem 2. T h e o r e m 3. A s s u m e the assumptions (A1)-(A3), (A~), (As) are valid. Then there exists a sufficiently small positive el such that ]'or 0 < ~ < ~1 system (3.3) has an exponentially attracting T-periodic solution (z -- h(v, q(r, ~) ), y = p(r, q(v, e), e)) satisfying Ih(~", q(r, e))] < 5.

406

Klaus R. Schneider

From T h e o r e m 3 we get immediately C o r o l l a r y 2. Under the assumptions of Theorem 3 the equilibrium point (z = 0, y = 0) of system (2.1) is strongly vibrationally stabilizable with respect

to the slow component z. We illustrate T h e o r e m 3 by considering the van der Pol equation with large d a m p i n g [8]. In that case, it can be represented by the singularly p e r t u r b e d system dz

dt dy e-~=z+y-

(5.1)

y3 .

It is well-known that system (5.1) has for 0 < r << 1 a unique exponentially stable relaxation oscillation [8]. T h e corresponding closed curve in the (z, y)phase plane contains the origin as unique equilibrium point which is unstable. Our goal is a strong vibrational stabilization of the z-component near the origin by applying an additive high-frequency control. We consider the control system

dz a (~) ~ - - y -'[- - - C O S dt dy y3. e--~=z+y--

(5.2)

Introducing the fast time 7" and the new coordinate x by from (5.2) dx

--

=

--g

z =

x +

sin

r

we get

y,

dr

dy

d"-T = a s i n r + x + y

(5.3)

y3. -

For Ixl sufficiently small, the differential equation

d__._yy= x + y - y3 dr has three equilibria ~-1 < Y~' < Yz where yZ 1 and y~ are hyperbolic stable equilibria which are located near - 1 and 1 respectively. Consequently, for sufficiently small lal and Ixl, the second differential equation in (5.3) has two T-periodic solutions p~_l(r) and p~(r) which are exponentially attracting and satisfy [p~l(r) - Y~-l[ < ~ and [p~(r) - Y~I[ < ~ respectively, where is a small n u m b e r [9]. Therefore, according to T h e o r e m 3, there exists a

Vibrational Control of Singularly Perturbed Systems

407

sufficiently small el such t h a t for 0 < e _< el system (5.3) has two exponentially attracting T-periodic solutions (x_ 1(% e), Y-1 (7-, e)), (xl (% e), Yl (% e)). Therefore, system (5.1) can be strongly vibrationally stabilized with respect to the slow component. Note, if we want to stabilize (5.1) by the linear multiplicative control : --y + -- COS

dt dv e-~ = z + y - - y a,

Z, (5.4)

then computer experiments indicate that the equilibrium point (z -- 0, y = 0) of (5.1) cannot be vibrationally stabilized that way.

6

Acknowledgment

The author acknowledges stimulating discussions with V.V. Strygin.

References 1. Baillieul, J., Lehmann, B. (1996) Open-loop control using oscillatory inputs. In: The Control Handbook, Ed. W.S. Levine, CRC Press, Boca Rata.n, 967-980 2. Bellman, R., Bentsman, J., Meerkov, S.M. (1983) Vibrational control of systems with Arrhenius dynamics. J. Math. Anal. Appl. 91, 152-191 3. Bellmann, R.E., Bentsman, J., Meerkov, S.M. (1986) Vibrational control of nonlinear systems: vibrational stabilizability. IEEE Trans. Automat. Contr. AC-31, 710-716 4. Bellman, R., Bentsman, J., Meerkov, S.M. (1985) On Vibrational Stabilizability of Nonlinear Systems. J. Optim. Theory Appl. 46, 421-430. 5. Bogoljubov, N.N., Mitropolskij, Ju. A. (1974) Asymptotic methods in the theory of nonlinear oscillations, (in russian). Nauka, Moscow 6. Cinar, A., Deng, J., Meerkov, S.M., Shu, X. (1987) Vibrational stabilization of a chemical reactor: an experimental study. IEEE Trans. Automat. Contr. AC-32, 348-352 7. Fenlchel, N. (1971) Persistence and smoothness of invariant manifolds. Math. J. Indiana Univ. 21, 193-226. 8. Grasman, J. (1987) Asymptotic methods for relaxation oscillations and applications. Springer-Verlag, New York 9. Hale, J.K. (1980) Ordinary differential equations. 2nd ed. Krieger Publishing Company, New York 10. Meerkov, S.M. (1980) Principle of vibrational control: theory and applications. IEEE Trans. Automat. Contr. AC-25, 755-762 11. Meerkov, S.M., Shapiro, G.I. (1976) Method of vibrational control in the problem of stabilization of ionization-thermal instability of a powerful continuous C02 laser. Automat. Remote Contr. 37, 821-830

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12. Osovets, S.M. (1974) Dynamic methods of retention and stabilization of plasma. Soviet Phys. Uspekhi 112, 6370-6384 13. Sanders, J.A., Verhulst, F. (1991) Averaging methods in nonlinear dynamical systems. Springer-Verlag, New York 14. Shapiro, B., Zinn, B.T. (1997) High-frequency nonlinear vibrational control. IEEE Trans. Automat. Contr. A C - 4 2 , 83-90 15. Wiggins, St. (1994) Normally hyperbolic invariant manifolds in dynamical systems. Springer-Verlag, New York

R e c e n t A d v a n c e s in O u t p u t R e g u l a t i o n o f Nonlinear Systems A n d r e a Serrani 1, A l b e r t o Isidori 12, C r i s t o p h e r I. Byrnes 1, a n d Lorenzo Marconi 3 1 Department of Systems Science and Mathematics Washington University St.Louis, MO 63130 USA [email protected], it - ChrisByrnes@seas. wustl, edu Dipartimento di Informatica e Sistemistica University of R o m e Via Eudossiana 18, 00184 Rome, Italy isidor i@giannutri, caspur, it 3 Dipartimento di Elettronica Informatica e Sistemistica University of Bologna Via Risorgimento 2

40136 Bologna, Italy imarconi@de is. unibo. s

A b s t r a c t . This paper presents some of our recent results in output regulation theory for nonlinear systems. We show how some of the difficulties inherent to the problem of nonlinear output regulation have been tackled and solved. In particular, we have proposed a methodology for internal model design, based on passivity theory, which enables to enlarge the class of nonlinear systems for which semiglobal robust regulation is possible. The same technique is also instrumental in solving the global output regulation problem for a restricted class of nonlinear systems. Furthermore, endowing the internal model with an adaptation mechanism, we give a solution to the longstanding problem of output regulation in presence of parametric uncertainties in the model of the exosystem. An interesting application of the proposed methodology to the design of an autopilot for a VTOL aircraft is introduced and briefly discussed.

1

Introduction

T h e t h e o r y of o u t p u t r e g u l a t i o n for nonlinear system has experienced a vigorous growth in the last decade. Beginning with the seminal work of Isidori a n d Byrnes [4], the p r o b l e m of letting the o u t p u t of a nonlinear s y s t e m a s y m p t o t ically t r a c k or reject t r a j e c t o r i e s g e n e r a t e d by a fixed e x t e r n a l a u t o n o m o u s system has been the s u b j e c t of intensive investigations. T h e t y p i c a l f o r m u l a tion of the p r o b l e m considers a nonlinear s y s t e m of the form

= ,~) = f(~, h(~,~~,, ~~),

(1)

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with state x E X C ]Rn, control input u E ]Rm, exogenous input w E W C ]Rr and tracking error e E ]Rm. In these equations, /~ E ]Rp represents a vector of parameters whose value m a y not be accurately known. It is assumed t h a t the exogenous input is generated by an a u t o n o m o u s dynamical system of the form

= s(w),

(2)

which is supposed to be neutrally stable (i.e. Lyapunov stable in forward and backward time). It is also assumed t h a t f ( x , w, u, il), h(x, w, I~), s(w) are sufficiently smooth functions of their a r g u m e n t s and f(0, 0, 0,ju) = 0, h(0, 0, it) = 0, s(0) = 0. T h e problem of nonlinear output regulation aims at finding a controller modeled by equations of the form

= ,7(~, e) u = eft)

(3)

with state ~ ~ S ~ C W', in which '7(~, e) and e(~) are sufficiently s m o o t h functions of their arguments, 7/(0,0) = 0 and 0(0) = O, such that, in the forced closed loop system

ic = f ( x , w, 0(~),/~) = ~(~, h(~, ~, ~)) (v = s(w) ~=0,

(4)

for every initial condition (x(O), ~(0), w(O), p(O)) in some neighborhood S of (0, O, O, 0), the response (x(t), ~(t)) (a) is bounded, and (b) such t h a t lim e(t) = O.

t--CO0

Since the initial conditions of the plant, the exosystem and the controller are allowed to range on some open neighborhood of their respective origins, we deal with a problem of local regulation. Moreover, since the property of asymptotic regulation is required to be persistent in spite of small p a r a m e t e r variations, the former is referred to as the problem of structurally stable output regulation. It is now well understood t h a t a solution of the above p r o b l e m exists only if there is a controlled-invariant submanifold of the state space in which the error vanishes, as dictated by the solvability of the following equations

0--W-

s(w) = / ( . ( w ,

~), c(~, ~), ~, ~)

(5)

o = h(.(~,~),~,~) in the unknowns ~r(w,/~) and c(w,/~), which are C k functions (for some large k) defined in a neighborhood of (0, O) and such t h a t zr(O,/~) = 0 and c(O, p) =

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411

0. As shown in the cited paper by Isidori and Byrnes [4], solvability of these equations (which constitute a nonlinear equivalent of the celebrated equations of Francis and Wonham [2]) is a basic necessary condition for the solution of a problem of output regulation. Once the solvability of equations (5) has been established, the solution of the problem of structurally stable regulation is obtained as the parallel connection of two separate controllers of the form =

+ Ne

/~im : 7(~1)

(6)

and

~0 = K~, + Le Ust = M~o.

(7)

The first controller, referred to as the internal model, has the role of generating asymptotically the input c(w,lz), while the second one locally exponentially stabilizes the interconnection of the plant and the first controller [1]. This methodology is commonly known as internal-model based control. Since the appearance of the cited works, researchers have begun to address the problem of achieving both robust and semi-global regulation. The former refers to the problem of designing a controller achieving regulation for values of the unknown parameters ranging on some fixed compact set, and the second to the problem of guaranteeing a domain of convergence for the regulation error which includes arbitrary compact sets. While it has been recognized that a careful design of the internal model provides the key ingredient to solve the first problem, the issue of designing a unit, which replaces the role of subsystem (7), capable of securing the required convergence properties on a possibly large domain is indeed rendered more difficult by the fact that the controlled plant is interconnected with an internal model which is only neutrally stable. To simplify the task of designing such a controller, restrictive assumptions on the zero dynamics of the plant have been considered. The work of Khalil [5] deals with the case of relative degree equal to the dimension of the state space (i.e., the case of trivial zero dynamics) and utilizes a technique developed earlier by Esfandjari and Khalil to design an error-feedback controller in which the components of the internal state are estimated by means of a "saturated" high-speed observer. Extensions of this result in case the zero dynamics possess certain properties of "input-to-state stablility" (in the sense of Sontag [12]) have been presented in subsequent papers [6,3]. A lot of research efforts has been devoted to the task of weakening these assumptions. In the next sections, we begin presenting a recent result that shows how the problem of robust semi-global regulation can be solved if the zero dynamics of (8) are globally asymptotically and locally exponentially stable [10]. Then, we briefly turn our attention to the problem of achieving global robust regulation. We show that, if the system is globally linearizable by output injection and its zero dynamics are globally exponentially stable,

412

A. Serraniet ~.

then global regulation is possible [9]. In both cases, a fundamental role is played by the passivity property of the internal model. A critical aspect of the design of the internal model, which has been set aside for a long time, is the necessity of knowing exactly the parameters of the exosystem. It is well known that, if the parameters of the exosystem (2) and those of the internal model do not match exactly, a sizeable steady-state error may occur. New recent approaches to the design of internal models have shown that also this kind of "sensitivity" can be eliminated and that the accurate knowledge of the parameters of internal model is no longer a requirement [11]. Finally, as a non-academic application of nonlinear regulation theory, we present the design of an autopilot for the autonomous landing of a VTOL aircraft on a ship whose deck is subject to oscillations due to high sea conditions [7].

2

Robust

semi-global

output

regulation

We consider systems whose model admits a global relative degree with respect to the regulated variable as output, and for which a global solution to the regulator equations is known to exists. In the single-input, single-output case, we concentrate our attention to systems diffeomorphic to the normal form

= fo(z, x l , w , ~ ) E1

~

X2

...

(8)

~,

=

e

= xl-q(w,~),

f.(z,x~,...,x,,w,,)+b(,)u

and driven by a linear, neutrally stable exosystem of the kind ~b = Sw.

(9)

Suppose a v-dimensional internal model = q~+ Ne

(10)

Uim = / . ~ ,

in which (O, N) is a controllable pair and (O, F) is an observable pair in companion form, is available for (8). Then, the problem in question reduces to that of rendering attractive an invariant manifold of the form M={(z,x,~):z=r

x=~(w,p),

~=r(w,p)}

for any initial condition in an a priori fixed set and for every p ranging on some given set 9 . The main difficulty is that, once we change coordinates as 5=z-r

~=x-~(w,p),

~=~-r(w,p),

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413

the plant (8) augmented with (10) has a zero dynamics which reads as

=

Clearly, the augmented system will be at most critically minimum phase, and every attempt to stabilize the equilibrium (~', s ~) = (0, 0, 0) using feedback from the partial state ~ (which corresponds to the error e and its derivatives up to order 1) will fail. A way to circumvent this difficulty is to replace the stabilizing term N e in (10) with a more general function of the vector ~. In this case, if the zero dynamics of the original plant have a globally asymptotically (and locally exponentially) stable equilibrium at s = 0, it can be shown that there exists a change of coordinates which puts the system in a form for which a control that achieves semiglobal regulation can be found. The control law in question uses a combination of high-gain and low gain feedback from the partial state ~ and the state ~ of the internal model. The partial state ~: is then replaced by suitable estimates generated by the saturated high-speed observer of Khalil (see [10]). To highlight the structure of the controller, suppose that the equilibrium 5 = 0 of the system r

-

5 = f0(~, 0, w,~) is globally asymptotically stable, and locally exponentially stable, uniformly with respect to the initial condition w(0) of the exosystem and the parameter /J. Then, it is well known that for the system z --- f0(5, ~1, w,p)

(11)

driven by the chain of integrators 9.. ~r-- 1 ~

(12) 1)

there exists a parameter-dependent linear feedback of the form V = V(~', k ) : = - k r - l b o x l

-

kr-2bix2

.....

kbr-2xr-i

such that the closed loop system (11)-(12) is semiglobally asymptotically stable in the parameter k. The term N[s - v(~, k)] is then used in place of N e in (10). The properties of the matrix 4i of the internal model, on the other hand, ensure that there exists a matrix M such that the matrix 4~ + e N M is Hurwitz, for any arbitrarily small positive constant e. It can be shown that there exists a m a t r i x / / s u c h that the manifold ~,={(z,x,~):z=~(w,~,),

~,=,~(w,~,),

~=r/v(w,~,)}

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A. Serrani et al.

is a globally defined, invariant output-zeroing manifold for (8)-(10), under the control u = Uim. Then, the additional control u = - K [ } r - v(~, k)] + e K M ~ , where K is a positive constant, and e, k and K are suitably chosen, renders the manifold in question attractive. Moreover, the domain of attraction can be enlarged to include any given set of initial conditions s ~(0), ~(0), w(0), p, adjusting the parameters e, k and K . Once the partial state ~ has been replaced by estimates provided by the observer of Khalil, the controller has the following final form / / = P r / + Qe

~. =sat(,) = q~ + N [ H # - V(fl, k)] u = F~ - K [ H f / - v(f/, k)] + e K M ~

(13)

where P, Q and H are suitable matrices.

3

Global

robust

regulation

For a specific class of minimum-phase systems, it is indeed possible to design a controller yielding global robust regulation. The class of systems in question, modeled by equations of the form = f ( x , y, w, M) -F g ( x , y, I~)u --- h ( x , y , w , p ) e = U - q ( w , ~,)

(14)

is diffeomorphic through a parameter dependent change of coordinates and a state augmentation of the kind il = A~I + B u to a system having the following structure: = F ( p ) z + G(y, w, p ) y = H ( p ) z + K ( y , w, p)y + b(p)~l i1 = A11 + B u .

(-,1..-~176

In the above equations, the matrices A and B are chosen as

0

A =

-A~ .--

0

0

....

0

0

9.-

0

A~-2

0

0 i

-A~-i

(15)

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415

in which Ai > 0 for i - 1 , . . . , r - 1 and r > 2 is the relative degree of (14). The matrix F ( p ) is assumed Hurwitz for every p E 9 , and this restricts systems (14) to be exponentially minimum-phase. Assuming that a global solution z = ~ ( w , p ) , y = q ( w , p ) and u = c(w,l~ ) of the regulator equations for (14) exists, and that an internal model of the form (10) has been found, the design of a global regulator can be accomplished through an iterative procedure. The crucial point is the existence of a recursive change of coordinates for the states of the dynamic extension which puts the system in a form suitable for a global backstepping-like design, with feedback from the partial state (e, ~, r/) only. Therefore, the need for an observer of the derivatives of the error is completely avoided. The interested reader is referred to [9] for further details.

4

Adaptive

internal

model

While internal-model based control schemes efficiently address the problem of tracking/rejecting those exogenous inputs that can be generated by a fixed autonomous finite-dimensional dynamical system, it is also true that a precise model of the exosystem must be available, to be replicated in the control law. This limitation becomes immediately evident in the problem of rejecting a sinusoidal disturbance of unknown amplitude and phase. An internal-model based controller is able to cope with uncertainties on amplitude and phase of the exogenous sinusoid, but the frequency at which the internal-model oscillates must exactly match the frequency of the exogenous sinusoid: any mismatch results in a nonzero steady-state error. In a recent paper [11], we have shown how it is possible to overcome this limitation, by means of a scheme that adaptively tunes the parameters of an internal model of fixed structure, so to have convergence of the natural frequencies of the internal model to those of the exosystem. The crucial step towards the accomplishment of this goal is a suitable parameterization of the internal model, which has been inspired by the work [8] and we regard as a "canonical" one. Suppose we are given a parameter-dependent internal model of the form

= q~(a)~ + NO(e, e(l) . . . . , e (~-1)) Uim = F ~

in which 0(e, e ( 0 , . . . , e ('-1) is a smooth function of the regulated error and its derivatives, (~(cr), N) is a controllable pair and (~(a), F) is an observable pair for every value of the unknown parameter a ranging over some compact set L'. Given any arbitrary Hurwitz matrix F, and any vector G such that the pair (F, G) is controllable, there exists a unique, nonsingular solution Ma satisfying -

FM.

=

at.

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A. Serrani et al.

Then, we can change coordinates, and consider an internal model of the form

v

= (F+G~,)r = ~.

NO(e,eO),... , e ( ' - l ) )

(16)

Since a is unknown, appealing to the principle of "certainty equivalence", we replace LVa in (16) with an estimate ~t, governed by an adaptation law of the kind

~r = ~,(~, 0), to obtain

= (F + G~])( + NO(e,eO),...,e(r-U)

(17)

Uim = ~ t ~ .

Once this has been accomplished, a robust feedback controller from the error e and a fixed number of its derivatives and the state of the internal model (~, ~) can be designed, to achieve boundedness of all trajectories and regulation to an output-zeroing manifold of the form .M, = { ( z , z , ~ , ~ t ) : z = r

x=w(w,p),

~=M~,r(w,p)}.

Moreover, it can be shown that, if the initial conditions of the exosystem are such that all modes of F + GLVa are excited, limt--,oo Lb(t) exists, is equal to the true value kV~, and the convergence rate is ultimately exponential. T h e saturated high-speed observer of Khalil can be used to generate estimates of the error and its derivatives, and the resulting controller can be implemented using feedback from the error e only.

5

Autonomous

landing of a VTOL

aircraft

We conclude presenting a non trivial application of o u t p u t regulation theory. T h e problem is that of designing an autopilot for a Vertical Take Off and Landing (VTOL) air vehicle, required to s m o o t h l y land on a sea vessel undergoing a pitch motion. The task of s m o o t h landing is accomplished in two phases: first, the goal of the autopilot is to synchronize the aircraft with the ship at a vertical distance H above the landing deck. Clearly, the vertical offset H is introduced in order to to prevent crashes between the aircraft and the ship due to negative vertical errors which can occur during this phase. In the second phase, the offset H is allowed to gracefully decay to zero so that the aircraft can land smoothly. The challenging aspect of the problem, particularly evident in the first phase, is that no explicit knowledge of the reference trajectory (i.e. of the deck position) is available: only the tracking errors and possibly their derivatives are provided by passive sensors lodged

Recent Advances in Output Regulation of Nonlinear Systems

417

on the aircraft. The reference signal, on the other hand, can be modeled as linear combination of a fixed number (say N) of sinusoidal functions of time whose frequencies, amplitudes and phases are unknown, but range within fixed closed intervals. In view of this, the problem in question can be approached as a regulation problem in which the exosystem is uncertain, and therefore must be tackled using the methodologies presented in the previous sections. In the following, we briefly outline some basic features of the problem. The reader interested in the details is referred to [7]. A simplified model of a VTOL aircraft is given as follows Xl ~ X2

~ = _ sin(Oi) h

+ cos(01) 2sin(~)

Yi = y2

y2 = - cos(Oi) h

(18) + sin(Oi) ~

F

- g

= ff cos(a)F where M denotes the mass of the aircraft, J the m o m e n t of inertia about the center of mass C, l the distance between the wingtips and g the gravitational acceleration. The control inputs are the thrust directed out the bottom of the aircraft, denoted by T, and the rolling moment produced by the torque F, acting at the wingtips, whose direction with respect to the horizontal body axis is given by some fixed angle a (see figure 1). As expected, xi,

T

Fig. 1. Forces acting on the aircraft. Yi and 01 represent respectively the horizontal and vertical position of the center of mass and the roll angle of the aircraft with respect to the horizon,

418

A. Serrani et al.

18

Adaptation turned on

2O

16 ~

18

1.5

14

16

1

14

~

~

0.5

~12

10 17~ ~ , , ~ 16 ~- 8 15

-0.5

2O 411 Time(~)

i8

9 6 13 411

60

4 2 0

50 Time

Fig. 2.

i

100

(~e)

O0

50

J

100 Time (sec)

150

Vertical, lateral and roll error

while x2, y~ and 02 the respective velocities. T y p i c a l u n c e r t a i n t i e s which the a u t o p i l o t have to deal with are given by the value of the m a s s M (and thus of the m o m e n t of i n e r t i a J ) and of the angle a . T h e internal m o d e l unit m u s t c o m p e n s a t e for b o t h the uncertainties in the signal to be t r a c k e d and the p a r a m e t r i c uncertainties of the model. In view of the previous discussion, the control T is assigned as the s u m of two t e r m s T = T~m + Tst representing respectively the o u t p u t of the internal m o d e l unit and an e x t r a t e r m which, along with F = Fst, is used to globally stabilize the zero-error m a n i f o l d . In this p a r t i c u l a r case, a challenging task is the c o m p u t a t i o n the s t a b i l i z i n g feedback: as a m a t t e r of fact, the coupling a renders the p l a n t m o d e l nonm i n i m u m phase with respect to the o u t p u t s x a n d y, a n d this c o m p l i c a t e s any classical control design based on s y s t e m inversion. In our case, t h e design of the stabilizing unit has been successfully accomplished e m p l o y i n g a high gain control for the i n p u t Tst a n d a saturated control law for the i n p u t /'st. We conclude this section presenting some s i m u l a t i o n results which show t h e performance of the r e g u l a t o r in case the reference signal (deck position) is given by the s u p e r p o s i t i o n of two c o m p l e t e l y unknown sinusoidal functions of time. Figure 2 shows the vertical t r a c k i n g error a n d the l a t e r a l / r o l l error. To stress the i m p o r t a n c e of correctly setting the right frequencies in the internal m o d e l and the effectiveness of the a d a p t a t i o n law, we have run the s i m u l a t i o n , up to t i m e t -- 50 sec, w i t h the a d a p t a t i o n law t u r n e d off and with a internal m o d e l whose n a t u r a l frequencies are m i s m a t c h e d with those of the exosystem. A t t i m e t = 50 sec the a d a p t a t i o n law is t u r n e d on. Note

Recent Advances in Output Regulation of Nonlinear Systems

419

t h a t , after t i m e t = 50 sec the s t e a d y - s t a t e error quickly decays to zero, and the aircraft is synchronized with the l a n d i n g deck. A t t i m e t = 100 sec, when the vertical, l a t e r a l a n d roll errors have b e c o m e negligible, the vertical offset H is let to decay to zero, and aircraft lands s m o o t h l y on the l a n d i n g deck.

References 1. C.I. Byrues, F. Delli Priscoli, A. Isidori, W. Kang, Structurally stable output regulation of nonlinear systems, Automatica, 33: 369-385, 1997. 2. B.A. Francis, W.M. Wonham, The internal model principle of control theory, Automatica, 12: 457-465, 1976. 3. A. lsidori, A remark on the problem of semiglobal nonlinear output regulation, IEEE Trans. on Automatic Control, A C - 4 2 : 1734-1738, 1997. 4. A. Isidori, C.I. Byrues, Output regulation of nonlinear systems, IEEE Trans. Autom. Control, A C - 3 5 : 131-140, 1990. 5. H. Khalil, Robust servomechanism output feedback controllers for feedback linearizable systems, Automatica, 30: 1587-1599, 1994. 6. N.A. Mahmoud, H.K. Khalil, Asymptotic regulation of minimum phase nonlinear using output feedback, IEEE Trans. on Automatic Control, A C - 4 1 : 14021412, 1996. 7. L. Marconi, A. Isidori, A. Serrani, Autonomous vertical landing on an oscillating platform: an internal-model based approach, submitted. 8. V. O. Nikiforov. Adaptive non-linear tracking with complete compensation of unknown disturbances. European Journal of Control, 4:132-139, 1998. 9. A. Serrani, A. Isidori, Global robust output regulation for a class of nonlinear systems, Systems and Control Letters, 39: 133-139, 2000. 10. A. Serrani, A. lsidori, L. Marconi, Semiglobal output regulation for minimumphase systems, Int. J. Robust and Nonlinear Control, 10:379-396, 2000. 11. A. Serrani, A. Isidori, L. Marconi, Semiglobal nonlinear output regulation with adaptive internal model, submitted. 12. E.D. Sontag, On the i n p u t - t o - s t a t e stability property, European J. Contr., 1: 24-36, 1995.

Sliding M o d e Control of the Prismatic-prismatic-revolute Mobile R o b o t w i t h a Flexible Joint* Hebertt Sira-Ramirez 1 Centro de lnvestigaci6n y Estudios Avanzados del IPN (CINVESTAV-IPN) Departamento Ingenierfa El~ctrica, Secci6n de Mecatr6uica Avenida I.P.N. # 2508 Col. San Pedro Zacatenco, A.P. 14-740 07300 M~xico D.F., M~xico h a ira@mail. r invest av. m x

A b s t r a c t . A sliding mode controller is proposed for the regulation of the prismaticprismatic-revolute (PPR) mobile robot equipped with an underactuated arm coupled to the robot main body by means of a flexible joint. The system, which happens to be differentially flat, can then be robustly controlled using a combination of the sliding mode control approach and exact tracking error linearization of prescribed off-line planned trajectories facilitated by the flatness property of the system.

1

Introduction

A finite dimensional nonlinear multivariable system is said to be differentially flat if it is equivalent, by means of endogenous feedback (the feedback does not need variables which are foreign to the system) to a linear controllable system in decoupled Brunovsky's form . Flat outputs are defined as a set of independent variables, whose cardinality equals that of the control input set, which completely parameterize the system state variables and control inputs. In other words, all system variables are differential functions of the fiat outputs. This means that they are functions of the fiat outputs and of a finite number of their time derivatives. Many nonlinear systems of practical interest turn out to be differentially fiat and, hence, linearizable by means of endogenous feedback. Theoretical and application developments of Differential flatness can be found in the several articles by Prof. M. Fliess and his colleagues [3], [4], [5]. Flatness has also been extended to linear and nonlinear delay differential systems and to systems described by linear partial differential equations. Even though flatness can be advantageously combined with m a n y nonlinear controller design techniques, like backstepping and passivitybased control (see, respectively, Martin et at [6] and Sira-Ramirez [9]), a fiat This research was supported by CINVESTAV-IPN and by the Consejo Nacional de Ciencia y Tecnologia (CONACYT), under Research Grant # 32681-A

422

H. Sira-Ramfrez

system is more naturally controlled by means of exact linearization. However, exact linearization is based in exact cancellation of systems nonlinearities and these invariably depend on system parameters. Exact linearization is thus known to exhibit a lack of robustness with respect to parameter uncertainty and unmodelled external signals affecting the system behavior. Sliding mode control, on the other hand, enjoys great popularity due to its simplicity and enhanced robustness with respect to unmodelled perturbation input signals and parameter uncertainty. The fundamental developments of this interesting controller design technique were mainly carried out in the former Soviet Union where a wealth of scientists contributed to its development during the years. We refer the reader to the book by Utkin [10] where fundamental developments and interesting application examples can be found. In this article, a dynamic nonlinear multivariable sliding mode controller is proposed for the trajectory tracking error regulation of the PrismaticPrismatic-Revolute (PPR) mobile robot equipped with an underactuated arm which is coupled to the robot main body by means of a flexible joint. The P P R robotic system, treated by Reyhanoglu et al in [7] from the viewpoint of non-integrable dynamic constraints, has been shown to be differentially flat, in an article by Sira-Ramirez [8], and therefore to be equivalent by means of dynamic state feedback to a set of decoupled controllable linear systems. Comparisons between the feedback performance obtained from the exact linearization approach and the passsivity-based control approach, were carried out for the P P R robot in a paper by Espinoza et al [2]. The flatness property and its intrinsic linearizing endogenous feedback option is here advantageously combined with sliding mode control for the robust feedback regulation of several prescribed maneuvers for the mobile P P R robot. This article is organized as follows: Section 2 deals with the modeling aspects of the P P R robot from a Lagrangian viewpoint and the verification of the flatness property of the system. Section 3 develops a sliding mode controller in terms of general trajectory tracking tasks for the mobile robot. Section 4 is devoted to present simulation results and evaluates the robustness of the performance of the proposed controller for a typical maneuver. The last section presents the conclusions and suggestions for further research.

2

2.1

T h e P P R m a t h e m a t i c a l m o d e l a n d s o m e of its properties Derivation of the PPR robot model

Consider the P P R mobile robot, shown in Figure 1, with 02 = 0 + r We denote by (x, y) the vertical projection of the end effector position (of mass m) on the plane of coordinates, (X, Y), on which the robot main body (of mass M) translational motions take place. We refer, however, to (x, y) as the

Sliding mode control of PPR robot

423

arm's tip coordinates. The geometric features of the system readily reveal the following set of relations between the (x, y) coordinates and the main body center of gravity coordinates (Xb, Yb). x = r e + I cos(0~) ::r ~ = &B - l/~usin(02) y = yB + l s i n ( 0 ~ )

102 r

~ = yB +

T r

.,

\- ._..iv

/

/

/

\~

x

o

"-

.

[

"'-. .........4

Fig. 1. The PPR mobile robot.

Within a Lagrangian dynamics viewpoint, we take as generalized coordinates the vector;

q = [zB, YB, O, 0~]T With reference to these coordinates, the kinetic energy function for the system, is given by

T ( ~ , ys, ~, y, O) = I ( M ~

+ M~)~ + m x ~ + m ! ) 2 + I~J~)

This expression can also be written, more compactly, as T(q, q) = 89 with

D(q) =

M +m 0 0 M+m 0 0 -mlsin(O~) m/cos(02)

0 - m l sin(O~) ] 0 mlcons(O~) I I IJ | 0 rnl ~ J

424

H. Sira-Ramirez

On the other hand, the torsion spring, assumed to be linear, is regarded as physically acting between the angular orientation of the body, 0, and that of the arm, 02. The potential energy is then given by V ( q ) = 1 K ( 0 - 02)2 = lqTKq with

K=

0 0 0 K O-K

Thus, the Lagrangian of the system is obtained as,

The vector of generalized external forces is seen to be

Q = [FI,F2, T,O]r where F1 and F2 are the forces applied to the robot's main body center of gravity in order to achieve translational movement of the robot on the plane. T is the torque applied to the base revolute portion of the mechanical system, thus creating angular movement, through the flexible joint, of the robot's arm. Applying the Euler-Lagrange equations, the model for the robot is seen to be of the following general form,

n(q)~ + W(q,q) + Kq = Q where the term of Coriolis and centripetal forces,

-ml0~ cos(e2) W(q,~l) =

-mlO~ sin(02) 0 0

can be factored as

[~00 -mid,.cos(02) 0 0 -m10202 sin(02) C(q, ;~) =

oo

o

O0

0

4

Sliding mode control of PPR robot

425

Note that the matrix -D(q,(1) = D(q) - 2C(q, gl) is skew-symmetric. This property can be exploited in passivity-based regulation schemes of the PPR robot (see [2]). The system is therefore characterized by the following implicit state space model (M + m)~B -- ml sin(0~)02 -- mlO~ c o s ( 0 ~ ) = F1 ( M + m)~B + mlcos(02)O~ - mlO~ sin(02) = F~ IO + K(O - 0~) = T ml=O= - mlsin(O=)~B + mlcos(O~)ftB -- K(O - 02) = 0

2.2

(1)

Flatness property of the PPR robot model

The PPR robot model (1) is easily seen to be differentially fiat. Indeed, the fiat outputs are given by the main body center of gravity position coordinates, XB, YB, and the orientation angle, 92, of the robot arm. Indeed, all variables in the system (i.e. states and control inputs) are expressible as differential functions of the fiat coordinates, (XB, YB, 0~). 0=02+~

1

[ m l 2 0 ~ - mla:B sin(02)+ ml~lB cos(0~)]

F1 = ( M + . ~ ) ~ - m t sin(0~)//~ - ,.10~ cos(0~) F~ = (M + m)//B + mt cos(0~)//~ - mld~ sin(0~) r = -ml~

-

~ ml

i[

sin(02) + . ~ t ~ cos(02) + (S + ,-l~)/~2 + ~: mt~0~4) -

-

(x(~)-/}~i?o + ~/~)B + 202y(Ba)) sin(0~)]

(2)

Note that the end effector position is also expressible in terms of the fiat outputs, by means of X = xB +lcos(O~),

y=yB+lsin(0~)

The differential parameterization (2) allows one to compute explicitly off-line suitable reference trajectories for all state variables of the system in terms of desired fiat output reference trajectories. 2.3

Invertibility

of

the differential parameterization

The structure of the dependence of the control inputs on the fiat output time derivatives reveals that, while the torque control input T depends up

426

H. Sira-Ramlrez

to fourth order time derivatives of all the fiat outputs, the forces F1 and F~ involve at most second order time derivatives of only two fiat outputs. The multivariable input-fiat output relation is, therefore, not an invertible one in the sense that the higher order derivatives of the fiat outputs are not in a one to one relationship with a possible set of independent control inputs. A dynamic extension , or "prolongation" is therefore needed on the first two control inputs in order to achieve a desirable input-fiat output "decoupling". The required one-to-one relation would then read F1 = (m + M)x(~ ) - rnl [0( 4 ) - 6(02)2t/~] sin(02) -ml

[4020~3)+ 3(0212- ( 0 ~ ) 4]cos(02)

: (m + M)y(~ ) - rnl [40~0~a) + 3(~/2)~ -(02)4] sin(0u)

+~ [o~')- 6(0~)'~,]cos(o,) T = (I + ml~)02 +

',T-%

-

Considering F1 and F; as new control inputs, we can rewrite the above relation as

[ rn+M 0

0 -mtsin(02)] rx(~)] / [y(A )

m + M

--rnlsin(O~) m/cos(O~)

mlcos(02)

mlU

J ko ')

=

+ ml [40~07 ) + 3(0~) 2 - (0u) 4] cos(02) + me

[4ti20~a) + 3(02) 2 - (ti2)4] sin(02) + 6ml(ti~)~0~ cos(0~) + k~

{ - ~(1 +ml~)02 + m / ( [ K _03] ~B + t~'~). + 20~y(~)) sin(0~)

The global invertibility of the matrix in the left hand side of the previous expression implies that a suitable (global) state-dependent input coordinate transformation reduces the system to the following decoupled set of linear controllable systems in Brunovsky's canonical form,

x(~)=,l,

y(~)=v~, 0~4 ) = v 3

(3)

Sliding mode control of PPR robot

427

with vl, v2 and va defined in the obvious way. 2.4

Control

objectives

It is desired primarily to execute a feedback controlled translational motion of the P P R robot, along with some requirements on the arm orientation. Generally we would have available a specification of the nominal motions as a set of functions: x~(t), y~(t) and O~(t) characterizing, respectively, the desired main body movement and the desired orientation of the arm. In some other examples, it is desired to follow a particular trajectory characterized by a prescribed body movement and the body orientation; z~(t), y~(t) and 0* (t), while, as in [7], it is additionally required that the robot's arm actual angular position, 0~, evolves at the nominal value of zero during the execution of the trajectory tracking maneuver. In this last kind of maneuvers it is implied that no vibrations are allowed which are due to revolute joint flexibility during the maneuver. It is clear, from the flatness property, that any desired maneuver, regardless of its particular specifications, can always be translated to a required nominal maneuver, exclusively given in terms of suitable trajectories for the flat outputs. For this reason, we concentrate our controller design efforts in achieving asymptotic trajectory tracking for the flat output coordinates (xB, YB, 02).

3 3.1

Sliding

mode

feedback

controller

design

An exact tracking error linearlzation approach

Consider the following discrete sets of real constant coefficients,

Assume that each particular described set constitutes a Hurwitz set. In other words, they are associated with stable polynomials, defined in terms of the complex variable A, which are of the followingform

p(.)(A) ----~A4 + ~3(.)A 3 -4- /~2(.)A2 "4-fll(.)A q" ~0(.) d the associated differential polynomial, We denote by p ()(M) p(.)(~) :

d4

d3

d2

d

P r o p o s i t i o n 1. Given a prescribed trajectory, {x~(t), y~(t), O~(t)} for the flat outputs of the PPR robot (1), the following dynamic feedback controller, achieves closed-loop asymptotic exponential tracking of the given path.

428

H. Sira-Ramfrez

=

(m + M)v. - ml [vo~ - 6(t}~)~t/~] sin(O~) -ml [40~0~3) + 3(02) 2 -(t}~) 4] cos(e2)

=

(m + M)vy - ml [4020 (a) + 3(0~) 2 - (t~2)4] sin(02)

T = (1 + .-,t~)~ +

I

~

~-[ml vo2--ml (v~:+ [K

0~] ~13+02ijB+ 202y(ff)) sin(02)

+ml(vv+ [@--O~]~lB--O'2"B--202x(~')cos(02)] with d2

.

13x

-~.(~B Vy =

- ~ox(~

a~ . -d-izu~(t) - ~(v~ -~(0~

V07t

d ,

- ~B(t))

~o~'(t)

) -

d .

- -~v~(t)

- ~'~(t))

d3 . "~uB(t))

-

d2 . "~yB(t))

) - ;3o~(uB - u'~(t) )

- ~.~(o~ (~) - ~o~'(t)) - ~o~'(t))

- ~(0~

- ~o~(0; - ~d02~

, .))

- ~oo~(o~ - o z ' ( t ) )

Proof Let ex, eu and e02 denote, respectively, the tracking errors, xo -x~(t), YB y*B(t) and 02 - O~*(t). The proposed controller yields the following closed-loop tracking error dynamics

[ Morn

0 -raisin(02)] [ [p~(~)]e~ ] M + m mlcos(02) [pv(~)J eu = 0 [-mlsin(O~) mlcos(O~) rnl~ [Po~(~)] eo2 Since the above matrix is non-singular, it follows that x

ex = O~

y

e y = O,

02

eo~ = 0

and therefore e,, ey and e02 asymptotically exponentially converge to zero.

Sliding mode control of PPR robot

429 []

The linearizing controller derived above is based on exact cancellation of the system's non-linearities. The performance of the linearizing controller deteriorates when unmodelled external perturbation inputs and uncertain parameter perturbations affect the system. In the next section we still propose to take advantage of the simplicity of a linearizing controller but this time endowed with robustness features borrowed from sliding mode control. 3.2

A p r o p e r t y o f sliding m o d e c o n t r o l

Let cr be a scalar quantity and W be strictly positive constant. Suppose ~(t) is an unknown perturbation signal with uniformly bounded magnitude for all t ~ O, i.e.

T>~oXl~(t)l < B The following proposition is a standard result in sliding mode control (see Utkin [10]). P r o p o s i t i o n 2. Given any arbitrary initial value or(O) of or, the closed-loop trajectories, or(t), of the discontiuous perturbed dynamics, = - W sign cr + ~(t)

converge to zero, in finite time, T, if and only if W > B. Moreover, T is upper bounded by the quantity

I~(0) I W-B [] The sliding mode trajectories of the scalar function a are, therefore, robust with respect to unmodelled additive bounded perturbation input signals. Our results below are based on this fundamental fact. 3.3

A property of high-gain control

A widely recognized disadvantage of sliding mode controllers is the, so-called, "chattering phenomenon" by which the high frequency "bang-bang" control input excites unmodelled parasitic dynamics and causes oscillatory motions in the system's directly actuated variables, thus degrading the closed-loop performance. A popular technique, which still has fast stabilizing features but which lacks the degree of robustness of sliding mode control, consists of replacing the discontinuous controller by a high gain controller. The following result establishes the salient features of such substitution in fundamental terms.

430

H. Sira-Ramlrez

P r o p o s i t i o n 3. Given any arbitrary initial value a(O) of the scalar quantity,

a, the closed-loop trajectories, a(t), of the "high-gain" controlled perturbed dynamics, ff

= -w

-

-

I~1+,

+ ~(t)

crosses, in finite time T, the boundary of a neighborhood of zero, of radius R, from whose interior the trajectory never escapes, if and only if W > B. Moreover, T and R satisfy T < l a(0) I -

R -

w-B'

e W/B-t

Proof Consider the Lyapunov function candidate V(a) = 89 s. The time derivatives of V(a) along the closed-loop perturbed trajectories of cr(t) satisfy,

f'(~)--~=-w I~]q-,

+~r

I ; / + i +l~llr

l

=-(1~1+ Thus I) is negative everywhere except in the interior of the ball given by

Be

B = {~ : I,~1 -<W ---u--~} thus the trajectories of a approach B from the outside, and are constrained by B when they start from the inside. Note that the boundary of the ball/3, does not contain system trajectories since, on the boundary the closed-loop system satisfies: & = - B s i g n ~r+~(t) and ab" = - I ~ r l ( B - ~ ( t ) s i g n or) < 0. The boundary of/3 is not an invariant set. Hence, any trajectory starting outside /3 crosses its boundary and, thus, B is reachable in finite time. Integrating the differential equation for a and uniformly upper bounding Ir by B and the quantity, I~(t)l/(l~(t)l + e), by 1, yields the time estimate T. [] 3.4

Dynamic sliding mode control of the PPR

mobile robot

Consider the following sets of constant coefficients,

{~o~,~,~},

{~oy,~,~}, {~oo~,~o~,~2o~}

Sliding mode control of P P R robot

431

As before, we assume that each set constitutes a Hurwitz set, i.e. they build stable polynomials of the following form, p(.)(A) = A3 + a~(.)A ~ + al(.)A 1 + ct0(.) '~ the associated differential polynomial, We denote by p (.)(~7)

p(.)(d).~ = d 3

d2

d

Define, as before, the set of tracking errors e~ = xB - x*(t), ey = YB - y~(t) and e02 : 02 - 0~ (t). Consider a set of sliding surfaces az, ay and as2, comprising a sliding surface vector, given by expressions of the form

Note that if the sliding vector components ~x, ~y and ~0~ are indefinitely constrained to be zero, by means of a control action, the corresponding tracking errors e~, ey and e0 asymptotically exponentially converge to zero. 4. Given a desired nominal trajectory {x*B (t), Y*B(t), O~(t)} for the flat outputs of the robot (I), the following d y n a m i c sliding mode feedback controller, achieves asymptotic exponential tracking of the given path.

Proposition

[e~ = (m + M ) v . - rnl [vo~ -

-ml [4o=e +

6(~J=)202]sin(eD

=- (o=)'] cos(e=)

F~ = (m + M)vy - ml [40=e~ 3) + 3(0~) = - (02)'] sin(0~)

T = (I + ml~)g~

+ ~I [ m l 2 v o ~ - m l ( v ~ + [ K - O ~ ) ~ n + O 2 i ) B

+ 2t~2y(ff))sin(#2)

with v~ =

d•

d3

x~(t)

- ~x(x~

~ - d-V~(t))

d2

- ~,~(~

-ao~(~B - dx'B(t)) -- W~ sign ax

,

- d-~(t))

432

H. Sira-Ram/rez

~ =

d4 "

-,~2~,(y~)

-~o~(~

-

-g-gw,(t)

,,o~ = ~ o ~ ' ( t ) -

d .

~vB(t))

~ " ~-V~B(t))- ~ ( ~

-

w~

-

d2 "'t'" ~/5~B~ ))

~isn ~

~o~(od ~) - ~ o ~ ' ( t ) ) -

~,o#~ - jd28 g ~*'t'" ~ ))

-,~oo~(Y~ - ~ o ~ ' ( t ) ) - wo~ ~i~ ~o~ (6)

where W~:, Wy and Wa2 are, sufficiently large, strictly positive constants and the function "sign" stands for the signum function. Proof The proposed dynamic discontinuous feedback controller yields the following closed-loop sliding surface vector dynamics

M+

M + rn

mlcos(O,) I

L-ml sin(0~) ml cos(O~)

ml~

d-"~ ~

;: + Wv signcr v |

J Lz {[po~(z)] ,,~ } + w,~ sign ~,~ j

=0 i.e.

m

0

-mlsin(02)l [ ~..+w.

.+m

sig.~

lcos(02) ! / cr,+W sign

I-raisin(02) mlcos(02)

ml 2

1

[

=0

J [ d ae~ + Wo~ sign ae~J

It follows that d ~-/ar + W~ sign o'~: = O,

d ~O'y + Wy sign ~ry = O,

d

"~as~ + We~ sign ao2

= 0

and therefore a~, cry and c% converge to zero in finite time. As a consequence, the tracking errors ex, e v and e02 asymptotically exponentially converge to zero. []

4

Simulation

results

We considered a mobile P P R robot with the following physical parameters: m=3Kg,

M:10Kg,

k--0.5N-m/rad,

Sliding mode control of PPR robot J=0.1N-m-s2/rad,

433

1=0.5m

We tested the performance of the dynamic sliding mode controller (5), (6), for the tracking of a planned nominal trajectories of the flat output variables, (xl, (t), Yi~(t), 0; (t)). 4.1

T r a c k i n g a circular t r a j e c t o r y ( a r m aiming at a fixed p o i n t )

A circular path of radius R was specified for the main body center of mass coordinates (xB, YB). The corresponding circle defined in the plane (X, Y) with center located on the Y axis, at the point (0, R),

x*m(t) = Rsin(wt),

y*B(t) = R(1 - cos(wt)),

R = 2 m, to = 0.1 rad/s

It was desired to track the circular path while the robot arm was required to aim, at all times, inwards the circle, directly pointing towards the fixed point (0, R) in the plane (X, Y). This demand implied that the nominal arm angular orientation, 02, had to satisfy, 02 = wt + ir/2. The nominal value of the body orientation angle, 0, computed, according to the first equation in (2), turns out to be, o = t o t + rc

71"

ml [~Bcos(tot) + YB sin(tot)] = tot + -~

(7)

i.e., the body orientation angle 0 coincides with the arm orientation angle 02. The torsion spring, thus, exercises no restoring torque along the nominal path. In order to avoid chattering and undesirable oscillations we proceeded to change the hard switch in the sliding mode controller by a high gain "soft" switch. Simulation results, illustrating the robot change in position and the arm position, with initial conditions starting significantly off the planned trajectory, are given in Figure 2. The sliding mode controller parameters including the high gain switch parameters were set to be Wz = Wy = 2, Wo2 = 1, e = 0.05, while the sliding surfaces defining polynomials were chosen to be of the form: A3 + (2~to'` + a)A 2 + (2~to'`a + to,~)a + toga = (a 2 + 2~to'`A + to~)(A + a) i.e., we specified the tracking errors linearized closed-loop dynamics by a characteristic polynomial with one real and two complex eigenvalues located in the left portion of the complex plane. We chose: a = 2,{ = 0.8,to'` = 1 for the main body positions closed-loop dynamics, and a = 1, { = 0.8, to,, = 1 for the arm angular orientation closed-loop dynamics. Figure 2 shows the time evolutions of the system generalized variables XB, YB, 0, 8~ respectively compared with their nominal trajectories. In this

H. Sira-Ramirez

434

[m]21

xs(t),xs*(t)

/!-.....\.\

Ol

[m] ,]Ys(t),Ys*(t}

,, ......

/ \ ...... "

0

20

40

60

80

0

time [s]

20

40

60

40

60

40

60

40

60

[rad]l~ 02(t),O*2(t) 0

tin] ;] y

.~t

20

40

60

80

0

time [s]

~ x(t}--~

0

20

40

80

time [s]

[N]2t

"~t ~Fl(t)

60

80

0

time [s]

[N]~ ~t) 0

20

80

time [s] I-

20

so

time [s]

[N-!I T(t) 20

40

60

80

0

time [s]

20

80

time [s]

Fig. 2. State variables, end effector position and control inputs responses for circle tracking maneuver

figure we also show the three control inputs. Figure 3 depicts the motions of the main body and of the end effector in the plane. The control system successfully tracks the prescribed trajectory. A robustness test was also carried out to check for the effect of unmodelled sustained oscillatory perturbations which were also allowed to directly act on the underactuated arm position dynamics. For these simulations we used the following perturbed model of the P P R robot,

sin(O2)0~-

(M + m)~s - ml mlO~ (M + m)~lB + mlcos(e=)02 - mlO~

cos(e2) = F~ sin(e2) = F= + 0.5 r/(t)

Ig + K ( e - O 2 ) : T

ml202 - ml

sin(02)~B + ml COS(02)~iB-- K(O - 02) = 0.1 ~/(t)

(8)

with 7/(t) = sin(t). Figure 4 shows the perturbed evolutions of the system generalized position variables zB, yB, 0, 02 compared with their nominal trajectories, z*s(t), Y*B(t),

Sliding mode control of P P R robot

Y,. [m]

435

,'"L

)x

/'

)/ [m]

X F i g . 3. Circle tracking maneuver, main body and end effector position trajectories.

... .... .xs(t),~*(t)

0

20

40

[m], 'yB(t)'yB*(t)

f

60

80

0

20

40

so

time [s]

[rad],o;~i~0 ( t ) , e * { t ) ~ _ ~ ~ o

[rad],ii~. %(t),82*(t )~.~_~_~

20 40 60 80

o

20 40 60 80

time [s]

'!1

o

20 40 60 80 time [s]

0

20

40

80

time [s]

60

80

time [s]

time [s]

-:t o 0

20 40 so 80

time [s]

20

40

60

80

time [s]

F i g . 4. State variables, end effector position and control inputs responses for perturbed circle tracking maneuver.

436

H. Sira-Ramfrez

9" (t), 95 (t). The three control inputs are shown to absorb the unmodelled os-

cillatory perturbation in a quite efficient manner. The actuated body position coordinates are practically unaffected by the unmodelled perturbation while the controller still manages to reasonable regulate the unactuated coordinate, 92, towards the prescribed trajectory with small oscillations (see Figure 5).

Y 4

[ml

z"-

3.

2-

1-

f J

0

[ml g

X Fig. 5. Circle tracking maneuver, main body and end effector perturbed position trajectories.

4.2

Tracking a circular trajectory (arm aiming tangentially to the path)

It was also desired to track the same circular path specified above while the robot arm was required to aim, at all times, tangentially to the traversed circle defined in the plane (X, Y). The nominal arm angular orientation, 92, satisfied, then, 92 -- wt. The nominal value of the body orientation angle, 9, computed, according to the first equation in (2), turns out to be,

ml R~ 2 9=~;t+-~--~Tr ml [~ncos(wt) +ijB sin(wt)] = ~t + - - K

(9)

i.e., the body orientation angle 9 has a constant offset with respect to the arm orientation angle/92 thus creating, on the torsion spring modeling the flexible joint, a constant restoring torque of value m l l ~ 2. The numerical values given yield a constant torque of 0.03 [N-m] with a constant offset of 0.06 [rad]. Figures 6 and 7 depict the evolution of the unperturbed motions of the mobile robot with the prescribed maneuver.

Sliding mode control of P P R robot

[m],~l

.......

0

20

YB(t),Ys*(t)

xB{t),xB*(t). ",,,

40

mli1

/,

60

80

20

[rad] ii~ 9(t)~Y'(t) ~ 40

60

0

40

60

80

F,(t) 0

20

40

20

40

60

[s]

~ 20

80

time

80

time

0

60

[s]

[rad]~1

+-JJ 20

40

[s]

time

0

437

0

_

_

F

d

t

)

.

20

80

0

20

time [s]

time [s]

_

40

60

80

time [s] EEE~

[N-m],!!{ T(t) -04~ ~60

80

40

60

80

time [s]

F i g . 6. State variables, end effector position and control inputs responses for circle tracking maneuver with tangential arm pointing.

Y4-[m] 3-

~/

~,ctory

2-

O-

~\~

y(x) '~ " ' ~

( ,..i.. [m]

• F i g . 7. Circle tracking maneuver, main body and end effector position trajectories with tangential arm pointing.

438 4.3

H. Sira-Ramfrez Point to point regulation along a straight line segment

Consider the motion represented by a straight line segment with prescribed initial point given by the coordinates, (xB0, YB0), and terminal point given by, (XBl, YB!) in the (X, Y) plane. The robot starts with zero velocity and arbitrary orientation off the line segment. While aligning the arm, it is required to reach the starting point of the line segment (XBo,YBo) and park there, then to proceed to follow the prescribed line segment during a finite period of time Its, tf] and, finally, park again at the specified final point (xB/, yB!). It is further required to execute a pointing maneuver of the arm which takes it along the same direction of the robot orientation angle coinciding with that of the prescribed line path. For the desired maneuver we prescribe off-line a polynomial spline of the B6zier type for each of the flat output coordinates, x*8(t), YB(), * t and 0~(t). The orientation angular trajectoryfl~ (t), has a different time interval specification than that of the translational variables trajectories x~ (t), y~ (t). The nominal displacement and orientation variables were specified as

x~(t)

XB0 + (~Bs ...

XBO) \ t l - t, }

(t-t,

~ \tl-t,}

...

o;(t) =

,"

,2 \ t l - t, } + ~3 \ t l - t, )

"'

~2\t1_t,/+~kt1_t,)

~ ~]

J

\tl-t,) (t-t,'~ ~]

r6 \ t l - t i ]

o20 + (o2I -o20) (\ tlo~ -t---t-'~t,o~~

-I-r3( t--t2~ ~ ~ ... \tlo~-t,o2/

re

~

~2 \ t;-~ -T,o~

~ t-t'~ ~+1 \tlo---t,---o2 ] J

We chose a nominal trajectory for the angular orientation 02 which tried to orient the arm from the beginning of the maneuver in a period of time [t~o~,t]o2] which preceded the transfer maneuver interval [ti, tf]. The desired nominal body and arm orientation angles are the same as that of the line arctan[(yBf -- YBO)/(XB! -- XB0)] = 7r/4 [rad]. The polynomial coefficients were chosen to be: rl = 252, r2 = 1050, r3 = 1800, r4 = 1575, r5 = 700, r6 = 126 and ti = 10 [s], tf = 25 [s], rio2 = O, tfe~ = 10, (XBI,YBf) = (8.5,8.5),

(x~0, y~0) = (0.5, 0.5)

Sliding mode control of PPR robot

439

Figures 8 and 9 depict computer simulations illustrating the performance of the previously designed feedback controller for significant initial deviations from the prescribed path and from the prescribed orientation angle for both the robot body and the unactuated arm. These initial conditions, at time t -- to, were set to be

xB(to) = 2.0 [m] yB(to) = 3.0 [m] O~(to) = rr [rad]

,o.i

[m] 7. XB(t),xB*(t{f .................... [m] -5

[rad]41 -5

0

5

10 15 20 25 30 35 40

~0(t),0*(t) 0

5

O(to) =

~ [rad]

lO.i!yB{t),yB*(t -5

0

5

10 15 20 25 30 35 40

time [s] time [s] [rad]~-~e2(t),e2.(t )

10 15 20 25 30 35 40

time [s]

[m] 'i,i x(t),y(t) f

-5

0

5

10 15 20 25 30 35 40

time [s]

[N] 2s:~F ~

-5

0

5

10 15 20 25 30 35 40

o5 0

5

10 15 20 25 30 35 40

-5

0

5

10 15 20 25 30 35 40

-5

5

10 15 20 25 30 35 40

time [s]

time [s]

0

time [s]

time [s]

Fig. 8. State variables, end effector position and control inputs responses for a rest to rest maneuver in a fine segment.

5 Conclusions A dynamic sliding mode controller has been developed for the P P R mobile robotic with a flexible appendage. The proposed controller design method heavily relies in the differential flatness of the system, which is quite helpful in facilitating the off-line trajectory planning aspects which achieve a desired motion maneuver in the plane. To avoid the classical chattering behavior

440

H. Sira-Ramlrez

of sliding mode controlled mechanical systems, we carried out the implementation of the derived controller with the aid of a high gain saturation function replacing the troublesome switch. The simulation results show that the dynamic high gain controller practically retains all the nice qualitative characteristics of sliding mode control. Reasonable tracking accuracy and unmodelled perturbation rejection features were also present in the high gain controller.

Y

initiiii ram,,

tm] ~

_~ /+

'; ":,

m~eu~er,..l

0-

[m]

-1

• Fig. 9. Portion of the line segment tracking maneuver: Main body and end effector position trajectories.

References 1. Bailheul, J. (1993) Kinematically redundant robots with flexible components. IEEE Contr. Syst. Mag., 13, 15-21. 2. Espinoza-P6rez, G., Sira-Ramlrez, H., and Rios-Bolivar, M. (2000) Regulation of the Prismatic-Prismatic-Revolute Robot with a Flexible Joint: A combined passivity and flatness approach (submitted for publication). 3. Fliess, M., LSvine, J., Martfn, Ph., and Rouchon, P. (1992) Sur les syst~mes nonlinSalres diff~rentiellement plats. C. R. Acad. Sci. Paris, 315, 619-624. 4. Fliess, M., LSvine, J., Martin, Ph., and Rouchon, P. (1995) Flatness and defect of nonlinear systems: Introductory theory and examples. Int. J. of Control, Vol. 61 , pp. 1327-1361, 1995. 5. Fliess, M. L~vine, J. Mart/n, Ph. and P. Rouchon, (1999) A Lie-Bs approach to equivalence and flatness. IEEE Trans. on Auto. Control, 44, 5, 922937. 6. Martfn, P. Murray, R. M., and Rouchon, P. (1997) Flat systems, Proc. of the 4th European Control Conference, Brussels, Belgium, 211-264.

Sliding mode control of P P R robot

441

7. Reyhanoglu, M., van der Schaft, A., and McCalmroch, N. H., and Kolmanovsky, I. (1999) Dynamics and Control of a Class of Underactuated Mechanical Systems. IEEE Trans. on Auto. Control, 44, 9, 1663-1671. 8. Sira-Ramlrez, H. (2000) Control of a P P R Robot with a Flexible Arm. (submitted for publication) 9. Sira-Ramlrez, H. (2000) Passivity vs flatness in the regulation of an exothermic chemical reactor, European J. of Control, 6, 3 (to appear). 10. Utkin, V. (1978) Sliding Motions in the Theory of Variable Structure Systems, MIR Publishers, Moscow.

The ISS Philosophy as a Unifying for Stability-like Behavior

Framework

E d u a r d o D. Sontag Department of Mathematics Rutgers University New Brunswick, NJ 08903 s o n t a g @ c o n t r o l , r u t g e t s . edu

http ://~w. math. rutgers, edu/~ sont ag

A b s t r a c t . The input to state stability (ISS) paradigm is motivated as a generalization of classical linear systems concepts under coordinate changes. A summary is provided of the main theoretical results concerning ISS and related notions of input/output stability and detectability. A bibliography is also included, listing extensions, applications, and other current work.

1

Introduction

In this talk, I discuss the "input to state stability" way of thinking a b o u t nonlinear stability questions. I will be very informal - - the expository paper [68] should be consulted for more details and precise s t a t e m e n t s of results as of 1998, and several more recent citations are provided later. Consider the general "port" picture

=

A

y

I

W

--

here v and w are external signals, x the internal state. We s t u d y conditional (asymptotic) stability of v, w. There are two desirable, and c o m p l e m e n t a r y , features of stability:

9 asymptotic: "v small :O w small" - - where "small" m a y be interpreted as "-+ 0 when t -+ +cx~", "bounded", or via an e - J definition.

9 transient: "overshoot depends on initial state" - - with fading effect of

~(0)

=

x~

Our definitions a t t e m p t to c a p t u r e these two aspects.

444

Eduardo D. Sontag

Pictorially: U

W

't

't

T h e "magnitude" of a signal might be e.g.: 9 norm: Iw(t)l 9 error: [ w ( t ) - Wdesired(t)[ 9 distance to a set A: Iw(t)la = dist ( w ( t ) , . 4 ) - - e . g . . 4 = periodic orbit, a s k w ( t ) --+ A as t ~ ~

but in this presentation, we restrict ourselves to norms. (The literature usually deals with more general cases. For instance, results on internal stability are often given for [w(t)1.4. This generality allows considering issues such as full-state observer design, in which the relevant concepts concern stability with respect to the "diagonal" set .4 = { ( x , x)} where the states of the plant and observer coincide.) Specifically, let us consider i/o systems U(')

and various choices of v and w. Three central theoretical concepts for linear systems ic=Ax+Bu,

y=Cx

(to be generalized) are as follows: 1. Internal Stability ( i n p u t to s t a t e ) : v = u, w = x. 2. External Stability ( i n p u t to o u t p u t ) : v = u, w = y. 3. Detectability ( i n p u t a n d o u t p u t t o s t a t e ) : v = ( u , y ) , w = x . We will refer to them as the f u n d a m e n t a l triad. [

external stability

\

1

[

/

internal stability

detectability

]

The ISS Philosophy as a Unifying Framework for Stability-fike Behavior

445

I n t e r n a l S t a b i l i t y means that A is a Hurwitz matrix, i.e.: x(t) -4 0 for all solutions of x = A x , or equivalently, that x(t) --4 0 whenever u(t) -4 O, and moreover one has the explicit estimate

Ix(t)l ~< fl(t)lx~ where

= Ile'all

0 and 7

=

IIBll

[leall f0 ~176

ds

and Ilull~ = (essential) sup norm of u restricted to [0,t]. For t large, x(t) is bounded by 7 llulloo, independently of initial conditions; for small t, the effect of initial states m a y dominate. Note the superposition of transient and asymptotic effects. Internal stability will be generalized to "ISS" later, with the linear functions of Ix~ and Ilulloo replaced by nonlinear ones. External S t a b i l i t y means that the transfer function is stable or, in terms of a state-space realization, that an estimate as follows holds:

ly(t)l ~

/~(t)lx~

where 7 is a constant and /3 converges to zero (fl m a y be obtained from the restriction of A to a minimal subsystem). Note that even though we only require that y, not x, be "small" (relative to Ilull~), the initial internal states still affect the estimate in a "fading m e m o r y " manner, via the fl term. (For example, in P I D control, when considering the combination of plant, exosystem and controller, the overshoot of the regulated variable will be determined by the magnitude of the constant disturbance, and the initial state of the integrator.) External stability will generalize to "lOS". (Zero-)Deteetability u(t) =

c~(o - 0 ~

means that the unobservable part is stable i.e., u(t) - o ~

~:(t) ~ o as t ~

or equivalently: u(t) ~ o ~ u(t) - 4 o ~

x(t) -4 o

and can be also expressed by means of an estimate of the following form:

Ix(t)l ~ ~(t)lx~ +~111ulloo +'r~ Ilyll~ where 7i's are constants and fl converges to zero (now fl is obtained from a suitable matrix A - L C , where L is an observer gain). Zero-detectability's nonlinear version will be "IOSS".

446

Eduardo D. Sontag

T h e components of this triad are interrelated: external stability

&

detectability

~

internal

stability

- - this is a routine exercise in linear systems theory and obvious intuitively: 9 If internally stable, then x --+ 0 for all u --+ 0, so in particular this happens when C x ( t ) --+ 0 (detectability), and it always holds t h a t y ( t ) = C x ( t ) -+ 0 (i/o stability). 9 Conversely, if u --+ 0 then y ~ 0, (by external stability) and this then implies x -~ 0 (by detectability). Let us turn to the nonlinear generalizations. These generalizations will be so that, in particular, the above equivalence still holds true.

Input-to-State I

Stability

ext" stability

]

I int. stability

detectability

]

(/

We consider systems of the form :

f(x,u),

y = h(x)

evolving in finite-dimensional spaces ]~'~, and we suppose t h a t inputs u take values in ~ m and outputs y are ~P-valued. An i n p u t is a measurable locally essentially bounded u(.) : [0, cx~) --~ ~"~. We employ the notation I~1 for Euclidean norms, and use I1~11, or I1~11~ for emphasis, to indicate the essential s u p r e m u m of a function u(.). The m a p f : li~n x ~ , n __+ ~,, is locally Lipschitz and satisfies f(0, 0) = 0. T h e m a p h : I~ n --+ ~P is locally Lipschitz and satisfies h(0) = 0. T h e internal stability property for linear systems a m o u n t s to the " L ~ -+ L ~ finite-gain condition" that

Ix(t)l < clx~

+ c sup lu(s)l se[0,tl

holds for all solutions (assumed defined for all t > 0), where c and A > 0 and appropriate constants. W h a t is a reasonable nonlinear version of this? Two central characteristic of the ISS philosophy are: (1) using nonlinear gains rather than linear estimates, (2) not asking a b o u t exact values of gains but

The ISS Philosophy as a Unifying Framework for Stability-like Behavior

447

instead asking qualitative questions of existence: a "topological" vs. a "metric" point of view. (The linear analogy would be to ask "is the gain < (x)?" "is an operator bounded?") Our general guiding principle m a y be formulated thus:

notions of stability should be invariant under (nonlinear) changes of variables. By a change of variables in 11~l, let us mean here any transformation z = T ( x ) with T(0) = 0, where T : ~ t _.+/~t is a h o m e o m o r p h i s m whose restriction Tla,\{0) is a diffeomorphism. (We allow less differentiability at the origin in order to state elegantly a certain converse result later.) Let us see where this principle leads us, starting from the "Loo -+ Loo finitegain condition"

Ix(t)l < clx~ e-xt + c sup lu(s)l ,e[o,tl and taking both state and input coordinates changes x = T(z), u = S(v). For any input u and initial state x ~ and corresponding trajectory x(t) = x(t,x~ we let x(t) = T(z(t)), u(t) = S(v(t)), z ~ = z(0) = T - t ( x ~ For suitable functions a_.,~, ~ E/Coo, we have:

~(Izl) _< IT(z)l < ~(Izl) Vz 9 m" IS(v)l _< ~(1~1) V~ ~ ram. The condition Ix(t)l _< clx~

~_(Iz(t)l) < ce-X'~(Iz~

csup,e[o,tl lu(s)l becomes, in terms of z, v: + c sup ~(Iv(s)l) ,e[0,t]

v t > 0.

Using again "x" and "u" and letting/3(s,t) := ce-Xt-5(s) and 7(s) := c~(s), we arrive to this estimate, with/3 E / C s 7 E/Coo: _~(Ix(t)l) ___ /3(Ix~

+ 7(llulloo) .

(For a n y / C E function/3, there exist t~l, mz E/Coo with ([67])

/~(r,t) < ~ ( . ~ ( , - ) e - ' )

Vs, t

so the special form of/3 adds no extra information.) Equivalently, one m a y write (for different/3, 7)

Ix(t)l < /3(Ix~

+ 3'(llulloo)

or one m a y use "max" instead of "+" in the bound. A system is input to state stable (ISS) if such an estimate holds, for some /3 E/Cs 3' E/Coo. More precisely, for each x ~ u, the solution x(t) = x(t, x ~ u) is defined for all t > 0, and the estimate holds.

448 2.1

Eduaxdo D. Sontag Asymptotic

Gain Characterization

For u = O, the e s t i m a t e reduces to l~(t)l _< x~(l~~ so ISS i m p l i e s t h a t the unforced system k = f ( x , O) is ( a s y m p t o t i c a l l y ) s t a b l e (with respect to

x = 0). A n ISS system has a well-defined asymptotic 9ain: there is s o m e 3' E / C ~ so t h a t , for all x ~ and u: lim Ix(t,x~

< 3'(llulloo) -

#--x(0)

x(O

j

J

A far less obvious converse holds:

("Superposition principle for ISS") A system is ISS if and only if it admits an asymptotic 9ain and the unforced system is stable.

Theorem.

T h i s result is nontrivial, and constitutes the m a i n c o n t r i b u t i o n of the paper [73], which establishes as well m a n y other f u n d a m e n t a l c h a r a c t e r i z a t i o n s of the ISS property. T h e p r o o f hinges u p o n a r e l a x a t i o n t h e o r e m for differential inclusions, shown in t h a t p a p e r , which relates global a s y m p t o t i c s t a b i l i t y of an inclusion x E F(x) to global a s y m p t o t i c s t a b i l i t y of its convexification.

2.2

Dissipation

Characterization

of ISS

A s m o o t h , proper, and positive definite V : it~n --+ iI~ is an IS'S-Lyapunov

function for x = f(x, u) if, for s o m e 3', a E/Coo, V(x,u) = VV(x) f(x,u) <_ - a ( ] x ] ) + 3 ' ( l u l ) i.e., one has the dissipation i n e q u a l i t y

v(~(t~)) - v(~(t~))

<__

~ ( u ( s ) , ~,(s)) as 1

vx, u

The ISS Philosophy as a Unifying Framework for Stability-like Behavior

449

along all trajectories of the s y s t e m , with "supply" function w(u, x) -- ~ ( l u l ) -

~(Ixl) T h e following is a f u n d a m e n t a l result in ISS theory: Theorem.

[69] A system is ISS if and only if it admits an ISS-Lyapunov

function. (Sufficiency is easy: a differential i n e q u a l i t y for V provides an e s t i m a t e on

V(x(t)), and hence on [x(t)[. Necessity follows by a p p l y i n g a converse Lyapunov t h e o r e m for uniform G A S ([45]) over all Ildlloo < 1, to a s y s t e m of the form x = g(x, d) = f(x, dp(l~l)), for an a p p r o p r i a t e "robustness m a r g i n " p E /Coo. T h i s is in effect a s m o o t h converse L y a p u n o v t h e o r e m for locally Lipschitz differential inclusions.) 2.3

I S S is N a t u r a l

for Series Connections

Consider a cascade connection of ISS s y s t e m s

.~ =

f(z,x)

=

g(x, u)

(the z system is ISS with x as input).

Pick matching (ef. [83]) I S S - L y a p u n o v functions for each s u b s y s t e m :

?~(z, ~) _< o(1~1) - ,~(Izl) v~(~, u) < ~(1~1) - 2o(1~1). Then, W(x, z):= Vl(z)§ V~(x) is an I S S - L y a p u n o v function:

w(;~, z) _< ~(lul) - e(l~l) - o,(Izl) and so a cascade of ISS systems is ISS.

2.4

Generalization

t o S m a l l Gains

In p a r t i c u l a r , when u = 0, one o b t a i n s t h a t a cascade of a G A S and an ISS system is again G A S . More generally, one m a y allow i n p u t s u fed-back with "small gain": if u = k(z) is so t h a t Ik(z)l < 0 - 1 ( ( 1 - e)c~(Izl)), i.e.

~i(lul) _< (1 - ~)~(1~1) then

W(~,, z) < -O(l:~l) - eo~(Izl) and the closed-loop system is still G A S .

450

Eduardo D. Sontag

Even m o r e generally, under suitable conditions on gains ( S m a l l - G a i n Theorem [27] of J i a n g , Praly, and Teel) the closed loop s y s t e m o b t a i n e d from an interconnection of two ISS s y s t e m s z = f ( z , z, u) a n d k -- g ( z , z , v), is itself ISS with respect to (u, v).

2.5

Series Connections:

An

Example

As a simple i l l u s t r a t i o n of the cascade technique, consider t h e a n g u l a r mom e n t u m s t a b i l i z a t i o n of a rigid b o d y controlled by two torques a c t i n g along principal axes (for instance, a satellite controlled by two o p p o s i n g j e t pairs). If w = (wl, o32, o33) is the a n g u l a r velocity of a b o d y - a t t a c h e d f r a m e w i t h respect to inertial coordinates, and I = diag(I1, I s , / 3 ) are t h e p r i n c i p a l m o m e n t s of inertia, we o b t a i n the equations:

I& =

-o33

0

~2

--031

Iw +

v.

We assume Is :# /3; then, i n t r o d u c i n g new s t a t e and i n p u t c o o r d i n a t e s via (Is-I3)xl = Ilo31, x2 = w~, x3 = o33, I ~ u ~ = ( / 3 - I 1 ) o 3 1 o 3 3 + v t , and I 3 u z = (I1 - I2)o31w~ + v~, we o b t a i n a s y s t e m on ~ z , with controls in ]R~: E1 = X2X3 X2 = Ul

X3 =

U2 .

T h e n the following feedback law globally stabilizes the system: u1 :

- x I -- x 2 -

u2 :

-x3

x 2 x 3-.}- v 1

.-b x ~ --b 2 x t x 2 x 3

.-b v2

when vl = v2 - 0. T h e feedback was o b t a i n e d a r g u i n g in this way: with z2 : = xl + x2, z3 : = x3 - x2t, the s y s t e m becomes: xl ------x~ q- ~(xl, z=, z3) z= ------z2q-vl z 3 = --z3q-v2.

The ISS Philosophy as a Unifying Framework for Stability-like Behavior

451

The xl-subsystem is easily seen to be ISS, because d e g ~ a < 2 and hence the cubic term dominates, for large xl. Thus the cascade is also ISS; in particular, it is GAS if vl - v2 --- 0. (We also proved a stronger result: ISS implies a global robustness result with respect to actuator noise.) 2.6

Generalizations of Other Gains

ISS generalizes finite L ~176 ---> L ~ gains ("L 1 stability") but other classical norms often considered are induced L 2 --~ L 2 ( " H ~ " ) or L 2 --~ L cr ("H2"). Nonlinear transformations starting from "H~"

/0'

Ix(s)l 2 ds <_ clx~ 2 + c

/0'

lu(s)l 2 ds Vt > 0

lead to (for appropriate comparison functions):

I'

I'

a(l~(s)l) d. < ,~(Ix~ +

~(lu(s)l) ds Vt ___0.

T h e o r e m . There is such an "integral to integral" estimate if and only if the

system is ISS. The proof of this unexpected result is based upon certain known (and nontrivial) characterizations of the ISS property; see [67]. On the other hand, ,%2 __~ L~,, stability:

[x(t)l < c l x ~ - x t + c

Z'

[u(s)l 2 ds for all t > 0

leads to (for appropriate comparison functions):

~_(Ix(t)l) _< fl(l~~

"y(lu(s)l)ds

+

forallt>O.

This is the ilSS (integral ISS) property to which we'll return later. 2.7

Remark: Reversing Coordinate Changes

The "integral to integral" version of ISS arose, in the above discussion, from coordinate changes when starting from L2-induced operator norms. Interestingly, this result from [16] shows that the reasoning can be reversed: T h e o r e m . Assume n ~s 4,5. If ~ = f ( x , u ) is ISS, then, under a coordinate

change, for all solutions one has: t

2

~o Ix(s)[ ds ~- lx~

t

~

'U(S)I2 ds.

452

Eduardo D. Sontag

Similarly, global exponential stability is equivalent to global asymptotic stability. (Center manifold dimensions are not invariant, since coordinate changes are not necessarily {2~ at 0.) The cases n = 4, 5 are still open. A sketch of proof is as follows. Suppose 5: = f ( x , u) is ISS. Pick a robustness margin p E/Coo,so t h a t 5: = f(x, dp(lx]) ) is uniformly GAS over all ]]dl[~ < 1 and let V be s proper, positive definite, so that

V V ( x ) . f ( x , dp(Izl)) <_ - v ( x )

Vx, d.

Suppose (see below) that we have been able to change coordinates so that V(x) = Ix[ 2. So, W(z) := Y(T-l(z)) = ]z] 2 with z = T(x). Then, whenever

lul_
we have

d Izl =/at = W(z) = y ( ~ ) _< - y ( ~ ) So, if x E K ~ is so t h a t a ( r ) :=

= -Izl ~

]T(x)l _< x(P(l~l)),

and

max d ]z]2 ~dr I~l_
then:

dlzl 2 dt

-< - Izl~ + ,~(lul) = - I z l ~ + v

(v is the input in new coordinates) and integrating, one obtains f Izl 2 < Iz~ z + f Ivl ~. This gives the L 2 estimate as wanted. The critical technical step, thus, is to show that, up to coordinate changes, every Lyapunov function V is quadratic - - let us provide a sketch of the proof. First notice that the level set S := {V(x) = 1} is homotopically equivalent to S n - I (this is well-known: S • ~ _~ S because ~ is contractible, and S • is homeomorphic to I~ n \ {0} ~- ~n-1 via the flow of 5: = f(x,O)). Thus, {V(x) = 1} is diffeomorphic to ~n-1, provided n r 4, 5 (h-cobordism theory of Smale and Milnor; Poincard would give a homeomorphism, for n :/= 4). Finally, consider the normed gradient flow 5:_

vv(x)' IVV(~)l ~

and take the new variable z := v / v ( x ) o(x') where z ' is the translate via the flow back into the level set, and # : {V = 1} _~ {Izl = 1} is the given diffeomorphism. The picture is as follows:

The ISS Philosophy as a Unifying Framework for Stability-like Behavior

453

., e ( x )

..

~-~/Vffil

~

~1=1

(Actually, this sketch is not quite correct: one needs to make a slight adj u s t m e n t in order to o b t a i n also c o n t i n u i t y and differentiability at the origin; the actual coordinate change is z = "/(V(x))8(x'), so IV(z) = 7(Izl), for a suitable 7.)

3

Integral-Input

to S t a t e S t a b i l i t y

The "L 2 ~ L ~ " operator gain property led to ilSS:

~_(Ix(t)l) _< ~(Ix~

+

~(lu(s)l)

ds.

There is a dissipation characterization here as well. A smooth, proper, and positive definite V : ~ n __+ ~ is an ilSS-Lyapunov function for k = f(x, u) if for some positive definite continuous (~ a n d 7 E/Coo

~rV(x)f(x, u) <- -~(t~1) +~(lul)

v~ ~ ~n, u ~ ~m

- - observe that we are not requiring now ~ E }Coo. (Intuitive!y: even for c o n s t a n t u one may have l)" > 0, b u t "r(lul) E s m e a n s t h a t V is "often" negative.) A recent result from [5] is this: T h e o r e m . A system is ilSS if and only if it admits an ilSS-Lyupunov func-

tion. Since any/Cr function is positive definite, every ISS system is also ilSS, b u t the converse is false. For example, a bilinear system m

k = (A + Z u i A i ) x

+ Bu

i=1

is iISS if and only if A is a Hurwitz m a t r i x , but in general it is n o t ISS - 171 e.g., if B = 0 and A + ~i=1 u~ is not Hurwitz for some u ~ As a n o t h e r example, take k = - t a n -1 x + u. This is not ISS, since b o u n d e d i n p u t s m a y produce u n b o u n d e d trajectories; b u t it is ilSS, since V(x) = x t a n -1 x is a n iISS-Lyapunov function.

454

Eduardo D. Sontag

3.1

An Application ofiISS Theory

Let us illustrate the ilSS results through an application which, as a matter of fact, was the one that originally motivated much of the work in [5]. Consider a rigid manipulator with two controls: M +.'"+++

The arm is modeled as a segment with mass M and length L, and the hand as a point with mass m. Denoting by r the position and by 8 the angle of the arm, the resulting equations are: ( m r ~ + M L ~ / 3 ) O + 2mr§

= ~,

,~ - mr~ 2 = F

where F and r are the external force and torque. In a typical passivity-based tracking design one takes r

"=

F :=

--kd, O -- ~v, (O - Od) - - k d ~ § -- k w ( r

--

rd)

where r d and Od are the desired signals and the gains (kd~,...) are > O. For constant reference Od, rd, there is tracking: 9 -+ Od, 0 --+ O, and analogously for r. But, what about time-varying Od, rd .~ Can these destabilize the system? Yes:

there are bounded inputs which produce "nonlinear resonance" - - so the system can't be ISS (not even bounded-input bounded-state). The figures that follow show the "r" component of the state of a certain solution which corresponds to the shown input (see [5] for details on how this input and trajectory were calculated). Ul

2o

4o

6o

Jo

ioo

The ISS Philosophy as a Unifying Framework for Stability-like Behavior

455

j r

20

40

60

80

10

On the o t h e r h a n d , m a n y i n p u t s are not d e s t a b i l i z i n g - - how does one formulate q u a l i t a t i v e l y this fact? One way is by showing t h a t t h e s y s t e m is iISS. T h e closed-loop s y s t e m is 4-dimensional, with s t a t e s (q, q), q = (0, r) and u = (kp~0~, kp~d):

( m r s + M L 2 / 3 ) 0 + 2mr§

= Ul

--

kdl~

--

kplO

mi: -- mr~ s = us -- kd2~" -- kp2r To prove ilSS, we consider the mechanical energy V a n d note the following passivity-type estimate: d d-"~V(q(t), (l(t)) < - c l l 4 ( t ) l s + cslu(t)l 2 for sufficiently s m a l l cl > 0 and large cs > 0. In general, we say t h a t a s y s t e m is h-dissipative with respect to an o u t p u t function y = h(x) (continuous and with h(0) = 0) if for s o m e C ~ positive definite, p r o p e r V : ~ n _+ ]~, a n d for some % 0 as a b o v e

VV(x)f(x,u) <_ -~(h(x))

+7(lul)

Vx ~ ~ n , u ~ ~

and weakly h-detectable if, for all trajectories, y(t) = h ( x ( t ) ) --- 0 implies t h a t x(t) --+ 0 as t --> ~ . This is proved in [5]: T h e o r e m . A system is ilSS if and only if it is weakly h-detectable and hdissipative for some output h. W i t h o u t p u t q, our e x a m p l e is weakly z e r o - d e t e c t a b l e a n d dissipative, since u = 0 and q - 0 i m p l y q = 0. T h u s it is ilSS, as c l a i m e d .

3.2

Mixed Notions

Changes of variables t r a n s f o r m e d "finite L 2 gain" to an "integral to integral" property, which t u r n s o u t to be equivalent to ISS. F i n i t e gain as o p e r a t o r s

456

EduardoD. Sontag

between L v and Lq spaces, with p ~ q both finite, lead instead to this type of "weak integral to integral" estimate:

/0'

~_(l~(s)l)

ds <

,~(1~~ + ~

(/0'

7(lu(s)l) ds

)

for appropriate /Coo functions (note the additional "a"). See [6] for more discussion on how this estimate is reached, as well as this result: T h e o r e m . A system satisfies a weak integral to integral estimate if and only if it is ilSS. Another interesting variant results by studying mixed integral/supremum estimates:

~_(l~(t)l ___ /~(Ix~

+

71(lu(s)l)ds + 72(11u11~)

for suitable/3 E/C/: and ~,71 E/Coo. This result is also from [6]: T h e o r e m . The system ~ = f ( x , u) satisfies a mixed estimate if and only if it is ilSS.

4

Input/Output

Stability

I ext. st.bility I ~

I int. stability

(

detectability

I

The second component of the fundamental triad is input to output stability (lOS) for systems with outputs z = f ( x , u), y = h(x):

ly(t)l < ~'(l~~

sup

~e[o,t]

7(1~(~)1)

for all solutions, assuming completeness (for some /3 E K~s and 7 E /Coo). This is closely related to "partial stability" (if h is a projection, so y is a subset of variables), and "stability with respect to two measures". A dissipation (Lyapunov-) type characterization of this property is as follows. An lOS-Lyapunov function is a smooth V : ~ n __+ ~>0 so that, for some ai E/Coo, for all z E ]~n, U E ]~m:

~l(Ih(x)l) _< V(~) ___ ~(Ixl)

The ISS Philosophy as a Unifying Framework for Stability-like Behavior

457

and V(x) > c~3(N[) =~ V Y ( x ) f ( x , u) < 0. For systems that are bounded-input bounded-state stable, we have (see [77]): T h e o r e m . A system x : f ( x , u), y : h(x) is I O S if and only if it admits an IOS-Lyapunov function. 4.1

Motivation: Regulator

Theory

One m a y re-interpret this result as the existence of a new o u t p u t m a p = a ' ~ l ( V ( x ) ) which dominates the original o u t p u t (y < ~ and which is monotonically decreasing (no overshoot) as long as inputs are small. This is, in fact, one generalization of a central argument used in regulator theory (Francis equations). Let us sketch how IOS is motivated by regulator theory. (See the paper [76] for some more details.) In a regulator system, for each exogenous signal d(.) (a disturbance to be rejected, a signal to be tracked), the o u t p u t y(-) ("error") must decay to zero as t --+ or One assumes t h a t the exogenous signal is generated by an "exosystem" described by differential equations. For example, for linear systems (a nonlinear version is also well-known, cf. [23]) one studies the closed-loop system k=Az+Pw,

w=Sw,

y=Cz+Qw

seen as a system x = f ( x ) , y = h(x), where the extended state x consists of z and w, and the z-subsystem incorporates both the state of the system being regulated (the plant) and the state of the controller, and the equation ~b = S w describes the exosystem. (Later, we introduce inputs into the model.) For example, a second order system ~ - y = u + w under the action of all possible constant disturbances w leads to the conventional proportionalintegral-derivative (PID) controller given by a feedback law u(t) = clq(t) + c~y(t) + c3v(t), for appropriate gains cl, c2, c3, where q = f y and v = Y. Let us take cl = - 1 , c2 = c3 = - 2 . Viewing disturbances as produced by the exosystem ~b = 0, the complete system is q=y,

y=v,

~=-q-y-2v+w,

w=0

with output y, z = col (q, y, v). The routine way to verify the regulation objective is: one assumes that A is Hurwitz (after feedback) and that there is some m a t r i x /7 solving Francis' equations: II S = A/7 + P ,

O = C /7 + Q .

458

Eduardo D. Sontag

Consider the new variable ~ := z - I I w . The first identity for / / allows decoupling ~) from w, leading to ~ = A~). Since A is a Hurwitz matrix, one concludes that ~)(t) --~ 0 for all initial conditions. As the second identity for / / g i v e s that y(t) = C y ( t ) , one has the desired conclusion that y(t) -+ O. The key fact is that the new output 9 dominates the old (lYl < c I~)1) and (for some a E 1 ~ )

I~)(t)l < ~(iy(0)i),

vt > 0

i.e., the overshoot for this (also stable) output depends only on its initial condition. Note that a zero initial value ~)(0) implies ~) = 0 (initial state of the internal model and exosignal match), but this is false for the regulated variable. For example, in the PID regulator, ~) replaces q by q - w (internal model disturbance); but with e.g. x(0) = y(O) = v(0) = 0 and w(0) = 1, y ( t ) = 89 - t overshoots (even if y(0) = 0).

--

By a further modification (introduce a Lyapunov function for the ~) subsystem), we also have that ~)(t) can be defined so that it decreases monotonically. The significance of this interpretation is that, instead of output zeroing submanifolds, one considers two functions to be compared in amplitude, one corresponding to zero error, the other to a new and well-behaved output map. This "comparison in amplitude" (CIA) principle is a general theorem for nonlinear systems, via the results in [76] and [77]. The usual formulation, motivated by linear theory, includes no external inputs. Inputs allow studying the effect on the feedback system of exosignals not exactly represented by the exosystem model. The lOS property amounts to asking small steady-state error if the exosignal is "close" to the model. The paper [76] should be consulted for a "catalog" of variants of the lOS notion, and its companion paper [77] for the corresponding Lyapunov characterizations.

5

Zero-Detectability: [

ext. stability

IOSS ]

h

i int. stability

detectability

i

S

The third component of the fundamental triad, zero-detectability is typically defined by asking "u = 0 and y = 0 ::~ x ( t ) --+ 0 as t -+ oo" - - this is too weak a property for nonlinear systems: it is not "well-posed" (what happens if u, y ~ 0?).

The ISS Philosophy as a Unifying Framework for Stability-like Behavior

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More natural is input/output to state stability (IOSS):

Ix(t)l _< /3(l~~

+ sup "7(lu(~)l) + sup "r(ly(s)l) se[O,t]

se[O,t]

along all solutions (for some/3 E KS/:, 7 E/Coo) which results from the linear detectability estimate under coordinate changes. It implies, in particular, x --r 0 if both u, y -+ 0 as t --+ ~ .

u--+0 [

y-+0

The terminology IOSS is self-explanatory: formally, there is "stability from the i/o d a t a to the state".

5.1

Dissipation Characterization

of IOSS

A smooth, proper, and positive definite V : ~ n _.+ ]R is an IOSS-Lyapunov

function if, for some ai E/Coo,

~ v ( x ) f(~, u) < -al(l~l) + a~(lul) + a3(lyl) for all x E ~ n , u E ~m. This is from [35] and [36]: T h e o r e m . A system ~ = f ( x , u), y = h(x) is lOSS if and only if it admits

an IOSS-Lyapunov function. As a corollary, IOSS is equivalent to the existence of a norm-estimator: driven by the i / o d a t a generated by the original system, it estimates an upper bound on the internal state.

,[------] u

y D

[

z

I

, w

x

This is defined as a system ~ = g(z, u, y), w = e(z), whose inputs are the i / o pairs of the original system, which is ISS with respect to u, y as inputs (so that there is robustness to signal errors), and, for some p E KS and fl E KS/:,

Ix(t)l _ /3(Ix~176

+ p(Iw(t)l)

vt ~ 0

for all initial states x ~ and z ~ (See the paper [36] for the precise definition.)

460 6

Eduardo D. Sontag Comments

There are m a n y foundational directions still being explored. Let me s u m m a rize just a few of them: 9 Observers require a notion of incremental IOSS, not merely distinguishing from x = 0. This is a very appealing problem. See [75] for some preliminary remarks. 9 An ISS-like "globally m i n i m u m phase" property - - m e a n i n g that "the zero-dynamics are ISS" can be defined with no recourse to normal forms. (See [41] for a preliminary version.) 9 A c o m m o n generalization of lOSS and lOS is, for "regulated" and "measured" o u t p u t s w, y: Iw(t)l < /~(Ix~

+

sup 7(lu(s)l ) -F sup "r(ly(s)l) 9e[0,t] ,e[0,t]

along along all solutions, for appropriate comparison functions; moreover, one m a y allow the overshoot to depend on yet another fixed function of x ~ such as a distance to a set. Characterizations are now being worked out, cf. [21]. One might call this property " i n p u t / m e a s u r e m e n t to o u p u t stability" (IMOS). I have focused on basic theoretical constructs, instead of on applications, in this brief survey. The next section provides references to more work related to ISS-related theory and applications.

7

Additional

References

Textbooks and research monographs which make use of ISS and related concepts include [13,24,37,38,32,60]. After the definition in [63] and the basic characterizations in [69], the main results on ISS are given in [73]. See also [8,78] for early uses of a s y m p t o t i c gain notions. "Practical" ISS is equivalent to ISS with respect to c o m p a c t attractors, see [71]. Several authors have pointed out that time-varying system versions of ISS are central to the analysis of asymptotic tracking problems, see e.g. [87]. In [10], one can find further results on L y a p u n o v characterizations of the ISS property for time-varying (and in particular periodic) systems, as well as a small-gain theorem based on these ideas. Perhaps the most interesting set of open problems concerns the construction of feedback laws that provide ISS stability with respect to observation errors. Actuator errors are far better understood (cf. [63]), but save for the case of special structures studied in [13], the one-dimensional case (see e.g. [11])

The ISS Philosophy as a Unifying Framework for Stability-like Behavior

461

and the counterexample [12], little is known of this fundamental question. Recent work analyzing the effect of small observation errors (see [68]) might provide good pointers to useful directions of research (indeed, see [40] for some preliminary remarks in that direction). For special classes of systems, even output feedback ISS with respect to observation errors is possible, cf. [52]. Both ISS and iISS properties have been featured in the analysis of the performance of switching controllers, cf. [17] and [18]. Coprime factorizations are the basis of the parameterization of controllers in the Youla approach. As a matter of fact, as the paper's title indicates, their study was the original motivation for the introduction of the notion of ISS in [63]. Some further work can be found in [64], see also [14], but much remains to be done. There are now results on averaging for ISS systems, see [54], as well as on singular perturbations, see [7]. Discrete-time ISS systems are studied in [31] and in [29]; the latter paper provides Lyapunov-like sufficient conditions and an ISS small-gain theorem, and more complete characterizations and extensions of m a n y standard ISS results for continuous time systems are given in [30]. Discrete-time iISS systems are the subject of [2], who proves the very surprising result that, in the discrete-time case, iISS is actually no different than global asymptotic stability of the unforced system (this is very far from true in the continuous-time case, of course). In this context, of interest are also the relationships between the ISS property for a continuous-time system and its sampled versions. The result in [80] shows that ISS is recovered under sufficiently fast sampling; see also the technical estimates in [53]. Stochastic ISS properties are treated in [86]. A very interesting area regards the combination of clf and ISS like-ideas, namely providing necessary and sufficient conditions, in terms of appropriate clf-like properties, for the existence of feedback laws (or more generally, dynamic feedback) such that the system x = f(x, d, u) becomes ISS (or iISS, etc) with respect to d, once that u = k(x) is substituted. Notice that for systems with disturbances typically f(0, d, 0) need not vanish (example: additive disturbances for linear systems), so this problem is qualitatively different from the robust-clf problem since uniform stabilization is not possible. There has been substantial work by m a n y authors in this area; let us single out among them the work [81], which deals primarily with systems of the form k = f(x, d) + g(x)u (affine in control, and control vector fields are independent of disturbances) and with assigning precise upper bounds to the "nonlinear gain" obtained in terms of d, and [9], which, for the class of systems that can be put in output-feedback form (controller canonical form with an added stochastic output injection term), produces, via appropriate clf's, stochastic ISS behavior ("NSS" = noise to state stability, meaning that so-

462

Eduardo D. Sontag

lutions converge in probability to a residual set whose radius is proportional to bounds on covariances). In connection with our example from tracking design for a robot, we mention here that the paper [50] proposed the reformulation of tracking problems by means of the notion of input to state stability. The goal was to strengthen the robustness properties of tracking designs, and the notion of ISS was instrumental in the precise characterization of performance. Incidentally, the same example was used, for a different purpose - - namely, to illustrate a different nonlinear tracking design which produces ISS, as opposed to merely iISS, behavior - - in the paper [1]. Neural-net control techniques using ISS are mentioned in [59]. A problem of decentralized robust output-feedback control with disturbance attenuation for a class of large-scale dynamic systems, achieving ISS and iISS properties, is studied in [28].

Incremental ISS is the notion that estimates differences I x l ( t ) - x2(t)l in terms of K:s decay of differences of initial states, and differences of norms of inputs. It provides a way to formulate notions of sensitivity to initial conditions and controls (not local like Lyapunov exponents or as in [46], but of a more global character, see [3]); in particular when there are no inputs one obtains "incremental GAS", which can be completely characterized in Lyapunov terms using the result in [45], since it coincides with stability with respect to the diagonal of the system consisting of two parallel copies of the same system. This area is of interest, among other reasons, because of the possibility of its use in information transmission by synchronization of diffusively coupled dynamical systems ([56]) in which the stability of the diagonal is indeed the behavior of interest. Small-gain theorems for ISS and IOS notions originated with [27]; a purely operator version (cf. [20]) of the IOS small-gain theorem holds as well. There are ISS-small gain theorems for certain infinite dimensional classes of systems such as delay systems, see [79]. The notion of IOSS is called "detectability" in [64] (where it is phrased in input/output, as opposed to state space, terms, and applied to questions of parameterization of controllers) and was called "strong unboundedness observability" in [27]. IOSS and its incremental variant are very closely related to the OSS-type detectability notions pursued in [34]; see also the emphasis on ISS guarantees for observers in [49]. The use of ISS-like formalism for studying observers, and hence implicitly the IOSS property, has also appeared several times in other authors' work, such as the papers [19,47,55]. It is worth pointing out that several authors had independently suggested that one should define "detectability" in dissipation terms. For example, in [48], Equation 15, one finds detectability defined by the requirement that there should exist a differentiable storage function V satisfying our dissipation inequality but with the special choice a3(r) := r ~ (there were no inputs in

The ISS Philosophy as a Unifying Framework for Stability-like Behavior

463

the class of systems considered there). A variation of this is to weaken the dissipation inequality, to require merely

# o ~ v v ( ~ ) f ( x , u ) < ~a(lyl) (again, with no inputs), as done for instance in the definition of detectability given in [51]. Observe that this represents a slight weakening of our property, in so far as there is no "margin" of stability - a l ([xD. Norm-estimators are motivated by developments appeared in [26] and [57]. The notion studied in [62] is very close to the combination of IOSS and IOS being pursued in [21]. Partial asymptotic stability for differential equations is a particular case of output stability (IOS when there are no inputs) in our sense; see [90] for a survey of the area, as well as the book [58], which contains a converse theorem for a restricted type of o u t p u t stability. (We thank A n t o n Shiriaev for bringing this latter reference to our attention.)

References 1. Angeli, D., "Input-to-state stability of PD-controlled robotic systems," Automatica 35 (1999): 1285-1290. 2. Angeli, D., "Intrinsic robustness of global asymptotic stability," Systems Control Letters 38(1999): 297-307 3. Angeli, D., "A Lyapunov approach to incremental stability properties," submitted for publication, January 2000. 4. Angeli, D., and E.D. Sontag, "Forward completeness, unboundedness observability, and their Lyapunov characterizations," Systems and Control Letters 38(1999): 209-217. 5. Angeli, D., E.D. Sontag, and Y. Wang, "A characterization of integral input to state stability," IEEE Trans. Aurora. Control, to appear. 6. Angeli, D., E.D. Sontag, and Y. Wang, "Further equivalences and semiglobal versions of integral input to state stability," Dynamics and Control, to appear. 7. Christofides, P.D., and A.R. Teel, "Singular perturbations and input-to-state stability," IEEE Trans. Automat. Control 41(1996): 1645-1650. 8. Coron, J.M., L. Praly, and A. Teel, "Feedback stabilization of nonlinear systems: sufficient conditions and Lyapunov and input-output techniques," in Trends in Control, A. lsidori, ed., Springer-Verlag, London, 1995. 9. Deng, H., and M. Krsti~, "Output-feedback stablization of stochastic nonlinear systems driven by noise of unknown covariance," Systems Control Left. 39(2000): 173-182. 10. Edwards, H., Y. Lin, and Y. Wang, "On input-to-state stability for time varying nonlinear systems," submitted, February 2000. 11. Fah, N.C.S., "Input-to-state stability with respect to measurement disturbances for one-dimensional systems," Control, Optimisation and Calculus of Variations 4(1999): 99-121.

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12. Freeman, R.A., "Global internal stabilizability does not imply global external stabifizability for small sensor disturbances," IEEE Trans. Automat. Control 40(1996): 2119-2122. 13. Freeman, R.A., and P.V. Kokotovi'c, Robust Nonlinear Control Design, StateSpace and Lyapunov Techniques, Birkhauser, Boston, 1996. 14. Fujimoto, K., and T. Sugie, "State-space characterization of Youla parametrization for nonlinear systems based on input-to-state stabifity", Prac. 37th IEEE Conf. Decision and Control, Tampa, Dec. 1998, pp. 2479-2484. 15. Gr/ine, L., "Input-to-state stability of exponentially stabifized semilinear control systems with inhomogeneous perturbations," System ~4 Control Letters 38(1999): 27-35. 16. Grune, L., F.R. Wirth, and E.D. Sontag, "Asymptotic stability equals exponential stability, and ISS equals finite energy gain - if you twist your eyes," Systems and Control Letters 38 (1999): 127-134. 17. Hespanha, J.P, and A.S. Morse, "Certainty equivalence implies detectabifity," Systems and Control Letters 36(1999): 1-13. 18. Hespanha, J.P, and A.S. Morse, "Supervisory control of integral-input-to-state stabilizing controllers," Proc. of the 5 th European Control Conference, Karlsrtthe, September 1999. 19. Hu, X.M., "On state observers for nonlinear systems," Systems ~ Control Letters 17 (1991), pp. 645-473. 20. lngalls, B., and E.D. Sontag, "A purely input/output version of the IOS small gain theorem," in preparation. 21. Ingalls, B., E.D. Sontag, and Y. Wang, in preparation. 22. Isidori, A., "Global almost disturbance decoupling with stability for non minimum-phase single-input single-output nonlinear systems," Systems ~d Control Letters 28(1996): 115-122. 23. Isidori, A., Nonlinear Control Systems, Third Edition, Springer-Verlag, London, 1995. 24. Isidori, A., Nonlinear Control Systems II, Springer-Verlag, London, 1999. 25. Jiang, Z.-P., and I.M. Mareels, "A small-gain control method for nonlinear cascaded systems with dynamic uncertainties," IEEE Trans. Automat. Control 42(1997): 292-308. 26. Jiang, Z.-P., and L. Praly, "Preliminary results about robust Lagrange stability in adaptive nonlinear regulation," Int. J. of Adaptive Control and Signal Processing 6(1992): 285-307. 27. Jiang, Z.-P., A. Teel, and L. Praly, "Small-gain theorem for ISS systems and applications," Mathematics of Control, Signals, and Systems 7(1994): 95-120. 28. Jiang, Z.-P., F. Khorrami, and D.J. Hill, "Decentralized output-feedback control with disturbance attenuation for large-scale nonlinear systems," Proc. 38th IEEE Conf. Decision and Control, Phoenix, Dec. 1999, pp. 3271-3276. 29. Jiang, Z-P., E.D. Sontag, and Y. Wang, "Input-to-state stability for discretetime nonlinear systems," in Proc. 14th IFAC World Congress (Beijing), Vol E, pp. 277-282, 1999. 30. Jiang, Z-P., and Y. Wang, "A converse lyapunov theorem and input-to-state stability properties for discrete-time nonlinear systems," Automatica, to appear. 31. Kazakos, D., and J. Tsinias, "The input-to-state stability condition and global stabilization of discrete-time systems," IEEE Trans. Automat. Control 39(1994): 2111-13.

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53. Ne~i~, D., A.R. Teel, and E.D. Sontag, "Formulas relating /Cs stability estimates of discrete-time and sampled-data nonlinear systems," Systems and Control Letters 38 (1999): 49-60. 54. Ne~i~, D., and A.R. Teel, "Input-to-state stability for nonlinear time-varying systems via averaging", submitted for publication, 1999. 55. Pan, D.J., Z.Z. Han, and Z.J. Zhang, "Bounded-input-bounded-output stabilization of nonlinear systems using state detectors," Systems Control Lett., 21(1993): 189-198. 56. Pogromsky A.Yu., T. Glad and H. Nijmeijer "On diffusion driven oscillations in coupled dynamical systems," Int. J. Bifurcation Chaos 9(1999): 629-644. 57. Praly, L., and Y. Wang, "Stabilization in spite of matched unmode]]ed dynamics and an equivalent definition of input-to-state stability," Mathematics of Control, Signals, and Systems 9(1996): 1-33. 58. Rumyantsev, V.V., and A.S. Oziraner, Stability and Stabilization of Motion with Respect to Part of the Variables (in Russian), Nattka, Moscow , 1987. 59. Sanchez, E.N., and J.P. Perez, "Input-to-state stability (ISS) analysis for dynamic neural networks," IEEE Trans. Circuits and Systems L" Fundamental Theory and Applications, 46(1999): 1395-1398. 60. Sepulchre, R., M. Jankovic, P.V. Kokotovi~, Constructive Nonlinear Control, Springer, 1997. 61. Sepulchre, R., M. Jankovic, and P.V. Kokotovi6, "Integrator forwarding: a new recursive nonlinear robust design," Automatica 33 (1997): pp. 979-984. 62. Shiriaev, A.S., "The notion of V-detectability and stabilization of invariant sets of nonlinear systems," Proc. 37th IEEE Conf. Decision and Control, Tampa, Dec. 1998, pp. 2509-2514. 63. Sontag, E.D., "Smooth stabilization implies coprime factorization," IEEE Trans. Automatic Control 34(1989): 435-443. 64. Sontag, E.D., "Some connections between stabilization and factorization," Proc. IEEE Conf. Decision and Control, Tampa, Dec. 1989, IEEE Publications, 1989, pp. 990-995. 65. Sontag, E.D., "Remarks on stabilization and input-to-state stability," Proc. [EEE Conf. Decision and Control, Tampa, Dec. 1989, IEEE Publications, 1989, pp. 1376-1378. 66. Sontag, E.D., "Further facts about input to state stabilization", IEEE Trans. Automatic Control 35(1990): 473-476. 67. Sontag, E.D., "Comments on integral variants of ISS," Systems and Control Letters 34(1998): 93-100. 68. Sontag, E.D., "Stability and stabilization: Discontinuities and the effect of disturbances," in Nonlinear Analysis, Differential Equations, and Control (Proc. NATO Advanced Study Institute, Montreal, Jul/Aug 1998; F.H. Clarke and R.J. Stern, eds.), Kluwer, Dordrecht, 1999, pp. 551-598. 69. Sontag, E.D., and Y. Wang, "On characterizations of the input-to-state stability property," Systems and Control Letters 24 (1995): 351-359. 70. Sontag, E.D., and Y. Wang, "On characterizations of input-to-state stability with respect to compact sets," in Proceedings of IFAC Non-Linear Control Systems Design Symposium, (NOLCOS '95), Tahoe City, CA, June 1995, pp. 226-231. 71. Sontag, E.D., and Y. Wang, "Various results concerning set input-to-state stability," Proc. IEEE Conf. Decision and Control, New Orleans, Dec. 1995, IEEE Publications, 1995, pp. 1330-1335.

The ISS Philosophy as a Unifying Fi'amework for Stability-like Behavior

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72. Sontag, E.D., and Y. Wang, "Detectability of nonlinear systems," in Proc. Conf. on Information Sciences and Systems (CISS 96), Princeton, N J, 1996, pp. 1031-1036. 73. Sontag, E.D., and Y. Wang, "New characterizations of input to state stability," IEEE Trans. Aurora. Control 41(1996): 1283-1294. 74. Sontag, E.D., and Y. Wang, "A notion of input to output stability," Proe. European Control Conf., Brussels, July 1997, Paper WE-E A2, CD-ROM file ECC958.pdf, 6 pages. 75. Sontag, E.D., and Y. Wang, "Output-to-state stability and detectability of nonlinear systems," Systems and Control Letters 29(1997): 279-290. 76. Sontag, E.D., and Y. Wang, "Notions of input to output stability," Systems and Control Letters 38(1999): 235-248. 77. Sontag, E.D., and Y. Wang, "Lyapunov characterizations of input to output stability," S I A M J. Control and Opt., to appear. 78. Sussmann, H.J., E.D. Sontag, and Y. Yang, "A general result on the stabilization of linear systems using bounded controls," IEEE Trans. Autom. Control 39(1994): 2411-2425. 79. Teel, A.R., "Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem," IEEE Trans. Automat. Control 43(1998): 960-964. 80. Teel, A.R., D. Ne~i~, and P.V. Kokotovi~, "A note on input-to-state stability of sampled-data nonlinear systems," Proc. 37th IEEE Conf. Decision and Control, Tampa, Dec. 1998, pp. 2473-2479. 81. Teel, A.R., and L. Praly, "On assigning the derivative of a disturbance attenuation control," submitted, 1999. 82. Teel, A.R., and L. Praly, "Results on converse Lyapunov functions from classKL estimates," Proc. 38th IEEE Conf. Decision and Control, Phoenix, Dec. 1999, pp. 2545-2550. 83. Teel, A., and E.D. Sontag, "Changing supply functions in input/state stable systems," IEEE Trans. Aurora. Control 40(1995): 1476-1478. 84. Tsinias, J., "Sontag's 'input to state stability condition' and global stabilization using state detection," Systems Control Lett., 20(1993): 219-226. 85. Tsinias, J., "Input to state stability properties of nonlinear systems and applications to bounded feedback stabilization using saturation," E S A I M Control Optim. Calc. Var. 2(1997): 57-85. 86. Tsinias, J., "Stochastic input-to-state stability and applications to global feedback stabilization," Int. J. Control 71(1998): 907-930. 87. Tsinias, J., and I. Karafyllis, "ISS property for time-varying systems and application to partial-static feedback stabilization and asymptotic tracking," IEEE Trans. Automat. Control 44(1999): 2173-2184. 88. Su, W., L. Xie, and Z. Gong, "Robust input to state stabilization for minimumphase nonlinear systems," Int J. Control 66(1997): 825-842. 89. van der Schaft, A., "L2 Gain and Passivity Techniques in Nonlinear Control," Springer-Verlag, London, 1999. 90. Vorotnikov, V.I., "Stability and stabilization of motion: Research approaches, results, distinctive characteristics," Automation and Remote Control54 (1993): 339-397.

Control Design of a Crane for Offshore Lifting Operations Michael P. S p a t h o p o u l o s and D i m o s t h e n i s F r a g o p o u l o s System Dynamics and Control Department of Mechanical Engineering University of Strathclyde Glasgow G1 1X J, Scotland mps@mecheng, s t r a t h , ac. uk

A b s t r a c t . This paper presents a control system that reduces pendulation of suspended loads in offshore lifting operations. We develop a model of the ship crane together with a designed anti-pendulation arm. We derive two different types of models for the system and we apply non-linear and linear control design techniques. The non-linear design involves Liapunov analysis and the linear involves LQG and generalized predictive control. The advantage for the linear control is that the use of the vessel dynamics and the sea wave disturbances in the control design improves considerably the behavior of the controlled system

1

Introduction

T h e vast m a j o r i t y of crane systems used offshore have no m e a n s to c o m p e n sate for the m o t i o n s induced on the s u s p e n d e d load caused by the sea, wind and other external disturbances. These induced m o t i o n s t a k e the form of heave (vertical motion) and p e n d u l a t i o n of the load. Previous efforts on the c o m p e n s a t i o n for disturbances induced on the suspended load have resulted in passive systems i n c o r p o r a t i n g d a m p i n g elements tuned to c o m p e n s a t e for specific conditions of load mass and sea state. These however have a l i m i t e d success on the crane. T h e proposed active control s y s t e m a i m s at m a i n t a i n i n g the position of the suspended load within a sphere of o n e - m e t e r r a d i u s relative to an inertial frame of m e a s u r e m e n t . T h e control s y s t e m uses two different a c t u a t o r s to achieve this aim. In the vertical plane defined by t h e crane b o o m , a c t u a t i o n is provided by a swinging arm (anti-pendulation arm) pivoted at the t o p of the boom. A hydraulic piston is used to force this a r m to move. T h e cable holding the load is hanging from the lower end of this a r m . T h u s the b o o m is not used (in luI~ing) for p e n d u l a t i o n control, as n o r m a l l y the b o o m is a slow m o v i n g device whose effectiveness is further l i m i t e d at high b o o m inclinations. In the lateral direction the whole crane is r o t a t e d a r o u n d a vertical axis (slewing). T h e second a c t u a t i o n corresponding to the crane slewing m o t i o n r o t a t e s the base of the crane a b o u t a vertical axis.

470

M.P. Spathopoulos and D. Fragopoulos

In terms of measurements the instantaneous position of the load has to be calculated from the measurements of ship motions, suspension line inclination and boom angle, and from the geometry of the crane structure as well as the length of the line. In terms of the ship motions heave, roll and pitch are assumed measurable for this work. Consider the 3D model of a pendulum consisting of a point mass attached to a light inelastic cable with accelerating cable-top, moving in space. When this model is simplified by the assumption of small angle oscillations, the dynamics become decoupled so that the system may be described by the dynamics of two mutually perpendicular vertical planes. The 3D motion of the pendulum may then be described as the Cartesian combination of the motion in these two planes. Now since our objective is to contain pendulation, it is indeed desirable that the cable angle stays close to vertical, within a few degrees. This observation justifies considering the pendulation problem in one plane at a time. In this paper we preset aspects of modeling the crane in both planes using two type of models, the torque control model and the kinematic (angular velocity control) model. Using the torque model we apply the Liapunov technique for controlling the motion in the luffing plane. Linear control design methods, namely LQG and GPC, are considered for the linearised system based on the kinematic model. In terms of practical considerations, an important aspect is the saturation of the actuator both in angle and angular velocity. Another consideration is that the system needs to operate with different cable lengths and, since we derive controllers for a fixed length, a scheduling scheme to cover various cable lengths is needed. The linearised kinematic model can be used in both planes and the controller is required to provide angular velocity signals for the slewing and for the motion of the anti-pendulation arm. Simulation results, design issues and comparisons are included.

2 2.1

The

model

M o d e l i n g o f t h e c r a n e in t h e v e r t i c a l luffing p l a n e

The crane consists of a control arm attached with one end on the boom-tip, with the other end attached on a cable. At the lower end of the cable hangs a point mass (load). The cable is modeled like a fight string. The whole system is effectively a double pendulum. The particularity of the problem is that the attachment of the double pendulum (boom-tip) is accelerating in the vertical plane. Thus our objective is to use the actuation on the control arm in order to damp the pendulations in the load.

Control Design of a Crane for Offshore Lifting Operations

iz

zb

471

Boom-tip . . . . .

-

inlay'--

Y Fig. 1. Pendulum in luffing plane

Newton's laws of motion for a point mass, applied to the mass of the pendulum, yield rn9 = - F n sin 0 m~i = - r a g + FR cos 0

(1)

where F n is the reaction force (tension) from the cable. From the kinematics obtain y = 11 s i n e + l~ sin0 + Yb z = --ll cos r -- 1~ cos 0 + zb

(2)

where Yb, Zb are the coordinates of the boom-tip. The motion of the b o o m - t i p m a y be regarded as a disturbance. Take first and second order time derivatives of the above ~) = 11 cos Cq~ + 12 cos 0t~ + i2 sin 0 + ~)b = 11 sin r 1 6 +2 12 sin 0t~ -- i2 cos 0 + Zb

= 11 cos r 1 6 -2 11 sin r

(3)

+ l~ cos 00" - l~ sin 0t~2

+ 2i2 cos 0 t / + i~ sin 0 + ~)b ~/= 11 sin r

+ 11 cos d 5 2 + 12 sin 0t~ + 1~ cos 0t~~

+ 2i~ sin 0t~ - i2 cos 0 + ;/b

(4)

From (1) eliminate F n to obtain cos 0/) + sin 0~/= - g sin 0

(5)

472

M.P. Spathopoulos and D. Fragopoulos

Substitution of (4) into (5) obtain

120+11 c o s ( C - 0 ) r -11 s i n ( C - 0)4~2 +cos 0/)b + s i n Os +g sin 0 +

20i2 =

0(6)

The last equation is a fundamental relation. It links the evolution of r to that of 0(t), given the motion of the boom-tip. Note that the above relation is a kinematic one and is independent of the mass, m. So far no control is introduced. A. N o n l i n e a r t o r q u e c o n t r o l m o d e l Let the torque T, on the control arm, be the control input. T h e tension of the cable, FR, may be obtained from (1) and (4) FR = m ( - sin 8# + cos 0(J~ + g)) = m (11 sin(r - 0)r + 11 cos(r -- 0)r 2 + 1202 -

sin 0/)b + cos 0(J/b + g) - i'2)

(7)

The torque, T, on the control arm is given by

T = J~r + FRI1 sin(r - 0) + mapg s i n e + mapcos r

(8)

+ mapsin ~)Zb

where ma is the mass of the arm, Ja its m o m e n t of inertia relative to the pivot point and p is the distance of the centre of gravity of the arm to the pivot. Substitution for the expression for FR in the above obtain T = roll sin(r - 0) (11 sin(r - 0)r + 11 cos(r -- 0)r 2 + 1202 - sin O~tb + cos O(Jib + g) - i'2) + J~r + mapcos r

+ rnapsin r

+ g)-

(9)

Adding to the above roll cos(r - 0) multiplied by the left hand side of (6) leads to

T = Jar + m l l c o s ( r

+ 2 i ~ 0 ) + n , ll s i n ( r

+ (roll 'b map) (cos Cyb -at- sin r

+ g)) + ml~8.

('2t~ ~ - i ~ ) (10)

From here on we will take the length of the cable, 12, to be fixed (/2 = 0). Equations (6) and (10) are describing the motion of the crane, with the torque T serving as the input. Collecting terms together it is possible to put these equations in the standard form for mechanical systems

M(q)O + C(q, q) + K(q) + W(q)ib = B f

(11)

Control Design of a Crane for Offshore Lifting Operations where q = ( r 0) T is the coordinate vector, M(q) > 0, given by

(

Ja + .,l~

M(q) = \mlll2 cos(r - 0)

473

M(q) the inertia matrix with

m/ll~ cos(C-- e)) ml2~

(12)

C(q, q) a vector accounting tbr centripetal and Coriollis effects C(q,q) =

_r

sin(C-

O)rnlll2

(13)

K(q) a vector of elastic and gravitational terms

( (77111-J- ITlap)ff sin r

K(q) = \

m/~gsin0

B = (~)andf

"~ ]'

= T. The term

reference frame (boom-tip), with

(14)

W(q)i~ accounts for accelerations in the

ib =(ijb ~b) T and

( (roll + map) cos r (roll + map) sin r "~ m12 cos 0 rot2 sin 0 ) .

W(q)

(15)

Note that at r = 0, the mass matrix has determinant det (Me=e) = J~rnl~. This implies that in case the inertia of the arm, Ja, is negligible, the mass matrix losses rank on r = 0. This is of particular importance when considering the linearised model around the equilibrium position r = r --- 0 = 0 -- 0. To obtain a model of the form of (11) with variable cable length, the coordinate vector q needs to be extended to include l~ and also the hoist inertia needs to be included. B. L i n e a r i s e d t o r q u e c o n t r o l m o d e l Linearise around r = q~ = 0 = 0 : O, or q = q = 0 and ~ = O , f = 0 by keeping only the first order terms to obtain

M(O)~ + Koq + W(O)ib = B f with K0 =

(16)

((roll+ map)g 0 ) Notice that M(0) is singular whenever 0 mgl2 "

Ja = 0. From (16) we obtain

=

0

mllg + rnapg r _ -'-~-a m a p yb .. + i T Ja Ja

g = ~mg121. ( 0 - r..1 6mapgll 2

,

( mapll

(16a) 1~

11

T

(16d)

474

M.P. Spathopoulos and D. Fragopoulos

with state variable x = (Xl x2 x3 x4) T ---- ( r ~ 0 0 ) T . Notice t h a t ~b has less effect on the pendulation. Now instead of (16d) one m a y use the following 1 . 11 ;

= ~y-~-~Yb

1 .

(16b)

= -g0

(16c)

with state variable x -- ( x l x~ x3 x4) T = ( r r 0 ~))T, where (16b) is derived by differentiation of (2) over time, and (16c) is just the linearisation of (5). Using equations (16a), (16b), (16c) is preferred as besides its elegance it introduces the variable y as a state variable. Penalizing y improves the controlled system's performance when the L G Q design is considered. Note that in the actual crane there are restrictions in the torque T. C. N o n l i n e a r a n g u l a r v e l o c i t y c o n t r o l m o d e l Introduce the control as = u

(17)

Substitute in (6) and rearrange (/'2 -- 0) to obtain 120 = 11 sin(r - O)u 2 -- It cos(r - 0)ti - sin Og - cos O~/b -- sin Oiib

(18)

Equations (17) and (18) are the governing equations of m o t i o n for the pendulum. D. L i n e a r i s e d v e l o c i t y c o n t r o l m o d e l Linearise (2) and (18) around r ----0,9 = 0, y -- 0,4 ----0 and ~)b ----~b = u = 0 to obtain r (19)

= 1~1ii - l ~ l l l u - l~liib i)

=

-gO

which is of the form

Bl~lb +

= Ax +

B2u

with x = ( r 0 ~))T and

[ A B1 B2 ] =

0 0 l; 1

0-g

-

-l

2

(20)

0

T h e eigenvalues of this model are at s = {0, :t:jv/-g/12. It is worth mentioning again that the consideration of the third state ~) in (16c) and (19) improves

Control Design of a Crane for Offshore Lifting Operations

475

considerably the control performance when the LQG method is used, due to its direct involvement in the cost function. In the actual crane there are restrictions in r and r Note that linearisation about a constant hoisting velocity, i2 = v, results in an alteration of the third equation of (19) to //=

-go

- 2 v t ~ - l ( y - l l u - yb)

and the pole polynomial for the new state space model becomes s(12s 2 + 2vs + g). Thus when hoisting down (v > 0) a damping is introduced, when no hoisting is used (v = 0) the dynamics are marginally stable which is characteristic of conservative systems, and when hoisting up (v < 0) the dynamics become unstable.

2.2

M o d e l i n g o f t h e c r a n e in l a t e r a l m o t i o n

In this case we make the simplifying assumption that the cable top moves in a straight line as a result of slewing of the crane. Denote the slewing angle by a (positive in the clockwise sense to the upward vertical direction). Then the horizontal motion of the actuator is x -- Rc~, where the effective radius R, is the projection of the boom length on the horizontial plane. The non-linear

Z

zb~. Boom-tip --~

. . . . . . . . . . .

iuertially fixed frame

xb

~iv/ 12 m

Fig. 2. Crane in slewing direction

model for the above is obtained in [3] with modifications for accelerating frame. 1~r + 2i2r + (~ + #b) cos r + g sin r + ~/bsin r = 0

(21)

476

M.P. Spathopoulos and D. Fragopoulos

The linearization of the above around r = g) = i 2 = 0 yields 12r162

= 0

(21a)

This is the same as equation (19), derived for the luffing plane using the kinematic model, with the substitution of a for r r for 0, R for 11 and Xb for Yb.

3

Nonlinear control using Lyapunov vertical plane (Torque model)

analysis

for the

Here the boom tip will be assumed fixed in an inertial frame (Yb = ~/b = 0). Consider the total mechanical energy, H, of the double pendulum system defined by (6), (10)

H := ira(y2+ 2)+ IJa(b2+g(mll +map)(l-cosr + 9rnl2(1 -- cosO) 1

/o:

12/~2

\

21112r cos( r

1

Jaq52

+ 9(rnll + m~p)(1 - cos r + #rnl2(1 - cos O)

(22)

The double-pendulum system is energy conserving with the potential energy being velocity independent and the motion constraints being time independent. Thus the rate of increase of the mechanical energy equals the power input induced by external forces, see [4]. Thus

/;/-- CT

(23)

The above relation implies that the system from T to r is passive. This property is an important property often arising with mechanical systems and it will be exploited in obtaining Lyapunov stability. Consider the Lyapunov candidate function: V := H - g ( , , , l l + r.ap)(1 - c o s r

1

re(roll + ,'nov),'6 ~ L

(24)

with r > 0 which represents the kinetic plus a modified potential energy term of the system with the potential energy pertaining to the controlled variable 1 "2 9 Equivalently r replaced by the positive definite term proportional to 7r8 write

Control Design of a Crane for Offshore Lifting Operations

477

which is positive definite in r 0, r but only locally in 0. This type of L y a p u n o v function is the application to the double p e n d u l u m of the Lyapunov function proposed, in the more general setting of underactuated mechanical systems , see [2], [5]. Taking the time derivative of V and using (23) obtain = CT - g ( m l l + map) sin r162+ rg(mll + m a p ) r 1 6 2

(25)

Then ~" can be m a d e negative by taking T = -kor +g(mll + map)(sinr - re),

k0 > 0

(26)

resulting in

9 = -k0r

(27)

which implies that r --+ 0 as t ---r cx~. Note that in the above k0 m a y be any passive m a p between T and r To complete the stability proof, LaSalle's theorem is then used. We now state the m a i n result: T h e o r e m 1. Consider the control defined by (26) and = 0 with stable positions 0, 0 = (2n + 1)rr where n is

double pendulum described by (6), (I0) with the i)b = ~b = O. the system has equilibria f o r ~b = r = O, 0 = 2nTr and unstable equilibria f o r r = an integer.

P r o o f : Under (26), we know that (~ --+ 0 as t --+ oo. According to LaSalle's invariance principle it is enough to consider trajectories along which the following condition holds = 0

(28)

Along such trajectories we have r =const. and $ - 0. Substitute in (6) to obtain l~0 + g sin 0 = 0

(29)

which corresponds to the u n d a m p e d simple p e n d u l u m equation. T h e above yields 0 as an analytic function of 0. By successive differentiation it m a y be seen easily that O(t) is infinitely continuously differentiable, i.e. 0 E C r162Now (10), (26) and (28) yield l~ cos(r - 0)t~ + 12 sin(r - 0)t~~ + r p g r = 0

(30)

where p - 1 + ~m i x " Thus relations (28), (29) and (30) define the equilibrium (positive limit) set of the system. A contradiction a r g u m e n t will be now used to show that 0 = 0. Namely assume that 3t0 : t~(t0) r 0. Substitute in (30) from (29) to obtain - g cos(r - 0) sin 0 + 12 sin(r - 0)t~2 + r p g r = 0.

(31)

478

M.P. Spathopoulos and D, Fragopoulos

Differentiate the above over time and eliminate 02 from (29) and r from (28) to obtain

(-12 cos(r -/9)/~ ~ - 2g sin(r -/9) sin/9 - g cos(r - 2/9)) = 0.

(32)

From (31) and (32) eliminate O: /~ (-l~ cos2(r --/9) sin/9 + rp cos(r --/9)r -- sin(2r - 2/9) sin/9 -- cos(r --/9) cos(r - 2/9)) = 0

or equivalently (al cos(2r -/9) + bl sin(2r -/9) + (a= cos(2r - 3/9) + b2 sin(2r - 3/9)) = 0

(33)

with 1

cos r + cos 2r + 3 sin 2r

1 bl = ~ ( - 2 / 1 p r

.

s i n r + 3 c o s 2 r -- s i n 2 r

1 a 2 =

--~

b2 =

-

1 4

Equation (33) is of the form Oy(O) -- 0. Since 0(f0) # 0 obtain y(O) = O. Differentiating the equation y(9) = 0 three times obtain the following system of equations valid for some to and some values of/90 =/9(f0), vo =/9(t0) # 0,

v(o) = o 0re = 0 v(0) = aa

02y02 Oy .. ~(0) = a-~ + ~ o = 0 03Yo 3

v(~)(~ : ags

02Y00

+3ag~

(34) 0Y0(3) = 0 .

+ oe

With 0(t0) = -g/12 sin O0 and O(a)(t0) = -g/12vo cos 00 obtained from (29), the above is a linear homogeneous system of equations in al, bl, a2, b2. After removing dependent terms, the right hand side equations of (34) are equivalent to

v(Oo) = o ~(Oo) = o

Control Design of a Crane for Offshore Lifting Operations

479

(92Y(0o) = 0 Day (903 (0o) = O. The above is of the form Ax = 0 constant matrix Cl A= sl -ct -Sl

where x = (al bl a2 b)) T and A being a sl c3 s3 - c l 3sa - 3 c 3 -Sl -9c3 -9s3 ] cl 27s3 - 2 7 c 3 ]

/

where el = c o s ( 2 ~ - 0 ) , sl = sin(2~ - e), c3 = cos(2~ - 3e) and s 3 = cos(2~b - 3/9). The determinant det(A) = 192, a non-zero constant. T h u s x = 0 is the unique solution of the homogeneous equation. However this is not possible by the definition o f x le.g. a2 --- - 1 / 2 ) . Thus u0 = 0(t0) = 0 for all t0 E D. From the smoothness of 0(t) we conclude that/~(t) = 0. Substitution in (29) yields sin0 = 0 while substitution in (30) .yields ~b = 0. Thus, the possible trajectories contained in the invariant set at = 0 are, ~ =/9 -- 0 and = 0, /9 = nrr, with n an integer. From (29), odd n corresponds to unstable equilibria. T h e above analysis implies that depending on the initial conditions and effectively the initial value of V, the pendulum m a y come to rest after a n u m b e r of complete revolutions of the lower part. If however the initial energy function, V, is low enough no revolution in 9 is possible. In the above model the second link has been assumed to be a rigid rod. Thus in the case of a cable replacing the rod the validity of the above theory is limited to trajectories where the tension, FR, remains positive. A variant of the control law proposed above m a y be obtained by taking V as V = H - (1 - r)g(mll + l~ap)(X

-

-

COS•)

Note that the energy function V above is just the overall mechanical energy of the system when r = 1. A control law

T=-ko~+(1-r)g(mll

+map)gsin~,

ko > 0

results in (27). In the case r = 1, the control law corresponds to the d y n a m i c s of a passive damper. The resulting closed loop system has similar stability properties to those in Theorem 1 with the difference of the extra unstable equilibria for r = (2k + 1)rr and stable equilibria for ~ = 2kTr where kis an integer. The proof is similar to t h a t of Theorem 1 with an a m e n d m e n t required for proving the instability for r = (2k + 1)rr which m a y be done by considering small variations (linearisation) around those points.

480 4

M.P. Spathopoulos and D. Fragopoulos Linear

control

design

The previous Lyapunov analysis provides a control structure for stabilizing the plant under deterministic assumptions. This controller is of the PD type and the controller parameters need to be tuned. It would be of interest to develop controllers for the linearised plant that have considerable scope in attenuating noisy disturbance as are the waves. We use the linearised kinematic model since then, as shown in section 2.2, the dynamics of the lateral motion when linearised around an equilibrium point are identical to those of the luffing plane. In this section LQG (Linear Quadratic Gaussian) and GPC (Generalized Predictive Control) optimal control designs ([8], [9]) are considered using the linearised kinematic model (19), see [6], [7]. As a first attempt uncoloured (white) noise source is used to model the boom disturbances. Subsequently a dynamic disturbance model is used for modeling the effect of the waves. The latter approach is shown to be more successful.

4.1

LQG/GPC design using white d i s t u r b a n c e m o d e l

In this problem we cannot measure all the states (we have partial observations). As a result of the separation principle the LQG design is naturally decomposed into LQ optimal state-feedback control and Kalman filtering. Similarly to the LQG design the GPC design is decomposed into G P C and Kalman filtering, see [6]. For the LQG design we construct the error output vector, z, T

To improve the disturbance rejection of the system we remove the 0 weighting in the cost function. (The 0 weighting has the effect of reducing the gains and effectively de-tuning the controller). Also we decrease the control weighting, both for r and r and increase the y weighting. The observations, z, are defined as z = Cx + u where C=

0:)

01

with v being the measurement noise. Thus we avoid the problem of measuring the state variable y and the system is detectable, see [6]. The Kalman filter is started with no prior knowledge of the initial value of and will take at least 14s before a useful estimate may be obtained. After this tuning-in period the error is reduced to a ripple. As we see from model (19) the matrices are parameterized by the cable length, 1~ , which is variable in practice. Thus it would be appealing to obtain controller parameters can be parameterized by the cable length. As it turns out it is possible to scale the time and the

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481

various system variables to obtain a normalized model independent of the system parameters. Thus, a design obtained for the normalized plant may be directly converted to obtain one for any cable length, see [6], [7]. The scaling of time or equivalently scaling of the s-variable, implies that the response times is also scaled. However the damping ratio of the system will be preserved. An advantage of the use of white noise disturbance model is that no scheduling, when varying the cable length, is needed. The normalized system matrices are given below

[(i~ 4.2

LQG/GPC

u s i n g dynamic disturbance m o d e l

A dynamic model Wa, with state-space description:

ice = A e z e + Be~

0b = cbx~

(35)

is used to generate the disturbance, ~Jb, from a white noise process, ~. This models the disturbance process as a frequency spectrum approximation resulting from the ship motion when excited by waves. The disturbance process that affects the top of the boom of the crane is given by the model of the vessel under consideration in connection with the wave model based on the Jonswap spectrum model. Basically the wave height Hs and the frequency of the wave peak energy wp parameterize the wave model. The above disturbance model may then be combined with the crane model (19) to obtain an augmented model. Now, due to the disturbance model involved, the augmented model equations cannot be normalized so the controller depends (not simply parameterized) on the cable length l~ and on the disturbance model parameters H, and wp. A scheduling scheme has been considered to cover various cable lengths, see [6]. For the disturbance model the following three approximations have been considered. A. S e c o n d o r d e r a p p r o x i m a t i o n We present, for example, a second order approximation of the power spectrum of the vessel named Belos plus the wave model for wave height Hs = 2.4m, wp -- 1.02rad/sec. This is given by: w,~ =

0.17 s 2 + 0.198s + 0.9801

The frequency response of the approximated wave/ship spectrum (model) and the actual disturbance are shown in the figure 3.

482

M.P. Spathopoulos and D. Fragopoulos

B. H i g h e r o r d e r a p p r o x i m a t i o n For the entire operating region of the sea waves defined in terms of the wave peak energy and wave height and for the given vessel (Belos) this approximation is given by

wd =

,0.9s + 1.1-I3:l s 2 + 2r ;z +

+w~

+

where

~.

Poles (rad/s) .52 .72 .94 1.16

r .125 .15 .15 .11

~.

Zeros (rad/s) .59 .25 .88 .30 1.05 .30

The frequency response of the disturbance model is shown in figure 4. This model derives a more complicated controller however the advantage is t h a t there is no need to continuously m o n i t o r the wave pattern and schedule the controller as a function of the wave peak energy and wave height. Thus we only need scheduling in terms of the cable length. C. W o r s t c a s e a p p r o x i m a t i o n We calculate W~7 which, in a sense, is the approximation of a worst wave disturbance model for the operating region (0.5m < H, < 3.5m,

0.5rad/s < wp < 1.14rad/s).

This is given by 0.41(s + 0.6) W~ = s2 + 0.196s + 0.67

(36)

In order to find the above we first construct the optimal controller based on white noise assumptions as in section 4.1 and then we test its behavior for all possible waves in the bounded area under consideration. The worst behavior occurs under the worst case wave. After testing (perform optimization) we derived the worst sea wave to be: ( H , = 3 m and wp = .85 t a d / s ) . T h e second order approximation model given by equation (36) and the actual power spectrum of the ship (Belos) affected by the worst sea wave model (named wave response) are shown in figure 5. W~, in effect, is a b a n d pass filter and for the bounded operating wave frequency region it will produce the best results for the worst case disturbance. The model m a y result in a non-optimal performance when is compared to a disturbance model t h a t is designed for the appropriate wave frequency. However, it has the advantage

Control Design of a Crane for Offshore Lifting Operations

483

/

Ii& I/\ / "'tt

f

\

\

I /d d

./ 0,4

0.8

0.8

1

1.2

1.4

1.6

Fig. 3. Disturbance model: second order (x-axis : frequency (rad/s))

el

at

E

| O~

o4

OI

W

12

S4

11

Fig. 4. Disturbance model: 8 th order (x-axis : frequency (rad/s))

that there is no need to continuously monitor the sea wave pattern and to schedule the controller accordingly. The disturbance model order is reduced from an 8 th order approximation derived in the previous section to a 2 nd order. Thus the design of the controller is simplified.

5

S i m u l a t i o n results

Having obtained several controllers, the non-linear closed loop system is simulated using realistic ship and wave models. The Belos ship data are used from the Marintek report together with the JONSWAP wave model, see [1], [6] for significant wave height H s = 2.4rn. The response of the system is simulated for motion in the vertical plane. Moreover the boom tip is assumed to be moving horizontally. The following tables provide the standard deviation (sdv) and m a x i m u m values of the various system variables. Note that the angle r is limited 0.44 tad while r is limited to -I-0.40 rad/s. The first 30s of

484

M.P. Spathopou]os and D. Fragopoulos

3 2.5

2

1.5

I

0.5

0.4

0.8

0.8

1

1.2

1.4

1.8

Fig. 5. Disturbance model: worst case

data are discarded for the linear designs in order to obtain good estimates for all states. The simulation results for the LQG and the GPC designs are given in Table 1 where +wh, +2M, +8M or +wo correspond to designs with white noise disturbance, second order disturbance model, eighth order disturbance model and worst case disturbance model, respectively. At the end we include simulation results using the Lyapunov controller (26) for two different values of k0 and r = 1. The Lyapunov analysis holds only for one plane and it would be interesting to extend the results in the 3D plane (in the expense of rather complicated formulas) since the decoupling of the two planes holds only for the linearised models. Simulation results using the torque based finearised model (16a)-(16c) were not better than using the linearised kinematic model (19). Since the latter can be used for both planes we considered model (19) for the application. The objective of the control design is to keep the pendulation, the sdv and maximum (max) of y as small as possible. From Table 1, we can see: 1) The control results of LQG and GPC with the approximated disturbance model are better than the ones with white noise. 2) The control results of GPC may appear slightly better than that of LQG particularly when the max value of y and the values of the angle 0 are taking into consideration. 3) The control results of LQG and GPC +8M appear better than that of LQG and G P C +2M in the expense of higher order controllers. Also the results using the approximated worst case disturbance +wo are comparable to the ones using +8M. Thus, in practice, the use of +wo is preferable since it leads to simpler controllers. 4) The Lyapunov control design, appropriate for stabilizing a deterministic plant, does not perform better than the linear design using the approximated disturbance model. The lyapunov controller does not attenuate

Control Design of a Crane for Offshore Lifting Operations

485

so well the wave disturbances as the controllers t h a t incorporate the dist u r b a n c e model in the design. However, the Lyapunov controller requires less power. It must be noticed that if the overall horizontal m o t i o n of the crane top exceeds the actuator reach the system will fail to work within specifications a n d the performance m a y rapidly deteriorate. Design LQG+wh LQG+2M LQG+8M LQG+wo (~PC+wh GPC+2M GPC+8M GPC+wo Lyapunov k0 = 8 k0 = 3

u( m ) Sdv .120 .116 .063 .062 .136 .079 .049 .080

~b(rad/s) P(kW) r max Sdv max Sdv max Sdv max .581 .120 ..40 2.56 21.9 .124.44 .480 .125 .40 .125.44 2.67 23.3 .577 .137 .40 .138.44 3.08 25.9 .591 .138 .40 .139.44 3.32 26.4 .557 .126 .40 2.71 19.8 .131 .44 .351 .132 .40 3.15 25.1 .141 .44 .140.44 .317 .134 .40 3.12 21.1 .384 .142 .40 3.47 26.1 .151 .44

.240 .746 .125 .38O

.078 .29 .100 .40

1.23 7.90 1.97 14.8

.084 .28 .103.37

0 (rad) Sdv max .011 .040 .010.038 .008 .047 .006.044 .012 .051 .004 .025 .002 .009 .002 .012 .021 .074 .011 .040

T a b l e 1. Simulation results with 11 = 3.25m, l~ = 14m, R u n time: 1800s, Hs = 2.4m, aJp = 1.02 t a d / s , m = 5000 kE, m a = 200 kg, p -- 1.5m, yb(Sdv)= .42(m), y b ( m a x ) = 1.55(m) and

Ja = p2ma + ma l~.

6

Conclusions

T h e modeling of a crane for offshore lifting operations has been studied a n d various models have been derived. T h e problem of p e n d u l a t i o n control has been studied using Lyapunov technique for the n o n - l i n e a r model and linear techniques such as LQG and G P C for the linearised k i n e m a t i c model. For the latter the use of a p p r o x i m a t e d d y n a m i c models for the wave a n d ship disturbances improved considerably the results. A c k n o w l e d g m e n t : This work was s u p p o r t by the E u r o p e a n U n i o n contract T H E R M I E OG-00171-95.

References 1. Atkins Research & Development Dynamics of Marine Structures, CIRIA Underwater Engineering Group, Report UR8, 1977.

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2. F. Bonstany, PhD Thesis, Ecole de Mines, Paris, 1992. 3. M. Fliess, J. Levine, P. Rouchon, A Simplified approach of crane control via a generalized state-space model, Proc. IEEE Con/erence Decision and Control, Brighton, 1991. 4. H. Goldstein, Classical Mechanics, Addison Wesley, 1977. 5. J.J.E. Slotine and W. Li, Adaptive Manipulator Control: A Case Study, Proc. IEEE Int. Conference Robotics and Automation, Raleigh, N.C. 1987. 6. D. Fragopoulos and Y. Zheng, Pendulation control of an Offshore crane, Technical report, Dept. of Mechanical Eng., University of Stathclyde, 1998. 7. D. Fragopoulos, M. P. Spathopoulos and Y. Zheng, A pendulation control system for offshore lifting operations, Proc. of the IFAC l$th World Congress, Beijing, 1999. 8. D.W. Clark, C. Mohtadi and P.S. Tufts, Generalised predictive control Part I: the basic algorithm and Part II: extensions and interpretations, Automatica 23(2), pp. 137-160,1987. 9. P.J.Gawthrop, H. Demircioglou and I.I. Siller-Alcala, I, Multivariable Continuous-time Generalized Predictive Control: A state space approach to Linear and Non-linear systems, CSC-98001 report, University of Glasgow, 1998.

N e w Theories of Set-valued Differentials and N e w Versions of the M a x i m u m Principle of Optimal Control Theory* H6ctor J. Sussmann Department of Mathematics Rutgers, the State University of New Jersey Hill Center--Busch Campus 110 Frelinghuysen Road Piscataway, NJ 08854-8019, USA sussmann@hamilt on. r u t g e r s , e d u , h t t p ://www. math. r u t g e r s , edu/~ sussmann

1

Introduction

The purpose of this note is to announce two new theories of generalized differentials--the "generalized differential quotients," abbr. GDQs, and the "path-integral generalized differentials", abbr. P I G D s - - w h i c h have good open mapping properties and lead to general versions of the m a x i m u m principle. In particular, we use GDQ theory to s t a t e - - i n Theorem 5 - - a version of the m a x i m u m principle for hybrid optimal control problems under weak regularity conditions. For single-valued maps, our GDQ theory essentially coincides with the one proposed by H. Halkin in [4], but GDQ theory applies as well to multivalued maps, thus making it possible to deal with non-Lipschitz vector fields, whose flow maps are in general set-valued. The results presented here are much weaker than what can actually be proved by our methods. More general versions, involving systems of differential inclusions, are discussed in other detailed papers currently in preparation. The GDQ concept contains several other notions of generalized differential, but does not include some important theories such as J. Warga's "derivate containers" (cf. [9]) and the "semidifferentials" and "multidifferentials" proposed by us in previous work (cf. [7]). For this reason, we conclude the paper by giving, in w a brief sketch of the definition of our second t h e o r y - - t h e P I G D s - - t h a t contains that of GDQs as well as the other theories mentioned above. Research supported in part by NSF Grant DMS98-03411-00798 and AFOSR Grant 0923.

488 2

H~ctor J. Sussmarm Notational

preliminaries

A set-valued map is a triple F --= (A, B, G) such t h a t A a n d B are sets and G is a subset of A x B. If F -= ( A , B , G ) is a set-valued m a p , we say t h a t F is a set-valued map from A to B. In t h a t case, we refer to the sets A, B, G as the source, target, and graph of F , respectively, a n d write A = S o ( F ) , B = T a ( F ) , G = G r ( F ) . I f x E S o ( F ) , we write F(x) = { y : ( x , y ) E G r ( F ) } . T h e set D o ( F ) = {x E S o ( F ) : F(x) r 0} is t h e domain of F . If A, B are sets, we use S V M ( A , B) to denote the set of all set-valued m a p s from A to B, a n d write F : A > )B to indicate t h a t F E S V M ( A , B). If F t and F2 are set-valued m a p s , then the composite F~ o F1 is defined iff Ta(F1) : So(F~) and in t h a t case: So(F2 o El) def So(F1) Ta(F2 o F1) d___~fTa(F2) G r ( F 2 o F1) def {(x, z ) :

(3y)

((x, y) E G r ( F 1 ) , (y, z) E G r ( F ~ ) ) } .

If A is a set, then I n denotes the identity map of A, t h a t is, the triple ( A , A , AA), where A a = { ( x , x ) : x E A}. T h r o u g h o u t this paper, the word "map" always s t a n d s for "set-valued m a p . " T h e expression "ppd m a p " stands for "possibly p a r t i a l l y defined ( t h a t is, not necessarily everywhere defined) o r d i n a r y ( t h a t is, single-valued) m a p , " and we write f : A----> B to indicate t h a t f is a p p d m a p from a set A to a set B. A time-varying ppd map from a set A to a set B is a p p d m a p from Ax~toB.

A cone in a real linear space X is a n o n e m p t y subset C of X such t h a t r.c E C whenever c E C, r E ~ and r > 0. We use l~ to denote the set of strictly positive integers, a n d write 2~+d-----eriNU{0}. If n E 25+, r E ~ , and r > 0, we use ] ~ (r), ]]~ (r) to denote, respectively, the closed and open balls in ~'~ with radius r. We write ]]]O,II~ for il~ (1), I ~ (1). If k E l~I and M is a m a n i f o l d of class C k, then T M and T*M denote the tangent and cotangent bundles of M , so T M and T*M are manifolds of class C k - l . If x E M , then T~:M a n d T~M denote the t a n g e n t and cotangent spaces of M at x. 3

Regular

maps

If X , Y are metric spaces, then SVMcomp(X,Y) will denote the subset of S V M ( X , Y ) whose m e m b e r s are the set-valued m a p s from X to Y t h a t have a c o m p a c t graph. We say t h a t a sequence {Fj}jr of m e m b e r s of SVMeo,~p (X, Y) inward graph-converges to an F E SVMcomp (X, Y ) - - a n d write Fj ~ + F - - i f for every open subset /2 of X • Y such t h a t G r ( F ) C 12 there exists a jn E i~l such t h a t G r ( F j ) C_ /2 whenever j > j n .

Set-valued Differentials and the Maximum Principle of Optimal Control

489

D e f i n i t i o n 1. Assume that X , Y are metric spaces. A regular set-valued map from X to Y is a set-valued m a p F E S V M ( X , Y ) such t h a t 9 for every compact subset K of X , the restriction F [ K of F to K belongs to SVMcomp (K, Y) and is a l i m i t - - i n the sense of inward graphconvergence--of a sequence of continuous single-valued m a p s from K to Y. We use R E G ( X ; Y) to denote the set of all regular set-valued m a p s from X to Y. (> It is easy to see that if F : X --4 Y is an ordinary (that is, single-valued and everywhere defined) map, then F belongs to R E G ( X ; Y) if and only if F is continuous. It is not hard to prove the following T h e o r e m 1. Let X, Y, Z be metric spaces, and suppose that F belongs to R E G ( X ; Y ) and G belongs to R E G ( Y ; Z ) . Then the composite map G o F belongs to R E G ( X ; Z). (>

4

Generalized

differential

quotients

(GDQs)

D e f i n i t i o n 2. Let m, n be nonnegative integers, let F : ~ ' ~ ) >~'~ be a setvalued map, and let A be a n o n e m p t y c o m p a c t subset of ~n• Let S be a subset of ~m. We say that A is a generalized differential quotient (abbreviated "GDQ") o f f at (0, O) in the direction of S, and write A E GDQ(F; O, 0; S), if for every positive real number (i there exist U, G such that 1. U is a compact neighborhood of 0 in ]~,n and U n S is compact; 2. G is a regular set-valued m a p from U M S to the (f-neighborhood A 6 of A in lI~n• 3. G(x). x C_ F(x) for every x E U f3 S. (>

I f M , N a r e C l m a n i f o l d s , ~ E M , ~ E N , SC_M, a n d F : M > >N, then we can define a set GDQ(F; i:, ~1;S) of c o m p a c t n o n e m p t y subsets of the space Lin(T~M, TyN) of linear maps from T~M to TuN by picking coordinate charts M ~ x - + ~ ( x ) E ~ m , N ~ y - + q ( y ) E II~n--where m = d i m M , n = d i m N - defined near i:, ~ and such that ~(x) = 0, q(y) = 0, and declaring a subset A of L i n ( T , / , T#N) to belong to GDQ(F; ~, ~); S) if n~/(9) o A o D~(2) -~ is in GDQ(q o F o ~ - 1 ; 0 , 0 ; ~ ( S ) ) . It turns out that, with this definition, the set GDQ(F; ~:, Y; S) does not depend on the choice of the charts ~, 77. In other words, the notion of a GDQ is invariant under C 1 diffeomorphisms and makes sense intrinsically on C 1 manifolds. The following facts about G D Q s can be verified.

490

Hdctor J. Sussmann

1. If M, N are C 1 manifolds, 9 E M , U is a neighborhood of ~ in M , F : U --4 N is a continuous map, F is differentiable at Z, ~ = F(~), and L = D F ( ~ ) , then {L} E G D Q ( F ; ~, fl; M ) . 2. If M , N are C 1 manifolds, 9 E M , U is a neighborhood of ~" in M , F : U --4 N is a Lipschitz continuous map, .0 = F(~), and A is the Clarke generalized Jacobian of F at ~, then A E G D Q ( F ; ~, ~; M ) . 3. (The chain rule.) If Mi is a Gq manifold and x i E M i for i = 1,2,3, SI C_ Mi, Fi : Mi ) )Mi+l, and Ai E G D Q ( F i ; ~ i , ~ i + l ; S i ) for i = l , 2 , and either Ss O U is a retract of U for some compact neighborhood U of ~s in Ms or F1 is single-valued, then As o A I E G D Q ( Fs o F1; i: x , ~:z ; $1). 4. (The product rule.) If M1, Mz, N1, Ns, are C 1 manifolds, and, for i = 1,2, ~i E Mi, fli E Ni, Si C Mi, Fi : Mi ) )N i, and Ai belongs to G D Q ( F i ; i:i, Yi; Si), then A1 x As E G D Q ( F t • F~; (~l,~s), (~l,~s);S1 • Ss). 5. (Locality.)If M, N, are C 1 manifolds, ~" E M , /) E N, and, for i = 1,2, Si C_ M , Fi : M ) ) N , and there exist neighborhoods U, V of i:, .0, in M, N, respectively, such that (U • V ) O G r ( F 1 ) = (U • V ) n Gr(Fs) and U A S 1 = U N S s , then GDQ(FI;~,~I;S~) = GDQ(F~;I:, ~j;Ss). It is easy to exhibit maps that have G D Q s at a point ~ but are not classically differentiable at 9 and do not have differentials at 9 in the sense of other theories such as Clarke's generalized Jacobians, Warga's derivate containers, or our "semidifferentials" and "multidifferentials". (A simple example is provided by the function f : ~ --+ ~ given by f ( x ) = xsin 1 / x if x 5s 0, and f(0) = 0. The set [ - 1 , 1] belongs to G D Q ( f ; O, 0; ~ ) , but is not a differential of f at 0 in the sense of any of the other theories.) In addition, GDQs have the following directional open mapping property. T h e o r e m 2. Let m, n be nonnegative integers, and let C be a convex cone in I~m. Let F : ~ m ~ )~n be a set-valued map, and Iet A E G D Q ( F ; O , O ; C ) . Let D be a closed convex cone in ]~n such that D C_ I n t ( L C ) U {0} f o r every L E A. Then there exist a convex cone A in ]~n such that D C_ I n t ( A ) U {0}, and positive constants g, ~, having the property that (I) if y E A and IlYll < and y E F(x).

~, then there exists an x E C such that Ilxll < ~IlYll

Moreover, the cone A and the constants g, tr can be chosen so that the following stronger conclusion holds: (II) if y E A and Ilyll = ~ < ~ then there exists a compact connected subset Z~ of (C n ~ (~)_) x [0, 1] such that (0, O) E Zy, (x, 1) E Zy f o r some x belonging to C n lt~ ( ~e ), and r y ~ F ( x ) whenever 0 < r < 1 and ( x , r)

zv.r Associated to the concept of a G D Q there is a notion of " G D Q approximating multicone to a set at a point" :

Set-valued Differentials and the Maximum Principle of Optimal Control

491

D e f i n i t i o n 3. If X is a finite-dimensional real linear space, a convex multicone in X is a n o n e m p t y set of convex cones in X . If M is a manifold of class C 1, S C_. M and z E S, a GDQ approximating multicone to S at x is a convex multicone C in T~M such t h a t there exist an m E Z+, a m a p F : R m ~ ~M, a c l o s e d convex cone D i n l ~ r n , a n d a A E GDQ(F; O, x; O), such that F ( D ) C_ S and C = {LD : L E A}. (> If X is a finite-dimensional real linear space, then X t denotes the dual of X . If S is a subset of X, the polar of S in X is the set S~ = {yE X*: y(x) _< 1 whenever x E S}. If C is a cone in X , then C ~ is a closed convex cone in X t , and C~r = { y E X t :y(x) < 0 whenever x E C } . W h e n it is clear from the context what the space X is, we will write C • rather than C:~. We remark, however, t h a t if C is a cone in a linear subspace Y of a linear space X, then C # and C x~ are different objects, and this distinction will be crucial in the statement of our main result (cf. the definition of "multiplier," Def. 10). If C is a convex multicone in X , the polar of {7 is the set C• = C l o s ( U { c •

:C EC}),

so s177is a (not necessarily convex) closed cone in X t .

5

Discontinuous

vector

fields and

their

flows

If n E 1~, we use B(I~n), BL:(~ n , ~ ) , to denote, respectively, the *,-algebra of Borel subsets of ~ n and the product a-algebra I3(]~n) | Lebesgue(~). We let .A/'(]Rn,]R) denote the set of all subsets S of ]~n x R such that IIn(S) is a Lebesgue-null subset of the real line, where/-/n is the canonical projection ]~n x ]~ ~ (x,t) -+ t E ]~. Finally, we use Bf~,(IRn,~) to denote the ~r-algebra of subsets of ~ n x/I~ generated by B/: (~n, J R ) U N ( ~ n, ~ ) . It is then clear t h a t B(]R n x ~ ) C B/:(]Rn,]~) C B/~e(~n,~), and both inclusions are strict. D e f i n i t i o n 4. Let n, m E Z+, and let f be a ppd m a p from ~ n x ~ to ]Rm. 1. We say t h a t f is locally essentially Borel• Lebesgue measurable, or locally BEe (]~n, ]~)-measurable, if f - l ( U ) O K E BEe (]~n, ]~) for all open subsets U of ~ m and all compact subsets K of D o ( f ) . AAe,l~ x l~l~ 2. We use "-'B~: ~'~n • IR; ~rn) to denote the set of all locally Bs measurable ppd maps from ~ n • ]R to ]~rn. 3. We call f locally integrably bounded (LIB) if for every c o m p a c t subset K of D o ( f ) there exists an integrable funtion ~ 9 t --~ ~(t) E [0, +oo] such that IIf(x, t)ll < ~0(t) for all (x, t) E K. 0

492

H6ctor J. Sussmann

If n, m 9 Z+, f : ]~n x ~ ----> ]~m, I is a c o m p a c t interval, and S C_ D o ( f ) , we write ~-s(f, I) to denote the set of all curves ~ 9 C ~ ; ~ n ) such that (((t), t) 9 S for all t 9 I. We write Z ( f , I) for ~Do(f)(f, I), I f r ~ -4 ~ + U {+(x~} is a function, we use ~s~(f, I) to denote the set of all curves ~ 9 ~ s ( f , I) such that

l i m s u p [l~(t) - (_(t-)I[ ~ r t~,~

(1)

t -- t

for almost every { 9 I. e tno c (]~. x ~ ; ~ m ) , i i s a F a c t 1 If n, m 9 Z+, f 9 Ad~' compact interval, and ( 9 3 ( f , I), then the function I 9 t ~4 f(~(t), t) 9 ~,n is measurable.

Ifn, m 9 Z+, f 9 .h4b' etoc ~ (~ n X]~;l~m), I is a c o m p a c t interval, and f is locally integrably bounded, then we can define a m a p 7),1 : I x I • S ( f , I) -4 ~ m by letting

jfa t 7),I (a, t, () = ifaEI,

f(((s), s) ds

(2)

t 9149

D e f i n i t i o n 5. If n, rn 9 Z+ and f : ~'~ • ~ - - - - > integrally continuous if

AAe ,loc { ~

n

I~ m, we call f locally

]l~ ; ]l~ rn ) ,

2. for every compact subset K of Do(f) and every c o m p a c t interval I there exists an integrable funtion ~ 9 t -4 r 9 [0, +oo] with the property that IIf(x,t)ll < r for all (x,t) 9 If and the restriction to ZCK(f,I) of the m a p 7),I is continuous. (> By taking coordinate charts, it is easy to see t h a t the concept of a "locally integrally continuous time-varying ppd section f : M x]l~----> E " is well defined if M is a manifold and E is a vector bundle over M . D e f i n i t i o n 6. Let M be a manifold of class C 1.

1. A time-varying vector field (abbreviated " T V V F " ) on M is a ppd m a p f : M x ~----> T M such that f ( x , t ) E T x M whenever (x,t) 9 D o ( f ) . 2. We use T V V F ( M ) to denote the set of all T V V F s on M . 3. If f 9 T V V F ( M ) , a trajectory (or integral curve) of f is a locally absolutely continuous m a p ( : 1 - 4 M , defined on a n o n e m p t y subinterval I of II~, such that the conditions (((t), t) 9 D o ( f ) and ((t) = f(~(t), t) are satisfied for a.e. t 9 I.

Set-valued Differentials and the Maximum Principle of Optimal Control

493

4. We use Traj (f) to denote the set of all trajectories of f , and Traj r (f) to denote the set of all ~ ~ Traj (f) whose domain is a c o m p a c t interval. 5. For (z, b, a) E M x R x ~ , we define

9 /(m,b,a) -- {~(b) : ~

~ Traj

The m a p 4 i J ' : M x / 1 ~ x ~ 6. For each (b,a) E ] ~ x ~ , w e

(f) : ~(a) =

(3)

x }.

~ ~Mistheflowoff. define a maP~b/,~ : M

r = ~ / ( x , b , a ) for z ~ M . The maps r maps of f . F a c t 2 Let M be a manifold of class

M

~ >M by letting ~ > M are the flow 0

and assume f E T V V F ( M ) . Then the flow maps ~]b,a satisfy the identities qdaa = IM and ~]c,b o ~ !b,a ---4~Y - c,a , if a,b, c E ~ and a 0, ~" E Ii~n, and v is a n o n e m p t y subset of ~m, we define

trL~,v,~(t ) = sup{ d i s t ( f ( x , t ) , v) : x E ]~n, ii x _ i:]l < / 3 } , so or!..... Z is a function on /I~ with values in [0, oo]. (We take the value of the right-hand side to be zero if the set of those x E ~'~ such that II~: - ~11 < and f ( x , t) is defined is empty.) If I is a subinterval of ~ , 5: E ~ n , and fl > 0, we define

s~,I: = {(~,t): 9 ~ ~,11~-~11 <~,t EI}.

(4)

A.~e,tocl~n F a c t 3 Assume that n, m E Z+, f belongs to ,.,t3z ~ • ~ ; ~m), z E ]~'~, v C ~m, and v # 0. Let I be a nonempty subinterval of]~, and assume that fl > 0 and S~:,t,~ C_ D o ( f ) . Then ~ry,e,v,~ is measurable on I whenever

0<~<~.

0

We let C(1) be the set of all cones in ~ . Then C(1) has exactly four members, and "C q C(1)" is an alternative way of saying that C is one of the four sets {0}, [ 0 , + o o [ , ] - oo,0], ]~. I f C E C(1), t E Ii~, and h > 0, we use f + C ( h ) , t + C, to denote the sets {{+ r : r E C, Iv I < h}, {~+ r : r e C}, respectively. D e f i n i t i o n 7. Assume that n , m E Z+, f belongs to "A"Ase z, l ~ /TI~ ~~ n • ~ ; ~ m ) , v C/l~m, v # 0, (~,t~ E ~ n • ~ . Let C E C(1). 1. We say that v is an approximate limit set o f f at (~, D along c , and write v E App-lim~e,

t_.~f,t_fEcf(Z, t),

494

HdctorJ. Sussmann if (s t-) 9 Int~.x(~+c)(Do(f)) and lim (h,/~)-.+(o,o), h>o, ~_>o

1 ~+

C(h)

tr/,e,v,#(t) dt = 0 .

2. We say that a vector v is an approximate limit value o f f at (~,t~ along C, and write v = app--lim~_.~, t~,t_~.Cf(X, t), if {v} 9 A p p - - l i m ~ ,

t..+f,t_~ecf(x, t).

<>

We say that (~, t~ is a point of approximate continuity along C if f ( ~, t-) = app--lim~ ~e,t ~t,t_tec f ( X, t ) . We use the expressions "approximate right limit," "approximate left limit," "approximate limit," "point of approximate right continuity," "point of approximate left continuity," "point of approximate continuity," respectively, as alternative names of "approximate limit along [0, +cr [ ," "approximate limit along ] - cx~,0]," "approximate limit along ]~," "point of approximate continuity along [0, +cx~ [ ," "point of approximate continuity along ] - cx~,0]," and "point of approximate continuity along R." By taking coordinate charts, it is easy to see that all these concepts are well defined if M is a manifold, E is a vector bundle over M , and f is a time-varying ppd section f : M x ~ - - - - > E. The following lemma gives a useful sufficient condition for a point (x, t) to be a point of approximate continuity of a time-varying vector field. To state the lemma, we first introduce the obvious one-sided analogues of the usual notions of a Lebesgue point and a point density. * if I C_ ~ is an interval, and C E C(1), then a C-Lebesgue point of a locally integrable function qo : I -~ ~ U {+oo} is a point t E I such that ko(t-)[ < ~ , i + C(]z) C I for some positive number ]z, and lim 1 ~ + ~o(t) - to(t-) h-.0+ h c(h)

d t = O.

9 If E C It~ is a measurable set, a point of C-density of E is a point t E E such that lim h - l m e a s ( ( i + C ( h ) ) \ E ) = 0 . h-+0+

L e m m a 1. Assume that n, rn E Z+, C E C(1), C7~ {0}, (~,t~ e ~ n x ~ , and f E "AAe,toC/~n "m: ~ x ]~;~m) Assume, moreover, that (~,i) belongs to Intit-x(~+c) (Do(f)), and there exist positive numbers h, fl, an integvable function ~ : t + C(h) ~ RU { + ~ } , and a measurable subset E o f f + C(tt), such that (a) the set S = S~,~+c(~),Z is contained in Do(f), (b) llf(x,t)ll< ~(t) for all (x,t) E S,

Set-valued Differentials and the Maximum Principle of Optimal Control

495

(c) t is a C-Lebesgue point of ~, (d) f is a point of C-density orE, (e) limz..,~,t..,f, teEn(f+c) f(x,t) = f(~',t-). Then ($,t-) is a point of approximate continuity o f f .

7

<>

Variational g e n e r a t o r s

I f ~ : [a, b] -+ ~'~ is a continuous curve, and a > 0, we use Tn(~, a) to denote the " a - t u b e about ~ in ]R'~," that is, the set

7-" (~, ~) d~f {(x, t ) : x ~ ~ ' , a < t < b, Ilx - ~(t)ll _ a } . D e f i n i t i o n 8. Let n, m E 2~+, and let f be a p p d m a p from ~ n • ~ to ~ m . Let a, b, ~ be such that a, b E ~ , a < b, and ~ e C o ( [a, b] ; ~ n ). A variational generator for f about ~ is a measurable set-valued m a p A : [a, b] ) )It~mxn with c o m p a c t convex n o n e m p t y values such t h a t there exist ka, 5, k having the following three properties: 1. kA : [a, b] -+ [0, q-cx)] is integrable and such t h a t

sup {IILll : L e A(t)} < kA(t) for all t E [a,b]; 2. (i > 0 and Tn(~,&) C Do(f); 3. k = {k~}0
k":ia, b]-~[0,+~],

for 0 < a < ~ ,

such that lirr~0 f : k~(t)dt = 0 and sup(min {]IA~ (x,t, L )II:L E A (t)}: I i x - ~ ( t ) ] ] < a}_< aka(t) for all t E [a, b] and all • E ] 0, ~], where

AY~(x,t, L) d--~ff(x,t) -- f ( ~ ( t ) , t ) -- L - ( x - ( ( t ) ) . We use VG(f, ~) to denote the set of all variational generators of f a b o u t the curve ~. If n, m, f , a, b, (, A are as in Definition 8, we use F(A) to denote the set of all measurable single-valued selections of A. Then /'(A) is a n o n e m p t y convex weakly compact subset of L 1 ([a, b]; ~ , n • We now specialize to the case when m = n. If L belongs to LZ([a, b]; ~n• we let ML be the fundamental m a t r i x solution of the linear time-varying equation ~ / = n(t). M. T h a t is, ML is a continuous m a p from [a, b] • [a, b] to ~'~ • that satisfies

ML(t,s) = 1 ~ . +

L(r) .ML(r,s) dr.

(5)

496

Hdctor J. Sussmann

F a c t 4 If B C Ll([a,b],~ nxn) and 13 is bounded, then the map -

is continuous.

f3~eak~L_~MLeCO([a,b]•215

<)

If n, m, f , a, b, ~, A are as in Definition 8, and m = n, we define

.A4(A) ~ f { ML : L E F(A) } C_ C~ [a,b] • [a,b] ; ]~n• F a c t 5 A4(A) is nonempty and compact. In particular, if t , s E [a,b] then the set

./t/[t,s(A) d-~f{ M L ( t , s ) : L E F ( A ) }

(6)

is a nonempty compact subset o f ~ nxn.

8

<)

Differentiation of flows

We let C(2) be the set of all cones in R 2 that are products C+ x C_, where C+ E C(1) and C_ E C(1). Then C(2) has exactly sixteen members. If X_, X+ are finite-dimensional real linear spaces, M is a set of linear maps from X_ to X+, and v+, v _ , are nonempty subsets of X _ , X+, we define a set [ M ; v + , v _ ] of linear maps from X_ x ~2 to X+ by letting [ M ; v + , v _ ] be the set of all maps [M;v+,v_], for all M E M, v+ E v+, v_ E v _ , where [M; v+, v_] is the linear m a p from X_ x I~ 2 to X+ given by [M;v+,v_](V, fl, a ) ~ f M

" v + fl . v+ - a . M . v_ .

It is then clear that if M , v+, v _ are compact, then [ M ; v + , v _ ] is compact. The following result is the general theorem on G D Q differentiation with respect to the state and the endpoints, along a trajectory ( E Traj c(f), of the flow ~ ! generated by a time-varying vector field f . The basic requirements are local integral continuity, the existence of a variational generator, and the existence of approximate limits of f at the endpoints of ~. T h e o r e m 3. Assume that Q is a manifold of class C 1, f is a time-varying ppd vector field on Q, a,b E I~, a <_ b, C + , C _ E C(1), C = C+ x C_ E C(2), and C b'a = (b, a) -{- C = (b -s C+ ) x (a + C_).

Assume, I. f is 2. A is 3. v+, that

in addition, that locally integrally continuous, a variational generator for f along ~, v_ are nonempty compact convex subsets of T((b)Q, T((a)Q, such

v+ E App-limx_,~(b),t~b,t-beC+f(x, t ) , V_ E App-lim~-.((a), t - . a , t _ a e c _ f ( x , t)-

Then the set [.h4b,a(A); v+, v_] belongs to G D Q ( ~ f ; (~(a), b, a), ~(b); O x C b ' a ) .

Set-valued Differentials and the Maximum Principle of Optimal Control

497

T h e o r e m 3, c o m b i n e d with the chain rule, yields a result for p r o d u c t s of flows. To s t a t e this result, we first define, if M 1, . . . , M m C ~nXn and V~., . . . , v + m , v _ 0, . . . , V m_- - 1 C_~ n , a s e t

[ i 1,

.

.

, M. m.; v. ~ , .

, V +,~ ~ V _0~ . . .

,

v_,2-1]

of linear m a p s from ~ n x ~ m + l to ~ n . Precisely, this set consists of all the m a p s [ m l , . . . , Mm; v~,..., v~', : , . . . , ~-11, for all M 1 E M 1 , . . . ,

M m E i m, ~k E v ~ , . . . ,~7 s v~, vo E vO, . . , ~ - ~ E v~-1, given by [M1,...,mm;v~.,...,v~,v~

am,...,~~ I n- - 1

do=,pm,o

v + ::r

- :p

,ovo_ +

k=l

where pt,k _=M t . Ml-1 . . . . . M k + l if k,gEl~l, 1 < k < g < m . Also, if f = ( f l , . . . , fro) is an m - t u p l e of t i m e - v a r y i n g vector fields on ]~n, we define the product flow map by letting 4~f(z, a m ,

9 ..,a

1

o~def

,a ] =

(

= \ e la=,a~_,

f,,-,

0 #am_l,am_:}

0 ...

o 4',:' , : ) (z)

for (x, a m , . . . , a 1,a ~ E ~ n x R m+~.

Remark 1. T h e concept of a v a r i a t i o n a l generator also m a k e s intrinsic sense on manifolds. T h e i m p o r t a n t new p o i n t is t h a t now A has to be taken to be a section along ~ of an a p p r o p r i a t e bundle. We m a k e this precise in the only case t h a t will be used here, namely, when st is an " a u g m e n t e d p p d vector field" on a m a n i f o l d M , t h a t is, a t i m e - v a r y i n g p p d section of the bundle T M x 1~. In t h a t case, we let E M be the vector bundle over M whose fiber EM(X) at each x E M is the p r o d u c t JI(VF(M)) xT~M, where J~(VF(M)) is the space of 1-jets at x of s m o o t h vector fields on M . T h e n , if a, b, ~ are such t h a t a, b E ~ , a < b, and ~ E C ~ [a, b] ; M ), a variational generator for f about ~ is a m e a s u r a b l e set-valued m a p A : [a, b] ~ )EM such t h a t 9 A(t) is, for a.e. t, a c o m p a c t convex n o n e m p t y subset of EM(~(t)) such t h a t ~rl,0(v) = fl(~(t),t) for every v E A(t), where ~rl,0 is the canonical projection from J~ (VF(M)) to J~ and f l is the T M c o m p o n e n t of f ; 9 locally, relatively to suitable c o o r d i n a t e charts, there exist kA, &, k having the three properties of Definition 8 . W i t h this definition of v a r i a t i o n a l g e n e r a t o r T h e o r e m 3 r e m a i n s true on m a n ifolds. A more detailed discussion o f the invariant definition of v a r i a t i o n a l g e n e r a t o r s is given in S u s s m a n n [6]. <>

498

H&tor J. Sussmarm

Theorem4. Let n, m, a . , a ~ m, f l , . . . , f m , f, ~x., . . . ,~m, C, C, C o , . . . , C m, be such that n E Z+, m E N, a . = (a~. ,. .. ,a ~ e ~ m + l , C~

,Cm

Er

m x C m-1

c = c ~" = a . + c =

andf=(fl,...,fm) 1. 2. 3. $. 5.

E

(T V V F ( ~

x~

, x m,

a.o < a.1 < _ . . . < _ a T

x . . . x C I xC~

(a~+Cm) •215 (a~176 n

.Assumethat,

fori=l,...,m,

f i is locally integrally continuous; ~ e C~ [a~-~,a~] ; ~ " ) N T r a j , ( f ' ) , ~ ( a ~ - x ) = x/.-1, and ~i.(ai.) = xi.; A i is a variational generator f o r f i along ~i, v~., vi_-1 are nonempty compact convex subsets o f ~ n such that V i+ E App-lim~,..,,~,,:, t_.+ai.,t_aiecifi(x, t ) ,

v/fi x E App-limr~r,:-~ , t . . + a : - ' , t _ a i - l e c , - , f i ( x , t ) . Let M i = . A / l a ! , a , - , ( A i ) , f o r i = 1 , . . . , m . Then the set [M1,...,Mm;v~.,...,vT,v~

(

belongs to G D Q ~ f ; ( x . , a .

9

,... ,a.),x.

~-1]

)

,1R x C .

(>

A G D Q m a x i m u m principle

T h e o r e m 3, together with the directional open m a p p i n g theorem 2, i m p l y a version of the m a x i m u m principle that contains and improves upon several previous s m o o t h and nonsmooth versions, for vector field systems as well as for differential inclusions and systems of differential inclusions. Moreover, one can also allow " j u m p maps," and obtain a "hybrid" version. We state this more general version directly but, for simplicity, we only discuss the vector field case. For our restricted purposes, let us define a hybrid optimal control problem to consist of the specification of a finite sequence ( 2 2 1 , . . . , 22u) of "ordinary control systems," together with "Lagrangians" L 1 , . . . , L u for 221,... , 22u, "switching constraints" S l, . . . , S u, "switching cost functions" W1, . . . , W~,, and "time sets" T_1 , T~, T_2 , T ~ _ , . . . , T_~, T~. Precisely, each 22i is a triple S / = (Qi, Ni, Fi) consisting of a state space Qi, a controller space H i, and a controlled dynamics, that is, a p a r a m e t r i z e d family F i = {Fi~}neu, such that, for each 7/ E L/i, F~ is a p p d time-varying vector field on Qi. Each L i is a family {Lni } , e u ' of ppd functions L,i : Qi x ]~ ----> ]~. For each i E { 1 , . . . , p}, the switching constraint S i is a subset of the p r o d u c t Qi x ]~ x Qi~l x ~ , where "iScl '' m e a n s "i + 1" if i < p, and "1" if i = ~.

Set-valued Differentials and the Maximum Principle of Optimal Control

499

The switching cost functions are functions ~0i : Qi x IR x Qi-i-t x IR --+ IR. T h e time sets T [ , T~ are subsets of IR. A controller is a/~-tuple r / = ( r / l , . . . , rf') E H 1 x --- x//~'.

A trajectory for a controller t / = ( r / t , . . . , r/0) is a / J - t u p l e ![ = ( ( ~ , - - - , ( ~ ' ) with the property that, for each i, (i is an absolutely continuous curve in Oi, defined on a compact interval I(~ i) = In_ ((i), a+((i)], that satisfies the conditions ~i(t) ~ Do(F~,) and ~i(t) = Fi~,(~i(t)) for a.e. t E I((1), and is such that

~(~) e S ~, where

~(t)~f(~i(t),t)

and

cri (~)d_~f(~i(a+ (~i)), ~ig-i (a_ (~i-i-1))).

A trajectory-control pair (abbr. T C P ) is a pair (~, tl) such that t/is a controller and t~ is a trajectory for t/. A T C P (l[, r/) is admissible if, for each index i, a-(~ i) E T i , a+(C) E ~ , and the functions I(~ i) 9 t -~ L i (~i(t)) E IR are a.e. defined--that is, ~i(t) E Do(L~.) for a.e. t E I ( ~ i ) - - a n d Lebesgue integrable. The cost of an admissible T C P (l~, 1/) is the number

J(~, ")d-----elE# fa+(,') LT/'i(~i(t)) dt i=1 a-(~i)

-~ E ~ 9 ( o . i ( ~ )

)

i=1

An optimal TCP is an admissible T C P (~, r/) such that J(~, r/) < J(~', r/') for every admissible T C P (~', tf). For each i, we define the L-augmented dynamics to be the family

of ppd maps from Qi • ~ to TQ i • ~ given by

~i(q,t) = [F~(q,t)] [i'~(q,t)j

for

(q,t) E Qi • ~ .

Now assume that H1. (~,r/) = ((~1 . . . . ,~f,), (r/i,... ,r/~,)) is an admissible TCP, and a i_ = a _ t(ci~ i x i_ = ~ i ( a i ) , i _ i (a+), i , , , a +=a+(~;), x+--~ for i = 1 , . . . ,p. H2. ! [ 1 , . . . , A l' are variational generators f o r / ~ , , . . . , F~, along ~l . . . , C'. H3. C i are, for i = 1 , . . . ,/~, ~r E { + , - } , cones in/R such that a ia E Inta~+c~Ti~

and

~i def~ i " -:a = txa, a~,)

is a point of approximate continuity o f / ~ , along C~,.

500

Hdctor J. Sussmann

H4. For each i E { 1 , . . . , p}, Ci is a convex multicone in 7". ~ Q+i • Ts._~, Q ~ l where, for a E { + , - } ,

R~ d~r{{0} if C~ ; {0}, =

A if C~

{0},

Q~ ~e=,Q, • (a~ + R~). H5. s is a G D Q approximating multicone for the restricted switching set

at the switching point p i = ai(~) = (.~., ~i-f-t). H6. For each i E { 1 , . . . , / t } , / 2 i is a subset of the dual space

(that is, of

~ (Q i x ~ )

x T ~ _ , +.~ ,tt~i-i-t .~ x ~

/)'

, or, equivalently, of

__(T(-z;,-~'J")(Qi x A x Qi-i-t x 11~))t),_ and /2 i belongs to the generalized differential quotient GDQ(~i; pi, ~i (pi); Q~. x Q/._+t). H7. Por each i E { 1 , . . . ,p} and each r/ E N i, the time-varying map /~ is locally integrally continuous. H8. Each control system ~/ is invariant under time-interval substitutions. (That is, if r/,~ E/4 i, ( E C~ [a,b] ; Qi) n T r a j ~ (_b7 0 )i , and J is a compact subinterval of [a,b] such that (~(t),~) ~ Do(F~) for t e J, then there exists a controller 0 E / / i such that F~(q,t) = Fio(q,t) whenever q E Q i t E [a,b],t q~ J, and Fio(q,t) = F~(q,t) whenever q E Qi, t E J.) We now define the notion of a "multiplier" along (~, r/), and what it means for a multiplier to be "Hamiltonian-maximizing." For i = 1 , . . . ,/J, and ~" E/4 i, we define the Hamiltonian

H~ :T*Q i

x ~ x ]~----:;> ]~

by letting

Hir

Ao) = A. F~(q,t) - AoLir

D e f i n i t i o n 9. If H1-H8 hold, then a multiplier along (~,11) is a triple (~b, r x) with the property that: 9 ~b is a p-tuple ( r , Cu) such that each r is a field of covectors along ~i (that is, r is a map from the interval [a/._, a~_] to T*Q i, such that ~bi(t) * i belongs to T~,(t)Q for every t E In/_, a~.]); 9 x i s a 2p-tuple (~t_ , ~ . ~u+ l ~ such that x ai E R ai whenever i belongs , ~u to { 1 , . . . ,p} and ~r E { + , - } ; ~...

-~

Set-valued Differentials and the Maximum Principle of Optimal Control

501

is absolutely continuous and satisfies the adjoint differential in-

* each r

clusion -(bi(t) E [ r 1 6 2 * r E~andr * for each iE { 1 , . . . , p}, r

.fi(t)

tE[ai_,a+],i.

for a.e.

{ + , - } , the switching conditions

r ~r

i .k ~ i e (c~)R~

+ (cq~,,

hold, where

~ = (-r --~" i

=

--

T(~.~.,z,j.,)(Q

x i ~_h ~+ ~ w ,/,~*~ai*~ _~'- - - h i 4 ~ 1 , t 1, + - -

~x ~ x

-

Qi4.1

x

-

I~) ,

and

h/a =

~ (x'a, r (a/a), a~~, r H o,

Remark 2. The switching conditions take a more familiar form in the case of "fixed switching times" (that is, when the sets T~ consist of the single points aia) or of "totally free switching times," that is, when the Ta* are equal to the whole real line It~ or, more generally, are neighborhoods of the n oi. . Indeed, in both cases we can take C~ = Ria, and then (C~)R~ i J. = {0}. It follows that the i vanish, and the switching condition becomes

Ka

(-r

' '(a+),h+~,r

_ Er ~*~)

i+

(c')~,.

Suppose, in addition, that either I. we are in the fixed switching times case and the switching conditions are . i " i (x+, xi_"+I) E 5,0, where each S~i is a subset of Qi • Qi~rl , or

II. we are in the free~ switching times case, .the switching conditions are of the form (xi+,xi_+1) E Sio, where each S~ is a subset of Qi x Qi+t, the switching cost functions ~i do not depend on the times, and the times a iv are required to satisfy a+i = ai-i-1 _ . Then, in case I, each Ci will be a multicone in the product

r~,+O ~ x

c)i4-1 x { o } , { o } x ,~~ .,_+,~

so the switching condition will not impose any restriction on the h~. On the other hand, in Case II each Ci will be a multicone in the set

. r)i4-1 x ~ : r = {(v,r,w,s) e T , , + Q i x ] ~ x ~ ~ ,+,~

s}.

502

H6ctor J. Sussmarm

Hence all the m e m b e r s (6, § ~b, ~) of C~ will satisfy ~ + ~ = 0. Moreover, the fact that the ~oi do not depend on the times implies that we can choose the f2i to have vanishing time components. It then follows that the t i m e - p a r t of the switching condition becomes the familiar requirement t h a t h~_ --h~4.1 , that is, the condition that the H a m i l t o n i a n should not j u m p at the switchings. D e f i n i t i o n 10. If H1-H8 hold, and (~b, r x) is a multiplier along (~, r/), we say that (~b, r ~) is Hamiltonian-maximizingif, for every i E { 1 , . . . , p}, the inequality H~ (~i (t), r t, r _< H ,i, (~i (t), r (t), t, r i holds whenever ( E U i, t ~,_j] a i_, a +[, and continuity of ~T]l i and F~

(~(t),t)

is a point of a p p r o x i m a t e

D e f i n i t i o n 11. If (~b, r ~) is a multiplier along (~, r/), we say that (~b, r ~:) is nontrivial if it is not true t h a t r = x [ -- x~. = . . . . xu = x ~ . _ 0 and all the functions r are identically zero. (~ 5. If H1-H8 hold, and the pair (~, TI) is optimal, then there exists a nontrivial Hamiltonian-maximizing multiplier along (~, 7). (~

Theorem

By taking p = 1, Theorem 5 can be shown to include the classical "nonhybrid" s m o o t h and nonmsooth versions of the m a x i m u m principle given, for example, in Pontryagin et al. [5], Berkovitz [1], Clarke [3,2]. In t h a t case, the switching condition of Definition 9 becomes the transversality condition. When the augmented vector f i e l d s / ~r/'i are of class C l, one can take

A~(t)= ( ~(~(t),t)},

(7)

and the adjoint differential inclusion becomes the classical adjoint equation. On the other hand, if the function x --+ F i ( x , t ) is differentiable at ~i(t) for almost all t, then one can still take ./li to be given by (7), and A~ is a variational generator, provided that the differentiability of x -+ / ~ . (x, t) at ~i(t) has an obvious integral uniformity property with respect to t. So T h e o r e m 5 is in fact stronger than the classical versions, even in the setting of single-valued differentials. In addition, when the F~, are Lipschitz continuous on some tube a b o u t the reference trajectory, with an integrable Lipschitz constant, then one can take 0 stands ./1 9(t ~ : o be 0 / ~ ,,t(~ ' (t)), where F~: - ' r ]"s t h e m a p . x c F ~ -, L x , t ) , a n d ' .... for Clarke generalized Jacobian. Moreover, m ~ue ~1~ .... ltz case one can often take the /~i to be smaller than the Clarke generalized J a c o b i a n (for

Set-valued Differentials and the Maximum Principle of Optimal Control

503

example, equal to the classical differential, when it exists), so even in the Lipschitz case T h e o r e m 5 often yields a stronger conclusion than the usual n o n s m o o t h results. T h e o r e m 5 also applies to problems where the vector fields are only continuous with respect to the state (in which case the flow m a p s are set-valued) and to problems with discontinuous vector fields. An i m p o r t a n t class of such problems arises from differential inclusion systems. As long as the inclusions under consideration are almost lower semicontinuous, then there exist sufficiently m a n y integrally continuous selections to make our theorem applicable. All these applications will be discussed in a subsequent paper.

10

Proof

of Theorem

5

It is clear that we can assume, without loss of generality, that i

i i xi~-i (x+,a+, _ ,a~"-+ 1 ) = 0

for

i=l,...,/~.

(8)

We make this assumption throughout our proof. For each i E { 1 , . . . , p}, we let X i denote the space of all continuous fields of covectors along ~i, so the m e m b e r s of ,~,i are the m a p s * i E T~,(t)Q

[ a i , a~_] ~ t -+ r

such that r is continuous as a m a p from [ai_,a~.] to T*Q i. Then ~l"i is a Banach space. If i E { 1 , . . . ,~}, we use 1]i to denote the set of all pairs (~,t) such that a i_ < t < a~_, ~ E /1 i and (~(t),t) is a point of a p p r o x i m a t e continuity of /7~, a n d / ~ . We then write 13 to denote the set of all triples (i, ~, t) such that i E { 1 , . . . , m } and (~,t) ~ 12i. If )'V is a subset of ]2, define ~Pw to be the set of all multipliers

(~, r ~) = (r

r

r ~_, ~ , . . . , ~_, ~ )

along (~, r/) such that g

r + ~(llr

+ I~-I + I~1) = 1

i----1

and (&) the inequality

H~ (~ (t), r (t), t, ~0) _<~ , (~ (t), r (t), t, r holds whenever (i, ~, t) E W ,

(o)

504

H~ctor J. Sussmann

Then ~Pw is a compact subset of the product space X~X

lxX ~X...XX

~x~+l.

Our goal is to prove that ~Pv • 0. It is clear that, if ~'V1,... ,YVk are subsets of V, then LPW~u...u~,Yk -- ~W~ r l . . .

~w~ 9

Therefore, if we prove that

(*) Ow # 0 whenever W is finite, then we will have shown that

{~W }WCV,W finite is a family of nonempty compact subsets of ,Y that has the finite intersection property. Since ~v = A {!Pw : W _C V, W finite } it will follow that Cv r 0, proving our conclusion. So it suffices to prove ( * ) . For this purpose, we fix a finite subset YV of V, and write kV = U~=IW' , where kV' C_ {i} • Y'. Write k~;i = {i} • 1/V', so

W i C_ Y i . We introduce the cost-augmented state spaces

O~ %r Oi • ~, together with the cost-augmented time-varying vector fields

F~,r 9 TVVF(Qir defined by Do(P~,r : { (q, r, t) 9 Q~: (q, t) 9 Oo(F~) ADo(L~) },

F~,r

r, t) = [[L~(q,t) F~(q, t)] i f ( q , r , t ) 9 Do(F~).

(Here we are using the canonical identification of T(q,r) (Qi x ~ ) with TqQ i • and writing the members of TaQ i x ~ as column pairs. The above formula defines F~,c(q,r,t ) as a member of TqQ i x ~ , so F~,c(q,r,t ) belongs to T(q,r)(Qi x N). Therefore F~, c is indeed a time-varying vector field on Q~.) Then an integral curve of F~,r is a locally absolutely continuous curve I 9 t ~ ~r defined on an interval I, such that

= (~(t), A(t)) 9 Qi,

Set-valued Differentials and the Maximum Principle of Optimal Control

505

(a) ~ is an integral curve of F~, (b) the function I E t --+ L~(~(t),t) is a.e. defined and locally integrable, and (c) A(t) = A(s)

+Jtn~(~(u),u)du for all s,t E I.

(In other words, ~c consists of an integral curve ~ of F~ together with a "running cost" function A along ~.) We also introduce the

cost-augmented variational generators Aic, defined by Aic(t) = Ai(t)

x

{0}.

Precisely: 9 If Qi is ~ n or an open subset of ~ n , so that Q~ is an open subset of ]~n • ~ = ~ n + l , then .4i(t) is a subset of ]~(n+Uxn, whose m e m b e r s are (n + 1) • n block matrices L=

[ L]

LEI~ nxn ~ E I K lxn

so the adjoint differential inclusion is equivalent to the assertion t h a t

-r

L(t)]

e(t) J

= [r162

c.e

for some measurable selection t

L(t) =

[[n(t)]e(t)

o f A i. In that case, the set A~(Q is a subset o f ~ (~+l)• bers are the square (n + l) • (n + 1) block matrices

whose m e m -

suchthat Lc= [~] E.4i(t).Theadjointdifferentialinclusion, with the statement that r

is constant, is equivalent to the assertion that

for some measurable selection t-->

of ~i, where

together

:

rL/,)00] Le(t )

= [r162

-

506

H~ctor J. Sussmarm

* If Qi is a manifold, then the above description of the nature of z[i and A~ and their relation to the adjoint equation remains true locally, in coordinates, and can be made valid globally, in an intrinsic way, as explained in R e m a r k 1. We now let (~ be the cost-augmented version of (i, obtained by initializing the running cost to the value 0 at time ai_.. T h a t is, ( i : [ai__,a~.] --+ Q~ is the curve given by

~t(t) = [r

t,v(t)

,

where Ai(t) =

L 0,

(s), s)

If u E N, we use N + , , to denote the nonnegative orthant of N~, that is, the set of all row vectors e = ( e l , . . . , eL,) E NL, such t h a t e j >_ 0 for j = 1, . . . , u. For e = ( e l , . . . , e , ) E Nu, we write

lel

%f

I~I +-"+

Is,,l.

In particular, i f e E N+,~,, then JeJ = et + -.. + eL,. If r > 0, we use 7~u(r) to denote the u-dimensional simplex

P . ( r ) % f { ~ ~ ~ + , . : I~1 < r } . For each i, we let u i be the cardinality of 14,'i . We choose once and for all an ordered ui-tuple W ~ =

(

i i (q,t~)

,

(r

,...,

( ( u~, , t u , ) )

(10)

such that the times t} satisfy

~'~ _< t'~ _<... _< t'~,, (The ordered ui-tuple and 14;i is the set {((~,t/1) , ((~,t/~) , . . . , ( ~ , , t ~,).}. i W i is of course uniquely determined by the set 1/W in the special case when 14;i has no "repeated times"--i.e., if ((, t) E I'Vi, ((', t) E 14]i implies ( = ('.) Also, we write t~ = ak, t~,+l = a~_. We let § be the minimum of all the nonzero members of the set { /t j + l - t j : j =iO , . . .

,

ui

, i=l,...,p

}

.

We then define, for each i, afline functions

~,(§

~ ~'

, ~-;(~') e

inductively for j = 1 , . . . , u i + 1, and prove inductively that the inequality t~i < _ r ~ ( e , )" _< t 5i + 4 + - - - +

e ji_ l

if

e i = ( e li, 9 . . , ~ , i)

E ~,.,(§

( 11)

Set-valued Differentials and the Maximum Principle of Optimal Control

507

holds i f j E { 1 , . . . ,vi}. The construction is as follows. First, we define v~(e i) = t~, so (11) is trivially true when j = 1. Next, assume that rj(e i) has been defined for some j E l~l such that 1 _< j < v i, and (11) holds. If tj+li is equal t o tj,i then we let rj+l (r = 7.j(el) + e~.. If t}+ 1 > t}, then we define rj+ 1 (e i) = t~.+l. It is then clear that, in both cases, e i --~ rj+l (e i) is an affine function, and (11) holds if j is replaced by j + 1. We complete the definition by letting riv,+i ~[ei/, = aS-" It is clear that rj(0) = tji for all i, j. It follows from the construction that the inequalities rj(~') + ~ < rj+,(~')

02)

hold for j = 1 , . . . ,v ~ and e i E Pv,(~). Indeed, (12) follows clearly from the definition of rj+l(ei ) if t}+ 1 = t~.. If t}+ 1 > t}, then (12) follows because (11) implies that

~j(,') + ~] <_ t} + ~'~ + ... + 4-1 + 4 < t} + I-'1 < t j i+ P <

i i i = U+I(~). t~+l

The inequality (12) implies that, if we write

then for each i, {Ij (ei)}je{t ..... v,} is a familiy of pairwise disjoint subintervals of [ai,a~+], has length ej' and z } ( ~ ' ) c _ [tj,tj 9 such that I}(e') " ~ ~+ § j = 1 , . . . , v i. We let v, Zi(~i) ~ f U ij(~i), j=l

so Zi(~ i) has measure I~il: Write Q,. = Qi x R~. x Ri._ x ] ~ v ' . Fix i, and define set-valued maps i i Oj : Q.

, i " "Qc

inductively, for j = 1 , . . . , v i + 1, as follows. First of all, we let

O~(z,~+,~_,~) = ,~;',r ~t~,a'

+a_

(z,0)

We then define

o}+~(z,~+,~_,~)=

e;+;),),,;(,)+,oe;(,)+,,,,;(~)

(O}(z,~+,~_,~))

for j = 1 , . . . , v i - l, and

0~,+~ (~, ~+,~_, ~) =

o;+.+,.:,(~)+,

o 9 v(.)+~ ,,.:,(~)(o~,(z,~+,~_,~)).

508

HEctor J. Sussmann

We let 0 ~ ' def____O vl i +

.

(In other words: we make a "packet of needle variations of 7/i," by substituting the controller (j for r/i on the interval [vj(e), vj(e) + r for j = 1 , . . . , ui; then, using the new control--which depends on e as a p a r a m e t e r - - w e move in the cost-augmented state space Q~ by initializing the state component at z and the running cost component at 0 at time a i_ + (~_, and then following integral curves of the new dynamics up to time a~. + (~+, thus obtaining, for each value of z, (~+, c~_ and ~, a not necessarily unique point in Q~; then O ~ ( z , o%, o~_,~) is the set of all points that can be obtained in this way. T h e fact that the endtimes are a~r + a + and a i_ + c~_ rather than a i+, ai... means that we are also making "variations of the endtimes;" the fact that the initial condition is z rather than x~ means that we are m a k i n g "variations of the initial state" as well.) Write

X i = ~ir = (x i_ O) X i'1 = ( X L ai_)

Y' = ,% ~(x'") --

.

3.

--

'

X +i ----~c(+) s a s = (~ i (a+),.k i i ( a+i) ) , x+i'l : 9

y.~ =

iX i t

i

+,

ai +1 ,

i,l

dx+ ),

G'

3.

x: = ~rI 1 = (d(tj), ;r x j ,1 = (xj,t~), i~ =

Yj =

F;,

),

FC;,r

),

z': X!,~ x~3 '~

i

i

i

( x ~i , %i , , it j ) ,i

(xj,t~,tj,tj+,) .

It is then clear that O #i (x_i , 0, 0, 0) = {X~_} and Oji ( x i_ , 0 , 0 , 0 ) j = 1 , . . . ,v i. We now let

K i=QixCi+xC

i xP~,,(§

= {Xj}

for

(13)

and c o m p u t e G D Q s Dj E GDQ(Oj;(xi_,O,O,O),Xj;K

i)

inductively, by applying Theorem 3 and the chain rule.

(14)

Set-valued Differentials and the Maximum Principle of Optimal Control For j = 1 ,u i, we let Dj be the set of all linear m a p s A i measurable selections Lr of A~, where '

" " "

j,L~,

509 for all

3"--I

~,~

~,~+,.-,~

~,_.

-

_

M~,_ da ML~ (t(, ai ) Mad, = ML~(t ,t

,

and

and prove by induction on j that (14) holds for every j E { 1 , . . . , u'}. First of all, let A~ : Q~ x ~ x II~ x ~ u ' --+ Q~ x ~ x ~ be the m a p

(z,,~+,~_,~) Then

o~

~ ( ( z , O ) , t ~ , a L + ,~_).

o A'~.

= ~<,,o

The m a p A~ is of class C 1. Therefore, if B~ is the linear m a p B~ : T.,._ Qi x ~ x ~ x ~,,, --~ TxL Q i x ~ x given by

B~(~,.+,._,~)

= (~,0,._),

then

{B~} E GDQ(A~;(xi ,O,O,O), (Xi_,t~,a/ );Ki). Let

K~ = O, ~ • ( d + c ~) x (a'_ + c _ ) . Theorem 3 tells us that a m e m b e r A/1 of is given by A/I

Clearly,

=

[Mq,o._ (Ari . {F~,i r

AjK i C_K~,

GDQ(r162 (XL, t], as X[; K[) i,1

i )}, {F~,,r

i,1

)}]

so the chain rule applies, and we can conclude that

D~ = A~ o Bt e GDQ(O~; (xi._, O, O, 0), X~; KI). Now assume that j E { 2 , . . . , ui}, and we have shown t h a t D~_ 1 belongs to GDQ(O~_I; (xi._, O, O, 0), Xj_I; Ki). Let

A~:Qix~x~x~,.,~

>:,qix~x~x~

be the set-valued m a p that sends (z, a + , a _ , ~) to the set

oj_l(z,

~+, ~_, ~) • {~j_, (~)+~3._1} x {~j-1 (~)} x {~j(~)}

510

H&tor J. Sussmann

Let.4~. : Q ~ x ~ x ~ x ~ ) )Q~• the point (Z, rl, r2, r3) to the set i

~b,c(Z, rl,r2) • {r3} x {~I}. Then

o~. = ~:,,c o ~. o A~.. It follows from the inductive hypothesis that, if B~ is the set of all linear maps B~,Lr : T~, Q i x R x IR x ]~., -+ Tx;_, Q~ x ~ x ~ x ~ , for all measurable selections Lc of A~, where

9 ' i B~,~(~) = ~j_~(~) + ~_~ - tj_~, B~,~(~) ~j_~(~)- t j' - i , =

9

i

B~,4(~) ----7j(x) -- tj, then

B~ E GDQ(A}; (xi_, 0, 0, 0), X~'_ai; Ki). ~ .

Theorem 3 implies that, if B~ is the linear map

[~j : Tx;_, Qir x

~ x ~ x

R - + Txj_, Qir

x ~ x l~

given by then

{/~} E G D Q ( A~ji , .X ~ _i , 3I , X ~ _i ,I2 , Q. r

i

x ~3).

Finally, Theorem 3 also implies that, if/~j,Lr is the linear map

hj,Lr : Txj_~ Qri x ~ • l~-+ Tx;Qr given by B~,~r

i ~ , ~ ) = ~ 5 ~ + i~,~_~. (v - ~ U - ~ ) ,

and ]3~. is the set of all/3j,Lr for all measurable selections Lr of A~, then F* Bj~ E GDQ(4~ ,',r x j _i,~i , X j ; Q r i

x

]~).

Then the chain rule implies that

B~^'oh} oB~9 e O n Q(o;;i ( x i _, 0,0,0),x? i a ; g i )

Set-valued Differentials and the Maximum Principle of Optimal Control

511

Clearly, 1~i~. o B~~i o B~' is the set o f all m a p s Bj,Lr o B~~i o Bj,Lr , for all Lr E F(Air G i v e n a p o i n t (v, o%, or_, e) E T x L Q~ x ~ x ]ir x ]Ir we have Bj,Lr (Y, O~..I., O~_, E) = (V, P l , P2, P 3 ) ,

B j ( V , p l , p 2 , p 3 ) = ( V + (Pl - P 2 ) ~ i j - I , P a , P l ) ,

and

where v = A~._X,L.(V, ~+,._,.),

p~ = B ; ", ~ ( , ) ,

p~ = Bj.,~(~)),

p~ = Bj,.(~)

It follows that

~,

o~o

~_,~)=^,

-

,

~))

____B~,Lr " ( ( V + (Pl -- P2) ~ ji- I , i

= PaY] + M j 5 _ I " W , where W -- V -[-(to I - p 2 ) r j i _ 1 - p l Y j i _ l .

Since Pl

-

P2 : e j - t a n d Pl = r~ l ( e ) - t ji _ 1 -b e j - 1 , we find

w = v + ~s_l~ ~ x - plYj-1 = A}_I,L~(V,~+,~_,~) --"

j--l,--

+ ~ j - l ~ _^~' j-~

- ;1~'_1 i

--O~_

"

k---1

+(~-j_l(~) - t j' _ l ) Y j _~l + E j - 1 y~i - 1 - P l y~j - 1 j-2 k=l

+(,-j_x(~) - t}_~ + ~j_~)~'_~ + ~_~r

- ~'_~) - p ~ ' _ ~

j-2

-- ~:_~,_. (~- o_~) § Z~M:_~,~. ~ k=l

+(.j_x(~) _ '

_

+ ~j-IZj_I j--1

:

M ~ _ 'I ,

.

(~ ~-~) Z~,;-~,~ ~'~ _

+

i

i

k=l

Therefore

'

'(

)

j-1 k=l

.

P3, Pl))

.

512

Hdctor J. Sussmarm

Then ^i ~i " (Bj,Lr o Bj o B),Lr

i , a+, a _ , ~) = pzYj + Mj,j_ 1 9W j-1

=

a yi

9

Mi

"

tj)Yj

k=l = Z },Lo

a+, ._,

.

It follows that o

Bj,Lr

:

D~,

so (14) holds. Now that we have proved that (14) holds for all indices j 9 { 1 , . . . , v~}, we know in particular that

D~, 9 GDQ(O~,; (x~, O, O, 0), X~,; Ki).

(15)

We let D~ be the set of all linear maps A i# , L r Lc of A~, where

for all measurable selections Vi

, a + , a _ , e )-aef Ai#,LctV, = a + Y r i + M+,_i 9 (9 _ a_Y_i ) + Z e k M ~ _ k . Z~' , (16) k----1

and we let aL)

.~_d-----efzv~Lc(a~.

aari d e f :~~r tai ,v,+,k='V, Lot +,t~).

and

We will prove that

D# 9 GDQ(O#, (xL, 0, 0, 0), X+, K i ) . i

-~

i

.

'

i

.

(17)

Let A~ :Qi x ~ x ~ • ) )Q~• sends (z, a+, a_, r to the set

Oi~,(z,a+,a_,r Let.4~:Qi x~x~x]~

i x {r~,(r

i x {vv,(r

x {a~' q-c~+}.

) >Q~ x ~ x ~ b e t h e s e t - v a l u e d m a p t h a t

(Z, rl,r2,r3) to the set 4)F~'.o(Z, rl,r2) x {r3} x {rl}. Then O~ = 4)F; ',r o A# o A~ . 9

~i

"

It follows from (15) that, if B~ is the set of all linear maps

B~,L r :T~:LQi x R x ~ x ~ , , ~

Tx;,Q~ x ~ x ~ x R ,

sends

Set-valued Differentials and the Maximum Principle of Optimal Control

513

for all measurable selections Lr of A~, where ----

z~vi,Lr

(y, 0~+,

,

,

,

,

B~,~ (~) = 7-.',(~) + ~., - O , B~ , ~(~)

=

'

r~;, (~)

t ~' , ,

-

then

B ~ E GDQ(Ai#;(x~ O, O, O~ X i'a" K i) ~ .

Theorem 3 then implies that, if B~ is the linear map

h i : Vx:, Qir x ~ x ~ • ~---~ Tx:, Qir x il~ x II~ given by

[ ~ ( V , pl,p2,p3) = (V + (Pl - P~)Y'~,,P3, Pl), then {B i } E GDQ(-~i#;X ta Xi'2"Qir x ~3) V*

,

Vl

7

Finally, Theorem 3 also implies that, if/}~,Lr is the linear map ^i B#,L~ : Tx:, Q ir x ]~ x ~ -+ T~:uQr

given by

h~#,Lr

ai, a2) = a,Y~_ -t- M+,,,;

9(V - a~Y~,)

and B~ is the set of all J~#,Lr for all measurable selections Lr of A~, then ^ " B~# E G D Q ( r F ' ,, , ~

i,2

i i ' ,i ,X+;Qr x (a~_ + C+) x ~) .

It is easy to verify that (.4~ o A~)(K i) C_ Qic x (a~. + C~_) x R ) . Therefore the chain rule implies that ~i i B^ "~ o B# o B~#' E G D Q ( O #i , . (x_, 0, 0, 0), X~3; K i ) . ^

,

~.

.

Clearly, B~ o B~ o B~t is the set of all maps i

for all Lc E F(A~). Given a point (v, a+, a_, ~) belonging to the product Tx.- Q~r x R x ~ x ~ , , , we have B~,~o (~, ~+, ~_, ~) = (V, p,, p~, p~),

514

H~ctor J. Sussmann

and B-~~ (v,p~,p~,p~) = ( v + (p~ -

p~l?~,,p~,p~)

where

v = Z~,,Lr ' (v, a + , a _ , e) ,

= B ~ , ~, ( ~ ) ,

pl

B~,~(~)),

p2 =

It follows that ^i

~'

~

= p3Y~_ + , ,,V/i +,,,, . W ,

where

W=

V + (pl-p2)

(~) -

Since pl - p~ = ev, and pl = ~ ,

,

vi-PlY~.,.

t~, + ~v', we find

P"--I

(

= M~.,_ . ~ ) - a - Y i -

)+E

ekM~,',k . Z , k _ p l y 2 i

k=l

+(~,i (~) -

t .i , ) Y ; ,i +

Eul g^~i , lls_ t

= M ~ , , _ - ~ ) - a _ Y i_ + E

i i e k M ;i, , k . Z~-PlY~,,

k=l

-t~_~

+ ~slYj_~

- E,)

Vi--I

(

= Mi~",- " v - a - Y - i

)+

.

E

ek. ~,',k" Z~

k=l

i ( e ) - - t iv, -[- sv, - Pl)Y2,i + +(T~;,

~v i Z /

Vi

=M~,,_.(~-._Y'_)+E~M'~,,~.Zt. k=i

Therefore v i

M+,~,,

9

W = M~.,,,

.

v i _

. (~_a_y_i)+ E

i ~k ]~f2,i,k

k=l v i

= M'

9 (~ - ~ _ ~ )

+ ~M~,~. k=l

Z~

P3 :

Or+ .

Set-valued Differentials and the Maximum Principle of Optimal Control

515

Then A~

~.

.

=

M~.,~,, . W b, i

= a + Y r + M~,_ . ( 9 - a _ Y i _ ) + E e k M ~ , k "

Zik

k=l

----- A ~ , L r

Ot.{., O ~ _ ~ e ) .

It follows that

13~,Lr o/3~ o B~,Lr = D ~ ,

so (17) holds. We have thus shown t h a t for each i E { 1 , . . . ,/J}, the set D~ of all linear

maps Ai#,z, defined by (16), for all measurable selections Lr of Aic, is a GDQ of O~ at ( (xi_, O, 0,0), xi+) along the set K i aefinea by (13). We now combine all the O~t into a single "grand m a p "

O:Q.

~ )Q# x i ~ ,

where

Q. = Q l . x . . . x Q.~, and Q#=Q~• Roughly speaking, if p E Q . , then p = ( p l , . . . ,PU), where pi = ( z , ai + , ~i _ , e i)

~

for

Q~. (18)

i= 1...,/~.

Then each pi gives rise to one or several points Oi#(pi). So to each pi there correspond one or several terminal points w i and terminal Lagrangian costs ta, as well as a terminal time a+i + a +i , an initial time a i_ + a i , and an initial state z i. In particular, this gives rise to "switching points" ai E Q ~ , defined by a i = ( w , , a9+ +i

Ot~., ' Z z"-+ l

, a i_~ ' l + a i + l" ) .

(19)

Moreover, p also gives rise to a cost a0, given by P

P

i=1

i=1

9 " i

i zi4i,ai-i-1

..l_oti+l).

(20)

We will define O(p) to be the set of all/~ -t- 1-tuples

(al,... ,~,a0)

~ Q#

x I~

obtained from p in this way. The precise definition is as follows. Let p = ( p l , . . . , p , ) e Q. = Q~ x . . . Q.~.

(21)

516

H~ctor J. Sussmann

9 Define z ~,

o~i+,

' a~_,

~i

by means of (18), so

z i E Qi

i ,~+

,

ai E R - ,

ER+,

_

~i

E~..

Then O(p) is the set of all # + 1-tuples (21) such that, for some wIEQt,...,w

~EQ~,~ 1EIR,...,~ ~ER,

the conditions (19) and (wi,g i) E Oi#(p i) hold whenever i = 1 , . . . ,/~, and (20) is satisfied. This completes the definition of of 19. Let

p = (p,...

,p"),

where/5 i = (x/_, 0, 0, 0) for i = 1 , . . . , p. Then

o(f) = {(a, e0)}, where

=(dl,... ,@#), di=(x~.,a+,x_

,a_

j

for

i=l,...,/l,

# i=l

(Recall that we are assuming that (8) holds.) Let K=K

l •215

K ~'.

We now write down a G D Q

D E GDQ(O;f, (~, #0); K ) .

(22)

For this purpose, we first define a linear m a p Zlbc,w, for each ~-tuple Lc -- ( L c l , . . . , L c~) of measurable selections L~ of A~ and each /a-tuple oJ = ( w l , . . . ,w u) such that cai E f2i for i -- 1 , . . . ,/J. We then let D denote the set of all maps Ar.r of p-tuples.

for all possible pairs (Lr

Let u = (u 1,

, u") e VrQ.

Write tti

= (V i , 0 ( +i , 0 ~ _i , ~ i ) ,

so u i E Tp, Qi, for each i. Let s i = ( ~ , . . . ,

E~,) for each i.

Let Lr = ( L ~ , . . . , Lr~) be a , - t u p l e of measurable selections L / of A i. Write

L e,(~)

-

Set-valued Differentials and the Maximum Principle of Optimal Control

517

Then define , a L ~ , ~ ( u ) = (s 1

s~,so)

where (qi Ot+,i Vi+l,Oti+l)

r vi

r

+._ .(~' -

_r_') + ~ + y ~

,

j=l

so =

+ i=1

Then it is easily verified that (22) holds. Now let 7 be a smooth real-valued function on the manifold Q # such that 7(~) = 0 and 7(a) > 0 whenever o" # ~'. Let

Ca = c l x . . . x c ~x]-c~,0]. Define a subset ,Sr~t,# of Q# x ~ by letting

L,,,.#% ~ {(a, a0) E Q# x ]~ : o" E Sre.,t,# , aO + 7(0") < a0}, where

s , , , . a %r SL,t •

• sr

Then C# is a G D Q a p p r o x i m a t i n g multicone to S~e,t,# at the point (d, &0). Clearly, if p E K is such that (r E &eat,# for some then p , a give rise to an admissible trajectory-control pair a0. If a 5/: d, then a0 < #0, contradicting the optimality of Moreover, the fact that (r E , ~ t , # also implies t h a t optimality of (~, r/) tells us t h a t a0 = &0- Hence

(a, a0) E O(p), (~,//) with cost (~, r/). So o" = #. ao < &0, so the

O(K) n ,~,,,,# = {(~, &0)}. It then follows from the transversal intersection theorem that the multicones D ( K ) and C# are not transversal. Therefore there exists a nontrivial linear functional ~ E T(*tr,oo)(Q # • ~ ) such that 1. there exist C 1 E C 1 , . . . , C ~ E C~ such that

(~,(sl,...

,sU,r)) > O

whenever s 1 E C l , . . . ,s u E C u, r < O;

518

H~ctor J. Sussmann

2. there exist p-tuples

Lc = ( L c l , , Lr~) E/"(A~ x . . . x F(Ac"), = (~x ,~") E Y2x x x g?u such that

if, zaL~

_< o

for all u E K. Now write

(23)

= (~1,... , ~ " , - r where ~i E To~ Q~ for i -- 1 , . . . , p, and r Then

I

r

E ]~.

I

(24)

and, for each i, whenever

<~',8i> > 0

(25)

s i E C i.

Since Q~ = Q~. x Qi._+l, we can write =

lri, r

r2)

(26)

,

(27)

and ~i

(O.)~

i

i

i

where -

"

*

i

i

i

rr*

-

/"li~- 1

7r 2 E

RL~I

and * Qi , w iI E T x~.

i o E R + i, wl,

~i-i-1 , w i2 E T*x,j.xw

Now pick u = ( u t , . . . ,u u) E K ,

and write u i = (v i, ~ . , aL, ei). Define the s i, q', q~, and so as above. Then p

y~<~i, ~i) _ r i=l

_
i

0~2, 0 E

Ri4-1 _

9

Set-valued Differentials and the Maximum Principle of Optimal Control

519

SO P

(28)

" i i + rri~ai~-l~ (~b]'qi)+(~i2'vi+l)+ ~rl~r+ - ] < r176176

E( i=1

Let

~i = [ ~ i , - r

~

~ = [ ~ , o],

~i = [~i, o1,

~ = [ ~ , o],

r =

,

= ,~, - r

(30)

~i = ~i - r

(3,)

~-~ = ~;" - r

'

(321

Then (28) says t h a t "i i ((~2~,qi) q-((bi~,v'+l)-l-rrla+ + 7ri2oti'~ - I ~] <--r

It

i=t

+ Ep( f f f i , s ' ) ) ,

i=1

i=1

that is E

( ( ( ~ ] ' q i ) - ~b~

+ (r

9"

'

v ' + l ) + rrz~

'

+

-

i=1

_<

i=1

which can be rewritten as

i----1 i i -1- i ~ i + l <_r E ( w ~ , qi) + (w~,9 v i) + Wt,oCt+ + W2,o,~_ , i=1 i,e.~ as #

""

-i i

#~ i~-1~ < 0 .

(33)

i=1

We now extract information from (33) by m a k i n g special choices of u, i.e., of the vt,9 a +i , ai_, ei. First, we write

Lc(t)= L[L(t) O0]' e(t)

Hdctor J. Sussmann

520

define

6i (t) = (bil 9 ML, (ai+, t)

(34)

for i = 1 , . . . , p, and observe that 6i is absolutely continuous and

d (,~'(t)) =

Lc(t) c.e..

Therefore, if we write ~i(t) = [r

ai(t)],

(35)

then

r

= -r

and hi(t) - O. Since r

v(t) - ~ ( t ) . e(t)

= !b~ = [r162

we can conclude that

(36)

ai(t) -- - r SO

r

+ r

= -r

that is

-r

[L(t)]

Since t --~ [ g(t)

= [r162

[L(t)] [ e(t) I

D

is a measurable selection of .~i we have shown that

[

gfi(t)

is absolutely continuous

]

(37)

and

-r

e [r162

A~(t)

a.e.

t

(38)

i.e., that r is a solution of the adjoint differential inclusion. Next, fix a value i0 of i, let i' = i0~- 1 choose all the ~_ 's, a+ 's, and e's e .~ual to zero, and l e t v I9 = 0 f o r , # 9, , v 9, i" ' = v e T , , : Q |" ' . T h e n q " ' = M L ( ( a + i ' . .a. i. ~ . ~ i and ~ = 0 if i 7s i'. So (33) tells us that ~i" ~i" "io ~i" (~l,q >+<~2,v ><0,

that is,

((b~'. MLc(a+,a_) + "r176

_< O,

(39)

so that
+ ~ o , 9 ) _< 0.

(40)

Set-valued Differentials and the Maximum Principle of Optimal Control

521

Since @ = [@ -

o]

(40) says that (r

9i

.t

io

(a'_) + r176 - Cow 2 ,v) < O.

(41)

Since v is an arbitrary vector in Tz,,Q i', we have established that ~o = r

_ r162

(42)

Since io was an arbitrary index in the set { 1 , . . . ,/~}, and i I = ioq-1, we have in fact shown that r

= r

- r

if i E { 1 , . . . ,/~}.

On the other hand, the fact that r imply !b~ = r

+ r

i

i

= [r162

and r

= r

if i e { 1 , . . . , p } .

Since #~ = rr~ - r ~1 = r

(43)

o and #~ = rr~ - r

- r (44)

i

we have

~"

+ 7r]

(45)

~. + ~

(46)

and

~9 = r

It follows from (43), (44), (45), and (46), that =

where

~q, r

~)

~ti _ (--r

=

r

+ r w/,

(a/_'i'l),--~'~).

Then (25) says that ~ i _ Coati E (Ci) • .

(47)

Next, we fix a value i0 of i, let i' = i0+1, choose all the v's, a+'s, and e's equal to zero, and let a/_ = 0 for i ~ i', a i'_ = ct_ E Ci'_. T h e n i' "' q*"' = --ct_ML~,(a+,a~ )

~ "' 9 F~

',,( x '"' , a _i ~) ,

and ~ = 0 if i r i'. So (33) tells us that

-a_(r

"i'

" "' ~i' -io , ML, j (a'+, a'..) . F~,, (x~,"' a~)) + 7r 2 . a _ <_ O,

522

H&tor J. Sussmann

that is, i~ ~i' "' - - a _ ( r ^i'1 (a_) -F~,, (x*._, a t"') ) + ~r~~ 9c~_ < 0,

or, equivalently, . . (x<, . . a*._)) + a_r - a _ ( r I. (.a t.) . . <',,

that is,

(~,o

~_.

e (x_, e a_) r +~.~o "" 9a _ < _ O ,

h") <_ o.

_

Since this is true for all sufficiently small c~_ 6 C~, we conclude that

~?-h" e (cf) l Since io was an arbitrary index in the set { 1 , . . . ,p}, and i' : ioq-1, we have in fact shown that

-i hi-it 7ru_ 9 (C/..4-1)• whenever i E { 1 , . . . , # } .

(48)

Given any i 9 { 1 , . . . ,/a}, let iZ1 be the unique index io 9 { 1 , . . . ,p} such that ioq-1 = i. Define

xi_

=Tr

-i-1 _ hi 2

(49)

Then (48) says that

I

~- 9 (c~-)~

whenever

i 9 {1,... ,.}.

] (50)

Next, we fix once again a value i0 of i, choose all the v's, ~_ 's, and ~'s equal to zero, and let (~. -- 0 for i # io, a ~ = a+ 9 C~~ Then r and ~ = 0 if i # io. So

9

_-- c~+<,o (X!O, a~2), ~io

(33) tells

'

us that

-io io o%(r ~io ,F~,o(X + , a +io) ) + T r ~io 1 .c~+ < 0,

that is,

.+
-,o (~+, ,o a+)) ,o + ~1 -,o ""+ < O, 9G,o

or, equivalently, .

a+(r ~ that is,

.

F~~

.

.

.

.i

o

+ a + r 1 7 6 o ( x ~ , a ~ ) + ~r1 9~+ _~ 0,

~+. (~o + h~) < 0.

Since this is true for all sufficiently small a _ E C~~ we conclude that ~rl +

E

9

Set-valued Differentials and the Maximum Principle of Optimal Control

523

Since i0 was an arbitrary index in the set { 1 , . . . ,p}, we have in fact shown that /r/1 + h~ 9 (6"~) • whenever i 9 { 1 , . . . , p } .

(51)

Define ~+ = ~r~ " + h+. ~

(52)

Then (51) says that [

.~ 9

(C~) x

whenever

i 9 { 1 , . . . ,/J}.

[

(53)

Since

_~

i i =-~++h+

- T -' r y = _~,_ -

cnd

h ~_ ,

(47) implies

[ (--r

+

h~., r

i_ -hi_) 9

r

(Ci)X . [

(54)

Then (50), (53) and (54) show that the switching conditions hold. The next step is to fix a value i0 E { 1 , . . . p } and a J0 E { 1 , . . . ,vi~ choose all the v's, a_'s, and a+'s equal to zero, and let el __ 0 for i # i0, and e i~ = ( e l , . . . ,e~,o), where ej = 0 i f j # jo, ejo = e, with 0 < r < P. Then

r

--_ eML,2 (a~. ,t;~ . Zj~ ,

and • = 0 if i r i0. So (33) tells us that r162 ML,2 (a~., t;:) . Zj:) <_ O, that is,

9(el ~ .ML•o (a~,t;~

Z~~ <

O,

or, equivalently, ~.~,,o.,o~ Zjoo) < o XT"

~ jo

j ,

__

9

Since this is true for all sufficiently small nonnegative r we conclude that

(r

z?) _<0.

Since i0 and j0 are arbitrary, we have shown that

(~'(t~.), zj) < o whenever i E { 1 , . . . p } , j E { 1 , . . . ,vi~

(55)

524

H6ctor J. Sussmann

Now, given any i, j , (55) says t h a t i ^i (r^i (t~), Yj) < (~'(t~),~'),

that is, (r

'

i , i F~j(~ (tj),t~)) - r i i i (tj),t~))i < (r i (tj),F~,((

i

i (~ i (t j), t j),

i t~), COL,,i (~i (t~),

or i i (t./) i ,r HQ(,~

i r t~,

<_

H . ,i

i r (t j), tj,i (~ i (t j),

r

We have thus established that the 3it + 1-tuple .. , . . . , - , +1 satisfies all the conditions of the definition of ~ w , except possibly for the normalization condition (9). On the other hand, if we let 7 be the left-hand side of (9), then 7 ~ 0. If we show that "t > 0, then we can divide the ~i, ~ : and 40 by % and obtain a new multiplier for which (9) holds. Hence the conclusion t h a t r r 0 will follow if the prove that 7 :~ 0. Assume t h a t 3' = 0. Then r = 0, and the equalities ~i_ = *;~_ = 0 and r (a~.) = 0 hold for all i. It then follows from the adjoint differential inclusion that, for every i, r = 0 for all t E [hi_, a~.]. In particular, the definitions of the h~: imply that h +i = h i = 0 for a l l i . Then (49) and (52) imply that /r~ = ir~ = 0 for all i. Then (31) and (32) imply (since r = 0) that rr~ : lr~ : 0

for all

Moreover, since r

~=0

and

i.

-- 0 and r

~=0

(56) = 0, (43) and (44) imply that

for all i.

(57)

Then (26), (56) and (57) imply that ~i = O

for all

i.

(58)

Then (23) implies that ~ = 0, contradicting the nontriviality of ~. This contradiction shows t h a t 7 :~ 0, and our proof is complete. 11

Path-integral

generalized

differentials

If n , m E Z+, a : [ 0 , 1 ] - + ~ " is a Lipschitz function, and h : [0, 1] -+ ~ m • is integrable, we use h * a to denote the "chronological product" of h and a, that is, the absolutely continous function fl:[0, 1]--+]I:Lm given by

fl(t) =

h(s) . &(s) ds .

Set-valued Differentials and the Maximum Principle of Optimal Control

525

Let n G Z+, and let S be a subset o f ~ n. We write .A(S) to denote the subset of C~ [0, 1] ; ~'~ ) consisting of all absolutely continuous curves a : [0, 1] --+ ~ n such that or(0) = 0 and &(t) E S for almost all t G [0, 1]. If C is a convex cone in ~n, and r > 0, we write C(r) =- {v E C : Ilvll < r}. D e f i n i t i o n 12. Let n, m E 2~+, let F : ~ n ) >It~m, and let C be a closed convex cone in II~n. We say that A is a path-integral generalized differential of F at (0, O) in the direction of C, and write A E P I G D ( F , C), if A is a n o n e m p t y c o m p a c t subset of ]~rn• and for every positive real n u m b e r there exists a number R E ] 0, cx~ [ with the property t h a t ( # ) for every r G ] 0, R] there exists a m a p

G E REG(A(C(r));C~

1]; I[~m xn ) • ]~rn)

such that ~b)(#:a) h(t) E A ' and I]v[]
References 1. Berkovitz, L. D., Optimal Control Theory. Springer-Verlag, New York, 1974. 2. Clarke, F. H., The Maximum Principle under minimal hypotheses. SIAM J. Control Optim. 14, 1976, pp. 1078-1091. 3. Clarke, F. H., Optimization and Nonsmooth Analysis. Wiley Interscience, New York, 1983. 4. Halkin, H., Necessary conditions for optimal control problems with differentiable or nondifferentiable data. In Mathematical Control Theory, Lect. Notes in Math. 680, Springer-Verlag, Berlin, 1978, pp. 77-118. 5. Pontryagin, L. S., V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mischenko, The Mathematical Theory of Optimal Processes. Wiley, New York, 1962. 6. Sussmann, H. J., An introduction to the coordinate-free maximum principle. In Geometry of Feedback and Optimal Control, B. Jakubczyk and W. Respondek Eds., M. Dekker, Inc., New York, 1997, pp. 463-557. 7. Sussmarm, H. J., Multidifferential calculus: chain rule, open mapping and transversal intersection theorems. In Optimal Control: Theory, Algorithms, and Applications, W. W. Hager and P. M. Pardalos Eds., Kluwer, 1998, pp. 436487.

526

H~ctor J. Sussmann

8. Sussmarm, H. J., A maximum principle for hybrid optimal control problems. In Proc. 38th IEEE Conf. Decision and Control, Phoenix, AZ, Dec. 1999. IEEE publications, New York, 1999, pp. 425-430. 9. Warga, J., Optimization and controllability without differentiability assumptions. SIAM J. Control and Optimization 21, 1983, pp. 837-855.

Transforming a Single-input Nonlinear S y s t e m to a Strict F e e d f o r w a r d F o r m via Feedback Issa A m a d o u Tall 1,2 and W i t o l d R e s p o n d e k 1 1 Laboratoire de MathSmatiques INSA de Rouen 76131 Mont Saint Aignan, France tall@imi, insa-rouen, fr wresp@lmi, insa-rouen, fr

Departement de Math~matiques et lnformatique Universit~ de Dakar, Senegal

A b s t r a c t . We study the problem of transforming a single-input nonlinear control system to a strict feedforward form via a static state feedback. We provide checkable necessary and sufficient conditions to bring the homogeneous terms of any fixed degree of the system into a homogeneous strict feedforward form. If those conditions are satisfied, this leads to a constructive procedure which transforms the system, step by step, into a strict feedforward form. We explain our method by showing how it works for terms of degree three. We also illustrate it by analyzing threedimensional systems.

1

Introduction

Consider a single-input nonlinear control s y s t e m of the form

• : ~ = F(~, u), where ~(.) E ~ n and u(.) E R . We say t h a t the s y s t e m / / is in (resp. in strict feedforwardforTn) if we have

feedforward

form

)

9

~,~ = F,,(~,~,u).

~, =

Fn(u)

A crucial p r o p e r t y of systems in strict feedforward form is t h a t their solutions can be found by quadratures. Indeed, knowing u(t) we i n t e g r a t e Fn(u(t)) to get ~ , ( t ) , then we integrate Fn_l(~n(t), u(t)) to get ~ n - l ( t ) , we keep d o i n g t h a t , and finally we integrate F l ( f 2 ( t ) , . . . , ~n(t), u(t)) to get f l (t).

528

lssa Amadou Tall and Witold Respondek

Notice that, in view of the above, systems in strict feedforward form can be considered as duals of fiat systems. In the single-input case, flat systems are feedback linearizable and are defined as systems for which we can find a function of the state that, together with its derivatives, gives all the states and the control of the system [4]. In a dual way, for systems in strict feedforward form, we can find all states via a successive integration starting from a function of the control. Another appealing feature of systems in (strict) feedforward form is that we can construct for them a stabilizing feedback. This important result goes back to Teel [16] and has been followed by a growing literature on stabilization and tracking for systems in (strict) feedforward form (see e.g. [5], [11], [13], [17], [3], [12]). It is therefore natural to ask which systems are equivalent to (strict) feedforward form. In [10], the problem of transforming a system, linear with respect to controls, into (strict) feedforward form via a diffeomorphism, i.e., via a nonlinear change of coordinates, was studied. Very recently, a geometric description of systems in feedforward form has been given in [2]. In the present paper we give necessary and sufficient conditions for transforming a system, affine with respect to controls, to a strict feedforward form. In our study we use a very fruitful method proposed by Kang and Krener [8] and then followed by Kang [7]. Their idea, which is based on classical Poincar~'s technique for linearization of dynamical systems (see e.g. [1]), is to consider the action of homogeneous transformations of any fixed degree on the homogenous part, of the corresponding degree, of the system and then to analyze the action of the feedback group on the system step by step. The paper is organized as follows. In Section 2 we give basic definitions. In Section 3 we introduce a strict feedforward normal form and give a result stating that any system that is feedback equivalent to a strict feedforward system can be transformed to that normal form. We recall m-invariants of homogeneous transformations in Section 4. Our main results are given in Section 5. The first result states that if a system is feedback equivalent to a strict feedforward system, then the first nonlinearizable term must be strict feedforward when brought to a Kang normal form. The second result gives necessary and sufficient conditions for feedback transformation into a strict feedforward form of the homogeneous part (of any fixed degree higher than the degree of the first nonlinearizable term) of the system. In Section 6 we illustrate our results by showing how the proposed method works for systems that can be transformed into a strict feedforward form up to degree 3. Finally, in Section 7 we analyze feedforward systems on ]R3, in particular, we calculate the codimension of the class of systems on IR3 that are feedback equivalent to a strict feedforward form.

Strict Feedforward Form

2

529

D e f i n i t i o n s and N o t a t i o n s

Consider a nonlinear control s y s t e m

where ~(-) E ~ n a n d u(-) E ]~. We say t h a t E is in feedforward form, or t h a t it is a feedforward system, if for a n y 1 < j < n we have

fj(E,) = fj(E,j,... ,~n) and gj(,~) = gj(~j,... ,~n). We say t h a t S is in strict feedforward form, or t h a t it is a strict feedforward system, if for any 1 < j < n we have

fj(~)---- fj(~j+l,"',~n)

a n d gj(~) =gj(~j+l,'",~n)-

A p p l y to the s y s t e m ,U a feedback t r a n s f o r m a t i o n o f the form

r :x = r = ~(~) + ~(~)v. T h e t r a n s f o r m a t i o n F brings s to the s y s t e m

~:

~

=

](~) + ~(~)v,

whose d y n a m i c s are given by

] = r ( I + g~) = r (g~), where for any vector field f and any diffeomorphism r we denote (r

= dC(C-X(x)) 9f ( r

In the p a p e r we will be interested in t h e following question. Problem:

When is the system Z feedback equivalent to a strict feedforward

system ? All objects, i.e., functions, m a p s , vector fields, control systems, etc., are considered in a n e i g h b o r h o o d of 0 E ~ n a n d assumed to be C ~ 1 7 6 Let h be a s m o o t h function. By co

h(x) = ht0l(x) + hill(x) + ht~](x ) + . . . = ~ htk](x) k=O

we denote its Taylor series expansion a r o u n d zero, where h[k](x) s t a n d s for a homogeneous p o l y n o m i a l of degree k.

530

Issa Amadou Tall and Witold Respondek

Similarly, for a m a p r of an open subset of ~ n to ~ n (resp. for a vector field f on an open subset of]~ n) we will denote by r (resp. by f[kl) the t e r m of degree k of its Taylor series expansion at zero, i.e., each component 0~.kl of r [kl (resp. fJk] of f[k]) is a homogeneous polynomial of degree k in x. Denote also ~i = ( x l , . . . , xi). Together with the system E:

~ = f(~)-t-g(~)u,

we will consider its Taylor series expansion 57~ : ~ = F~ + Gu + ~--~(f[k](~) + glk-ll(~)u),

(1)

k-~2

where F = ~ ( 0 ) and G = g(0). We will assume throughout the paper that

f(0) = 0 and g(0) # 0. Consider the Taylor series expansion F ~ of the feedback transformation F given by r~:

= r = T~ + E;~ ctkl(~) u = a(~) + Z(~)v = K ~ + Lv + ~"~~

+

fllk-ll(~)v),

(2)

where T is an invertible matrix and L ~: 0. Let us analyze the action of F ~ on the system S ~176 step by step. To start with, consider the linear system

d = F~ + Cu. Throughout the paper we will assume that it is controllable. It can be thus transformed by a linear feedback transformation of the form yl : x=T~

u = K ~ + Lv to the Brunovsk~ linear part (F,G), to the Brunovsk~ Krener [8], [7] and F~:

canonical form (A,B), see e.g [6]. Assuming that the of the system S ~176 given by (1) has been transformed canonical form (A, B), we follow an idea of Kang and apply successively a series of transformations

x = ~+ r u = v + ~lm~(~) + z l m - l l ( ~ ) v ,

for m = 2, 3 , . . . . A feedback transformation defined as a series of successive compositions of tim, m = 1, 2 , . . . will also be denoted b y / ~ because, as a formal power series, it is of the form (2). We will not address the problem of convergence and we will call such a series of successive compositions a formal feedback transformation.

Strict Feedforward Form

3

531

Strict Feedforward N o r m a l F o r m

Since the linear part (F, G) of the system ,U~176 , given by (1), is controllable, we can assume, without loss of generality, that the system is in the form oo

(f[ml(~) + g[m--Z](~)U),

S ~176 : ~ = A~ + Bu + E

(3)

rn~--9

where (A, B) is in Brunovsk~ canonical form. Recall that, as proved by Kang [7], see also [14], any nonlinear system of the form (3) can be put, via a feedback transformation, to the following normal form, which we will call Kang normal form,

2Y~F : ~ = Ax + Bv + ~

f-[ml(x),

rn=2

where for any m > 2, n

-~](xl

xi) ' "

if l < j < n

2

- - -

(4)

if n - l < _ j < _ n .

0

It is natural to ask whether it is possible to bring a system, that is feedback equivalent to a strict feedforward system, to a Kang normal form (4) which would be simultaneously strict feedforward, that is, which would satisfy

Although, this is always possible for the first nonlinearizable term, see Proposition 2 below, in general the answer to the above question is negative. For this reason we will introduce the following notion. D e f i n i t i o n 1. A strict feedforward normal form is the system

SSFNF : X = Ax + By +

][ml(x), m=2

such that for any m > 2,

f!ml[x ~ = f cm,jxr~+l + ~-~=j+2x~P~,~.-~](xj+t, 9"" ,xi)

[0

if 1 < j < n - 2 if n - l < j < n ,

where Cmj E ~. Usefulness of strict feedforward normal form is justified by the following resuit. T h e o r e m 1. The system 2S~176 given by (1), is feedback equivalent to a strict feedforward system if and only if it is feedback equivalent to a system in strict feedforward normal form.

532

4

Issa Amadou Tall and W/told Respondek m-Invariants

Consider the following feedback t r a n s f o r m a t i o n r m : x = ~ + C/m](~)

/l : U -Jr" o([rn](~) --}-z[m--l] (~)V.

Observe that each transformation F m, for m > 2, leaves invariant all homogeneous terms of degree smaller than m. We will call F m a homogeneous feedback transformation of order m. We will recall invariants, found by Kang [7], of the action of F m on the following system S tin] : ~ = A~ + B u + f[m](~) + glm-~](~) u + O(~, U) m + l .

(5)

Let us define

Xri n-1 (~) = adiar

--1-gtrn-1](~)).

Following Kang [7], we denote by a [mlt,/(~) the homogeneous part of degree m-2of

CAt-1 [ x m 7 ~, Xm-ll i-2 Jlw.-,+2 ' where C = ( 1 , 0 , . - -

w / = {~ e ~ " l

, O) and the submanifolds Wi are defined as follows:

~+~ = . . . .

~. = 0 } .

The functions a[mlt'i(~), for 2 < i < n m-invariants of ~,[m].

1, 1 < t < n - i ,

will be called

The following result of Kang [7] asserts t h a t m-invariants a[m]t,i(~) are complete invariants of homogeneous feedback and, moreover, illustrates their meaning for the n o r m a l form E[NmF ] . Consider two systems: X '[m] , given by (5), and ~,[m], given by ~,[m] : x = A x + By + ]Im](x) + ~ [ m - l l ( x ) v + O(x, v) m+l. Let

{a [m]t'i : 2 < i < n - - 1 ,

l
{5[m]t'i : 2 < / < n - l ,

l
and

denote, respectively, their m-invariants.

}

Strict Feedforward Form Proposition

533

1. The m - i n v a r i a n t s have the following properties:

(i) the systems S [m] and ~,[m] are equivalent via a homogeneous feedback o f order m, modulo higher order terms, i f and only i f a[ra]t,i = fi[mJt,i f o r any 2 < i < n - 1 (ii) the rn-invariants

--NV:

and any l < t < n - i;

~[m]t,i of the

system

~ = A~ + B~ + f~J(~) + O(~,

~)m+~

where tirol(x) is o f the f o r m (4), are given by a[,~lt,i _

02 n[m-~] ,O X 2 n - - i + 2 x n - i + 21-'i'n-i + 2 ( xn --i+21

f o r any 2 < i < n - - 1

5

Main

Let "

and any l < t < n -- i.

Results

consider the oystem (3) and let m0 be the largest integer j + 1 such

t h a t all d i s t r i b u t i o n s D k = s p a n

g, a d f g , . . . , a d !

g , for 1 < k < n - 1,

are involutive m o d u l o t e r m s o f degree j - 1. As proved by K r e n e r [9], this means t h a t the system Z ' ~ , given by (3), is feedback linearizable up to degree m0 - 1. Since a system ',~, e a r i z a b l e up to degree m is also feedback equivalent to a strict feedforward ~ ~tem up to the s a m e degree, we can a s s u m e w i t h o u t loss of generality t h a t the s y s t e m L "~ is in strict feedforward n o r m a l form up to degree m0 - 1 (see T h e o r e m 1), t h a t is, it takes the form mo~

= A~ + B u +

~ lh["](~) + f[-~o](~) + g[mO-l](~)u

z ~ c~ :

m=2

(6)

+O(L u) m~ where, for any 2 < m < m0 - 1, we have h [.m ] ( ( ) =

{o m , j ( jm-I'-1

if l _ < j < n--- 2 if n - l < _ j < _ n .

P r o p o s i t i o n 2. Necessary conditions f o r the system (6) to be feedback equivalent to a strict feedforward s y s t e m are that the following relations L A ~ - . B a[rn~

: 0

hold f o r any O < s < n - 3 and any l < r < t < n - s - 2 .

534

Issa Amadou Tall and Witold Respondek

In other words, the above result says t h a t if a system is feedback equivalent to a strict feedforward system, then the first nonlinearizable term m u s t be strict feedforward when transformed to a K a n g normal form. As we will see in the sequel, the situation gets different when we proceed to higher order terms. Now, let us assume that the system E cr , given by (3), is in strict feedforward normal form up to degree m0 + 1 - 1, t h a t is that X'~ takes the form too+l--1

=A~+Bu+

too+l--1

E h[m](~)+ E f-[m](~) m=2 m=mo

S~:

"~-ftrno't'll(~) "~- gtrno't'l-1] (~)U "~ O ( ~ , u ) m~

(7) ,

where for a n y m 0 < m < m 0 + l - 1 ,

[ +

--j~P~'i-2](~J+l'" ' ' , ~ i )

=/o

if l < j < n - 2 if n - l < _ j < _ n .

Put to = 0 and, for any 0 1 and any 1 ~ k _< n, consider the vector field yktm+ll = ~ + 1

0 0~k + LA~kn+l

0

+'"

• rn-~-~+l

0

[too+tit,,+2 the (m0 +l)-invariants of the homogeneous system and denote by a k,i ,i

: ~ = A5 + Bu +

YkIt-i+1]

In the sequel we will need the following notations. Put A={(s,t)

EN•

l
0<s
Moreover, for any i satisfying 0 < i < l - 1, we denote by i. the smallest nonnegative integer such that for any j < i we have tj < ti,. The main result can be stated as follows. 2. The system S, ~176defined by (7)-(8), is feedback equivalent, up to order mo + l, to a strict feedforward system if and only if there exist real constants al,l,~t,2,'" ,O'l,t~ such that for any (s,t) E A and any 1 < r < t

Theorem

LA.-~ B

a [rn~

-

~Z . ~ i=0 kmtl. +1

o"t,k a k,i [m~

/ = 0. ]

(9)

Strict Feedforward Form

535

Notice that (9) is an invariant way of expressing the fact that I--1

a[m~

--

Z

ti+l Z

r

[m~

'

s+2

'r176 + / - 2l : "~t,s

i=0 k=tl, +1

where ()[m~ ,~t,s

mo+l-2] ~[c, t + l , " " , ~ n - s ) are homogeneous polynomials = rWt,s

of degree m0 + l - 2 depending on the indicated variables only.

6

Strict

Feedforward

Systems

up to Order

3

To illustrate our results we consider in this Section the problem of when we can bring a given system to a strcit feedforward form up to terms of degree 3. Consider the system oo

Z ~ : ~ : F ~ + e u + Z ( f [ k ] ( ~ ) + g[k-1](~)u). k=2

By an appropriate feedback transformations (a linear one followed by a quadratic one) we can bring the linear part of the system to the Brunovsk~ form and its quadratic form to the Kang-Krener form [8]. T h a t is, without loss of generality, we can assume that the system takes the form .~:

~ : A~ + B u + j~2l(~) + f[3l(~) + g[2](~)u -t- O(~, u) 4,

(10)

where (A, B) is in Brunovsk3) canonical form and { )-'~in_j+2 aj,i~ 2

if 1 < j < n - 2

j~2] (~) =

(11) 0

if n - l ~ _ j < n ,

where aj,i E ~ . Notice that the result of [8], giving the above normal form, implies that any nonlinear system with controllable linearization is feedback equivalent, up to order two, to a strict feedforward system. The system (10)-(11) is feedback equivalent, up to order 3, to a strict feedforward system if and only if there exists a transformation F ~3 which brings it to a strict feedforward normal form up to order 3. Such a transformation can be viewed as a superposition of transformations 1-2 and F 3. Since the transformation F ~ must preserve the fact that second order terms are strict feedforward, it is necessarily of the following form (see [15], where F ~ is said to preserve structurally the strict feedforward normal form up to order 2)

r2 : y = ~ + Ct~](~) u = w + ~t:](~) + ~[ll(~)w,

536

Issa Amadou Tall and Witold Respondek

given by

r [2]

= LA~(r

.tZl( )

b 2,j+l~j+l 2

_Lae(r

fl[1](~) = _LB(r where b2,j, for 1 < j _< n, are real constants to be fixed later. The transformation /,2 defined above takes the system (10)-(11) to the following one ,U:

9 = A y + B w + hD](y) + ~=](y) + [f~2], r

_ c3h[2] r COy

_.~_f[3](y) _1_g [ 2 I ( y ) w "4- O ( y , W) 4, where, for any 1 ~ j < n - 2, we have = --b2,j+lYj+l.

In order to define the transformation/,3, we will express it as the composition /,3 o/,a, where /,~ is given by xt X2

= Yl Y2-

~ Oy .r

LJ-i~ x j + l ---~Yj+I _ vs ' J Ay Oy "~ v w + V/-.-~i= ' n 1 L Ay " - i ~ Oy r

The transformation/,2 composed with F 3 brings the system (10)-(11) to S':

d~ = A x + B y + h[2](x) + f z ] ( x ) + [f~z], r +f[a] (x) + g[2l(x)v + O ( x , v) 4.

(12)

Now let us notice that there exists a transformation/,3 taking the homogeneous part of degree 3 of the system (12) into a strict feedforward form if and only if the Kang normal form of that homogeneous part is in feedforward form. Indeed, this is true by Proposition 2, if that homogeneous part is nonlinearizable and it is immediate if that part is linearizable. In order to analyze the Kang normal form of the homogeneous part of degree 3 of (12), let us calculate 3-invariants of (12). Define the vector field y]21

2 cO

= Xk-~X k + Ln:cX~

~

.n-k

+1

2 cO

+ ' ' " + L a x Xk

cOXn

.

Strict Feedforward Form

537

Consider the 3-invariants

{a[3]t's+2 : (S,t) EArl} of the system # 3 ] : ~ = Ax + By + / [ 3 ] ( x ) + g[2l(x)v + O(z, v) 4 and the 3-invariants

[3]t,s+2 : (s,t) E A}

%0

of the system

Z,k[3:] ,0:

& = A x + B y + [fl:], r [2]] + O(x, v) 4.

We conclude the following observation (which follows also from Theorem 2). C o r o l l a r y 1. The system (10)-(11) is feedback equivalent, up to order 3, to a strict feedforward system if and only if there exist real constants ~rl,a, for 1 < k < n, such that

LA--.S

l l,,+A)

4. a[3lt's+2 -- k=l ~'~ al'aak'~

= O,

for any (s,t) E z5 and any r ~_ t. In other words, the last condition says that n

a[3lt,s+2(x)

2_. al,kak,[3]t,s+~.. o (x] = -~[~l, r162 ~

- -

,x,~_~),

k----1

where the homogeneous polynomials/-)[1l " ~ t , $ depend on the indicated variables only. [3It,sT2,

Actually, we can distinguish two cases. If/[~](x) = 0, then a~, o I z) = 0 and the system (10)-(11) is feedback equivalent, up to order 3, to a strict feedforward system if and only if there exist polynomials ,~[11 ~,t,$ such that at3lt's+2(x)

--

~'~t,s

" " 9 , x._s).

(13)

In fact, in this case the system is feedback linearizable up to order 2 and in order to be feedback equivalent to a strict feeforward system up to order 3, the terms of order 3 must be in strict feedforward form when brought to a Kang normal form (compare Proposition 2). if the latter normal form contains a term of order 3 which is not strict feedforward, no feedback transformation can annihilate that term. On the other hand, if the condition (13) is satisfied,

538

Issa Amadou Tall and Witold Respondek

then, in order to bring the system (10)-(11) to a strict feedforward form up to degree 3, we can choose F ~ = id, F ~ = id, and F~ which transforms (f[3l, g[~]) of (10)-(11) to its Kang normal form. In the other case, that is when f-[~l(x) :/: 0, the system (10)-(11) is feedback equivalent, up to order 3, to a strict feedforward system if and only if there exist real constants (r~,~, for 1 < k < t~, and linear polynomials o[ll "1~t,$ satisfying Q[~] t+$ = Ql~](xt+l, .-. a[3lt,s+2

tl

~

x,~_,) such t h a t

[3]t,s+2

-- 2..4 Crl,kak, 0

:

[~[1]

(14)

~t,s,

k=l

where tl denotes the largest integer j such that 1 _< j _< n - 2 and ]Jj~](x) ~: 0. In this case, in the definition of F ~ we take b~,~ = -(rl,a, for 1 < k < tl, where a l , k are given by the formula (14). We complete it with F 3 followed by F~3, the latter being defined as the transformation bringing ([f-Is], r + f[3l,g[2]) of (12) to its Kang normal form. This successive composition of F ~, F~3, and F 3 annihilates, if necessary, t~ terms of order 3, which are not in strict feedforward form.

7

Strict

Feedforward

Systems

on R a

Consider the system L" on ~3 whose linear part is controllable. Bringing its linear part to the Brunovsk~ form and its quadratic and cubic parts to the Kang-Krener form we get, c o m p a r e (10)-(11), ~1 = ~2 + a1,~3~ + j~3](~) + O(~, u) 4 ~ = ~3

where .~3](r = r162

+ e2~2 + e3~3) and e,, e2, e3 9 ~ .

We consider the transformation F ~ given by, c o m p a r e Section 6, yl = ~1 + F 2 : y~ = ~2 + 2b~,l~l~ + b ~ , ~ y3 = ~3 -4- 2b~,1~22 + 2b2,1~1~3 + 2b~,2~2~3 + b2,3~32, whose action on the first c o m p o n e n t of the system (15) gives

ul = 4~ + a l , ~ + 2 6 1 , ~ ( ~ + a ~ , ~ ) + ~31(~) + 0(~, u) 4 = Y2 - b2,2~ + a 1 , ~ 3 + 2al,~bl,~1~23 + j~3] (~) + O(~, u) 4 = Y2 -- b2,2y~ "4- al,2y] - 2a1,2bl,2yly23 - 2al,2y~(2b2,2y2 4- b3,2Y3)

(15)

Strict Feedforward Form

539

Define the following transformation F 3 = Yl -- f~a p[2]dy 2

zl

F 3 : Z2 = Y 2 "~- o [3] ( yl, z3 ---- .~2

y2)

v=~3. The composition of F 2 with F 3 takes the system (15) into the following normal form xl "- x2 - b2,2z~ + a,,2z32 -I- .f~3](z) + O ( z , v) 4 x2 -- x3 X3 ~

'U~

where .f~l(~) = ~ ] ( ~ )

_ 2.~,~b~,~

- 2.~,~(2b~,~

+ b~,~).

We thus

conclude that the system (15) is feedback equivalent, up to order 3, to a strict feedforward system if and only if

/~31(~) _ 2 a l , 2 b l , 2 ~

- 2a~,2~(2b2,~ + b2,3~3) = R~310(~2, ~3),

(16)

i.e., if and only if either al,2 ~ 0 or al,2 -- 0 and Q -- 0. To compare this result with general considerations of Section 6, let us calculate the set of 3-invariants [3]t,s-b2

: (s, t)

al, 0

A}

of the system

Et3l 1,0

:

~

A~ + Bu + [~21, y[~l] + O(~,

~)',

which, for n = 3, reduces to al,[3]i,2 0 , and the set of 3-invariants

(~[3]t,,+2 : (8, t) ~ A} of the system

~t3]: ~=A~+Bu+~31(~)+O(~,u)4, which, for n = 3, reduces to ~[311,2. Recall that C = (1, 0, 0). Since )~21(~) = a~,2~, we have

al,[311,2 0 :

Cad~B[fl2l, y[2l] = _4a~,2~l.

The term f-J3] is in Kang normal form; therefore ~3] satisfies (16) if and only if we have ~[311,2 _ 0 2 ~ 31 j_ n [ l ] . O:c~3 = 4ai,~b2,ixl ~ "~i,0 = -~

[311,2. n [ l ]

- ~l,0,

540

Issa Amadou Tall and Witold Respondek

where

61[ 1] "~ 1,0

= Q~,~)(x~, xz) is a linear form of z~ a n d z3 only. T h i s gives 9

a[311'2(z) + ~

[311 2 ,

,

,-~[11 ' x

t z) = wl,0t ~, z3).

T h e last f o r m u l a coincides with (13), if f121 is linearizable, a n d with (14), for o'1,1 = -b2,1, if f-J2] is not linearizable. This result implies t h a t the class of systems t h a t are feedback equivalent, up to order 3, to a strict feedforward form is a union of a n open set, given by al,u ~ 0, and of a set of codimension 2, given by al,~ = 0 a n d r = 0. Notice t h a t the l a t t e r is c o n t a i n e d in the set of s y s t e m s t h a t are feedback linearizable up to order 2. In what follows, we will consider the nonlinearizable case, t h a t is al,~ ~ 0. Assume t h a t the s y s t e m (15) is feedback equivalent, up to order m, to a strict feedforward s y s t e m , i.e, it is of the form

(17) ~3 =

U,

where the t e r m s f~kl(~), for 2 < k < m, are in strict feedforward n o r m a l form, t h a t is f~] and the

(,~) = ci,,~ + ,~ p[k- ~](,~2,,~3) term

y ra+l

g~m+l](~) is in K a n g n o r m a l form, t h a t is

=

p ra- ]

Like in the case of t e r m s of degree 3, we look for a t r a n s f o r m a t i o n F
E ra :

yl = ~1 + bra,l~l~ y~ = ~2 + m b , , , l ~ lr a - - I ~2 + bra,2~2m c m - - 1q3 d" Y3 ~3 + m ( m - - 1 ) b r a , l ~ n - 2 ~ ~-1- 7 h a m,lq~ rn-1

+mbm,2~2

m

~3 + bm,3~3

and a t r a n s f o r m a t i o n _rra+l. For the general case, where all F k, for 2 < k < m - 1 are considered, we refer the reader to [15].

Strict Feedforward Form

541

The action of F m on the first component of the system (17) gives Yl -- ~2 "4- Ek%2 j~k] (~) -4- rrtbm,l~ n-1 (~2 "4- f~l2] (~)) -}- j~m+l] (~) -4- 0 ( ~ , It) m+2 = y2 -- bm,2y~ + Ekm-_~ f~k](y) + r n a l , 2 b m , l y F _ l y ~ +2b2,2y2(rnbm, l y F - l y~ + bm,2y~m ) _ 2a1,2y3 ( m ( r n - 1)bm,lyl"~-2 y~ __ br.,3y~rn ) + ~m+ll(u) + O(y, .)m+2 + mbm,l~F-~Y3 + ~Ttbm,~am - 1 Y3 1= Y2 -- bm,~y2m + E kj=~2k ] ( y ) m -- rna,,uOm,lyl"

m-1 Yw

+y3p ['q (yl, y2) + y]ff[m-~] (y~, y3) + ~m+ll (y) + O(y, U) m+2 . Define the transformation x l : Yl - fYo 2 tllmldy2 F m+l : X3 = V--~X

2 2 3 .

Then the composition of lowing one J: l

--

Fm

with

F m+l

takes the system (17) into the fol-

_ mal,20rn,lX 1

+~r

X3

x3) + ~m+11(~) + O(~, v)~+~

22 : X3 d:3 : v. We thus conclude (compare T h e o r e m 2) that the system (17) is feedback equivalent, up to order m + 1, to a strict feedforward system if and only if there exists a real number bm,t and a polynomial function p}m-1] such that p [ m - 1 ] = p } m - 1 ] ( x 2 , x 3 ) and f~lm+1](X) = ( m a l , 2 b r n , l X r ~ - l

.-{- p ~ r n - 1 ] ( x ~ , x 3 ) ) x g .

Notice that it follows from the last formula that the class of systems, whose terms of order m + 1 can be brought via feedback to a strict feedforward form, contains a component of codimension (m('~-l) _ 1). This is to be c o m p a r e d with re(m+1) 2 , which is the codimension of the orbit of systems whose terms of order rn + 1 are feedback linearizable. Of course, both codimensions grow to infinity together with m but there are "more" systems that are feedback equivalent to a strict feedforward form.

References 1. V.I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, Second Edition, Springer-Verlag, 1988.

542

Issa Amadou Tall and Witold Respondek

2. A. Astolfi and F. Mazenc, A geometric characterization of feedforward forms, in Proc. MTNS'2000, Perpignan, France, 2000. 3. S. Battilotti, Semiglobal stabilization of uncertain block-feedforward systems via measurement feedback, in Proc.o] NOLCOS'98, Enschede, the Netherlands, 1998, pp. 342-347. 4. M. Fliess, J. L6vine, P. Martin, and P. Rouchon, Flatness and defect of nonlinear systems: Introductory theory and examples, International Journal o] Control, 6 (1995), pp. 1327-1361. 5. M. Jankovic, R. Sepulchre, and P. Kokotovic, Constructive Lyapunov stabilization of nonlinear cascade systems, IEEE Trans. Automat. Control, 41 (1996), pp. 1723-1735. 6. T. Kailath, Linear Systems, Prentice-Halt, Englewood Cliffs, N J, 1980. 7. W. Kang , Extended controller form and invariants of nonlinear control systems with single input, J. of Mathem. Systems, Estimat. and Control, 4 (1994), pp.253-256. 8. W. Kang and A.J. Krener, Extended quadratic controller normal form and dynamic feedback linearization of nonlinear systems, SIAM J. Control and Optim., 30 (1992), pp.1319-1337. 9. A.J. Krener, Approximate linearization by state feedback and coordinate change, Systems Control Letters, 5 (1984), pp. 181-185. 10. A. Marigo, Constructive necessary and sufficient conditions for strict triangularizability of driftless nonholonomic systems,in Proc. 34 th CDC, Phoenix, Arizona, USA, 1999, pp. 2138-2143. 11. F. Mazenc and L. Praiy, Adding integrations, saturated controls, and stabilization for feedforward systems, IEEE Trans. Automat. Control, 41 (1996), pp. 1559-1578. 12. F. Mazenc and L. Praly, Asymptotic tracking of a reference state for systems with a feedforward structure, Autornatica, 36 (2000), pp. 179-187. 13. Sepulchre R., Jankovi~ M., and Kokotovi~ P. Constructive Nonlinear Control, Springer, Berlin-Heidelberg-New York, 1996. 14. I.A. Tall and W. Respondek, Transforming nonlinear single-input control systems to normal forms via feedback, in Proc. MTNS'2000, Perpignan, France, 2000. 15. I.A. Tall and W. Respondek, Feedback equivalence to a strict feedforward form for nonlinear single-input systems, in preparation. 16. A. Teel, Feedback stabilization: nonlinear solutions to inherently nonlinear problems, Memorandum UCB/ERL M92/65. 17. A. Teel, A nonlinear small gain theorem for the analysis of control systems with saturation, IEEE Trans Aurora Control, 41 (1996), pp. 1256-1270.

Extended Active-passive Decomposition of Chaotic Systems with Application to the Modelling and Control of Synchronous Motors R a n j a n Vepa Department of Engineering Queen Mary and Westfield College Mile End Road London E1 4NS, United Kingdom r . vepa@q~v, ac. uk

A b s t r a c t . In this paper an alternate method of Active-Passive Decomposition of chaotic systems is extended to establish a practical and useful decomposition method. The extended Active-Passive Decomposition is useful in synchronising a whole class of coupled chaotic systems. The method is shown to be useful in characterising both chaotic and non-chaotic synchronous motors. It is then shown that the decomposition may be exploited constructively to design synchronising controllers for such motors.

Keywords: Non-linear control, Synchronisation, and transformers

1

Passivity, Gyrators

Introduction

Chaotic systems are generally characterised by sensitive dependence on initial conditions. Given this s i t u a t i o n , it was considered t h a t the s y n c h r o n i s a t i o n of the o u t p u t s of two identical chaotic s y s t e m s is not really feasible. P e c o r a and Carroll [17], who were able to show t h a t s y n c h r o n i s a t i o n of the o u t p u t s of two chaotic systems was, in fact, feasible. T w o s y s t e m s m a y be said to have synchronised if a functional relationship exists between the s t a t e s of the two systems: Synchronisation m a y be considered to be a form of feedback control to achieve perfect m o d e l following t y p e b e h a v i o u r in t h a t the process m a y be m a d e to track a desired t r a j e c t o r y . It is this p a r t i c u l a r i n t e r p r e t a t i o n we shall focus on in this p a p e r . One of the a p p r o a c h e s to s y n c h r o n i s a t i o n is based on a special d e c o m p o s i t i o n of the s y s t e m into two s u b s y s t e m s . T h e d e c o m p o s i t i o n is used as a basis for constructing an a p p r o p r i a t e m o d e l t h a t is capable of producing a reference trajectory. We shall first briefly present this d e c o m p o s i t i o n known as Active-Passive Decomposition, and present a s y s t e m a t i c a p p r o a c h for Extended Active-Passive Decomposition.

544

Ranjan Vepa

Consider an uncontrolled n dimensional dynamical system in the form, dx (t)

dt

_ F (x (t))

where F (x (t)) is a vector field of the vector x (t). This autonomous system is rewritten a s a non-autonomous system, dw(t) _ F ~ o ( w ( t )

z(t))

dt

where w (t) is the new state vector corresponding to x (t) and z (t) is some vector valued function of time that augments w (t) and is given by, either z ( t ) = V ( w ( t ) ) or : v (z ( t ) , w (t)). Kocarev and Parlitz [9] introduced this method of decomposition and they termed it "Active-Passive Decomposition" based on the characterisation of the conditional Lyapunov exponents of the subsystems. The term "ActivePassive Decomposition" has connotations of Passivity which is a completely different concept in control theory. In this paper we shall therefore propose an alternate method of Active-Passive Decomposition based on the notion of passivity. We will then apply non-linear controller synthesis concepts and show how such a decomposition is useful in achieving synchronisation. Apart from the fact that synchronising controllers may be implemented in a whole class of chaotic systems they are also the basis for designing adaptive systems, where the convergence of the scheme is often proved by the use of passivity concepts.

2

Active-Passive Systems

Decomposition

of Controlled

Chaotic

Electrical networks with only components that are resistive, inductive or capacitive are said to be passive and passivity means that the network can absorb or store energy but cannot generate it. A passive electrical network is characterised by an impedance matrix that is said to be positive real. For a characterisation of positive real transfer functions the reader is referred to Baher [1]. For a further analysis of the passivity interpretation of systems and the characterisation of the matrix transfer functions the reader is referred to Slotine and Li [20]. To generalise the concept of passivity to non-linear systems the notion of hyperstability is introduced. For a system to be hyperstable, the linear part must necessarily be positive real. For a complete discussion of hyperstability the author is referred to [19]. Through the work of Willems [22] the concept of hyperstability may be considered as a special case of dissipativity; i.e. when the supply rate is assumed to be bilinear. Although in the contexts of

Extended Active-Passive Decomposition of Chaotic Systems

545

non-linear control no distinction is made between the concepts of Passivity and Hyperstability in the context of circuit theory, passivity and positive real transfer functions are normally associated with linear circuits and Hyperstability with nonlinear circuits. Willems also introduced the notion of a storage function which could be considered as a L y a p u n o v function. It must be said that the use of Passivity techniques in control is not new. The concept of passivity has been used in Adaptive control over number of years. (Landau [11]).Its explicit usage in the design of global stabilizing control laws has been advocated by Isidori [8] and Ortega et. al. [15]. Ortega et. al. [16] also explicitly exploited the connection between the Lagrangian fomalism in Dynamical Systems and the the Passivity approach, van der Schaft [21] treats passivity based control as an interconnection property of Hamiltonian Systems and introduces the notion feedback passivity. This notion is implicit in the work of Hill and Moylan [6]. P r o g r o m s k y [18] dealt with a class of controlled synchronisation problems and showed when conditions for feedback passifability are satisfied the conditions for controlled synchronisation are also satisfied. There a n u m b e r of applications in engineering such as combustion process, Chemical processes, Hydraulic and other fluid-flow based control systems, Aero and hydroelastic systems which are essentially non-hamiltonian and non-dissipative. T h e y often also include either a gyroscopic c o m m p o n e n t , a circulatory component or even an active or energy generative conponent. T h e method of Active-Passive decomposition is essentially an "inverse" interconnection m e t h o d that seeks to exploit this passivity property in an explicit and constructive way. We are now in a position to apply the concepts of passivity and hyperstability to chaotic systems. Consider a controlled n dimensional dynamical system in the form, dx(t) _F(x(t))+G(x)u dt where V (x (t)) is a vector field of the vector x (t) = [v T (t) w T (t)] T. Decompose the system into two subsystems, d v (t) - F~ (v (t) , w (t)) + G~ (v (t) , w (t)) u dt

dw (t)

d------/--- F~ (v (t) , w (t)) + G~o (v (t) , w (t)) u y (t) = h (w (t))

where v (t) = [x~ ( t ) , xz (t), x~ (t) ..... , x~ (t)] ~" and w (t) = [~k+~ (t), xk+~ (t), xk+~ (t) ..... , xn (t)] T .

546

Ranjan Vepa

When the w-subsystem is hyperstable with respect to y(t), then the decomposition of the system will be referred to as Active-Passive Decomposition. It must be said that Active.Passive Decomposition based on the class of hyperstable systems may be very restrictive. However by introducing the concept of Output Hyperstabilizability the conditions on the w-subsystem may be relaxed. If the w-subsystem is Hyperstable with the introduction constant gain linear output feedback it is said to be Output Hyperstabilizable. In fact a further relaxation of the conditions imposed on the w-subsystem is possible when it is not Output Hyperstabilizable. If there exists an invertible linear autonomous dynamic compensator, with a stable inverse, such that the wsubsystem is stabilizable with the introduction of dynamic output feedback, then the direct sum of the w-subsystem and the inverse of the state-space model of the dynamic output feedback, is Output Hyperstabilizable. In the case when the output feedback is static, the direct sum reduces to a feedforward gain. Although such decompositions strictly do not fall into the category of Active-Passive Decomposition, we shall categorise them also as active passive decompositions. In [3] the property of Output Hyperstabilizability is implicitly applied to design an adaptive controller. Thus the method of Active-Passive Decomposition proposed here, provides a relatively flexible method of decomposing a chaotic system. It is important to note that such decompositions need not be unique and a particular system may be decomposed into an active and a "passive" subsystem in more than one way. The choice of a particular decomposition is dictated by the application and the following example illustrates the point. E x a m p l e 1: C h u a ' s c i r c u i t The well known chaotic system that goes by the name of Chua's circuit is considered here. It is a relative simple circuit with five passive elements and a piecewise linear resistor. Chua's circuit is one of the simplest paradigm's for studying the onset of chaos in electrical circuits. An example of a passive network with a positive real impedance matrix, is the circuit which is obtained by replacing the non-linear element in Chua's circuit by a linear resistor. The governing equations of the circuit in terms of the input and output voltages and the current in the inductor are,

LiL J

1 R2

iL

where GI = 1 and G = ~-. Taking Laplace transforms and applying Kirchoff's laws to the two-input two-output circuit it can be verified that the impedance matrix, Z (s), relating the input and output voltages to the input and output currents is,

v2

LVout

L~out

Extended Active-Passive Decomposition of Chaotic Systems

547

is a positive real matrix transfer function. The governing equations of Chua's circuit with the non-linear resistive element are, C2.~21 = -

LiL J

-G

0

G

1

v2

1 R~

ig

-

where G = ]~ and the current-voltage relationship of the non-linear element is given by,

inr

1

= ff (Vl) = ~710U1 "{" ~ (Tnl -- m0) [Ivl

+ vbpl- Iv1 -

vbpI],

and the points Vl = +Vbv = q--E define the break points at which there are changes in the slope of the current-voltage relationship. If we let, v = vz, w = [v2 i2]T, the system decomposes into an active (v) and a passive subsystem (w). E x a m p l e 2. E x a m p l e o f t w o c o u p l e d v a n d e r P o l o s c i l l a t o r s Consider a chaotic system comprising of two coupled van der Pol oscillators. The governing equations are given by, = A~v + An~ + A ~ w , ~k = Aww + Anw + Awvv + Bu,

0 [ O lc~1 ] ' A n ~ o = - [

2

w 2 ( w l 0+ a v t ) ~ ] ,A~o~ = [ ? a ~ ]

, w i t h B - - - [01] and

n = [0 x]. Based on the generalised concept of Active-Passive decomposition the above decomposition also constitutes an Active-Passive decomposition of the coupled system; i.e. there exists a constant D such that the system is output hyperstabilizable with respect to the augmented output, Y2a (t) = H w (t) + Du. In fact, it can be shown, that in the above case, there exists such a scalar constant and is given by D = K - 1 . Thus the modified non-linear w-subsystem with feedforward is Output Hyperstabilizable and the above decomposition constitutes an Active-Passive decomposition of the coupled system.

548 3

RanjanVepa Characterisation of Coupling, The Gyrator, Transformer and Mutually Coupled Coils

the

Ideal

It must be said that the issue of the characterisation of the coupling has not been given much attention in the treatment so far. Traditionally in circuit theory there have three principal passive elements, the resistor, the capacitor and the inductor. Non-linear resistors may generally be classified as being either active or passive. However in the case of inductors and capacitors there is a need to introduce another element, the ideal gyrator or simply the gyrator. It may be used to characterise ideal transformers and by appropriate cascading techniques can be used to represent mutually coupled inductances (and mutually coupled capacitances as well). These will aid the cascading and coupling of various passive and active elements that will form the basis of the Extended Active-Passive decomposition of certain non-linear systems. Yet the ideal gyrator and the ideal transformer are important coupling elements as they do not absorb any real power. Apart from the ideal gyrator, the ideal transformer and the shunt impedance, we may construct other models of real coupling, by suitably connecting one, two or three independent impedances to an ideal transformer.

4

Application Synchronous

to the Control Motors

of Permanent

Magnet

A primary example of a real system that can be modelled in terms of an ideal gyrator, an ideal transformer and other passive elements is the permanent magnet synchronous motor. A number of interesting models of synchronous machines may be developed by introducing various levels of approximations (see for example Hammons and Wining [5]). As we have a greater interest in isolating the principal characteristics of the models rather than studying the dynamics of synchronous machines, we choose a rather simplified and yet non-linear model for the purposes of this example. An interesting application of the extended Active-Passive decomposition is to permanent magnet synchronous motors. The importance of this example stems from the fact that these motors are fast replacing the conventional actuating elements in such safety critical applications as aircraft flight controllers. Mukhopadhyay and Malik [14] present an early application of optimal control theory to the control of synchronous motors. To design a non-linear controller a suitable approach is to construct a control Lyapunov function. (see for example Freeman and Kokotovic [4]). In the literature on the control of these archetypal motors, almost all authors (see for example Lin and Chiu [13] and Hsien, Sun and Tsai [7] and the references quoted there) with the exception of a few, (see for example [3]) adopt a linearization or quasi-linearization technique. One other non-linear technique reported is

Extended Active-Passive Decomposition of Chaotic Systems

549

based on input-output linearisation (Baik, Kim and Youn [2]). Linearization is often unnecessary step and results in an undue and excessive use of control effort.

Example 3: S y n c h r o n i s a t l o n a n d c o n t r o l o f P e r m a n e n t M a g n e t S y n c h r o n o u s motors In this typical example we present an approach to modelling and decomposition of the motor dynamics followed by an exact control strategy based on synchronisation. The governing equations may be written in the form,

o1

0 LqJ " ~ [ i q j = - [ 0 o

r,< l _

LL~ 0 JLiq]

r ol riA R J [iqJ

rol + iv<,] [oJ

LVqj

J& + B w = Td - TL, Td -- 3Piq 2 where id is the current in the direct axis, iq is the current in the quadrature axis, Ld is the inductance in the in the direct axis, Lq is the inductance in the quadrature axis, R is the stator resistance, P is the number of pole pairs, w is the rotor speed, ~b is the flux of the rotor magnets, va and vq are the stator voltages in the direct and quadrature axes, J the moment of inertia of the rotor, B the viscous friction coefficient, TL the load torque which is assumed constant and Td is the driving torque. The load torque, TL, may be expressed as, TL = Bwo

3Piqo 2 and the equation of motion for the rotor may also be written as, J& + B (~ - wo) -

3 P (iq - iqo) 2

An interesting feature of this model is the fact that under certain circumstances, when the stator resistance R is relatively low, it results in a chaotic response (see for example Z. Li, Bo Zhang and Zongyuan Mao [12]. It has the form of Silnikov's third order model that is given by, :~ = ~ r x - p y + P ( x , y , z ) = px+ cry+Q (x,y,z) i: = Az + R ( x , y , z )

where the point (0, O, O) is a fixed point of the saddle-focus type. The system admits a homoclinic orbit and conditions for the choatic break-up of the orbit may be established by Melnikov's method. Further the gyratic nature

550

Ranjan Vepa

of the coupling between the rotor and the electromagnetic sub-system is now apparent from the above form of the equations. In fact the above third order model is representative of most synchronous machines and for this reason the permanent magnet synchronous motor was chosen as the basis for this example. A typical control strategy is to maximise the driving torque or current in the quadrature axis and minimise the current in the direct axis. It is interesting to note that such a design strategy tends to make the motor less stable and potentially more chaotic. In fact it appears that for extracting the maximum torque from these motors they must be designed to be almost chaotic and subsequently stabilised, optimally, by feedback control. When the two axis currents are not available for feedback and only the rotor speed is available as a measurement (measured by a tacho generator for example), a convenient control strategy is to construct a model of the electromagnetic sub-system that will synchronise with it. The model then takes the form,

[~,, o 1 ,, r,,,,.,,1 rR ol r,,,,.,1 0 LqJ ~ Liq,~J : - LO RJ LiqmJ -'-~~,,"r0 L -,.,,,loj,.[!"'1.~,,,,,_ ,:,,.,.,[o,..,,,,..,1- ,.[v,, ., v], , ., where the flux of the model rotor magnets and the model stator voltages are assumed to be different from those of the motor. Defining the current and stator voltage error vectors as,

eq =

iq

r,,.,.l=_r L Vqm

L ~q'~ J ' L Vqe j

vq

the dynamics of the error then takes of the form,

0 LqJ "

ro-L,0

-P"'/L,~

Leg] -

J L,~qJ-

L 0 RJ Leq]

rvd l LVq~j

which is conveniently a linear system. Synchronisation can be guaranteed provided the control inputs can be found such that the error vector tends to zero as the time, t, tends to infinity. The problem of designing a controller for the motor is reduced to one of designing a linear model synchronising controller and a non-linear control law for the motor. In the case of synchronous machines this error vector is inherently stable and is the main reason for the stability of two machines connected in parallel, mentioned earlier. We adopt a sequential or backstepping Lyapunov approach. A suitable control Lyapunov function for the plant is, =

L2

Extended Active-Passive Decomposition of Chaotic Systems

551

We assume the following control laws of the form, vd = vdo - Kd (w - wo) a n d vq --" vqo - Kq (W -- COO).

By writing the non-linear governing equations in terms of perturbations about the position of equilibrium, it can be shown that, Vp = - J B

(w - ~0) 2 - R L d i ] -- R L q (iq - iqo) 2

provided, vao = - P w o L q i q o ,

Vqo = Riqo + P w o O

and - -

~

Thus the desired equilibrium position is uniformly asymptotically stable. The method assumes that the parameters of the model are known apriori. When this is not the case an adaptive approach must necessarily be adopted (see for example Kristic, Kanellakopoulis and Kokotovic [10]. The technique is briefly outlined. As a first step we introduce a Load torque-speed o b s e r v e r which is on one hand a state estimator for the rotor speed and a parameter estimator for the load torque. The load-torque equation now takes the form, TL = Bwo

3Piqo 2

where w0 is an unknown but constant parameter. The governing equations of the motor now take the form, Ld 0 ] a

0

ia

R 0

id

[,. ,.0] [0.] [,q-,qo]

f O --Lq~ " -P~ [-Lq~•176 "31"[Vq :dRiqo] -P~ LLd 0 j fLiq ~.diqo] and Jdo = - B

(w - wo) + 3 P (iq - - iqO) ,(OO = O. 2

The load torque-speed observer is assumed to be of the form, J~ = -B

3p

(do - ~o) + ~

(iq - - iqo) -- K . , ( ~ - ~ )

~0 = A'~ [(~ - ~) + Q (~0 - ~0)] where K~, KT and Q are the observer gains which are assumed to be strictly positive. The second equation yields the parameter adaptation rule. Defining the observer error as, er

~0

CJO

'

552

RanjanVepa

the error dynamics is given by, d ~ = - B (e~ - e~-) - K,~e~, ~r = - K r [e~ + Qe~].

To select the observer gains we choose the observer Lyapunov function,

vo=

~ + ~-7C~,.

It follows that, ~2 Vo = - % 2 ( B + K,~) - e.~ (BQ)

and the observer error is uniformly asymptotically stable. To design the synchronising model control laws, we begin with the certainty equivalence control laws, Vd = VetO-- Kd (& -- if;O) a n d Vq = Vqo -- Kq (& - d;o).

The control laws are expressed as, va = vao - K a (~ - ~ o ) + K a (e,~ - e ~ . ) ,

Vq = VqO -- Kq (w - ~o) -F I(q (ew - e r ) .

To simplify the problem without losing generality we assume that all the model parameters are known and that, r = ~5m. The non-linear nature of the problem results in the control laws being coupled at this stage and the classic observer-controller separation principle is only partially valid. The model and observer control laws are not independent of the controller gains. Nevertheless coupled model control laws of the form, [ : : : ] ..~ p(~..l_ew ) [ O _ ~ ) m ] + p e w [ 0 d - L q ]

[!drn]

rt~-mj o 1-r ts,2 s,, s~2j s1,,1 Leqj r<<,l_ LKqj r,<<,l (e~-eT) "'<'" and a modified load torque-speed observer coupled with a load torque parameter adaptation rule, of the form, Je,o = - B (e~ - e~.) - K,~e,, - Kaid,~ -- Kq (iqm - iqo) B. K r e~. = - B [e,o + Qe~] + Is

+ Kq (iqm - iqo)

that will ensure that the model and observer errors are uniformly asymptotically stable are established based on the Lyapunov approach. The corresponding control Lyapunov function is the sum, V = Vm + lip + Vo where Vp and Vo are as defined earlier and Vm is given by, L2

L~ e2

Extended Active-Passive Decomposition of Chaotic Systems

553

The control gains are chosen to ensure that, II < O. In case some of the model parameters are not known with certainty, an adaptive approach is adopted estimate these parameters. This sequential method we have adopted here to construct the Lyapunov function is the backstepping procedure outlined by Kristic, Kanellakopoulis and Kokotovic [10]. The above example illustrates that the backstepping procedure is complementary to the extended active passive decomposition proposed in this paper, thus illustrating the usefulness and the application of the proposed decomposition scheme.

5

Conclusions

Based on the concept of passivity and positive real transfer functions, an alternate approach to Active-Passive Decomposition of chaotic systems is introduced. Loss free gyrators are distinguished from lossy passive systems. Thus the basic theme of the paper is that a typical chaotic system m a y be decomposed to cascade of subsystems based on their passivity characterisation. The concept is generalised to include a whole class of systems that are not necessarily passive or hyperstable but are merely Output stabilizable by an invertible dynamic compensator. The importance of the generalisation stems from the fact that there are a whole class of non-linear systems that are stabilizable by output feedback or dynamic compensation and a further subclass of systems that could be linearized by dynamic output feedback. Such systems could be "transformed" to passive sub-systems by suitable compensation, feedback and feedforward. The decomposition permits the design of synchronising subsystems with guaranteed exponential convergence of the error to zero. The decomposition is complementary to the backstepping procedure for constructing control Lyapunov functions. In fact the Lyapunov functions for the plant were constructed by implicitly taking advantage of the passivity properties of the system. It is this complementary feature of the decomposition that makes it a practically useful technique and this fact is demonstrated in the paper by illustrating a complete design example.

References 1. H. Baher, Synthesis of Electrical Networks, Chapter 2, John Wiley and Sons, 1984, pp. 19-46. 2. I. -C. Baik, K. -H. Kim, M. -J. u Robust nonlinear speed control of PM synchronous motor using adaptive and sliding mode control techniques , IEE Proc.-Electr. Power Appl., Vol. 145, No. 4, July 1998, pp. 369-376. 3. G. Espinoza-Perez and R. Ortega, An output feedback globally stable controller for Induction Motors, IEEE Trans. on Aut. Contr. AC-40(1), pp. 138-143.

554

Ranjan Vepa

4. R. A. Freeman and P. V. Kokotovic, R o b u s t N o n l i n e a r C o n t r o l D e s i g n : S t a t e S p a c e a n d L y a p u n o v T e c h n i q u e s , Birkh~user, 1996. 5. T. J. Hammons and D. J. Winning, Comparison of Synchronous machine models in the study of the transient behaviour of electrical power systems, Proc. lEE, Vol. 118, No. 10, Oct. 1971 pp. 1442-1458. 6. D. J. Hill and P. J. Moylan, The Stability of Non-linear Dissipative Systems, IEEE Trans. Auto. Contr., vol. 21, pp-708-711, 1976. 7. T. -L. Hsien, Y. -Y. Sun and M. -C. Tsai, H ~ control for a seusorless permanent-magnet synchronous drive, IEE Proc.-Electr. Power Appl., Vol. 144, No. 3, May 1997, pp. 173-181. 8. A. Isidori, N o n - L i n e a r C o n t r o l S y s t e m s II, Springer-Verlag, 1999, Section 10.7-10.9, pp. 42-74. 9. L. Kocarev and U. Parlitz, Generalized Synchronisation, Predictability, and Equivalence of unidirectionally coupled dynamical systems, Phys. Rev. Lett., vol 76, no. 11, 1996, pp. 1816-1819. 10. M. Kristic, I. Kanellakopoulis and P. V. Kokotovic, N o n l i n e a r a n d A d a p t i v e C o n t r o l D e s i g n , John Wiley and Sons, New York, 1995. 11. I. D. Landau, R. Lozano and M. M'Saad, A d a p t i v e C o n t r o l , Springer-Verlag, 1997, Appendix C Passive (Hyperstable Systems) pp 515-531. 12. Z. Li, Bo Zhang and Zongyuan Mao, Strange Attractors in Permanent-magnet Synchronous Motors, Proc. IEEE Int. Conf. On Power Electronics and Drive Systems, PEDS'99, Hong Kong, pp. 150-155. 13. F.-J. Lin and S. -L. Chiu, Robust PM synchronous motor servo drive with variable-structure model-output-following control, IEE Proc..Electr. Power Appl., Vol. 144, No. 5, September 1997, pp. 317-324. 14. B. K. Mukhopadhyay and O. P. Malik, Optimal control of synchronous-machine excitation by quasilinearisation techniques, Proc. [EE, Vol. 119, No. 1, Jan. 1972, pp. 91-98. 15. R. Ortega, A. Loria, P. J. Nicldasson and H. Sira-Ramirez, P a s s i v i t y b a s e d c o n t r o l o f E u l e r - L a g r a n g e s y s t e m s , Springer-Verlag, Berlin, Communication and Control Engineering Series, 1998. 16. R. Ortega, A. Loria, R. Kelly and L. Praly, On Output-feedback Global Stabilisation of Euler-Lagrange Systems, Int. J. on Robust and Nonlinear Control, Special Issue on Mechanical Systems, Ed. H. Nijmeijer and A. van der Schaft, vol. 5, No. 4, pp. 313-374, July 1995. 17. L. M. Pecora and T. L. Carroll, Synchronisation in Chaotic Systems, Phys. Rev. Left., vol. 64, no. 8, 1990, pp. 821-824. 18. A. Yu. Pogromsky, Passivity based design of synchronizing systems, Int. J. of Bifurcation and Chaos, vol. 8, no. 2, pp. 979-998, 1998 19. V. Popov, H y p e r s t a b i l i t y o f A u t o m a t i c C o n t r o l S y s t e m s , SpringerVerlag, 1973. 20. J. J. E. Slotine and W. Li, A p p l i e d N o n l i n e a r C o n t r o l , Chapter 4, Prentice Hall International Inc., 1991, pp. 126-156. 21. A. van der Schaft, L2 G a i n a n d P a s s i v i t y T e c h n i q u e s in N o n l i n e a r C o n trol, Second Edition, Springer-Verlag, Berlin, Communication and Control Engineering Series, 1999. 22. J. C. Willems, Dissipative Dynamical Systems, Archive for Rational Mechanics and Analysis, vol. 45, pp. 321-393, 1972.

On Canonical Decomposition of Nonlinear Dynamic Systems Alexey N. Zhirabok Far Eastern State Technical University Vladivostok, 690600, Russia zhirabok@mail,

ru

Abstract. A problem under consideration is to obtain a canonical decomposition of nonlinear dynamic systems. This decomposition generalizes a well-known decomposition in the linear case. An algebraic approach based on a special mathematical technique which generalizes the pair algebra of Hartmanis and Stearns is developed to study this problem. This algebraic approach makes it possible to find out that this decomposition is similar for both the discrete-time and continuous-time systems.

1

Introduction

Here is a well-known [5] canonical decomposition of linear dynamic systems that contains four subsystems (Fig.l): $11 - - controllable and observable, E01 - - uncontrollable and observable, $10 - - controllable and unobservable, and E00 - - uncontrollable and unobservable. Such a decomposition in nonlinear control theory can be obtained using geometric methods (see e.g. [3]) in the following way. Consider the minimal distribution, closed under the Lie bracket, that contains all vector fields f(., u) corresponding to constant controls u of the dynamics -- f(x($), u(g)),

y(t) -'- h(~c(t)).

(1)

If it is of constant rank, then it gives a decomposition into accessible and nonaccessible parts of the system. Similarly, the minimal codistribution that contains the differentials of all observations, invariant under the Lie derivative with respect to all vector fields f(-, u), gives - - again under the constant rank assumption - - a decomposition into observable-unobservable parts. Now, taking simultaneously two mentioned above decompositions, the decomposition in question can be obtained. In this paper we suggest another approach to produce a similar decomposition for nonlinear systems described by equations (1) in the continuous-time case and by the equations

x(t + 1) = f(x(t), u(t)),

y(t) = h(x(t))

(2)

556

A.N. Zhirabok

in the discrete-time case. H e r e z E X C R n,uEUCR m,yEYCR l,fis a nonlinear vector function such t h a t a solution to equations (1) exists and h is a vector function. For these systems the symbol S will be used. This approach is based on algebraic m e t h o d s and it gives a decomposition shown in Fig.2. In contrast to the linear case, it can have some additional connections (marked as dotted lines) which correspond to the features of these subsystems: controllable and observable, uncontrollable and observable, and so on. Notice that, in general, the terms "controllable" and "observable" correspond to the inputs and outputs of these subsystems. In the linear case they reflect controllability and observability properties. At the same time, terms "uncontrollable" and "unobservable" describe corresponding properties exactly. Here is a description of the nonlinear decomposition (to be specific, consider the discrete-time case):

$11 : x l l ( t + 1) = f l l ( x 1 1 ( t ) , z 0 ~ ( t ) , u ( t ) ) ,

Sol : x~

+ 1) = y~176

S10: xx~ + 1) = .f~~176

z ~ (t), x~

z~176 u(t) ),

S00: x~176+ t) =/~176176176z~ y(t) = hl(zll(t), z~ The task is to find the functions f11, f 0 1 fl0, f00 and h'. To solve this task, a special m a t h e m a t i c a l technique (so-called algebra of functions) will be used. It generalizes the pair algebra proposed by H a r t m a n i s and Stearns [2]. T h e algebra of functions has been developed by Zhirabok and Shumsky [8] and is used to solve some problems of system theory [8,9]). At first, consider main results of this technique according to [8].

2

Algebra

of functions

Vector functions with the domain X are elements of this technique including some binary relations, operations, and operators. 1. Partial ordering relation <: for some vector functions a : X ~ S and fl : X ~ T denote a _< /~ if p a = /3 for some function p : S ---+ T, i.e. ~u(a(z)) = / 3 ( x ) for all z E X where S and T are some sets. W h e n a < /3 a n d / 3 < a , denote a ~ / 3 . For example, if a = [zl Ix2]T a n d / 3 = z t z 2 , then a ~/3; if a = [xl + •2[Z2] T a n d / 3 = [XlIZ2] T, then a --~/3; if a = [xl[z2x3] T and/~ = [xlz2lx3] T, then neither a __
O n Canonical Decomposition

557

If the functions a and /3 are linear a n d a ( x ) --" Ax, fl(x) : Ux for some m a t r i c e s A a n d B, then a < / 3 (or A < B) when N A = B for s o m e m a t r i x Y . Clearly, A < B iff rank(A) = rank[ATI BT] a n d A -~ B iff rank(A) =

rank(B). 2. O p e r a t i o n s x and U: a x ~---- max(gig _< a , g

a U/3 = m i n ( g l a _< g,/3 < g). T h e functions a x / 3 and a It/3 exist for a r b i t r a r y functions a a n d / 3 because the set of all p a r t i t i o n s on X is a l a t t i c e [1]. It is clear t h a t a x / 3 < a and a x / ~ _< /3, therefore a x / 3 = [o T[/3T]T. By analogy, a < a U/3 a n d / 3 < a U/3. Thus, each c o m p o n e n t o f t h e function a U/3 d e p e n d s on b o t h c o m p o n e n t s of the functions a a n d / 3 . T h i s rule can be used to c a l c u l a t e the function a U/3 by h a n d in s o m e s i m p l e cases: e.g., if a = [XlIX2X3] T and /3 = [XlX2lX3] T, then a II/3 = x l x 2 x a . In t h e general case, some certain differential equations shc,uld be solved [8]. I r a ( x ) = Ax a n d / 3 ( x ) = Bx, then A x B =: [ATIBT]T; m a t r i x N corresponding to t h e function # = aU/3 can be o b t a i n e d as follows. Let [QIP] be a m a t r i x of m a x i m a l r a n k such t h a t [QIP][AT[BT] T ---- 0, then Y = QA = - P B . It can be shown also t h a t N is a m a t r i x of m a x i m a l r a n k such t h a t each row of N linearly d e p e n d s on rows of the m a t r i x A and rows of the m a t r i x B, therefore the m a t r i x N corresponds to s p a n ( A ) N span(B). 3. Binary relation A : for a given function f , the function a and differentiable function /3 form a pair, i.e. (a,/3) E A, i f / ~ ( a ( x ) , u ) = (d/3/dx)f(x,u) - or /~(~(x), u) = /3(f(z, u)) in the d i s c r e t e - t i m e case - - for s o m e function /~ : S x U ~ T a n d for all z E X , u E U. In the linear case, when ~: = F x + G u or x(t + 1) = F z ( t ) + Gu(t) for some m a t r i c e s F a n d G, ( A , B ) E A if rank(A) = rank[ATI(BF)T]. T h e relation A is of s e c o n d a r y i m p o r t a n c e , it is used only for the o p e r a t o r s m and M definition. 4. O p e r a t o r s m a n d M : r e ( a ) is a function satisfying the follow] ; conditions: (~, m(~)) e ~,

V(o,/3) e z~ -+ re(o) _/3.

Thus, m ( ~ ) is a m i n i m a l function with which the function a forms a pair. M(/3) is a function satisfying the following conditions: (M(/3),/3) e A,

V(a,/3) 9 A - ~ ~ _ M(/3).

Thus, M(/3) is a m a x i m a l function f o r m i n g a pair w i t h / 3 . T h e o p e r a t o r M can be c a l c u l a t e d as follows: if/3 is a scalar function a n d d

~-~xf(x,u)= E a i ( x ) b i ( u ) i=l

(3)

558

A.N. Zhlrabok

(or f l ( f ( x , u ) ) = ~'~ff=l ai(x)bi(u) in the discrete-time case) and functions bl,b~,...,bd are linearly independent, then M ( ~ ) = al • a2 • ... • ad. If fl = fll x / ~ x ... x ill, then M(fl) = M(fll) x M ( • ) x ... x M(flt). In the general case, to calculate the operator m , certain differential equations should be solved [8]. To obtain a linear solution in the form m ( a ) ( x ) = Q x for some matrix Q of maximal rank, the algebraic equation Q f ( x , u) = p ( a ( x ) , u) for any vector function p should be solved. In the linear case, M ( B ) = B F and re(A) = Q where [QIN][FTIAT] T = 0 and [QIN] is a matrix of maximal rank. If the matrix F is nonsingular, then re(A) = A F -1. Main properties of the relation <, operations, and operators are:

1. a < ~ (

)ax#-~a~

~au#-~;

2. i f a _< J and fl <_ J, then a U , 8 < J;

3. if a < fl, then m ( a ) _< m(/3); 4. M(a x/~) _~ M(a) x M(/~); 5. M(m(a)) _> ~,

3

Algebra approach

m(M(/3)) _< ft.

of functions

and

differential

geometric

Consider some relationships between the algebra of functions and the differential geometric approach using [3]. Let a : X ) S and fl : X ) T be some differentiable vector functions and ai be some component of the function a, then dai = ~'~j=l(Ooti/Oxj)dxj n is a differential one-form corresponding to this component. Let St and B be codistributions spanned by the one-forms dat, da~, ..., dak and dill, dj32, ..., dill where k and l are numbers of the components of a and fl respectively (in this case it can be said that the codistributions St and B correspond to the functions a and fl). It has been shown that sum of St and B corresponds to a x fl and the intersection St [7 B corresponds to a II ft. If a < fl, i.e. fli = ~ia for each i and some function 4 , then k

j=l

hence dfli E A and B C_ A. On the other hand, let B C A, i.e. arbitrary one-form dfli E B can be represented as dfli = ~]j=l k cij - (x)dotj where cij(x) are some functions. Therefore, the function fli can be represented via a l , a2, ..., ak, i.e. fli = Ji (a) for some function Ji, or a < fli for each i which implies a < j3. Therefore, if B C A , then a
On Canonical Decomposition

559

Consider a system described by the equation = f ( x ) + g(x)u

(4)

where f is a differentiable vector function and g is a differentiable matrix function. Let/3 be a scalar function; it follows from (3) that in this case M(fl) = (dfl/dx)f x ((dfl/dx)g) T.

(5)

Let A be a codistribution invariant under the dynamics (4), i.e. L! (A) C A and Lg(A) C A where L! and Lg are the Lie derivatives. Consider the k inclusion L f ( A ) C_ A which implies rl(dai) = ~ j = l cij(x)daj for some functions cij, i = l, 2, ..., k. Since ai is a function then

Ly(dai) = dL! (ai) = d((dai/dx)f) and therefore (dai/dx)f _> a; by analogy, ((dai/dx)g) T > a. These inequalities are equivalent to ( d a J d x ) f x ((doq/dx)g) T >_a or M ( a i ) > a according to (5). Since M ( a ) -~ M ( a t ) • M(~2) • ... x M ( a p ) , then M ( a ) _> a, or

(a, a) 9

za.

Hence, if the codistribution A is invariant under the dynamics (4), then ( a , a ) 9 A; it can be shown that if ( a , a ) 9 A, then the codistribution A corresponding to the function a is invariant under the dynamics (4). By analogy, it can be shown that the inclusions L / ( B ) _C A and Lg(B) _C A are equivalent to (a,/3) 9 A and A is (h,f)-invariant under the dynamics (4) iff

(a

x

h, a)

9 A.

As follows from (5) and the equality L1(a ) = (d~,/dx)y, the operator M for the dynamics (4) is similar to the Lie derivative L I. The Lie bracket does not have any analogy in the algebra of functions and the operator ra does not have analogy in the differential geometric approach. Thus, for the system (4) and differentiable functions, there are some relationships between some main tools of the algebra of functions and the differential geometric approach. The algebra of functions has following advantage: in some cases, the function f is not required to be differentiable and this makes it possible to analyze systems with nondifferentiable nonlinearities. As it has been shown in Section 2, in the linear case the algebra of functions reduces to the well-known matrix operations; the differential geometric approach reduces to such linear geometric objects as a space, subspace, kernel, and so on [3]. Attempts to extend the differential geometric approach to the discrete-time systems were initiated by Nijmeijer in [6] and continued in [4,7]; in particular, certain Lie algebras associated with the system were introduced in [4]. The algebra of functions is more natural and simpler in this case because it includes the relation A adequately reflecting the discrete properties of the

560

A.N. Zhirabok

system. Really, if (a,/3) E A, then /J(~(x(t), u(~))) = f l ( f ( x ( t ) , u(t))) = fl(x(t + 1)) by definition of A. It can be shown t h a t in the discrete-time case the algebra of functions is equivalent to the pair algebra developed in [2], and it can be used for analysis of finite a u t o m a t a described by Boolean equations. It is clear that the differential geometric approach is impossible to use in this case.

4

f-invariant

functions

Function a is said to be f-invariant if (a, a) E A. It can be shown t h a t if ( a , a ) E ,4, then a < M ( a ) and r e ( a ) < a; if a < M ( a ) , then r e ( a ) < a and (a, a) E A. For the system (2), these three relations are equivalent. To m a k e it simple, consider a class of the continuous-time systems for which these relations are equivalent. By analogy with [3], the importance of f-invariant functions is that the system (1) or (2) projects onto a lower dimension system. Let ~b be f-invariant function with k components, i.e. (6, ~b) E A. By definition of the relation A, k-dimensional vector x ~ = ~b(x) satisfies the equation

x'(t) = (dr

u(t)) =/'(r

u(t)) = :'(x'(t), u(t))

in the continuous-time case and equation x'(t + 1) = r

u(t)) = f'(r

u(t)) = f'(z'(t), u(t))

in the discrete-time case for some function f ' . In these cases, there is a h o m o m o r p h i s m ~b : L' ---+ S ' . There are two f-invariant extreme functions which play i m p o r t a n t part in a decomposition. T h e first function a* is a m i n i m a l function satisfying the functional inequalities p __
r e ( a * ) _< a*

where p is a minimal function satisfying the equality

(OlOu)(dpldz)l(x, u) = o or

(o/ o.)(p(f(x, .))) = o for the system {I) or (2) respectively. T h e function a* is f-invariant for all discrete-time systems and for the considered c o n t i n u o u s - t i m e systems.

On Canonical Decomposition

561

T h e o r e m 1. [8]. Minimal function a* satisfying the inequalities p <_ a* and re(a*) < a ~ can be obtained as follows: let s ~ = p, O/i+l =

O/i II m ( a i ) ,

i = 0, 1, ...;

when a k+l ~ a k for some k, then a* = a k. If a* is not trivial, i.e. a* ~ const, then the system S can be decomposed as shown in Fig.3 - - by analogy with the decomposition of an uncontrollable system in the linear case [5]. Really, since ( a * , a * ) E A, then a function f* exists such that a * ( f ( ~ , u ) ) = f * ( a * ( x ) , u ) for all ( x , u ) E X x U (to be specific, the discrete--time case is considered). T h e function f* does not depend on the control u: since p < a*, then a* = Jp for some function J and

(a/Ou)(f* (a* (x), u))) = (a/Ou)(~* (f(z, u))) =

(a/ au)(,~p(f(~, u) ) ) = (d,f/ dp)(a/au)(p(f(~:, u) ) ) = O. To obtain the function f*, there should be considered x*(t + 1) = a*(x(t + 1)) = a * ( f ( x ( t ) , u ( t ) ) )

(or x ' ( t ) = (da*/dx)(f(x(t),u(t))) for the system (1)) and the vector x should be replaced with z* = a*(x); it is possible due to the condition (~*,~*) ~ za. A state vector of the subsystem 27. is defined as x. = a . ( x ) where a . is a maximal function satisfying the condition ~* x a . ~ e, e is an identical function. In contrast with the linear case, the subsystem 27. can be uncontrollable. To obtain a description of 27., there should be considered x . ( t + 1) = a . ( f ( x ( t ) , u(t))) (or ~. ---- ( d a . / d x ) ( f ( x ( t ) , u ( t ) ) ) for the system (1)) and x should be replaced with x* and x.; it is possible due to the condition a* x a . ~- e. As it follows from Fig. 3, if a* ~ const, then the system 27 is uncontrollable. The second function f3* is a m a x i m a l function satisfying the functional inequalities fl* < h , fl* < M(/3*). The function fl* is f-invariant in both the discrete-time and c o n t i n u o u s - t i m e Cases.

T h e o r e m 2. [8]. Maximal function fl* satisfying the inequalities fl* < h and fl* _< M(fl*) can be obtained as follows: let flo = h, fli+l = fli x M(fli),

when flk+l ~_ /3k for some k, then fl* =/3 k .

i = 0, 1, ...;

562

A.N. Zhirabok

If Z* is not trivial, i.e. D* ~ e, then the system S can be decomposed as shown in Fig.4 - by analogy with the decomposition of unobservable system in the linear case [5]: since (fl*,D*) e A, a function f0 exists such that fl*(f(:c,u)) = f~ for all (:c,u) e X x U. The function f0 can be obtained by analogy with the function 1". Since/~* < h, then hOD* = h for some function h ~ and y = h~ ~ where :c0 = fl.(:c). Find the maximal function Z. such that /~* x/3, _~ e; this function forms state vector :Co = D.(:c) of the subsystem S0. A description of So can be obtained by analogy with the subsystem S . .

5

Canonical

decomposition

Denote eli the homomorphism S ~ L/j, i, j = 0, 1; the function !bij determines state vector x ij = r (x) of the subsystem Sij. T h e o r e m 3. The following equalities hold:

r

= a , UD',

r

= a* UD*,

r

= a , UD,,

r

= a* UD,.

P r o o f : Notice that the operation U gives a function which has properties of both addends (in particular, the function r = a . i ID, gives uncontrollability and observability properties). Since D* and a* are f-invariant, their sum r is also f-invariant. Therefore, a function f ~ and a vector x ~ exist such that

x~

+ 1) = f~176

(or ~:~ = 1~176 for the system (1)). Both functions I* and f ~ do not depend on the control u because the equality r = a* U D* implies r = Da* = Dip for some functions D and ,f. Since D* forms the functions r and r the subsystems $11 and S01 are obtained as a result of decomposing of Z ~ Therefore it is possible to let h' ~- h ~ Since the function D* is f invariant, then the control u and state vector x ~ are inputs of the subsystem S1~, and other inputs are absent. It is evident that this corresponds to the description of L ] I in Section 1. By analogy, it can be shown that the vector x ~ is the only input of the subsystem E00. The functions f11,101, f~o, foo can be obtained on the basis of the functions r162 r r by analogy with the function f*.

6

Conclusion

A canonical decomposition of nonlinear dynamic systems has been obtained . It should be noticed that presented results are similar for both the continuous-time and discrete-time systems. This similarity is a consequence of an algebraic character of the used mathematical technique.

On Canonical Decomposition

9~ ' I 1

H'

,y

hp

.

I

.E'1o

Eoo Figure 1

"

U

Z:u

t I I

r

i

'I

I "

Z'1o

I I I L _~

.~'oo

'

1 Figure 2

{ y

563

564

A.N. Zhirabok

Z'*

1 1

I I h.

I '1

~"

i

I

,y

I

-I

Figure 3

u

"[

S~

_1 -[

So

,

y

Figure 4

References 1. Gratzer, G. (1978) General lattice A.eory. Academic-Verlag. Berlin 2. Hartmanis, J., Stearns, R. (1966) Algebrai(- tructure theory of sequent.~a machines, Prentice-Hall, New York 3. lsidori, A. (1989) Nonlinear control theory. An Introduction. Springer-Verlag, Berlin 4. Jakubczyk B., Sontag, E. (1990) Controllability of nonlinear discrete-time systems: a Lie algebraic approach. SIAM J. Control and Optimization. 28, 1-33. 5. Kwakernaak, H., Sivan, R. (1972) Linear optimal control systems. John Wiley and Sons Inc., New York 6. Nijmeijer, H. (1982) Observability of autonomous discrete time nonlinear systems: a geometric approach. Int. J. Control. 36, 867-874. 7. Nijmeijer, H., Van der Schaft A. (1990) Nonlinear dynamic control systems. Springer-Verlag, New York 8. Zhirabok, A.N., Shumsky, A. Ye. (1993) A new mathematical techniques for nonlinear systems research. Proc. 12th World Congress IFAC, Sydney, Australia. 3, 485-488. 9. Zhirabok, A.N., Zhukov, A.M. (1995) On canonical forms of nonlinear dynamic systems. Proc. 3d European Control Conf., Roma, Italy. 1,236-241.

New Developments in Dynamical Adaptive Backstepping Control Alan S.I. Zinober, 3ulie C. Scarratt, Russell E. Mills, and All Jafari Koshkouei Department of Applied Mathematics The University of Sheffield Sheffield S10 2TN, United Kingdom A. Z i n o b e r @ s h e f field, ac. Ilk

Abstract. We consider a number of dynamical adaptive backstepping control algorithms for the class of observable non-minimum phase nonlinear continuous uncertain systems (triangular and non-triangular), as well as systems with disturbances which can be converted to the parametric semi-strict feedback form. Nonlinear, sliding and second-order sliding control laws are developed. Adaptive backstepping algorithms with tuning functions, and modular parameter identification approaches are presented. BACK, a Maple symbolic algebra package, has been developed as a tool for the design of dynamical adaptive nonlinear controllers for regulation and tracking tasks.

1

Introduction

The control of nonlinear systems is very important because there are m a n y practical applications in which it is inappropriate t'or the nonlinear equations to be assumed approximately linear. In particular, if there are some unknown parameters, uncertain dynamics, and external and internal disturbances, classical methods do not provide suitable control. Adaptive control is usually employed for the regulation of uncertain systems, when no information is available about the bounds of the unknown parameters. In the 1990s a new family of adaptive control algorithms, using the backstepping approach, was developed [11,16]. This control scheme allows the systematic design of adaptive controllers for triangular nonlinear systems containing unmatched parametric uncertainty. The various backstepping control design algorithms [10,11,16] compiled in [17], provide a systematic framework for the design of tracking and regulation strategies suitable for large classes of nonlinear systems. These static adaptive backstepping (SAB) algorithms enlarged the class of nonlinear systems controlled via a Lyapunov-based control law to uncertain systems transformable into the parametric strict feedback (PSF) form and the parametric pure feedback (PPF) form. In general, local stability is achieved for systems in the PPF form, whilst global stability is guaranteed for systems in the

566

Alan Zinober et al.

PSF form [11]. These two forms can be seen as special structural triangular forms of nonlinear systems which are adaptively input-output linearizable with the linearizing output. A more general algorithm has been developed by Rios-Bolivar et al [20]-[25], which allows one to design dynamical adaptive controllers following an input-output linearization procedure based upon the backstepping approach with tuning functions [16], and is applicable to both triangular and nontriangular uncertain nonlinear systems. An alternative algorithm for the synthesis of dynamical adaptive backstel> ping (DAB) controllers has been proposed by R/os-Bolivar [19] and RiosBolivar et al [23]. It has been shown that a dynamical adaptive backstel> ping control designed using the DAB algorithm overcomes the limitations of the static adaptive backstepping algorithm design of Kokotovi~ et al. This broadens the applicability of the baekstepping technique to systems with nontriangular forms which are not transformable into either the parametric pure-feedback (PPF) form or the parametric strict-feedback (PSF) form. This algorithm allows one to apply the backstepping approach to a higher class of uncertain nonlinear systems, satisfying observability and minimum phase conditions, which may be in either triangular or nontriangular canonical forms. Its application to the dynamic adaptive regulation of uncertain nonlinear chemical processes has been reported in [23]. A simple and popular robust approach to the deterministic control of uncertain systems is the sliding mode control technique, which is based upon the special behaviour of variable structure systems in the so-called sliding regime. Sliding mode control is usually synthesized by means of high-frequency discontinuous regulation signals, which result in bang-bang control inputs with noticeable chattering in the controlled system responses. The chattering, associated with underlying discontinuous control policies, has traditionally been regarded as a serious drawback for the efficient regulation of certain types of continuous systems. This feature is particularly important in the regulation of either mechanical or chemical processes because discontinuities cannot be allowed in the actuator behaviour, and abrupt changes of the regulated variables are usually not tolerated. Second-order sliding techniques and continuous sliding approximations obviate this problem. Dynamical SMC policies, based on state- and control input-dependent manifolds, provide outstanding features in comparison with traditional discontinuous feedback design schemes utilizing only state-dependent surfaces. Both continuous control input signals and substantially smoothed chatter-free trajectories have been shown to be some of the several advantageous properties exhibited by the use of such input-dependent sliding manifolds. Dynamical feedback controllers have greatly improved the applicability of discontinuous feedback strategies leading to asymptotic stabilization and tracking in nonlinear systems. Applications of this approach to mechanical and electromechanical systems have been reported by Sira-Ramirez and Delgado [29]. The feedback regulation of nonlinear chemical processes without uncertainty, via

New Developments in DAB Control

567

dynamical discontinuous compensators synthesized for systems in the Fliess generalized observability canonical form, has been addressed by Sira-Ramfrez and Llanes-Santiago [30]. The control input signals thus obtained are substantially smoothed in comparison with their corresponding static alternatives. In order to provide robustness in the presence of undesirable disturbances, a combined DAB-SMC design algorithm has been proposed by Rfos-Bolivar et al [27]. The method has also been extended to second-order sliding which has certain advantageous properties, namely the prevention of chattering, higher accuracy and a significant simplification of the control law. The adaptive sliding backstepping control of parametric semi-strict feedback (SSF) systems with disturbances has been studied by Koshkouei and Zinober [12,13]. The method ensures that the error state trajectories move on a sliding hyperplane. A sufficient condition for the existence of the sliding mode in Rios-Bolivar and Zinober [22,24] is not required. If a plant has unmatched uncertainty, the system may be stabilized via state feedback control [6]. Some techniques have been proposed for the case of plant containing unmatched uncertainty [7]. The plant may contain unmodelled terms and unmeasurable external disturbances, bounded by known functions. The classical backstepping method has been extended to this class of systems [12] to achieve the output tracking of a dynamical reference signal. Extensions to modular designs have been recently proposed. The availability of symbolic algebra software has allowed the development of useful toolboxes and packages for the systematic analysis and design of feedback control systems. For instance, some of the toolboxes and packages developed thus far include analysis and control design for affine and nonaffine systems [8,9], modelling and nonlinear control design [5], and analysis and design based on flatness [28]. They simplify the use of systematic and recursive control design methods enabling the design of stabilizing controllers to be carried out more efficiently. A MATLAB Toolbox of some backstepping algorithms was developed by Rios-Bolfvar and Zinober [22,24]. Recently a MAPLE package has been written to assist in the design of a variety number of backstepping algorithms. This paper is organized as follows: Sections 2 and 3 present the class of nonlinear system and outline the generalized backstepping algorithm. In Section 4 the sliding mode control extension is developed, and second-order sliding is presented in Section 5. Modular designs are discussed in Section 6 and SSF systems are studied in Section 7. Section 8 describes some features of the MAPLE package B A C K and gives an example of the application of the dynamical adaptive backstepping algorithm. A comparative example is given in Section 9.1 and an SSF example in Section 9.2. Conclusions are presented in Section 10.

568

2

Alan Zinober et al.

The Nonlinear System

Consider a single-input single-output nonlinear system with linearly parameterized uncertainty /

\

(1)

=

where x E ~'~ is the state; u, y E ~ the input and output respectively; and 9 = [~1,..., 8p] T is a vector of unknown parameters, f0, go and the columns of the matrices r ~P E ~n• are smooth vector fields in a neighbourhood R0 of the origin x = 0 with f0(0) = 0, g0(0) r 0; and h is a smooth scalar function also defined in R0. The control goal is for the output to track adaptively a bounded desired reference signal yr (t) with smooth and bounded derivatives up to n-th order. It is assumed that the system has relative degree p _< n. The procedure proposed here for systems in nontriangular form becomes equivalent to the design of Krstid et al [17] when the output corresponds to a linearizing function of the system, i.e. the relative degree is equal to the system order, p -- n.

3

Dynamical Adaptive Backstepping Control Design

The Dynamical Adaptive Backstepping (DAB) algorithm proposed in [20] is based upon a combination of dynamical input-output linearization and the adaptive backstepping algorithm with tuning functions [16]. Since it has been developed in a general context, without the use of canonical forms, its applicability to both triangular (PSF and PPF forms) and nontriangular systems is guaranteed, but it requires that the controlled plant be observable and m i n i m u m phase. The observability condition is required to guarantee the existence of a local nonlinear mapping which transforms the plant into a convenient form of the error system, as shown below. The need for the minimum phase property is to allow the applicability of the systematic algorithm presented here, and to guarantee stability of the closed-loop system. This general algorithm includes as a particular case the adaptive backstepping algorithm with tuning functions developed for systems in PSF and P P F forms [16]. The steps leading to the design of the dynamical adaptive compensator follow an input-output linearization procedure in which, at each step, a control dependent nonlinear mapping and a tuning function are constructed [19,26]. The parameter update law and the dynamical adaptive control law which stabilize the controlled plant, are designed at the final step. In order to characterize the class of nonlinear systems for which this procedure is applicable,

New Developments in DAB Control we set up a nonlinear m a p p i n g by considering the o u t p u t n - 1 time derivatives as follows

= ~

y(t)

569

a n d its first

fo(x) +~(x)0 + (go(~)+ ~(~)0)~

(~)

Due to the presence of the u n k n o w n p a r a m e t e r vector 0 we rewrite (2)

= ~

fo(x) +~(~)~+ (go(x)+ ~(~)#)u + ~ l ( e - #)

(3)

where ~ is an estimate of 0, and the vector wl is defined as

(4) In other words, (3) m a y be rewritten as

(5) with

fo(~) + ~(~)# + (go(z) + ~,(~)~)u

(6)

The second time derivative of the o u t p u t is

~(.)o)~]

_a (L~)ax[/o(x)+~(,)o + (go(x)+ +

at~

(7)

which can be rewritten as

= ~ ( ~ , ~, ~, ~) = ~ ( x , ~ , u,u) + ~ ( 0 - #)

(s)

with

o~

Io(x) + ~(x)# + (go(x) + e(x)#)u +

0~

(9)

570

Alan Zinober

et al.

and

~

O (f-)h) (~(x) + u#(x)) -

(10)

0~

By proceeding successively in this manner, we obtain the j-th time derivative of the output

y(J) = s

0, u,/L,...,

u (j-:))

(11)

= z:"{(x, o, u, u , . . . , u(~-~)) + ~oj(o -

O)

with

0 (z{-l)0~[10(~)+~(x)~+(g0(~)+e(~)0)~] +

O----~-

k=0

Ou(k)

(12)

and

wj -

,13,

Ox

The expression (11) is valid if the relative degree is one. The general expression for systems with well-defined relative degree, i.e. 1 _< p _< n, has the form

r

= c { ( x , 0, u, a , . . . , u(~-~))

(14)

with

-t

O~

~=o

Ou(k)

(i~)

In other words, the time derivatives of the output are obtained by the application of the following recursivety defined operator z. ~ = h(~:)

(16)

New Developments in DAB Control

571

J--P-l O (f~J--l) u(k+l ) -~

0~

Ou(h)

k=o

which also characterizes the control dependent nonlinear mapping

FL~I z = z(~,0,~,...,u(n-p-')) = / ~. /

(:7)

Lz~-'J A s s u m p t i o n 1 System (1) is locally observable, i.e. the mapping (17) sat-

isfies the rank condition rank 0~(.)

(18)

in a subspace R1 C Ro C ~'~. A s s u m p t i o n 2 System (I) is minimum phase in R1 C Ro C ~n. For observable minimum phase nonlinear systems of the form (1), the general problem of adaptively tracking a bounded desired reference signal Yr (t) with smooth and bounded derivatives can be solved through the DAB algorithm summarized as follows: DAB Algorithm Coordinate transformation z , = y - yr(t) = h(~ zk = s

(19)

- yr(t)

- y(~-')(t) +~_,(.),

2 < k <

with

Wk)= ah(k-~)Oh(k-') [ 9 a~ ~k + o---7--Io + ~

+ (ao + e~)~

k-p-~ o~(k_~) a~(k_~) + ~ ov~ vi+i+ 0---7--

i=l

~8]~(k-l)

Oak_l~

u~(z))

1 (20)

(21)

Alan Zinober et al.

572

k-o-t Oak-1

+ Zi = l

Oak-1

Oak_~

o~--i-v,+~+-N-'-~+0----7-

(22) k

~=

rE~[z~

(23)

i----1

P a r a m e t e r u p d a t e law 0 ---- 7". :

FWTz

= I"

w T ...

w

(24)

z

Dynamical adaptive compensator ~)1 :

Y2

~)2 ---- V3

(25) 9

v~-~ =

1 (0h(r-,)

[ 0~_,~

- z.-1

k 0v~_p + 0v~_p ]

+y!.)- \ (o~(~-~) o~._~ [ 0x + ax / f0+~ti Oh ("-l)

+(go + ~O)vl

Oan-i

at

at

"-~ (o~(~-~) o~._1) - ~i = 2 o~ z,r~ O0 +

-

,--~

~

~

~

+ ---~,)v,+~

- c.z.

with vl = u, the ci's constant design parameters and F = /-T :> 0 the adaptation gain matrix. The control u is obtained implicitly as the solution of the nonlinear time-varying differential equation (25). The following steps lead to the general DAB algorithm summarized above: S t e p 1. Define the output tracking error as zl = u - y.(t)

= h(~)

- yr(t)

(26)

New Developments in DAB Control

573

whose time derivative is given by ~1 = h(1)(x, 0) - ~,(t) = 0--; fo + ~ 0 + (eo + ~ 0 ) u

- ~r(t)

(27)

If the relative degree p with respect to u is greater than one,

oA (go(x) + ~(x)O) = 0 Om

(28)

For the sake of generality, it is assumed here that the relative degree p is greater than one. Nevertheless, this algorithm is also applicable to systems with p = 1. By adding to and subtracting from the actual value of the parameters 8 their estimated values 8, (27) can be rewritten as

~1 = h(1)(x, 8) - y,(t) + w1(8

(29)

-#)

with

~(~)(~,~)

0h

: ~(So(x) + ~(~)0)

(30)

Oh wl = ~xxr

(31)

Consider the quadratic Lyapunov function

vl = ~z; + 89

r-1(8_0)

(32)

where F = /-T > 0 is a matrix of adaptation gains. The time derivative of V, is

= 21 (h(')(~, ~) - ~.(t)) + (8 One can achieve V1 = relation

o)~r-~(-D + r~Tz,)

(33)

-clz~ with cl a positive scalar design constant if the

h(1) (x, 0) - Y, (t) = - e l zl

(34)

is satisfied exactly and the parameter tuning function 8 "- 7"1

:

r~dTZ 1

(35)

The expression (34) represents a desired algebraic relation (including an embedded virtual control) for which effective stabilization of the output tracking error would be possible in combination with the estimation update law (35).

Alan Zinober et al.

574

However, since (34) is not valid in practice and vl is not considered as an update law but rather as the first tuning function, the deviation is defined to be the second error variable, i.e. z: = s

(36)

0) - u~(t) + ~

with

al = clzz

(37)

The closed-loop form is

(38)

il = - c l z l + z~ + w 1 ( 6 - O) and V1 ---- - c 1 z2 -+. Zl Z2 -~ (0 -

0)TF-I(--b

--]- T1).

(39)

By proceeding in this way successively, one obtains the following j-th generic step which characterizes the steps prior to the explicit appearance of the control input in the transformed dynamical system. S t e p j (2_< j <_ p - 1)

~

+~(0-~)+\ aO + ~ ]

Oaj_l

(40)

with

]~(J)(z,O,t)-

O]~(J-~) Ox (f~

wj

~

~x

§ T )

O]~(.i-~) O]~(.i-~) 00 rj + Ot gt(x)

(41) (42)

and rj the corresponding tuning function defined at this step. By augmenting the Lyapunov function

vj = v~_~ + ~

= ~ls

0) i----1

and the time derivative is

j-1

VJ-'~ -- E i=1

ciZ~ "~

(O--o)T/~-I(--[~.9ff Tj_I

.~ I'~OjTZj)

(43)

New Developments in DAB Control

575

(a],(J-') o~_,) (o_.~) O~ + OO

+ z~ \

j-1

+

0]~(i_1)

+

j-1

(o -

+ z~ z~_l + ho)(x,0,0 - y!~)(0 + + -Oc~j-lv --~ j + ~(fo

0c~j_ i

o--7-

+ ~0)]

(44)

The parameter estimate error (0 - t}) can be eliminated from Va- by choosing the update law 0 = r j = r j _ ~ + rwjz~

(45)

Noting that :

"

k

O-- rj-1 = O-- rj + vj -- rj-1 = O-- vj +_PwTzj,

(46)

one can rewrite @ as j-1

=

- F_c,4

+

(o - O ) T r - ' ( - O

+

,-5)

iml

"t- L

Zi

+~

C~O

"~-

-- "l'j)

z ' - - - N - + ,=3

+ hu)(~, o, t) - vr "l-

~,-~)r~2

+ ~(fo

+ ~)

Oaj-1 Oaj-1 O0 7"j --~ ~ --~ Zj_ 1

(47)

One can achieve ~ = -Y'~-i=lJ ciz~, with the ci's being positive scalar design constants, if rj is the update law and the relation

zi +

O0

+Ez, i=3

(A + CO) +

O0 ]

rq+

Oo~j71 Oat- 1 O0 '-j + ---gi-- + zj-1 = -c~zj

(48)

Alan Zinober et al.

576

is satisfied. Since (48) is not valid at the outset, its deviation is taken as the (j + 1)-th error variable

zj+~ = i~(j) (x, O, t) - y(~J)(t) + aj (x, O, t)

(49)

with

(

j/=~21 0]~(i_ 1)

j-1

O0~i_1 ~ OOlj

_

+

O0~j_ i

1

(50)

(fo(x) + ~(~)0) + --N-- + cjz~

Then the closed-loop form for ;?j is

~=-zs-l-cjz~+zj+~+~j(o-8)+\

oo + oo ]

- (~z,~+~-~,ah('-l) J-~za~'-l~r~T,:oO 3 )

(51)

and J = - ~c,z i---1

~, + zjz~+l + (o - O F r - ~ ( - o

+ .j) +

(52) Now the steps containing the control input and its derivatives are summarized in the following generic step. Stepk.

(p
()~k-1

C~k-I

~ = ,~(~)(~,o, ~,,..., ~(~-'),0 - ~(~)(~)+ - - - 5 / - + ~ - ~ k-p

C%'k-1 U(i)+ "~k(e--O) + -W-~a~-~[So + ~'o + (go + ~o)u] + V" ~.. au(,-~) i:1

+ (~a---7- + -~ ~ ]

(b-,~)

(53)

with

fi(k)(~,~,~,...,u(~-p),t)_

o,o-~) oti

~-k +

a~(~-~) [So + ~,o + (go + ~0)u] o---~

New Developments in DAB Control

+ k-p ~ Oh(~-')~(~) + -Oh(k_l) Ou(i-1)

i=l

,~k = \

o~,

577

(54)

Ot

+ --5g-~ ) (e + ~,.,)

(55)

and rk the tuning function defined at this step. By augmenting the Lyapunov function k

v~ = v~_, + 89d = -~ ~.= d +

(o - o) T r-'(o - 0)

(56)

and its time derivative is k--1

v~ = - E c,4 + ~o- o T F - , ( -

o + ,-~_, + ,~4z~)

i----1

(o/,/~-,/ o~,_,) (0 ,~) +

z,

o0

,=3 z , - b - U )

( b - .~_,) + k-p

zk zk-i + h(k/_ y!k) +

Oak-1

O0 Tk + ~

Oak-i

u(i)

Ou(~-l-------5'

i=1

o~_, + -gZo~_, [5 + ~0 + (eo + ~0)u]] +-N-

(57)

The parameter estimate error (0 - 0) can be eliminated from % by choosing the update law

(58)

0 = rk = r k - i + Y ~ z k .

However 7-k will instead be used as a new tuning function. Thus, noting that -

"

0 - rk-(=

k

0 - rk + r~ - r ~ - i = 0 - ~-~ + F w ~ z ~ ,

I)k can be rewritten as k-1

i=1

f;-,

oJ,<,-,)

+ 2__.z,-- N [,~=~ O

'

oo,_,]

+ Z z, (b- .,) i=3 aO ,/

(59)

578

Alan Zinober et al. [ (k-/=~ Oh(i_l)

k-i ~ - ~Cgai_i~ 1

z,

~(k) y!k) -

0C~k- 1

)0c~ki-I i u(i) + + k-eK.. z.., 0u(

TO~k-i+ z k - , ] ]

i=l If vk is the update law and the relation k-,

0s

k-1

0ai-l~

Zk-13ff (i~--~2z` 0----~--~ E Zi T Oak_l

Oak-1

[5 +

00/k+ k-p~_. Oak-1 uO ) + _ _ i=1 OU(i-1)

Ot

+

/1r

(60)

"3Ct~(k) -- yCk)

(eo +

1 _ --c~z~

(61)

is satisfied, then Vk = - ~ik=t ciz~ with the ci's being positive scalar design constants. However, since (61) is not valid at the outset, its deviation is taken as the (k + 1)-th error variable zk+~ = ],(~)(~, ~, u , . . . , ,,(k-,), t) - u~(k)

-

+,~k (~, #, u , . . . , u(h-o), t (62)

with

Olk=Zk-l-~- \i--2 ai - -E X I T )"~-/ n-i3( ' o T o 0 k - p Oak_~ u(i) + O~k-1 [/0 + E 0u(i-1) ~ L +r i=1 Oak-i Oak-i O0 rk + ~ + ekzk

+ (go +

~)u] + (63)

We obtain the closed-loop form

Oh(k-i) O~k-l~(e-rk) -

k-i (i_~

O~(i_i) k-i v-- 0c~i-1 z'T + 2 ~ z ` ~

) Fw T

i=3 and

k (4 = - ~

i=l

c , d + z~z~+l + (o - O ) ~ ' r - l ( - b +

~-~)

(64)

New Developments in DAB Control (~-~ +

zi

0 ]~(i- 1) 00

aa,-l~ (~_ ~) +~z'T]

579

(65)

i=3

i=2

Step n. The design of both the actual update law and the dynamical adaptive output tracking controller is completed at this final step. Using the definition

(62) aO/. _

~,=i,(")(~,~,~,...,~("-~),t)-u!")(t)+

]~

+

1

o--T-+

0C~n

-- 1

a~

~"

Io + ~0 + (ao + ~),, + ~ 0u(i-z) a~,_~ u(') + ~,(o - e) i=1 +\

(oh("-~) o,,._~) (~_ ~.1 ~ + a~

(66)

with ]g")(x,

O, u , . . . , u("-P), t) -

0~(--1)

a~

T. +

0~(--1)

a----~--[So+ ~ + (go+

"-p 0~(.-I)

+ Z

i=1

cgu( i - 1 )

("-1)

(Mn

(o~

a~(--1)

u(i) + ~

~),,] (67)

Ot

+ 0~,_i~ a~ ] (~ +~u)

(68)

and r n the tuning function defined at this final step. Augmenting the Lyapunov function

(69) and its time derivative is n--1

vo = - E ~ , z ~ + (o-~)~ F - ~ ( - ~ + ~o_~ + i ~ : z ~ i=1

+z.\ -~ + .-i

0~(i_i)

oO ] .-i

Oai-1'~ ._p

_O_a nu _

Oan - 1 i=l

1

(i)

580

Alan Zinober et al. 1

[So + ~o + (go + ~~

+ - ~ + ~

(70)

At this final step one can eliminate the parameter estimate error ( 0 - 0) from V. with the update law 2

(71)

0 = rn =- rn-1 + F w T z n = F W T z

where the regressor matrix W T is composed of the regressor vectors as tbllows

w~=

[~ ~ ... ~].

(72)

Then, noting that ~-

r.-i

(73)

= O" - r . + rn - vn-1 = O" - v . + F W nT Z n ,

Vn can be rewritten as

i=1

{~n--~

0h( i-1 )

~

+ ~/_, z , - - + L z , - ~ l i=2

O0

Oo~i-l~

i=3

(b-~-,~)

/

" - P 0 a . - 1 u(i) + O a . - 1

+ E

/=1

Ou(i-l)

0----~ + z . - 1

]

(74)

In order to achieve

9 = 9. = - ~ c , z ] < o

(75)

i---1

one must make the bracketed term multiplying z. equal to - c . z n , i.e.

n-i

O~(i_i)

0c~._1

+ -N-

.-i

O~i-i'~

"-- P Oc~n_l u(i) + _O0~n_ _ l

+ }2 ou(~-l) i=l

o~ "" =

-c,~z.

(76)

New Developments in D A B Control

581

The control function u can be obtained implicitly as the solution of the nonlinear time-varying differential equation defined by (76). Note that the control law (76) can be rewritten in the form of the d y n a m i c a l controller (25) by replacing the control input u and its derivatives u, f i , . . , by the extended state variables vl, v~, v3,. 99 respectively.

3.1

Analysis of Stability and Convergence

The overall closed-loop error s y s t e m has the form s

= Azz + W(O

(77)

O)

-

k

(78)

0 .-~ I N W T z

where A~ has the following form 9- c l

0

1

-1 0

...

-c2 1 + 0~,3 . . . - 1 - 02,3 --C3 99.

A z ~-

:

0

--02,n--i

--03,n--1

0

--t.O2,n

-Oa,n

0

0

02,n-1

02,n

03,n- 1

03,n

"..

:

:

...

--Cn--I

1 -~- On--l,n --Cn

... -1

-

On-l,n

(79)

with

o/,(,-,) o,,j =

o---7 +

oo ]

(80)

The off-diagonal elements of the matrix A z have a skew-symmetric form. The system (77)-(78) is stable, since the relation

A~+A T=-2

ci 0 . . . 0 0 c2...0 : : -. :

(81)

0 0 ...c,~ yields

r = - ~ c,d

(82)

i=1

with the quadratic Lyapunov function V=~z

1 T

z

(83)

Alan Zinober et hi.

582

The stability of the equilibrium (z, ~) -- (0, 0), with /~ -- B - ~, has been established [19,20,27]. Since the time derivative of the Lyapunov function V along the solutions of (77)-(78) is nonpositive, uniform stability of the equilibrium (z, 0) = (0, 0) is guaranteed. Moreover, by virtue of the LaSalleYoshizawa Theorem, it follows further that, as t -+ oo,

c,z~ = 0

lim 'J = - lim ~ t -.~. ~

t--+~

(84)

i---1

This proves, in particular, that z(t), ~(t) --> 0 as t -+ co and, consequently lim [y(t) - y~(t)] = 0

(85)

t--+ oo

Then, from (77)-(78), B = 0 and

(86)

w ( o - ~) = o

From (72) and the definitions of the regressor vectors given in (55), it is seen that the components of the regressor matrix W depend in general upon

t,x,z,O,u,...,u("-p-l) W = P(t, z, 0, u , . . . , u(n-a-O)F(x, u) Oh

Oh(l)Ox Oh1 =

cox + -~x 0/*(n-l)

F(x, u)

(87)

ha,,-1

Ox

where

F(x, u) = ~(x) + ~(~)u

(88)

Noting that the matrix P(.) in (87) is the partial derivative of the observability mapping with respect to x, namely P(t,z,~,~,

~,(.-.-i)) = ""

r0~(.)] t~J

(89) ==~-,(~,. ...... r

which by Assumption 1 is nonsingular, (86) and (87) imply f ( x , u ) ( B - 0) = 0

(90)

on the manifold characterized by z = 0 and ~ -- 0. Moreover, if l i n ~ - ~ y ! 0 ( t ) = 0, i = l , . . . , n , holds, which is a common requirement of

New Developments in DAB Control

583

the tracking problem, the asymptotic stability of the equilibrium (x, 0, u) = (X, 0, U) is guaranteed from the definition of zi's in (62) if, in addition, rank[F(x, u)](~,u)=(x,v ) = p

(91)

The above facts prove the following theorem: T h e o r e m 1 The closed-loop adaptive system consisting of the plant with the dynamical controller defined by (76,) and the update law (71), has a locally uniformly stable equilibrium at (z,O) = (0, O) and limt~o~ z(t) = O, which means that asymptotic tracking is achieved, i.e. lim [y(t) - yr(t)] = 0

(92)

t--+ r

Moreover, if limt--,~ y(i) = 0, i = 1 , . . . , n and ranklE(X, U)] = p, the equilibrium (x, O, u) = (Z, O, U) is asymptotically stable. Theorem 1 guarantees local asymptotic tracking in general. Nevertheless global asymptotic tracking can be achieved if Assumptions 1 and 2 are satisfied globally.

4

Dynamical

Adaptive

Sliding Mode

Control

A particularly important aspect in regulation and tracking tasks for uncertain systems is robustness in the face of disturbances and unmodelled dynamics. In [21]-[27] solutions to this problem have been proposed, which are based upon the combination of the adaptive input-output linearization algorithm above and sliding mode control (SMC). It allows one to design dynamical adaptive sliding mode tracking controllers. The resulting control law achieves robust asymptotic stability with considerably reduced chattering. To provide robustness, the DAB algorithm can be modified for the design of dynamical adaptive output tracking controllers (see [26] for details). The modification is carried out at the final step of the algorithm by incorporating the following sliding surface defined in terms of the error coordinates cr = klZl + . . . + kn_lZn_l + Zn = 0 where the scalar coefficients ki > 0, i = 1 , . . . , n manner that the polynomial p(s) = kt + }~s + . . . + kn_ls n-~ + s n-1

(93) 1, are chosen in such a

(94)

584

Alan Zinober et al.

in the complex variable s is Hurwitz. Additionally, the Lyapunov function is modified as follows n-1

v= ~1 z T

+ ~1 2 + 1"z(~176

(95)

/=1

By differentiating (95) and substituting (96)

Zn = ~r - (kizi + . . . + k n - l Z n - 1 )

we can obtain the update law n-1

i=1 n--1

n--i

i=1

i=i

(97) and the dynamical adaptive sliding mode output tracking controller

z._l + h(-/(.) - ~-/(t)

( ~ z, 0h(i-i)

+ ~.(.) +

rL--1

,=1 \

oO + oo ](~n-")

n--1

+~

k i ( - z i - 1 - ciz~ +

zi+i)

i=1

= - ~ ( a + flsign(cr))

(98)

with tr > 0, fl > 0 and c~n defined by 4.(.)

_

o~,~_1 a~,~_l [ ] a---E-r,,+ ~ fo + ~ + (go + ~'~)~ n--p--1

+

~

i=1

~Oln_l au0-i)

U ( i ) Jr -(~ ( -l ' n _ 1

0t

(99)

New Developments in DAB Control

585

This dynamical adaptive sliding mode control yields rt,--1

i=1

= -- [ Z l Z 2 . . . Z,,_I] Q [ Z l Z 2 . . . z n _ l ] r - ,~,2 - , ~ l ~ l

(100) with a simpler

O =

.

c

00 l

.

Lk~ k~

(101)

k._~+c~_~

which is a positive definite matrix. This guarantees asymptotic stability of the system. Therefore the sufficient condition for the existence of the sliding mode in [26] has been removed by using (96).

5

Dynamical Adaptive Backstepping Second-Order Sliding Mode Control

The DAB design technique has been extended by incorporating a secondorder sliding approach [34] and is called the Dynamical Adaptive Backstepping Second-Order SMC (DAB-SOSMC) algorithm. Some improvements have been made and this new algorithm is presented here. The main advantages of the DAB-SOSMC algorithm are the prevention of chattering, higher accuracy and a significant simplification of the control law. The DAB algorithm is extended to incorporate second-order sliding in the following way: Suppose that we have just completed step n - 1 of the DAB algorithm, i.e. we have obtained the closed-loop value of zn- z, and defined a new error variable zn. We halt the DAB design at this step and define a new final step that incorporates the second-order sliding mode: S t e p n. After Step n - 1 of the Dynamical Adaptive Backstepping algorithm the transformed system is Zl :

- - C i Z l "b z2

q- Wl(~ -- ~)

~=-zz-c2z2+z3+~(O-~)+~

-

Alan Zinober et al.

586

& = -zk-1

c~zk + z~+~ +~k(O - b)

-

(~

- \i=~ zi

Z n - - 1 "= - - Z n - 2

-- Cn--lZn-1

+\

(lo2)

"Jr" Z n Jr ~ d n - l ( O -- 0)

+a~]

~

-

oi,(,-,) "-' a~,_, ~ r~# a~ + ~= zi a~ ]

(2,= z ' - - d - +

Ez'--~

)

with n--1

(103) i=1

The Lyapunov function, V n - 1 is defined as n--1

v._, = ~ ~ 4 +

(o - O)Tr-'(o - O)

(lo4)

i=1

and its time derivative as n--1 i=1 o-'

§

\i=~

zi..~

Ob

+ Ez,-~-) ,:~

+(o- o)-~-' (-~ +,-,,_,)

(105)

where the final error coordinate, z,~, is defined as Zn ~--- h ( n - - t ) ( X , O , t t , . - . , ~ t ( n - - p - - 1 ) , t )

-- y r( n - l )

+~,_~(=, b, ~,..., uI--p-,), 0

(lo6)

We now define our sliding surface to be a c o m b i n a t i o n of the final two error coordinates Yl = zn + C Z n - l = O ,

c > O

(107)

New Developments in DAB Control

587

and extend the Lyapunov function as follows (108) Differentiation gives n--1

v.

= - ~

c,z~ + z , , _ l z n + (e -

0)Tr-I(-~ + ~._1)

i=l

(109)

+ yi~)i

By taking the derivative of k'n along the sliding manifold described by .the equation Yi -- 0, i.e. zn = - c z n - 1 , and defining the update law to be 0 -7"n-- 1 j w e have n--1

r

:

- ~,

c,z:- czL1

(110)

i=1

If we can steer Yl to zero in finite time, the overall closed-loop error system has the form ---- Az.~ --I- I~(9 - ~)

(111)

2

(112)

8 = FITvT5

where 5 : [ z i , . . . ,

Zn-1]T, and

-ci 1 -1 -c2 0 -i-~,3 Az --

:

0

. ..

1 --[- ~;2,3 - 9 9

:

0 ~;2,n-1

-ca

...

~3,n-i

:

"..

:

(113)

0

-~,n-2

-q3,n-2 ... 1 + qn-2,n-i

0

--~2,n--1

--~3,n-i

...

.Cn--i

--

C

with

(0,~ (i-i)

Oai-i

]

Fw T

(114)

Alan Zinober et al.

588

The off-diagonal entries of A, yield a skew-symmetric matrix. The form of Az is important for the stability of the system since the relation

s

+~

= -2

cl

0 ...

0

0

c2 ...

0

.

.

.

"

] (115)

.

0 0 ...c._l+c yields (110). By a proof analogous to that used in [20,27] for the DAB system, it can be shown that the stability of the overall system is guaranteed and that asymptotic tracking is achieved. 5.1

Second-Order Subsystem

The design of the control law and its implementation are now discussed. The sliding surface Y l = Zn "}- C Z n - 1 ~-- 0

results in the following second-order subsystem u~(t) = u~(t) 9~(t) = kl(~, 0, u , . . . , u(n-p)) _ y!n+~) _ cu(,)

(116)

+ k s ( z , O, u . . . . , u(n-P))O + O~ k3(z, O, u,. . ., u("-P))O

+k4(x, t~, u , . . . , u(~-P))u (n-~ Setting X = (x,O,u,.'',U(n-P),Yr,'..,Y!"+I)), rewritten as

y,(t) = u~(t) ~l~(t) = H(x) + flo(x)w(t)

the above system can be

(117)

with y2(t) not determinable due to the presence of uncertainties, and w(t) = U(n--P+I)

For the sake of simplicity, we make the following assumptions on H(X) and 8o = ~ o ( x )

Assumption 3 ]H(x)] < H,~ ,

Hm > 0

(118)

New Developments in DAB Control

589

Assumption 4

O
(119)

Bt,Be>0

The control problem now becomes that of finding a control law w ( t ) such that Yl (t), Y2(t) are steered to zero in a finite time in spite of the uncertainties. This problem has been solved in [4] where it has been shown that the control w ( t ) can be implemented as a bang-bang control, switching between two extreme values - - W u a ~ : , + W M a ~ . Alternatively twisting second-order sliding control could be used. The classical switching logic for a double integrator ( H ( x ) = 0, B1 = Be = 1)is

--WMa. w(t)

I y1(t) - - ~ _

=

"J-WMax

w,,o. }

1y2(t)ly~(t)l y=(t)ly=(t)lNyl(t) u wMo.

<

0}

y l ( t ) < - - l2 y~(t)ly2(t)l w,~~ "1, j IwI Yl(t) = - 2

(12o)

l y2(t)ly2(t)l A y l ( t ) > O} WM..

This classical switching logic is dependent on both Yl (t) and y2(t). However, it is possible to express the logic in terms of yl (t) only which, by assumption, is available for measurement. A suitable algorithm [4] is Algorithm 1 1. Apply the DAB algorithm until the end of Step n - l , computing z l , . . . , z, (i.e. ~ 0 ) , . . . , h ( - - l ) , a l , . . , a , _ 0 ; 2. Select the sliding surface as Yl = z , + c z n - t = 0; 3. To the system Yl =Y2 y~ = H ( x ) +

,~o(x)~(O

where w ( t ) = u ( " - p + I ) , X = [x,O, u, . . ., u( " - p ) , y r , . . ., y(n+O], H(.),fl0(') are suitable functions to be bounded above, apply the sub-algorithm:

(a/Set

(0, 11n (o,

(b) Set YMa= = yl(0); Repeat, for any t > 0, the following steps: i. If [Yt(t) - 89 -- Yt(t)] > 0 then set a -- a* else set o~---1. ii. If y l ( t ) is an extremal value then set YMa~: = yl(t); iii. Apply the control law 1 w ( t ) = - - a W M = = s i g n { y l (t) -- 5 y M , = } (121) until the end of the control time interval.

590

6

Alan Zinober et al.

Modular Design Approach

Modular designs are inspired by traditional parameter estimation techniques. The core idea of the modular approach is the separation of the design of the control law from that of the parameter update law. This separation of the design leads to two distinct modules; a controller module and an identifier module; that are independent of each other. This independence or modularity of the controller-identifier pair enables the combination of any stabilising controller with any identifier. Krsti6 et al [17] have proposed a modular design based upon their traditional backstepping design technique that leads to the design of a controller module suitable for use with any identifier. This combined modular backstepping design differs from the traditional bazkstepping design by its treatment of the parameter estimate error ~ and its derivative/~. In traditional backstel>ping designs the effects of ~ and 0 in the z-subsystem are cancelled by the controller. In the modular backstepping design these terms are considered to be disturbance inputs and their destabilizing effects compensated by the introduction of nonlinear damping functions. The modular backstepping design exhibits significant advantages over that of the traditional backstepping design. It removes the built-in interaction between the controller and identifier, which can lead to complexity in the control law, thus giving rise to a comparatively simpler controller. This separation of the design allows use of any identifier which can guarantee the boundedness of the disturbance inputs/~ and 8; a significant advantage over traditional backstepping designs which are restricted to Lyapunov-based update laws. The design also achieves improved transient performance arising from the inclusion of nonlinear damping functions. It is these advantages over the traditional adaptive bazkstepping design that provide the motivation for this section. The design by Krsti~ et al [17] is applicable only to systems in triangular (PSF, PPF) form. Here we extend the modular design approach to systems in nontriangular form and propose a design based upon the DAB algorithm (see Section 3).

6.1

Controller Module

Consider the single-input single-output observable minimum phase nonlinear system of the form (1) with relative degree p < n. The control goal is for the output to track adaptively a bounded reference signal Yr (t) with smooth and bounded derivatives. We develop a Dynamical Adaptive Modular Backstepping (DAMB) algorithm for the design of a controller that can be combined with any identifier as follows:

New Developments in DAB Control

591

S t e p 1. Define the output tracking error as (122)

Zl = h(x) - y~ (t) whose time derivative is given by

Oh

~ = ~ [fo + ~ e + (g0 + ~ ) ~ ] - y~(~)

(123)

If the relative degree p with respect to u is greater than one,

Oh O--x (go(x) + ~(x)6) = 0

(124)

For the sake of generality, it is assumed here that the relative degree p is greater than one. Nevertheless, this algorithm is also applicable to systems with p = 1. By adding to and subtracting from the actual value of the p a r a m eters O their estimated values 8, the first error coordinate can be rewritten as

~I = ~(1)(~, ~) _ yr(t) + ~ I ( 0

- ~)

(125)

where

Oh ]z(1)(x,~) = ~xx (f0(x) + k~(x)~) Oh wl = ~xx~(x)

(126) (127)

Consider the following quadratic Lyapunov function V1 = l z ~

(128)

T h e time derivative of Vt is = zl (],(1)(x, ~) - yr (t) + ~ (e - 0))

(129)

We require that V1 be nonpositive, so we set

h(~)(~,O) -

y~(t)

=

-clz~

-

~1~12zl

(130)

where gl[w1[2zl is a damping term introduced to c o m p e n s a t e for the effect of the p a r a m e t e r estimate error ~. The expression (130) represents a desired relation which, if satisfied, would enable us to achieve stabilization of the output tracking error. However, since our relation is not valid in practice, we define the second error variable as z2 = ~t(1)(x, ~) - y~(t) + ~

(131)

Alan Zinober et al.

592 where

ai = CiZi Jr t~ilwil2zi

(132)

This gives the closed-loop form of zi as

~ = - c l z l - ,~ll~l=z~ + z2 + ~ g

(133)

f4 = - c ~ z ~ + z~z2 - ~11~12z~ + ZlWl#

(134)

and

If z2 = 0, the derivative of the Lyapunov function would have the form

71 = -ClZ~ - ,~11~112z~+ z ~ 1 0

(135)

By completing squares we obtain =

--1 -c~z~ - ~ wl zl - 12 ~ # I2 + 4--47,1#12

< -clz~

+

a--~-~I#1=

(136)

giving zl as bounded whenever ~ is bounded.

Step 2.

~2=]~(2)(x,~,t)-ijr(t)+

(fo + ~P#)+ - ~ + w2~+ ~2~

(137)

with

0]~(1) a]~(1) ~(2)(~,b,t) = -~-(/o + ~o) + a--7

{ a],(I) a~11

(138)

w~

= \ a.

+ a, ] m(~)

(139)

~2

= ((9]~0) \ a~

+ -0al ~ - ]~

(140)

We augment the Lyapunov function

V2=V,+~z~=~(z~+z~)

(141)

New Developments in DAB Control

593

whose time derivative is

+~

z~ +/,/~/_ ~(t) + ~CqO~l(fo + e~) + - -

Ot

(142) We now have not only a ~-dep.endent disturbance term w20, but also a 9dependent disturbance term ~20. We introduce the nonlinear damping terms -g~[w212z~ and -r/~ ]~21~z~ to counteract the effects of both the disturbance k

terms ~ and 9, and define the relation zi

0al (gt = - c ~ z ~ - ~ b : [ ~ z ~ - ~ [~2l~z~

-f- h (2) -- ijr(t) -[- E - ~ - ( f o q-ff'~) "[-

(143)

This expression is not valid at the outset and represents only a desired relation. For this reason we define its deviation to be our next error variable. (144) where

~ =z~+

~zi ( f o + g , ~ ) + - - - ~ +Oca2 zi2 + , ~ 2 [ ~ 2 ] U z u + ~ 2 [ S ~ 1 2 z 2

(145)

giving the closed-loop form ~2 = - z x - c~z~ + zz - ~2[~212z2 - ~2]~2[~z~ + w~0 + ~2~

(146)

and

(147)

+ z 2 z 3 + z1,~10 + z 2 ~ 2 0 + z ~ 2 a

If z3 = 0, the derivative of the Lyapunov function would give v2 = - c l z ?

- ~2z~ - ~ l l ~ l l ~ z ~ - ~ l ~ P 2 z ~

+z1~10 + z ~ f i + z ~ By completing the squares, we obtain

- .21r

~z~ (148)

594

Alan Zinober

e t al.

1

(1 1)

< - c l d - c2z~ +

~

+ ~

I#1= +

I~l2

(149)

which !mplies that zl and z2 are bounded whenever the disturbance inputs

and 0 are bounded. Step j.

(3 _< j _< p - 1) cOo~j _

~j = ],(~)(x, ~,t) - ~!5)(t) +

(/o + eo) + - - ~

i

+ ~5~+~

(15o/

with

~u)(x, ~i, t) - ahu-1) cOx (fo -Fm#) -F ahu-1) cO-----t---

(151)

( cO]~(5-1) cO~5_11

~5=\

cOx + cOx ]~,(x)

(152)

( cOh(J-~) cO~5-~~

(153)

We augment the Lyapunov function 1 j Vj = Vj_, + l z ] = ~ E z ~

(154)

i----1

giving j--1

j-1

j-1

j--1

i:1

i---1

i:2

i~--i

+~,~,O+zj i:2

[

zs_~ + ~(5) - y~(5)(~) +

(fo+~O)

cO(~ 5 - 1 +T +~5#+ ~50]

(155)

To ensure the Lyapunov derivative is nonpositive, we choose

zs-~ +]~(5)(~,~,r162

9

+ a__~_(fo+e~)+ = -c5z5 - ,r

cOaS- 1

cOr

- ~51~51=z5

(156)

However, since (156) is not valid from the outset, we set its difference to be the error variable

zs+~ = ],(J)(x, ~,~) - ~(5)(~) + ~j

(157)

New Developments in DAB Control

595

where

Oaj _ 1

Oa.i- 1

+tcjlwjl2zj + rSl~jl2zj

(158)

which gives the closed-loop form

~j = -zj-1-cjzj-,r

+ zj+l +~jO+~jO

(159)

and then J =

-

J

J

E c44 E ~,l~,l~ E,,mJ ~z~ -

i=1

-

4=1

i----2

J

J

+zjzj+, + E z,w,O + E zi~ib 4=1

(160)

4=2

which, if zj+l = 0, can be interpreted as

4=1

This ensures

4=1

4=2

zl,..., zj are bounded whenever 0 and 0 are bounded.

The following steps deal with the control input and its derivatives: Stepk. (p< k
k-p Oak-1 u(i) +

~ = h(~)(x,~,,,..., u(~-~),t) - gk)(t) + ~_, ou(4-1)

aO~k_ 1

ot

4=1

Cgak_, [fo + ~O + (gO+ ~O)u] + wkO+ ~k'O +---g-2-~

(162)

with ]~(k)(x, ~, u , . . . , u (k-p), t)

-

-a}'(k-') Ox

[So + ~0 + (go + ~)u] +

k - p O}gk_1) 4=1

wk =

au(4-1) u

(i)

Oh(k-l)

+

(O& k-l) + "--~---x O~k-l~ a ~ )

(163)

0----7--

(gr + ~u)

(164) (165)

596

Alan Zinober

et al.

We again augment the Lyapunov function k

1

v~ :

v~_, + ~z~ = ~ E z~

(166)

i----1

which has the time derivative k-1

k-1

k-1

k-1

v~ = - E ~,4 - E ~,I~,l ~4 + E ~,ml ~4 + E ~,~,~ i=1

i=2

i=l

i=1

k-1 [ k-p O0lk_l _ u + F_, z,~,~ + z~ z~_~ + ],(~)- y?)(t)+ F_, _o,,(,-~) i:2

(i)

i:1

+---ST-~~ [So + ~O + (go + ~O)u] + - - o r

+~0 + ~ ]

(167)

Set --CkZk -- '~kl~kl2Zk -- ~kl~kl~zk : Zk-~ + ]~(k) _ y<))

k-p

Oak-1 u(i) Oak-1 +--~-~~ [So + ~0 + (eo + ~0)~] + 2_, o~-~-,l + --or 9

x--,

(168)

i----I

Since (168) is not valid at the outset, its difference is taken to be the (k+ 1)-th error variable zk+l = ]~(~)(z,O,u,...,u(k-P),t)

- y(k) + o ~ ( x , O , u , . . . , u ( k - P ) , t )

(169)

where ak = zk-1 +

,,~~

[So + ~,o + (go + ~0),,] +

ouI,-,)~ ,~I,) + i=I

~Otk - 1

o----V- + ckzk + '~kl~kl~z~ + ,lkl~l~zk

(170)

This gives the closed-loop form ~

= -zk-1

- c~z~ - ~ l w k l 2 z k

--

~kl~kl2zk + zk+l + wh~+ ~k~

(171)

and k

k

k

v~ : - F~ c,4 - E ~,1~,1~4 - E ~,1~,1~4 i=1

i=1

i=2 k

k

i=l

i=2

(172)

New Developments in DAB Control

597

which, if zk+, = 0, can be interpreted as k

k

k

v, _< E.,z,' + E

+E

-

i--1

(173)

i=2

i=1 ~ i

ensuring z l , . . . , Zk are bounded whenever 0 and 0 are bounded. Step n. In this final step the design of the dynamical adaptive output tracking controller is completed.

~. =/,(')(~,o,u,...,u('-,),O-yp)(t) +

~OLn - -

i

a---WOa,,_: u(i)

+---~'--x Oc~,~_i [f0 + g~0 + (g0 + ~0)u] + ~

--8u(i_:)

i=1

+~j+ :j

(174)

where

i,(")(,,,~,~,...,d"-,),t)- a~(~-:~ a ~ [So + ~o + (go +,.o)~] n-P Oh(n-:) u(O

+ ~i = l a~(,-:~ wn= \

~xx

+

ah("-U

+

(175)

o----7-

Ox ) (~" +pu)

(176)

Augmenting the Lyapunov function we obtain 1 T V. = Y.-I + 89 = :z z

(178)

whose time clerivative is n--I

n--1

n--1

n--1

n--1

Vn : - E e l z2 - E ~il~il2z?- E ~il~il2z? "~-E zi~iO "j- E zi~i~ "~ i=l

i=1

i=2

i--1

0an-i

z. z~_: + hi")- y!~) + - - 7

n--p

+ ~ i=1

+ ~

i=2

0c~n-i u(i)

ou(,-,)

Alan Zinober et al.

598

For V. to be nonpositive we set

-c.z. -

~.l~.l~z. - ,.l~.l~z. = z._l + 4(") - u~(")

0~.-1 u(i) +

+--G--~ O~ [fo "4-gt# + (go + ~oO)u]-}- ~

0u(i_l )

OOln_ 1

0t

(180)

i=1

The control function u can be obtained explicitly as the solution of the above equation, giving 1

u("-p) =

[-z._l + u!")(t)

OU(n_p_l ) "-[- OU(n_p_l )

Ox

+

Ox ]

.-p-1 (0s --

Zi=I

&'.-l ~

O,a(i_l)

Oh ("-l) Ot

U (i)

-t- O U ( i _ I ) ]

Oa.-i Ot

c.zn (181)

The closed-loop form of the final error coordinate is ~n - - - z n - ~

-

c.z.

- ~.l~.l~z.

+ ~ . # + ~.0

- ,.l&12z.

(182)

giving n i=1

i=1

/=2

i=1

+ ~ zd~

(183)

i=2

which, upon completion of the squares, gives

r

c,z~i=l

~,

-

+

i=1 -

,i

-

+ Y~

i=2

_< -

i=2

ciz~ +

i=1

2 +

i=i

~1012 i=1 1012

T]i

(184) i=2

New Developments in DAB Control

599

and hence the boundedness of the state of the error system z is guaranteed whenever the disturbance inputs 0 and 0 are bounded. Moreover, since Vn is quadratic in z, it is possible to show that the boundedness of z is guaranteed when 0 is square-integrable but not bounded. This observation is needed for modular designs with passive identifiers where 0 cannot be guaranteed a priori to be bounded. The state of the error system satisfies

s = Az (z, O, t)z + W(z, O, t)O + Q(z, O, t)O

(185)

where

[-%181

1 --C2

--

.

8 2

1

. .

...

- c 3 - 83 ...

Az(z,O,t) =

: 0

O

0

0

"..

:

. . 9 --On

--

(186) 8n

with s~ = ~ l ~ i l 2 + ~ l ~ l

(187)

~

and

W(z,~,t) =

6.2

~2

,

Q(z, ~,t) =

(188)

Identifier Module

We have designed a control law which is independent of the update law. This independence means that we can combine our control law with any identifier that can guarantee independently the boundedness of our disturbance inputs t~, O. We further require that the identifier used has a slow rate of adaptation since the controller in our modular design does not cancel the effects of 0 and 0 but instead compensates for them by the introduction of nonlinear damping functions. There are several types of identifiers which can independently guarantee that the parameter estimate error and its derivative are bounded and are hence suitable for use with our controller. Such identifiers include passive identifiers,

Alan Zinober et al.

600

and traditional gradient. These identifiers require the use of observers for their implementation and result in simpler update laws than those generated using the traditional backstepping design technique. It is this simplification and choice of update laws along with a simpler controller, that make the modular backstepping design a more versatile technique than the traditional backstepping design. There are several types of identifiers which can independently guarantee that the parameter error and its derivative are bounded. These include passive identifiers and swapping-based identifiers reproduced here for completeness (Krstid et al [17]), and least-squares-based identifiers [31]. Passive identifiers

Consider the parametric z-model

=A~(z,O,t)z+W(z,O,t)O+Q(z,O,t)~

(189)

it has been shown by Krstid et al [17] that if the term Q(z, O, t)O was not present there would be strict passivity from the input 0 to the output W(z, O, t)z. To eliminate this term the observer

= A~(z,O,t)~ + Q(z,O,t)~

(190)

is introduced and its error defined to be e= z- ~

(191)

The observer error is governed by an equation driven by

= A~(z,O,t)~+ W(z,O,t)O

(192)

which,_as mentioned above, possesses a strict passivity property from the input 0 to the output W(z,O,t)e. Therefore it is possible to choose the zpassive identifier

0 = rW(z,O,t)e,

r = rT > 0

(193)

T h e o r e m 2 ( z - P a s s i v e ) The closed-loop adaptive system consisting of the plant (I), controller (181), observer (190), and update law (193) has a locally uniformly stable equilibrium at (z, 0, e) = (0, 0, 0) and lim z(t) = lim e(t) = O. So t.--+ r162

t

(194)

lim [y(t) - y,. (t)] = 0 t --4-oo

and asymptotic tracking is achieved. Moreover, if tli_.m y!i)(t) = O, i = 0 , . . . , n - 1, and F(O, O) = O, then lim x(t) = O. t --4"O0

New Developments in DAB Control

601

S w a p p i n g i d e n t i f i e r s Swapping schemes convert dynamic parametric models into static ones enabling the use of standard parameter estimation laws. Krsti6 et al showed that it is possible to convert the parametric z-model

= Az (z, ~, t)z + W(z, ~, t)~ + Q(z, ~, t)b

(195)

by application of the following filters

~T = Az (z, ~, t)[2 T + W(z, 0, t)

(196)

~2o = Az (z, 0, t)Y2o + W(z, ~, t)~ - Q(z, ~, t)~

(197)

into the linear (in ~) model -"

Z

"~ ~'~0 - -

(198)

f'~T~

which can be written in the form

~_..f~T~ "4- ~"

(199)

allowing the use of either the gradient or least squares update laws. Since is an exponentially decaying signal governed by = A~ (z, ~, t)~,

~ e ~'~

(200)

the update laws may be either normalized or not. The gradient update law is defined as:

"

O=F

Y2e

F=

FT

>0,

t~>0

(201)

and the least squares update law is defined as: 0 = F 1 + ~,tr{Y2rF~}

/~= -FI+

~H2T ~tr{/2TF/2 }F,

(202)

F(0) = F(O) T > 0, ~ _> 0

(203)

where setting v = 0 gives non-normalized update laws. Standard parameter estimators cannot guarantee that 0 is bounded unless they use normalized update laws. For this reason, normalization is common in traditional adaptive linear control. However, in this design, normalization is not necessary because the nonlinear damping incorporated into the error system guarantees that the filter state /2 is bounded even if its input, the regressor W, grows without bound. Therefore, 0 is bounded which means that nonlinear damping acts as a form of normalization. Both the nonlinear damping and the update law normalization slow down the adaptation. Slow adaptation is, as previously mentioned, desired since the effects of ~ are not cancelled in this controller (unlike the tuning function design).

602

Alan Zinober et al.

the signals in the closed-loop adaptive system (182) consisting of the plant (1), dynamical controller (181), 1~lters (196), (197) and either the gradient (201) or least-squares (2O2) update law, are uniformly bounded, and lim z(t) -- lim e(t) -=- 0. This means, t --4. O0 t "-~"O0 in particular, that asymptotic tracking is achieved: Theorem 3 (z-Swapping Scheme)All

lim [y(t) - yr (t)] = O.

(204)

t --+ o o

Moreover, if tfirn y!C(t ) = O,i = 0 , . . . , n - 1 and F(0,0) = O, then lim x(t) = O. $,-400

7

Parametric

Semi-Strict

Feedback

Systems

(SSF)

Consider the semi-strict feedback form (SSF) [12,13,33]

ki=Xi+l+~oT(xl,x2,...,xi)O+~i(x,w,t), ~,, = f(~) + a(~),, + ~o,r(~)o + ,7,,(~, w, t) y~xl

l
(205)

where x = Ix1, x2,. 9 xn] is the state, y the output, u the scalar control and ~ i ( x l , . . ., xi) G ~P, i = 1 , . . . , n, are known functions which are assumed to be sufficiently smooth. 0 E ~P is the vector of constant unknown parameters and qi (x, w, t), i = 1, . . . , n, are unknown nonlinear scalar functions including all the disturbances, w is an uncertain time-varying parameter. A s s u m p t i o n 5 The functions qi (x, w, t), i = 1 , . . . , n are bounded by known positive functions h i ( x 1 , . . . x i ) E ~P, i.e.

1,7~(x, w,t)l ~ h ~ ( x l , . . . ~ ) ,

i = 1,...,n

(206)

As before, the output y should track a specified bounded reference signal yr(t) with bounded derivatives up to n-th order. If a plant has unmatched uncertainty, the system may be stabilized via state feedback control [6]. Some techniques have been proposed for the case of plants containing unmatched uncertainty [7]. The plant may contain unmodelled terms and unmeasurable external disturbances, bounded by known functions. First, a classical backstepping method will be extended to this class of systems to achieve the output tracking of a dynamical reference signal. The sliding mode control design based upon the backstepping techniques is then presented in Section 7.2.

New Developments in DAB Control 7.1

Backstepping

603

Algorithm

We first follow a backstepping approach which differs from Koshkouei and Zinober [12,13]. The functions that compensate the system disturbances, are continuous. S t e p 1.

Define the error variable zl = xl - y~ then

~1 = x2 + ~oT(x~)O + ql(X, w,t) - Or

(207)

From (207) h = x~

+~T~+

~(~,

~ , t ) - 9~

+~T#

(208)

with wl(xl) = ~ol(xl) and t~ = 0 - 0 where 0(t) is an estimate of the unknown parameter 0. Consider the stabilization of the subsystem (207) and the Lyapunov function 1

V~(z~,O) = ~z 1 + #T T'-X#

(209)

where F is a positive definite matrix. The derivative V1 is

Vl(zl,O) = zl (xu § Define ci =

Fwizi.

§

y,) §

F-l (Fwlzl -- "O) (210)

Let

#~ = .~ (.~, ~, t) + ~ =

-~0

+

9~ - ~1zl -

h'~zl

(211)

hlizll + •

with ci, a and e positive numbers. Define the error variable z= = x~ - - l ( = ~ ,

=

0, t) - y,

h~zl

x2+'oTO+cLzl--iJ,+

(212)

n

Then ~1 = - c l z ~ + z2 + ~T~ + , l ( x , ~ , 0

-

h~zl

(213)

hllzll + ~e -at

and ~'l is converted to

<_

+ n

_o)

Alan Zinober et al.

604

S t e p 2.

Consider the second L y a p u n o v function

V2(zl, z:, O) = V~ + ~z~

(214)

Then

n = - C l z2 --~ -~-f~e -at -~- z2 (Zl --~x3 --~03TO n ( Oall )

OaI

+ ,j2(~,~,t)- j~-~~t~,~,t)

Oai

ot

o~l z2

0a~000- 9"(t)) + t~TF -I (v2 - ~) where 032 = ~ T ( x l , x 2 ) -

(215)

and r~ = rl +F032z~ = F (031zl + 03~z~).

~(xl)

Let

hi z2 ox~ "}- ~ e-at h2z2

Oa 1

\ax, }

Oal

h~lz~l + ~ - o ' + -~T~ ~ + a-~- ~ Oal

+ - - ~ +//r(t)

(216)

with c2 > 0 and z3 = x 3 - / 3 2 = x3 - a2 - ~.. T h e n

(

oo,

h~ -

h~

h2[z2[+ ~-e- ' t +

n

~

hz Z2oxx

(oa~)~)

~_e-.t

H- n

-bo a ' (~: - 0)

Step k

(1 < k < n - 1).

) \Ozl/

z~ (217)

The t i m e derivative of the error variable Zk is

k- i a ~ k - I

~k = xk+1 + 03~o- ~-~-j~

~,+,

O~k- i ~ + ~k

a~ '

i=1

_y(21(t) +03#~ oa,_~ at

(218)

New Developments in DAB Control

605

where k-1 O0lk_l

t

OXi

i:l

h~

k--1 ( O 0 1 k _ l ~ 2

(k = hklzkl + "--e -at n

hi l ~ Z k l

i=1

(219) + n~-e-at

k-1 Oak-1 ~k = rlk -- E ~ .i i=1

Define

Zk+l = X k + l -- ~k = X k + l -- Otk -- y!k) w h e r e k-1

Oxi xi+l i----1

Oak-i + with

ca

(kzk + - -

o---i-

+ oo

"~

Fwk

~ zi+i o6 ]

ki---1

> 0. T h e n the time derivative of the error variable

~k = - z k - 1

- ckzk + z k + l + w T a + & - (~zk

oo,)

+

zi+l ~

Zk

(22o)

is

00 (221)

Fwk

\i=1

The time derivative of

k

9~ < - ~

Vk is

c, zg + z~ z~ + ~ +

k(k + 1)ee_at 2----7-

+

~TF_I(

"~ -

~)

i=l

+\~=~

0--7-z~+1 ( r k - a )

(222)

since k

Tk :

7"k_ 1 -1- FCOkZ k :

F2wizi i=1

S t e p n.

Define Zn = Xn -- fin--1 = Z n -- Oln-1 -- y!n)

(223)

Alan Zinober et al.

606

with a n - 1 obtained from (220) for k = n. Then the time derivative of the error variable zn is 1

OC~n-

O~._ -~

i

i

_

Oc~,,_

i

i=1 + ~.~(~, t)~ + ~.

-

yr

(2241

where ~n (x, 0) is defined in (219) for k -- n. Extend the Lyapunov function to be

v. = v._~ + ~1z .2

(225)

The time derivative

<-

,=,

c~4 + (" +1)~ -~

- 7

.-2

fa,~i

"~

-.___ (226)

where r . = 7-.-i + F w T z .

(227)

if we select the control

1

[

~.--1

Oa._,

O~.-i

.-2

Oai ~ Fw.

\i----1 +u~(") - r

(228)

with c. > 0. Selecting 0 = rn, 0 is eliminated from the right-hand side of (226). Then 1 9. < - W . + ~ e ( . + 1)e - a '

(229)

n clz~, and with W . = ~1 )--]~i=l v. - v.(o) ___ -

Z

w . + 4 - + 1) (1 - e - o ' ) / ( 2 a )

(23o)

New Developments in DAB Control

607

Therefore O<

W. < Vn(O)+e(n+l)(1-e

-~')/(2a)

and lim tl t Wn < Vn(0) + e(n + 1)/(2a) < oo t.--+ oo ,10

Since W n is a uniformly continuous function, according to the Barbalat Lemma, limt_+r W,~ = 0. This implies that l i m t . . ~ zi = 0, i = 1, 2 , . . . , n, and limt-+~ (y - yr) = 0. R e m a r k 1 When there is no u n k n o w n p a r a m e t e r 0 in the s y s t e m equation, one can attain the tracking p e r f o r m a n c e directly. T h e n (229) becomes

f~. < -cV. + ~1 ( n +

1)e_at

(231)

with O < c < m1
,(n

+

1) (e_O,- e_o, ) +

0 < v. < 2--~-c)

limt-~o~ Vn = 0 implies that l i m t ~ lirnt_~ (y - Yr) = 0.

7.2

V.(0)e-"

(232)

zi = O, i = 1, 2 , . . . , n. This guarantees

Sliding Backstepping Control

The adaptive sliding backstepping control of SSF systems has been studied by Koshkouei and Zinober [12,13]. The controller is based upon sliding backstepping mode techniques so that the state trajectories approach a specified hyperplane. The sufficient condition (for the existence of the sliding mode) given by Rios-Bolivar and Zinober [23,22,24], is no longer needed. To provide robustness, the adaptive backstepping algorithm can be modified to yield adaptive sliding output tracking controllers. The modification is carried out at the final step of the algorithm by incorporating the following sliding surfdce defined in terms of the error coordinates ~r : k l z l + .. 9+ kn - l z,~- i + z,~ : 0

(233)

where ki > 0, i : 1 , . . . , n - 1, are real numbers. Additionally, the Lyapunov function is modified as follows

i "-~

v.=~z?+~ i=l

1~ + l(o - ~)rF_~( ~ ~)

(234)

608

Alan Zinober et al.

The time derivative is I7n = Qn_l + o& rl--1

1

<_ - ~ ciz~ + ~e(n -

1)e -~. +

z._:z.

i=1 n-1

O~n-1 0

O~n-1

T ^ i--1

0---[-+~,~ +

i=2

+k~ z ~ - c ~ z ~ + m - h ~ l z t l + ~ e _ ~ t /

ki

Zi_l

cizi

zi+l

~i -

~izi

-z

fi

+

O0

rwiEz,+,~)

-

zi+, ao i r

l=1 n-2

wo i=l

\i=1

-S:z,+,~176

/J

(235)

with vn = r n - 1 + F~r

Wn +

(

kiwi i=1

)]

(236)

Setting 0 = rn, 0 is eliminated from the right-hand side of (235). From (233) we obtain Zn

=

cr

-- k l Z l

-

k2z 2 -

...

-

]r

(237)

_,

Substituting (237) in (235) removes the need for the sufficient condition for the existence of the sliding mode [22,23] and n-1

v~ _< - ~ c,4 - z~_, (k,~, + k~z~ + . . . + k~_lZn_,) i=1

+ ~(n i=l

-

1)~-~

+ ~ [ z . _ , + f(x) + g(x)u + w.~0 O0

Ot

+ ~n -

New Developments in DAB Control

+kt

h~zl ) hllzll+~e_at +711

z2-clzl-

rt--1 Q + E ki

-

609

zi-I

cizi +/i+l

-

+ ~i -

~izi

i=2

O0 ] F

\i=1

- E zi+, N

COn "lt- #

kicoi

-

(To -

(238)

i----1

We now obtain the update law n--,

0 : ~o : To_~ + r~(co. + E kicoi) i=l .-1 n-1 i=1

i=1

and the adaptive sliding mode output tracking controller

u----

1

g-~5[

-- z"-l-- f(x)--coTnO-t-

ooO_, ~

+V(")+ .-1

(

-~-~ki

Oc~n-1

O0 vn + E

(

kl - c l z l + z 2 -

-zi-l-cizi+zi+l-r

"-~ i----1

Oa.-t

ax-~ xi+1

h z,)

hllzll+~e_at Oai-i

O~ (Tn-q)

i=2

+

/=1

z,+,oo/rwi

-.w~-

+k,=~ ~ Zi+l ~~ 1 7 6 r

K + ~k,.i i=1

COn

+Ekicoi i=1

sgn(~)

/

(240)

/

where kn = 1, K > 0 and W > 0 are arbitrary real numbers and

i-100lk_ l Oxj hi,

l~i = hi + E

1< i< n

j=l Then substituting (240) in (235) yields

~',, = v,,_, +~,~

(241)

Alan Zinober et al.

610

< - [zlz~... z,-1] Q [zlz2... zn_l] T - K I~1 - w ~ 2 1 + ~ e ( n - 1)e -~t

(242)

with Q as defined in (101). Let # , = [z, z2 ... Zn-,]Q[zl z~ ... zn_I]T+K Icrl+Wcr 2. Then, similarly to (229), we have

(/,~ < - # n 3- (n 3- 1)_______~ee_at 2

(243)

which yields

fo'

v. - v.(o) <_ -

#.as + ,(n + 1) (~ - e -o')/(2a)

(244)

Therefore

0 <_fO t #rids <_ V.(O) 3- (n + 1)e (1 - e -at)/(2a) and lim f t #,~ds < V,(O) 3- e(n 3- 1)/(2a) < o~

t--+ooJo

From the Barbalat Lemma, limt--,oo # n = 0. This implies that limt--,oo zi = 0, i = 1 , 2 , . . . , n and limt~c~cr = 0. Particularly, lirnt.-,oo ( x l - y ~ ) = 0. Therefore, the stability of the system along the sliding surface a = 0 is guaranteed. However, if e is a sufficiently small positive number and a suitably large, V < 0. Also, the design parameters K, W, ci and ki, i = 1,..., n - 1, can be selected to ensure that V < 0. 2 One can apply a different procedure at the n-th step.

Remark

: ~

1 [

-z._,-f(x)-w.~0+

+v!") +

0~,,_1

n--1 I

- ~

+

ki

\1--i

o~

-

kl

.-1

0~._,00 7. + ~ 0~._,0x,~i+, i=1

- c a z l + z2 -

z~_~ - e i z i + z i + ~ -- r

lye

hllzll+ ~e-~ OOli-1

oo,

~ Zi+I-"'-z-bY/r~' +~,=, ao} r

-Ksgn(a) - ( W + ~.~=lkiui) a ]

( 7 . -- ~i)

"-'

W. + ~ kiwi

(245)

New Developments in DAB Control

611

with kn = 1, K > 0 and W >>0 arbitrary real numbers and i-1

h~ /Yi

8

h M+

MAPLE

n

+,_..,S-" j=l

Symbolic

h~

hjl, l+ Algebra

1
(246)

n

Design

Package

A MAPLE design package has been developed to allow the straightforward generation of the backstepping algorithms for particular systems. The most general class corresponds to uncertain observable minimum phase systems, which may be in triangular or nontriangular form, and also the SSF form. Triangular systems in PSF or PPF forms are particular subclasses of linearizable observable systems. The DAB algorithm may be used for the design of nonadaptive controllers for nonlinear systems without uncertainty by specifying null matrices 9 and gt. Also, the combined DAB-SMC algorithm, the combined DAB-SOSMC algorithm the SSF algorithms and modular algorithms may be implemented via our MAPLE symbolic algebra package, B A C K . The package, developed in the MAPLE 5 Release 5 symbolic algebra environment, implements the various algorithms, for the synthesis of tracking and regulating adaptive (and non-adaptive) controllers; requiring a minimum of effort by the user. It has the following features: 9 automates the backstepping control design procedures, enabling straightforward use of the algorithms without expert knowledge of the techniques 9 does not use transformations into triangular canonical forms 9 contains an observability checker 9 gives the user the option of generating MATLAB or C ++ code programs for computer simulation of the closed-loop systems 9 automatically generates a text document containing the system model, the necessary error coordinate transformation, the control law, parameter tuning functions (if adaptive) and the sliding surface (if selected). The types of controllers designed by B A C K problems include:

for regulation and tracking

* static and dynamical non-adaptive linearizing controllers for deterministic systems 9 static and dynamical adaptive backstepping controllers for uncertain systems 9 robust static and dynamical backstepping combined with sliding mode control (SMC) or second order sliding mode control (SOSMC) for uncertain systems

612

Alan Zinober et al.

The outputs generated by the B A C K package are the feedback control law, the coordinate transformation placing the system into the error coordinates, the parameter update law for uncertain systems, the sliding surface for the combined backstepping-SMC and backstepping-SOSMC designs, and the text equation file. If requested, MATLAB or C ++ code programs are produced for simulation purposes. The user needs to provide the nonlinear functions of the mathematical model of the system written in the general form (1), and the symbolic desired output. When Yr is constant the controller is a regulator, otherwise Yr is a timedependent function and the controller is designed for tracking tasks.

8.1

Illustrative Example

Consider the third order uncertain nontriangular system xl = - x l + 8(x2x3 + 1) x~ = x3 + u x3 = -xl

(247)

- x 3 + ~x~

where O is an unknown scalar parameter. This system is not transformable into the PSF or the PPF form. Nevertheless, it is globally stabilizable to the equilibrium point (x, ~) = (O, 0, -O, 0) by choosing y = x2 as the output. This output ensures that Assumptions 1 and 2 in Section 3 are satisfied globally. The system model is entered in MAPLE using the tbllowing commands: with(back); f:=matrix(3,1,[-xl,x3,-xl-x3]); g:=matrix(3,1,[0,1,0]);

h:=vector(l,[x2]); yd:=vector(l,[O]); phi:=matrix(3,1,[x2*x3+l,O,x2"2]); psi:=matrix(3,1,[O,O,O]); The observability of the system can then be checked by using observe(f,g,h,yd,phi,psi); and the dynamical adaptive control law is found by running the appropriate algorithm, depending on the user's requirement. For example, for the Dynamical Adaptive Backstepping Algorithm, with the option of first order sliding:

backdsmc(f,g,h,yd,phi,ps•

New Developments in DAB Control

613

For the Dynamical Adaptive Backstepping Algorithm with second order sliding: backsosmc(f,g,h,yd,phi,psi);

In both cases, after invoking the desired program, the user is asked to choose from several options. These include whether simulation files are needed, which format is required and the filenames. The system transformation, control law and tuning functions will then be printed both in the MAPLE worksheet and the equation file.

9

Practical

Examples

A continuously stirred tank reactor is used to illustrate the DAB-SMC and DAB-SOSMC algorithms, and then an example of an SSF system is presented.

9.1

Comparative Example: Continuously Stirred Tank Reactor

Consider the following nonlinear third-order dynamic model [15,25] of a Continuously Stirred Tank Reactor (CSTR) in which an isothermal liquid-phase, multicomponent chemical reaction takes place xi = 1 - (1 + Dal)Xi + Da2x~9 ks = D a i x i - x2 - (D.2 + Da3)x~ + u

(248)

~3 = Da3x2~ -- x3 y=x3

with * xi: normalized concentration CA/C.4F of a species A * x2: normalized concentration C B / C A F of a species B 9 x2: normalized concentration C c / C A F of a species C 9 CAF: the feed concentration of the species A (tool 9m - t ) 9 u: the ratio of the per-unit volumetric molar feed rate of species B, denoted by NBF, and the feed concentration CAF, i.e. u = N B F / F C A F 9 F : volumetric feed rate (m3s - t ) 9 D,1 = k l V / F constant parameter 9 Da~ = k 2 V C A F / F constant parameter 9 Da3 = k a V C A F / F constant parameter 9 V: the volume of the reactor (m 3) 9 kl, ks, k3: first order rate constants (s -1)

Alan Zinober et al.

614

The system has a constant stable equilibrium point, for every constant volumetric feed rate value u = U, which is located in a minimum phase region of the system [29]

X1-

I + Da2X~ 1 + D~I "-1+

U+I+D,,1 ] l+Dat 2 (D.2 + D~,3 + D,,1D.3)

1+4

X2 = (1 + D.1) X3 : Da3X~

The operating region of the system is, of course, the strict orthant in ~3, where all concentrations are positive. In other words,

X = { x E ~ 2,

s.t.

x~>0

for i = l, 2, 3}

We assume that the constant parameters Dal, Da2 and Da3 are all constant but unknown. Thus, system (248) can be rewritten as ~l = 1 - xl + ~oT(xl, z2)0 x2 = - x 2 + u + ~oT(xl, x2)O x3 = - x 3 + ~,T(x~)0 y=

(249)

xs

with 0 = [01 05 03]T = [Dal Da~ Das]T the unknown parameter vector and

~T=[-~I ~ 0] ; ~ = [ ~

-~

-~]]

; ~=[0

0 ~]

Both the DAB-SMC algorithm and the DAB-SOSMC algorithm can be applied to system (249) to synthesize a dynamical adaptive controller for its robust regulation. D A B - S M C A l g o r i t h m Applying the DAB-SMC algorithm, we synthesize a dynamical adaptive SMC compensator for the regulation of system (249): Coordinate transformation zl = y - X s = x s - X 3

z~ = - x 3 + ~T(~2)0 + cizl

z~ : ,~(~,~) + -b-~-Ou Sliding surface = k l z l + k~z~ + z3 : 0

(25o)

New Developments in DAB Control

615

P a r a m e t e r u p d a t e law 0 = r3 = ~-z + Ftr(kl~a + k ~

(251)

+ wa)

with wT = (c, - 1)~T(x2) + Oa

0~T(xl,x2)

T.

cga

o~T~

"

(252)

~(~, ~) = ~ - (~,

ou r

- 1)~3 - ~ - ~

+ ~r~

+~vTF(zl~v3 + z 2 ~ ) + c~z~ D y n a m i c a l a d a p t i v e S M C law

1[

a~r ~

- ~1 - (k~ + z 2 ) ~ ( ~ 3

- ~) - kl(-clzl

+ z~)

c9x2

-~0

a~ - k~(-zl

[ tg~

- ~

C92~T ~ x

+ z3) - b~:~ (1 -

~)

~Ct

-- (' --O0 + u--oX2 ) r3 -- ~(cr +/~sign(a))

J

(253)

where/" = F T > 0 is a diagonal matrix containing the adaptation parameter gains. The output y = x3 converges asymptotically to the desired value X3. D A B - S O S M C a l g o r i t h m The combined DAB-SOSMC algorithm can be applied to synthesize a dynamical adaptive discontinuous controller for the robust regulation of system (249). C o o r d i n a t e transformation zl = y - X a = x 3 - X 3 (254)

616

Alan Zinober et al.

o~ +

T

+

Sliding s u r f a c e y l = za + cz2 = 0

(255)

P a r a m e t e r U p d a t e Law

= /1 zl.~oa + z2

(cl - 1)~oT + -~'-z vcp2)]

(256)

where F = F T > 0 is a diagonal matrix containing the adaptation parameter gains. D y n a m i c a l a d a p t i v e S O S M C law Using the algorithm and noting that n -- 3, p = 2 we have = --o~WM~.sign

Yl -- ~Yl~o~

(257)

the control guarantees Yl and y~ to be bounded and dependent upon yl (0) and y~(0). S i m u l a t i o n s Computer simulations were performed using both the DABSMC and DAB-SOSMC designed control laws for the robust regulation of a CSTR with the following "unknown" parameters Da~ = 3 . 0

;

Da2=0.5

;

D ~ a = 1.0

The desired equilibrium, corresponding to a constant value of u given by U = 1, is obtained as X~ =0.3467

;

X~=0.8796

;

Xa=0.7737

whilst the design parameters for the DAB-SMC law were selected to be c1=2,

c~=l,

kl=l,

F=213 , n=2,

/3=1

and for the DAB-SOSMC law cl = 2 ,

c2=1,

k=4,

F=213,

WMa~:=500

New Developments in DAB Control

617

Fig. 1 shows the DAB-SMC controlled CSTR output responses, whilst Fig. 2 depicts the DAB-SOSMC controlled responses. It can be seen that the DABSMC controlled responses exhibit good transient performance to the equilibrium point, whilst achieving parameter convergence and very small control chatter. The DAB-SOSMC controlled responses also exhibit very good transient performance and parameter convergence. In comparison with the DABSMC algorithm it can be seen that the DAB-SOSMC algorithm removes chattering completely from the control law whilst the control is simpler, achieving a significant reduction in the control computation.

Concentration of the species xl, x2, x3 1.21 1

Estimates of Dal, Da2, Da3 3

/

2i

0.8 0.6 0.4 0.2

5 t

10

0

5 t

10

Sliding surface

Control input, u 0.5

2

0 1.5

-0.5 1

0.5 0

-1 5 t

10

-1.5

0

5 t

10

Fig. 1. DAB-SMC Controlled responses of the Isothermal CSTR

9.2

SSF System

Consider the second-order system in SSF form ~1 = X2 "[- X l ~ "~

~

=

u

AXl2 cos(Bxlz2)

(258)

where A and B are considered unknown but it is known that IA[ < 2 and IB[ _< 3. We have hi = 2z~

618

Alan Zinober et at. Estimates of Dal, Da2, Oa3

Concentration of the species xl, x2, x3 3 J

2

0.8 0.6 0.4 0.2

0

0 0

10

5 t

5 t

10

Sliding surface

Control input, u 2.5

0.2 0 1.5

-0,2

1

-0.4

0.5

-0,6

0

5 t

10

-0,8

0

5 t

10

Fig. 2. DAB-SOSMC Controlled responses of the Isothermal CSTR

zl = z l

--Yr

z~ = x~ + x16 + clzl +

4x~zl

hllzll + ~e -~t

- ilr

4x4zl 0J1 = X l

C~al 0./2 = - - ~ Z I Xl

T2 = F (~izi + ~2z~)

r = 2~

o_~

0:el Z2

~_=

\~-~/

Then the control law (228) becomes

~ = - ~ - c~z~ - ~ 0

i)al + y(r2) -r + gOai ~ - ~ + -h-a~l - ~ + --~-

(259)

Simulation results showing desirable transient responses are shown in Fig. 3 withgr = 0 . 4 , a = 10000, e = 0 . 0 1 , F = 3 . A = 2 , B = 3 a n d e l =c2=20. Alternatively, we can design a sliding mode controller for the system. Assume that the sliding surface is o" -- klzl +z2 = 0 with kl > 0. The adaptive sliding

New Developments in DAB Control

619

mode control law (240) is

u : (c, kl

1) zl

-

balx2

-

k ~

-

o3~

~h~lzll+~e-ot -w0"-

+ ~

Oal_

Oal y(2)

+ -~-'~ + -5- +

g+~+

a~l h~sgn(0")

(260)

where r~ = F ( Z l o 3 1 -~- 0"(o3 2 -~- kio3i)). Simulation results showing desirable transient responses are shown in Fig. 4 with Yr = 0.05sin(2~rt), ki = 1, K=5 W=0, a-l,e=0.001, F=0.003, A=2, B=3andci=c2=10.

0.4~

x~(t)

x2(t)

0.5 t

0.5 t

0.2

0

0

Parameter e6tlmate

-I -1.5 "

0'.5

Fig. 3. Responses with nonlinear control for SSF system

10

Conclusions

A number of backstepping control techniques have been studied, both with and without sliding, for the class of observable non-minimum phase nonlinear continuous uncertain systems (triangular and non-triangular), using a dynamical backstepping approach. Adaptive backstepping algorithms with tuning functions and the alternative modular parameter identification approach have been presented. Systems, which can be converted to a parametric semi-strict feedback form, with disturbances and unmodelled dynamics have also been considered.

620

Alan Zinober et al. x2(t)

xl(t) 0.05

0.5

0

-0.5 -0.05

0

5

10

-1

0

t

Parameterestimate

Control action 20

0,3

100

5

10

t

0.3001

0.3

5

10

-10

t

5

10

t

Fig. 4. Responses with sliding control for SSF system

Nonlinear, (first-order) sliding and second-order sliding control laws have been employed. In the sliding backstepping approach, the controller is designed so that the trajectories remain on a specified sliding surface. Backstepping sliding mode control benefits from the advantages of both adaptive backstepping and robust sliding approaches. Second-order sliding provides more accurate sliding and prevents undesirable chatter motion. Secondorder sliding and the modular approach yield simpler control laws. B A C K , a Maple symbolic algebra package, has been developed as a tool for the design of dynamical adaptive backstepping nonlinear controllers for regulation and tracking tasks. Some examples have been presented to illustrate the practical application of the various control algorithms. Further research work is being undertaken in the areas of second-order sliding, the modular approach and extending the design of controllers for SSF systems.

References 1. Bartolini G., A. Ferrara, L. Giacomini and E. Usai (1996) A combined backstepping/second order sliding mode approach to control a class of nonlinear systems, Proc. IEEE International Workshop on Variable Structure Systems, Tokyo, Japan, 205-210

New Developments in DAB Control

621

2. Bartolini, G., A. Ferrara and E. Usai (1997) Applications of a suboptimal discontinuous control algorithm for uncertain second order systems, Int. J. Robust Nonlinear Control, 7, 299-320 3. Bartolini, G., A. Ferrara and E. Usai (1997) Output Tracking Control of Uncertain Nonlinear Second-Order Systems, Automatica, 33, 2203-2212 4. Bartolini, G., A. Ferrara and E. Usai (1998) Chattering avoidance by secondorder sliding mode control, IEEE Trans. Automat. Contr., 43, 241-246 5. Blankenship, G. L., R. Ghanadan, H. G. Kwatny, C. LaVigna and V. Polyakov, (1995) Tools for integrated modelling, design, and nonlinear control, IEEE Control Systems Mag., 15, 65-79 6. Corless, M., and G. Leitmann (1981) Continuous state feedback guaranteeing uniform ultimate botmdedness for uncertain dynamical systems, IEEE Trans. Automat. Contr., 26, 1139-1144 7. Freeman, R. A., and P. V. Kokotovid (1996) Tracking controllers for systems linear in unmeasured states, Automatica, 32, 735-746 8. de Jager, B. (1996) Symbolic analysis and design for (nonaffiX-Mozilla-Status: 0009 systems, Proc. 13th IFAC World Congress, San Francisco, F, 289-294 9. Glumineau, A. and L. Graciani (1996) Symbolic nonlinear analysis and control package, Proc. 13th IFAC World Congress, San Francisco, F, 295-298 10. Jiang, Z. P. and L. Praly (1991) Iterative designs of adaptive controllers for systems with nonlinear integrators, Proc. 30th IEEE Conference on Decision and Control, Brighton, UK, 2482-2487 11. Kanellakopoulos, l., P.V. Kokotovid and A. S. Morse (1991) Systematic Design of Adaptive Controllers for Feedback Linearizable Systems, IEEE Trans. Automat. Contr., 36, 1241-1253 12. Koshkouei, A. J. and A.S.I. Zinober (1999) Adaptive Sliding Backstepping Control of Nonlinear Semi-Strict Feedback Form Systems, Proc. 7th IEEE Mediterranean Control Conference, Haifa 13. Koshkouei, A.J. and A.S.I. Zinober (2000) Adaptive Output Tracking Backstepping Sliding Mode Control of Nonlinear Systems, Proc. 3rd IFAC Symposium on Robust Control Design, Prague 14. Koshkouei, A.J. and A.S.I. Zinober (2000), Adaptive Backstepping Control of Nonlinear Systems with Unmatched Uncertainty, submitted to CDC2000, Sydney 15. Kravaris, C. and S. Palanki (1988) AIChE Journal, 34, 1119-1127 16. Krsti6, M., I. Kanellakopoulos and P.V. Kokotovi6 (1992) Adaptive Nonlinear Control without Overparametrization, Systems and Control Letters, 19, 177185 17. Krsti6, M., I. Kanellakopoulos, and P.V. Kokotovid (1995) Nonlinear and Adaptive Control Design, New York: John Wiley & Sons 18. Marino, R. and P. Tomei (1995) Nonlinear Control Design, London: PrenticeHall 19. Rios-Bolivar, M. (1997) Adaptive Backstepping and Sliding Mode Control of Uncertain Nonlinear Systems, PhD dissertation, Applied Mathematics, University of Sheffield 20. Rios-Boh'var, M., H. Sira-Ramirez and A. S. I. Zinober (1995) Output Tracking Control via Adaptive Input-Output Linearization: A Backstepping Approach, Proc. 34th IEEE CDC, New Orleans, 2, 1579-1584

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21. Rios-Bol/var, M. and A.S.I. Zinober (1994) Sliding mode control for uncertain linearizable nonlinear systems: A backstepping approach, Proc. 3rd IEEE Workshop on Robust Control via Variable Structure and Lyapunov Techniques, Benevento, Italy, 78-85 22. Rios-Bol/var, M. and A.S.I. Zinober (1997a) Dynamical adaptive backstepping control design via symbolic computation, Proc. 3rd European Control Conference, Brussels 23. Rios-Bol/var, M. and A.S.I. Zinober (1997b) Dynamical adaptive sliding mode output tracking control of a class of nonlinear systems, Int. J. Robust and Nonlinear Control, 7, 387-405 24. Rios-Bolfvar, M., and A.S.I. Zinober (1998) A symbolic computation toolbox for the design of dynamical adaptive nonlinear control, Appl. Math. and Comp. Sci., 8, 73-88 25. Rios-Bolivar, E.M. and Zinoher, A.S.I. (1999) Dynamical adaptive sliding mode control of observable minimum-phase uncertain nonlinear systems, in "Variable structure systems, sliding mode and nonlinear control", ed K.D. Young and 0. (~zgiiner, Springer-Verlag, London, 211-235 26. Rios-Bollvar, M., A.S.I. Zinober and H. Sira-Ramirez (1996) Dynamical Sliding Mode Control via Adaptive Input-Output Linearization: A Backstepping Approach, in: Robust Control via Variable Structure and Lyapunov Techniques (F. Garofalo and L. Glielmo, Eds.), Springer-Verlag, 15-35 27. Rios-Boh'var M., A.S.I. Zinober and H. Sira-Ramirez (1997) Dynamical Adaptive Sliding Mode Output Tracking Control of a Class of Nonlinear Systems, Int. J. Robust and Nonlinear Control, 7, 387-405 28. Rothfuss, R. and M. Zeitz (1996) A toolbox for symbolic nonlinear feedback design", Proc. 13th IFA C World Congress, San Francisco, F, 283-288 29. Sira-Ramlrez, H. and M. Delgado Passivity Based Regulation of Nonlinear Continuous Processes (to be published in Advances in Control) 30. Sira-Ramlrez, H. and O. Llanes-Santiago (1994) Dynamical Feedback Strategies in the Regulation of Nonlinear Chemical Processes, IEEE Trans. Control Systems Technology, 2, 11-21 31. Slotine, 3-J. E. and W. Li (1991) Applied Nonlinear Control, Prentice-Hall, London 32. Sontag, E. D. (1989) Smooth stabilization implies coprime factorization, IEEE Trans. Automat. Contr., 34, 435-443 33. Yao, B. and M. Tomizuka (1997) Adaptive robust control of SISO nonlinear systems in a semi-strict feedback form, Automatica, 38,893-900 34. Zinober, A.S.I., J.C. Scarratt, A. Ferrara, L. Giacomini and E.M. Rios-Bollvar (1999) Dynamical adaptive first and second order sliding mode control of nonlinear non-triangular uncertain systems, Proc. 7th IEEE Mediterranean Control Conference, Haifa

List of PARTICIPANTS Second NCN Workshop: Nonlinear Control in the Year 2000

O. Ren~e Pierre-Antoine Dirk Andrei Diego Marcelo Claudio Nnaedozie Edouardo

Alessandro Victor Miguel Andrea Radhakisan Iyad Alfonso John S. Antonio Georges Nils Christiane Pierre Antonio Guido Pierre-Alexandre Claudio Ugo Claude Jochen Christopher I. Francesca Gr~goire Madalena Yacine Stephen Jesus Patrizio Paolo Fritz Fahio Jean-Michel Maria Belen Brigitte Gilney Jos6 A. Claudio Raffaella Domitilla Emmanuel SeRe

ABIB ABSIL AEYELS AGRACHEV ALONSO ALTAFINI ANEKE ARANDA-BRICAIRE ASTOLFI AYALA AYALA BOTTO BACCIOTTI BAHETI BALLOUL BANOS BARAS BARREIRO BASTIN BERGLUND BERNARD BERTRAND BICCHI BLANKENSTEIN BLIMAN BONIVENTO BOSCAIN BREZINSKI BROECKER BYRNES CERAGIOLI CHARLOT CHAVES CHITOUR CHOU CLEMENTE-GALLARDO COLANERI COLETTA COLONIUS CONTICELLI CORON D'AMICO D'ANDREA-NOVEL DAMM DE DONA DE PERSIS DE SANTIS DEL VECCHIO DELALEAU DIOP

Ren6e.Abib@ univ-rouen.fr absil@ montefiore, ulg.ac.be [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] vayala@socompa, ucn.cl [email protected]

[email protected] [email protected] [email protected] [email protected] [email protected] abarreiro@uvigo,es [email protected],ac.be berglund@epfl,ch Christiane, Bernard @dg 12,cec.be bertrand @lss.supelec.fr [email protected] [email protected],nl Pierre-Alexandre.BUman @inria.f r [email protected] boscain @sissa.it Claude.Brezinski@ univ-lille1.fr jbroe@ physik3.gwdg.de ChdsBymes@ seas.wustl.edu ceragiol@ calvino.polito.it [email protected] sontag@ hilbert.rutgers.edu ychitour01 @cybercable,fr chou@caor,ensmp.fr jesus @mailhost, math. utwente, nl colaneri@ elet.polimi.it paolo@iange,cnr, it coloni us @math, uni- augsburg .de [email protected] Jean-MicheI.Coron@ math.u-psud,fr [email protected] andrea@caor,ensmp.fr damm@lss,supelec.fr [email protected],ac.be depersis @zach.wusU.edu [email protected] .it ddomitilla @hotmail.com delaleau@lss,supelec.fr [email protected]

Mahmoud Ludovic Moez Thomas Michel Fabienne Halina Kenji Martha Jean-Paul

Denis Lats Manuel Viet Mohamed A. Boumediene Simon Guido Christian Henri Salim

Ernest Alberto Sebastien Bronislaw Frederic Mohamed

C~dric Philippe Georgia Matei Christopher

Balint Philipp Ali Jafari Alexander Mohsen Frangoise

loan D. Beatrice Laurent Laura Jean Antonio Johannes Alan F. Frangois Delfim F.

Nicolas Lorenzo Joerg

Alessia Riccardo Richard

ELLOUZE FAUBOURG FEKI FLIEGNER FL1ESS FLORET FRANKOWSKA FUJ'IMOTO GALAZ-LARIOS GAUTHIER GILLET GRONE GUERRA HAGENMEYER HAMMAMI HAMZI HECKER HERRMANN HINTZ HUIJBERTS IBRIR IRVING ISIDORI JACQUET JAKUBCZYK JEAN JEROUANE JOIN JOUAN KALIORA KELEMEN KELLETT KISS KOHLRAUSCH KOSHKOUEI KRISHCHENKO LAKEHAL-AYAT LAMNABH1-LAGARRIGUE LANDAU LAROCHE LAVAL LEVAGGI LEVINE LORIA LOWIS LYNCH MALRAIT MARADO TORRES MARCHAND MARCONI MARECZEK MARIGO MARINO MARQUEZ

[email protected] ludovic.faubourg @sophia.inria.fr [email protected] T. Flieg her @maths, bath, ac. uk fliess@cmla,ens-cachan.fr

floret @lss.supelec.fr [email protected] fuji@ robot.kuass.kyoto-u.ac.jp galaz@lss,supelec.fr [email protected] [email protected] gruene@ math.uni-frankfurt.de [email protected],pt [email protected] r [email protected] [email protected] [email protected] [email protected] [email protected] h.j.c,[email protected], uk [email protected] Ernest.Irving @wanadoo.fr [email protected] jacquet@ mat. ua, pt jakubczy@panim impan.gov.pl [email protected] [email protected] Cedric.Join @cran. u-nancy.f r Philippe.Jouan@ univ-rouen.fr g,[email protected] mkelemen@ele,etsmtl.ca [email protected] [email protected] [email protected] ajafari@ sheffield.ac.uk [email protected] lakehal@ Iss.supelec.fr [email protected] [email protected] [email protected] laval@suniut 1.iutv,univ-paris 13.fr [email protected] [email protected] [email protected] Ioewis@erssl 1.et.tu-dresden.de alanl@erssl 1.et.tu-dresden.de malrait@ cas.ensrnp,f r [email protected] marchand @lagep.cpe.fr [email protected] [email protected] [email protected] Marino@ing uniroma2.it marquez@ Iss.supelec.fr

Philippe Denis Frederic Mouhiba Russell E. Raul J. Gerard Claude Luc Hugues Paloma Philippe Mariana Henk Hom~:re C~dric Romeo Lucia Elena William Ewa Mich~le Marco Paulo S. Nicolas Benedetto Nicolas Alessandro Jean-Baptiste Jean-Francois Richard Laurent Christophe Alban Jose Cesareo Anders Martin Witold Eugenio Hugo Lionel Pierre Joachim Yuri L. AndrOs Giovanni L. Andrey V, Julie C. Jacquelien MA. Kurt Klaus R, Gerhard Rodolphe Andrea

MARTIN MATIGNON MAZENC MF.ZGHANI MILLS MONDRAGON MONTSENY MOOG MOREAU MOUNIER MOYA MULLHAUPT NETTO NIJMEIJER HKWAWO NOUILLANT ORTEGA PALLOTTINO PANTELEY PASILLAS-LEPINE PAWLUSZEWICZ PELLETIER PENGOV PERElRA DA SILVA PETIT PICCOLI PINFIELD PISANO POMET POMMARET POTHIN PRALY PRIEUR QUADRAT RAIMUNDEZ RANTZER RAU RESPONDEK ROCHA RODRIGUEZ ROSIER ROUCHON RUDOLPH SACHKOV SAEZ-SCHWEDTT SANTOSUOSSO SARYCHEV SCARRATT SCHERPEN SCHLACHER SCHNEIDER SCHREIER SEPULCHRE SERRANI

martin@cas,ensmp.fr [email protected] [email protected] [email protected] r,e,mills@ sheffield.ac.uk [email protected],uk montseny@ popcsc.laas.fr Claude. [email protected] r [email protected] mounier@ief,p-sud,fr [email protected] rnulhaupt@cas,ensmp.fr netto@lss,supelec.fr [email protected] [email protected] [email protected] [email protected] [email protected] elena.panteley@ mageos.com [email protected] epaw@ cksr,ac.bialystok,pl mpelleti@ u-bourgogne.fr [email protected] [email protected] petit@cas,ensmp3r [email protected] pnp@svl,co.uk pisano@diee,unicait pomet@sophia,inria.fr [email protected] Richard. Pothin @ircyn.ec-nantes.f r [email protected] [email protected],fr [email protected] cesareo@uvigo,es rantzer@ control.lth.se [email protected] wresp@lmi,insa-rouen.fr eugenio@ mat.ua.pt [email protected] LionelRosier@ math.u-psud.fr [email protected] rudolph@erssl 1,et.tu-dresden.de [email protected] [email protected] uva.es santosuosso@ ing.uniroma2.it [email protected] j.c.scarratt@ sheffield.ac,uk [email protected] kurt @regpro.mechatronik.uni-linz.ac,at schneider@wias-berlin,de g.schreier@ ieee,org [email protected] [email protected]

Anion Fatima Herbertt Edouardo D. Marco Michael Gianna Christophe Hector J. Issa A. Alborto Emmanuel Ton J.J. Arian J.

Rajah Paolo Eric Fabian Dirk Vladimir M. Igor Pierlugi Qinghua Alan

SHIRIAEV SILVA LEITE SIRA-RAMIREZ SONTAG SPADINI SPATHOPOULOS STEFANI SUEUR SUSSMANN TALL TESI TRELAT V A N DEN BOOM VAN DER SCHAFT VEPA VETTORI WALTER WIRTH WOLLHERR ZAKALYUKIN ZELENKO ZEZZA ZHANG ZINOBER

[email protected] [email protected] [email protected] sontag @hilbert.rutgers.edu spadini@ poincare,dma.unifi.it [email protected],ac.uk [email protected] [email protected] sussmann@ math.rutgers.edu [email protected] atesi@dsi,unifi.it [email protected] vdboom @control-lab.et.tudelft.nl a,j, vanderschaft@math,utwenteonl

[email protected] [email protected] [email protected] r fabian@ math,uni-bremen.de [email protected] vladimie@ zakaLmccme.ru [email protected] [email protected] [email protected] A Zinober@ sheffield.ac,uk

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New Directions in Nonlinear Observer Design 552pp: 1999 [1-85233-134-8] Vol. 245: Garulli, A.; Tesi, A.; Vicino, A. (Eds) Robustness in Identification and Control 448pp: 1999 [1-85233-179-8] Vol. 245: Aeyels, D.; Lamnabhi-Laganigue,F.; van der Schaft, A. (Eds) Stability and Stabilization of Nonlinear Systems 408pp: 1999 [1-85233-638-2] Vol. 247: Young, K.D.; OzgOner, U. (Eds)

Variable Structure Systems, Sliding Mode and Nonlinear Control 400pp: 1999 [1-85233-197-6] Vol. 248: Chen, Y.; Wen C.

Iterative Learning Control 216pp: 1999 [1-85233-190-9] Vol. 249: Cooperman, G.; Jessen, E.;

Michler, G. (Eds) Workshop on Wide Area Networks and High Performance Computing 352pp: 1999 [1-85233-642-0] Vol. 250: Corke, P. ; Trevelyan, J. (Eds) Experimental Robotics VI 552pp: 2000 [1-85233-210-7]

Vol. 255: Leonessa, A.; Haddad, W.H.;

Chellaboina V. Hierarchical Nonlinear Switching Control Design with Applications to Propulsion Systems 152pp. 2000 [1-85233-335-9] VoI. 256: Zerz, E. Topics in Multidimensional Linear Systems Theory 176pp. 2000 [1-85233-336-7] Vol. 257: Moallem, M.; Patel, R.V.;

Khorasani, K. Flexible-link Robot Manipulators 176pp. 2001 [1-85233-333-2] Vol. 258: Isidori, A.; Lamnabhi-Lagarrigue, F.;

Respondek, W. (Eds) Nonlinear Control in the Year 2000 Volume 1 616pp. 2001 [1-85233-363-4]

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Perspectives in Robust Control (Lecture Notes in Control and Information Sciences)

Nonlinear Control in the Year 2000, Volume 2 (Lecture Notes in Control and Information Sciences)

Nonlinear Control in the Year 2000: Volume 1 (Lecture Notes in Control and Information Sciences)

Singular Control Systems (Lecture Notes in Control and Information Sciences)

Networked Control Systems (Lecture Notes in Control and Information Sciences)

Ensuring Control Accuracy (Lecture Notes in Control and Information Sciences)

Nonlinear Control Systems: An Algebraic Setting (Lecture Notes in Control and Information Sciences, Volume 242)

L2-Gain and Passivity Techniques in Nonlinear Control (Lecture Notes in Control and Information Sciences)

Advances in the Control of Nonlinear Systems (Lecture Notes in Control and Information Sciences)

Feedback Control, Nonlinear Systems, and Complexity (Lecture Notes in Control and Information Sciences)

Assessment and Future Directions of Nonlinear Model Predictive Control (Lecture Notes in Control and Information Sciences)

Nonlinear and Adaptive Control: NCN4 2001 (Lecture Notes in Control and Information Sciences)

Advances in Communication Control Networks (Lecture Notes in Control and Information Sciences, Volume 308)

Linear and Nonlinear Iterative Learning Control (Lecture Notes in Control and Information Sciences) (v. 291)

Nonlinear Analysis and Synthesis Techniques for Aircraft Control (Lecture Notes in Control and Information Sciences)

Nonlinear Control Systems: An Algebraic Setting (Lecture Notes in Control and Information Sciences)

Control of Indefinite Nonlinear Dynamic Systems: Induced Internal Feedback (Lecture Notes in Control and Information Sciences)

Nonlinear Model Predictive Control: Towards New Challenging Applications (Lecture Notes in Control and Information Sciences)

Nonlinear Control of Wheeled Mobile Robots (Lecture Notes in Control and Information Sciences)

Control Theory in the Plane (Lecture Notes in Control and Information Sciences)

Multidisciplinary Research in Control: The Mohammed Dahleh Symposium 2002 (Lecture Notes in Control and Information Sciences)

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Topics in Time Delay Systems: Analysis, Algorithms and Control (Lecture Notes in Control and Information Sciences)

Control Theory in the Plane (Lecture Notes in Control and Information Sciences)

Recent Advances in Learning and Control (Lecture Notes in Control and Information Sciences)

Robustness in Identification and Control (Lecture Notes in Control and Information Sciences)

New Directions and Applications in Control Theory (Lecture Notes in Control and Information Sciences)

Limited Data Rate in Control Systems with Networks (Lecture Notes in Control and Information Sciences)

New Directions in Nonlinear Observer Design (Lecture Notes in Control and Information Sciences)

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Finite Zeros in Discrete Time Control Systems (Lecture Notes in Control and Information Sciences)

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