Lecture Notes in Control and Information Sciences Editor: M. T h o m a
217
Franco Garofalo and Luigi Glielmo (Eds)
Robust Control via Variable Structure and Lyapunov Techniques
~ Springer
Series A d v i s o r y B o a r d A. Bensoussan • M.J. Grimble • P. Kokotovic • H. Kwakernaak J.L. Massey • Y.Z. Tsypkin
Editors Professor Franco Garofalo Facolta di Ingegneria di Benevento Universita di Salerno Corso Garibaldi, Palazzo Bosco, 82100 Benevento, Italia Professor Luigi Glielmo Dipartimento di lnformatica e Sistemistica Universit~ di Napoli "Federico II" via Claudio 21, 80125 Napoli, Italia
ISBN 3-540-76067-9 Springer-Verlag Berlin Heidelberg New York British Library Cataloguing in Publication Data Robust control via variable structure and Lyapunov techniques. - (Lecture notes in control and information sciences ; 217) 1.Coatroltheory 2.Automaticcontrol l.Garofa]o, Franco II.Glielmo, Luigi 629.8'312 ISBN 3540760679 Library of Congress Cataloging-in-Publication Data A Catalog record for this book isavailable from the Library of Congress 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-Verlag London Limited 1996 Printed in Great Britain 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 69/3830-543210 Printed on acid-free paper
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Preface Engineers, whatever their fields of action, are "naturally" interested in robust design techniques because the working and durability of their pieces of work is always jeopardized by mutable and unpredictable environments. Traditional sectors of the engineering sciences use to acknowledge the ignorance--on the physical system, its parameters and interactions with the outside world--at the end of the design stage by using "safety factors." The last decades instead, and hence the younger areas of engineering, have seen a dramatic development of both sophisticated mathematical modelling techniques, and new and advanced computational tools. Since a complete description of systems remains still impossible, or impracticable, the designer tries to find a suitable mathematical description also for the unknown or "uncertain" parts of the system at hand, and to use such information during the design process, not just at its end or in the validation phase. The price to be paid for that is often a more complicated technique of design, but this can now be faced with powerful and easily available means of computations. The above approach, applicable to all areas of engineering, is fully exploited in the realm of control engineering whose foundations are laid on mathematical modelling. For this reason in recent years robust control theory and application have received considerable attention and developed along several directions. A large part of today's fervour on robust control research is focussed on those techniques which utilize the Variable Structure Control method and Lyapunov's Second Method, and constitute the backbone of the so-called "deterministic" control of uncertain systems. As well known, the two methods are closely related but, strangely enough, their respective research areas are quite separated and meetings between their researchers are not frequent. For this reason--and following the success of an analogous meeting organized in Sheffield, UK, by Alan Zinober in 1992--an IEEE Workshop on Robust Control via Variable Structure CdLyapunov Techniques was held in Benevento, Italy, in September 1994. The chapters in this book are enlargements and upgrades of some of the papers presented there. The criteria which guided us in the choice of the papers were essentially those of covering a large spectrum of recent research and introducing the most innovative ideas. As a result the reader will find a survey on control Lyapunov functions, new structures of sliding mode controllers with a discussion on higher order sliding rhodes, new techniques for the design of direct and indirect adaptive controllers, an introduction to the geometric theory of "flat" systems, controllers for plants with component-wise bounded inputs, robust design via linear matrix inequalities and polytopic covering, a discussion on the use of piece-wise linear Lyapunov functions, and some issues regarding dissipativity and absolute stability. To make the book readable and useful, the authors were asked to render their contributions as self-contained as Dossible.
Vln
Randy Freeman and Petar Kokotovid in Chapter 1 review some of the existing results on the Lyapunov design of robustly stabilizing feedback laws for uncertain nonlinear systems. Using the concept of a robust control Lyapunov function, the authors present robust and adaptive backstepping tools and demonstrate how they can be used in systematic design procedures. In Chapter 2--by Miguel Rios-Bolivar, Alan Zinober and Herbertt SiraRam irez--the ideas of non-overparameterized adaptive backstepping algorithm and sliding mode control are combined to provide robustness, even in the presence of unknown disturbances, for both control regulation and output tracking of a class of uncertain nonlinear systems. The effectiveness of the approach is illustrated through computer simulations. Zhihua Qu and Joseph Kaloust in Chapter 3 consider the systematic construction of continuous controllers for uncertain systems which do not satisfy any of the usual structural conditions imposed for stabilizability and easy design, such as the matching conditions, the conditions of being equivalently matched, and the generalized matching conditions. The "recursive-interlaced" procedure provides a way of determining stabilizability while proceeding with robust control design. The procedure is illustrated on two classes of nonlinear uncertain third order systems. In Chapter 4--written by Giorgio Bartolini, Antonella Ferrara and Tullio Zolezzi--a form of discontinuous state feedback is utilized to achieve exponential tracking for a nonlinear uncertain system. The control system is modelled as a controlled differential inclusion. The delicate issue of sliding mode in sampled-data systems is dealt with by Wu-Chung Su, Sergey Drakunov and Umit Ozgiiner in Chapter 5. The authors describe a new method of implementing discrete time sliding mode control for sample-data systems developed to reduce the thickness of the "boundary layer" around the sliding surface, to alleviate the effect of uncertainties, and to remove the undesired chattering in sliding mode. Three classes of sampleddata systems with uncertainties are investigated; namely, linear, nonlinear, and stochastic systems. By proper consideration of the sampling phenomenon in the control design, one can maintain the states in the vicinity of the sliding surface with accuracy up to at least O(T2), O(T2), and O(T½) respectively, T being the sampling period. Another issue of importance for sliding mode controller is the presence of actuators and measurement devices which leads--as described by Leonid Fridman and Arie Levant, authors of Chapter 6--to the "natural" replacement of regular sliding modes by higher order sliding modes. This is also true when a continuous actuator-like dynamic controller is used in order to avoid real discontinuity of the system. Definitions and examples of these higher order sliding modes are given and compared with other known control theory notions, and their stability is discussed. Finally the authors present an example of a third order sliding algorithm with finite time of convergence. An adaptive discontinuous controller is studied in Chapter 7--by Gene Ryan--to address a servo problem for a fairly general class of nonlinearly perturbed, single input, single output, linear systems with nonlinear actuator char-
1X
acteristic (encompassing hysteresis and dead-zone effects). The reference signal may be chosen from a broad set of signals and a single controller is constructed which guarantees asymptotic tracking of the reference signal for every system belonging to the given class. A possible solution to the problem of robust identification will be found in Chapter 8--written by Tamer Ba~ar, Garry Didinsky and Zigang Pan. The systems considered are linear in the unknown parameters and subjected to unknown disturbances. The authors study the performance attained by this class of algorithms, both theoretically and through simulations, and show that it is superior to that attained by other competing algorithms. The comparative study has been conducted in the absence of control, as well as when a certaintyequivalence controller is in the feedback loop. In Chapter 9, Martin Cortess and George Leitmann present continuous controllers for a class of nonlinear uncertain systems when each control input is subject to an upper bound on its magnitude or norm. The proposed controllers ensure that all closed loop state trajectories which originate in a bounded region exponentially converge to a desired neighborhood of the origin with a desired rate of convergence. In Chapter 10 the authors--Franceseo Amato, Franco Garofalo, Luigi Glielmo and Alfredo Pironti--initially consider a linear system whose matrices depend, as the ratio of multiaffine polynomials, on a vector of uncertain parameters. They show how the problem of synthesizing both state and output feedback controllers, guaranteeing quadratic stability of the closed loop system, can be converted into an optimization problem involving a set of Linear Matrix Inequalities (LMIs). When the dependence on parameters does not fall in the above category, they propose to use an algorithm which, under quite general assumptions, covers the image of a nonlinear function by that one of a multiaffine function; this algorithm applied to the given system, at the price of some conservatism, transforms the quadratic stabilizability problem into an LMIs feasibility problem. The conservatism of the procedure can be reduced at will by using another algorithm which improves, at the price of an heavier computational burden, the goodness of the covering. Chapter l l - - b y Franco Blanchini and Stefano Miani--provides a detailed survey of recent results concerning robust analysis and synthesis for uncertain systems via piecewise linear (polyhedral) Lyapunov functions. After illustrating some basic properties of the polyhedral functions in robust stability analysis, the problem of robust synthesis is considered, providing equations which state necessary and sufficient conditions for a polyhedral function to be a Lyapunov function, and describing a procedure for the generation of polyhedral Lyapunov functions. Finally the authors discuss several (nonlinear) stabilizing controllers which can be associated to polyhedral Lyapunov functions, such as linear-variable structure controllers, minimum effort controllers and bangbang (i.e. piecewise constant) controllers. Numerical examples illustrate the described methods. In Chapter 12 Michel Fliess, Jean Ldvine, Philippe Martin and Pierre Rouchon introduce, in a differential geometric framework, fiat systems, an im-
portant subclass of nonlinear control systems. Utilizing the infinite dimensional geometry developed by Vinogradov and coworkers, flat systems are identified as those systems which are equivalent (in a sense to be specified) to a controllable linear one. The interest of such an abstract setting relies mainly on two points. First, the above system equivalence is interpreted in terms of endogenous dynamic feedback. Second, strong accessibility admits here a straightforward interpretation: the absence of non-triviM first-integrals. The presentation is as elementary as possible and illustrated by many examples, among them the VTOL aircraft. Lev Rapoport, author of Chapter 13, discusses asymptotic stability of selector-linear differential inclusions. While there are various approaches to attack this problem, most of them, based on quadratic stability approach or frequency-to sufficient conditions only. This chapter contains some step toward the derivation of a constructive criterion (i.e. necessary and sufficient conditions). The central idea is the relation between the absolute stability and the existence of the periodic motions of some specific kind. At first this approach is extended to the general two-dimensional selector-linear inclusions. Next the criterion is proven to be valid for the three-dimensional case. The general case of arbitrary dimension is briefly discussed also. Finally, Chapter 14--by Andrey Savkin and lan Petersen--considers a problem of absolute stability for a nonlinear uncertain system with structured uncertainty. An S-procedure for nonlinear systems is established and using this, the chapter shows that the absolute stability problem is equivalent to an absolute stability problem for a nonlinear uncertain system with unstructured uncertainty. The chapter also introduces a new notion of structured dissipativity for nonlinear systems. This notion extends the standard notion of dissipativity in that it allows for multiple supply functions. The chapter shows that a nonlinear uncertain system containing structured uncertainties will be absolutely stable if and only if a corresponding structured dissipativity condition is satisfied. It is also shown that structured dissipativity is equivalent, to the standard notion of dissipativity with a parameter dependent supply function. Here we would like to thank the contributors for their efforts in complying, without complaining, to our requests. A special thank goes to our Neapolitan colleagues th'ancesco Amato and Alfredo Pironti who helped us on various occasions during the preparation of the book. A final thank is for Christopher Greenwell of Springer-Verlag, London, for his help and patience. Napoti, April 1996 Franco Garofalo Luigi Glielmo
Table of Contents
List of Contributors 1
.............................
Approaches to Robust Nonlinear Control
xvii ...........
1
Randy A. Freeman and Petar V. Kokotovid 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 2
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Control Lyapunov Functions . . . . . . . . . . . . . . . . . . . . Robust Control Lyapunov Functions . . . . . . . . . . . . . . . Inverse Optimality . . . . . . . . . . . . . . . . . . . . . . . . . Robust Backstepping . . . . . . . . . . . . . . . . . . . . . . . . Adaptive Backstepping . . . . . . . . . . . . . . . . . . . . . . . Recursive Design . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . .
D y n a m i c a l Sliding M o d e C o n t r o l via A d a p t i v e I n p u t - O u t p u t Linearization: A Backstepping Approach ............
1 1 2 3 6 9 10 12
15
Miguel Rios-Bollvar, Alan S. I. Zinober and Hebertt Sira-Ramlrez 2.1 2.2
2.3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combined Backstepping-SMC Design for Uncertain Linearizable Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 The Adaptive Backstepping Algorithm . . . . . . . . . . 2.2.2 SMC Design via Backstepping . . . . . . . . . . . . . . Adaptive Dynamical SMC Design for Uncertain Nonlinear Systems 2.3.1 Dynamical SMC Design in Output Tracking Problems .
15 16 16 18 20 20
Xll
2.3.2 2.3.3
2.4 2.5
Adaptive Input-Output Linearization via Backstepping A d a p t i v e D y n a m i c a l S M C Design in O u t p u t T r a c k i n g Problems .......................... Design E x a m p l e : T h e B u c k - B o o s t C o n v e r t e r . . . . . . . . . . . Conclusions .............................
23 26 29 33
A Generic Lyapunov Procedure t o D e s i g n R o b u s t Control f o r N o n l i n e a r U n c e r t a i n S y s t e m s : I n t r o d u c i n g Interlacing into 37 Recursive Design ...........................
Zhihua Qu and Joseph Kaloust 3.1 3.2 3.3
3.4 3.5 3.6
4
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Uncertain Systems ......................... 3.2.1 Basic A s s u m p t i o n s . . . . . . . . . . . . . . . . . . . . . Problem Formulation ........................ 3.3.1 T h e Need for a Generic P r o c e d u r e . . . . . . . . . . . . 3.3.2 R e c u r s i v e - I n t e r l a c i n g Design P r o c e d u r e . . . . . . . . . 3.3.3 A Class of L y a p u n o v F u n c t i o n s . . . . . . . . . . . . . . 3.3.4 S t e p - b y - S t e p Design for t h e M o t i v a t i n g E x a m p l e 3.3.5 A G e n e r a l i z a t i o n of t h e Design E x a m p l e . . . . . . . . . F e a t u r e s of t h e Design P r o c e d u r e . . . . . . . . . . . . . . . . . Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nonlinear Uncertainty
Tracking
via
Discontinuous
. .
....
. . . .
Feedback
37 40 42 46 46 50 51 53 61 64 66 70
under
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
Giorgio Bartolini, Antonella Ferrara and Tullio Zolezzi 4.1 4.2 4.3 4.4 4.5 4.6
5
Introduction and Problem Setting ................ T r a c k i n g C o n t r o l v i a Differential I n e q u a l i t y . . . . . . . . . . . Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples ........................... . . . Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions and Further Developments ..............
Implementation of Variable Structure Data Systems .............................
Control
75 76 78 79 81 82
for Sampled-
87
Wu-Chung Su, Sergey V. Drakunov and limit ()zgiiner 5.1 5.2
5.3
5.4
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear Systems ........................... 5.2.1 E x o g e n o u s D i s t u r b a n c e s . . . . . . . . . . . . . . . . . . 5.2.2 S y s t e m P a r a m e t e r V a r i a t i o n s . . . . . . . . . . . . . . . 5.2.3 C o n t r o l Coefficient M a t r i x V a r i a t i o n s . . . . . . . . . . Nonlinear Systems ......................... 5.3.1 M a t c h e d D i s t u r b a n c e s . . . . . . . . . . . . . . . . . . . 5.3.2 C o n t r o l Coefficient V a r i a t i o n s . . . . . . . . . . . . . . . Stochastic Systems ......................... 5.4.1 C o n t i n u o u s M e a s u r e m e n t . . . . . . . . . . . . . . . . .
87 89 91 92 92 94 95 95 97 98
Xlll
5.5 5.6
6
5.4.2 Discrete M e a s u r e m e n t . . . . . . . . . . . . . . . . . . . E x p e r i m e n t a l E x a m p l e - O ( T 2) Sliding Mode C o n t r o l on a Flexible S t r u c t u r e . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
H i g h e r O r d e r S l i d i n g M o d e s as a N a t u r a l Control Theory ............................
Phenomenon
99 t01 104 in 107
Leonid Fridman and Arie Levant 6.1 6.2
6.3
6.4 6.5 6.6 6.7 6.8 7
Introduction ............................. Definitions of Higher Order Sliding Modes ........... 6.2.1 Sliding Modes on Manifolds . . . . . . . . . . . . . . . . 6.2.2 Sliding Modes with Respect to C o n s t r a i n t F u n c t i o n s . . Higher Order Sliding Modes in Control Systems ........ 6.3.1 Ideal Sliding . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Real Sliding a n d F i n i t e T i m e Convergence . . . . . . . . Higher Order Sliding Modes a n d Systems with D y n a m i c A c t u a tors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S t a b i l i t y of Second Order Sliding Modes in S y s t e m s with Fast Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E x a m p l e s of Second Order Sliding Modes . . . . . . . . . . . . Sliding Modes of Order 3 and Higher .............. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
107 109 t10 110 112 112 115 t 17 118 120 127 129
An Adaptive Servomechanism for a Class of Uncertain Nonlinear Systems Encompassing Actuator Hysteresis . . . 135
Eugene P. Ryan 7.1 7.2
7.3
7.4
Introduction ............................. Formulation ............................. 7.2.1 Class 7~ of Reference Signals . . . . . . . . . . . . . . . 7.2.2 System Class S . . . . . . . . . . . . . . . . . . . . . . . T h e A d a p t i v e Strategy . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 T h e S e r v o m e c h a n i s m . . . . . . . . . . . . . . . . . . . . 7.3.2 S c a l i n g - I n v a r i a n t N u s s b a u m F u n c t i o n s . . . . . . . . . . Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 C o o r d i n a t e T r a n s f o r m a t i o n . . . . . . . . . . . . . . . . 7.4.2 T h e Feedback-Controlled System . . . . . . . . . . . . . 7.4.3 T h e Main Result . . . . . . . . . . . . . . . . . . . . . .
A New Class of Identifiers for Robust Parameter and Control in Uncertain Systems ................
135 136 136 137 139 139 140 141 141 141 143
Identification 149
Tamer Ba~ar, Garry Didinsky and Zigang Pan 8.1 8.2
I n t r o d u c t i o n and P r o b l e m F o r m u l a t i o n . . . . . . . . . . . . . . P a r a m e t r i z e d Families of M i n i m a x Identifiers . . . . . . . . . . 8.2.1 F S D I identification . . . . . . . . . . . . . . . . . . . . . 8.2.2 N P F S I identification . . . . . . . . . . . . . . . . . . . . 8.2.3 Prefiltering-based identification . . . . . . . . . . . . . .
149 152 152 t55 160
XIV
8.3
8.4
A Comparative Study ....................... 8.3.1 A c o m p a r i s o n of s t r u c t u r e s . . . . . . . . . . . . . . . . 8.3.2 A c o m p a r i s o n o f t h e r a t e o f convergence . . . . . . . . . 8.3.3 P e r f o r m a n c e u n d e r a controller . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exponential
Convergence
Component-Wise
Bounded
for
Uncertain
Controllers
Systems
162 162 162 164 170
with
.............
175
Martin Corless and George Leitmann 9.1 9.2 9.3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem Statement ......................... A L y a p u n o v Sufficient C o n d i t i o n for E x p o n e n t i a l Convergence 9.3.1 P r o o f o f T h e o r e m 9.4 . . . . . . . . . . . . . . . . . . . 9.4 E x p o n e n t i a l S t a b i l i z a t i o n of t h e N o m i n a l S y s t e m . . . . . . . . 9.5 A s s u m p t i o n s on U n c e r t a i n t y . . . . . . . . . . . . . . . . . . . . 9.6 U n c o n s t r a i n e d C o n t r o l l e r s a n d G l o b a l Convergence . . . . . . . 9.6.1 Exponential Stabilization ................. 9.7 C o n s t r a i n e d C o n t r o l l e r s a n d M a i n R e s u l t . . . . . . . . . . . . 9.7.1 S p e c i a l Case: k l i -= 0 ................... 9.8 R e l a x a t i o n of A s s u m p t i o n s . . . . . . . . . . . . . . . . . . . . 9.9 S t a b i l i z a t i o n of S p a c e c r a f t R o t a t i o n a l M o t i o n . . . . . . . . . . 9.9.1 N u m e r i c a l S i m u l a t i o n R e s u l t s . . . . . . . . . . . . . . . 9.10 A p p e n d i x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Quadratic Stabilization o f U n c e r t a i n Linear S y s t e m s
....
175 176 178 179 182 183 184 185 185 187 187 189 191 192 197
Francesco Amato, Franco Garofalo, Luigi Glietmo and Alfredo Pironti 10.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Q u a d r a t i c S t a b i l i z a t i o n v i a S t a t e F e e d b a c k of a Class o f UncertMn Linear Systems ........................ 10.3 Q u a d r a t i c S t a b i l i z a t i o n v i a O u t p u t F e e d b a c k . . . . . . . . . . 10.4 A G e n e r a l P r o c e d u r e . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 A n A l g o r i t h m to Cover t h e I m a g e of a N o n - M u l t i a f f i n e F u n c t i o n b y t h e I m a g e of a Multiaffine F u n c t i o n . . . . 10.4.2 S o l u t i o n o f P r o b l e m 10.3 in t h e N o n - M u l t i a f f i n e Case 10.4.3 A R e f i n e m e n t P r o c e d u r e . . . . . . . . . . . . . . . . . . 10.5 C o n c l u s i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 P i e c e w i s e - l i n e a r Functions in R o b u s t Control
.........
197 198 202 203 204 206 208 210
213
Franco Blanchini and Stefano Miani 11.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 P r e l i m i n a r y results a n d n o t a t i o n s . . . . . . . . . . . . . . . . . 11.2.1 N o t a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 S y s t e m s u n d e r c o n s i d e r a t i o n a n d m a i n definitions . . . 11.2.3 L y a p u n o v f u n c t i o n s a n d i n v a r i a n t sets . . . . . . . . . . 11.3 S y n t h e s i s o f p o l y h e d r a l L y a p u n o v f u n c t i o n s . . . . . . . . . . . 11.4 R o b u s t a n a l y s i s v i a piecewise-linear functions . . . . . . . . . .
213 215 215 216 218 223 226
XV
11.4.1 T r a n s i e n t analysis . . . . . . . . . . . . . . . . . . . . . 226 11.4.2 D e t e r m i n a t i o n of the largest d o m a i n of a t t r a c t i o n . . . 229 11.5 R o b u s t l y stabilizing control laws . . . . . . . . . . . . . . . . . 231 11.5.1 Linear p r o g r a m m i n g control . . . . . . . . . . . . . . . . 232 11.5.2 Piecewise-linear robust, stabilizing control . . . . . . . . 233 11.5.3 B a n g - b a n g control for s i n g l e - i n p u t c o n t i n u o u s - t i m e s y s t e m s 2 3 4 11.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 12 A L i e - B ~ i c k l u n d A p p r o a c h t o D y n a m i c F e e d b a c k E q u i v a l e n c e and Flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
Michel Fliess, Jean Ldvine, Phihppe Martin and Pierre Rouchon 12.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Five i n t r o d u c t o r y examples . . . . . . . . . . . . . . . . . . . . 12.2.1 T h e V T O L e x a m p l e . . . . . . . . . . . . . . . . . . . . 12.2.2 T h e V T O L with a model of the a c t u a t o r s . . . . . . . . 12.2.3 T h e inverted p e n d u l u m e x a m p l e . . . . . . . . . . . . . 12.2.4 A n implicit m o d e l of p e n d u l u m . . . . . . . . . . . . . . 12.2.5 T h e Huygens oscillation center . . . . . . . . . . . . . . 12.2.6 C o n c l u s i o n . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 I n f i n i t e - d i m e n s i o n a l differential g e o m e t r y . . . . . . . . . . . . 12.3.1 Infinite n u m b e r of coordinates . . . . . . . . . . . . . . 12.3.2 C h a n g e s of coordinates a n d Lie-Bgcklund m a p p i n g s . . 12.3.3 F l a t n e s s . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 I n t e r p r e t a t i o n of equivalence in t e r m s of feedback . . . . . . . . 12.4.1 Trivial residual d y n a m i c s . . . . . . . . . . . . . . . . . 12.5 C o n t r o l l a b i l i t y a n d first integrals . . . . . . . . . . . . . . . . . 12.5.1 O n the strong accessibility property . . . . . . . . . . . 12.5.2 A general definition of c o n t r o l l a b i l i t y . . . . . . . . . . . 12.5.3 C o n t r o l l a b i l i t y of flat systems . . . . . . . . . . . . . . . 12.5.4 A linear controllable d y n a m i c s is fiat . . . . . . . . . . . 12.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
245 246 246 246 248 249 249 250 250 250 255 258 260 262 263 263 264 264 265 266
13 A s y m p t o t i c S t a b i l i t y a n d P e r i o d i c M o t i o n s o f S e l e c t o r - L i n e a r Differential Inclusions . . . . . . . . . . . . . . . . . . . . . . . . 269
Lev B. Rapoport 13.1 P r o b l e m S t a t e m e n t . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 M a i n Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 T h e T w o - D i m e n s i o n a l Case . . . . . . . . . . . . . . . . 13.2.2 T h e T h r e e - D i m e n s i o n a l Case . . . . . . . . . . . . . . . 13.2.3 T h e n - D i m e n s i o n a l Case . . . . . . . . . . . . . . . . . .
269 270 274 277 284
14 S t r u c t u r e d D i s s i p a t i v i t y a n d A b s o l u t e S t a b i l i t y o f N o n l i n e a r Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
Andrey V. Savkin and Ian R. Petersen 14.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
287 289
X~I
14.3 S - P r o c e d u r e for N c n f i n e a r Systems . . . . . . . . . . . . . . . . 14.4 T h e M a i n Result . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 A b s o l u t e S t a b i l i t y . . . . . . . . . . . . . . . . . . . . . . . . .
291 296 299
List of C o n t r i b u t o r s Dr. Francesco Amato Dipartimento di Informatica e Sistemistica Universit~ di Napoli "Federico II" via Claudio, 21 80125 Napoli, Italy e-maih amatoOdian, dis. unina, it Professor Giorgio Bartolini Dipartimento di Informatica, Sistemistica e Telematica Universit~t di Genova via dell'Opera Pia, l l A 16145 Genova, Italy e-maih giobOdist, dist. unige, it Professor Tamer Ba~ar Coordinated Science Laboratory University of Illinois 1308 West Main Street Urbana, IL 61801, USA e-mail: t b a s a r © i s a a c s , c s l . uiuc. edu Professor Franc© Blanchini Dipartimento di Matematica ed Informatica Universitk degli Studi di Udine via delle Scienze 208 33100 Udine, Italy e-maih blanchini©uniud, it Professor Martin Corless Department of Aeronautics and Astronautics Purdue University West Lafayette, IN 47907, USA e-maih corless©ecn, purdue, edu Garry Didinsky Coordinated Science Laboratory University of Illinois 1308 West Main Street Urbana, IL 61801, USA Professor Sergey V. Drakunov Department of Electrical Engineering Tulane University New Orleans, LA, USA e-maih drakunov©mailhost, t c s . t u l a n e , edu
xviii Dr. Antonella Ferrara Dipartimento di Informatica, Sistemistica e Telematica Universit~ di Genova via dell'Opera Pia, l l A 16145 Genova, Italy e-mail: ferraraGdist, dist. unige, it
Professor Michel Fliess Laboratoire des Signaux et Systbmes C.N.R.S.-Supdlec, Plateau de Moulon 91192 Gif-sur-Yvette, France e-mail: fliess©lss, supelec, fr
Professor Randy Freeman Department of Electrical and Computer Engineering Northwestern University Evanston, Illinois, USA e-mail: freeman@leo, eecs. nwu. edu
Professor Leonid Fridman Samara Architecture and Building Institute Samara, Russia e-mail: fridman©icc, ssaba, samara, ru Professor Franco Garofalo Facolt~ di Ingegneria di Benevento Universit~ di Salerno Corso GaribMdi, Palazzo Bosco 82100 Benevento, Italy e-mail: g a r o f a l o C d i s n a , d i s . unina, i t Professor Luigi Glielmo Dipartimento di Informatica e Sistemistica Universit~ di Napoli "Federico II" via Claudio, 21 80125 Napoli, Italy e-mail: glielmo@disna, dis. unina, it
Joseph Kaloust Department of Electrical and Computer Engineering University of Central Florida Orlando, FL 32816, USA e-mail: j ok©engr, ucf. edu
x/x Professor Petar V. Kokotovi~ Center for Control Engineering and Computation Department of Electrical and Computer Engineering University of California Santa Barbara, CA 93106, USA e-mail: petar@seidel, ece. ucsb. e d u Professor George Leitmann College of Engineering University of California Berkeley, CA, USA e-mail: g l e i t © c o e , b e r k e l e y , e d u Professor Arie Levant Institute for Industrial Mathematics Beer-Sheva, Israel e-mail: levant~bgumail, bgu. ac. i l Jean L~vine Centre Automatique et Syst~mes ]~cole des Mines de Paris 35 rue Saint-Honore 77305 Fontainebleau, France e-mail: levine©cas, ensmp, f r Philippe Martin Centre Automatique et Syst~mes I~cole des Mines de Paris 35 rue Saint-Honore 77305 Fontainebleau, France, e-maih martin©cas, ensmp, f r Dr. Stefano Miani Dipartimento di Elettronica e Informatica Universit~ degli Studi di Padova via Gradenigo, 6/a 35131 Padova, Italy Professor Umit ()zgiiner Department of Electrical Engineering The Ohio State University Columbus, OH, USA e-mail: umit©ee, eng. ohio-state, edu
.KX
Zigang Pan Coordinated Science Laboratory University of Illinois 1308 West Main Street Urbana, IL 61801, USA Professor Ian R. Petersen Department of Electrical Engineering Australian Defence Force Academy Campbell, 2600, Australia e-maih irp©ee, adfa. oz. au Dr. Alfredo Pironti Dipartimento di Informatica e Sistemistica Universith di Napoli "Federico II" via Claudio, 21 80125 Napoli, Italy e-mail: piront i©disna, dis. unina, it Professor Zhihua Qu Department of Electrical and Computer Engineering University of Central Florida Orlando, FL 32816, USA e-maih quz©engr, u c f . edu Professor Lev B. Rapoport Institute of Control Sciences Profsoyuznaya 65 Moscow 117806, Russia e-maih lionGashtech, msk. ru Dr. Miguel Rios-Bollvar Departamento Sistemas de Control Escuela de Ingenierla de Sistemas Universidad de Los Andes M4rida 5101 Venezuela e-mail: m. rios@sheffield, ac. uk
Pierre Rouchon Centre Automatique et Syst~mes l~cole des Mines de Paris 60 bld. Saint-Michel 75272 Paris cedex 06, France e-mail: rouchon©cas, ensmp, f r
XX
Dr. Eugene P. Ryan School of Mathematical Sciences University of Bath Claverton Down Bath BA2 7AY, UK e-mail: epr©maths, bath. ac. uk Dr. Andrey V. Savkin Department of Electrical and Electronic Engineering and Cooperative Research Center for Sensor Signal and Information Processing University of Melbourne Parkville Victoria 3052, Australia e-maih savkin©ee, mu. oz. a u Professor Hebertt Sira-Ramirez Departamento Sistemas de Control Escuela de Ingenierla U.L.A. Avenida Tulio Febres Cordero Universidad de Los Andes M4rida 5101 Venezuela t-mail: isira©ing, ula. ve Dr. Wu-Chung Su Department of Electrical Engineering National Chung-Hsing University Taiwan Republic Of China Professor Alan S. I. Zinober Applied Mathematics Section School of Mathematics and Statistics The University of Sheffield Sheffield S10 2TN, UK e-mail: a. zinober©sheff ield. ac. uk
Professor Tullio Zolezzi Dipartimento di Matematica Universit~ di Genova via L. B. Alberti, 4 16132 Genova, Italy
1. A p p r o a c h e s to R o b u s t N o n l i n e a r Control R a n d y A. F r e e m a n and P e t a r V. K o k o t o v i d 1.1 I n t r o d u c t i o n In the last decade, major progress has been made in the development of a geometric theory of nonlinear feedback systems (as summarized in (Isidori, 1989; Nijmeijer and van der Schaft, 1990)). Combined with the well-established Lyapunov stability theory, these geometric results form a solid foundation on which to build systematic design procedures. To become practical, such procedures must overcome two important obstacles. First, they must expand the geometric methods to incorporate uncertainties in the system models. Second, they must deal with the crucial shortcoming of the Lyapunov approach, namely, the lack of tools for the systematic construction of Lyapunov functions. In this chapter we survey recent advances in the latter direction. We present some basic tools and illustrate how they can be combined in recursive design procedures. The references in this chapter will be presented primarily at the end of each section; they are intended to be representative rather than exhaustive.
1.2 C o n t r o l L y a p u n o v F u n c t i o n s Concepts from Lyapunov stability theory are suitable for analysis, but how can they be converted into design tools? One possibility is to explicitly express the dependence of the derivative ~)(x, u) of a candidate Lyapunov function V(x) on the control variable u, and then show that V can always be made negative by a judicious choice of u. This leads to the definition of the control Lyapunov function (clf) and its robust counterpart, the robust control Lyapunov function (rclf). We will briefly discuss the relationship between the existence of such functions and the stabilizability of nonlinear systems. We also discuss the relationship between the relf and an associated differential game; this gives us a recipe for choosing a feedback law which is optimal with respect to a reasonable performance criterion. We are interested in designing continuous state feedback laws u = k(x) which drive the state x to the origin x = 0 from any initial condition. Our goal is to achieve global uniform asymptotic stability (GUAS) of the origin, or, when this is not possible, global uniform stability and ultimate boundedness (GUSUB) with respect to some positively invariant residual set containing the origin. We define three types of stabilizability in the order of their desirability. We say a system is globally asymptotically stabilizable (GAS) when there exists
a feedback law which renders the origin GUAS. We say a system is globally practically stabilizable (GPS) when for every ¢ > 0 there exists a feedback law which renders the system GUSUB with respect to a subset of an c-ball around the origin. Finally, we say a system is globally stabitizable (GS) when there exists a feedback law which renders the system GUSUB. These three types of stabilizability have robust counterparts RGAS, RGPS, and RGS, all defined when stability or boundedness holds for any admissible uncertainty. A control Lyapunov function (clf) for a nonlinear system & = f(x, u) is a C I, positive definite, proper function V(x) for which there exists a bounded s e t / 7 such that x~H
~
3u such that V V ( x ) . f ( x , u ) < O.
(1.1)
Thus V is a clf when for every x ~ H we can find some value of the control u which makes V negative. At first it would seem that finding a clf for the system is tantamount to solving the stabilization problem. However, several issues remain to be addressed once a clf is found. One of these issues is whether or not a continuous feedback law u = k(x) can be found to satisfy (1.1). For this we must place additional restrictions on the system and on the clf. If we assume our system is aj]ine in the control, then the existence of a clf implies that the system is GS. If in addition we have /7 = {0}, then the existence of a clf implies that the system is GPS. In order to guarantee asymptotic (rather than practical) stabilizability, our clf must also satisfy a small control property (scp) which says that small values of u satisfy (1.1) whenever x is small. Thus for systems affine in the control, the existence of a clf (with/7 = {0}) satisfying the scp is equivalent to GAS (necessity comes from converse Lyapunov theorems). Once a clf is known, a stabilizing feedback law u = k(x) can be found analytically or numerically. But how do we find a clf for a given system? For globally feedback linearizable systems, we can use the linearizing coordinate transformation and find a clf by solving the linear Lyapunov matrix equation. For broader classes of systems, we can find a clf using recursive procedures such as backstepping (discussed below). In general, however, the question of how to find a clf remains open. The basic theorems relating stabilizability to the existence of clf's are found in (Artstein, 1983; Tsinias, 1989; Sontag, 1989a). The connection between clf's and asymptotic controllability is described in (Sontag, 1983). Some sufficient conditions for the existence of clf's are given in (Tsinias, 1990a).
1.3
Robust
Control
Lyapunov
Functions
We now extend the clf concept to systems with uncertainties. For reasons mentioned above, we only consider systems which are affine in the control: tr~
= f(x,.,w)
=
(1.2) i=1
Here u = ( u l , . . . , u,~) is the control and w is a disturbance representing the system uncertainties We assume the disturbance takes its values in some compact set W; in general we allow W to depend on x in a continuous manner. A robust control Lyapunov function (rclf) for the system (1.2) is a C 1, positive definite, proper function V(x) for which there exists a bounded set H such that x ~H
~
3u such that m a x [ V V ( x ) wEW
f(x,u,w)] < 0.
(1.3)
Thus V is an relf for our system when for every x ~ / 7 we can find some value of the control u which makes V negative regardless of the value of the disturbance w E W. The basic theorem for rclf's is similar to the one for clf's: the existence of an rclf for the system (1.2) implies RGS; if in addition we have H = {0} then the system is RGPS; and if in addition the scp is satisfied then the system is RGAS. Note, however, that the scp is even more restrictive when uncertainties are p r e s e n t - - i t implies that the uncertainties vanish as x approaches the origin! Therefore, for systems with uncertainties we often deal with RGS or RGPS rather than RGAS. In the case where we have RGS but not RGPS, the shape and size of the achievable residual set depend on the rctf. By adjusting the rclf we can sometimes arbitrarily reduce the size of the residual set in some directions by allowing it to peak in other directions. How do we find an rclf for a given system? One might start by finding a clf for the nominal system (e.g., w = 0) and then checking to see if the clf is also an rclf. It is obvious from (1.1) and (1.3) that this method works when w enters the system equations at exactly the same place as u; this is commonly known as the matching condition. This method also works if the mismatched uncertainties are sufficiently small. In general, however, we cannot obtain an rclf from just any ctf for the nominal system. We will show in Sections 1.5 and 1.7 how backstepping can be used to find rclf's for systems with large mismatched uncertainties. The basic theorems relating robust stabilizability to the existence of rclf's have been recently proven in (Freeman and Kokotovid, 1996). The method of finding an rclf for a system satisfying the matching condition by using any ctf for the nominal system can be found for example in (Corless and Leitmann, 1981; Barmish et al, 1983). The analysis showing that this method also works when mismatched uncertainties are sufficiently small can be found in (Chen and Leitmann, 1987). A more general matching condition under which one can use any elf for the nominal system is given in (Chen, 1993).
1.4 Inverse Optimality Our design is not over once we find an rclf; we are still faced with the choice of an appropriate feedback law. Usually at least one feedback law is known (perhaps obtained from the construction of the rclf), but there is no guarantee that it will provide adequate performance. We would prefer to choose a feedback law according to some reasonable optimality criterion, but how do we achieve
optimality while maintaining the negativity of V? It turns out that every rclf is an optimal upper value function of an associated differential game with a reasonable cost functional. We can therefore use this game to obtain an optimal feedback law which renders I / n e g a t i v e as desired. The calculation of the optimal feedback law can be formulated as a static nonlinear programming problem involving V, the system equations and constraints, and adjustable design parameters. If we follow this procedure for choosing the feedback taw, then the cost functional of the game gives us some measure of the performance of the closed-loop system. These results can be found in (Freeman and Kokotovid, 1996), and they represent a solution to the inverse optimal control problem (surveyed in (Glad, 1987)) in a game setting. Other authors have also used the construction of a cost functional as part of the design of a stabilizing feedback law; see for example (Sontag, 1989) for the state feedback case and (Tsinias, 1990b) for the output feedback case. Homogeneity properties of the system are exploited in (Hermes, 1991) in the construction of a cost functional. Instead of constructing a cost functional as a step in the design procedure, a more common framework is to directly pose the optimal control problem for a given cost functional. One such formulation leads to nonlinear H¢¢ control as described for example in (Isidori and Astolfi, 1992) and the references therein; these results are based on the game theory approach to linear H ~ control as described in (Ba~ar and Bernhard, 1991). The nonlinear H¢~ framework requires the solution of a Hamilton-Jacobi type partial differential equation, the solvability of which may not be feasible other than locally. Consequently only local stability results are usually obtained in this framework. We can illustrate the main point of inverse optimality by means of an elementary example. Suppose we wish to robustly stabilize the first-order system =
(1.4)
-x3+u+wx,
where u is an unconstrained control input and w is a disturbance input taking values in the interval [-1, 1]. A robustly stabilizing state feedback control law for this system is u
=
(1.5)
x3-2x.
This particular control law is the one suggested by feedback linearization (Isidori 1989; Nijmeijer and van der Schaft, 1990), and it indeed renders the solutions to the system (robustly) globally asymptotically stable. However, it is an absurd choice because the term x 3 in (1.5) represents control effort wasted to cancel a beneficial nonlinearity. Moreover, this term is actually positive feedback which increases the risk of instability. It is easy to find a better control law for this simple system, but what we desire is a s y s t e m a t i c method for choosing a reasonable control law given an rclf for a general system. One approach would be to formulate and solve an optimal robust stabilization problem with a cost functional which penalizes control effort. For the system (1.4), the cost functional J
=
i
x 2 + u 2 dt
(1.6)
is minimized (in the worst case) by the optimal feedback law u
=
x
x - • j x 4 _ 2x2 j 2 .
(1.7)
The control laws (1.5) and (1.7) are plotted in Figure 1.1. The optimM control law (1.7) recognizes the benefit of the nonlinearity - x 3 and accordingly produces little control effort for large x; moreover, this optimal control law never generates positive feedback. However, such superiority comes at the price of solving a steady-state Hamilton-Jacobi-Isaacs (HJI) partial differential equation, a task feasible only for the simplest of nonlinear systems. Indeed, for a general system and cost functional of the form =
f(z,u,w),
J =
//
L ( z , u ) dt,
(1.8)
[L(x,u) + VV(x). f(x,u,w)],
(1.9)
the steady-state HJI equation is 0 = rain max W
where the value function V(x) is the unknown. For an appropriate choice of the function L(x, u) in (1.8), a smooth positive definite solution V(z) to this equation (1.9) will lead to a continuous state feedback control u(x) which provides optimality, stability, and robustness with respect to the disturbance w. However, such smooth solutions may not exist or may be extremely difficult to compute. The main point of inverse optimality is that a known rclf for a system can be used to construct an optimal control law directly and explicitly, without recourse to the HJI equation (1.9). Indeed, in (Freeman and Kokotovid, 1996) we provide a formula which generates a class of such optimal control taws and which involves only the rclf, the system equations, and design parameters. The control laws given by our formula are called pointwise min.rwrm control laws, and each one inherits the desirable properties of optimality. For example, the simplest pointwise rain-norm control law for the system (1.4) is u
=
x - 2z 0
when x 2 < 2, when x 2 > 2.
(1.10)
This control law is compared to the optimal control law (1.7) in Figure 1.2. We see that these two control laws, both of which are optimal with respect to a meaningful cost functional, are qualitatively the same. They both recognize the benefit of the nonlinearity - x 3 in (1.4) and accordingly expend little control effort for large signals; moreover, these control laws never generate positive feedback. The main difference between them lies in their synthesis: the pointwise rain-norm control law (1.10) came from the simple formula we provide in (Freeman and Kokotovid, 1996), while the control law (1.7) required the solution of an HJI equation. In general, the pointwise min-norm calculation is feasible but the HJI calculation is not. After we find an rclf V for our system and choose some preferred feedback law based on V, we can implement the control and achieve stability as desired.
2
I
I
I
I
I
I I
!
1.5
I
(1.5).
-
(1.7) . . . .
1
:
:
0.5 u
*'*''*'"''"'*" "*'*
0
, .........i
-0.5 -I -1.5 -2 -4
I
II
I
-3
-2
-i
I
t
I
I,,
0
1
2
3
4
X
Figure 1.1
A comparison between the control laws (1.5) and (1.7)
However, we must still ask whether or not there is a better rclf for our system. We will see below t h a t our choice for an rclf can be crucial for the system performance, and the question of finding a "good" rclf remains largely open. A case study on the effects of different choices of rclf's and feedback laws on the system performance can be found in (Freeman and Kokotovi~, 1992b).
1.5 Robust B a c k s t e p p i n g We will use a variety of design tools to find a stabilizing feedback law for the following second-order single-input system:
z: =
x2 + w : x 3,
(1.11)
z2
u+w~.
(1.12)
=
The disturbance w = (w:, w2) represents the uncertainty in the system, and our choice of a design tool depends on what we know about w. If we assume knowledge of a compact set W such that w(t) E W for all t ~ 0, then we can use the robust backstepping tool of Section 1.5. We will later modify the backstepping tool for the case where the set W exists but is unknown. If w is an unknown constant, we can use the adaptive backstepping tool of Section 1.6. In this section we assume w(t) G W for all t > 0, where W = [ - 1 , i] 2. The design involves two m a j o r steps: first, we must find an rclf for the system; second, we must choose a feedback law according to the rclf. We construct an rclf by first treating w: as a matched uncertainty: we pretend the state z2 is the control and consider only the first state equation (1.11). The uncertainty w: is matched with respect to x~, and so we can use a clf V1(Xl) for the "nominal" system ~1 = z2 as an rclf for the first state equation (1.11).
2
I
1
t
I
I
I
I
(1.10) - -
1.5
(1
t
7) . . . .
0.5 u
0 -0.5 -1 -1.5 }
t
I
-3
-2
-1
-2 -4
I
t
I
I
0
1
2
3
X
F i g u r e 1.2
A comparison between the control laws (t.10) and (1.7)
1 2 and we obtain An obvious choice for V1 is Vl(xl) = 7Zl, m~xV1
m ~ x ( x l x 2 + w l x 4)
=
=
zlx2+x 4
(1.13)
which can clearly be m a l e negative (for Xl nonzero) by choice of x~. We next need to find a smooth feedback law kl(x~) for x2 such that (1.13) is negative definite. A simple choice is x2 = k l ( x l ) = - x l - x 3. This choice will determine the shape of the rclf for the entire second-order system, and it is clearly nonunique. We therefore emphasize the role of kl(xl) as a design choice to be adjusted to achieve desired performance. After we design V~(x~) and kz (xl) according to the above preliminary step, we are ready to construct an rclf V for the entire second-order system. We will took for a function ~/~(xl,x2) so that V = V1 + V2 is an rctf for our system. Our choice for V2 will be based on the following calculation: max w
<
max ~ + max ~2
<
z,z2 + x 4 + maxV2
<
- x ~ + Xl [x2 -- kl(xl)] + max"/2.
w
w
(1.14)
w
The second line follows from (1.13), and the third line shows the discrepancy between x2 and our choice for kl (x 1)- From this calculation we see that a simple 1 X ~ - kl(xl)] ~, and with this choice we obtain choice for V~ is V2 = 7[ maxV w
<
- x ~ + xl [.2 - kl(xl)] + Ix2 - k~(x,)] u (1.15)
where kl can be explicitly calculated as a function of xl, x2, and wl. Clearly the right-hand side of (1.15) can be made negative by choice of u for all (xl, x2) ¢ (0,0), and it follows that V is an rclf for our system with / / - - {(0,0)}. We thus conclude that the second-order system (1.11)-(1.12) is RGPS. One can show that if w2 were not present, the scp would hold and we would therefore conclude that the system is RGAS. We have found an rclf and thus determined that our system is stabilizable. The second major step in the design is to choose a stabilizing feedback law u --k ( x l , x2) according to the rclf. This step can be accomplished by manipulating the expression (1.15) using Young's inequality (or completing the squares) and then deciding on k by inspection; however, this method does not necessarily lead to a desirable feedback law (cf. Section 1.4). Alternatively one could solve the static nonlinear programming problem in (Freeman and Kokotovi6, 1996) to obtain an optimal k. Our above choice for the rclf was perhaps the simplest one, but it turns out that it can lead to feedback laws with unnecessarily large local gains in critical areas of the state space. Thus even if we choose an optimal feedback law according to the above rclf, the closed-loop behavior can be poor. One can try to improve performance by scaling V1 and V~, but the quadratic nature of V2 as chosen above is an obstacle to any significant improvement. To construct a better V2, we take advantage of our knowledge of the middle term in (1.14). In certain regions of the state space where this middle term is negative, we are free to choose V2 = 0 and thus make V independent of u. Such a "flattened" rclf allows for greater flexibility in the choice for a feedback law, and as a consequence the required local gain can be greatly reduced. The proper choice for the relf is therefore cruciM in determining closed-loop performance. The terminology "integrator backstepping" was introduced in (Kanellakopoulos et al, 1992), and the technique has evolved from the early results (Tsinias, 1989; Byrnes and Isidori, 1989; Kokotovi6 and Sussmann, 1989). A generalization to the nonsmooth case is given in (Coron and Praly, 1991). What if the set W above is compact but unknown? In (Kanellakopoulos, 1993) it was shown that the above tools can be applied to find an rclf, and in this case the bounded set / / will be unknown. We can therefore conclude that the system is RGS, but for a given feedback law we will have no a priori knowledge of the size of the resulting residual set. As an illustration, suppose W = I-c, c]2 where c is an unknown positive constant. Then the calculation in (1.13) becomes max V1 -- x , x 2 ..t- cx 4. (1.16) w
Because c is unknown, we can no longer choose kl(xl) "- - x l - x l a as above; we must include a term which will dominate the c-term regardless of the value of c. It is clear from (1.16) that we need to choose k l ( x l ) - - x l - o ~ ( x l ) where c~(xl) is any smooth function which satisfies (~(0) = 0 and also xlc~(xl)/x~ -~ (x) as txlt --~ c~. One possibility is c~(x~) -- x~, and with this choice the calculation in (1.14) becomes
maxI/
_< x l x 2 + cx 4 + maxI/~
W
W
<
(1.17)
-x~-x~+cx4+xl[x2-kl(Xl)]+maxV2.
--
W
The combination of the first three terms in the right-hand side of (1.17) will be negative for xl sufficiently large, regardless of the value of c. Thus by choosing V2 as above (either the quadratic one or the flattened one), we obtain an rclf for our system but with u n k n o w n / / . What has been accomplished can be described in terms of input-to-state stability (ISS) with the disturbances wt and w2 regarded as inputs. The relationship between the size of the residual set and c is determined by the ISS gain of the system, and this gain can be manipulated by adjusting the rclf and the feedback law. If we decrease the ISS gain (usually by increasing the control effort), we can reduce the effect of the disturbances on the system.
1.6 Adaptive Backstepping If we know more about the system uncertainties, we can make the stabilizing feedback laws less conservative and use smaller gains. A common situation is when all uncertainties occur as unknown constant parameters appearing linearly in the state equations; then we can modify the backstepping tool to include adaptation. In this way we can achieve asymptotic stabilization even without knowledge of bounds on the uncertainties. To illustrate, suppose Wl and w2 in the system (1.11)-(1.12) are constant unknown parameters. In this section we wilt design a feedback law u = k ( x l , x 2 , 3 1 , 3 2 ) and parameter update laws for 31 and 32 using an adaptive backstepping procedure. The aim of this prgeedure is to cancel (rather than dominate) the uncertain terms appearing in V. We first pretend the state x2 is the control and consider only the first state equation (1.11). If we choose V1 = ~x 1 2 1 + ~(wl 1 A -- wl) 2 we obtain ~71 =
(1.1S)
x l x 2 + w l x 4 + (31 - Wl)CZl.
We next find a control law kl(xt, 31) for the control x2; a "certainty equivalence" choice would be kl = - x l - 31x 3 (exact cancellation would occur if wl = 31). Substituting,
1/1 =
- x ~ + (3, - wl)('~'1 - x~) + Xl [x2 - k l ( x l , 31)].
(1.19)
We could choose an update law ~1 = x 4 to eliminate the second term in (1.19), but this would lead to overparameterization. We instead call the would-be update law x 4 a tuning function and proceed with the design. To keep the design simple, we will choose the quadratic form for 1/2 (rather than the flattened form): V2 = ~[ 2 - k l ( x l , 31)] 2 + ~(w21 ~ _ w2)2. We compute V2 as follows: =
-
301 (- +
+
-
(1.20)
10 where ]¢1 can be explicitly calculated as a function of xl, x2, wl, wl, and $1, with the unknown parameter wl appearing linearly. We now take V = V1 + V2 and add (1.20) to (1.19) to obtain --X 2 -~ IX2 -- k l ( X l , W l ) ] (~t -}- x 1 -~ w 2 - ]~1)
-~
+ (~1 - ~1)($1 - ~ ) + (~2 - ~ ) $2.
(1.21)
At this point we would like to use part of the control u to cancel the matched terms xl + w2 - ]¢1. However, these terms contain the unknown parameters wi and thus cannot be exactly cancelled. We instead "cancel" these terms using certainty equivalence, i.e., wherever we would like to use wi we instead use wi. Because the unknown parameters wi appear linearly, the use of certainty equivalence produces errors which can be factored into the terms in (1.21) containing w l - wl and w2 - w2. Thus setting u - v + certainty equivalence terms, we obtain
=
-~
+ [ ~ - k1(~1, ~1)] v + (~1 - wl) ($1 - Zl) + (w2 - w2) ($2 - / ~ )
(1.22)
where the functions/31 and/32 contain no uncertainties. We can thus eliminate the last two terms in (1.22) by choosing update laws $1 --/~1 and $2 =/32. We then choose v so that V is nonpositive and Y < 0 whenever (Xl, x2) ¢ (0, 0) (for example, v -- - x 2 + kl). We conclude that the closed-loop fourth-order system is globally uniformly stable, and moreover that the states xl and x~ converge to zero. The adaptive backstepping tool evolved from the extended matching condition of (Kanellakopoulos et al, 1991a), and it is the main topic of the recent text (Krsti~ et al, 1995).
1.7
Recursive
Design
We can use the robust backstepping tool of Section 1.5 to find stabilizing controllers for systems of the form ~2
=
~2 + ¢ l ( x , u , t ) ,
(1.23a)
=
x3 + ¢ 2 ( x , u , t ) ,
(1.23b)
= u+¢n(x,u,t). (1.23c) Such a system is a chain of integrators perturbed by uncertain nonlinearities ¢i. The following assumption on the uncertainties ¢i allows us to apply the backstepping tool: for each ¢i we assume knowledge of a bounding function P i ( X l , . . . , xi) such that [¢i(x,u,t)[ _< ; i ( x l , . . . , z ~ ) (1.24)
11 for all x, u, and t. The restriction here is that the i-th bounding function pi depends only on the states xl through xi; we call this the strict feedback dominance condition. We first show that the above uncertainty structure can be formulated in terms of a disturbance w and a compact set W as required in the rclf definition. Indeed, we let w = (wl . . . . , w~,) where wi = ¢i(x, u, t) for each i. The set W will be of the form W = 14] × ... z W~ where Wi = [-p~, Pi]. From (1.24) we see that each set Wi is a function of the states xl through xi. The resulting set W(x) is compact for each fixed x. We construct an rclf for the system (1.23) by repeatedly applying the backstepping tool of Section 1.5. We first pretend x2 is the control and design an rctf V1(xl) and a feedback law x2 = kl(Xi) according to the first state equation (1.23a). We then construct a function V2(xl,x2) as in Section 1.5 such that V1 + V2 is an rclf for the second-order system (1.23a)-(1.23b) with x3 now acting as the control variable. We then design a stabilizing feedback x3 = k~(xt, x2) according to V1 + V2. We repeat this procedure with the addition of each integrator, at each step constructing a function V/ and a feedback law ki, until we obtain an rclf V = V1 + - - - + V~ for the entire system (t.23). Finally, we choose a stabilizing feedback law u = k(x) according to the rclf V. The strict feedback dominance condition on the uncertainties guarantees t h a t this recursive procedure is well-posed. At each step in the recursive procedure, we have plenty of freedom in our choices for Vi and ks, and these choices will be crucial in determining the performance of the system. For example, if we use the flattened form for each V/instead of the quadratic form, we can obtain a feedback law with much lower local gains (especially when n is large). Such a feedback law can provide improved performance with less control effort as compared to a feedback law based on the quadratic ~ . The flattened rclf was also instrumental in solving the problem of robustness with respect to unknown state measurement disturbances. By using the flattened ibrm we can design a feedback law k(x) such that the system with control u = k(x +d) is (globally) ISS with respect to the input d (here d represents state measurement error due to inaccurate sensors or the use of an observer). This additional robustness is important when exact ~state measurements are unavailable. The above design procedure first appeared in (Freeman and Kokotovid, 1992a; Marino and Tomei, 1993), and a recursive sliding mode version is given in (Slotine and Hedrick, 1993). A comprehensive version is described in (Qu, 1993), where it is shown that the strict feedback form is a nonlinear extension of the generalized matching condition of (Thorp and Barmish, 1981). The design using a flattened rctf and its associated advantages are described in (Freeman and Kokotovid, 1993a). The modification of the design to achieve robustness with respect to state measurement error is given in (Freeman and Kokotovid, 1993b). Suppose each uncertainty ¢i in the system (1.23) can be written as
12
¢i(x,u,t)
=
oTTi(Xl,...,xi)
(1.25)
where 0 is a vector of unknown constant parameters and each ~/i is a known smooth function satisfying 7/(0) = 0. Note that similar to the bounding functions Pi of Section 1.7, the i-th function 7/ depends only on the states xl through xi. Systems with such uncertainties are said to be in parametric strict feedback form. For these systems we can repeatedly apply the adaptive backstepping tool of Section 1.6 to obtain a stabilizing controller, just as we repeatedly applied the robust backstepping tool in Section 1,7. At each step we call the would-be parameter update law a tuning function and propagate it through the rest of the design. The resulting controller consists of a feedback law plus one update law for each unknown parameter, so that the dynamic order of the controller is the same as the number of unknown parameters. The original adaptive backstepping design procedure appeared in (Kanellakopoulos et al, 1991b) along with geometric conditions for the existence of a coordinate transformation into parametric strict feedback form. The overparameterization in this original design was cut in half in (Jiang and Praly, 1991) and completely eliminated by the introduction of tuning functions in (Krstid et al, 1992). In an alternative parameter estimation approach (Praly et al, 1991), we first apply the tools of Section 1.5 to obtain a feedback law which guarantees boundedness in the absence of adaptation. We can then add parameter estimators to the system and use the parameter estimates in the control law to drive the state to the origin. Such estimation-based adaptive schemes have been developed in (Krstid and Kokotovid, 1993, 1994). All of these adaptive designs are described in detail in (Krstid et al, 1995).
1.8 Concluding Remarks We have reviewed only several of many concepts, tools, and procedures currently employed in the design of robustly stabilizing feedback controls. The rclf appears to be a unifying concept for these designs. Robust and adaptive backstepping are effective tools for constructing both Lyapunov functions and stabilizing feedback laws for strict feedback systems. The flexibility of these tools creates the opportunity for performance improvement, such as with the flattened rclf. A natural further step is to address the issue of optimality, and we have mentioned some recent advances in that direction.
References Artstein, Z., 1983, Stabilization with relaxed controls, Nonlinear Analysis, 7(11), 1163-1173.
13 Barmish, B. R., Corless, M. J., Leitmann, G., 1983, A new class of stabilizing controllers for uncertain dynamical systems, SIAM Journal on Control and Optimization, 21,246-255. Ba~ar, T., Bernhard, P., 1991, H~-Optimat Control and Related Minimax Design Problems, Birkh~iuser, Boston. Byrnes, C. I., Isidori, A., 1989, New results and examples in nonlinear feedback stabilization, Systems and Control Letters, 12,437-442. Chen, Y. H., 1993, A new matching condition for robust control design, Proceedings American Control Conference, San Francisco, California., 122-126. Chen, Y. H., Leitmann, G., i987, Robustness of uncertain systems in the absence of matching assumptions, International Journal of Control, 45, 15271542. Corless, M. J., Leitrnann, G., 1981, Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems, IEEE Transactions on Automatic Control, 26, 1139-1144. Coron, J.-M., Praly, L., 1991, Adding an integrator for the stabilization problem, Systems and Control Letters, 17, 89-104. Freeman, R. A., Kokotovid, P. V., 1992a, Backstepping design of robust controllers for a class of nonlinear systems," Proceedings of the IFAC Nonlinear Control Systems Design Symposium, Bordeaux, France, 307-312. Freeman, R. A., Kokotovid, P. V., 1992b, Design and comparison of globally stabilizing controllers for an uncertain nonlinear system, in Isidori, A., Tam, T. J., (eds.), Systems, Models, and Feedback: Theory and Applications, 249264, Birkh~iuser, Boston. Freeman, R. A., Kokotovid, P. V., 1993a, Design of 'softer' robust nonlinear control laws, Automatica, 29(6), 1425-1437. Freeman, R. A., Kokotovid, P. V., 1993b, Global robustness of nonlinear systems to state measurement disturbances," Proceedings IEEE Conference on Decision and Control, San Antonio, Texas, 1507-t512. Freeman, R. A., Kokotovid, P. V., 1996 (to appear), Inverse optimality in robust stabilization, SIAM Journal on Control and Optimization, 34. Glad, S. T., 1987, Robustness of nonlinear state feedback--A survey, Automatica, 23(4), 425-435. Hermes, H., 1991, Asymptotically stabilizing feedback controls and the nonlinear regulator problem, SIAM Journal on Control and Optimization, 29(1), 185-196. Isidori, A., 1989, Nonlinear Control Systems, Springer-Verlag, Berlin, second edition. Isidori, A., Astolfi, A., 1992, Disturbance attenuation and Hoo-control via measurement feedback in nonlinear systems, IEEE Transactions on Automatic Control, 37(9), 1283-1293. Jiang, Z.-P., Praly, L., 1991, Iterative designs of adaptive controllers for systems with nonlinear integrators, Proceedings IEEE Conference on Decision and Control, Brighton, UK, 2482-2487. Kanellakopoulos, I., 1993, 'Low-gain' robust control of uncertain nonlinear systems, Technical Report UCLA/EE/CL 930224, University of California, Los Angeles.
14 Kanellakopoulos, I., Kokotovid, P. V., Marino, R., 1991a, An extended direct scheme for robust adaptive nonlinear control, A utomatica, 27(2), 247-255. Kanellakopoulos, I., Kokotovid, P. V., Morse, A. S., 1991b, Systematic design of adaptive controllers for feedback linearizable systems, IEEE Transactions on Automatic Control, 36(11), 1241-1253. Kanellakopoulos, I., Kokotovi6, P. V., Morse, A. S., 1992, A toolkit for nonlinear feedback design, Systems and Control Letters, 18, 83-92. Kokotovid, P. V., Sussmann, H. J., 1989, A positive real condition for global stabilization of nonlinear systems, Systems and Control Letters, 13, 125-133. Krstid, M., KanellakopouIos, I., Kokotovid, P. V., 1992, Adaptive nonlinear control without overparameterization, Systems and Control Letters, 19,177t85. Krstid, M., Kanellakopoulos, I., Kokotovid, P. V., 1995, Nonlinear and Adaptive Control Design, John Wiley & Sons, New York. Krstid, M., Kokotovid, P. V., 1993, Adaptive nonlinear control with nonlinear swapping, Proceedings IEEE Conference on Decision and Control, San Antonio, Texas, 1073-1080. Krstid, M., Kokotovid, P. V., 1994, Observer-based schemes for adaptive nonlinear state-feedback control, International Journal of Control, 59, 1373-1381. Marino, R., Tomei, P., 1993, Robust stabilization of feedback linearizable timevarying uncertain nonlinear systems, Automatica, 29(1), 181-189. Nijmeijer, H., van der Schaft, A. J., 1990, Nonlinear Dynamical Control Systems, Springer-Verlag, New York. Praly, L., Bastin, G., Pomet, J.-B., Jiang, Z.-P., 1991, Adaptive stabilization of nonlinear systems, in Kokotovid, P. V., editor, Foundations of Adaptive Control, Springer-Verlag, Berlin, 347-434. Qu, Z., 1993, Robust control of nonlinear uncertain systems under generalized matching conditions, Automatica, 29(4), 985-998. Slotine, J. J. E., Hedrick, K., 1993, Robust input-output feedback linearization, International Journal of Control, 57, 1133-1139. Sontag, E. D., 1983, A Lyapunov-like characterization of asymptotic controllability, SIAM Journal on Control and Optimization, 21(3), 462-471. Sontag, E. D., 1989, A 'universal' construction of Artstein's theorem on nonlinear stabilization, Systems and Control Letters, 13(2), 117-123. Thorp, J. S., Barmish, B. R., 1981, On guaranteed stability of uncertain linear systems via linear control, SIAM Journal on Control and Optimization, 35, 559-579. Tsinias, J., 1989, Sufficient Lyapunov-like conditions for stabilization, Mathematics of Control Signals and Systems, 2, 343-357. Tsinias, J., 1990a, Asymptotic feedback stabilization: a sufficient condition for the existence of control Lyapunov functions, Systems and Control Letters, 15, 441-448. Tsinias, J., 1990b, Optimal controllers and output feedback stabilization, Systems and Control Letters. 15. 277-284.
. D y n a m i c a l Sliding M o d e Control via A d a p t i v e I n p u t - O u t p u t Linearization: A B a c k s t e p p i n g Approach M i g u e l Rios-Bolivar, A l a n S. I. Zinober and H e b e r t t Sira-Ram~rez 2.1 I n t r o d u c t i o n Output tracking and regulation problems of linear and nonlinear systems with parametric uncertainties and unmodelled dynamics, have been widely studied in recent times. These studies have been usually carried out from practical perspectives and based upon adaptive or robust control schemes. An outstanding approach has been developed in the new generation of backstepping control design methods (Kokotovid, 1992). The original adaptive backstepping procedure was developed in a series of enlightening contributions by Kanellakopoulos et al (1991, 1992), Kokotovid et al (1992) and Krstid et al (1992, 1994), providing a systematic framework for the design of regulation strategies suitable for a large class of state linearizable nonlinear systems exhibiting constant, but unknown, parameter values. In fact, when the controlled linearizable plant belongs to the class of systems transformable into the parametric-strict feedback form, this approach guarantees global regulation and tracking properties (Kanellakopoulos et al, 1991). A very appealing aspect of the ba.ckstepping design method is that it provides a systematic procedure to design stabilizing controllers, following a step-by-step algorithm, which may be implemented by using symbolic algebra software. The success of this approach has motivated an increasing interest to extend the early contributions, and, as a result, various applications has been reported (Dawson et al, 1994; Sira-Ramfrez et al, 1995b). Moreover, a combination of the adaptive backstepping algorithm with Pulse Width Modulation (PWM) has been proposed by Sira-Ramfrez et al (1995a), which has been applied on DC-DC power converters. Similar approaches, considering combined backstepping-Sliding Mode Control (SMC) have been proposed by Sira-Ramfrez and Llanes-Santiago (1993), Rios-Bolfvar and Zinober (1994). Karsenti and Lamnabhi-Lagarrigue (1994) analyzed the nonlinear parameterization case and proposed their combined scheme for a class of nonlinear system transformable into the non-pure parametric feedback form. Rios-Bolfvar et al (1995a) obtained dynamical adaptive sliding mode output tracking controllers via dynamical input-output linearization and a backstepping algorithm. All
16 these approaches have been developed to obtain robust adaptive control in the presence of unknown disturbances and when no information is available about bounds of the uncertain parameters. These combined schemes have been successfully used in the regulation and tracking control of both continuous and discontinuous control systems. Furthermore, a more direct algorithm has been proposed by Sira-Ramlrez et al (1995b, 1995c), which achieves non-overparameterized adaptive controllers by implementing the fundamental ideas related to the adaptive backstepping algorithm without explicit system transformation into the parametric-pure or parametric-strict feedback forms considered by Kanellakopoulos et al (1991) and Krsti6 et al (1992). This new algorithm is based upon an adaptive inputoutput linearization procedure, under conditions of parametric uncertainty, and allows the design of static controllers for linearizable systems and dynamical controllers for minimum phase systems. The dynamical adaptive controllers designed using this procedure arise from considering nonlinear transformations depending on the control input and its derivatives, to place the system into the error coordinate state space. This aspect is particularly important when the approach is used in combination with either PWM or SMC strategies because the resulting controllers achieve robust asymptotic stability with considerably reduced chattering (Sira-Ramirez et al, 1995b, 1995c; Rios-Bollvar et al, 1995a). This chapter is devoted to the presentation of novel combined adaptive backstepping-SMC schemes to design both static and dynamical adaptive controllers for nonlinear systems with parametric uncertainties entering linearly in the system dynamics. Section 2.2 outlines a combined scheme to design static controllers for linearizable nonlinear systems. Section 2.3 is devoted to dynamical adaptive SMC design for minimum phase systems, and Section 2.4 presents an application of the dynamical adaptive SMC strategy to the output tracking control of the average model of the Buck-Boost power converter.
2.2 Combined Backstepping-SMC Design for Uncertain Linearizable Nonlinear Systems In this section we consider the adaptive SMC design for the class of uncertain linearizable nonlinear systems transformable into the parametric-strict feedback form, by first considering the non-overparameterized algorithm proposed by Krsti~ et al (1992).
2.2.1
The
Adaptive
Backstepping
Algorithm
This algorithm is a systematic procedure to design adaptive controllers through a step-by-step procedure. It interlinks at each step, a change of state coordinates, a tuning function for the uncertain parameters, and a function for stabilizing a virtual system with a known Lyapunov function. The class of systems
17
to which this scheme applies is restricted to systems which can be transformed into the parametric-strict feedback form
Jci =
Xi+l+¢T(xl,...,xi)O,
l
(2.1a)
x~ = ¢0(x) +¢T(x)0 +30(X)U
(2.1b)
where x E IR ~ is the state, u E IR is the input, 0 E IRv is the vector of unknown constant parameters, ¢0 and the components of ¢i E IRv, 1 < i < n, are smooth nonlinear scalar functions, and/30 (x) # 0 for all x E IR~. In the regulation problem the backstepping technique employs the error
coordinates Z1
----
Z2
Zr~
Xl,
~(x~, 0),
~2 _-
Xn
-
-
(2.2)
Otn-i(Xl,...,
~n--l,0),
c q ( x i , . . . , xi, 0), for _< i <_ n, are stabilizing functions and 0 is a tuneable estimate of the unknown parameters, in order to regulate xi to x~ -- 0 and stabilize the corresponding equilibrium x ~
where
x 81 Xe i+t
= __ --
O, -¢~iT 0,
•• ---
-~T(o,
(2.3a) i=l,...,n-1
~
, -¢di_i)0)0.
--¢e10,...
(2.35)
The non-overparameterized design procedure proposed by Krsti~ et al (1992) is a recursive algorithm in which, at its i-th step, an i-th order subsystem ~k
=
--zk-x--ckzk + zk+l--
8--0
~k
8~
--
k-2 + ~ Zj+l Oaj rwk, O0 j=l
i
< k < i - 1,
i-10ai-1 (xj+l +¢To)
(2.4a)
0a,-lb,
(2.4b)
j=l
where the ek's are positive constants, F = F T > 0 is a matrix of adaptation gains and
(
)
i-1 OOti_i ¢" j=l
is stabilized with respect to the Lyapunov function
4----1
(2.5)
18 by the design of a stabilizing function O~i :
- - Z i - - 1 - - CiZi ÷ Z
j=l
OOli--1 " OOli-1 ~'-'='-----Xj+t -i-
Oxj
~ri
kj=l
Zj+ 1
" F - - 0^T
]
~i
(2.7)
and a tuning function i
(2.8) j=l
At the final step, the full z-system is stabilized with respect to the Lyapunov function 2
2
-
by designing the control n - 100ln _ 1
---- -~ol --zn-l - c~zn - ¢o + Z
~Ozj zj+l +
OOln - 1
O0 rn
j=l
( \j=l
(2.1o) 00
and the update law n j=l
to make Vn nonpositive. Parameter convergence feature of this method is expressed in Theorem 1 (Krstid et al, 1992), which establishes that the parameter estimates converge to their true values when r = rank[¢Ti,...,¢T] is equal to the number of uncertain parameters, and therefore the equilibrium x = x e, 0 = 0 is globally
asymptotically stable. 2.2.2 SMC
Design via Backstepping
It is well-known that SMC is a robust control scheme applicable to a broad variety of practical systems. It is based on the design of a discontinuous Variable Structure Control (VSC) law such that the system trajectories point towards a sliding surface S(t) and, once on the surface, remain on it generating a sliding mode (Utkin, 1992). SMC has been successfully applied to power systems, electric motors, robot manipulators, etc., and provides robustness in the presence of disturbances and parametric uncertainties. Here we show that the adaptive backstepping technique provides a simple way to obtain straightforwardly an
19 adaptive sliding surface, on which a sliding mode for uncertain linearizable systems may be generated. Let us consider the zn-coordinate of the error coordinates in (2.2)
By using the expression (2.7) in (2.12), for i = n - 1, one obtains S
Z~
~-~
(0~_2
oak_
k=l
)
00 n--2
+
co_1
0o6~-2
"~
Oa,~_2
(2.13)
k=l
We note that the expression (2.13) could be considered an adaptive sliding surface in the original state space, and a sliding mode may be generated on it. Moreover, the LaSalle invariance theorem allows us to guarantee that, on the invariant set towards which the state (z(t), O(t)) converges, the following conditions are satisfied (Krstid et al, 1992)
z(t) = 0;
~(t) = 0,
(2.14)
and consequently s ( t ) : = z~ (t) = 0;
(2.15)
; ' ( t ) = ~. = 0.
Thus, the adaptive backstepping controller guarantees convergence of state trajectories towards the sliding surface. Now, we must modify the control law in order to keep state trajectories on this surface in a sliding mode. We can achieve a sliding mode on this surface by using the following combined control law
u
=
r~--I
i__ /3o - z ~ - l - k s i g n S - ¢ 0 + E 0 a ' ~ -j=l I
@
Zj+ 1
\j=1
~F
oT
0~_i
_ OXj Xj+I ~- _ 03
"in
Wn
00
where k is a positive constant and sign(.) is the discontinuous signum function signS=
1 -1
if S > 0 , if S < 0 .
(2.17)
This control law guarantees global stabilization because the time derivative oi the Lyapunov function
20
~/~ =
- ~ ~ z ?3 - klSI j=l
(2.1s)
is negative definite. The sliding regime induced by the proposed control law would in practice exhibit a chatter motion associated with the discontinuous nature of the control input, as a consequence of infinitely fast changes between two different control values. This chatter motion may be eliminated by using the continuous approximation developed by Burton and Zinober (1986) instead of the signum function in (2.16), obtaining the smoothing control law S
1 U
-
--
-~0 -z._l-klSt+~ +
r~--I
0~,-1
0~.-1
¢0+~=1~ o~ ~+1+--08 r.
ZJ+I 08
(2.19)
where 3 -+ 0 + is the smoothing parameter.
2.3 A d a p t i v e D y n a m i c a l S M C D e s i g n for Uncertain Nonlinear Systems In this section we present a generalized design method to synthesize adaptive dynamical SMC for uncertain nonlinear systems. This approach is based upon an adaptive input-output linearization procedure, developed from a backstepping perspective, and is suitable for a class of input-output dynamically linearizable systems. The design method developed here considers a more general class of uncertain nonlinear systems, avoids explicit transformations to parametric-strict or parametric-pure feedback forms, and the adaptive backstepping algorithm proposed by Krstid et al (1992) forms a particular case. We concentrate here on the output tracking problem of nonlinear systems with unknown parameters, and the results obtained are valid locally in the state space.
2.3.1 Dynamical SMC Design in Output Tracking Problems Nonlinear controlled systems have been extensively studied from a differential geometry viewpoint, and, more recently, a differential algebra approach has been proposed by Fliess (1989) for the study of these systems using a unified treatment based on implicit ordinary differential equations. From this perspective, generalized canonical forms for linear and nonlinear systems have been
21
introduced, which consider nonlinear transformations depending on the control input and its time derivatives (Fliess, 1990). In this setting new and efficient dynamical feedback controllers can be developed for stabilization and output tracking problems of nonlinear systems. We describe here the design of dynamical compensators synthesized from SMC techniques (Sira-Ramfrez, 1993) and using the Fliess Generalized Observability Canonical Form (FGOCF). Consider the single-input single-output nonlinear system
~: = y =
(2.20a)
f(x, u), h(x),
(2.20b)
where x E IR'~ is the state, u E ]R the control input, y E IR the output, h(x) a smooth function on IR'~ with h(0) = 0, and f ( x , u) a smooth vector field on IRn+l. We assume that (2.20) is a nonlinear system with a well-defined relative degree r, i.e. 1 < r < n. The analysis of the observability of (2.20) can be based on the n - 1 time derivatives of the output equation
h dh
Y
q(~, ~) := y(n-1)
(2.21)
d(~-l)h
with the differential operator d defined by
d h = \ ~ ] ~+ ~
f
(2.22)
and the observability map q(x, ~) depending upon the first n - 1 derivatives of the input
ft = [u, i~,. . ., u('~-l)] T .
(2.23)
System (2.20) is called locally observable ifq(x, ~) has a regular Jacobian matrix with respect to the state, that is
(~)~ rank Oq(x, fi) _ rank 0z
= ~,
(2.24)
d,~_l (0_h~T ~,Ox]
which guarantees that (2.20) can be transformed, by use of the observability map
= q(x, ~) into the equivalent observability canonical form (FGOCF)
(2.25)
22
~2
~3~
(2.26) ~rt-- 1 =
c(~,u,...,u('~)),
Y where m = n - r is a constant scalar coinciding with the dimension of the zero dynamics. Defining the tracking error function z ( t ) as the difference between the actual system o u t p u t y ( t ) and a desired b o u n d e d o u t p u t reference signal y~ (t), with b o u n d e d derivatives y(O(t), i = 1 , . . . , n, (2.27)
z ( t ) = v ( t ) - y, (t),
we o b t a i n z(i)(t)
=
~i+1 - y(/)(t);
z(n)(t)
=
~,~-y('O(t)=c({,u,
0 < i < n-
1,
it,...,u('~))-y(n)(t).
(2.28a) (2.28b)
Then, defining zi = z (i-1), i = 1 , . . . , n, as c o m p o n e n t s of an error vector z, we express the tracking error s y s t e m (2.28) in F G O C F as Zl Z2
~-~ Z2, = Z3,
(2.29) Zn-
~
Zn ,
~,
=
c(~r(t) + z,~,<...,~(r~))
Yz
=
zl,
i
- v~°)(t),
with ~r(t)
=
col [ y r ( t ) , y ~ l ) ( t ) , . . . , y ~ - l ) ( t ) ] ,
(2.a0a)
z
=
col[zl,z2,...,z,].
(2.30b)
Under the assumption t h a t the condition oc(~(t)
+ z, ~, . . ., u ( ~ ) )
&(m)
# 0
(2.31)
is satisfied locally, no singularities need to be considered for designing a dynamical SMC. The sliding surface is specified in terms of the o u t p u t tracking error coordinates as o" = z,~ + c , ~ - 2 z , . - t + . . . + c l z 2 + COZl = 0
such that the polynomial
(2.32)
23 s n - 1 + cn_2 s n - 2 + . . . + c l s + co = 0
(2.33)
in the complex variable s is Hurwitz. Finally, the dynamical SMC n-1
'"
= gr
c i - l z i + l - ~(o" + Wsign(cr))
(t) - E
(2.34)
i=1
can be obtained in terms of an implicit ordinary differential equation with discontinuous right hand side. The ideal sliding d y n a m i c s , obtained from the invariance conditions cr = O, & r / d t = O, is non-redundantly defined by
Z2
=
Z3~
(2.35) •
~
Zn-1
-- E
n-I
i = 1 Ci--lZi:
which exhibits an asymptotically stable motion towards the origin of the error coordinates. Consequently the output tracking error function zl converges asymptotically to zero, as desired. This design method is applicable to perfectly known nonlinear systems. We next consider an adaptive version of this procedure for the case of nonlinear systems with parametric uncertainties.
2.3.2 Adaptive Input-Output Linearization via Backstepping The following adaptive input-output linearization procedure is based upon a backstepping-like algorithm for uncertain nonlinear systems. Consider single-input single-output uncertain nonlinear systems of the form
i,
=
f(x,
y
=
h(x),
(2.36a) (2.36b)
where x E IRn is the state, u E IR the control input, y E IR the output, 0 E IR? an unknown constant parameter vector, h ( x ) a smooth fimction on IR~ with h(0) = 0. f and the columns of 7 E IR~xp are smooth vector fields on IR'~+I. We assume that (2.36) is an observable systern with well-defined relative degree, and all the state variables are available for measurement. T h e control objective is to drive the system output y(t) to track asymptotically a desired reference signal y~ (t). We also assume that y~ (t) and its derivatives up to order n are bounded and sufficiently smooth functions of t. In most of the previous work addressing the adaptive backstepping algorithm, strict and pure feedback forms have been used, which can be considered as particular cases of (2.36). Recently, contributions reported by Sira-Ramlrez
24
et at (1995b), and Rios-Bolivar et al (1995a, 1995b), consider nonlinear systems without these particular forms, with a more direct adaptive algorithm being used to design adaptive backstepping controllers. This baekstepping-like algorithm is based upon an adaptive input-output linearization procedure and its generalization can be summarized as follows.
S t e p 1. By defining the tracking error function zl = y - yr(t) = h ( x ) - yr(t), we obtain the first subsystem Zl = ~ x [ f ( x , u) + 7(x, u)O] - i/~(t)
(2.37)
which can be rewritten as
~
_
~[.:(~, u) + -/(~,u)o] -/,~(t) + (o -
=
h(~)(~,o,~) - ~ ( t ) + (0 - 0)%~,
(2.38)
with Oh
htl)
:=
~[f(~,.)
+ ~(x, u)~],
~1
=
:(~,u)(Oh~
\ ~ ]-
•
(2.39) (2.40)
For the sake of clarity we assume that the relative degree of (2.36) p is one. Nevertheless, the algorithm presented here can be also generalized for p > i (see (Rios-Bolivar et al, 1995b)). The subsystem (2.38) can be stabilized with respect to the Lyapunov function
Vl = ~z? + 1(o - 0) 2 r-*(e - o)
(2.41)
whose time derivative is
We can achieve I71 = -z~ with the tuning function k
0 = rl = F z l w l ,
(2.43)
if the following expression is satisfied ]~(1)(x, O, it) - Yr (t) = - Z l .
(2.44)
Since (2.44) is not valid, we choose its difference as the second error coordinate Z2 : h ( 1 ) ( x , O, ~t) -- y r ( t ) ~- Z 1
to obtain the closed-loop form
(2.45)
25
h =-zl +z2+(O-O)r~l
(2.46)
and, taking (2.43) as our first tuning function, the actual time derivative of V1 yields
(/1 = --zi2 + zlz2 +
+ rl
.
The second term in ~] is compensated at the next step by following the fundamental ideas of the adaptive backstepping algorithm. Note that, in contrast to the adaptive backstepping algorithm developed by Krstid et at (1992), the design parameters ci's are not used at intermediate steps of the algorithm to avoid involving their products in the original state space, at subsequent steps. So, the resulting controller exhibits better transient performance and convergence properties. S t e p k (2 < k < n - 1). By applying this procedure successively, we obtain the recursively defined adaptive state coordinate transformation zk
:=
h(k-1)(x,O,u,...,u(k-2),t)--y!k-1)(t) + ak-l(x, O, u , . . . , u (k-a), t),
Oh(k-n ]~(a)
_
N
(2.4s)
T" Oh(k-l) k ( f + 7 O) + cO-----~F E z~wi i=I _ O]~(k-1)
k - 1 O i l ( k _ 1 ) U (i) + _
+ ~i=1 Ou(i-l------~ Oai-~ O[k
:
Zk _ l +
Zi
Oh(i-n
a---~ + E
\ i=3
Zi GO-----~
+ i=i
i:i
•/Oh(k-l)
:
<(,,u)[
xr~W k
i=2 k-2
k-1
~
(2.49)
Ot '
Ou(i_i)
O~k_l~ T, +
o,
+ ~0~k-1 (f +~T0)
~k-1 +zk, at
(2.50)
(m51)
)
k 0
:
Tk :
(2.52)
I'EZiWi, i=1
where 0 is an estimate of the unknown parameter vector and F = F T > 0 is a matrix of adaptation gains. So, after the (n - 1)-th step, the transformed system h
=
- Z l + z2 - ( 0 - 0 ) r ~ l ,
~2
:
- z I - z~ + z3 - (0 - o ) ~
(2.53a) + Oh~(O - m,
00
(2.Sab)
26
2:k
--Zk-1--Zk
----
"71-Z k + l - -
(O--o)To2k
Oh(i_i)
~,~
k-1
(Oa(k-1)
-
O0~,_ 1 ~
h('~)(x,O,~,,...,u('~-l),t)-
~'~)(t)
(2.5ac)
+ ~ , ( x , 0, ~,...,u('~-~),t)
~o), +(0-0)%,+\(0~(o-') -~ + 0~o_~(0_ oO / n--1
k 0
(2.536)
(2.53e)
Vn--1 = F E ZiW,, i=1
:
with ]~(~)(.), c~(.) and ~n defined according to previous expressions for k = n, is obtained by considering the Lyapunov function n--1
v,,_~ =
~z~ 1
+ ~1 (0
- o)TF-I(O
(2.54)
-- 0),
i----1
whose time derivative is given by
v~-i
=
- ~
z~ + zozo_l +
z,-ki=2
/----1
+ Z z'-~ ~0
-
'----3
+ (0 - o ) ~ r - ~ ( - O + ~ _ , ) .
(2.55)
Note that, from (2.53) and extending the Lyapunov funtion to incorporate the error coordinate z,~ at the final step, various adaptive controllers can be obtained. We next present the final step of the algorithm to design an adaptive dynamical SMC strategy. 2.3.3
Adaptive Dynamical Tracking Problems
SMC
Design in Output
Step n. We now design an adaptive dynamical SMC to stabilize the transformed system (2.53). The sliding surface is specified in terms of the error variables zi o" = c l z l + c2z2 + . " + c , ~ - l z n - 1 + z• = 0, (2.56) where the scalar coefficients ci > 0, i = 1 , . . . , n - 1, are chosen in such a manner that the polynomiM
p(s) = cl + e2s + . . . + c~-1~ ~-2 + ~n-~, in the complex variable s, is Hurwitz.
(2.57)
At this final step, we extend the Lyapunov function as follows
t 2
1
n-1
2
12
1
(2.58)
i=1
l'he time derivative of V~ is
~-~
--~ - - E Z 2 + Z n _ l Z n
(~
+
i=1
Zi
8]~(~-~)a~
ki----2
n-~
Oaf_i)(~-r~_~)
+ EZi---~
~
i=3
i=1 + G[h(n)(X,O, t t , . . . , U ( n - 1 ) , t ) -- y~n)(t) + O~n(X,O, t t , . . . , t t (n-2),t)
r~-I
(l~h (n-l)
+\
N
o_1
(
O(~n--1
(~ Tn) q- E Ci(--Zi--1-
+--~
-
Zi q- Zi+I)
i:, _
- ZNc ~, =\ ,
+ o0 )
To eliminate (0 - 0) from Vn, we choose the update law n-1 i=1 n-1
n-I
i=1
i=1
(2.60) and noting that
n--1 - ~-1--
~
- ~o-, = r~(~
+ E
i=1
we rewrite ~zn as ..~
n-1 -- g Z2 + Zn_lZn i=1
+ ~[h(~)(~, O,~,..., ~(~-~),t) - v!~)(t)
ci~,),
(2.61)
28 n-1
+ ~ . ( x , 0, u, ..., ,(.-2), t) + Z
c,(-z,_l - z, + z,+1)
i=1
i=3
°_1
/
o.,:1
fo,~(,-1)
-Ec'[,=l
i=1
-~ + oo i °°-''1
- E,=I c, [Ej__~z~ -oo + ~=~Ezj-7
r~,
(2:62)
and finally the adaptive dynamical SMC is specified in terms of the implicit ordinary differential equation -~(o" + Wsign(o')) =
+ ,~,,(x, o, u , . . . , u('-~), t) + ~
c~(-z~_t - z~ + z~+l)
i=1
o +Ez,-~
- kiz' ti=2 n-1
(9
r ,~o+Ec,~,
i=3
i=1
(Off(i__1) OOti_)_l
-~c,\,__1 N
+-flu 0,,-~,)
"-' { ~E z ~ =o/,(~-1) + ~ oo,~._1) ~5=~ oo 2..,zj--aT r~,,
-Ec, i=1
(2.6a1
j=3
to obtain
n-1
9. : - ~
z2 + z._lz~ - ~ 2 _
~Wl~l.
(2.64)
i=1
To prove the asymptotic stability let us rewrite I)'n as 9. = -zT Qz -
(2.65)
aWl¢[
with 1 + xc~
...
t~ClCn-1
t£Cl
t£ClC2
., -
KC2Cn-
I£C2
1
= ~clc,-1
...
l + xc~_l
~Cl
"'"
- - 5 q- t~cn-1
1
The principal minors of Q have the value
-½ + xc,-1 x
29 d
1 + ~y~c/~ > 0;
1 < d < n-
1.
(2.66)
i=1
Thus, a sufficient condition on the design parameters to achieve asymptotic tracking can be obtained from det(Q)=-4
1(
1:) tl--2
+x
l+c,_l-~Ec
>0.
(2.67)
i----1
So Vn < - z T Q z < 0 and therefore, since lim zl(t) = y(t) - yr(t) = O,
(2.68)
asymptotic tracking is achieved. The convergence of the state trajectories towards the sliding surface can be established from the LaSalle invariance theorem.
2.4 Design Example: The B u c k - B o o s t Converter Consider the average model of the Buck-Boost converter defined on the input inductor current xl and the output capacitor voltage x2
with
d~l =
01(1- #)x2 + 04#,
:b2 = y =
-02(1-#)xl xl,
1 01=T;
1
(2.69a)
-03x2,
(2.69b) (2.69c)
1 E 03=Rc; 04=T,
(2.70)
where L, C and R are respectively the inductance, capacitance and resistance values of the circuit components, while E is the constant external voltage source. These four circuit components define the set of unknown parameters 0. Note that (2.69) is not transformable into the parametric-pure or parametricstrict feedback forms, by parameter-independent state coordinate transformations, therefore adaptive baekstepping design, under conditions considered by Krsti5 (1992), is riot applicable. The control input function p takes values in the interval [0, 1] and the inductor current xl is the regulated output function. For/t : U constant, with 0 < U _< 1, the equilibrium values are readily obtained as
0304U XI(U)
- - 0102(1 -- U ) 2 '
X2(U) --
04U 01(1 - U)"
(2.71)
We are interested in driving the input inductor current Xl to follow a smooth trajectory between two operating equilibrium points X1, X~
30
yr(t) =
{xl
X; +(X1-X~)exp(-k(t-tl)
2)
0 < t < ~1,
t>_tl.
(2.72)
In order to implement the algorithm presented in the previous section, we rewrite the Buck-Boost model as
J:l x2 y
= =
7TI (xl,x2,#)O, "yY(~I, ~2,#)0, xl,
=
(2.73a)
(2.73b) (2.73c)
with
0 71 72
= = =
[01
02
03
04] T,
(2.74)
[(l-#)x2
#iT,
[0
--x2
0 0 --(1--#)x1
(2.75) (2.76)
0]T.
Firstly, we define the tracking error function Zl := Xl - Y r (t) and, applying the first step of the algorithm, obtain the following: (2.77) (2.78)
~1 = - z l + z2 + (0 - 0 ) T w l ,
v~ = - z ~ + zlz~ + (o - o ) ~ r - l ( - b
+ ~),
(2.79)
with ~
rl
:=
~(~1,~2,~),
(2.80)
=
Fzlwl.
(2.81)
At the second step, we define the sliding surface ~r = clzl + z2 and the aug1 2. The time derivative of V2 is mented Lyapunov function V2 = 171 + ~c~
- z [ + zlz2 + (0 - O ) r r - ' ( - 0 + ~1 + r~(~2 + c1~1))
+ ~ [~
+ ~1~0+ t 0 , - 0 , ~ / . - y~ - ~
+ ~,t-z, + z~/] (2.82)
with o~2 = 3'1 + 01 (1 -/~)72.
(2.83)
To eliminate (0 - t~) from ~z2 we choose the update law k
0 = ~2 = ~1 + r~(~2 + c1~1) = r [ z l ~ l + ~(~2 + c1~1)].
(2.84)
The control function # can be readily obtained as the solution of the following nonlinear time-varying differential equation
31 Tracking performance
Voltage capacitor
25
-1(
f-.. 20 -1! 15 -2(
10 5
t[ms] 0
2
t[ms]
-25
4
2
0
Control input
4
Sliding surface 10 o
0.8
-10
0.6
-2o 0.4 t[ms] 0
2
4
6
-30
t[ms] 2
0
4
6
Figure 2.1 Dynamic adaptively regulated tracking and capacitor voltage evolution of the Buck-Boost converter, control input function and adaptive evolution of the sliding surface.
1
-- (3 4 --?lX2 ) [ -- 0T022 -- "~ITT2 "t- Yr -~ Yr - c l ( - z l + z~) - ~ ((r + Wsign(~r)) ]
(2 ss)
and ~r2 yields v2 = - z ~
+ z~z~ - ~ 2
_
~Wl~l.
(2.86)
The sufficient condition on the design parameters to guarantee asymptotic
tracking is 1 ~(1 + ci) > ~.
(2.87)
An important advantage arises from the dynamical adaptive sliding mode control: the output tracking error function zl(t) asymptotically approaches zero with substantially reduced chattering. In order to test the robustness of the proposed scheme with respect to external perturbation inputs, we used in simulations a perturbed model including
32
Thetal hat
Theta2hat
15000
10000
10000
8000
5000
6000
0
t[ms] 0
2
4
6
4000
/ t[ms]
0
Theta3hat
X
5.4
5000
4
Theta4hat
104
0
5.3 -5000 5.2 -10000 -15000
t[ms] 0
2
4
6
5.1
t[ms] 2
0
4
Perturbation noise signal
0
F i g u r e 2.2
2
4
6
Parameter estimates and perturbation noise signal.
33 an external stochastic perturbation input u, additively influencing the external source voltage E, that is Xl
=
0 i ( 1 - # ) x 2 + (04 + u)#,
(2.883)
22 y
= =
--02(1 -- #)Xl -- 03X2, xl.
(2.88b) (2.88c)
The following nominal values of the parameters were used: C = 181.82 #F, L = 0.27 mH, R = 2.44~2, E = 14.667 Volts. These values yield the model parameters 01 = 3 . 6 x 103 , 0 2 = 5 . 5 x 103 , 0 3 = 2 . 2 5 x 103 , 0 4 = 5 2 . 8 × 103 , whereas the design parameters were cl = 3, ~ = 15, W = 10, F = / 4 . Figure 2.1 depicts the dynamic adaptively regulated tracking of the inductor current xl for a smooth transition between X1 = 22.5 amps and X~ = 10 amps, corresponding to U = 0.6 and U = 0.4695 respectively, as well as the time evolution of the controlled capacitor voltage x> The regulated output variable xt(t) is seen to exhibit asymptotic tracking to the desired reference input y~(t). The figure also shows the control input function and the time evolution of the adaptive sliding surface. Figure 2.2 shows the estimated parameter values 0 obtained from the updating law, and an example of the perturbation noise input.
2.5 C o n c l u s i o n s Combined adaptive backstepping-SMC schemes to design both static and dynamical adaptive controllers for uncertain nonlinear systems have been presented. It has been shown that, for the class of nonlinear systems with parametric uncertainties entering linearly in the system dynamics and transformable into FGOCF, adaptive dynamical SMC may be designed via an adaptive input-output linearization procedure, based upon the fundamental ideas of the backstepping algorithm. This control design method is applicable to uncertain nonlinear systems without explicit transformations to parametric-pure or parametric-strict feedback forms. The proposed controlled strategy has been applied to the output tracking control of the average Buck-Boost converter model and it has been shown to be remarkably robust with respect to external stochastic bounded perturbation input signals.
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Rios-Bolfvar, M., Zinober, A. S. I., Sira-Ramirez, H., 1995a, Sliding mode output tracking via backstepping for uncertain nonlinear systems, Proceedings European Control Conference, Rome, 699-704. Rios-Bolfvar, M., Sira-Ramfrez, H., Zinober, A. S. I., 1995b, Output tracking control via adaptive input-output linearization: A backstepping approach, Proceedings IEEE Conference on Decision and Control, New Orleans, 2, 1579-1584. Sira-Ram~rez, H., 1993, A Dynamical Variable Structure Control Strategy in Asymptotic Output Tracking Problems, IEEE Transactions on Automatic Control, 38,615-620. Sira-Ramlrez, H., Llanes-Santiago, O., 1993, Adaptive Dynamical Sliding Mode Control via Backstepping, Proceedings IEEE Conference on Decision and Control, 1422-1426. Sira-Ramfrez, H., Garcia-Esteban, M., Zinober, A. S. I., 1995a, Dynamical Adaptive Pulse-Width-Modulation Control of DC-to-DC Power Converters: A Backstepping Approach, submitted for publication.
35 Sira-Ramlrez, H., Rios-Bolivar, M., Zinober, A. S. I., 1995b, Adaptive inputoutput linearization for PWM regulation of DC-to-DC power converters, Proceedings American Control Conference, 1, 81-85. Sira-Ramfrez, H., Rios-Bot~var, M., Zinober, A. S. I., 1995c, A Nonoverparameterized Backstepping Controller Approach for the PWM Stabilization of DC-to-DC Power Converters, International Journal of Robust and Nonlinear Control, to appear. Utkin, V. I., 1992, Sliding Modes in Control Optimization, Springer-Verlag, Berlin.
. A Generic Lyapunov P r o c e d u r e to Design R o b u s t Control for Nonlinear Uncertain Systems: Introducing Interlacing into Recursive Design Zhihua Qu and Joseph Kaloust 3.1 Introduction Control of nonlinear uncertain systems has been becoming more and more a research focus since dynamics of most systems in applications are nonlinear and partially known. In order to achieve better performance, possible unknown dynamics must be considered in control design, and nonlinear dynamics must be compensated. Robust control theory of nonlinear uncertain system currently under development is to provide us what classes of uncertain systems are stabilizable and what are their robust controllers. The common technique used in design is the Lyapunov direct method since it works universally for all systems. The technique involves two key components: finding functions that bound the magnitude (or Euclidean norm) of uncertainties, and searching for a Lyapunov function and then determining a stabilizing control in the process of stability proof. Steady progress has been made in robust control theory for nonlinear systems. The first class of uncertain systems that can be stabilized as a whole is characterized by the matching conditions, defined by Gutman (1979) and by Corless and Leitmann (1981). Two robust controllers were proposed to achieve global stability, the discontinuous rain-max control (Gutman, 1979) yielding asymptotical stability and the continuous state feedback control (Corless and Leitmann, 1981) guaranteeing stability of uniform ultimate boundedness. If a system has uncertainties not satisfying the matching conditions, the conventional method is to leave the unmatched uncertainties out of the control design process. That is, design robust control only for the known part of dynamics and the matched uncertainties, and then hope that the control provides enough stability margin to cope with the mismatched uncertainties, see the references (Barmish and Leitmann, t982; Chen, 1987; Chen and Leitmann, 1987; Qu and Dorsey, 1991; and Qu, 1992b). This approach works only for a small class of systems, for instance, it was shown in (Qu and Dorsey, 1991; and Qu, 1992b) that, if the nominal system can be stabilized with arbitrarily large convergence rate, the system with arbitrarily large unmatched uncertainties can be stabilized. The main thrust of research has been to loosen the matching conditions
38 and to consider unmatched uncertainties in control design. Recently, two more classes of nonlinear systems not satisfying MCs are shown to be globally stabilizable. One of them is the class of systems with the matched uncertainties and the so-called equivalently matched uncertainties (Qu, 1992a). The equivalently matched uncertainties are those that can be treated as matched uncertainties in a Lyapunov argument; thus, the design procedure and robust controller are basically the same as those for systems satisfying the matching conditions. The other class of uncertain systems, defined by the so-called generalized matching conditions (Qu, 1993), has the property that, under a series of coordinate transformations, a system in the class is decomposed into a chain of cascaded subsystems which satisfy locally the matching conditions. The series of state transformations is used to design robust control recursively through a nonlinear mapping, which is in essence the same as the backstepping procedure (Kannellakopoulos et al, 1991). Among the existing results, the generalized matching conditions and the recursive robust control design procedure are most applicable since many electrical-mechanical systems inherently have the cascaded structure. However, there are many more systems that do not possess this type of structure. For those systems, the recursive (or backstepping) procedure does not work since interconnections other than cascade cannot be handled by one-direction, either backward or forward, recursion design. We will show that, although some form of recursion must remain, a successful design procedure must be both programmed and integrated with respect to the system such that sophisticated, nonlinear, large, and potentially uncertain interconnections within the system can be taken into account. This chapter searches for a method of designing robust controller for uncertain systems not belonging to any of the existing stabilizable classes. Especially, we are interested in systems with (uncertain) dynamics not satisfying the generalized matching conditions since those dynamics are intricate interconnections. As a continuation and expansion of the results in (Qu, 1994 and 1995), we introduce here a new constructive procedure for designing robust controller without imposing any structural conditions on systems or uncertainties. The procedure consists of major recursion steps in which the final outcomes are characterized by translational state transformations. The transformations allow us to design fictitious robust controllers as intermediate results in order to design the actual robust control at the final stage of the procedure. We choose to formulate the new procedure using recursion as major steps in order to compare it easily with the recursive procedure in (Qu, 1993). What is new and different about the procedure in this chapter is that, in each recursion step, there are five minor steps where so-called interlacing designs take place. These interlacing designs allow the designer in a fully programmed fashion to consider the system dynamics as a whole (so as to consider interconnections in design) and to coordinate fictitious control designs in such a way that uncertainties not matched in any sense are handled from the beginning in control synthesis and stability analysis. In the interlacing steps, sub-Lyapunov functions for subsystems are chosen judiciously since the Lyapunov functions are used not only for establishing stability
39 but also for generating fictitious controllers. That is, the new procedure takes full advantage of the only two choices a designer has: Lyapunov function and control structure. It is worth noting that the same word "interlacing" has a different meaning in Kannellakopoulos et al (1991) where it denotes simply a sequence of state transformation. Using the generic procedure, dynamic structure is fully explored by the newly defined forward and backward interlacing steps, stabilizability can be revealed, robust control can be designed, and no a priori structural assumption on system dynamics is needed. The procedure also includes the recursive backstepping as special cases. It is also worth mentioning here several results on recursive design. The basic idea of recursive design is to decompose nonlinear systems into subsystems that are of first-order integrators and whose couplings are either feedback or feedforward but not both. Stability analysis and control design for systems containing an integrator were studied in (Byrnes and Isidori, 1989; and Tsinias, 1989). As shown in (Kannetlakopoulos et aI, 1991; Marino and Tomei, 1993; Qu, 1993; and Slotine and Hedrick, t993), global stabilizing adaptive or robust controls have been found through a multi-step backward recursion for nonlinear, cascaded systems. It has been shown in (Mazene and Praly, 1994) that a version of feedforward recursion design exists for the class of high-order feedforward systems in which there is no feedback coupling, local dynamics are Lyapunov stable, and feedforward couplings are of higher order. The feedforward recursive approach has been demonstrated to be useful in proving asymptotic stabilizability of so-called null controllable linear systems via saturated control in (Lin and Saberi, 1993a; Teel, 1992a; and Teel, 1992b) and of so-called partially linear composite systems (Lin and Saberi, 1993a; Saberi et al, 1990). From both theoretical and application point of view, the proposed procedure has several outstanding features. First, it is applicable without any a priori requirement on location or functional form, or size of uncertainties. Neither it requires that the nominal system be asymptotically stable as did in the case of the matching conditions (for the purpose of finding Lyapunov function). Second, the procedure is systematic and constructive, and the objective of every minor and major step is defined explicitly and intuitively so that practicing engineers can apply it to their systems by following the examples in this chapter. Third, for systems satisfying either the matching conditions or the generalized matching conditions or having only equivalently matched uncertainty, the new procedure reduces to the existing design procedures respectively. Thus, the new procedure includes the recursive (or baekstepping) procedure as a special case. Forth, the procedure can be applied to most systems and, in the event that it fails to produce a stabilizable controller, the designer has the results of either to modify the fictitious control designed or to probe and conclude stabilizability. That is, because not all uncertain systems are stabilizable, the proposed procedure provides a generally applicable mean of determining stabilizable classes and stabilizability of uncertain systems. Fifth, a resulting robust control is always continuous and guarantees global stability in terms of either asymptotic, exponential, or uniform ultimate bounded stability. Finally, the procedure is
40 applicable in its simplified version to known nonlinear systems for which there has been any general design problem reported neither.
3.2
Uncertain
Systems
An uncertain system under our consideration is assumed to be described by the following vector ordinary differential equation: = ~ ( x , u, v,t),
(3.1)
where x E IRn is the state, u E IR"u is the control, and for some prescribed compact set S C IRp, v E S is the vector of (time-varying) uncertain variables. Model (3.1) covers all systems whose dynamics are modeled by ordinary differential equations and is therefore a symbolic expression. To facilitate stability analysis and control design, an equivalent but more specific model can be developed as follows. First, decompose the state vector x into state subvectors x / E IRnj , j = 1 , . . . , m. Accordingly, let the variables vl, where i = 1 , . . . , m and whose domains are some prescribed compact sets Si C IRp~, represent the uncertainties in subsystems. Second, let the i-th element in vector function .T(x, u, v, t) be denoted by - ~ ' i ( X l , • . • , Xi, Xi+I, Xi+2,..., Xrn, 7t, vi,t). Finally, define functions Fi(.), Gi(.) and Hi(.) to be
F i ( x ~ , . . . , z i , vi,t)
H~(~,...,~m,~,.i,t)
=
•~ i ( X l , . . . , Xi, O, O,...,O,O, vi,t,),
=
:ri(Xl,. • •, Xi, Xi+l, Xi+2, - • -, Z,~, U, Vi, t) -- ~C'i(Xl,...,Zi,
Xi+l,O,...,O,O,t)i,t),
and
G i ( x ~ , . . . , z , Xi+l,V~,t)
=
,vi(z~,...,xl, x~+l,O,...,O,O,v,t) - . ~ ' i ( x l , . . . , :~, O, O , . . . , O, O, vi, t).
Then, generic model (3.1) can be rewritten to be
ici = ~,n
=
+ H1 (-1,..., *m, u, -1, t), Fi(xl,...,xi,vi,t)+Gi(xl,...,zi,xi+l,vi,t) + H i ( x l , . . . , X m , U, vi,t), i = 2 , . . . , m - 1, Fm(xl,...,xm,vm,t)+Gm(xl,...,Xm, U, Vm,t),
where, for j = 1 , . . . , m and k = 1 , . . . , m -
(3.2a) (3.2b) (3.2c)
1,
Pj(xx,...,xj,v~,t)
=
~(~1,..., x~, t) + ~ ( ~ 1 , . . . ,
Hk(xl,...,~m,U, vk,0
=
hk(xl, • •., xm, t)
+ A h k ( z l , . . . , x,~, u, vk, t).
~ , ~ , t),
41 In model (3.2), functions Fi(.) denote local dynamics of subsystems and feedback dynamics from subsystems 1 to i - 1; functions Gi(.) denote the cascaded structure within the system and therefore has the property that, for all i E { 1 , . . . , m } and ( x l , . . . , x i , v i , t ) ,
G i ( x l , . . . , x i , O , vl,t) = O; functions Hi(.) represent interconnected coupling other than cascading and feedback, namely, they are feedforward dynamics excluding those between two consecutive subsystems and therefore have the property that, for all ( x l , . . . , x ~ , v i , t ) and i E { 1 , . . . , m - 1},
Hi(x1,..., x~+l, 0 , . . . , O, vi, t) = O. The pairs fi(') and Afi(.), hi(-) and Ahi(.) denote the known and uncertain parts of Fi(.) and Hi(-), respectively. That is, fi(zl,..
., x i , t )
~-
Fi(Xl,...,xi,
vi,t)l
, v i .~o
z~ f i ( x l , hi(xl,
. . . , xi, yi, t)
:
Fi(~,...,~i,~i,t)
- :~(~,...,~,t),
. . .,Xm,t)
=
H i ( x l , . . . , X m , U , vi,t)
=o,
and z ~ h i ( x l , . . . , xm, u, ,~, t) = Hi(x1, • . . , xm, u, vi, t) - h d x l , . . . ,
z,~, t).
There may be possible uncertainties in functions Gi (') as well, but these uncertain parts if any are not denoted by separated functions. Note that there are no functions hm ( x l , . . . , zm, t) and Ahm(Xl,. . ., x,~, u, vi, t) presented in (3.2c) since they can be merged into functions fm ('), Afro ('), and Gin('). Using generic model (3.2), a complicated system is expressed in terms of a number of interconnected subsystems. The decomposition in deriving model (3.2) can always be done and is unique with respect to any given partition of the state z into substates ~:i. This method of system decomposition can be illustrated graphically by extending block diagram or signal flow graph to represent nonlinear mapping, as shown in Figure 3.1. In the figure, symbol S -1 represents the integral operator. What is important in the generic model are the groups of system dynamics partitioned by defining a specific input-output chain and by the separation of dynamics based on feedback and feedforward. As shown in Figure 3.1, substate xl can be viewed as the system output, substates xi (i ~ 1) are intermediate inputs and outputs, and u is of course the system input. Since no structural requirement is imposed, one can choose the chain to facilitate stability analysis and control design.
42 Fe~iforward xflox. / ; ...........................................
r--- .......
Su~ystem# m
~~ ~ g;,,
i •
I
Feedback
~
r .................................
~~
J£ ...........
"~
Su~vstem# 1
E
v
~1
i
r
i
i I
L 1
E
I
E
~
I
1
I I I
•
x,
t
' \ " "'
t
Feedback xxto x~t Figure
3.2.1
3.1
Generic
model
of uncertain
systems
Basic Assumptions
A system whose dynamics are described by the above model may contain significant uncertainties. Due to the nonlinear nature of the system, stability analysis and control design will be pursued using the Lyapunov direct method. To handle the uncertainties in the system, one often uses the so-called deterministic design approach. The Lyapunov-based deterministic approach usually involves three steps. First, find deterministic bounding functions for the magnitude of the uncertainties. Second, a Lyapunov function is searched and often chosen as that of the known dynamics of the system. Based on the bounding functions of the uncertainties and the Lyapunov function, a stabilizing control is designed through stability analysis of the whole system using Lyapunov's direct method. The resulting control, if exists, is called robust control. In this chapter, we will proceed with the robust control design using this deterministic approach. Before considering the system, designing robust controller, and proceeding, the following fundamental assumptions are always required.
Fundamental Assumptions: Assumption A1 In the dynamics of uncertain systems, function ~(.) in (3.1) is Carathdodory (continuous with respect to x and u and piecewise continuous with respect to t), and the vector of uncertain variables v is Lebesgue measurable.
43 A s s u m p t i o n A2 In the dynamics of uncertain systems, function :F(.) in (3. I)
is uniformly bounded with respect to t and locally uniformly bounded with respect to x, u, and v. It has been shown in Hale (1977) and Hale (1980) that Assumptions A1 and A2 are necessary to ensure maximum continuation of an absolutely continuous solution (provided that the control designed is also a Carath~odory function) and to establish stability results. Also, if the control to be designed is limited to be continuous, global existence of classical solutions of the system (3.1) is guaranteed. Uniqueness of solution can also be claimed once stability result is obtained. Although fundamental assumptions are important, these assumptions provide no help in determining whether there exists a stabilizing control and, if so, how to find it. The reason is because not all uncertain systems are stabilizable, which can easily show by counterexamptes. As a result, it is crucial to identify stabilizable uncertain systems and to design robust control for these systems. Up to now, there are no necessary and sufficient conditions of stabilizability reported on robust control of systems (3.1). Three sufficient conditions on robust stabilization were developed. These conditions, summarized below, provide several classes of stabilizable uncertain systems and present the need of further relaxation. M a t c h i n g C o n d i t i o n s The matching conditions (MCs) introduced by Gutman (1979) and by Corless and Leitmann (1981) represent the first result on robust control of nonlinear uncertain systems. Two robust controllers were provided to guarantee global stability. For a system in the form of (3.1), the MCs are defined as follows: Decompose the generic function F(.) as
F(x, u, 7, t) = f(x, t) +
f(x, 7, t) + B(x, t)u +
7, t)u,
then, A f ( x , 7?,t) and AB(x, 71,t) are the input-unrelated and input-related uncertainties satisfying
A f ( x , rl, t ) = B ( x , t ) A f ' ( x , rl, t), A B ( x , rl, t) = B ( x , t ) A B ' ( x , TI, t), ttAB'(x,~,t)tt < C < 1,
(3.3)
where II" I] denotes Euclidean norm, C is a constant, and A f ' ( x , rl, t) is normbounded. The essence of the MCs is that, under the MCs, the dynamics of an uncertain system becomes
d: = f(x, t) + B(x, t ) { A f ' ( x , rl, t) + [I + AB'(x, r/, t)]u}, in which the uncertainties enter the system in the same channel as the control input u. The condition on AB'(.) ensures that there is no singularity problem in control design. These features provide an intuitive assurance of robust stabilizability. In fact, it works the same way mathematically. By requiring
44 in addition that the nominal, uncontrolled system x = f(x, t) is stabilizable, one can easily complete a Lyapunov argument using Lyapunov function V(x, t) of the nominal system. In the argument, the time derivative of V(x, t) has the property that both the control u and the uncertainty Af'(x, ~, t) are multiplied by the same factor ~7TV(x, t)B(x,t), thus, robust control can be designed to compensate the uncertainty Af'(.) using its size bounding function. It has been shown in applications that the MCs have a limited use since the uncertainties are required to be in the same channel as control input. Two extensions have been recently introduced to relax this limitation. E q u i v a l e n t l y M a t c h e d U n c e r t a i n t i e s Equivalently matched uncertainties (EMUs) as defined in Qu (1992a) are those uncertainties which contributions in Lyapunov argument can be factorized such that the terms in l}'(x, t) associated with the uncertainties have the common factor v T v ( x , t)B(x, t). The EMUs provide us a class of uncertainties that do not satisfy MCs but can be handles as if the MCs hold. This approach reveals that the key step mathematically in the robust control design is to make Lyapunov argument work by constructing common multiples (function factors) between terms associated with input and uncertainties. Such a treatment is also pivotal in another approach of robust control design, the generalized matching conditions (GMCs). G e n e r a l i z e d M a t c h i n g C o n d i t i o n s The GMCs were defined for nonlinear systems in (Qu, 1993) as follows. For a system in the form of (3.2), the GMCs imply in essence hi(.) =
= O,
nl < n2 <_ ... <_nm <_ nrn+l,
and, with z,~+i = u for notational simplicity, xi+, gi(xi,..., xi, xi+i, ~, t) > O, for a l l ( x i , . . . , x i + l , r l i , t ) , i E { 1 , . . . , m } . The last condition is on the ability of steering xi at both positive and negative directions by changing the sign of xi+l and by making xi+l large enough if xi+l were control variable. This condition is a generalization of that on ZIB'(.) in (a.a) for non-affine systems or subsystems and therefore necessary. The most important condition in the GMCs is that hi(.) = Ahi(-) = 0. Under this condition, the system can be decomposed of m subsystems connected in a series from the m-th subsystem to the first subsystem. The only interconnections between subsystems are feedbacks in the chain, that is, only from system i to j with i < j. No feedforward between subsystems in the chain is permitted. This structure allows one to design robust control recursively. That is, view the inputs of subsystems as fictitious control variables, and perform control design backwards in the chain (from the first to the m-th subsystem). Once fictitious controls are designed, translational transformations of state variables are introduced. Mathematically, these transformations define the "error signals"
45 to achieve decoupling and therefore form common multipliers in l) recursively. Intuitively, the new variables defined by the transformations make uncertainties locally matched with respect to fictitious inputs of subsystems. In some literature, this type of recursive design procedure is referred as backstepping procedure (Kannellakopoulos et al, 1991). It is worth comparing the above three sets of conditions on uncertain systems. It is obvious that both the GMCs and EMUs include the MCs as special case. Thus, an uncertain system is stabilizable if it satisfies either GMCs or EMUs. The GMCs have much wider applications than the EMUs, especially in mechanical-electrical systems such as robotic manipulators, power systems, servo systems, etc. This is because most of these systems contain electrical dynamics, actuators, and mechanical motions which subsystems physically satisfy the structure required by the GMCs. The GMCs and EMUs may not be applied together since the state transformations used in the GMCs form the new state and therefore hardly keep uncertainties not covered by the GMCs be EMUs even though they may be EMUs originally. The ultimate objective of robust control theory of nonlinear uncertain system is twofold. First, what classes of systems in the form of (3.2) but not satisfying any of existing structural conditions can be stabilized or controlled. Second, find procedures under which robust control u can be systematically designed. The key issue in the design is the search of Lyapunov functions and their associated robust controllers (which may be different for achieving various types of performances). Since control design will not be successful without stabilizability, these two problems cannot be separated. The design procedure proposed in this chapter provides a means to determine stabilizability and to find control. As wilt be shown in the applications of the proposed procedure, for systems whose uncertainties or dynamics consists of both Fi(.) and Hj (.) for some i and j, stabilizability can be guaranteed if there is balance between the magnitudes of Fi(.) and Hi(.) in the sense that Fi(.) and Hi(.) cannot be of large size and arbitrary simultaneously. It should be noted that, unlike the previous results in terms of structural conditions, the new balance condition just mentioned is not on locations of uncertainties but on their sizes. In fact, this type of balance condition is necessary for stabitizability. To see this conclusion, consider the system
21
=
A t l z l + x2 + Alax3,
25
=
A~lxl + A2~x2 + x3,
(3.4a) (3.4b)
23
=
u,
(3.4c)
where uncertain t e r m s Z~ij are independent but bounded by constants Cij > O. To verify that the system is not stabilizable for many sets of constants Cij, consider the simplest case that the uncertainties are time-invariant and stateindependent, in which case the transfer function between u and xl is x l (s) u(~)
A 1 3 ( s - z522) + 1 -
~(s - z~11)(~ - z ~ 2 ) - z~21~"
46 We see from the transfer function that, if 31I and A21 and A2~ are not small in magnitude and if z~13 is large in size, unstable pole-zero cancellation will occur and controllability will be lost. The unstable, uncontrollable pole-zero cancellation implies that the system cannot be stabilized. Stabilizability in terms of controllability can be guaranteed if a size balance between the two groups of uncertainties, {311 , 321, 322} versus A13, is required. Loosely speaking, the size balance condition is that, if the bound on z513 is large, the bounds on {All, A21, 322} should be small enough and vice versa. For nonlinear systems, the above argument using transfer function is no longer valid. It is the proposed generic design procedure that make it possible to derive stabilizability conditions and stabilizing controls.
3.3 P r o b l e m F o r m u l a t i o n Robust control design based on existing structural conditions excludes many important engineering problems. To design robust control for these systems not satisfying MCs or GMCs or EMUs, a generic design procedure, called recursive interlacing procedure, will be introduced. The procedure will be shown to include the ones used in preceding sections as special cases and to be applicable to uncertain systems without imposing any strong structural condition. To avoid unnecessary complications while establishing the generic nature of the procedure, we shall focus our attentions on two third order vector systems. The discussions of the generic procedure are organized into four subsections. First, by studying a class of third order systems, we reach the conclusion that a generic design procedure should have backward recursive and forward recursive steps. Tile specific illustrative example is used to reveal main ideas of the generic design procedure. Then, the recursive interlacing procedure is introduced, followed by selection of Lyapunov functions. Then, detailed design of robust control is proceeded with for the illustrative example. Finally, it is shown that the results on the illustrative example can be applied to the second example with only minor modifications even though the second example contains several additional uncertainties. 3.3.1
The Need for a Generic
Procedure
The generic procedure is rooted in backward recursion since recursive designs have been successful in dealing with cascaded systems. Thus, the key to have a more general design procedure is how to handle large uncertainties beyond the scope of the generalized matching conditions, namely, functions hi (.) and Ah~(.) in model (3.2). These connections are often strong, complicated and possibly uncertain, and the design procedures of robust control discussed earlier do not apply in the presence of these dynamics. To be specific, we use the following example to see the impact of hi (.) and Ah~(.) on control design and to motivate the need and characteristics of the generic approach.
47 E x a m p l e 3.1
Consider the third-order vector system given by
21 22 ~3
= = =
AfI(xl,vl,t)xl xs, u,
+x2 + Ahi(xl,x~,vz,t)x3,
(3.5a) (3.5b) (3.5c)
where xi E IR~', u E IR~', and A f ~ ( x l , v l , t ) and A h i ( x l , x 2 , v l , t ) are nonlinear uncertainties bounded by known, well defined functions. We assume for simplicity that nl = n2 = n 3 = n 4 : n/. It is obvious that system (3.5) does not satisfy the GMCs or the concept of EMUs or the feedforward structure. The following recall is obvious but crucial: a stabilizing control u must make x3 converge to a stabilizing control law for the first two state equations of (3.5). Since the state is of higher dimension than the control any successful design requires some scheme of recursive mapping. As under the GMCs, recursion steps in design will be instrumental in directing the search for appropriate Lyapunov functions. Denote fictitious controls for x2 and x3 by x d and x d, respectively. Then, dynamic equations of system (3.5) in terms of transformed state variables z1 = xl and zi = xi - x~ (i = 2, 3) become:
zl
i2
= =
= =
Af~(xl,vl,t)xl+xd+z2+Ah~(xl,x2, vl,t)x3 A f ; ( z l , v l , t ) x l + x d + z2 + A h ~ ( x l , x 2 , v i , t ) z 3 + h (xl, x2,
xs-x
,
(3.6a)
(3.6b) (3.6c) (3.6d)
The transformed dynamics (3,6) lead to three important observations: (i) Fictitious control x d should compensate for matched uncertainty A f t ( x 1 , v l , t ) x l . However, due to the existence of term Ah~(.)xa, backstepping (or recursive) design cannot be applied as it is. If the recursive design is used, x d must compensate for all uncertainties in the first subsystem including Ah~ (.)x3. Hence, x~ (designed for x~) would be function of not only xl but also x2 and x3; consequently, x~ would become a function of u, and in turn 5:~ would be function of iL. Under the standard recursion, this continuing chain of bazkward functional dependence in fictitious control design would result in endless argumentation of the state space model. As one of the four ideas of the generic design approach, fictitious control is allowed to be function of all state variables, but fictitious controls with backward functional dependence are selected so that state argumentation, if any, will be of finite order. The method taken is that dynamics beyond GMCs are compensated not by fictitious controls themselves but by choices of sub-Lyapunov functions. T h a t is, for subsystem (3.6b), fictitious control x~ will be designed to directly compensate only for A f t ( x 1 , vl, t)xl but not for Ah~(xl, x2, Vl, t)x3.
48 (ii) No matter how we choose z~, it must be at least a function of xl in order to stabilize the nominal system of equation (3.6b). Concentrating on ~d in (3.6c) and substituting (3.6a) for ~1 = kl, we can rewrite (3.6c) as z2 = - ~
+
x2A"" z2,vl,t) I - O Ox---~zanl~zl,
x 3 + o t h e r terms.
(3.7)
The second term on the right hand side of (3.7) contains uncertainty and therefore, the design of fictitious control z~ is possible only if the uncertainty associated with z3 does not make the matrix d
I - ~Ox2 A h f1~"z 1,z2, vl,~) singular or, equivalently, only if the uncertain matrix d
Ox~ Oxi A h ' it"z 1,x2,vl, t)
is smaller in Euclidean norm than one, the norm of known dynamics associated with z3. This implies that, in order to continue the backward recursive design of fictitious controls, the designer must look ahead to assess the impact of uncertainties beyond the GMCs on the subsequent subsystems. The impact can be controlled by proper selection of the current fictitious control which we will call the admissible fictitious control. Specifically, a nonlinear gain function is introduced to control the magnitude and partial derivatives of the current fictitious control, and its selection process is made explicit by the backward interlacing steps. The backward interlacing (multiple-step if necessary) steps together with the concept of admissible fictitious control and its gain function are the second main idea of the generic design procedure. Backward interlacing implies that the fictitious control z d must be selected according to the size of bounding function of Ah~(.) (which is beyond the scope of the GMCs). Thus, to derive a closed form robust controller, one must find a gain function satisfying certain properties in terms of bounding functions of dynamics (including uncertainties) that are beyond the GMCs. This is distinct to the generic design procedure since the procedure is applicable to uncertain systems without any structural condition. In contrast, robust control can be found under the GMCs or for the EMUs using merely bounding functions of uncertainties. (iii) It follows from (3.6b) that, no matter what is the choice of z d, the second to last term on the right hand side belongs to the EMUs with respect to u in subsystem (3.6), and thus can be compensated, which is made explicit by the so called forward interlacing steps. Unlike the EMUs made under the GMCs, this type of EMUs is not between two sequentially numbered
49 subsystems. Uncertainty decomposition performed from (3.6a) to (3.6b) and its associated forward interlacing steps are the third new idea of the generic design scheme. In equation (3.6b) obtained by the decomposition, the last term Ah'~(.)xda is the only remaining uncertainty not matched in any sense. However, the dependence that the unmatched uncertainty is proportional to Xa d provides an avenue for achieving stabilization by judiciously choosing a Lyapunov function associated with xad, called admissible Lyapunov function, such that the negative definite term produced from the design of xad dominates in size the unmatched uncertainty associated with xad. Again, this process will be made explicit by forward interlacing steps. The practice of having a generic Lyapunov function and then selecting an admissible one for size domination over unmatched uncertainties is the fourth idea of the generic design procedure. In comparison, any positive definite, differentiable function of local, transformed state is an admissible Lyapunov function under the GMCs.
In summary, in order to take into account in control design (uncertain) dynamics not satisfying the GMCs, a constructive design procedure can be developed. The procedure will consist of backward recursive steps whose final outcomes are fictitious controls and sub-state transformations. What's new is that, in each recursion step of the procedure, there are several sub-steps where new interlacing designs take place. These interlacing steps allow the designer to take full advantage of the only two choices he/she has: Lyapunov function and control structure. In each recursion, while the designer considers the current subsystem, he/she also bears the rest of the system in mind by both "looking back" (backward interlacing) and "looking forward" (forward interlacing). By looking back, a gain function is selected so that the resulting fictitious control is admissible in the sense that recursion process can be continued, and dynamics not satisfying the GMCs are decomposed. By looking forward, fictitious control and generic Lyapunov functions are chosen so that dynamics not satisfying the GMCs are compensated. Namely, all dynamics of the system are considered in design in an integrated and fully programmed fashion. Compared to the previous design procedures, this design procedure imposes no restrictions on either locations or types of (uncertain) dynamics, it can be used not only to determine stabilizable systems and their globally stabilizing robust controls but also to reveal instabilizabitity (note that, since uncertainties are allowed everywhere, the system under study may be unstabilizable), and it includes both recursive designs as a special case. In addition, no assumption is needed on stability of the nominal system or on existence of Lyapunov function. These features will be demonstrated shortly by studying a class of third-order vector systems. With this basic explanation of the ideas underlying the generic design procedure, we proceed with its description and applications.
51)
3.3.2
Recursive-Interlacing
Design
Procedure
The generic procedure, called recursive-interlacing procedure, is a fully programmed integration of four components: recursion (or backstepping), forward interlacing, backward interlacing, concepts of admissible fictitious controls and Lyapunov functions. For systems in the form of (3.2), the procedure is as follows: R e c u r s i o n S t e p 0: Apply transformation zl = zl - x d, where z~ is a given desired, differentiable trajectory for output tracking. For regulation, z d = 0. R e c u r s i o n S t e p i (1 <: i < m): For the i-th subsystem of state zi, determine admissible fictitious control zd+l (with xdm+t ----zm+~ ----u) and admissible Lyapunov function ~ ( z l , . . . , zi), and apply substate transformation zi+l -- Xi+l - xd+t through the following five sub-steps. S u b - s t e p i.h Stabilization of the dynamics satisfying GMCs in the /th subsystem. Assume first that, in the equation of ~i, the part of dynamics not satisfying the GMCs be zero and that xi+l be the control variable. Then, use a generic Lyapunov function Vi(zl,..., zi) to find the set of all stabilizing controls, denoted by X(~+1)1 , for the remaining dynamics. S u b - s t e p i.2: Forward interlacing to stabilize the EMUs. Again assume that zi+l be the control variable, determine in X(~+1)1 the set of all stabilizing controls, denoted by X~+1)2, such that all the EMUs associated with z~ in the preceding sulJsystems are also compensated. S u b - s t e p i.3: Backward interlacing to select an admissible fictitious cond • d d trol zi+ 1. Pink xi+ 1 E Xii+l)2 such that, if zi+2 were the control variable, if xj with j <_ i + 1 are fixed, and if xi+3 = -.. = z,n = u = 0, there is no singularity for designing xi+2 in the remaining dynamics of zi+l (to be derived) in the sense that the lumped matrix pre-multiplying xi+2 is either positive definite or negative definite. This can be done by checking and comparing dynamics associated with z~+2 in both equations of ~ and z~+l. S u b - s t e p i.4: Forward interlacing to determine an admissible sub-Lyapunov function V/. Function P~(zl,..., zi) should be chosen such that, not only ~ has a negative definite, radially unbounded term of [[zi[[ under zd+l, but also a proper fraction (for example, 1 / ( r e + l ) ) of the sum of negative definite terms in ~ ' = l VJ (generated under zd+l, j = 1 , . . . , i ) dominates in magnitude all unmatched uncertainties associated with zd+l (but not those with z d, k > i + 1) in )-~j'=l VJ" S u b - s t e p i.5: Backward interlacing to decompose uncertainty and to perform recursion (if i < m; exit if i - m). In the expression of ~ (or i ~i=1 VJ and zi), decompose all dynamics (including uncertainties)
51
beyond the GMCs into EMUs and uncertainties associated with x~ by variable substitutions of xk = zk + x d for i + 1 < k < m, where x d are to be designed in the subsequent recursion steps. Then, derive the dynamics Zi+l by sub-state transformation zi+t = xi+l - x~+~. Continue with i replaced by i + 1. 3.3.3
A Class
of Lyapunov
Ftmctions
In applications of the proposed procedure, Lyapunov function of the following generic form can be used:
Vi(Zl,...,zi) : l f ~i(llZl]l,...,llzill) dllzill2 ,
(3.8)
where f denotes the operation of taking indefinite integration except that the arbitrary constant C commonly equipped with indefinite integration is excluded, and ~i(-) is a positive function in the sense that, for some constant ~i > 0, ~i(') _> ~i" Function/3i(-) should also be chosen to be differentiable and locally uniformly bounded with respect to its arguments. If expression (3.8) is adopted, selections of ~/}(-) mentioned in the procedure should be done in terms of choices of ~i('). Note that dl[zill 2 -- 2 z [ d z i = 2 ~
z. dz.,
/=1
where zi~ denotes the/-th element of the substate zi. Denoting the functional relationship between zj and ziz (j < i) by zj = Zj,it(zit), we can rewritten (3.8) as
x~(zl,..., z~) =
~i(llZl,.(s)tl,..., IlZ~,.(s)ll)s ds.
(3.9)
Since functional relationship Zj,it depends on a specific trajectory and is therefore unknown, integration defined by (3.8) or (3.9) can not be calculated explicitly. That is, functional expressions other than integral expression (3.8) or (3.9) do not exist explicitly for sub-Lyapunov functions Y}(.) except for V1(z~), which marks another distinct feature of the generic design approach. To show positive definiteness of Lyapunov function (3.8), we introduce the well known, scalar inequality that, for a continuous function ~(.),
ooY~(s) d s > O
Vy#0
if~(y)y>Oforally#Oandif~(O)=O,
This inequality can be established by the mean value theorem in calculus as: there exists w E (0, y) (for y > 0) or w E (y, 0) (for y < 0) such that, whenever y#0,
/0
~(s) d~ = ~ ( ~ )
/0
ds = ~ ( ~ ) v = [ ~ ( ~ ) ~ ] ±
W
> 0.
52 The same inequality can also be shown graphically since ((y)y > 0 implies that the image of function ~(y) lies entirely in the first and third quadrants. Applying the above inequality with ~(y) = 9g(llzl,~(y)ll,
..., ttzi,gt(y)tt)y,
we know that all terms in the summation of (3.9) are positive definite with respect to their arguments. That is, sub-Lyapunov function ~ is positive definite with respect to zi. This and other useful properties of the Lyapunov function (3.8) are summarized by the following lemma in which properties (ii) and (iii) can be shown by the same scalar inequality. L e m m a 3.2 Let sub-Lyapunov function Vi be defined by (3.8). Then, (i) Lyapunov function m
V:=~ i=l
is positive definite with respect to the transformed state z. (ii) V is decrescent and radially unbounded. (iii) In a hyperball B(0, r) of radius r centered at the origin of the state space of z, V is locally uniformly ultimately bounded with respect to r.
The properties summarized in Lemma 3.2 are needed for concluding stability results using the following lemma. L e m m a 3.3 Suppose that, for any bounded set f2' E IRn, function F : IR'~ x IR + -+ IRn maps f21 x IR + into bounded sets in ]Rn. Let V ( x , t ) be a scalar function with continuous first order partial derivatives and M be a closed set in IR~. Then, (i) /f l)(x,t) _< O f o r a l l x E M c and i f V ( X l , t l ) < V(x2,t2) for all t2 >_ tl > to, all xl E M and all x2 E M~, then every trajectory of system (3.1) which at some time is in M can never thereafter leave Me. (ii) If, in addition to the conditions in (i), (/(x, t) >_ 0 for all x E 1Rn and t > to and ~Z(x,t) < - e < 0 for all t > to and x E My, then every trajectory of system (3.1) will enter and then stay inside Me in a finite time. (iii) If, in addition to the conditions in (i), the set M is bounded and V(x, t) is radially unbounded, then every trajectory of system (3.1) is globally uniformly ultimately bounded with respect to the set Me.
The above two lemn~tas will be used to conclude stability in a Lyapunov argument.
53
3.3.4 Step-by-Step Design for the Motivating Example To illustrate the generic design procedure, we shall apply it step-by-step to the motivating example, example 3.1. As will be shown, stabilizing the class of third-order vector uncertain systems in the form of (3.5) only requires two assumptions. The first is on the existence of bounding function for uncertainty Ah~ (.). A s s u m p t i o n A3 There exists a known function fib1 (') that is uniformly bounded with respect to time and locally uniformly bounded with respect to xl and x2 such that, for all (xl, x2, 1)1,t),
tl~hi(xl, x~, t)1, t)t t ~ Phi (Xl, X2, t). By the properties of function ph, ('), we can assume without loss of any generality that a scalar, differentiable function a(.) exist such that, for all
(xl, x2, t),
Ot(Xl, X2)
~
max{2,36ph,(xl,x2, t)},
~2(x,,x2 ) > 36 -
tUllx
144(
lt
~a(xl,xu) > -g-phi
(
Oc~
(3.103) d?a
1
)
Oa ; )
~=+gltZlllU gZTz~
(3.lOb)
'
,
(3.10c)
2
~4(xl,x~ ) >_ 4tlzltt 2 0@1
(3.10d)
It is obvious from the above inequalities that, no matter what is Phi ('), function o~(-) can be found by choosing it large enough and nonlinear enough. As an example, ifph~(Xl,X=,t) = c1(1 + x~ + x~) for some constant cl > 0, function a(.) can be selected to be OI(Xl,X2) = C2(1 "~ X2 -~- X2),
where constant c2 is chosen to satisfy the following inequalities: c~>_2ClC~+4c~+2cz
and
c~>__32clc~+16cl.
(3,11)
It is worth noting that choice of function a(.) is not unique and is worth recalling from Qu (1993) that bounding functions can be made differentiable by enlargement. Another assumption, given below, is on the size of uncertainty Af~ (.). A s s u m p t i o n A4 There exists a known function Pfl(') that is uniformly bounded with respect to time and locally uniformly bounded with respect to xl such that, for all (xl, vl,t),
54
In addition, it is assumed that bounding function p/~ (xl, t) be differentiable and satisfy the following inequality: 1
pf, (xl, t) _< 4a(xl, x2)"
(3.12)
Function a(.) defined in terms of bounding function Ph~ (') is the gain function explained in observation (ii) in the previous subsection. Its role is to make fictitious controls admissible through nonlinear saturation, and it enables us to derive an actual, closed-form robust controller. It follows from the discussions about system (3.4) that a condition in the form of (3.12) is necessary for stabitizability, which is what one finds out by applying the proposed recursiveinterlacing design procedure. Inequality (3.12)implies that, if uncertainty Ah~ has large magnitude, the magnitude of uncertainty Af~ (.) must be limited to be small, and vice versa. (3.12) may not be necessary, of (3.12) is necessary. The condition (3.12) on gain function and on relating bounding functions of two uncertainties is found by following step-by-step the design objectives of the recursive interlacing design procedure, as proceeded below. R e c u r s l o n S t e p 0: Form output tracking error zl = xl - x d. For simplicity, we choose x d = 0. Thus, the first subsystem becomes =
f;(xl,
.1,t)zl +
+
hi(Zl,
(3.13)
R e c u r s i o n S t e p 1: Determine sub-Lyapunov function and fictitious control for system (3.13). S u b - s t e p 1.1: Stabilization of the dynamics satisfying GMCs. Setting dynamics not satisfying the GMCs in the first subsystem zero reduces the first subsystem to be
~1 = Af~(xl,vl,t)xl + x2. Thus, for stabilization of locM and feedback dynamics, the set of all stabilizing controls for x2 is
Xdl
=
{wl(zl,x2, x3, t) ElR~'13~/l(.)ElC~ :
bZl [wl + AfI(Xl, vl, t)zl] < -'~l(llz~ll), Vzl, x~, x3 •
a ~'
,
where V1(zi) is a generic sub-Lyapunov function whose choice will be made later. S u b - s t e p 1.2" Forward interlacing to compensate for the EMUs. Since there is no preceding subsystem (i = 1), this step becomes triviM, and %~2 = Xdl •
OO
S u b - s t e p 1.3: Selection of an admissible fictitious control by backward interlacing. The dynamics associated with x3 in zl and 22 are /,h](.)x3 and x3, respectively. For a generic choice of fictitious control xd(zl, x2, x3) E X~, the dynamics associated with xa and satisfying the GMCs in transformed dynamics of z2 = x2 - x2d must be
I-
OX2
t
5-;7,/`h~(~, ~ , ~ , t ) -
~3 :=
B~(z~,~2,~,~,t)~, (314)
where I is the identity matrix of dimension n t. ~Ib make fictitious control design possible for x3 in the next recursion, matrix B2 (.) + B~ (.) must be positive definite. An admissible fictitious control that guarantees positive definiteness and belongs to X~2 is 1
xl -
1
zl,
(3.15)
ctt Zl , X2 )
under which, by (3.10),
1B
5( 2 + B~) = I+
Ahl + [Ahl]T
2ct
--
(Ah.~TOct
Oct T ~-£+-~i~
1[ ~oct oct] 2ct2 (/,hi) ~ + ~ ~ _ >
ct
ct~llzlll
?-~zz phi +
1
~I.
~
I (3.16)
Loosely speaking, a fictitious control in the form of (3.15) is chosen to reduce the impact, of uncertainties beyond the GMCs on subsequent subsystems by utilizing a proper gain function ct(,). In the case that ct(.) is a also function of x3, the partial derivative
ax~ azl x will be function of x3, but it does not have to be considered in (3.14) since it can usually be bounded by a function only of zl and x2. S u b - s t e p 1.4' Determination of admissible sub-Lyapunov function V1(za) of generic form (3.8) with i = 1. It follows that
s~ = ~ + ( ~ - ~ ) + a h i ( ~ l , x~, ~ , t ) ~ and that, under x2d in (3.15),
56
~>1(zl) =
Zl(llzlll)zTh
Zlz~nf~(~,,,1, t)z~ - l n ~ . it~lt 2 + n l . ~1~(~ _ ~ )
:
C~
+ Zl " zT Ah~ " x3 <_ - ~ n ~ . IIZlll~ + nl . z ~ ( ~ - ~ ) + ~1. z~ ahl . ~ . Since i = 1, there is no preceding subsystem, and hence no forward interlacing is needed. Thus, sufficient conditions for l/](z1) to be admissible are that, for constant ~-1 and for all (zl, 12), ¢~l(HzllI) >/71 > O, - -
and
lim inf d-+~ IlzaN>_d Ilx211 fixed
n~(NzlN)Nz~N~ a(Zl,X2)
= oo.
(3.17) Under the above condition, 1)1 has a negative definite, radially unbounded term of IIz~N. There are infinite choices of/71(.), and /71 (Ilzl N) _> a(zl, x~) with z2 being some constant vector is the most obvious choice. A specific choice of/71(.) is left to the reader. S u b - s t e p 1.5: Backward interlacing to decompose uncertainties and to perform state transformation. Making variable substitutions xk = zk + x d for k = 2, 3 in 1)1 yields (z~) _<
-
3~.
Nz~ II2 +/31"
zTlz2 -t-/~1"zTAhl "z3 + j31" zT
A h l . x d"
(3.18) The above process decomposes uncertainties into EMUs and unmatched uncertainty that will vanish as x~,d vanishes, which is the key to compensate uncertainties beyond the GMCs. Applying state transformation z2 = x2-x d with x d defined by (3.15), we can derive the second, transformed subsystem from (3.13): ~2
= {[11_
:=
al.gzl ( ~ 7 1 ) T ] z ~ f ~ z l +
A=(zl, z2) + B=(zl, z2, vl,t)z3.
[ 1 i _ _~7z1 ~c3~'~ T]
(3.19)
R e c u r s i o n S t e p 2: Select a sub-Lyapunov function and a fictitious control for system (3.19). S u b - s t e p 2.1: Stabilization of the dynamics satisfying GMCs. Since there is no dynamics beyond the GMCs in subsystem (3.19), no dynamics are excluded in this step. It is obvious that the set of stabilizing fictitious controls is
57
= {w2(zl,z2,x3, t) E]Rn' 13~/2(.)EKoo, 3(2>0 : a~2 [ B ~ 2 + d~] < -~(llz~ll) + ~ , V ~ , ~ 2 , z ~ , ~
c n~~'
,
V2(z1,z2)
where is in the form of (3.8) with i = 2, and constant e2 is a design parameter. S u b - s t e p 2.2" Forward interlacing to compensate for the EMUs. It follows from (3.I8) that the EMUs with respect to z2 in ~)1 is II)zTz2 or, mo,e conservatively, /31 (llzl II)llz~II IIz~ll Thus, it becomes apparent that the set of all stabilizing controls xaa, 2~2 (which is a subset of X~I if the EMU terms are first bounded and then compensated), is
/31(11Z,
I
{w2(zl,z~,xa,t) E ]Rn' ] 3 ~/2(') E ~z, 3e2 > 0 : /~2(ltZtlI, ItZ211)Z T
Z2 /31 [B2W2-FA2-F I[Zlt1~22 ]
< --~(II~11) + ~
Vz~, ~, z~, ~ e m"' ~. )
S u b - s t e p 2.3: Selection of an admissible fictitious control by backward interlacing. Since there is no uncertainty beyond the GMCs in system (3.19), every choice in A~2 is admissible. The total dynamics to be compensated by x~ can be bounded as
A2
9~(IIz~II) IIz~H~
91(llzlll)
+
ltzlll
_< IIz~ll X1 + ~ +
IlZllt
ac~ IIz~ll+ ~2(11~111,11z211)ttzltt ilzlll ~ a~ 2
Pft + ltzill ~2 + --~ + p}t
58
:= IIz~ll~2(zl, z2) + ilzlll ~21(zl, z~) := ~2(z~, z2),
(3.20)
in which transformation z2 = x2-x2d and inequality a2+b 2 >_ 2ab are used to render differentiable function ~r21(') and ~r22(.) It follows from the backward recursive design that, for system (3.19) with matched uncertainties bounded by functions in (3.20) and (3.16), a simple choice of fictitious control in X[2 is
X3d = -2 [~2 + ~2(~1, z2)z~ +
~2(z~, z~)ll~2(z~, 22)115o.2~(zl, z2)ll~ll 1J (3.21) 11~2(z~,z2)tl 3 + ~
where #2(z1,z2) = /)2(llZlll,IIz211)~r21(Zl,Z2)llzlllz2. The selection of x3d is made such that
v2 = ~2. z ~
<_ -~1. llz~llllz~ll-~. 11~2112+~ + ~ . z~B2. ( ~ - ~ ) .
(3.22) S u b - s t e p 2.4: Selection of admissible Lyapunov function V2(zl, z2) by forward interlacing. Our objective is to judiciously choose scalar function ~2(11z111,llz2lf) such that, in addition to making V2 have a negative definite and radially unbounded term of Ilz211, the unmatched uncertainty associated with x3~ in V1 is dominated in magnitude by a fraction of the negative definite terms in V1 and V2. To this end, let us first combine (3.18) and (3.22) and obtain
zT Ah/1. x d3 + ~ . z y a h i . ( ~ - ~g) + ~ . ~ z ~ . (xa - ~ ) . (3.23) It follows from (3.21) and (3.20) that unmatched uncertainty in (3.23) can be bounded as
<
2 ~ . IIz~ll "pn, (llz~ll + ~ ( z ~ , z2)llz211 + 5r21(Zl, Z2)IIZlI[)
× Ozl
-I~
/ +
8~lph~ o(]- + 2~pa~ [1 + 0"22(Zl, z~)]llz~illlz211
~----~--
~IHZlH2[8ph~Pfl -~-8ph~O~p2f~~-8ph~(~+ 4--~3HZ1H2
59
×
+ 1
Z~+
2
+ ~llz~ll {64Z, p~, [1 + ~ ] ~ . }
-<
14~'~llzltt2/sV +
sz~] 9~ / + ¼11z~tP[2~6z,&4
:-- llI,21 z T ~ h ~ xglfl, in which III. Ill denotes a bounding function of the argument, and property (3.10) is applied. Uncertainty compensation through size domination can be achieved if/~2(.) is chosen such that, for m = 3, 1
IIl~(llzlll)zTahi(x~'x2'~'t)xd3111 < -Eg-i
(1
9111z~ll~+~llz211~
)
It is obvious that sufficient conditions on ~ ( . ) to validate the above inequality are that, for all (zl, z2) and for a given constant ~_~ > 0,
~(tlz~ll, t1~11) >_ max {~2,64~1 ph~ a, 256flt p~, c~}.
(3.24)
Using this choice of Lyapunov function, we have
+ ~ . ~ [ A h l . (z3 - ~3~) + ~2. zT2B2 " (x3 -- x~).
(3.25) S u b - s t e p 2.5: Uncertainty decomposition and state transformation by backward interlacing. Applying variable substitution x3 = z3 + x~ to (3.25) yields
vl+v2 <-~-~29111z1112-3 ~Z~II~II ~ -F~2"I-Zl "Z r1 A h 'I. z a + ~ 2 . z ~ B 2 _
~
.z3.
(3.26)
By direct calculation, one can get
Ox~ ~3
= :=
OXl As(z1, z2, z3, vl, t) + u.
Ox2
(3.27)
R e e u r s l o n S t e p 3: Choose Lyapunov function and robust control for system (3.27). Since all uncertainties in subsystem (3.27) and expression (3.26) are either matched or equivalently matched, and since subsystem (3.27) is the last, analysis in the sub-steps becomes much simpler, as summarized below.
30 In S u b - s t e p 3.1, one can easily find that the set of stabilizing controls for local and feedback dynamics is
Ul = •~'d41= (W3(Zl,Z2,Z3,t)~]a n' ] ~'~3(')~]~e~, ~£3 > 0 : 0g3rt~3 + A3] < -~3(11z311) + ~3, Vzl, =5, z=, =3, =3 ~ ~ ' } . In S u b - s t e p 3.2, the set of stabilizing controls for both dynamics satisfying GMCs and the EMUs can be shown to be, if Lyapunov function l/~(-) is in the form of (3.8) with i = 3,
H2 = X~2 = {w3(zl,z2, z3,t) ElR~'1373(.)EK~,
3e3 > 0 :
< -~3(tl~II) + ~3, v z~, ~ , z~, ~3, z3 ~ ~ ' }, Since i = m = 3, S u b - s t e p s 3.3 and 3.5 become trivial; that is, any choice in/42 is an admissible robust controller, and no more state transformation is needed. Since there is no unmatched uncertainty associated with u, S u b - s t e p 3.4 becomes trivial, that is, all positive choices of/33(') yield admissible Lyapunov functions. As shown in (Qu, 1993), robust control u can be selected to be
, 3 ( ~ , z~, z3)
u:-z3-
11.3(
f
~ z3J'
(32s)
where
~3(zl, z~, z3)
= ~3(zl,z2, z3)za,
~3(zl, z2, z~) = flIA3tZt =
][]A31[[+ ~
0~
"Pn, ' IIz~ll +
~"
IIIB2Ill- Ilz~ll,
(Pl,'tl=~l[+ Ilx21l+p.~" II=all)+ 7=2 "llx3ll
and
This robust control is selected such that
= (/1 -t- V2 -t- V3 < -~---~fllllzllt 2 - 3~2Hz2H2-f1311z3H2 +e2 -t-e3.
(3.29)
Stability can be readily concluded from (3.29). Since three negative definite terms in (3.29) are radially unbounded with respect to Hzilt respectively, V < 0
61 holds outside the hyperball B(O, rl) for some rl > 0. Recalling from Lemma 3.2 that V is both locally uniformly bounded and radially unbounded, we can conclude that there exists a hyperball B(0, r2) with r2 _> r~ such that V(z) > V(z') holds for all z' E B(O, r~) and z ~ B(0, r2). Applying Lemma 3.3, we know that transformed state z is uniformly ultimately bounded. Finally, as proceeded in (Qu, 1993), it follows from definitions of transformed state z and fictitious controls that the original state x is uniform bounded and uniform ultimate boundedness. Once the bounding functions of uncertainties are given, the ultimate bound on the state can be determined but a general expression is too complicated to find. In summary, we have the following result: T h e o r e m 3.4 Suppose that uncertainties Ahi(. ) and AI~(.) in uncertain system (3.5) satisfy Assumptions A3 and A4. Then, system (3.5) is uniformly
bounded and uniformly ultimately bounded under robust control (3.28) if functions c~(.), /~1(') and [32(') are chosen to satisfy conditions (3.10), (3.17) and (3.23) (which can always be done).
3.3.5 A G e n e r a l i z a t i o n of t h e D e s i g n E x a m p l e To further elaborate on the nature of the generic design procedure, let us consider the following third order vector system which contains more uncertainties. E x a m p l e 3.5 51
Consider the system
=
x2
A f ~ ( x l , v l , t ) x l ÷ x ~ + A h ~ l ( X l , X 2 , Vl,t)x3, --- Af~l(Xl,v2,t)xt + Af~2(xl,x2, v~,t)x2÷ x3 ,
53
=
Afa(xl,x2, x3, v3, t)
+ u,
(3.30a) (3.30b) (3.30c)
where xi, u E lR n'. Compared with Example 3.5, the system has three additional uncertainties, Aft1(.), Aft2(.), and A f3(.). To show the generic nature of the design procedure, we choose to proceed with control design with minimum deviation from that of Example 3.5. Specifically, we assume that uncertainties Ah~(.) and z~f[ (.) be bounded as specified by Assumptions A3 and A4 and that nonlinear gain function ~(x, t) be selected to be (3.10). The additional uncertainties are assumed to satisfy the following assumption. Again, as shown in system (3.4), inequalities of form (3.12) and (3.31) are necessary. A s s u m p t i o n A5 There exist known functions P]~I (') and p/~ (.) that are uniformly bounded with respect to time and locally uniformly bounded with respect to xl and x2 such that, for all (xl,x~,v2,t),
Ilzhf;l(xl,v2,t)lt ltAg2(xl,x2, v2,t)ll
IlaZ~(~,~,~a,-~,t)ll
<_ pf~l(x,,t), _< pf~(xl,x2,t),
<_ pi~(x~,~,~,t) .
62
In addition, it is assumed that bounding functions PJa~(Xl, t) and PSi2(Xl, x2, t) be differentiable and satisfy the following inequality: 9
pf~,(xl,t) <_ 64a(xl,x2)'
9
PS':(xl'x2't) < 32c~(x~,x2)"
(3.31)
Step-by-step design is followed to construct robust controller for system (3.30) satisfying Assumptions A3, A4 and A5. R e c u r s i o n Steps 0 a n d 1: Since the first subsystem is identical for (3.5) and (3.30), all sub-steps are the same except that, in Sub-step 1.5, dynamics equation should be
~2
:
-t-[Af2~-t-lI-~
\Oz,,I J
+Ii+~Ah~_ 1
I'(0o,'~ r
j
OR T
:= A,(~,, ~,) + B=(z,, ~=,.~,t).~.
(a.32)
R e c u r s i o n Step 2: Dynamic equation (3.32) is different from (3.19) only in the definition of vector A2(-). Thus, sub-steps 2.1 and 2.2 remain to be the same, and fictitious control xad in sub-step 2.3 is of form (3.21) except that the bounding function for the dynamics in (3.32) must be rederived as /~1(][zi ]D [[zllt z ~ [A2 + Z2(IlztN, N~211) <
0c~
[ ip , W -~[[ZlH l{~z 1 Pf,
Pfui
Zl(Nz~II) [[Zli[ ggTz~ C9~ ][X2[[ + Z2(iizII[, ]lZ2l[) Ilzlll
Jr<
(3.33)
1 ( ~Pfl
~}- ~llz~I{
1
~ ml -~- pf~ NZ~II + PS=~NZ2N [°°It ) ~llz2II + ~,Iz, ll + ~llzflI 2 ]l~ ~1(11z111) IIz~ll
<
IIZ2ii Pf2a -'i- g "l- 7 -["
(~Ol 2
1
63
~ :=
÷
IIz~llo~(z~,z~) + IlZlllO~(q, z~).
9~(llzlll, llz~tl)/ (3.34)
Because bounding functions O'21(" ) and c~22(') are changed, bounding function on unmatched uncertainty associated with x3d (in Sub-step 2.4) becomes
li~,. zTnh'~, x~il < =
2A. ilz, li • p~, (ilz~ll +
[
~=(z~, z=)iiz~ii + ~=l(Z,, z2)iiz, ll)
(
Allzlll 2 8phlP/1 + 8phlveP}l + 8phlP/a, + 8fib1 P/a2 + -~
+2'~IIzlii2 ~ 2)+ 8#~ph~]~J + 291ph,[1 + ~(Zl, z~)lllzlilliz2il -< ]---~/3111zl]I218Ph'P:'+ 8Ph'c~P~' + SPh~PI=' + 8Ph~ ( p]~= + 2-c~
+ lllz~II= {64/3tp~ 1[1 + ~,2]=c~}
8/~lphlO~] 1 r.
<
1..,,z1:
:=
IlIV,.z~h~-~gII{.
[
(~-Jc1) ' oil
In this case, inequality
IIl~(llz~ll)zr Ah'~(<'~2"o*'t)xglll <- 7 7 - f
9~11z~112+ ~=llz=ll=
holds with m = 3 if t32(-) is chosen such that, for all ( q , z2) and for a given constant -~2 > O,
~2(liz, il,llz=ll)>max{_~=, 2 5 6 k . p h , .~, 294/31.p~,.a}.
(3.35)
Using Lyapunov function (3.8) with/~(.) defined in (3.35), we can show that inequality (3.25) holds. Therefore, Sub-step 2.5 is the same except that dynamics of za are given by
~3 = Af3(.) +
d
c~x3/A ~-~ ~ . ~:1 .zl + z~ + Ah~
Oz3 : A ", :=
'.3)
+ A f ~ .x~ + xa) + u
A3(zl, z2, z3, v,,t) + u.
(3.36)
64
R e c u r s i o n S t e p 3: Note that the only difference between (3.27) and (3.36) is the definition of vector Aa(.). It follows that robust control u should be given by (3.28) except that ]IIAaHI "-
Pfa d- CgXl (Ph " [[XlH-{- I]x2H d-ph,. Hx3H) +
Ox2
" (P.r,, " llxlll + p.r~," I1'~11 + II~atl).
By the same argument as that for Example 3.1 we can conclude the following theorem.
Suppose that uncertainties Ah~(.), zif~(.) and Aft(.) in uncertain system (3.30) satisfy Assumptions A3, A 4, and A5. Then, system (3.30) is uniformly bounded and uniformly ultimately bounded under robust control (3.28) with the above modification on IllAaI[[ if functions a(.), ~1(') and/~2(') are chosen to satisfy conditions (3.10), (3.17) and (3.35) (which can always be done).
T h e o r e m 3.6
3.4 Features of t h e D e s i g n P r o c e d u r e As shown in the step-by-step designs of the two illustrative examples, the recursive interlacing design procedure has the following features and therefore is generic. (i) No a priori structural assumption on system dynamics is needed. System dynamics are fully explored and taken into account by the forward and backward interlacing steps leading to a robust control. Objectives of all steps of the procedure are defined explicitly. (ii) No assumption about stability of the nominal system or its Lyapunov function is required. (iii) All existing structures (MCs, GMCs, EMUs and feedforward structure) and their control designs are subsumed as special cases. (iv) The constructive nature of the procedure sheds light on the stabilizability of the system. In terms of design steps, the recursive interlacing design procedure are different from all previous design procedures in the following aspects: (i) Since stabilizability is no longer guaranteed, the designer has to find out stabilizability through a careful application of the procedure. (ii) As before, a sequence of sub-state transformations introduce design variables, fictitious controls. However, fictitious control for the i-th subsystem
OO
cannot be selected only based on the dynamics of the i-th system. A gain function must be selected to limit if necessary the impact of dynamics beyond GMCs on the rest of subsystems through the fictitious control. (iii) Due to the nature of gain functions, control efforts of fictitious controls may be limited. If so, output tracking error cannot be made uniform ultimate bounded with respect to any given measure on accuracy. (iv) Decomposition of dynamics not satisfying GMCs must be performed so that compensation can be achieved by one or multi-step forward interlacing. (v) The existence of uncertainties not matched in any sense demands sophisticated choices of Lyapunov functions. Specifically, non-quadratic Lyapunov functions of integral form (3.8) are used. As shown by the design examples, the procedure is applicable as tong as the following simple conditions hold: ~i <_ n2 <_ --" _< n m <_ nrn+l, and
xi+lgi(xl,..., xi,
xi+l,
vi,t) xi+t
is a positive or negative definite function of with xm+l = u for all i = 1 , . . . , m. Compared with the GMCs, intricate interconnections in the forms of functions hi(.) and z~hi(.) are allowed, which makes the condition on dimensions of subsystems insignificant. The only real restriction remaining is that the uncertainties presented in gi(') a r e sign invariant, and the %action 9i(') does not have singularity with respect to xi+l. Although the same condition is required in robust control for systems under the generalized matching conditions, the key difference here is that the order of numbering subsystems can be rearranged by rewriting or recollecting the functions fi ('), Afi ('), hi (.), and z3hi(.). T h a t is, we can manipulate the known dynamics of the system and search for a numbering sequence for subsystems in order to satisfy the condition, while, due to the fixed structure, there is no remedy if the condition fails in the GMCs. In the design process, the functions hi(.) and z~hi(.) represent dynamics beyond the scope of the GMCs. Both these two functions are first temporarily disregarded in the i-th recursion step and then compensated in latter steps. The difference between treating the known dynamics hi(.) and the uncertainty Ahi(.) is that function hi(-) can be compensated directly since it is known, while function Ahi(.) can only be compensated after it is bounded and then replaced by its known bounding %nction. The functions fi(') and Afi(.) are handled in a similar fashion. One should explore and utilize in the design the properties of functions fi(-) and hi (-), but for generality, no assumption is made in the design procedure on these known dynamics. It is apparent that, if Afi(.) = Ahi(.) = 0 for all i, the procedure is readily applicable to known nonlinear systems in a simplified version since
66 known dynamics can be compensated directly, and the words "robust" and "uncertain" can be removed. If h~(.) = z2h~(.) = 0 for all i, all the interlacing steps in the generic procedure become trivial and sub-Lyapunov functions can be chosen freely, which shows that the procedure includes the recursion (or bazkstepping) design procedure for systems satisfying the GMCs as a special case. Also, the procedure includes those for systems with MCs or EMUs as special cases.
3.5
Simulation
To demonstrate the effectiveness of robust controls designed using the recursive interlacing design scheme, simulation results of the design examples will be provided. Simulations were done using SIMNONQ. Example 3.1 was simulated under the following choices: • Uncertainties and their bounding functions are chosen to be 1
A f ~ ( x l , v l , t ) = pf~(xl,t) = 288(1 + z I + z~)' and
Ah~(xl, x2, vl, t) = Ph~ (xl, x2, t) = 1 + x~ + x~. * Gain function a is chosen as a = 36phi(z1, x2). * Initial conditions are zi(0) = t, for i = 1,2, 3. • Lyapunov functions are chosen to be such that
& fll = l + z ~ ,
j32 = 64fllap~l[1+ ~22]2,
~3 = T0"
• The control design parameters are e2 = 1.0 and ¢3 = 1.0. The simulation results are shown in Figures 3.2 and 3.3 for the state variables and control, respectively. As mentioned in the control design, the choice of f13 affects the magnitude of robust control u. To demonstrate that, Example 3.1 was also simulated with f13 = 1, another valid choice. Figures 3.4 and 3.5 shows that, compared with the control given in Figure 3.3, there is a significant increase in magnitude of the robust control u. Because of the increased control effort, the system converges somewhat faster to the origin. Simulation was also performed for time varying uncertainty. Specifically, uncertainty Ah~ (.) in Example 3.1 was chosen to be
hi(xl,
,1,t) =
+
sin(2t) +
+ cos(St)G
which is also bounded by Ph~ = 1 + z~ + z22. As shown in Figure 3.6 and 3.7, the convergence of the output state is slow due to the fact that the system is under continuous excitation of the time varying functions in uncertainty Ah~.
67
0.8,
0.6
0.2
0
0.5
1
~ .5
2
2.5
1
O.B
0,8
0*4
0.2
0
--0.2
0
o.s
F i g u r e 3.2
1 .~ s=cl
1
2
2.5
3
Example 3.1: Trajectories of x, and x2 when #3 = ~32/10
0.5
0
--0.5
-.1
--1.5
0.~
1
1.5 Robust
control
2
2.5
3
2
2.5
3
U
2 0 ~2 --4
--8
--10
/
--12 --14 --16
0
F i g u r e 3.3
0.5
1
1.5
Example 3.1: Trajectories of x~ and control when ]33 =
~2/10
68
0.8
0.6
0.4
0,2
0
--0,~ 0
0.5
t
1.5 State
vsrl~ble
2
2.5
3
x2
I o.9
o.s o.7 0.6
0.4 o.~ o.2 o.1
o~
F i g u r e 3.4
Example 3.1: Trajectories of xl and x2 when/~3 = 1
O.S
0
--o.s
--1
--1 .S
-2;, x
--1-5
0.5
1
o.m,
1
s ~ 1 . 5B
2
2.5
3
t ,5
2
2.~
3
I0 B
0
=eta
F i g u r e 3.5
Example 3.1: Trajectories of x3 and control when Ha = 1
69
1
0.8
o.~
0~4
0.2
00
20
40
flO
8'0
100
120v=
140
160
180
120
140
160
180
8 ~ J
St=te
v~rl=blo
x2
1.2
1
0.8
.~ o.6 0.4
0.2
0
20
Figure 3.6
40
60
80
=ec=
100
Example 3.1: Trajectories of xl and x2 under TV uncertainty
St~t~
vm¢i~bl~
x3
1 0.8 0.6 0.4 0.2
--0.2
~0.6 --0.8 --t
20
~0
60
80
8~ca
100
120
~0
--4 --6 --8 --10 --12 --1,4, 20
4O
60
8O
tOO
t20
140
160
180
==c8
F i g u r e 3.7
Example 3.1: Trajectories of x3 and control under TV uncertainty
70
1.2
1
0.~
0.6
0.4
0
qt~5 8 ~ B
~tlte 1.2
vatri~ble
x2
I
1
O.a
O.e
0.4
0.2
o~
t
Figure 3.8
1 ,~ ,,t,c
2
2.5
3
Example 3.5: Trajectories of xl and x2
Simulation of Example 3.5 was done using the same choices as those for Example 3.1 but with the additional uncertainties 1 A f ~ z = z S f 2 2 = 12or' and
Afa=x32.
It can be verified that inequality (3.31) holds. The simulation results of the state variables and control are given in Figures 3.8 and 3.9, respectively.
3.6 C o n c l u s i o n s The recursive-interlacing design procedure is presented in which design objectives of recursive and interlacing steps are explicitly defined. By following the procedure, the designer is guided to choose controls and Lyapunov functions. As the final result, the designer will have either a stabilizing robust control with stability proof completed or, if no control can be generated, certain measure about instabilizability of the system under consideration. It is new and important that the procedure does not impose any a p r i o r i structural assumption in the study of robust stabilizability and control design. The recursive interlacing design is developed based on backward recursion and also contains feedforward interlacing steps which are for compensating EMUs and for properly choosing Lyapunov functions. Other new ideas con-
71
1
0.5
0
--0.5
--3
--15
--2
0.5
1
1.5 8 ~ 8
2
2
o~g ........
1
1.5
2
2.5
10
5
0
--5
~10
- 1~ / ~ -
-- .....
F i g u r e 3.9
3
Example 3.5: Trajectories of x3 and control
rained in the procedure include admissible fictitious control, Lyapunov functional, and decomposition of dynamics that do not satisfy existing structural conditions. Hence, the proposed design procedure naturally includes the existing ones as special cases. Despite its widest applicability, the procedure remains to be intuitive. Two third-order vector systems are shown to be robustly stabilizable. Future research should be directed to reveal the full potential of the procedure by searching for more classes of stabilizable uncertain systems and by concluding in a nonlinear setting the necessity of a successful application of the procedure oil stabilizability.
References Byrnes, C., Isidori, A., 1989, New results and examples in nonlinear feedback stabilization, Systems and Control Letters, 12,437-442. Barmish, B. R., Leitmann, G., 1983, On ultimate boundedness control of uncertain systems in the absence of matching assumptions, IEEE Transactions on Automatic Control, 27, 153-t58. Chen, Y. H., t987, On the robustness of mismatched uncertain dynamical systems, ASME Journal of Dynamic Systems, Measurements and Control, 109, 29-35.
72 Chen, Y. H., Leitmann, G., 1987, Robustness of uncertain systems in the absence of matching assumptions, International Journal of Control, 45, 15271542: Corless, M. J., Leitmann, G, 1981, Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems, IEEE Transactions on Automatic Control, 26, 1139-1144. Gutman, S., 1979, Uncertain dynamical systems--A Lyapunov min-max approach, IEEE Transactions on Automatic Control, 24, 437-443. Hale, J. K., 1980, Ordinary Differential Equations, Robert E. Krieger Publishing Company, Inc., 2nd edition. Hale, J. K., 1977, Theory of Functional Differential Equation, Springer-Verlag, New York. Kannellakopoulos, I., Kokotovid, P. V., Morse, A. S., 1991, Systematic design of adaptive controllers for feedback linearizable systems, IEEE Transactions on Automatic Control, 36, 1241-1253. Lin, Z., Saberi, A., 1993a, Semi-global stabilization of partially linear composite systems via feedback of the state of the linear part, Systems and Control Letters, 20, 199-207. Lin, Z., Saberi, A., 1993b, Semi-global exponential stabilization of linear systems subject to 'input saturation' via linear feedbacks, Systems and Control Letters, 21,225-239. Marino, R., Tomei, P., 1993, Robust stabilization of feedback linearizable timevarying uncertain nonlinear systems, Automatica, 29, 181-189. Mazenc, F., Praly, L., 1994, Adding an integration and global asymptotic stabilization of feedforward systems, Proceedings IEEE Conference on Decision and Control, Lake Buena Vista, FL, 121-126. Qu, Z., t992a, Global stabilization of nonlinear systems with a class of unmatched uncertainties, Systems and Control Letters, 18, 301-307. Qu, Z., 1995, Robust control of nonlinear uncertain systems without generalized matching conditions, IEEE Transactions on Automatic Control, 40, 14531460. Qu, Z., 1994, A new generic procedure of designing robust control for nonlinear uncertain systems: Beyond the back-stepping design, Proceedings IEEE Workshop on Robust Control via Variable Structure ~ Lyapunov Techniques, Benevento, Italy, 174-181. Qu, Z., 1993, Robust control of nonlinear uncertain systems under generalized matching conditions, Automatica, 29, 985-998. Qu, Z., Dorsey, J. F., 1991, Robust control of generalized dynamic systems without matching conditions, ASME Journal of Dynamic Systems, Measurements and Control, 113,582-589. Qu, Z., 1992b, Robust control of a class of nonlinear uncertain systems, IEEE Transactions on Automatic Control, 37, 1437-1442. Saberi, A., Kokotovid, P. V., Sussmann, H. J., 1990, Global stabilization of partially linear composite systems, SIAM Journal on Control and Optimization, 28, 1491-1503.
73
Slotine, J.-J. E., Hedrick, J. K., 1993, Robust input output feedback linearization, International Journal of Control, 57, 1133-1139. Teel, A., 1992a, Global stabilization and restricted tracking for multiple integrators with bounded controls, Systems and Control Letters, 18, 165-171. Teel, A., 1992b, Semi-global stabilization of minimum phase nonlinear systems in special normal forms, Systems and Control Letters, 18, 187-192. Tsinias, J., 1989, Sufficient Lyapunov-like conditions for stabilization, Mathematics of Control Signals and Systems, 2, 343-357.
. N o n l i n e a r T r a c k i n g via Discontinuous Feedback under Uncertainty Giorgio Bartolini, Antonella Ferrara and Tullio Zolezzi 4.1 Introduction and P r o b l e m Setting In this chapter we consider asymptotic tracking problems for nonlinear control systems, described by ordinary differential equations, subject to deterministic uncertainty on the dynamics. This means that the dynamics are not available for the controller in an exact form: only explicit bounds on the terms of the dynamics are known. Hence, the control system is modelled as a controlled differential inclusion, which is obtained making appropriate use of such bounds. We shall thereby consider the differential inclusion
x(t) E F(t, x(t), u),
t > O,
(4.1)
Ix(0)l bounded by a known constant, where x(t) E ]RN is the (locally absolutely continuous) state variable, u E ]RM is the control variable to be chosen from a control region modelling the control constraint u e U
(4.2)
where U is a given closed set in IRM. In (4.1),
F : [ 0 , +oo) x IR'J × U 2~ IRN is a given measurable multifunction, explicitly known to the controller, which possesses Carath~odory selections
f = f(t,x,u)
E F(t,x,u)
for almost every t _> 0, and every (x, u). Any such f wilt be referred to as an uncertain system dynamics. We assume that the state variable x(t) is available to the controller. A fixed linear time-invariant model is also given as follows
y(t) = Ay(t) + By
(4.3)
with y E IRN, v E IRM, y(0) fixed, t ~> 0. Here y is the model state and v the control.
76 The control objective can be expressed as follows: given a > 0 and any fixed model trajectory (v, y) solving (~.3), find a state feedback control law
u*=u*(t,z) ~ U such that any uncertain dynamics f[t, x, u*(t, x)] corresponding to u* leads to states x(t) fulfilling the tracking condition [y(t) - z(t)[ < (const)e -at,
t _> T
(4.4)
for some time T > 0 independent of x(O). The special form (4.3) of the model is selected here for the sake of simplicity. Actually, any trajectory y = y(t) whose dynamics is explicitly known will do in our approach.
4.2
Tracking
Control
via Differential
Inequality We follow the approach used in (Macki et al, 1990) and (Paden and Sastry, 1987) with some basic modifications, as sketched in this section. Pick any N × N constant matrix C whose eigenvalues have real parts less than or equal to - a , and consider
s P
= =
y-zeCtco, A y + B y - CeCtco
(4.5a) (4.5b)
(having dropped the dependence of y and x on t, for the sake of simplicity), where co is a fixed vector of IRN. Let N
V(x) = E
[xi[,
x e IllN.
(4.6)
i=1
We would like to choose the control law u* in such a way that the following inequality is obtained
dv[s(t)] <
~k ~
(4.7)
for almost every t _> 0 such that s(t) ~ O, for some fixed constant k ¢ 0. By standard reasoning, if (4.7) holds, then for some T > 0, we get
s(t) = o,
t>T
yielding the required asymptotic behaviour (4.4). Expanding (4.7), we obtain
77
z'}(t)
:
z ' ( A y + B,, - I ( t , ~, u ' ) - C ~ % o )
=
z ' b - I ( t , ~, u')] < - k 2
for any z in the subdifferential OV[s(t)]. Hence, we require u* E U fulfilling z'I(t, x, u*) > z'p + k s
for some uncertain dynamics f. Suppose that for every model trajectory (v, y) there exists k ~= 0 such that for almost every t > 0 and x with s ~ 0 we can find some point u* = u*(t, x) E U (depending on the model trajectory as well) which solves the following inequality inf{z'w : w •
F[t,x,u*(t,x)]}
> z'p+k 2
(4.8)
for any z • OV[s(t)]. Then, (4.7) is obtained once the tracking condition (4.4) is fulfilled. Of course, the above calculations are only formal, and we need to be more accurate. As shown in Macki et aI (1993), care should be exercised when using naively the above approach. The reason is the following. In general, the feedback control law u* solving (4.8) will be a discontinuous function of x. Hence, any uncertain dynamics f = f ( t , x(t), u*[t, x(t)]) corresponding to it will be discontinuous too. Then, the problem arises of how to interpret the differential inclusion (4.1) with u = u*. The pointwise equality (a.e.)
x(t) = f ( t , x ( t ) , u * [ t , x ( t ) ] ) ,
t >_ 0
(4.9)
is a very restrictive condition (even impossible to hold). We can interpret (4.9) in the sense of Filippov (see (Filippov, 1988) or (Utkin, 1992)), which is customarily done in the setting of variable structure control theory due to the physical interpretation of the control system (4.9), (Utkin, 1992). However, if x is a Filippov state corresponding to the discontinuous feedback u*, and x(t) belongs to the discontinuity locus of u* for a set of positive measure, the convexification procedure involved in the definition of Filippov solution may well destroy sign conditions like (4.8) which are (after all) based on the (naive) pointwise interpretation of (4.9). Our work is devoted to overcome this contradiction, establishing tracking conditions based on a suitable modification of the above approach. This will be achieved either by a different choice of the function V, or by a modification of (4.8). A more general approach, obtained by possibly different choices of V, is sketched below. Fix the desired trajectory (v, y) solving (4.3). We consider a smooth function s = s(t,~) : [ o , + ~ ) x ~ N _+ mJ"
78 (depending on (v, y) as well) such that whenever ¢ = x(t) is any locally absolutely continuous function on [0, +c~) and
s(t)=s(t,z(t))=O,
t>_T
then (4.4) is fulfilled. Note that the choice of s in (4.5) is of course a particular case.
In (Paden and Sastry, 1987; Bartolini and Zolezzi, 1988, 1991) we find further examples of how to effectively choose s in order to achieve tracking control or asymptotic linearization of an uncertain nonlinear control system via the differential inequality (4.7) and variable structure control methods. Assume that there exists a constant
k#0 such that for almost every t > 0 and every x with s(t,
,) ¢
0
we can find some point u* E U which verifies
sup{z'sx(t,x)w : w e F[t,x,u*(t,x)]} ~ - k 2 - z ' s t ( t , x )
(4.10)
for any z E OV[s(t, x)]. Then, (4.7) holds and we obtain the tracking condition (4.4). Of course the approach we described from (4.5) is a particular case, and (4.10) contains (4.8) as a special instance.
4.3
Results
Here we state two theorems which make rigorous the approach sketched in the previous section. Proofs and applications can be found in (Bartolini and Zolezzi, 1995). Let . 1/~ v(x) =
*i
,
x e
i----1
Since V V is continuous off the origin, an approximation argument yields the following result. T h e o r e m 4.1
Let F be independent of t. Let u*=u*(x)
~ V
be such that inf{s'(t)w : w E F[z,u*(x)]} > s'(t)p+k2[s(t)[
79
for almost every t >_ O, every x and s(t) • O. Then, (4.4) is verified by every Filippov state x in [0, +oc) correspondin 9 to any uncertain dynamics f[x, u* (x)]. Now, let V be given by (4.6). Given any uncertain dynamics f and any measurable feedback
u*=
u,
consider the set
c : N {el
co
f*(t, B(x,5) \ N) : 5 > O, measN = 0}
where 'el co' denotes the closed convex envelope, f*(t, x) = fit, x, u*(t, x)] and B(x, 5) denotes the ball around x of radius & Let G* be the union of these sets G taken as f runs among all Carath~odory selections of F. T h e o r e m 4.2 Suppose that there exists u* E U such that condition (4.I0) is fulfilled with F[t, x, u*(t, x)] replaced by G*. Then (4.4) is verified by every Filippov state in [0, +oc) corresponding to any uncertain system dynamics
fit, x, u*(t, x)]. Summarizing, the previous results give sufficient computable conditions, namely the pointwise inequality (4.10), such that the tracking condition is verified against any uncertainty acting on the given control system.
4.4 Examples As a first example, we generalize to uncertain systems the application reported in (Paden and Sastry, 1987, Section 3) making use of Theorem 4.2. Given the desired trajectory with y E C 2 ([0, +oe)), we consider the uncertain system O = g(o, o) + N(O, O)u, lul _< M (4.11) where 0 E IRm, N is a known positive definite continuous m x m matrix, M is a known constant and g is measurable, locally bounded and uncertain. A given multifunction is known, such that g(x) e G(x),
x e 11%2~'.
We assume that G is upper semicontinuous with compact values. Of course, G models the amount of uncertainty on the control system. Consider the error vector E = 0 - y and fix the matrix C as indicated in the previous sections. Then, let V be given by (4.6), and
80
s = E - CE.
(4.12)
Define
u*(t,z) = - h ( t , z ) VV[s(x)],
a.e. z E IR 2m
(4.13)
where the continuous positive function h will be chosen later. Now , apply Theorem 4.2 using x = (0,~)' (since the dynamics (4.11) involve 0). In order to show in detail how condition (4.7) is obtained, we compute for almost all t > 0 with s(t) 7i 0 the following quantity
= -ij + Cl~ + 0 E - 9 + C E + n(o, ~),
(4.14)
where L denotes the union of Filippov sets of g + Nu* as g ranges over all uncertain terms in (4.11). Remembering (Paden and Sastry, 1987, property 10 on page 75), one has, relying on the semicontinuity of G,
E -~)+ C E - hgOY(x) + co G(O, ~).
(4.15)
Then, by (4.7) and (4.15), for every z E OV(s)
= - h z ' N p + z'(-~j + CE, + w) for some p e OV(s) and w E coG(0, ~). Following (Baden and Sastry, 1987), choose z = q the closest point to the origin in OV(s), s ~ O, with respect to the norm induced by N. Then, q'Np > qtNq, ]q] > 1, thus
( / < - h q ' g q + ]q[ (]~)[ + ]C/~] + ]w[). It follows that (4.7) holds for some k ¢ 0 provided we choose h verifying a h > k 2 + (I/11 + ICE1 + w*) 2v/~,
(4.16)
where ~ is the smallest eigenvalue of N and w* = max{Iw I : w e G(0,~)}. All terms of (4.16) are known and available to the controller. The control constraint in (4.11) is met if the continuous function h is defined by taking equality in (4.16) and [ k2 + (l~)l + IC/~t + w * ) x / ~
2v/-~-~ _< a M .
The existence on the whole interval [0, +co) of Filippov's solutions to (4.11) with u = u* follows as in (Paden and Sastry, 1987, Appendix 2). As a second example generalizing the previous one, consider the uncertain system a(O)~ = g(O, ~) + u, luf < M (4.17) where 0 and g are as in (4.11), and a(O) is a rn x m uncertain matrix, which is everywhere continuous and nonsingular. A given multifunction
81 A : tR-~ _~ 1R~ is known such that a(O) E A(O) for any 0. We consider G, C, y, E, s, V as in the previous example and again
u* (t, x) = - h ( t , x) VV[s(x)],
(4.18)
h continuous and positive to be chosen later. Then, for almost every t > 0, with s(t) ~ 0 (4.19) ~7 = z'(--~l -- CE) Jr- z' a - l w - hz' a - l q for every z E cgV(s) and some w E coG(0, 0), q C cOV(s). Assume that a positive constant a is known such that for any uncertain matrix a, any eigenvalue of the symmetric part a(O) - t is greater than or equal to a for any 0. Then, choose z = q in (4.19). We obtain 9 _< 2,/~m (I/; + ICEI + w*) - ~h/2,
(4.20)
where w* is a known constant such that w* = max{Ia-twl : a(O) E A(O), w ~ a(0,0)}. Hence, (4.7) holds provided that we choose h verifying
~h/2 > k ~ + 2,/~(t/) I + ICEI + w*).
4.5 Simulations To provide simulation evidence we consider the uncertain control system
cos(el02) ) 4 + h cos 02 l + h cos 02
2 + k sin 02
q1+02 q2 sin01 1+03
02
+u,
with 01(0)=0,
01(0)=1,
02(0)=1, 02(0)=0, and h, k, ql, q2 uncertain parameters about which it is known that 0.5 < ql _< 1,
lq21_2,
th{ _< 0.5, Ikl <_ 0.5. The objective is to track the trajectory
(4.21)
82
0 i-cos(t)
I
0
1
2
,
3
Figure 4.1
Y=
~,,,
9 t[se¢] 10
cos t ) sin t '
01 - c o s t
(4.22)
t > 0.
Applying the results previously presented we obtain the control law u*=-asgn
] t~l+sint+Ol-cost 02-cost+O2-sint
[ ['
t>O
(4.23)
where a = L1 [1 + x/2([cost[ +[sint[ + 101 +sint[ + [t~2- cost[ + L2] with L1, L2 suitable constants (and 'sgn' denotes the sign). Simulation results are reported in Figures 4.1-4.5, which show quite satisfactory tracking results.
4.6 C o n c l u s i o n s and Further D e v e l o p m e n t s We have considered nonlinear control systems subject to deterministic uncertainties. Using a differential inequality approach we have obtained computable criteria yielding a discontinuous feedback control law which generates a prescribed tracking condition.
83
0'81 0,7
0"6i 1 0.5 t \ 62-sin(O I \
iii \
::! Z 0
1
2
3
4
F i g u r e 4.2
Ogt) 1'51
,
~ '
5
8
9tI~e~l~O
82 - sint
i
[
o.5! J
O~
-o.51
"1"511.5
-'1
-0~.5
F i g u r e 4.3
0
015
82 versus 81
01(t)
115
84
s1
T o.8~-
i -~
0.6l
1 "1
0'41 0.2
q 4
o! -0.21 -0.4i -0.6[r -0.8f
Figure 4,4 sl versus t
It
°16t °4i
1 t
o.2t 0
-0.4~[
J
-O.8)
J
o6!
t
Figure 4.5 s2 versus t
85 All the results dealt with in this chapter assume that the state variable is available to the controller. A topic which deserves further investigation is then that of output feedback. This problem can be briefly sketched as follows. Suppose that the state variable of the control system (4.1) is no longer available and the output is given by q = h(x)
(4.24)
for a given smooth mapping h : I~N --4 IRK . We guess that this problem could be solved by using an exponential observer z like the one introduced in (Bartolini and Zolezzi, 1986) corresponding to the choice of s and using the output feedback u* = u* (t, q). In this way the estimates of Bartotini and Zolezzi (1986) yield, for any unavailable state x tortesponding to the observer equivalent control
Is[x(t)]t
(const)Ix(0)- z(0)te
t
0
(4.25)
for a suitable constant c < 0. Then, if s is given by (4.5), we get the tracking condition (4.4). Here we need suitable assumptions about the size of the uncertainty allowed in (4.1) and about the existence of the observer's equivalent control, as related to the Filippov's solutions of the combined system made up of the control system and the observer dynamics. It is perhaps possible to use alternatively some results by Ciccarella et al (1993). Work about these topics is in progress.
References Bartolini, G., Zolezzi, T., 1986, Dynamic output feedback for observed variablestructure control systems, Systems and Control Letters, 7, 189-193. Bartolini, G., Zolezzi, T. 1988, Asymptotic linearization of uncertain systems by variable structure control, Systems and Control Letters, 10, 111-117. Bartolini, G., Zolezzi, T. 1991, Asymptotic linearization of multivariable uncertain systems, IMA Journal of Mathematical Control and Information, 7, 351-360. Bartolini, G., Zolezzi, T., 1995, Discontinuous feedback in nonlinear tracking problems, to appear on Dynamics and Control. Ciccaretla, G., Dalla Mora, M., Germani, A., 1993, A Luenberger-tike observer for nonlinear systems, International Journal of Control, 57, 537-556. Filippov, A. F., 1964, Differential equation with discontinuous right hand side, American Mathematical Society Translations, 42, 199-231. Filippov, A. F., 1988, Differential equations with discontinuous righthand sides, Kluwer.
86 Macki, J. W., Nistri, P., Zecca, P., 1990, A tracking problem for uncertain vector systems, Nonlinear Analysis Theory Methods Applications, 14, 319328. Corrigendum, 1993, same journal, 20, 191-192. Paden, B. E., Sastry, S. S., 1987, A calculus for computing Fitippov's differential inclusion with application to the variable structure control of robot manipulators, IEEE Transactions on Circuits and Systems, 34, 73-82. Utkin, V. I., 1992, Sliding modes in control optimization, Springer-Verlag, Berlin.
. I m p l e m e n t a t i o n of V a r i a b l e S t r u c t u r e C o n t r o l for S a m p l e d - D a t a Systems Wu-Chung Su, Sergey V. Drakunov and Umit Ozgiiner 5.1 Introduction The characteristic feature of a continuous time variable structure systems (VSS) is that sliding mode occurs on a prescribed manifold, or switching surface, where switching control is employed to maintain the state on that surface (Utkin 1978; Utkin and Young, 1979; Drazenovic, 1969; Drakunov and Utkin, 1992; Young and Ozgiiner, 1993). Since the theory has been originally developed from a continuous time perspective, implementation of sliding mode for sampleddata systems encounters severM incompatibilities due to limited sampling rate, sample/hold effect, and discretization errors. As a result, a direct translation of continuous time variable structure control design for discrete implementation leads to the chattering phenomenon in the vicinity of the switching surface. This chapter deals with the implementation of variable structure control for sampled-data systems by maintaining sliding mode in discrete time. Although a considerable amount of work has been done analyzing discrete time sliding mode, very little was directly addressed to the sampling issues. Milosavljevic (1985) studied the oscillatory characteristic (quasi-sliding) in the neighborhood of the discontinuity planes due to diseretization of continuous time signals. Existence conditions of quasi-sliding mode were derived as a discrete extension from the continuous time VSS theory. For control law design, Utkin and Drakunov (1989) proposed a definition of discrete time equivalent control that directs the states onto the sliding surface in one sampling period. To remain on the surface, the associated control appears to be non-switching. Subsequently, the theoretical basis was furnished with a formal definition of sliding mode for discrete time systems in the context of semigroups (Drakunov and Utkin, 1992). Related works from different prospectives can be found in (Sarpturk et al, 1987; Kotta, 1989; Furuta, 1990; Sira-Ramfrez, 1991; Kaynak and Denker, 1993; Bartotini and Ferrara, 1993). Sarpturk et at took a Lyapunov point of view for discrete time linear systems. It was asserted that the switching control be bounded in an open interval to guarantee convergence of sliding motion. This interval was later found to depend linearly on the distance of the state from the switching surface (Kotta, 1989) which suggested a nonswitching control when discrete time sliding mode is attained. Sira-Ramirez imposed the geometrical concept to general nonlinear SISO systems along with the exis-
88
\
\
Slidingsurfaces(x)=O. Ill
Figure 5.1
Discrete time sliding mode in sampled-data systems
tence issues (Sira-Ramfrez, 1991). Parallel results with continuous time VSS's were obtained. Subsequently, the formal definition of sliding mode for discrete time systems was established by Drakunov and Utkin (1992) in the context of semigroups. To implement discrete time sliding mode control law in compliance with the existence conditions, Furuta proposed the idea of sliding cone (sliding sector), where switching control takes place only when the states are out of the sector, while inside the sector, the control law remains continuous (Furuta, 1990). Kaynak and Denker (1993) used an ARMA model to characterize the control-sliding surface relationship and yielded a non-switching type of control with a predictive-corrective scheme. Bartolini and Ferrara (1993) incorporated adaptive control strategy to account for system uncertainties and designed control law in terms of a discrete time equivalent control. Our work has mainly extended the original idea of Utkin and Drakunov (1989) to more general problems with detailed analysis of the s.a.mple/hold effects. Preliminary results were reported in (Su, Drakunov and Ozgiiner, 1993; Drakunov, Su and Ozgiiner, 1993; Su, 0zgiiner and Drakunov, 1993) and the full development is presented here. The control objective is to maintain the states on the switching surface at each sampling instant. Between samples, the states are allowed to deviate from the surface instead of being constantly and exactly on the switching surface, even the equivalent state travels within a boundary layer of that surface (see Figure 5.1). It will be demonstrated that the thickness of this boundary layer can be reduced significantly by proper
89 consideration of the sampling phenomenon in the control design. Three classes of systems (linear, nonlinear, and stochastic systems) will be investigated individually. Robustness against both internal and external uncertainties will be considered. Furthermore, the chattering problem will be addressed. The chapter is organized as follows: in Section 5.2 we consider discrete time sliding mode in linear sampled-data systems. The intersample behavior as well as the matching condition will be examined. Robustness against system uncertainties are elaborated. Three types of uncertainties wilt be studied individually. The overall effects can be treated as a linear combination of the three. In Section 5.3 the results are extended to nonlinear systems. Based on similar assumptions, the control strategies can be directly translated from the linear case. Section 5.4 deals with linear stochastic systems. Sliding mode is defined in terms of a conditional probability function, given the o'-algebra generated by the measured state information. Both cases with continuous measurement and discrete measurement will be considered, rib minimize the deviation from the sliding manifold in the mean square sense, the optimal filtering problems will be solved. In each of the above, the chattering phenomenon will be effectively removed with robustness. An experimental example of a flexible structure vibration control is given in Section 5.5 to demonstrate the effectiveness of the proposed sampled-data sliding mode control technique. Finally, the conclusion section summarizes the chapter.
5.2
Linear
Systems
Consider the dynamic system with a prescribed switching surface = A x + B u + D f,
s = {x t s(z) = c x = 0},
(5.1)
(5.2)
where the state x C R '~, the control u E R "~, the disturbance vector f E R I, and the sliding surface vector s E Rm; A, B, D are constant matrices of appropriate dimensions; and the m × n matrix C is chosen so that the system, when traveling on 8, will achieve the desired sliding mode dynamics (Utkin and Young, 1978). We say the disturbance is "rejectable" if there exists a control u that can eliminate its effect instantaneously in the dynamic system. If the closed loop system behavior does not depend on f ( t ) , we say the controlled system is disturbance invariant. It is well known that the matching condition (Drazenovic, 1969) rank [B, D] = rank [B] (5.3) has to be satisfied for the system (5.1) to be disturbance invariant in sliding mode. The discrete time representation of the dynamic equation (5.1) is obtained by applying u through a zero-order-hold: x k ÷ l = ~Sxk + V u k + dk,
(5.4)
90 where 4~ = eAT, F = f [ e A~ d~ B, dk = f [ eA~DS((k + 1 ) T - A) dA. Note that the magnitude of F and dk are both O(T) if f(t) is bounded. The discrete time sliding mode control is to steer the states towards and maintain them on the surface $ at each sampling instant such that
sk = C x k = O ,
k ~ k~.
(5.5)
During the sampling interval k T ~_ t ( (k + 1)T, the state will deviate from $. The following lemma provides bounds on the deviation from the sliding surface. L e m m a 5.1 If the disturbance f(t) in (5.1) is bounded and smooth in the intersample interval k T < t < (k + 1)T, then sk = O, Sk+l = 0 imply s(t) = O ( T 2) during that sampling period. The O ( T 2) deviation is inevitable for sampled-data systems. It is the ultimate performance a sampled controller can attain for continuous time plants. Note that if sk = O ( T 2) and Sk +1 = 0 (T 2), the intersample value of the sliding surface vector is still O(T2). The sample/hold effect not only destroys perfect sliding mode as depicted above, it also invalidates the complete disturbance rejection property for the discrete time system (5.4). L e m m a 5.2 The sampled-data system (5.3) is disturbance invariant on the discrete time sliding manifold (5.5) only if the continuous time matching condition (5.3) holds. The above lemma states the necessity of the continuous time matching condition, however, the sufficiency is not confirmed. The zero-order-hold is applied to the control variables only. A similar "hold" does not take place in the disturbance channels. In general, the discrete time disturbance dk will not lie in the range space of the control coefficient matrix _F and hence cannot be rejected completely even if discrete time sliding mode occurs. Although perfect sliding mode and complete disturbance rejection are not possible in sample-data systems, one can still maintain the states in the vicinity of the sliding surface and retain satisfying disturbance rejection character by proper consideration of the sampling phenomenon in the control design. We will consider three different types of uncertainties that are important in affecting a dynamic system's behavior; namely, exogenous disturbances, system parameter variations, and control coefficient variations. It will be shown that these uncertainties can be rejected to at least O(T 2) accuracy and that the chattering phenomenon inhabiting in continuous time variable structure systems can be removed. Since the control in the sampled data system will show jumps from the continuous time point of view, the formal concept of continuity is not compatible with the notion of discrete time control. In order to make the analogy in between, let us introduce the following definition:
91 D e f i n i t i o n 5.3 The discrete time control law uk is said to be equivalent to discontinuous ff Auk = O(1), continuous if Auk = O(T), smooth ff A2uk = O(T~), where A denotes the backward difference operator 1 - z -1. To avoid a cumbersome description, the "continuity" of discrete time control to appear in the following text will refer to its equivalent meaning as defined above. 5.2.1
Exogenous
Disturbances
Consider the case when only the exogenous disturbance f ( t ) is present. We define the discrete time equivalent control by solving sk+l = 0 (Utkin and Drakunov, 1989), namely, from (5.4) and (5.5) it follows sk+l = Cxk+l = CCxk + CFuk + Cdk,
(5 6)
which leads to
u;q • -(cv)-Ic(¢xk
+
(5.7)
assuming O P is invertible. Note that u~ q is generally not accessible because it requires the value of dk, which depends on the future values of the disturbance f ( t ) from the present sampling instant k T to the next. one (k + 1)T. It is impossible to evaluate dk exactly unless f(t) is known. Nevertheless, with boundedness and smoothness assumptions imposed on f(t), dk can be predicted by its previous value dk-t (Morgan and Ozg/iner, 1985), which can be computed from (5.4). The error is dk-dk-1 :
lot
f(k +l )T-~, eAXDIT-XJk ¢(a) dz d,~ : O(T2),
assuming )~(t) is bounded. To approximate u keq , we define u kin = - ( C ] ' ) - l c ( ~ X k
q- dk-1).
(ss)
Taking the Taylor's series expansion of ~b yields the relationship C@xk = sk + ( A T + T- ~2 A . . 2 + . . . ) xk. Since the magnitude of C F and Cdk-1 are both O(T), in order that u~n be admissible, the magnitude of CCxk must also be O(T). In other words, xk must lie within an O(T) boundary layer of $ (or sk = O(T)) for u~n to be admissible. In order to define the boundary layer of the sliding surface, let us adopt the following notation: ST = {xk[u~n E U} is the O(T) boundary layer of S, where U is the set of admissible values of the control u. We propose the variable structure control law I uk =
in
uk u~ut
when xk E ST, otherwise.
(5.9)
92 where u~,ut = - ( C F ) - I C (~Xk + dk-t --sk + Ksgn(sk)). The positive definite matrix K will determine the stepsize for the state to approach the boundary layer ST. The magnitude of K has to be chosen small enough not to overshoot ST.
Theorem 5.4
For a linear sampled-data system (5.4) with the exogenous disturbance satisfying the matching condition (5.3), the effect of f(t) can be reduced to O(T 2) if the variable structure control law (5.9) is applied.
5.2.2 S y s t e m P a r a m e t e r Variations Consider a dynamic system subject to parameter variations
5: = (A + AA(t))x + Bu.
(5.10)
The matching condition for the uncertainty AA(t) is rank[AA(t)O, B] = rank[B]
Vt,
(5.11)
where 19 is an n × (n - m) matrix with the columns forming a basis of the subspace null(C) (Drazenovich, 1969). The unknown AA(t)x may be treated as a disturbance. The associated sampled-data representation is given in (5.4), where P
T
dk = /n eA;~z~A((k + 1)T - )~)x((k + 1)T - ,k) d~.
(5.12)
Assuming AA(t) is bounded, dk can be predicted by dk-1 with O(T ~) error, and we have the following theorem.
Theorem 5.5 For a linear system with parameter variations (5.10), the control (5.9) will lead to sk+l = O(T 2) if AA(t) satisfies the matching condition (5.11). Furthermore, the state will attenuate to zero asymptotically. 5.2.3
Control
Coefficient
Matrix
Variations
Systems with unknown control coefficient matrix variations have the following dynamic equation J: = Ax + (B + AB(t))u. (5.13) The matching condition is rank[AB(t), B] = rank[B]
Yr.
(5.14)
The system dynamic equation in discrete time is the same as (5.4), where
dk
= =
AFkuk,
/To
eA
aB((k + 1 ) T -
da.
93 Employing the control law uikn induces a dynamic feedback in Uk through dk_ 1. Since sk+l = C¢)xk + C ( F + AFk)ua, (5.15) the stability of sk implies stability in xk, which ensures stability in uk if C ( F + AFk) is invertible. Introduce the following notation:
eark = Z rk
-
Ark_l,
ak = cz
rk(cr) -I.
(5.16)
The quantity Ak can be regarded as a measure of variations in the control coefficient matrix with respect to the known one F. If A B ( t ) and A/)(t) are bounded, we have eark = O ( T 2) and zik = O(1). L e m m a 5.6 I*br the linear system with control coefficient matrix variation (5.t3), if A B ( t ) satisfies the matching condition (5.t4), and the states xk-1, xk are in the boundary layer ST, then the control u~n will lead to sk+l = --2Aksk + Ak-lsk-1 + 5k
(5.17)
with 5k = O(T2). Stability of s~ in (5.17) can be assured if Ak is small enough; that is, the variation of A B ( t ) is small with respect to B. Since the quantity 5k is O(T2), the state will enter the O ( T 2) boundary layer of $ if (5.17) is stable. T h e o r e m 5.7
I?~e state xk attenuates to zero asymptotically if
(i) The conditions in Lemma 5.6 hold. (ii) Equation (5.17) is stable. (iii) The m x m matrix C ( F + AFk) is nonsinguIar. If all three types of uncertainties are present in the dynamic equation, the variable structure control law (5.9) can still be employed to compensate the uncertainties simultaneously. In this case, dk is the lumped effect of all disturbances, the resultant sliding mode accuracy is s(t) = O(T~). As long as the states remain in the O ( T 2) boundary layer of the sliding surface, uikn becomes continuous and chattering in the control is removed. T h e o r e m 5.8 The control uikn is continuous in the sense of Definition 5.3 if s k - t = O(T2). Furthermore, uk approaches the continuous time equivalent control as the sampling rate approaches infinity.
uk
T--+ O
......~ ueq(t) = - ( C B ) - I C ( A x
+ Dr).
(5.18)
94
5.3
Nonlinear
Systems
In the previous sections, we have developed a discrete time sliding mode control method applied to linear sampled-data systems to alleviate chattering. Here, we extend it to nonlinear systems. We assume the nonlinearities and uncertainties are bounded and smooth, in this case the past values of the states can be utilized to predict the effect. This approach allows us to keep the state in the O(T 2) vicinity of the sliding manifold. Consider the nonlinear system
ic = a(x) + B(x)u + v(x, t)
(5.19)
with a prescribed sliding manifold S = {xl 8(x) = O},
(5.20)
where a(x) and B(x) are known differentiable functions with dimensions n × 1 and n × m respectively; v(x, t) is an n × 1 unknown vector-valued differentiable function. The disturbance matching condition is rank[B(x), v(x, t)] = rank[B(x)]
when x E S, t >_ 0.
(5.21)
Let G(x) = ° s ( x ) be the rn × n Jacobian matrix of s(x) so that the dynamics of the sliding surface vector can be expressed as ~(x) = G(x)a(x) + e(x)B(x)u + G(x)v(x, t).
(5.22)
Let us introduce the following notations:
dt,
fk
---- fk(k+l)T G ( x ) a ( x ) dt,
f~
:
F(k+I)T a(x) JkT
gk
__ fk(k+l)T G ( x ) B ( x ) d t ,
Bko
=
r(k+X)TB(x) dt, JkT
hk
=
h~ =
f(k+l)TG(x)v(x,t)dt,
(5.23)
fj(k+l)T V(z, t) dr. kT
Applying a zero-order-hold to the control and integrating (5.19) and (5.22) on the interval [(k + 1)T, kT] yields the discrete time representation
xk+l
=
xk
+
uk +
Sk+l
=
Sk -}" fk -}- Bk Uk -{- hk.
(5.24@ (5.24b)
The control objective is to keep the states xk on the sliding surface S in discrete time s(x~) = 0, for k > ks. (5.25) The following lemma is the nonlinear version of Lemma 5.2. 5.9 The sampled-data system (5.24a) is disturbance invariant on the discrete time sliding manifold (5.25) only if the continuous time matching condition (5.21) holds.
Lemma
95 5.3.1
Matched
Disturbances
Consider the system (5.19), in which the disturbance v(z, t) satisfies the matching condition (5.21). From (5.24b) it follows that the equivalent control is tt; q : -- B k l ( sk -t- A -~- hk ) ,
(5.26)
assuming Bk is invertibte. The magnitude of fk, Bk, and hk is O(T) if G(x), a(x), B(x), and v(x, t) are bounded, fk, Bk, and hk usually cannot be evaluated exactly since they depend on the future value of the state and the unknown disturbance. Nevertheless, with some mild smoothness condition on the functions G(x), a(x), B(x) and v(x, t), they can be approximated by the following expressions:
Fk =
G(xk)a(xk)T,
(5.27a)
Ok hk
a(,k)b(xk)T, sk+l - sk - Fk - / ) ~ .
(5.27b)
= =
(5.27c)
With (5.27), we obtain an approximation for the equivalent control (5.26) Ukin
- - O k l ( S k -~ Pk -t- h k - 1 ) ,
(5.28)
where 0k is invertibte by assumption. Similar to (5.8), Xk must lie in the boundk to be admissible. The proposed variable structure control dry layer ST for u in law is: uk = { -0k-1 (sk + Fk +/~k-1) --/3kl(Ksgn(sk) + Fk +/tk-1)
when x~n E ST, otherwise.
(5.29)
If the disturbance in dynamic system (5.19) satisfies the matching condition (5.21), then the effect of v(x,t) can be reduced to O(T 2) ff the control (5.29) is applied.
T h e o r e m 5.10
5.3.2 C o n t r o l C o e f f i c i e n t V a r i a t i o n s Consider the case when there is an unknown variation b(x) in the control coefficient = a(x) + B(x)u + b(~)u. (5.30) The matching condition is rank[B(x), b(x)] = rank[B(x)]
when x C S.
(5.31)
Following the same discretization procedure yields the discrete representation indicated in (5.24b) with
96 ha
-
Ba ua
ha
=
f(a+l)T G(x)b(x)dt. JkT
Introduce the following notation: ea~ eB k
eh~ e1~
= = -
-
=
Bk--[~k,
~
=
[~a--Bk-1,
B k -- B k - 1 ,
e~k
:
B k -- J ~ k - 1 ,
ha - ha-l, fk -- fa-1,
ehk ea~
= =
ha -- ha, A k - - Z~k-1,
(5.32)
where Ak =/3aB~ "1. It can be proved that Aa = O(1), eak = O(T), and e/k = O(T2), e~k = O(T2), where j = F, B, F, B, B. L e m m a 5.11 When the state is in the boundary layer ST, the following equalities are true: sk+l hk
= =
ha-- hk-i + a~(2Sk -- sk-i) + a~, --Ak[(Sk + fk + ha-i) + fl~(2Sk -- sk-i) +/?~],
(5.33) (5.34)
where c~ = O(T), c~ = O(T2), fl~ = O(T), fl~ = O(T2). The equalities in Lemma 5.11 can be obtained by direct substitution from (5.32) and (5.28) into (5.24b). Furthermore, a~
=
_2eB k/~- 1,
a~
=
-et~k B ~ l fk + eL + eDk f]kle['k + e~k_ 1 + e~Bfl(f'k-1
(5.35a) + Bk-lUk-1),
(5.35b)
e~k B~-1,
(5.35e)
--eio,~ -- ehk_xe~]k Bkl(e_pk -- / O k _ l ~ k _ l ) .
(5.35d)
They serve as preparation for the theorem stated below: T h e o r e m 5.12 For the nonlinear system with control coefficient matrix variation (5.30), the control (5.29) will lead to sk --+ O ( T 2) asymptotically if (i) The state lie in the boundary layer ST. 2.
(ii} The variation b(x) satisfies the matching condition (5.31) and b(x) is bounded. (iii) The following dynamic equation for Sk is stable: Sk+t = aOsk q- a~sk-1 q- a~sk-2 q- ~k,
(5.36)
97
where a2
= =
--2Ak -- 2Ak¢3~ + 2 ~ = --2Ak + O(T), ~_~ + 2 ~ - ~ + ~9~ + 2~_~
=
z~k_~ + O(T),
a~
:
~
5k
=
--AkeA -- ~ a k A - 1 + Aka~ + ~ak(Fk-2 + B k - 2 u k - ' j
1 ak
- ~
- ~-~9~
-
~
: O(T),
2 - AkJ~ + Ak-~flk_l + ~'~ = O(T2) .
If the sampling rate is sufficiently high, the dynamic equation (5.36) is reduced to second order Sk+l = - - 2 z ~ k S k -t- A k - l S k - 1 .
(5.37)
This agrees with (5.17), the result for linear systems. Stability of sk can be assured if Ak is small enough; that is, the variation of the control coefficient ~(z) is small with respect to B(x). If the O(T) terms in (5.36) are not negligible, the dynamic equation for sk becomes third order with an O ( T 2) input. Stability of (5.36) implies that sk -+ O ( T 2) asymptotically.
5.4 Stochastic Systems This section is devoted to discrete time sliding mode control for continuous time linear systems with stochastic disturbances. The disturbance is assumed to satisfy an Ito type stochastic differential equation (Bucy and Josef, 1987) dx
=
A x d t + B u d t + (D~dt + R d w j ,
d~
=
W~ dt + Q dw2,
(5.38a) (5.385)
where wl(t), w2(t) are independent standard Wiener processes. Here D~ represents the colored part of the noise and the formal expression r~d~--~-x the white ~ dt part. Without loss of generality, the autocovariance matrices of the white noises are assumed to be identity. We consider two possible state information channels: continuous measurement and discrete measurement. To coinpensate for the stochastic disturbances with a sampling controller, a mixed continuous-discrete type of filtering problem will be solved if continuous state information x(t) is used. On the other hand, the use of discrete state information xk will lead to a discrete time optimal filtering problem. In both cases, we apply the following assumption: A s s u m p t i o n 5.13
(i) W has asymptotically stable eigenvalues;
98
(ii) R is full rank; (iii) The processes Rwl(t) and Qw2(t) are uncorrelated.
5.4.1 C o n t i n u o u s M e a s u r e m e n t Consider the discrete time version of (5.38a)
zk+l = Cxk + Fuk + rlk + vk,
(5.39)
where r/k = f(k~+I)T ea((h+l)T-;gD~(.X) dA is the colored part of the disturbance and vh = f(k+l)T eA((h+I)T_~,)Rdwx(.X ) is a white noise. Since the white noise cannot be compensated, the control objective is to minimize the effect of the colored part r/h in the sampled-data system (5.39). Applying the control law with an estimation of Ok uh =
(cr)-icoh,
(5.40)
we obtain
sh+1 = C(rlk + vh -- Ok).
(5.41)
For uk in (5.40) to be admissible, it still requires that the state lie in the O(T) boundary layer ST. This boundary layer can be defined similarly as in the previous sections. Here, we only consider when Xk E ST. In the presence of stochastic disturbances, sliding mode is defined in terms of a probability measure. To minimize EIIsh+lll 2 given the observation of x(t) up to the instant kT, the estimate 0h should be picked as a conditional expectation Ok = E(~hlYhr), (5.42) where 5rt denotes a c~-algebra generated by the process z(t) ~'t = ~r{z(r)10 < r < t}.
(5.43)
Since wl(t) is a martingale (Bucy and Josef, 1987), the estimation rlk is obtained by taking a conditional expectation given the ~r-Mgebra ff~'kT
Oh = fk(:+l)TeA((k+l)T-~,)Dd)t ~(kT) = F.~k,
(5.44)
where the matrix ~ is of the form
F. =
/(
e Wx D dJ~,
(5.45)
and ~ is a conditionM expectation ~(t) = E(~(t)l:Pt).
(5.46)
99 Therefore, equation (5.44) suggests that in order to obtain the values of ilk we need a filter to calculate ~k- The following theorem provides the form of such a filter.
T h e o r e m 5.14 If Assumption 5.13 holds, the optimal estimate of ~k is ~k = zk + LRT xk,
(5.47)
Zk = Azk_j + ~k-1 + IIuk_l
(5.48)
where zk satisfies with A, 17 and ~k defined by: A
=
e HT ,
II
=
fo T
e HA
(5.49a) d£ M,
(5.49b)
eH(kT-)QNx()~) d,k,
(5.49c)
aT f 3 (( k - 1 ) T
where the matrices H, N, M, L are H
=
N
=
M
=
L
=
W - LRTD, - L R T A + W L R T - LRT D L R T,
(5.50a) (5.505)
- L R T B, p D T R(RT R) -2
(5.50c) (5.50d)
and P is the positive definite solution of the algebraic matrix Riccati equation P W T + W P - p D T R(RT R ) - 2 R T D p + QQT = O.
(5.51)
The filter (5.47) and (5.48) provides a way to calculate the conditional expectation of ~k given the information from the continuous time process x(t). Substituting the estimate into the equation (5.44) we obtain an optimal ilk as ilk = ~(zk + LRTxk).
(5.52)
The quantity Yk + vk -- ilk from the right hand side of (5.41) is a discrete-time white noise process with variance of order O(T). It means that the mean square deviation of Sk+l from zero is of order O(T½).
5.4.2
Discrete
Measurement
Suppose we only have access to the discrete state information; the c~-algebra generated by the discrete measurement Xk, x k - 1 , . . , is ~Ck = o'{Xjl0 < j < k}.
(5.53)
100
The discrete time representations of (5.38a) and (5.38b) to be considered are Xk+l
=
~)xk + T~k + Fuk + vk + uk,
(5.54a)
~k+l
=
~ k +qk,
(5.54b)
__
F(k+l) T
where vk f : eA(T-~)D f : e W ( ~ - r ) Q d w ~ ( k T + v) dA, and qk = JkT eW((k+l)T-~)Qdw2(A) are discrete white noises generated form the process dw2; the matrices ~ and T are !P -: e WT,
T = e AT
e-AX De W;~ d)~.
(5.55)
To minimize the effect caused by ~k, we apply the control law Uk = - - ( C r ) - l C ~ x ~ - ( c r ) -~ C r ~ ,
(5.56)
where ~k is the optimal estimation of (k given Jt'k ~ = E(~klJ:~).
(5.57)
From (5.54), the discrete measurements Xk and xk-1 yield the innovation process Yk-1 = Xk -- qhXk_l -- FUk-1 = T~k-1 + vk-1 + ~k-1. (5.58) Combining (5.58) with (5.54), one can construct a stationary Kalman filter with the process noise qk-1 in (5.54) correlated with the measurement noise (vk-1 + vk-1) in (5.58) through dw2. To obtain a standard Kalman filter formulation with uncorrelated noises, we rewrite equation (5.38b)
~k+~ = ~ k + 9/~-~yk + ~k,
(5.59)
where /~ a n d / ~ are the covariance matrices = E(qk(vk + vk)*),
R = E((Vk + Vk)(Vk -t- l/k)*),
and ~, qk are defined as
With (5.59) and the innovation process Yk, an optimal estimation for ~k is acquired (k .L
= =
IP~k-1 "t- f I R - l y k - 1 q- L(yk-1 -- T~k-1), p T T ( T P T T +/~)-1,
(5.60a) (5.60b)
P
=
~(P-
(5.60c)
p T T ( T p T T + R ) - ~ T P ) ~ T + 0,0 T,
where (~T is the autocovariance matrix of the discrete white noise ~ in (5.59). Applying the control law (5.56) yields =
+
+
- r
k).
The quantity T(~a - ~k) + Vk + ~k from the right hand side of (5.61) is also a discrete time white noise with variance of order O(T). Therefore, the mean square deviation of sa+~ from the sliding surface is O(T½).
101
5.5 E x p e r i m e n t a l E x a m p l e - O ( T 2) S l i d i n g Mode Control on a Flexible Structure This section deals with sampled-data implementation of sliding mode for a flexible structure control problem. We apply the sampled-data sliding mode control techniques to a flexible structure developed for Large Interconnected Vibration Experiment (LIVE) at the Ohio State University. The truss structure, constructed of hollow PVC tubes and Noryl nodes, is aligned on a vertically cantilevered configuration shown in Figure 5.2.
Cantilevered To Wall (
J
J
z
/ 5.6 m
)'
~
Acceterometers
/ / ,
/
ProofMass
~ Actuators
~] ..........
3r
Motors
Figure 5.2
LIVE truss structure
The physical properties, control hardware setups, preliminary dynamic modelling and experimental results were published by Redmill and Ozgfiner (1992). Extended works in sampled-data sliding mode control experiment are oresented here.
102 The dynamic equations of the vertically cantilevered truss can be modelled based on the Euler-Bernoulli beam formulation. The bending motion is described by a fourth order partial differential equation (PDE).
pAQ + E I Q " = 0,
(5.62)
where the 'dot' indicates differentiation with respect to time, and the 'prime' differentiation with respect to the spatial variable, E I is the bending stiffness of the truss, A is the cross-sectional area and p is the density. The independent variable along the length of the truss is z, t is time, and Q(z, t) is the flexural displacement. The boundary conditions (BC) with one fixed end and one with an end mass ML are given as
Q(O,t)
=
0,
Q'(O,t) -EIQ"(1, t) EIQ"'(1, t) - MLQ(/,t)
=
0,
(5.63a) (5.63b)
=
0,
(5.63c)
=
-u(t).
(5.63d)
To solve for Q(z, t), one can follow the standard Fourier method and convert the PDE into infinite assumed modes which are driven in parallel by the boundary control u(t). Each mode can be expressed as a second order underdamped oscillation equation:
Q(z,t) = Ei~=l qi(t)¢i(z) qi + 2{iwi(ti + w~qi = kiu(t), i = 1, 2 , . . . ,
,
(5.64) (5.65)
where ¢i(z) is the spatially dependent mode shape coefficient, qi(t) is the time dependent modal state, wi is the modal frequency, {i is the damping ratio, and ki is control gain for each mode. The system parameters are given in (Redmill and C}zgiiner, 1992) and (Su, Drakunov and Ozgiiner, 1992). Control action is obtained using proof-mass actuators with neglected actuator dynamics. Sensing of truss motion is accomplished through a piezoelectric accelerometer mounted on the structure to measure the relative velocity between the proof-mass actuator and the structure. Combining the dynamic equation (5.65) with the actuation and sensing mechanism, we come up with the following I/O relationship:
i(t) +
= 0(t) + E %1 y,(t), + wyyi(t) =
i = 1, 2 , . . . ,
(5.66) (5.67)
where y(t) is the velocity measurement obtained from the output of the accelerometer; y0(t) is a low frequency (,~ 0.1 Hz) measurement noise; yi(t), i = 1, 2,..., is the flexural velocity for each mode. With the system parameters wi, ~i, ki, we obtain a state space representation for each mode of interest
103
0.3 ~--
, FFT of the output in ~ 3 ~ 1 2
0 2i i • Ii
solid' Sliding mode control dash;d: No control
i l
L, 0
5
10
15
20 25 30 Frequency (ttz)
0.3 [
35
40
45
50
',-, Enlargement solid: Sliding mode control ; ',,
0.2
dashed: No control
01!
;
',
<",
0
F i g u r e 5.3
',--, 0.5
1 1.5 Frequency (Hz)
2
2.5
Frequency content: Sliding mode control (c = 1) vs No control
xi
=
y~ =
Aixi + Biu,
(5.68a)
Gzi,
(5.6Sb)
where xi = [qi (ti] T is an 2 x 1 modal state vector, u is the scalar control, Yi is the bending velocity of mode i, Ai =
[0
-w~
1]
-2~iwi
'
Bi =
[0] bi
'
and bici = ki.
(5.70)
There are infinitely m a n y possible choices for bi and ci satisfying (5.70). For simplicity, we choose
bi = Iv/~l,
ci = sgn(kd tv/~l.
(5.71)
The sampling frequency throughout this experiment is I00 Hz, which corresponds to sampling period T = 0.01 second. The experimental results presented here are for the first flextural mode only. To demonstrate the effectiveness oi the proposed control method, we take the following scenario: Part 1:0 < t < 2 Part 2 : 2 < t < 12 Part 3:12 < t < 20
S e t t l e m e n t - uk = O. Shake up the truss - uk = 0.3sin (wl • k T ) . Sliding m o d e control vs No control.
104
of the output in pa~ 3 (12 <,t < 20) for c=10,
0.3
dashed: No control solid: Sliding mode control
0.2
0.I
0
0
5
I0
15
20
25
30
35
40
45
50
Frequency (Hz) 0.3
.........,.....
/'\
dashed:No control
Enlargement
solid: Sliding mode contro~
0.2
0.1
0
o
Figure 5.4
0.5
l
1.5 Frequency (Hz)
2.5
2
Frequency content: Sliding mode control (c = 10)
vs
No control
The sliding surface for the second order vibrating system is chosen as s(x) --- [c
llxl
= cqi -4- 41 = O.
Experiments are conducted for c = 1 and c = 10. A saturation function is employed to limit the applied control uk = sat(u~q, 0.1). Figure 5.3 depicts the frequency content (FFT) of yk in Part 3 for c = 1. Similar result for c = 10 is shown in Figure 5.4, in which better performance is obtained.
5.6 C o n c l u s i o n We have investigated the sample/hold effect and taken it into account for discrete time sliding mode control design. We showed that the piecewise constant control law due to zero-order-hold has an inherent limitation in disturbance rejection. Even if the sampled-data system attains sliding mode in discrete time, the intersample behavior still deviates from the sliding manifold. The error is inevitable since a similar "hold" does not take place in the disturbance channels.
105 By using the proposed discrete time sliding mode control laws, the states can be maintained in an O(T 2) boundary layer of $ for deterministic systems and O(T½) for stochastic systems. The results are summarized in Table 5.6.
Linear Deterministic Systems ~: = A s + AA(t)x + Bu + AB(t)u + D f ( t ) Exogenous disturbances f(t) s(t) = O(T2), t >_ k~T
System parameter variations AA(t) Control coefficient matrix variations AB(t)
s(t) -+ 0 as t -~ s(t) -+ 0 as t -+ oe
Nonlinear Deterministic Systems
= a(z) + B(x)~ +b(x)~ + ,(x,t) Matched disturbances v(z, t) s(t) = o(T~), t >_ k~T Control coefficient matrix variations b(x) s(t) --~ O(T 2) as t --~ c¢ Stochastic Systems dx = Axdt + Budt + (D~dt + Rdw~) d~ = W~dt + Qdw2
Continuous measurement .Tt Discrete measurement ~-k Table 5.1
(EHskl[2)½ O(T½), k >_ ks (E[[sk]]2)½= O(T½), k > ks =
Sliding surface errors under different types of disturbances
The idea of a boundary layer enables one to smooth the control instead of using instantaneous switchings. Meanwhile, it does not compromise chattering with robustness as many other smoothing functions do (Young and Drakunov, 1992). The proposed methods are simple from an implementation point of view. It deals with both exogenous disturbances and internal uncertainties. Furthermore, the notorious chattering problem associated with the continuous time variable structure control is removed with considerable robustness. Acknowledgements. The first author was supported by an Ohio State University, Presidential Fellowship, while the other two were partially supported by AFOSR G r a n t ~ F49620-92-J-0460.
References Bartolini, G., Ferrara, A., Utkin, V. I., 1993, On robust adaptive control of mimo linear plants using discrete-time sliding modes, Proceedings IFAC World Conference, Sidney, Australia, 51-56. Bucy, R., Josef, P., 1987, Filtering for stochastic processes with applications to guidance, Chelsea Pub. Co., New York.
106 Drakunov, S. V., Su, W. C., 6zgiiner, f~., 1993, Discrete-time sliding-modes in stochastic systems, Proceedings American Control Conference, San Francisco, CA, 966-970. Drakunov, S., Utkin, V. I., 1992, Sliding mode control in dynamic systems, International Journal of Control, 55(4), 1029-1037. Drazenovic, B., 1969, The invariance condition in variable structure systems, Automatica, 5, 287-295. Furuta, K., 1990, Sliding mode control of a discrete system, Systems and Control Letters, 14, 145-152. Kaynak, O., Denker, A., 1993, Discrete-time sliding mode control in the presence of system uncertainty, International Journal of Control, 57(5). Kotta, U., 1989, Comments on 'the stability of discrete-time sliding mode control systems,' IEEE Transactions on Automatic Control, 34(9), 1021-1022. Milosavljevic, C., 1985, General conditions for the existence of a quasi-sliding mode on the swithing hyperplane in discrete variable, Automation and Remote Control, 46, 307-314. Morgan, R., Ozgiiner, U., 1985, A decentralized variable structure control algorithm for robotic manipulators, IEEE Transaction Robotics and Automation, 1(1), 57-65. Redmill, K., Ozgiiner, U., 1992, Experiment on a truss-panel structure, Proceedings American Control Conference, Chicago, IL, 2530-2534. Sarptiirk, S., Istefanopulos, Y., Kaynak, O., 1987, On the stability of discretetime sliding mode control systems, IEEE Transactions on Automatic Control, 32(10), 930-932. Sira-Ramfrez, H., 1991, Nonlinear discrete variable structure systems in quasisliding mode, International Journal of Control, bf 54(5), 1171-1187. Su, W. C., Drakunov, S. V., Ozgiiner, U., 1993, Sliding mode control in discrete time linear systems, Proceedings IFAC World Conference, Sidney, Australia. Su, W. C., Drakunov, S. V., Ozgiiner, U., Redmill, K., 1995, O(T 2) Sliding mode control on a flexible structure, Proceedings American Control Conference, Seattle, WA, 1318-1322. Su, W. C., Ozgfiner, U., Drakunov, S. V., 1993, Discrete time sliding modes in nonlinear sampled-data systems, Proceedings of the Thirty-First Annual Allerton Conference on Communication, Control, and Computing, Urbana, IL. Utkin, V. I., 1978, Sliding modes and their application in variable structure systems, MIR, Moscow. Utkin, V. I., Drakunov, S., 1989, On discrete-time sliding mode control, Proceedings of IFAC Symposium on Nonlinear Control Systems (NOLCOS), Capri, Italy, 484-489. Utkin, V. I., Young, K. D., 1979, Methods for constructing discontinuity planes in multidimensional variable structure systems, Automation and Remote Control, 39, 1466-1470. Young, K. D., Ozgfiner, U., 1993, Frequency shaping compensator design for sliding mode, International Journal of Control, 57(5), 1005-1019. Young, K. D., Drakunov, S., 1992, Sliding mode control with chattering reduction, Proceedings American Control Conference, Chicago, IL, 1291-1292.
. H i g h e r O r d e r S l i d i n g M o d e s as a N a t u r a l P h e n o m e n o n in C o n t r o l Theory L e o n i d F r i d m a n and A r i e Levant 6.1 I n t r o d u c t i o n Sliding modes are used in order to keep a dynamic system to given constraints with utmost precision. They are also insensitive to external and internal disturbances. These features are provided due to a theoretically infinite frequency of control switching. At the same time, any presence of real actuators and measuring devices may cause considerable change in the real system behavior. There are also many arguments for the use of continuous controllers as approximations for discontinuous ones. Both reasons lead to the replacement of regular sliding modes in real systems by some special ones. Consider the phenomenon more explicitly.
A. Presence of Fast Actuators Let the constraint be given by the equality of some constraint function cr to zero and the sliding mode ~r _-- 0 be provided by a relay control. Taking into account an actuator conducting a control signal to the process controlled, we achieve a more complicated dynamics. In this case relay control u enters the actuator, and continuous output variables of the actuator z are transmitted to the plant input (Figure 6.1). As a result discontinuous switching is hidden now in the higher derivatives of the constraint function (Utkin 1981, 1992; Fridman 1983, 1985, 1986, 1990, 1991, 1993).
B. Artificial Actuator-like D y n a m i c s One of the main known drawbacks of regular sliding modes is the so-called chattering effect which is exhibited by high frequency vibration of the plant. This vibration features an infinitesimally small vibration magnitude of the plant itself (in the state space) and finite or eventually even large vibration magnitude of the plant velocity. Such fast velocity changes require large forces to appear accompanied probably by a significant energy release. This may cause a system disaster. To avoid chattering several approaches have been proposed. The main idea is to change the dynamics in a small vicinity of the discontinuity surface in order to avoid real discontinuity and at the same time to preserve the main
108
z ,[
Plant
Actuator
Relay Controller
Figure 6.1
Control system with actuator
properties of the whole system. A transition to the new dynamics defined near the switching surface has to be sufficiently smooth. The idea is realized by insertion of some artificial actuator (Figure 6.2). This actuator may be a functional (Slotine and Sastry, 1983) or may have its own dynamics (Emelyanov and Korovin, 1981; Emelyanov, 1984). We are interested here in the latter case. The actuator installed (artificial or real) has mainly some fast dynamics. The faster the dynamics, the more accurate is the modeling of the original discom tinuous dynamics with all its advantages and disadvantages. However, from time to time a different mode, contiguous to the ordinary sliding mode, appears. The corresponding state and velocity vibration magnitudes both tend to zero when switching imperfections vanish. Moreover, the actuator dynamics do not need to be really fast for the existence of such a mode. Such modes were called higher order sliding modes (Levantovsky, 1985; Emelyanov et al, 1986a, b, c, 1990; Chang, 1990; Sira-Ramfrez, 1992a, b, 1993; Levant (Levantovsky), 1993a). Convergence to this special mode may be asymptotic (Fridman, 1983, 1985, 1986, 1990, 1991, 1993; Levantovsky, 1986; Emelyanov et al, 1990; Chang, 1990; Elmali and Olga~, 1992; Sira-Ramfrez, 1992a, b, 1993) or may feature a finite time as well (Levantovsky, 1985; Emelyanov et al, 1986a, b, c, t990; Levant (Levantovsky), 1993a). A higher order sliding mode (HOSM) is a movement on a special type integral manifold of a discontinuous dynamic system. It appears every time when some fast actuator-like dynamics are implanted instead of an ordinary relay in a variable structure system. Even if this mode is not stable, it plays the same role as integral manifold in ordinary differential equations. Thus, HOSM is an important natural phenomenon in control theory. In Section 6.2 the definitions of HOSM are presented. The difference between the definitions is discussed. The place of HOSM in control theory and connection between the sliding order and sliding accuracy are discussed in Sec-
109
zl
Plant
Real Actuator
Tz Artificial Actuator
X
Relay Controller
Figure 6.2
Control system with artificial actuator
tion 6.3. It is shown in Section 6,4 that HOSM emerges every time when we have dynamic actuators in sliding mode control systems. Section 6.5 is devoted to study of second order sliding mode stability in systems with actuators. Various examples of second order sliding modes are presented in Section 6.6. Section 6.7 deals with examples of higher order sliding modes, an example of third order sliding algorithm with finite convergence time is presented.
6.2 Definitions of Higher Order Sliding M o d e s Regular sliding mode features few special properties. It is reached in finite time which means that a number of trajectories meet at any sliding point. In other words, the shift operator along the phase trajectory exists, but is not invertible in time at any sliding point. Other important features are that the manifold of sliding motions has a nonzero codimension and that any sliding motion is performed on a system discontinuity surface and may be understood only as a limit of motions when switching imperfections vanish and switching frequency tends to infinity. Any generalization of the sliding mode notion has to inherit some of these properties. A definition developed by Drakunov and Utkin (1992) deals with the properties of the shift operator and allows, thus, extension of the definition to dynamic systems of even completely different nature. The definitions developed below utilize the other properties mentioned. In many cases both definitions are satisfied (not only in the ease of the regular sliding mode].
110 Let us remind first what are Filippov's solutions (Filippov, 1960, 1988) of a discontinuous differential equation
= v(x), where x C IR~, v is a locally bounded measurable (Lebesgue) vector function. In this case, the equation is replaced by an equivalent differential inclusion
~ v(x). In the particular case when the vector-field v is continuous almost everywhere, the set-valued function V(x) is the convex closure of the set of all possible limits of v(y) as y -4 x, while (y} are continuity points of v. Any solution of the equation is defined as an absolutely continuous function x(t), satisfying the differential inclusion almost everywhere. The following definitions are based on the works by Levantovsky (1985), Emelyanov et al (1986a, b, 1990), Levant (1993a).
6.2.1 Sliding M o d e s on Manifolds D e f i n i t i o n 6.1 Let L be a smooth manifold. Set L itself is called the first order sliding set with respect to L. The second order sliding set is defined as the set of points x C L, where V(x) lies entirely in tangential space TL to manifold L at point x. D e f i n i t i o n 6.2 It is said that there exists a first (or second) order sliding mode on manifold L in a vicinity of a first (or second) order sliding point x, if in this vicinity of point x the first (or second) order sliding set is an integral set, i.e. it consists of Filippov's sense trajectories. Let L1 = L. Denote by L~ the set of second order sliding points with respect to manifold L. Assume that L2 may itself be considered as a sufficiently smooth manifold. Then the same construction may be considered with respect to L2. Denote by L3 the corresponding second order sliding set with respect to L2. L3 is called the 3-rd order sliding set with respect to manifold L. Continuing the process, achieve sliding sets of any order. D e f i n i t i o n 6.3 It is said that there exists an r-th order sliding mode on manifold L in a vicinity of an r-th order sliding point x E Lr, if in this vicinity of point x the r-th order sliding set Lr is an integral set, i.e. it consists of Filippov's sense trajectories.
6.2.2 Sliding M o d e s with R e s p e c t to Constraint Functions Let a constraint be given by an equation c~(z) = 0, where ~ : ]R~ -4 lR is a sufficiently smooth constraint function. It is also supposed that total time
111 derivatives along the trajectories o', &, ~ , . . . , o"(~-1) exist and are single-vMued functions of x, which is not trivial for discontinuous dynamic systems. In other words, this means that discontinuity does not appear in the first r - 1 total time derivatives of the constraint function o'. Then the r-th order sliding set is determined by the equalities
(6.1) Here (6.1) is an r-dimensional condition on the state of the dynamic system. D e f i n i t i o n 6.4 Let the r-th order sliding set (6.1) be non-empty and assume that it is locally an integral set in Filippov's sense (i.e. it consists of Filippov's trajectories of the discontinuous dynamic system). Then the corresponding motion satisfying (6.1) is called an r-th order sliding mode with respect to the constraint function or. To exhibit the relation with the previous definitions, consider a manifold L given by the equation ~r(x) = 0. Suppose that o',/r, ~ , . . . , o"(~-2) are differentiable functions of x and that rank{Vo', V / r , . . . , Vo"(~-2)} = r - 1
(6.2)
holds locally (here rank V is a notation for the rank of vector set V). Then Lr is determined by (6.1) and all Li, i = 1 , . . . , r - 1 are smooth manifolds. If in its turn Lr is required to be a differentiable manifold, then the latter condition is extended to rank{Vc~, V / r , . . . , V~r (r-l)} = r. (6.3) Equality (6.3) together with the requirement for the corresponding derivatives of o" to be differentiable functions of x will be referred to as the sliding regularity condition, whereas condition (6.2) will be called the weak sliding regularity condition. With the weak regularity condition satisfied and L given by equation o" = 0 Definition 6.4 is equivalent to Definition 6.3. If regularity condition (6.3) holds, then new local coordinates may be taken. In these coordinates the system will take the form Yl =~r, ~)1 =Y2, ... 9~-1 =Y,'; =
= Cen
P r o p o s i t i o n 6.5 Let regularity condition (6.3) be fulfilled and r-th order sliding manifold (6.1) be non-empty. Then an r-th order sliding mode with respect to the constraint function cr exists if and only if the intersection of the Filippov vector-set field with the tangential space to manifold (6.1) is not empty for any r-th order sliding point.
112 Proof. The intersection of the Filippov set of admissible velocities with the tangential space to the sliding manifold (6.1), mentioned in the proposition, induces a differential inclusion on this manifold. This inclusion satisfies all the conditions by Filippov (1960, 1988) for solution existence. Therefore manifold (6.1) is an integral one. Let now ~r be a smooth vector function, a : IRn ~ IRm, e = ( a l , . . . , am), and also r = ( r l , . . . , rm), where ri are natural numbers.
Definition 6.6 Assume that the first ri successive full time derivatives of ~ri are smooth functions, and a set given by the equalities O'i=&i=(ri=...=cr
}ri-1)
=0,
i= 1,...,m,
is locally an integral set in Filippov's sense. Then the movement mode existing on this set is called a sliding mode with vector sliding order r with respect to the vector constraint function cr. The corresponding sliding regularity condition has the form rank{V~ri,..., V ~ } r ' - l ) li = 1 , . . . , m } = rl + . . . + r.~. Definition 6.6 corresponds to Definition 6.3 in the case when rl . . . . . rm and the appropriate weak regularity condition holds. A sliding mode is called stable if the corresponding integral sliding set is stable. Remarks. (i) These definitions also include trivial cases of an integral manifold in a smooth system. To exclude them we may, for example, call a sliding mode "not trivial" if the corresponding Fitippov set of admissible velocities V ( x ) consists of more than one vector. (ii) The above definitions are easily extended to include non-autonomous differential equations by introduction of the fictitious equation i = 1
6.3 Higher Order Sliding Modes in Control Systems Single out two cases: ideal sliding occurring when the constraint is ideally kept and real sliding taking place when switching imperfections are taken into account and the constraint is kept only approximately. 6.3.1
Ideal
Sliding
All the previous considerations are translated literally to the case of a process controlled
113
where x E lR'~, t is time, u is control, and f, ~r are smooth functions. Control u is determined here by the feedback u = U(t, x), where U is a discontinuous function. For simplicity we restrict ourselves to the case when cr and u are scalars. Nevertheless, all statements below m a y also be formulated for the case of vector sliding order. Regular sliding modes satisfy the condition that set of possible velocities V does not lie in tangential vector space T to the manifold v- = 0, but intersects with it, and therefore a trajectory exists on the manifold with velocity vector lying in T. Such modes are the main operation modes in variable structure systems (Emelyanov, 1967; Utkin, 1977, 1981, 1992; Itkis, 1976; DeCarlo et al, 1988; Zinober, 1990) and according to the above definitions they are of the first order. When a switching error is present the trajectory leaves the manifold at a certain angle. On the other hand, in the case of second order sliding all possible velocities lie in the tangential space to the manifold and even when a switching error is present, the state trajectory is tangential to the manifold at the time of leaving. To see connections with some well-known results of control theory, consider at first the case when
= a(x) + b(x)u,
v- = v(x) E IR,
u C IR,
where a, b, v- are smooth vector functions. Let the system have a relative degree r with respect to the output variable v- (Isidori, 1989). This means that Lie derivatives Lbv-, L~,L,~V-,..., LbL~-2o " equal zero identically in a vicinity of a given point and LbL~-lv- is not zero at the point. The equality of the relative degree to r means, in a simplified way, that u first appears explicitly only in the r-th full time derivative of v-. It is known that in this case ~(i) = L~v- for i = 1 , . . . , r - 1, regularity condition (6.3) is satisfied automatically and also 0~o"(r) = LbL~-la ~ O. There is a direct analogy between the relative degree notion and the sliding regularity condition. Loosely speaking, it m a y be said t h a t the sliding regularity condition (6.3) means that the "relative degree with respect to discontinuity" is not less than r. Similarly, the r-th order sliding mode notion is analogous to the zero-dynamics notion. The relative degree notion was originally introduced for the autonomous case only. Nevertheless, we will apply this notion to the non-autonomous case as well. As was already done above, we introduce for this purpose a fictitious variable xn+l = t, ~ + 1 = 1. It has to be mentioned that some results by Isidori will not be correct in this case, but the facts listed in the previous paragraph will still be true. Consider a dynamic system of the form
=a(t,x)+b(t,x)u,
~=v-(t,x),
u=U(t,x)
CIR.
T h e o r e m 6.7 Let the system have relative degree r with respect to the output function cr at some r-th order sliding point (to, xo). Let, also, the discontinuous
114
function U take on values from sets [K, ~ ) and ( - c o , - K ] on some sets of non-zero measure in any vicinity of any r-th order sliding point near point (t0, x0). Then this provides, with sufficiently large K, for the existence of r-th order sliding mode in some vicinity of point (to, xo). Proof. This theorem is an immediate consequence of Proposition 6.5, nevertheless, we will detail the proof. Consider some new local coordinates y = ( y t , . . . , y,~), where yl -= cr, y2 = & , . . . , Y~ = ~r(~-l). In these coordinates manifold L~ is given by the equalities Yl = Y2 = "'" = Y~ = 0 and the dynamics of the system are as follows:
Yr-t
=
Yr,
y~
=
h(t,y) + g(t,y)u,
=
(6.4) g(t,y) ¢ O, =
Denote Ueq = - h ( t , y)/g(t, y). It is obvious that with initial conditions being on the r-th order sliding manifold Lr control u = Ueq(t, y) provides for keeping the system within manifold L~. It is also easy to see that the substitution of all possible values from I - K , K] for u gives us a subset of values from Filippov's set of the possible velocities. Let IUeql be less than K0, then with K > K0 the substitution u = U~q determines a Filippov's solution of the discontinuous system which proves the theorem. The trivial control algorithm u = - K sgn ~r satisfies Theorem 6.7. Usually, however, such a m o d e will not be stable. It follows from the proof above that the movement in the r-th order sliding m o d e is described by the equivalent control method (Utkin, 1977), on the other hand these dynamics coincide with the zero-dynamics (Isidori, 1989) for corresponding systems. There are some recent papers devoted to the higher order sliding mode technique. The sliding mode order notion by Chang (1990), Elmali and Olgaq (1992) seems to be understood in a very close sense (the authors had no possibility to acquaint themselves with the work by Chang (1990)). The same idea is developed in a very general way from the differential-algebraic point of view in the papers by Sira-Ramlrez (1992a, b, 1993). In his papers sliding modes are not distinguished from the algorithms generating them. Consider this approach. Let the following equality be fulfilled identically as a consequence of the dynamic system equations (Sira-Ram~rez, 1993): p ( ~ ( r ) , . . . , / ~ , ~, x, u ( ~ ) , . . . , / t , u) = 0.
(6.5)
Equation (6.5) is supposed to be solvable with respect to cr(r) and u (k). Function c~ m a y itself depend on u. The r-th order sliding m o d e is considered as a steady state ~r - 0 to be achieved by a controller satisfying (6.5). In order to achieve for v" some stable dynamics
115 Z = ~(r-1) + alo.(r-2) + ... + ar_lcr = 0, the discontinuous dynamics = -sgn ~
(6.6)
is provided. For this purpose the corresponding value of c~(~) is evaluated from (6.6) and substituted into (6.5). The obtained equation is solved for u (k). Thus, a dynamic controller is constituted by the obtained differential equation for u which has a discontinuous right hand side. With this controller successive derivatives c~,..., ~(~-1) will be smooth functions of closed system state space variables. The steady state of the resulting system will satisfy at least (6.1) and under some relevant conditions also the regularity requirement (6.3), and therefore Definition 6.4 will hold. ttence, it may be said that the relation between our approach and the approach by Sira-Ramfrez is a classical relation between geometric and algebraic approaches in mathematics. Note that there are two different sliding modes in system (6.5), (6.6): a regular sliding mode of the first order which is kept on the manifold Z' = 0, and an asymptotically stable r-th order sliding mode with respect to the constraint ~ = 0 which is kept in the points of the r-th order sliding manifold cr = / r =/~ = ... = ~(~-1) = 0.
6.3.2 Real Sliding and Finite Time Convergence Remind that the objective is synthesis of such a control u that the constraint cr(t,x) = 0 holds. The quality of the control design is closely related to the sliding accuracy. In reality, no approaches to this design problem may provide for ideal keeping of the prescribed constraint. Therefore, there is a need to introduce some means in order to provide a capability for comparison of different controllers. Any ideal sliding mode should be understood as a limit of motions when switching imperfections vanish and the switching frequency tends to infinity (Filippov, 1960, 1988). Let e be some measure of these switching imperfections. Then sliding precision of any sliding mode technique may be featured by a sliding precision asymptotics with c -+ 0.
Definition 6.8 Let (t, x(t, e)) be a family of trajectories, indexed by ¢ E IR ~, with common initial condition (to, x(to)), and let t >_ to ( o f t E [to, T]). A s s u m e that there exists tl > to (or tl E [t0,T]) such that on every segment [t',U], where t' >_ t,, (or on [tl, T]) the function c~(t, x(t, ~)) tends uniformly to zero with c tending to zero. In this case we call such a family a real sliding family on the constraint c~ = O. We call the motion on the interval [t0,tl] a transient process, and the motion on the interval [tl, cx~) (or It1, T]) a steady state process. D e f i n i t i o n 6.9 A control algorithm, dependent on a parameter c E IR ~, is called a real sliding algorithm on the constraint ~ = 0 if, with e -+ O, it forms a real sliding family for any initial condition.
116 D e f i n i t i o n 6.10 Let 7(~) be a real-valued function such that 7(e) -~ 0 as c -+ O. A real sliding algorithm on the constraint o" = 0 is said to be of order r (r > O) with respect to 7(¢) if for any compact set of initial conditions and for any time interval [Tt, T2] there exists a constant C, such that the steady state process for t E IT1, T2] satisfies
It(t, ~(t, ~))1 < cb'(~)l ~.
In the particular case when 7(~) is the smallest time interval of control smoothness, the words "with respect to 7" may be omitted. This is the case when real sliding appears as a result of switching discretization. As follows from (Levant, 1993a), with the r-th order sliding regularity condition satisfied, in order to get the r-th order of real sliding with discrete switching it is necessary to get at least the r-th order in ideal sliding (provided by infinite switching frequency). Thus, the real sliding order does not exceed the corresponding sliding mode order. The regular sliding modes provide, therefore, for the first order real sliding only. The second order of the real sliding was really achieved by discrete switching modifications of the second order sliding algorithms (Levantovsky, 1985; Emelyanov et al, 1986a, b, c, 1990; Levant, 1993a). A special discrete switching algorithm providing for the second order real sliding was constructed by S u e t al (1994). Real sliding of the third order is demonstrated later in this chapter. Real sliding may also be achieved in a way different from the discrete switching realization of sliding mode. For example, high gain feedback systems (Saksena et al, 1984; Young et al, 1977) constitute real sliding algorithms of the first order with respect to k-1, where k is a large gain. Another example is adduced in Section 6.6 (Example 6.12). It is right that in practice the final sliding accuracy is always achieved in finite time. Nevertheless, besides the pure theoretical interest there are also some practical reasons to search for sliding modes attracting in finite time. Consider a system with an r-th order sliding mode. Assume that with minimal switching interval ~- the maximal r-th order of real sliding is provided. This means that the corresponding sliding precision Ic~l ,-~ v ~ is kept, if the r-th order sliding condition holds at the initial moment. Suppose that the r-th order sliding mode in the continuous switching system is asymptotically stable and does not attract the trajectories in finite time. It is reasonable to conclude in this case that with r --+ 0 the transient process time for fixed general case initial conditions will tend to infinity. If, for example, the sliding mode were exponentially stable, the transient process time would be proportional to r In r -1. Therefore, it is impossible to observe such an accuracy in practice, if the sliding mode is only asymptotically stable. At the same time, the time of the transient process will not change drastically, if it was finite from the very beginning.
117
6.4 Higher Order Sliding Modes with Dynamic Actuators
and Systems
Suppose that the plant has relative degree r with respect to the output function c~. T h a t means that we can describe the behavior of the first r coordinates of the control system in form (6.4). Assume that relay control u is transmitted to the input of the plant (Figure 6.1) by a dynamic actuator which itself has an l-th order dynamics. The behavior of the first l actuator coordinates and of the first r plant coordinates is described by the equations Yl = y ~ , . . . , y ~ - i = y~,
= h(t, y) + g(t, y)z , il
:
z2,...,z~-I
zt = p(t, y, z) + q(t, y, z)u,
g(t, y) # o, :
zz,
(6.7)
q(t, y, z) 7£ O.
(6.8)
where yl = o'. This means that the complete model of the sliding mode control system has relative degree r + l, and, therefore, the (r +/)-sliding regularity condition holds. According to Theorem 6.7, a sliding mode with respect to has to appear here, which has sliding order r + l . If the controller itself is chosen in an actuator-like form (Figure 6.2), the corresponding sliding order will be still larger. Thus, higher order sliding modes emerge every time when we have to take into account dynamic actuators in a sliding mode control system. Consider a special case when the actuator is fast. In this case equation (6.8) has the form #zl = p(t, y, z, #) + q(t, y, z, #)u, (6.9) where # is a small actuator time constant. In fact all motions in system (6.7), (6.9) have fast velocities in such a case~ There are two approaches to investigation of systems with such actuators: consideration of a small neighborhood of the higher order sliding set (Fridman, 1985, 1990), and transformation to a basis of eigenvectors. In both cases the complete model of the control system will be a singularly perturbed discontinuous control system. Fast motions in such systems are described by a system with an (r + l)-th order sliding mode. Following is some simple informal reasoning valid under sufficiently general conditions. Let an actuator be called precise if its output is used by the process controlled exactly as a substitution for the control signal. This means, in particular, that the dimensions of the actuator output and control (relay output in Figure 6.1) coincide. Also require for such an actuator that for any admissible constant control value the output of the actuator settle at the control value after some time and that this transient time be small if the actuator is fast. No chattering is generally observed in a system with a precise actuator, if the corresponding higher order sliding mode is stable. Indeed, let common
118
conditions on regular sliding mode implementation be satisfied for the process controlled. This means, in particular, that 8 = 0 implies u = Ueq where equivalent control ueq is a sufficiently smooth function of the state variables. Therefore, the output of the actuator inevitably tends to this smooth function while the process enters the higher order sliding mode a - 0 and the chattering is removed. However, if the fast actuator is not precise, fast motion stability in the higher order sliding mode is also to be required in order to avoid chattering (see remark at the end of the next section). On the other hand instability of the sliding mode of corresponding order r > 2 leads to appearance of a real sliding mode which is usually accompanied by a chattering effect, if the system with a fast stable precise actuator is considered (Figure 6.1). Indeed, suppose that there is no chattering, c~ ~_ 0 and the actuator output is stabilized at some slowly changing value. This is possible only if the relay output is constant or an infinite frequency switching of the relay output takes place. The latter means that cr - 0, also total derivatives of c~ of orders up to r - 1 equal zero and the higher order sliding mode is stable in contradiction to our conditions. On the other hand the actuator output will settle at the relay output value before the system leaves some small vicinity of the manifold c~ = 0. This prescribes the needed sign to 8 and prevents leaving this small vicinity of the manifold. Hence, the actuator output performs fast vibrations while the trajectory does not leave a small manifold vicinity.
6.5
Stability of Second Order Sliding Modes Systems with Fast Actuators
Consider
a
simple example of
a
in
dynamic system
Yl
=
Y2,
(6.10a)
Y2
=
ayl + by2 + c y 3 + k s g n y t ,
(6.10b)
~ai,jyj,
(6.10c)
Yi :
i = 3,...,n.
j=l
The second order sliding set is given here by equalities Yl = Y2 -- 0. Following Anosov (1959), we single out the exponentially stable and unstable CaSeS.
• E x p o n e n t i a l l y stable case.
Under the conditions
b < 0,
k < 0,
(6.11)
the set Yl = Y2 = 0, Icy31 < k is an exponentially stable integral manifold for system (6.10). • Unstable case.
Under the condition
119 k>O
or
b>0
the second order sliding set of system (6.10) is an unstable integral manifold. • Critical case. k<_O,
b
bk=O.
With c = b = 0, k < 0 the second order sliding set of system (6.10) is stable but not asymptotically stable, with c :fi 0 stability is determined by the properties of the whole system. Condition (6.11) is used in works by Fridman (1990, 1991, 1993) for analysis of sliding mode systems with fast dynamic actuators. Here is a simple outline of these reasonings. It is not required that actuators be precise. One of the actuator output variables is formally replaced by & after application of some coordinate transformation. Let the system under consideration be rewritten in the following form: Ftz
=
A z + Brl + D l x ,
(6.12a)
Cz+brl+ D2x+ksgn~,
(6.12b)
=
71,
(6.12c)
=
F(z, r/, s, x),
(6.12d)
tLil =
where z E IRm, x G IRn, r/,~r E 1R. With (6.11) fulfilled and Re Spec A < 0 system (6.12) has an exponentially stable integral manifold of slow motions being a subset of the second order sliding manifold and given by the equations
z=H(~,x)=-A-~D~x+O(~),
~=~=0.
Function H may be evaluated with any desired precision with respect to tile small parameter # . Therefore, according to Fridman (1990, 1991, 1993), under the conditions ReSpec A < 0,
b < 0,
k< 0
(6.13)
the motions in such a system with a fast actuator of relative degree 1 consist of fast oscillations, vanishing exponentially, and slow motions on a submanifold of the second order sliding manifold. Thus, if conditions (6.13) of chattering absence hold, the presence of a fast actuator of relative degree 1 does not lead to chattering in sliding mode control systems. For any chattering simulation in this case it is necessary to take into account some other factors like positive feedbacks (Fridman, 1990) and time delays (Fridman et al, 1992) or to consider systems with relative degree of actuator more than 1. Remark. Stability of the fast actuator and the second order sliding mode in (6.12) still do not guarantee absence of chattering if the actuator is not
120 precise. Indeed, stability of a fast actuator corresponds to the stability of the fast actuator matrix ReSpec ( A B) C b <0. Consider the system PZl
=
Zl + Z2 + 77+ DlZ,
#z2
=
2z2 + z3 + D 2 x ,
p~) = 2 4 z l - 6 0 z 2 - 9 7 + D 3 z + k s g n ~ r , & --- 7, ---- F(Zl, z 2 , 7 , o', x ) , where zl, z2, 7, a are scalars. The dimensions of the actuator output zl, z2, T/and the relay output sgn ~r are not equal, so the actuator is not precise. It is easy to check that the spectrum of the matrix is { - 1 , - 2 , - 3 ) and condition (6.11) holds for this system. On the other hand the motions in the second order sliding mode are described by the system #il
=
zl+z2+Dlz,
#i:2
-~
2z2 q- D 2 x ,
ic
=
F ( z l , z2, O, O, x).
The fast motions in this system are unstable and the absence of chattering in the original system cannot be guaranteed.
6.6 E x a m p l e s of Second Order Sliding M o d e s Without loss of generality we shall illustrate the approach by some simple examples. Consider, for instance, sliding mode usage for the tracking purpose. Let the process be described by the equation .= u,
x, u E IR,
and the problem is to track a signal f(t) given in real time, where If[, Ill, I]] < 0.5. Only values of z, f, u are available. The problem is successfully solved by the controller u = - s g n cr, cr = x - f ( t ) , keeping cr = 0 in the sliding mode of the first order. In practice, however, there is always some actuator between the plant and the controller, which inserts some additional dynamics and removes the discontinuity from the real system. With respect to Figure 6.1 let the system be described by the equation i~=z.
I~I
where z C IR is an output of some dynamic actuator. The input u of the actuator is the relay control u = - s g n cr. E x a m p l e 6.11 For example
Assume that the actuator has some fast first order dynamics. #~ = u - z,
where # is a small positive number. The second order sliding manifold L~ is given here by the equations
c~=x-f(t)
=0,
ir=z-f(t)=O.
Equality = l ( u _ z) - f ( t ) # shows that the relative degree here equals 2 and, according to Theorem 6.7, a second order sliding mode exists in the system, provided # < 1. The motion in this m o d e is described by the equivalent control method or by zero-dynamics, which is the same: from rr = / , = / i = 0 achieve u = p f ( t ) + z, z = 9/(t) and therefore
x = f(t),
z = ¢(t),
u = ttf(t) + z.
It is easy to prove (see Section 6.5) that the second order sliding mode is stable here with p small enough. Note that the last equality describes the equivalent control (Utkin, 1977, 1981, 1992) and is kept actually only in the average, while the first and the second are kept accurately in this sliding mode. Here and further the examples are accompanied by simulation results with
f(t)=O.O8sint+O.12cosO.3t,
x(0) = 0,
z(0) = 0,
where z is the actuator output,. The plots of x(t) and f ( t ) with /1 = 0.2 are shown in Figure 6.3, whereas the plot of z(t) is demonstrated in Figure 6.4.
E x a m p l e 6.12 One of the main ideas of the binary system tlheory (Emelyanov and Korovin, 1981; Emelyanov, 1984) is to insert some artificial fast dynamics in the switching process of relay system. This may be regarded as an installation of a fast actuator. For example, let
-z,
Iz! > lul,
where # is a small positive number. This is a slightly modified A , - a l g o r i t h m by Emelyanov and Korovin (1981). Having substituted u = - s g n c~ achieves the classical form of the algorithm with z being considered as a control:
122
1.897384E-01
\
! l!
\
\
i
\ -6.961547E-02
O.O00000E+O0
\ 5.988000
F i g u r e 6.3 Asymptotically stable second order sliding mode in a system with a fast actuator. Tracking: x(t) and f ( t ) .
7.630808E-01
"i
l!
il J ii i:
i r'..AI
o
i
$;$--
°1
-4.040195E-01 O.O00000E+O0
5.988000
F i g u r e 6.4 Asymptotically stable second order sliding mode in a system with a fast actuator: actuator output z(t).
123
1.941250E-01
,
\
ri
i
\
r!
\ -7.492492E-02
t 5.988000
0.000000E+00
Figure 6.5 Unstable second order sliding mode in a system with A,-controUer. Tracking: x(t) and f(t).
i=
1 --z, # 1 -~sgno',
]z I > 1,
(6.14) Izl_< 1.
Similarly to the previous example, a second order sliding mode is achieved here with II < 2. Simple calculation shows that the trajectories revolve around the second order sliding manifold in coordinates t, x, z. Algorithm (6.14) provides for appearance of a real sliding mode of the first order with respect to #-1. Simulation results for p = 0.01 are shown in Figures 6.5, 6.6. The chattering is obvious here.
E x a m p l e 6.13 The last actuator-like algorithm (6.14) may be modified in order to receive finite time convergence to the second order sliding mode. The aim is gained by twisting algorithms (Levantovsky 1985, 1987; Emelyanov et al, 1986a, b; Levant, 1993a)
=
-z, -5sgn~r, -sgn~,
crY>0, crd<_0,
1"1 > 1, Izl< 1, I z t < 1.
Derivative 8 has to be calculated here in real time. Having substituted the first difference of (r for &, achieves another version of the algorithm adapted for implementation:
124 1.000000 . . . .
.
.
..
"/" ]
": °.. .
,
•
.
.°
.°
• °.
,
.o , •
•
0
°°
•
: 3::: ,
"
"
I
"i "'.'" ." .... " " . . . . . . . , . .
""
,..'"',i
..
.
. :.'.'...'"'t'""" . . . . o..-
.
•
,.
-..
"l °
:
-1.000000
" , "
....
0.000000E+00
5.988000
Figure 6.6 Unstable second order sliding mode in a system with At-controller: control z(t) (values are taken at discrete times).
=
{ -z(t~), -5sgno,(t~), -sgnc~(tl),
Iz(ti)l > 1, c~(t~)A,r~> 0, Iz(t~)l < 1, c~(ti)Acq <_ O, Iz(t~)l _< 1.
Here ti < t < ti+l. This algorithm modification constitutes a second order real sliding algorithm, which means that the sliding accuracy is proportional to the measurement time interval squared. The corresponding simulation results are shown in Figures 6.7, 6.8.
E x a m p l e 6.14 Another algorithm (Emelyanov et al, 1990; Levant, 1993a) serving the same goal is the algorithm
z = -2v~sgn
~ + zl,
~
Z ~
~1 = { -sgnc~,
lzl > 1, lzl < 1.
The discrete switching modification of this algorithm also constitutes a second order real sliding algorithm. Its simulation results are shown in Figures 6.9, 6.10. Note that in the latter example only the weak regularity condition (6.2) holds. Examples 6.13 and 6.14 are representatives of large algorithmic families. Details and a number of other examples for second order sliding modes attracting in finite or infinite time may be found in (Emelyanov et al, 1986a, c, 1990; Levantovsky, 1985, 1986, 1987; Chang, 1990; Elmali and Olgaq, 1992; Sira-Ramfrez, 1992b, 1993; Levant, 1993a).
125
~y-', \ /.:
1.903470E-01
/i
f ! !
\
\
/
\ \
_/
-6.939553E-02
O.O00000E+O0
5.988000
F i g u r e 6.7 Second order sliding mode attracting in finite time: twisting algorithm. Tracking: x(t) and f(t).
3.919948E-01
;i j!
"-
i/i" i,~ " \
/
\
/
./
J
-1.077855E-01 0,000000E+00
5.988000
Figure 6.8 Second order sliding mode attracting in finite time: twisting algorithm. Control z(t).
126
1.884142E-01
,/
\
|
i I
o
\
-6.939553E-02
~.--0.000000E+00
/ 5.988000
Figure 6.9 Example 6.14: second order sliding mode attracting in finite time. Tracldng: x(t) and f(t).
3.732793E-01
I !
V ! :.
\
\
\ \
-1.077609E-01 0.000000E+00 Figure
6.10
Control z(t).
/ /
f
J
J 5.988000
Example 6.14: second order sliding mode attracting in finite time.
127
6.7 Sliding M o d e s of Order 3 and Higher Note, following the classical result by Anosov (1959), that for any ai,j, k ~ 0 t h e / - t h order sliding set Yl = Y2 = . . . . Yt = 0 in the system ~)1 =
Y2,
?)t- 1 ---- Yt, il~ = i=1 al,j yj + k s g n y l , ~]i -~- 2__,j=lai,jYj, i = t + l,•..,n, is always unstable with 1 > 3. This leads to an important conclusion. Even a stable higher order actuator may insert additional chattering into the closed dynamic system. Whenever a possibility of using actuators with r-th order dynamics (r > 2) for first order sliding mode control systems is concerned, one has to search for stable attractors of a corresponding (r + 1)-dimensional fast dynamic system or to use some special control algorithms• E x a m p l e 6.15 Continuing the example series, let now the actuator have second order dynamics: = zl,
zl + 3azl + 2a~z = 2a2u,
where # = 1 / a --+ 0. Let also If(3)(t)[ < 0.5 be true. Calculation shows that o-(3) = - f ( 3 ) ( t ) - 3c~zl - 2c~2z +
20~2U.
It is easy to check (see Theorem 6.7) that, with tt < x/~/3, there is a Filippov's solution lying on set cr = & = /~ = 0, which corresponds to the third order sliding mode. However, it is unstable according to Anosov (1959). Certainly, an approximation of an ideal regular sliding mode is achieved with # --+ 0 (Figure 6.11). However, the actuator introduces here considerable chattering (Figure 6.12). The following is the first known example of a third order sliding algorithn~ with finite convergence time as well as of a third order sliding mode being attractive in finite time at all. E x a m p l e 6.16 A third order sliding a l g o r i t h m w i t h f i n i t e c o n v e r g e n c e time Define, determining by continuity when necessary,
128
1.925830E-01
i
\ \
-7.37o699E-o2
~ ~+ J 0.000000E+00
F i g u r e 6.11 degree 2: x(t)
5.988000
Unstable third order sliding mode in a system with actuator of relative
and fCt).
9.999989E-01
r
-.
% .
-
...
•
•
....
..
:.
•
.iL"
•
',"1:..,
-..'...."
..
• ,
o
-.
.
•
• "
o
•
°
......
..... •
:'L:. 1: : : "
•
"-.- . . r
-7.211555E-01
•
•
b ",,, • I ° ° ° * * • • .
0.000000E+00
:" ,
: 'l':'.i..~.
•.. I . ?+
•~°'
**
~+
+.
..'... -1"
,. *
•o J " •
*
• :.. ::. •
•
5.988000
F i g u r e 6.12 Unstable third order sliding mode in a system with actuator of relative degree 2: actuator output z(t) (values are taken at discrete times).
129 In order not to change the notation, variable z is used below as an actual control. Introduce also an auxiliary variable zl. The following algorithm provides for finite time convergence to the third order sliding mode
£1
----- Zl, = --15sgn (/~ -- ~(cr,&)).
The proof is provided by a sequence of simple calculations. It is necessary to check that • ¢(~r, &) is a continuous piece-wise smooth function; • Functions c~,/r 5 may be taken as new coordinates. There is a first order sliding mode on the manifold
(6.15) • The corresponding sliding motion is described by equation (6.15) which provides for finite time convergence to the origin o- = / r = 0. In a similar way a finite time convergence algorithm of an arbitrary sliding order may be constructed. The discrete switching modification of this algorithm =
zi(ti),
constitutes a third order real sliding algorithm providing for the curacy being proportional to the third power of the discretization The simulation results are shown in Figures 6.13, 6.14, 6.t5. It was r = 10 -a, 10 -4, 5 . 10 -5 and the sliding precision sup Icr[ = 2.8. 10 .9 , 2.7-10 - l ° was achieved.
sliding acinterval r. taken that 10 -~, 1.9.
Note that the sliding algorithms from Examples 6.13, 6.t4, 6.16 cannot be produced by the powerful differential-geometric methods by Sira-Ramfrez. At the same time these algorithms are beyond any doubt of practical interest. One of the present authors has already successfully applied such algorithms in solving avionics problems and constructing robust differentiators (Levant, 1993b). It has to be mentioned that these algorithms also provide for much higher accuracy than the regular sliding modes (Levant, 1993a).
6.8 C o n c l u s i o n s * Higher order sliding mode definitions were formulated. • It was shown that higher order sliding modes are natural phenomena for control systems with discontinuous controllers if the relative degree of the system is more than 1 or a dynamic actuator is present.
13o
1.884106E-01
/
\ \
o/
R~J
|mE m m m
-1.608659E-01 0.000000E+00 Figure 6.13
9.980000
Third order sliding mode attracting in finite time. Tracking:
x(t)
and
/(t).
1.516734E-01
It\ i\ 0
\
/~
\
/
,
\ -1.050930E-01
\
\,j
0.000000E+00 Figure 6.14
/
/
\\
i k
\
~J 9.980000
Third order sliding mode attracting in finite time. Control z(t).
131
2.850000E-01
/'\ /
J
i
/
J
;,
-1.380000E-01 0.000000E+00 F i g u r e 6.15
9.980000
Third order sliding mode attracting in finite time. Control derivative
zl(t) = ~(t).
• A natural logic of actuator-like algorithm introduction was presented. Such algorithms also provide for the appearance of higher order sliding modes. • Stability was studied of second order sliding modes in systems with fast stable dynamic actuators of relative degree 1. • A number of examples of higher order sliding modes were listed. Among them the first example was presented of a third order sliding algorithm with finite time convergence. The discrete switching modification of this algorithm provides for the third order sliding precision with respect to the minimal switching time interval.
References Anosov, D. V., 1959, On stability of equilibrium points of relay systems, Automaticai Telemechanica (Automation and Remote Control), 2, 135-149. Chang, L. W., 1990, A MIMO sliding control with a second order sliding condition, ASME Winter Annual Meeting, paper no. 90-WA/DSC-5, Dallas, Texas. DeCarlo, R. A., Zak, S. H., Matthews, G. P., 1988, Variable structure control of nonlinear multivariable systems: a tutorial, Proceedings of the IEEE, 76, 212-232.
132 Drakunov, S. V., Utkin, V. I., 1992, Sliding mode control in dynamic systems, International Journal of Control, 55(4), 1029-1037. Elmali, H., Olga~, N., 1992, Robust output tracking control of nonlinear MIMO systems via sliding mode technique, Automatica, 28(1), 145-151. Emelyanov, S. V., 1967, Variable Structure Control Systems, Moscow, Nauka, [in Russian]. Emelyanov, S. V., 1984, Binary Systems of Automatic Control, Moscow, Institute of Control Problems, [in Russian]. Emelyanov, S. V., Korovin, S. K., 1981, Applying the principle of control by deviation to extend the set of possible feedback types, Soviet Physics, Doklady, 26(6), 562-564. Emelyanov, S. V., Korovin, S. K., Levantovsky, L. V., 1986a, Higher order sliding modes in the binary control systems, Soviet Physics, Doklady, 31(4), 291-293. Emelyanov, S. V., Korovin, S. K., Levantovsky, L. V., 1986b, Second order sliding modes in controlling uncertain systems, Soviet Journal of Computer and System Science, 24(4), 63-68. Emelyanov, S. V., Korovin, S. K., Levantovsky, L. V., 1986c, Drift algorithm in control of uncertain processes, Problems of Control and Information Theory, 15(6), 425-438. Emelyanov, S. V., Korovin, S. K., Levantovsky, L. V., 1990, A new class of second order sliding algorithms, Mathematical Modeling, 2(3), 89-100, [in Russian]. Filippov, A. F., 1960, Differential Equations with Discontinuous Right-Hand Side, Mathematical Sbornik, 51(1), 99-128 [in Russian]. Filippov, A. F., 1988, Differential Equations with Discontinuous Right-Hand Side, Kluwer, Dordrecht, the Netherlands. Fridman, L. M., 1983, An analogue of Tichonov's theorem for one kind of discontinuous singularly perturbed systems, in Approximate Methods of Differential Equations Research and Applications, Kuibyshev University Press, 103-109, [in Russian]. Fridman, L. M., 1985, On robustness of sliding mode systems with discontinuous control function, Automaticai Tetemechanica (Automation and Remote Control), 5, t72-175 [in Russian]. Fridman, L. M., 1986, Singular extension of the definition of discontinuous systems, Differentialnye uravnenija (Differential equations), 8, 1461-1463, [in Russian]. Fridman, L. M., 1990, Singular extension of the definition of discontinuous systems and stability, Differential Equations, 10, 1307-1312. Fridman, L. M., 1991, Sliding mode control system decomposition, Proceedings European Control Conference, Grenoble, 1, t3-17. Fridman, L. M., 1993, Stability of motions in singularly perturbed discontinuous control systems, Proceedings IFAC World Conference, Sydney, 4, 367370. Fridman, L. M., Shustin, E., Fridman, E., 1992, Steady modes in discontinuous equations with time delay, Proceedings IEEE Workshop Variable Structure and Lyapunov Control of Uncertain Dynamic Systems, Sheffield, 65-70.
Isidori, A., 1989, Nonlinear Control Systems, second edition, Springer Verlag, New York. Itkis, U., 1976, Control Systems of Variable Structure, Wiley, New York. Levant, A., (Levantovsky, L. V.) 1993a, Sliding order and sliding accuracy in sliding mode control, International Journal of Control, 58(6), 1247-1263. Levant, A., 1993b, Higher order sliding modes, In International Conference on Control Theory and Its Applications: Scientific Program and Abstracts, Kibbutz Maale HaChamisha, Israel, 84-85. Levantovsky, L. V., 1985, Second order sliding algorithms: their realization, in Dynamics of Heterogeneous Systems, Institute for System Studies, Moscow, 32-43, [in Russian]. Levantovsky, L. V., 1986, Sliding modes with continuous control, in Proceedings All-Union Scientific-Practical Seminar on Application Experience of Distributed Systems, I. Novokuznezk, Moscow, 79-80, [in Russian]. Levantovsky, L. V., 1987, Higher order sliding modes in the systems with continuous control signal, in Dynamics of Heterogeneous System, Institute tbr System Studies, Moscow, 77-82, [in Russian]. Saksena, V. R., O'Reilly, J., Kokotovid, P. V., 1984, Singular perturbations and time-scale methods in control theory: survey 1976-1983, Automatica, 20(3), 273-293. Sira-Ramfrez, H., 1992a, On the sliding mode control of nonlinear systems, Systems and Control Letters, 19, 303-312. Sira-Ramfrez, H., 1992b, Dynamical sliding mode control strategies in the regulation of nonlinear chemical processes, International Journal of Control, 56(1), 1-21. Sira-Ramirez, H., 1993, On the dynamical sliding mode control of nonlinear systems, International Journal of Control, 57(5), 1039-1061. Slotine, J.-J. E., Sastry, S. S., 1983, Tracking control of non-linear systems using sliding surfaces, with applications to robot manipulators, International Journal of Control, 38,465-492. Su, W.-C., Drakunov, S., Ozguner, U., 1994, Implementation of variable structure control for sampled-data systems, Proceedings IEEE Workshop on Robust Control via Variable Structure 8J Lyapunov Techniques, Benevento, Italy, 166-173. Utkin, V. I., 1977, Variable structure systems with sliding modes: a survey, IEEE Transactions on Automatic Control, 22, 212-222. Utkin, V. I., 1981, Sliding Modes in Optimization and Control Problems, Nauka, Moscow, [in Russian]. Utkin, V. I.,1992, Sliding Modes in Optimization and Control Problems, Springei Verlag, New York. Young, K. K. D., Kokotovid, P. V., Utkin, V. I., 1977, A singular perturbation analysis of high-gain feedbazk systems, tEEE Transactions on Automatic Control, 22, 931-938. Zinober, A. S. I., editor, 1990, Deterministic Control of Uncertain Systems, Peter Peregrinus, London.
0
An A d a p t i v e S e r v o m e c h a n i s m for a Class of U n c e r t a i n Nonlinear Systems Encompassing Actuator Hysteresis
Eugene P. Ryan 7.1 Introduction Approaches to adaptive control can be classified broadly into two groups: those that--either implicitly or explicitly--exhibit some aspect of identification or estimation of the process to be controlled, and those that seek only to control. Here, we adopt the latter approach: see the monograph (Ilchmann, 1993) tbr a comprehensive overview. In common with many of its precursors in the area (see, for example, Byrnes and Willems, 1984; Corless, 1991; Corless and Leitmann, 1984; Cotless and Ryan, 1993; Helmke, Pr/itzel-Wolters and Schmid, 1990; Ilchrnann and Logemann, 1992; Ilchmann and Owens, 1990; Logemann and Zwart, 1991; MSartensson, 1985, 1986, 1987, 1990, 1991; Miller and Davison, 1989; Morse, 1984, 1985; Nussbaum, 1983; Ryan, 1990, 1991a, 1991b, 1993, 1995; Townley, 1995; Willems and Byrnes, 1984), the chapter is primarily concerned with demonstrating the existence--under weak assumptions--of a single controller that achieves some prescribed control objective for every member of the given class. This we do by explicit construction of one such universal controller. The main contribution is to demonstrate that (uncertain) input nonlinearities of a fairly general nature--encompassing, for example, hysteresis and dead-zone effects--can be tolerated (a treatment of output dead zones, via a different approach, is contained in (Tao and Kokotovid, 1993)). The approach adopted here is closely akin to that of (Ryan, 1992): however, we emphasize the fundamental distinction that, in the present paper, the class $ of systems is considerably more general (by virtue of the allowable input nonlinearities 9). In (Ilchmann and Ryan, 1994), a problem of `k-tracking is addressed: there, the distinguishing features are (a) the treatment of (bounded) output feedback noise and (b) the restriction to c o n t i n u o u s feedback strategies, parameterized by `k > 0 (a prescribed tracking error tolerance), which ensure that the tracking error e(t) = y ( t ) - r ( t ) is asymptotic to the prescribed interval [-`k, `k] CIR. In effect, in (Ilchmann and Ryan, 1994), continuous feedbacks are constructed that guarantee approximate tracking with prescribed asymptotic error bounds. Nonlinear actuator characteristics--of the type considered in the present chapter-can be handled within the latter framework of c o n t i n u o u s feedback strategies for ,k-tracking: these results are reported in (Ryan, 1994). In essence, the present
136 chapter can be viewed as a companion to the latter paper insofar as it considers exact (asymptotic) tracking in place of asymptotic A-tracking but at the expense of discontinuous feedback in place of continuous feedback. The mathematical framework is that of differential inclusions within which the analysis is essentially Lyapunov based.
7.2
Formulation
We consider a class $ of nonlinearly-perturbed, single-input, single-output, linear systems with nonlinear actuator characteristics: the general member, which we denote by ~ = (A, b, c, d, f, g), is given by
(v(t) = A w ( t ) + b[f(t, w(t) ) + v(t)] + d(t, w(t) ), w ( t o ) = w ° ~ IR k,
v(t) = g(t, u(O, ~t(.)), y(t) = c~(t),
(7.1)
~,(t) e ~a,
y(t) ~ ~ .
The output y(t) is available for feedback. The control signal u(t) drives an actuator modelled by g. The actuator may be a device with memory, that is, it may depend on the history ut(.) : s ~4 u(s), s <_ t, of the control signal, as is the case with hysteresis. For this class, we address a servo problem (with internal boundedness) of (7~, S)-universal tracking of reference signals r(.) of class 7~ (to be specified), posed as follows. Does there exist an adaptive output feedback strategy
,u(t) = u(k(0,y(t),~(t)), k(t) = K(k(0, y(t), ,(0),
"1
(7.2)
k(to) = k ° ~ Ia,
which guarantees that, for every (r(.), ,U) E 7~ × S and for each (to, w °, k °) E IR × ]R" x ]R, every solution (w(.), k(.)) of the initial-value problem (7.1), (7.2) has maximal interval of existence [t0, c~), is bounded, and is such that the output y(.) - cw(.) asymptotically tracks the reference signal r(.) in the sense that the error e(t) = y(t) - r(t) tends, as t -4 ¢¢, to zero? Whether or not the above problem is well posed evidently depends on the nature of the classes 7~ and $. 7.2.1
C l a s s 7~ o f R e f e r e n c e
Signals
As in (Ryan, 1992), for the class 7~ of reference signals we take those functions p : IR --4 IR that are absolutely continuous on compact subintervals of IR and are bounded with essentially bounded derivative p E L ~ (IR). Thus, we identify
137
u
,
y
ZES
(7~, S)-universal strategy
Figure 7.1
l~
rET~
General framework
7¢ as the Sobolev space WI,°°(IR), which we equip with the norm lI HI,~ given by
Ilpll,,o~ where
II I1~
=
Ilpllo~ + II/,11oo
denotes the usual norm on Z ~ ( m ) .
This class of reference signals is broad and includes, for example, scalar outputs from any stable linear system driven by L ~ inputs. However, as will be seen, our control strategy need not have recourse to dynamical systems (linear or otherwise) which may replicate the reference signals: in this sense, an internal model principle (Wonham, 1979) is not invoked in the chapter.
7.2.2
System
Class S
The class $ of systems 22 = (A, b, c, d, f, g) is implicitly defined through Assumptions 7.1-7.5 below. A s s u m p t i o n 7.1 /3 := cb is non-zero. A s s u m p t i o n 7.2 The linear system £, defined by the triple (A, b, c), satisfies the minimum phase condition: det
sI-A c
b ] 0 ] # 0
'¢s E C+
that is, £ is stabilizable, detectable and has no zeros in the closed right-halJ complex plane C+. Simply stated, Assumptions 7.1 and 7.2 imply that, for each system ~ = (A, b, c, d, f,g) E S, the triple (A, b, c) defines a linear, relative-degree-one, minimum-phase system system L; (with high-frequency gain of unknown sign). Next, we characterize the class of nonlinearities d (unmatched uncertainty) and f (matched uncertainty).
138
A s s u m p t i o n 7.3 d : 1R × IR n --+ IR n is a Carathdodory function and has the property that, f o r some scalar 6, tld(t, w)ll _< ~[1 + Ic~t] f o r almost all t and all w. A s s u m p t i o n 7.4 f : IR × lR n -+ IR is a Carathdodory function and has the property that, f o r some scalar ~ and known continuous function ¢ : IR n -+ [0, c¢), If(t, w)l < ~[llwll + ¢(cw)] f o r almost all t and all w. Thus, the disturbance d is "affinely" bounded, modulo an unknown scale factor 6 _> 0, by the function w ~-~ 611 + [cw[]. The function f is bounded, modulo an unknown scale factor a > 0, by the map w ~+ Ilwll + ¢(cw), where ¢ is known and continuous. This permits f to exhibit quite general dependence on the output variable: for example, if it is known only that f has polynomial (of unknown degree) dependence on y with (essentially) bounded t-dependent coefficients, then ¢ : y ~-~ exp(lyl) suffices. Finally, we make explicit our class of input actuator nonlinearities. We assume the existence of a non-empty-set-valued map u ~+ G(u) C IR satisfying Assumpion 7.5 below and such that every actuator characteristic is contained in the graph of G in the following sense: for all (t,~) E IR2 and every u(.) : lR --+ IR with u(t) = ~, e where, as before, at(.) denotes the restriction of u(.) to (-o%t]. A s s u m p t i o n 7.5 G is a continuous map from IR to the compact intervals o] IR with the property that, for some scalars F > 0 and 72 >_ 71 > O,
sgn( )G(e) c
c
e
I - r , v].
For example, Figure 7.2 below illustrates the graph (viz. the region enclosed by, and including, the uppermost and lowermost traces) of one such map G with sgn(~)G(~) = [71 [~[, 7~]~[] for all I~[ > F. Remarks. (i) In the present context of non-empty compact-valued maps from IR to its power set 2 ~t, G is continuous if it is both upper and lower semicontinuous at each ~ E IR: G is upper semicontinuous at ~ if, for every open interval I1 containing G(~), there exists an open interval/2 containing ~ such that G(I2) C I1; G is lower semicontinuous at ~ if, for every sequence ( ~ ) converging to and for every 7 E G(~), there exists a sequence (Tk) with 7k e G(~k) and % -+ 7 as k - + oo. (ii) The only requirement on the input nonlinearity g is that its characteristic is contained in the graph of one such (unknown) continuous G having the property that, for all sufficiently large I~t, the set sgn(~)G(~) is contained in the interval [71 I~1,721~1]:thus, the "positive and negative tails" of the actuator characteristic are required to be sector-bounded (with unknown bounds) in the first and third quadrants, respectively.
139
~ d i e n t
71
~Grap'h(G)
/
Figure 7.2 Graph(G) containing an admissible actuator characteristic with hysteresis and dead zone
(iii) We stress that the only a p r i o r i structural information available to the controller is knowledge of some continuous function ¢ assuring Assumption 7.4. tn particular, the controller is oblivious of the following parameters: the state dimension n, the non-zero "high-frequency gain" fl, the multiplier 5 in the bound on the disturbance d; the multiplier a in the bound on f, and the scalars F, 71, 72 associated with the unknown set-valued map G. We can now embed system (7.1) in the following differential inclusion J:(t) - A x ( t ) - bf(t, x(t)) - d(t, x(t)) E b G ( u ( t ) ) , ~(to) = x °,
(7.3)
y(t) = ~x(t)
the embedding being in the sense that every solution of (7.1) is a f o r t i o r i a solution of (7.3). Henceforth, the generalized model (7.3) is the focus of our attention: qualitative properties of solutions of (7.3) are inherited by solutions of every system of type (7.1) embedded in (7.3). We seek an adaptive strategy such that, for each (to, x °, k °) and reference signal r E 7~, asymptotic tracking of r is achieved by every solution of (7.3).
7.3 The Adaptive Strategy 7.3.1
The Servomechanism
Let u : IR --+ IR be any continuous function with the property that, for all 7 = (70, 7a, %) E IR3 with 70, %, 75 > 0, the associated function
-(~) > 70 t-(¢)t < 70 7b-(~), -(~) _<-7o
{ 7o.(~),
(7.4)
140
has the "Nussbaum properties":
f
v~ =
+co,
(7.5a)
lim~flf0~
=
-co.
(7.5b)
lim sup I
The proposed control strategy is given formally by (7.2) with
U(k, y, r) h'(k,y,r)
:=
.(k)[y-
r + (1 + ¢(y))sgn(y - r)],
:=
ly-rl[ly-rl+l+¢(y)].
Noting that the strategy is of discontinuous feedback form, for arbitrary r E 7~ and writing e(t) = y(t) - r(t) we interpret the control in the following generalized set-valued sense ~,(t) e O(t, e(t), k(t)) k(t) = l~'(t, e(t)),
k(to) = k °
f
(7.6)
with 0(t, e, k) := {.(k)[e + (t + ¢(e + r(t)))ul u e ¢(e)} C m, {+U,
¢:e~+
[-1, 1], e = 0 {-1},
k ( t , e) := lel[lel + 7.3.2
Scaling-Invariant
e> 0
i+
Nussbaum
e< I
¢(e + r(t))]. Functions
Whether or not the proposed servomechanism is well-defined clearly depends on the existence of functions ~ having associated functions (7.4) with properties (7.5): adopting the terminology of (Logemann and Owens, 1988), we refer to such functions as scaling-invariant. The next proposition, which is a straightforward modification (to include the role of the parameter 70) of the example in Section 4.2 of (Logemann and Owens, 1988), asserts that such scaling-invariant functions do indeed exist. P r o p o s i t i o n 7.6 Let 3' = (70,Ta,Tb) E IR3, with 70,7a,% > 0, be arbitrary and let v : 1R --+ IR be the function o ~+
~(o) := exp(Ob cos(}~O).
Then the associated function v~ : IR --+ IR, defined by (7.4), has the Nussbaum properties (7. 5).
141
7.4
Analysis
Let 7" (the reference signal to be tracked) be any fixed member of 7~. We first introduce a convenient state transformation.
7.4.1
Coordinate
Transformation
Let linear L : Et n -+ ~ n - 1 be such that kerL = imb. Under the coordinate transformation T : w ~ (z, y) = (Lw, cw) and writing e(t) = y(t) - r(t) (the tracking error), system (7.3) can be expressed as
i(t) = Alz(t) + A2e(t) + dl(t, z(t), e(t)),
]
~(t) - A3z(t) - A4e(t) - ~f2 (t, z(t), e(t) ) - d2(t, z(t), e(t)) E t3 G(u(t)),
(7.7)
(z(to), e(to)) = (z °, t°), where di : (t, z, e) ~+ L d ( t , T - l ( z , e + r(t))) + A2r(t), d2 : (t,z,e) ~+ c d ( t , T - l ( z , e + r(t))) - ~(t) + A4r(t), f ( t , T - l ( z , e + r(t))). A : (t,z,e) Note that, by Assumption 7.3,
IIdl(t,z,e)It <_ ~111+ I¢l], IId2(t, ~, z)tl <_ 5~[~ ÷ I~l], with
51 := 5llLll[1 + I[rlLoo]+ [[A2H[Lrlloo, ~ := ~ttc]l[x + IkIJoo] + (i + I1A411)IkNl,oo 7.4.2
The
Feedback-Controlled
System
Combining (7.6) and (7.7), the problem may now be reduced to an analysis of the asymptotic behaviour of solutions of the initial-value problem:
2(t) e F(t, x(t)), x ( t 0 ) = x ° = ( z °,e °,k ° ) E I R N,
I N:=n+l,
with the set-valued map (t, X) ~- (t, Z, e, ]¢) ~ f ( t , x) ~ I~ N defined by
(7.8)
142
F(t,x)
:=
Fl(z,e) x F2(t,x) x Fa(t,e),
Fl(z,e) F~(t,x)
:= :=
{Alz+A2e}+ {Aaz + A4e}
{~1EIR~-ll IIGII <51[1+1el]},
+ {el • IR I le, I _< ~[ll~-*(z, e + r(t))rl + ¢(e + r(t))]} + {e2 • IRI l~21 _< a211+ I~l]} + OC(O(t, e, k)), Fa(t,x)
:=
{k(t,e)}.
(Notation: for sets $1, $2 • IRP and scalar #, $1 + $2 denotes the set {sl + s21 sl • $1, s~ • S~} C IRP, and pS1 = {psi s • $1}.) Continuity of u, ¢, r and upper semicontinuity of ~b, together with compactness of its values, ensure that U is an upper semicontinuous map from IRa to the compact subsets of IR. Furthermore, continuity of G ensures that the composition
G o O: (t, e, k) ~ G((_/(t, e, k)) := U~eO(t,
~(0(t, ~, k), [-1,11) = a(O(t, ~, k)) is connected. We may now deduce that F is an upper semicontinuous map from IR x IRN to the non-empty, convex, compact subsets of IRN. Therefore, by Theorein 2.1.3 of (Aubin and Cellina, 1984) (see, also, Filippov, 1988), for each (to, x °) • IR x IRN, the initial value problem (7.8) has a solution (that is, an absolutely continuous function x(.) : [to, w) --+ IRN, with x(t0) = x °, satisfying the differential inclusion almost everywhere). Moreover, every solution has a maximal extension (Ryan, 1990). Before proving the main result, we state a technical lemma (a proof of which is contained in (Ryan, 1994)). L e m m a 7.7 Let (r,z r) • IR x IRn-l, w > r, I = Iv, w) and let-Bn_l(r) denote the closed ball of radius r > 0 centred at the origin in lit n-1. Suppose that 0 : I --+ IR is continuous, and R > O. If z : I --+ IRa-1 solves the initial-value problem
~ ( t ) - A l z ( t ) - A20(t) e - f i n _ l ( R [ l + lO(t)l]),
z(r) = z r,
then there exists a scalar # such that
f
iO(s)lllz(s)ll ds < p
l
10(s)l[1+ 10(s)l] ds
Vt • I.
143
7.4.3 T h e M a i n R e s u l t The main result may be paraphrased as follows: every maximal solution of the initial-value problem (7.8) (recall that at least one such solution exists) is bounded and so has interval of existence [t0, ec)) (finite escape times do not exist), the monotone increasing component k(.) has a finite limit and tile component e(.) (the tracking error) tends asymptotically to zero. T h e o r e m 7.8 Let x(-) = ((z(.),e(.),k(.)) : [t0,w) -+ IR g be a maximal solution of the initial-value problem (7.8). Then, i) W z
CX) ;
(ii) x(.) is bounded; (iii) lirnt~oo k(t) exists and is finite," (iv) e(t) ~ 0 as t --+ oc. Proof.
For almost all t E [to, w),
e(t)~(t)
<_ max{e(t)~'[ ¢ E F2(t, x(t))} < le(t)l(llA3tl + aN[T-tH)]lz(t)][ + le(t)f [([[A4[I + a/3tIT-]It)[e(t)[ + a/3¢(e(t) + r(t))
+ ZliT-11111rll ] + le(t)lS~[1 + [e(t)[] + max{e(t)¢[ ¢ e/3a(~r(t, e(t), k(t)))}. Writing el := ][A3[[-b aj3[[T-1][,
c2 :-----]IA4[[ q- a/JilT-l[[[1 q- [[r][oo] + ~3 q- ~2
we see that, for almost all t C [t0,w), e(t)t~(t)
5
C1]e(t) lHz(t)ll +
c2 ]e(t)] lie(t)] + ] + ¢(e(t) + r(t))]
+ max{e(t)¢ I ¢ e/3C(U (t, e(t), k(t))) }.
(7.9)
With a view to estimating the third term on the right hand side of the above inequality, we consider separately three exhaustive cases.
Case (a): ~,(k(t)) > F.
Define ~/a : =
{ ~, if/3<0 "~2, if/3 > 0
144 We will show that max{e(t)fft ~ E/3G(U(t,e(tl,k(t))} 1-
(7.10)
-i
If e(t) = 0, then (7.10) is obviously true. So suppose e(t) ~ O. Then (;(t, e(t), k(t)) is singleton-valued and, with slight abuse of notation, we will denote the modulus of its single element (non-zero) by IU(t, e(t), k(t))l and its sign by sgn(/_](t, e(t), k(t))). Clearly,
F < u(k(t)) <_ u(k(t))[fe(t)l + 1 + ¢(e(t) + r(t))] =
let(t, e(t), k(t))l.
Also
sgn(e(t)) = sgn(@, e(t), k(t))). Therefore, invoking Assumption 7.5, 711b'(t, e(t), k(t)) l < sgn(e(t))/3-1~ < 721U(t, e(t), k(t)))l for all ( E/3G(U(t, e(t), k(t))). Thus,
7I le(t))llU(t, e(t), k(t))l <_/3-%(t)~ <_ 721e(t)llU(t, e(t), k(t))h for all ~ E/3G(U(t, e(t), k(t))), from which (7.10) immediately follows. Case (b): u(k(t)) < - F .
Define 7b
/
72, 7i,
if/3 < 0 if/3 > 0
We claim that
max{e(t)¢l ¢ e ZV(~(t, e(~), k(t)))} _< ZTb ~(k(t))le(t)l [le(t)l + 1 + ~(e(t) + r(t))].
(7.11)
To show this, we argue in a manner similar to Case (a). Again, if e(t) = O, then (7.11) is self-evident. So suppose e(t) 7£ O. Then
F < -.(k(t)) < -.(k(t))[le(t)l + 1 + ¢(e(t) + r(t))] =
I~](t,e(t),
and sgn(e(t)) = -sgn(U(t, e(t), k(t))). Therefore, again invoking Assumption 7.5,
-7210(t, e(t), k(t))t < sgn(e(t))fl-lff < -71 tU(t, e(t), k(t))l for all ¢ E flG(U(t, e(t), k(t))), from which (7.11) may be deduced.
k(t))l
145 By upper semicontinuity of G and compactness C a s e (c): [~(k(t)) I < I". of its values, the image G([-F, F]) of the compact interval [ - F , F] is compact. Define
7* := max{Kil ¢ E O([-_r, F])} and write -~ := IZlb* + 7~r]. We will show that
(7.12) Again, if e(t) = 0, then (7.12) is self-evident. So suppose e(t) # O. If I/)(t, e(t), k(t)) I > F, then, by Assumption 7.5, we may conclude
Iq _< "721/~llU(t, e(t), k(t))l < 72]~lF[]e(t)i + 1 + ¢(e(t) + r(t))l for all ~; ~/~ C(D(t, ~(t), k(t)). If
tU(t,e(t),
k(t))l < F, then
le(t)ll~l <_ 7* Inile(t)l <_ 7* I/?lle(t)l[ie(t)l + 1 + ¢(e(t) + r(t))] for all ( E ,8 G(U(t, e(t), k(t)). Inequality (7.12) now follows. With 70 := F and the function ~ defined by (7.4), combining (7.10), (7.11) and (7.12) we now have
for all t E [t0,w), which, together with (7.9), yields
e(t)~(t) <_ c, ie(t)lllz(t)ll + [c2 + # + 9.~(k(t))]/~(t)
(7.13)
for almost all t E [to, w). We now focus on the first term on the right in (7.13). By (7.8), we have ~(t)
-
A,z(t)
-
A2~(t) E {¢, e ~"-'t t1¢*II < ~,[1 + le(t)l]}.
Therefore, by the lemma (with 0(-) = e(.) and R = 5t) we may conclude the existence of a scalar # such that
/t le(~)lllz(~)llds /,i le(~)l[1 + le(~)l]d~ <.
0
for all t E [to, w). Now,
le(~)l[1 + I~(~)l]
146
k(t) 0 < ½ 2(t) < 1
t j + Jk(r) [c3 +
(7.14)
for all r, t E [to, w) with t > r. Seeking a contradiction, suppose the monotone function k(.) is unbounded. Then there exists r E [t0,w) such that k(r) > 1. Dividing by k(t) >_ k(r) > 1 in (7.14), we may conclude that
Z f k ( t ) v~(~) dt~, 0 < constant + k - ~ Sk(T)
for all t e [r,w),
which, on taking limit inferior as t ~ w (k(t) -+ cx~), contradicts one or the other of properties (7.5) (specifically, if ~ < 0, then (7.5b) is contradicted; if ~ > 0, then (7.5a) is contradicted). Therefore, k(.) is bounded. By boundedness of k(-) and (7.14), we may conclude that e(.) is bounded. Since i(t) E Fl(z(t),e(t)) = { A l z ( t ) + A2e(t)} + Bn-l(5111 + le(t)l]) almost everywhere, we have
ltz(t)tl
<
llexp(Al(t-to))HHz°II
+
Ilexp(Al(t-s))ll[(llA211+ )l (s)l+ t]ds
for all t >_ to. Since o'(-.41) C e _ and e(.) is bounded, boundedness of z(.) follows immediately. We have now shown that the solution z : [t0,w) -4 IRN is bounded, and so w = oz. This proves assertions (i) and (ii). Assertion (iii) is an immediate consequence of boundedness and monotonicity of k(.). It remains to prove assertion (iv). First observe that, since ]~(t) _> e2(t) and k(.) is bounded, we may conclude that e(.) • L2([t0, c¢)). Secondly, by boundedness of r(.), x(.) and upper semicontinuity of F together with compactness of its values, there exists a compact set C such that F(t, z(t)) C C for all t • [t0,oc). Therefore, ~(.) • L ~ ( [ t 0 , ~ ) ) . It now follows (see, for example, Lemma 2 in (ayan, 1992)) that e(t) -+ 0 as t --+ c~.
References Aubin, J-P., Cellina, A., 1984, Differential Inclusions, Springer-Verlag, BerlinNew York. Byrnes, C. I., Willems, J. C., 1984, Adaptive stabilization ofmultivariablelinear systems, Proceedings IEEE Conference on Decision and Control, 1574-1577.
14{
Corless, M., 1991, Simple adaptive controllers for systems which are stabilizable via high gain feedback, IMA Journal of Mathematical Control and Information, 8,379-387. Cortess, M., Leitmann, G. 1984, Adaptive control for uncertain dynamical systems, in Blaqui~re, A., Leitmann, G., editors, Dynamical Systems and Microphysics: Control Theory and Mechanics, Academic Press, New York. Corless, M., Ryan, E. P., 1993, Adaptive control of a class of nonlinearly perturbed linear systems of relative degree two, Systems and Control Letters, 21, 59-64. Filippov, A. F., 1988, Differential Equations with Discontinuous Righthand Sides, Kluwer Academic Publishers, Dordrecht. Helmke, U., Priitzel-Wolters, D., Schmid, S., 1990, Adaptive tracking for scalar minimum phase systems, in ttinrichsen, D., Mgrtensson, B., editors, Control of Uncertain Systems, Birkhguser, Boston. Ilchmann, A., 1993, Non-Identifier-Based High-Gain Adaptive Control, SpringerVerlag, Berlin. Ilchmann, A., Logemann, H., 1992, High-gain adaptive stabilization of multiw~riable linear systems--revisited, Systems and Control Letters, 18,355-364. Ilchmann, A., Owens, D. H., 1990, Adaptive stabilization with exponential decay, Systems and Control Letters, 14, 437-443. Ilchmann, A., Ryan, E. P., 1994, Universal h-tracking for nonlinearly-perturbed systems in the presence of noise, (with A. Ilchmann), Automatica, 30, 337346. Logemann, H., Owens, D. H., 1988, Input-output theory of high-gain adaptive stabilization of infinite-dimensional systems with non-linearities, International Journal of Adaptive Control and Signal Processing, 2, 193-216. Logemann, H., Zwart, H., 1991, Some remarks on adaptive stabilization of infinite dimensional systems, Systems and Control Letters, 16, 199-207. Mgrtensson, B., 1985, The order of any stabilizing regulator is sufficient a priori information for adaptive stabilization, Systems and Control Letters, 6, 87-91. Mgrtensson, B., 1986, Adaptive Stabilization, PhD Thesis, Institute of Technology, Lund, Sweden. Mgrtensson, B., 1987, Adaptive stabilization of multivariable linear systems, Contemporary Mathematics, 68, 191-225. Mgrtensson, B., 1990, Remarks on adaptive stabilization of first order nonlinear systems, Systems and Control Letters, 14, 1-7. Mgtrtensson, B., 1991, The umnixing problem, IMA Journal of Mathematical Control and Information, 8, 367-377. Miller, D. E., Davison, E. J., 1989, An adaptive controller which provides Lyapunov stability, IEEE Transactions on Automatic Control, 34, 599-609. Morse, A. S., 1984, New directions in parameter adaptive control, Proceedings IEEE Conference on Decision and Control, 1566-1568. Morse, A. S., 1985, A Three Dimensional Universal Controller for the Adaptive Stabilization of Any Strictly Proper Minimum Phase System with Relative Degree Not Exceeding Two, IEEE Transactions on Automatic Control, 30, 1188-1191.
148 Nussbaum, R. D., 1983, Some remarks on a conjecture in parameter adaptive control, Systems and Control Letters, 3, 243-246. Ryan, E. P., 1990, Discontinuous feedback and universal adaptive stabilization, in Hinrichsen, D., M£rtensson, B., editors, Control of Uncertain Systems, Birkh~user, Basel-Boston. Ryan, E. P., 1991a, A universal adaptive stabilizer for a class of nonlinear systems, Systems and Control Letters, 16, 209-218. Ryan, E. P., 1991b, Finite-time stabilization of uncertain nonlinear planar systems, Dynamics and Control, 1, 83-94. Ryan, E. P., 1992, Universal Wl'°°-tracking for a class of nonlinear systems, Systems and Control Letters, 18, 201-210. Ryan, E. P., 1993, Adaptive stabilization of multi-input nonlinear systems, International Journal of Robust and Nonlinear Control, 3, 169-181. Ryan, E. P., 1994, A nonlinear universal servomechanism, IEEE Transactions on Automatic Control, 39, 753-761. Ryan, E. P., 1995, Universal stabilization of a class of nonlinear systems with homogeneous vector fields, Systems and Control Letters, 26, 177-184. Tao, G., Kokotovi6, P. V., 1993, Adaptive control of plants with unknown output dead-zones, Proceedings IFAC World Conference, Sidney, 6, 7-10. Townley, S., 1995, Simple adaptive stabilization of output feedback stabilizable distributed parameter systems, Dynamics and Control, 5, 107-123. Willems, J. C., Byrnes, C. I., 1984, Global adaptive stabilization in the absence of information on the sign of the high frequency gain, in Lecture Notes in Control and Information Sciences, 62, Springer-Verlag, Berlin-New York, 49-57. Wonham, W. M., 1979, Linear Multivariable Control: a Geometric Approach, Springer-Verlag, Berlin-New York.
. A N e w C l a s s o f I d e n t i f i e r s for Robust Parameter Identification a n d C o n t r o l in U n c e r t a i n S y s t e m s Tamer 8.1
Ba~ar, Garry Didinsky
Introduction
and Zigang Pan
and Problem
Formulation
Consider the linearly parametrized state dynamics:
=
A(x,u)O+b(x,u)+w,
x(0) = xo, t ~ T : = [ 0 , ~ ) ,
(8.1)
where 0 is an r-dimensional vector of unknown constant parameters (to be identified), x is the n-dimensional state vector, with Xo E Ill~ being an unknown initial state, u is a known input vector of dimension p, possibly governed by a controller policy , and w is an n-dimensional unknown £2 disturbance (written as, w E W). The (n x r)-dimensional matrix d(x, u) and the n-dimensional vector b(x, u) are jointly continuous in their two arguments, as well as in the time variable t, which we suppress in this formulation. Our first goal is to obtain a "good" estimate of 0 at each time t, denoted by 0(t), using the available information up to that time. This estimate should yield a good performance in the presence of the unknown disturbance w E W, in the sense that it should converge to the true value of the parameter vector, i.e.,
lim 0(t) = 0
t-+oo
(8.2)
and also show good transient behavior. Convergence to the true value may not always be possible due to tack of excitation of the parameters, and hence, we will also prescribe a "reasonable" persistency of excitation (PE) condition that will guarantee convergence in (8.2). The estimate 0 alluded to above will be determined by an estimator that uses the state x and the input u. The estimator will be denoted at time t by 5(t, ~(t)), where o(t) = {(z(s), u(s)), Vs <_ t} is the information available for this purpose. We let 5 E z2, where A is a class of admissible functions satisfying the standard requirements of measurability, adaptability (to the given input fields), etc. In accordance with our objective delineated above (asymptotic convergence to true parameter value, good transient behavior, insensitivity to unknown disturbances), we introduce a performance index that measures worst-case attenuation from additive disturbance and error in the initial estimate of 0 to the estimation error over an interval of interest. Formulating this in an £2 setting, we have (with 0(t) = 5(t, ~(t))):
150 (
L(5) =
sup
~
/
lie - OIIQ( )
(8a)
(ll lp + 10-
where I1' tin denotes an appropriate dimensional £2 norm, weighted by R, and I" IP denotes a Euclidean norm, weighted by P. We assume that Q(.) _> 0, and Qo > 0.0o is a known "bias" term which stands for some initial estimate of 9. Without any loss of generality, Q will be taken to be a matrix-valued function of (t, x[0,t], uio,t]). Our first objective can now be rephrased as finding an estimator that delivers a performance arbitrarily close to the minimum value of L(5), i.e., if 7* := inf L(5) 5Ez~
(8.4)
then we seek a 5~ E A such that L(5~) < 7 for 7 > 7". What we have here is in fact a differential game with 5 the minimizer, and (x0, t, w[0,~]) the maximizer and with kernel given inside the braces in (8.3). The disturbance attenuation level 7* is the upper value of such a game. Following Ba~ar and Bernhard (1995) for a similar set-up that arises in H a - o p t i m a l control, we now associate a soft-constrained cost function with this differential game: & ( 5 ; X o , Oo, wto,oo))
=
// -
- 721w(t)l 72(I0
-
2] dt
0O1 o),
(8.5)
which is parametrized by 7. The level 7" as defined by (8.4) is the smallest value of 7 > 0 for which the auxiliary differential game just defined in which 5 is still the minimizer and the triple (xo, O, w[0,~)) the maximizer, and Jr is the kernel, has upper value equal to zero. Toward defining the optimality of an estimator in terms of the auxiliary differential game defined by (8.5), let us first introduce the following terminology: For a fixed value of 7 > 0, "an estimator 5 is cost-bounding" if it satisfies the finiteness condition xo
sup Jr (5; Xo, 0o, w[0,~)) < c~. ~Oo,W[o,o¢)
Assuming that A has a non-empty subclass of cost-bounding estimators, we will say that an estimator (~n E A is minimax for a given attenuation level 7, if inf sup Jr(5; xo, 9, w[o,oo)) = sup J~(~r; xo, O, w[0,oo)). 5EZ~ Xo,8o,W[o,~) Xo,Oo,W[o,o,,) A more general objective can now be phrased as computing the least value of 7 > 0, denoted 7", such that Jr has a cost-bounding solution, and to find a minimax estimator policy, 5r for each 7 > 7*- Clearly the policy 5* defined earlier by (8.4) is given (whenever it exists) by lim%r. 5~ = 5*.
[bl.
We will obtain the solution to the identification problem formulated above (in a differential game setting) in several steps, all discussed in Section 8.2 of the chapter. First we enlarge the information set of the estimator by also including the derivative of the current value of the state, which yields a lower bound for the attainable disturbance attenuation level of the original problem (Section 8.2.1). Second, we modify the information set of the estimator in a different direction, by perturbing the system state by small unknown additive noise, and making available to the estimator only this noise-perturbed measurements. A complete family of identifiers for this perturbed problem are given in Section 8.2.2. The limiting performance attained by this identifiers is arbitrarily close to that attained by the one of Section 8.2.1, thereby justifying their use, for the original problem. Section 8.2 also contains expressions for a further simplified (approximate) estimator (obtained as a result of some singular perturbations analysis) which attains performance arbitrarily close to the others. In all cases we also have asymptotic convergence of the estimates to the true parameter value(s), under a persistence of excitation condition and some restrictions on the disturbances; proofs of these results can be found in (Didinsky et al, 1995). In the third subsection of Section 8.2 we present yet another class of parametrized identifiers, which uses the more familiar approach of "prefiltering" of the original dynamic system (so as to avoid using the derivative of the signal).
In Section 8.3, we provide a comparative study of these different identification schemes, with the first subsection discussing a structural comparison. This is followed in Section 8.3.2 by presentation of some simulation results (on a nonlinear system which exhibits finite escape) which show superiority of the approximate estimator of Section 8.2.2 on the one of Section 8.2.3, from the points of view of convergence, transient behavior, and robustness to noise. In Section 8.3.3 we briefly discuss the case when in (8.1) the control input u is not fixed, but will also have to be designed for stability and pertbrmance. We show through some simulation examples that if the identifier of Section 8.2.2 is used in a certainty-equivalence controller, the resulting system becomes asymptotically stable, while the identifier still achieves a certain level of disturbance attenuation. This controller turns out to be much superior over the one that uses prefiltered version of the original signal for parameter identification, as well as over a standard benchmark nonlinear adaptive controller designed using Lyapunov-based methods. The chapter ends with the concluding remarks of Section 8.4.
152
8.2
Parametrized
Families
of Minimax
Identifiers 8.2.1
FSDI
identification
We first assume that the state vector x is continuously differentiable, and include its derivative in the information set. We refer to this problem as the FSDI (full-state-derivative-information) identification problem. Let 7* be bounded and, given any 7 > 7", introduce the cost-to-come function (Didinsky and Ba~ar, 1992; Didinsky et al, 1993a, b):
:= fo' [Io -721~: - A(x, u)O - b(x, u)l 2] ds - 72(10 - 0Ot~o). Using the natural state dynamics for 0: = 0,
0(0) = 0,
this cost-to-come function can be rewritten in the simpler quadratic form
w., (t, o; xco,, ], Oto,~], uco,~]) -- -7 2 IO - O(t)I~(,) +
re(t)
where
=
A(x, u)TA(x, u) -- + Q ( x , u),
2(0) = Qo,
(8.6a)
$
---- Z - 1 A ( x , u ) T ( x - A ( x , u ) O - b ( x , u ) ) 1
+ -~Z-
=
1
Q(x, u)(0 - 0),
01Q(~,~) - 72 Ix -
A(x, u)(~
-
0(0) - 0o,
(8.65)
b(x, u)lL
m(0) = 0. (8.6c)
Using this representation, we immediately arrive at the following result (Didinsky et al, 1995): T h e o r e m 8.1
For the FSDI model, the following are true:
(i) 7* is bounded, if, and only if, there exists a 7 > 0 such that Z.~(t) > 0 for all t E T . (ii) Given any 7 < 7", or any finite 7 if 7" is not defined, equation (8.6a) does not admit a non-ne9ative definite solution. (iii) If T* is bounded, then for every 7 > 7", {0(t) = 0(t), t G T} is a minimax policy, where in simpler terms, and with 0(0) = 0o, .
0
=
Z - 1 A ( x , u)T(Jc -- A(x, u ) O - b(x, u)).
(8.7)
153
Proof. We first prove the necessity in (i). Assuming that 7" is finite, choose 7 > 7*. Given any t E T, let At be the class of estimator policies, which in addition to the FSDI information have access to the states 0 after time t. Obviously, A C At. Also clearly, an optimal strategy in At will choose its output after time t to be equal to 0. Hence, the following relationship will hold: 0 > inf
J~(5, xo,Oo,w[o,o~))
sup
- - 5E£L~ xo,Oo,w[o,¢¢)
=
inf sup sup { - 7 2 1 0 5[o,tl x[o,~l O
O(t)l~,(t ) + rn(t)}.
As the supremum over 0 in last expression above is finite, it implies that Z~(t) > 0. For a proof of sufficiency in (i), let 7 > 0 guarantee that K~(t) > 0 for all t E T. Choose the identification policy to be the "certainty-equivalence" policy 5(t, qt) = O(t), for all t E T. Then, we have the bound:
JlT[IO - 513 - "/21w12]dt - 7210o -
0o-21eo = w ~ (T, 0; x [ 0 m ] ,
0[0,T], U{0,T]) <
0
for all disturbances (xo, 0o, w[0,T]) and all T C T. Taking the limit as T 1" oc leads to the conclusion that the infinite-horizon cost is bounded from above by zero. Note that in addition to sufficiency in (i), this also proves (iii). Last, we prove (ii), by contradiction. Let 7 < 7", or be any positive value if 7" is infinite, and suppose that 2~(t) > 0, for all t E T. From (8.6a), it follows that
)J~(T) = Qo +
/0
ArAdt-
Qdt,
VT ¢ T.
As the first two terms on the RHS sum up to a positive definite matrix (since Qo > 0) and last term is nonpositive definite, it follows that Z~+~(T) > 0, for all T C T, for any e > 0. If "y* is finite, then the preceding result contradicts with the hypothesis that 2/* is the least admissible value of 7- If 7* is infinite, then the above again leads to a contradiction, since 7 + e leads to a positive definite L', and hence is an admissible attenuation level.
Remark. If we let 7 I" oe in (8.6a), (8.6b), the limiting filter is precisely the LS estimator for 0 (Sastry and Bodson, 1989), whereas if Q = A'rA and 7 = 7* = 1, the resulting filter is the LMS estimator. We now turn to performance evaluation of the FSDI identification scheme (8.6a), (8.6b). Theorem 8.2 below will say that with £2-bounded disturbances it is possible to identify the uncertain parameters O, provided that an appropriate persistency of excitation (PE) condition is satisfied. PE Condition lim /~min T--+oo
(/0
)
Q(x, tt) dE
: c~.
(8.8)
154 T h e o r e m 8.2 Consider the dynamic system (8.1), along with the identification scheme (8.6@ and (8. 7). Let all disturbances be £2, and the P E condition be satisfied. Then, if ~/* is bounded, for all 7 > "7*, 0 converges to the true value of O, i.e., lim 0(t) = 0.
t---)- oo
Proof. The proof follows the standard line of reasoning used in adaptive control
and identification (Sastry and Bodson, 1989). It will consist of two parts: First, it will be shown that lim )~min( ~ . ( r ) ) = oo,
V7 > 7*.
T--~oo
(8.9)
Fixing 7 > 7", l e t , = 1(3, + "~*) > 7" By definition, Z~ (T) = Oo +
//
AT A dt -
Q dt > O,
v r >_ O.
Now, after rewriting Z~(T) as X,(T)
=
1 f0T O dt - (12 Qo + £ T AT A dt - -~-ff
=
~(T)-
( 1~
,21)f0T Q dt
0e ) f 0 T Q dr,
and observing that (@ - 1_~) is negative, (8.9) follows from the PE condition. X
Second, let us introduce the positive definite function V(t, 0~) = 0T ~ (t)O~: where the error 0~ = 0.~ - 0 is generated by ~ . y = - Z ; 1 A T AO,y + Z ; 1 A T w ,
(8.t0)
The derivative of V(t, 0~) along the trajectory of (8.10) is I~T ~ t1(t,O~(t)) = --~-ffO.~QO~
Iw
- A(x,u)O.,12+lwl 2
(8.11)
which yields the bound (by integration of the RHS of (8.11)):
<
~(0)TQ&(0)+.£ []w]2 - [w - A(z, u)0~]2] ds
< ~(0FQo~(0) + -]o'lwl ~ ds. Since the disturbances belong to/22, it follows in view of conclusion (8.9) that lim t~ (t) = 0.
t --+oo
155
8.2.2 N P F S I identification Toward removing the dependence on x, we now introduce a n o i s e - p e r t u r b e d FSI (NPFSI) problem: The state-dynamics (8.1) are replaced by = A(y,u)8
+ b(y,u) + w,
(8.12)
x(O) = Xo,
and the measurement available %r identification purposes is y = x + ev,
(8.13)
where the n-dimensional £2 function v is the measurement disturbance, and c > 0 is a known (generally small) scalar. The value of e is chosen as to reflect our level of confidence (or inconfidence) in the knowledge of the state x, so that smaller values of c would signify higher levels of confidence. Note that on the RHS of the state equation, we have also replaced x by y; this has been done for reasons of tractability (but also such models do indeed arise in practice). Of course, the limiting case c -~ 0 would be the same if the RHS of (8.12) had instead x. The soft-constrained cost function (8.5) is a~cordingly modified to be
- ~IQ(y,~,)
J.~(5; Zo, 8°, w[o,~))
-'?(18--8Ot¢o 2
72(Iwi 2 + Ivt2)] +
I~o -
dt
- 2 xot~o),
(8.14)
where Po = !~I , and 20 is a known "bias" term. As it, was shown in (Didinsky et at, 1995), the NPFSI problem stated above may not have a solution unless the class of possible disturbances (w[0,o~), v[0,~), xo, 8) is restricted. Toward this end, given a prespecified set Y of time functions Y[0,oo), we introduce the set )a/(y) as: F~(Y) := { (w[0,~), v[0,cc), xo, 80) I The output Y[0,oo)
(8.t5)
generated by this quadruple belongs to the set Y}. The NPFSI problem defined on this set will now be called the "modified NPFSI problem," for which we denote the optimum disturbance attenuation level
by 7; (~). As in the FSI case, we first introduce the cost-to-come function:
t'l~(t,x,O;8[o,t),U[o,t),Y[o,t])
:=
sup
Jt(O[o,t);xo,w[o,t])
(8.16)
( Zo, ~,[o,~l) e w ( t,z ,e ;,?(t ) )
where jt
fot [IO(s) - o(8)l~(~,y,~) - ~2(10 - 0O1~o + lxo-
1 72(lw(s)t 2 + jly(s) ~O1~o)
-
X(8) I2)] ds (8.17)
156 and
w(t,
x, 0; ~(t)) := {(Xo, wto,tl) t x(t) = x, o(t) = o, (yto,tl, ~,to,~l) = ~(t)}.
Again, it can be shown that whenever it exists, the cost-to-come function admits a quadratic form, which in this case is:
[][ x
w~(t, x, o) = -3`~
~(t)
0
where =
[o A]
[1
--z~, 0 +
0
--
-
]l~
AT
0
0 1
-~Q(y,u)
+ re(t),
z(t)
[o o]
e2
o
~(t)
]
,,U
-Z
I 0
0 ] E, 0
2 b ~] : [o0 Ao][0]+[o] 1~1[,]0 (Y - ~) + I72 S _ 1 [ Q 0] ( 0 - 0 ) + e2
(8.18a)
[~
rn =
10 - 01o(~,~) ~2 72 - ~ly-
el 2,
rn(O) = O,
, (8"18b) (8.18c)
and initial conditions for the Riccati equation (8.18a) and estimator dynamics (8.I8b) are, respectively,
o ] and [~(0/ o Qo
s(o) = [ Po
0°]
We now have the following counterpart of Theorem 8.1. T h e o r e m 8.3 are true:
Given the modified N P F S I problem stated above, the following
O) 7~ (e) is bounded if, and only if, there exists a 7 > 0 such that Z~ (t) > 0 for all T, where ~ is the solution of (8.18a). (ii) Given any 3` < 3`y( ), or any finite 7 if 7~" (e) is not defined, equation (8. I8a) does not admit a non-negative definite solution on T . (iii) If 3`~.(e) is bounded, then for every 3' > 7~(e), and each t E T, O(t) = O(t) is a minimax policy, where 0 is generated by (8.I8b), i.e.,
[~]
o A
=[o o][ ~
b
±~_~
157
Proof. Let Z ~ be the solution to (8.18a) with 7 = oz (which exists on any finite time interval since it is a standard filtering Riccati equation). Then, for any t E T, and any 7 > 0, the solution to (8.18a) is given, whenever it exists, by z
(t) = z
(t) -
o
Q
Rewriting Z 7 in the form above, and using similar arguments as those used in the proof of statement (ii) of Theorem 8.1, we can readily arrive at statement (ii) of the present theorem. Proofs of (i) and (iii) are also similar to those of appropriate parts of Theorem 8.1, and hence are omitted. We will be particularly interested in the parametrized set: YT,~
:=
{Y[0,oo) I Z , ( t ) > 0, Vt e T } ,
(8.20)
where Zn is a solution of (8.18a). It easily follows that if the set Y~,~ is nonempty, then 7 ~ . , (¢) is bounded. We are now left with the question of whether the set Y~,~ contains sufficiently large classes of output functions to be of practical value. Toward this end, we introduce the following three classes of outputs:
£13A(L, C): This is the set of all waveforms Y[0,~) that satisfy the following two conditions for some positive constants L and C, and for all t, s E T: (i) HA(y(t)) - A(y(s))]le <_ Lit - sl, (ii) ]fA(y(t))ll~. <_ C.
£BT~A(L, C,p): This is the set of all waveforms Y[0,~) such that there corresponds to each element of this set a ~0[0,~) E £BA(L,C), with the property
iiA(y(t)) - A(~(t))H2 < p,
Vt e W.
£ B P p A ( L , C, p, ~; 7): This is the set of all waveforms Y[0,~) E £BpA(L, C, p) that satisfy the condition z F ( t ) > -fitI, for all t C T. Given any ~ > ~/*, and positive scalars L, C, c, -fi, and p, suppose that £BT~7)A(L,C,p,-fi;7) is non-empty, and -fi > 4Cp. Then, for any -fi - 4Cp < 2 C ~ ' the following hold:
T h e o r e m 8.4
(i) EBT)7)~(L,C,p,-fi;7) C Y~,~, and the filter (8.18a), (8.19) attains the disturbance attenuation level 7 for the modified NPFSI problem. (ii)
all disturbances in the set
lim 0(t) = 0. t ---+ c ~
C, p,
N £2, we have
158
Proof. It can be shown (Didinsky et al, 1995) that, for sufficiently small e > 0, there exists a positive definite solution Z to the Riccati differential equation (8.18a) for any measurement waveform that belongs to the set £,BT)T'~(L, C, p, ~;'y). Then, statement (i) follows from Theorem 8.3. Furthermore, the eigenvalues of Z associated with t~ tends to infinity as t --+ oo; the statement (ii) follows. Theorem 8.4 above has established the optimality of the filter described by (8.18a) and (8.19). But the computation of this filter is much more complex as compared with that of the filter given by (8.6a) and (8.7) in the FSDI case. To remedy this, we first note that 55~ can be partitioned as =
ZT
Z3
'
and use this in (8.18a) to arrive at the following decomposition: ~1
:
~I-
Z2
=
- Z I ( Z 2 + A),
ZI~I,
z3
=
-~A-
ATZ~- ~ Z , -
~UI(0) Z2(0) Za(0)
v1Q ,
= = =
71,1 0,
Qo.
It can be easily observed that Z1 = !~I , which implies that Z(t) > 0, for all t E T, if, and only if, 11~(t) := Za(t) -eZ=(t) T with Z2(t) > 0, for all t E T. It is useful to observe that H~ is the solution of the matrix differential equation
fI(t) = z T z2 -- -~gQ,
11(0) = Qo
(8.21)
and ~2 is obtained uniquely from the following linear matrix differential equation: Z2 = - l ( z 2 + A), ~2(0) = 0. (8.22) e Assuming that Z is positive definite,
Z_ 1 =
[ eI + e 2 ~ 2 1 1 - 1 ~
-eZ2H~ 1
_~;1~
1~$ ]"
In view of this, the NPFSI filter (8.18a) and (8.19) can be written approximately as the following filter strategy 5(t, U[o,t],~0,t]):
a(t,~t0,~, yt0,~l) = :
x "
=
o =
e,
(8.2aa) 1
A(V, u)O + b(y, u) + ~ ( y - a~); 1 ~F--I~
7e.:~
?2~T
~ty, J (~- ~);
a~(0) =
e(0) = 0o.
Xo, (8.23b)
(8.2ac)
Here we have identified Z2 with - A (to which it converges, outside the boundary-layer) and have ignored higher order terms of e in the expression for
159
Z - l ; furthermore, X ( is generated by the differential equation (8.6a) with x replaced by y. The inverse operation can further be avoided since E l - 1 satisfies the following Riccati differential equation: ~_~(~.~dF - 1 . ) = _ x F - l ( A ( y , u ) T
l 1; A(y,u)___~Q(y,u))Z(-
~.~F - 1 (0) ~---Qo- 1
(8.24) To study robustness properties of the approximate filter 5 defined by (8.23), we first introduce the set: Y'~,~: The set of measurement waveforms Y[0,~) for which the filter ~ attains the performance level 7 for the modified NPFSI problem on T. Obviously, by its definition, the filter 5 is robust and attains the disturbance attenuation level "7 on Y~,~. The interesting question that remains is how large this set is. To answer this question, let us consider the following set of measurements:
£ B p A ( L , C, ~): This is the set of all waveforms Y[0,~) G £BA(L, C) that satisfy the further condition that for some positive scalar ~, ~F(t) _> ~(t + 1)I, for all t E T. 1 Then, we have the following result (Didinsky et al, 1995). T h e o r e m 8.5 Consider the modified NPFSI problem, and let 7 > 7* where the latter was defined by (8.6). Then:
(i) There exists an co :> 0 such that, for all c E (0, co], the set £.BT)A (L, C, -fi) is a subset of Y--%e. Furthermore, the filter 5 defined by (8.23) achieves the disturbance attenuation level 7 for the modified NPF5'I problem. (ii) If in addition the disturbances w[0,~) and v[0,~) belong to £2, then the parameter estimate ~(t) converges to the true parameter value 0 as t -+ oo. Proof. Let ~ : : (x T, flT)T and ~ : : (~?T,oT)T. Define ~ : : ~ - ~ . Then, satisfies the following differential equation:
0
0
-lA(y,u)r [ Xo- o ] Oo Oo
0
'
Using the approximate filter 5 in tile cost function (8.14) results in the following expression: 1This is a "persistency of excitation" condition in this context, and the set introduced here is equivalent to ~ f 3 ~ A (L, C, O, "~).
160
J(zo, 0o, w[0,oo), v[0,oo)) := J(~; Xo, 0o, w[0,oo), v[0,~))
:
J;(i
o]+ 0
Qo
)
o 0
Q(y, u)
]
272~T
y-
÷
_ 72jwj2 _ . ~ j y _ &j2 dt. By a dynamic programming argument, the maximum of J is upper bounded by zero if the following Riccati differential equation admits a positive definite solution over the infinite horizon:
,gv~
[
_!i
A + [ -!ic
~F - 1AT
0
AT A ~ F -1
yTF-1AT
~F-1AT A yTF-1
1 1] [00] ]
-TA~ 0
~ +
1 0 VQ
Exploiting the smallness of the parameter e by using the singular perturbation theory, it can be shown that, for a sufficiently small values of e, for any y[0,oo) E £:BpA(L, C,-~), the matrix function ~ exists and is positive definite over the infinite horizon. This establishes statement (i). Furthermore, the eigenvalues of ~ associated with ~ tends to infinity as t -+ oo. Then, one can conclude parameter convergence if the disturbances are Z:2, which implies statement (ii). For details of these arguments, the reader should consult with (Didinsky et al, 1995).
Remark. As a by-product of our analysis, we have obtained a robust estimator which delivers asymptotically optimal performance even if the state measurement is corrupted by some small disturbance. Of course, the solution to the original problem can be recovered as a special case, by letting v - 0, or equivalently by replacing y by x in (8.235) and (8.23c). 8.2.3
Prefiltering-based
identification
In this subsection, we take a more classical approach toward designing an estimator that does not depend on ~. In particular, we will use the idea of prefiltered input and output (Sastry and Bodson, 1989), adapted to our context as follows: We first multiply both sides of (8.1) by e ~t, and second, integrate both sides with respect to time from 0 to t:
f0'
e~
dr =
f0'
e~[A(~, u)e + b(~, u) + w] dr.
(8.25)
161 Next, using integration by parts, we arrive at the following equality:
(8.26)
x(t) = A(t)O + b(t) + w(t), where
A
=
b = =
-A_A+A(x,u),
A(O)
=
O,
-~+b(~,~)+~,
~(0)
=
~o,
-),~+w,
w(0)
=
0.
Now, the algebraic relationship (8.26) can be viewed as providing a noisy measurement of 0, in a way similar to (8.1) was used in the FSDI case. In this set up, an estimator will be designed to attenuate the new measurement disturbance w[0,¢¢), and hence in (8.5) w will be replaced by w. To guarantee that w E/22, the filtering parameter A must be chosen to be strictly positive. As in the FSI case, let us introduce the cost-to-come function i4% (t, 0; x[o,t], O[o,t], u[o,t]) := riot [t0 - OtQ ~ - 72 Ix- ~i0-
~12] d s _ 7 2
2 (I0-0Otqo),
(8.27) which can be rewritten in the simpler quadratic form
w~ (t, o; xto,~ 1, olo,,3> uEo,~l) = -7 ~ to - ~(t)l~(t) + ~(t), where
rn
=
~Tf4_ 1Q, 72
S(O) = Qo,
=
Z-1AT(x-AO-b)+-~L~-IQ(t)(O-O),
=
tO - 01~ - ~ ~ I x - .No
- ~l ~,
(8.28a)
m(0) = 0.
0 ( 0 ) = 0 o , (8.28b)
(8.28c)
Using this representation, we have the following counterpart of Theorem 8.1: T h e o r e m 8.6 For the identification problem with the prefiltering, the following are true:
(i) 7" is bounded, if, and only if, there exists a ~/ > 0 such that Z.y(t) > 0 for all t E T, where Z.y is the solution to (8.28a). (ii) Given any 7 < 7", or any finite 7 if'f* is not defined, equation (8.28a) does not admit a non-negative definite solution. (iii) If'y* is bounded, then for every 7 > 7", {0(t) = 0(t), t E T} is a minimax policy, where in simpler terms, and with 8(0) = 0o, 0
=
~-IA(x,
u)~(x
- AO - ~).
(8.29)
162
The counterpart of Theorem 8.2, on the other hand, is the following. T h e o r e m 8.7 Consider the dynamic system (8.1), along with the identification
scheme (8.28a) and (8.29). Let all disturbances be ~.2, and the PE condition be satisfied. Then, if T* is bounded, for all 7 > 7", t~ generated by (8.29) converges to the true value of O, i.e., lira t~(t) = 0.
t--+cx~
8.3
A
Comparative
Study
In this section, we compare the NPFSI and prefiltering identification schemes from a practical/implementational point of view, i.e., we will compare the structures of the two estimation schemes and present some simulation results which show how the two perform numerically with and without a control input. It will be seen that on all counts which we found important, the NPFSI estimator exhibits superior qualities, and it outperforms the prefiltering estimator. 8.3.1
A comparison
of structures
One important factor in a realistic comparison of two filter structures is the number of internal states. Everything else being comparable, it would definitely be preferred to have an estimator of smaller dimensions (i.e., fewer internal states). From this point of view, the approximate NPFSI filter (8.23) fares much better. 2 It has internal states ~, 0, and Z F, which amounts to n + r + r 2 states, while the prefiltering estimator (8.28a)-(8.29) has states A, b, 0, and S, which amounts to (n. r) 2 + n + r + r 2 states. The number of states for the prefiltering estimator can be brought down to p + n + r + r 2 in the case of a linear system (where p is the dimension of input vector u), by first separately filtering x and u, and then using the outputs of these filters to construct A and b. Nonetheless, even in this special case, the dimension of the prefiltering estimator remains larger than that of the NPFSI estimator. 8.3.2
A comparison
of the rate
of convergence
A good estimator must have an acceptable convergence property. One way of testing this would be to simulate its behavior, when applied to a nontrivial system. Consider the following system with a finite escape time: 3
=
+ u,
=
(8.30)
2The approximate NPFSI filter is of lower dimension than the NPFSI filter (8.18a), (8.19); its performance, however, is comparable to that of the NPFSI filter when ~ is small, as shown
in Theorem 8.5. 3This specific example was suggested by Petar Kokotovid.
163
11 0.9[ 0.8
0.9
03 0.7
0.7[
0.6
0.6
0.5 0.4
0'5t 0.4
i./
0"3f 0.2 0.1 a)
6
!
8
0.3t0"2 I OJ o;
4
o
b)
F i g u r e 8.1 Comparison of performances of NPFSI approximate filter and prefiltering estimator when system has a finite escape. a) NPFSI approximate estimator,
b) Prefiltering estimator.
where we picked u = 0, 0 = 1 and x o = 0.1 (hence, the escape time is 10). A good estimator must identify the parameter 0 before the state x escapes to infinity. The reason for this test is that if an identification scheme can estimate the unknown p a r a m e t e r before the state escapes, then when the control is present it will have a good chance to stabilize the system (which is open-loop unstable). When we applied the two identification schemes mentioned above to syst e m (8.30), we found that the approximate N P F S I scheme:
= k
x
=
0
=
Z
=
x~0+u+
1(
-(1-~-2)Z2x
x-e),
4,
~(0)
=
Xo,
0(0)
=
0,
Z(O)
=
1,
(8.31)
resulted in finite-time convergence to the true parameter 0 = 1 before the escape. We found this property to hold consistently for sufficiently small values of e and for all "r _> ?'* = t. One such case is depicted in Figure 8.1(a) for = 0.01 and 3' = 1.2. The results for the prefiltering estimator:
164 k
0 L A
= =
z~(~-A~-~,), - A f i ~ + x 2,
0(0) fi,(O)
= =
0, O,
~(0)
=
~o,
z(0)
=
1,
.5'
b =
z
-~,+u,
=
(8.32)
were discouraging, however. We were unable to make the estimate 0 converge to the true parameter value for any values of 7 and A, even though in some cases it came close to 1 in a very small neighborhood of the finite escape time. Figure 8.1(b) depicts a typical response of the prefiltering estimator, here for values of A = 0.01 and ~/= 1.2.
8.3.3
Performance
under
a controller
Majority of the work in modern adaptive control is based on the certaintyequivalence principle (Sastry and Bodson, 1989; Narendra and Annaswamy, 1989). It stipulates that a successful controller can be build in two stages: First by designing an acceptable controller with full knowledge of the unknown parameters, and second by replacing 0 with an appropriate estimate. To also study some robustness issues, we take the following noise-perturbed version of the dynamics (8.30): = x~O + u + w,
x(O) = xo,
(8.33)
where again the true value of 0 is 1. For the first stage, we picked the performance index y =
/5
(x ~ - ~w ~) e~,
(8.34)
which describes a differential game with minimizer u and maximizer w, and with the value of 0 known. A controller policy #(x) that yields a finite upper value is found by solving the Isaacs inequality: f OV 2 m axl~-~-x(x O + # ( x ) + w ) + x 2 - 7 2 w
2
}
<0.
(8.35)
A simple solution here is V(x) = ½ax 2 and a corresponding controller is a
1
p(x; 0) = -x~O - kx, where a > 0 and k > 4-~ + -a" Hence, a certaintyequivalence controller is the policy u = p(x; O) = - x 2 0 - kx,
(8.36)
where t~ is some estimate of 0. Note that this controller adaptively linearizes system (8.33); i.e., when 0 converges to 0, it cancels out the nonlinear term x20, thus making the system linear, noise-perturbed.
~
0
~-
Gr~
¢,0
c:~C
o
0
°
~
c:r o
c:~ ,
~
r~
166
1.8I 2
1.4
t.2
1
i
'1
;"
0., 0.6
0.4 0.2 50
I00 a)
i50
b)
Figure 8.4 Certainty-equivalence controller with NPFSI approximate estimator under low frequency sine-wave noise input a) Parameter estimate, line).
b) State trajectory (solid line) and control input (dash-dot
When we used in the certainty-equivalence controller (8.36) the approximate NPFSI estimator (8.31) and the prefiltering estimator (8.32), we observed that the former consistently outperformed the latter: The best performance observed for the NPFSI filter was for small values of e (e = 0.01 was sufficient) and large values of 7 (see Figure 8.2); 4 the best prefiltering performance observed was for small A ()~ = 0.01 was sufficient) and smaller values of 7 (7* = 1) (see Figure 8.3). Both certainty-equivalence controllers were simulated for several initial conditions (Xo = 2, 5, 10) and for two sinusoidal inputs (w = sin(3t + 1), sin(0At + 1)). We found that larger initial conditions did not significantly alter the performances of the two estimation schemes after scaling. However, the frequency of sinusoidal disturbances did play an important role. As the frequency decreased, the performance somewhat deteriorated in both cases, but degradation was more pronounced in the prefiltering case (see Figures 8.4 and 8.5, which correspond to zo = 2 and w = sin(0At + 1)). For a further comparative study, we designed a nonlinear adaptive controller for the noise-free case, based on the quadratic Lyapunov function v(x,0)
=
1 ~ + l(t 9 - ~)~,
(8.37)
which led to 4This corroborates results in Didinsky a n d Ba,~ar (1994), where it was shown t h a t in t h e FSDI case t h e LS (~, = oo) e s t i m a t o r is o p t i m a l for control purposes.
167
ox 10"
1.8!
i ,,
¢
'L4 /
L
I
1.2 I I
-8i
!
O.B 0,6
4
0.4 0.2 a)
50
2OO
1® b)
i
150
200
F i g u r e 8.5 Certainty-equivalence controller with prefiltering estimator under low frequency sine-wave noise input a) Parameter estimate, line).
b) State trajectory (solid line) and control input (dash-dot
2,5!
3I i
-3}-
.,!i il i
I/ "6 'I=
2
4
6
;
1'0 1~2 1'4 1'6 18
al
F i g u r e 8.6 a) Parameter estimate. line).
lo 1'2 1~ 18 18
b)
Nonlinear adaptive controller with no disturbance b) State trajectory (solid line) and control input (dash-dot
168
f
/-
/q
30
]
2O 10 4
,
, i
I I
0 .10
\
I t
-5(
,;o
a)
Figure 8.7
I¢
b)
150
Nonlinear adaptive controller under low frequency sine-wave noise input
a) Parameter estimate, line).
b) State trajectory (solid line) and control input (dash-dot
u
=
-x2
0
=
- x 3,
-kx,
~(0) = 0.
(8.38a)
(8.385)
When disturbances were not present, this control law outperformed both NPFSI and prefiltering certainty-equivalence controllers (see Figure 8.6). However, it showed poor robustness properties in the presence of noise, as can be seen in Figure 8.7, where we used the same sinusoidal disturbances as in the NPFSI and prefiltering cases. Clearly, the NPFSI certainty-equivalence controller outperforms this standard nonlinear adaptive controller when noise is present. The observed parameter drift phenomenon can be partially resolved by introducing an additional nonlinear damping term in the controller. To see this, we simulated the system response under the control law u -- - z 2 ~ - k l x - k ~ x 5 for different values of k2. The closed-loop system then behaved much smoother for larger values of k2, but at the expense of large initial control action (see Figure 8.8 for system response when k2 = 0.2). To further compare the performance of the certainty equivalence controller using the approximate NPFSI filter with that of a Lyapunov based adaptive controller, let us consider the following nonlinear system: = x36 + u + w.
(8.39)
169
I
!i
/
I
",
~
i i
•
i
l
x
1
/
J t
~ I
I
p ,
t
i
E
I
J I I
-1
_2] 0
i 20
i 40
i 60
i 80
i
100
a)
120
i
i
1~3
160
180
200
-1C
i
i
r
i
t
20
40
60
80
100
i
120
140
1~
I~
I
200
b)
F i g u r e 8.8 Nonlinear adaptive controller with additional nonlinear damping term under low frequency sine-wave noise input a) Parameter estimate, line).
b) State trajectory (solid line) and control input (dash-dot
The objective here is not regulation, but to track a given reference trajectory, r ( t ) = sin(t). A certainty-equivalence control law for this purpose is given by (8.40)
u = -xat) - x + sin(t) + cos(t)
where t~ is the parameter estimate. Using the approximate NPFSI filter, the estimate ~i is generated by x:
=
e
=
2
=
x30+u+~- 1(x-e), -(1-
~ ) Z 2 x 6,
=
xo,
0(0)
=
0,
Z(O)
=
1.
(8.41)
For the Lyapunov based design, however, the dynamics for 0 would be as follows: 0 = x 3 ( x - sin(t)), 0(0) = 0. (8.42) With w = 0, we observed that the Lyapunov based adaptive design resulted in much slower convergence rate for the tracking error to diminish to zero, as compared with that of the NPFSI certainty equivalence controller (see Figs. 8.9 and 8.10, which correspond to x0 = 2). Clearly, the NPFSI certainty equivalence controller leads to a better transient performance (for tracking) in the
170
F'l
?"
i"
~
I
*,
".
t'~
'~
'~
f"
'~
J'l
"%
•
I
i !! i~ "~ li ! ' ' ! i i !! ii r! i~ i~ !! i i i
10~
10
~
Figure 8.9 filter
3o
4o
70
al
80
9o
100
-12I
o
10
20
3o
4o
~
60
70
8o
90
,00
b}
Tracking: Certainty-equivalence controller with NPFSI approximate
a) Parameter estimate, line).
b) Tracking error (solid fine) and control input (dash-dot
absence of disturbance. We have also observed similar behavior in the presence of disturbances; see Figures 8.11 and 8.12 for simulation results with the disturbance chosen as w(t) = 0.1 sin(0At + 1).
8.4 C o n c l u s i o n s After running through all factors of comparison for the evaluation of the two identification schemes, NPFSI came out as a clear winner: it has smaller dimension, better observed convergence rate, and better performance when used with a certainty-equivalence controller. In fact, we have not encountered any case where the prefiltering estimator outperformed the NPFSI estimator. NPFSI estimator has also exhibited superior robustness properties, and has outperformed a nonlinear adaptive controller, designed based on Lyapunov methods. There is one additional reason for preferring the NPFSI design over prefiltering, which is that, as shown in (Didinsky et al, 1995), the former can easily be extended to systems with time-varying parameters, whereas the same cannot be said for the prefiltering estimator. The observation that the controller design using the NPFSI identifier achieves better performance and robustness has been further justified theoretically in the recent papers (Didinsky and Ba~ar, 1994) and (Pan and Ba§ar,
171
I
o ~
i!l
1
0,5
i "~6u
I0
'!! ;!i,
i: ~: '!,! '~!/, i
i 1
20
60
40
N
70
iO0
b)
Figure 8.10 a) Parameter estimate, Une).
Tracking: Nonlinear adaptive controller
b) Tracking error (solid line) and control input (dash-dot
,2,,:,:,
-2
-2
-4
-4
-.6
~8
4
-I0
-10
-12
10
20
30
40
50 b)
60
70
80
90
100
-12I 0
,0
20
60
~
~
b)
60
7o
60
~
1=
F i g u r e 8.11 Tracking: Certainty-equivalence controller with NPFSI approximate filter under low frequency sine-wave noise input a) Parameter estimate, line).
b) Tracking error (solid line) and control input (dash-dot
172 3
2.G
2
'L5
0
0
1
i
~
i
r
,h
~
i,
k
20
30
40
50
60
70
80
90
a)
Figure 8.12 noise input
t00
i
i
i
i
L
i
i
i
i
10
20
30
40
50
60
70
80
90
100
b)
Tracking: Nonlinear adaptive controller under low frequency sine-wave
a) Parameter estimate, line).
b) Tracking error (solid line) and control input (dash-dot
1996). In the latter paper, it has further been shown for systems in parametricstrict-feedback form that the certainty equivalence design is robust (i.e., optimally disturbance attenuating) only when the system is of first order (as is the case with the examples considered in this chapter). For higher-order systems, the controller will have to be modified by also including covariance information in its structure in order to guarantee robustness with respect to exogenous disturbances. In the case of a persistently exciting reference signal, such an optimally disturbance attenuating controller converges to a certainty equivalent form only asymptotically as the covariance matrix goes to zero. One of the challenging works that remain today is a theoretical study of adaptive control design for uncertain systems with unknown high-frequency gains. The general objective here would be to obtain robust adaptive controllers based on the NPFSI identifier, which would feature superior disturbance attenuation and command following properties.
Acknowledgements. Various discussions on this topic with Petar Kokotovid of UCSB are gratefully acknowledged. Research leading to this work was supported in part by the U.S. Department of Energy under grant DE-FG-0294-ER-13939, and in part by the Joint Services Electronics Program under Contract N00014-90-J-1270.
173
References Ba~ar, T., Bernhard, P., 1995, H~-optimal control and related minimax design problems: a dynamic game approach, Birkh~user, second edition, Boston, MA. D idinsky, G., B a~ar, T., 1992, Design of minimax controllers for linear systems with nonzero initial states under specified information structures, International Journal of Robust and Nonlinear Control, 2, 1-30. Didinsky, G., Ba~ar, T., 1994, Minimax adaptive control of uncertain plants, Proceedings IEEE Conference on Decision and Control, Orlando, FL, 28392844. Didinsky, G., Ba~ar, T., Bernhard, P., 1993a, Structural properties of minimax controllers for a class of differential games arising in nonlinear H~-control, Systems and Control Letter.s, 21,433-441. Didinsky, G., Ba~ar, T., Bernhard, P., t993b, Structural properties of minimax controllers for a class of differential games arising in nonlinear H~-control and filtering, Proceedings IEEE Conference on Decision and Control, San Antonio, Texas, 184-189. Didinsky, G., Pan, Z., Ba~ar, T., 1995, Parameter identification for uncertain plants using H ~ methods, Automatica, 31(9), 1227-1250. Narendra, K. S., Annaswamy, A. M., 1989, Stable Adaptive Systems, PrenticeHall, Englewood Cliffs, NJ. Pan, Z., Ba~ar, T., 1996, Adaptive controller design for tracking and disturbance attenuation in parametric-strict-feedback nonlinear systems, Proceedings IFAC World Conference, San Francisco, CA. Sastry, S. S., Bodson, M., 1989, Adaptive Control: Stability, Convergence and Robustness, Prentice-Hall, Englewood Cliffs, NJ.
. E x p o n e n t i a l C o n v e r g e n c e for Uncertain Systems with Component-Wise Bounded Controllers M a r t i n Corless and George L e i t m a n n 9.1 I n t r o d u c t i o n A fundamental problem in the design of feedback controllers is that of stabilizing and achieving a specified level of performance from a system whose description contains significant uncertainty. One fruitful approach to this problem, especially when dealing with nonlinear time-varying systems, is based on the constructive use of Lyapunov functions; see (Corless, 1993a, 1993b; Corless and Leitmann, 1988, 1990; and Leitmann, 1990, 1993) for a survey of results in this area. Another practical problem is that of actuator saturation. In any control application, the actuators generating the control inputs have limited range; the inputs they generate are bounded in magnitude. Until recently, there has been little research published on this problem. Recent work includes (Blanchini, 1991; Corless and Leitmann, 1993, 1994; Dolphus and Schmitendorf, 1991; Franchek and Niu, 1993; Gutman and Hagander, 1995; Hached el al, 1990; Madani-Esfahani et al, 1988; Madani-Esfahani and Zak, 1987; and Soldatos and Corless, 1991). This chapter considers the problem of designing controllers for a class of nonlinear uncertain systems to ensure that all closed loop state trajectories which originate in a bounded region exponentially converge to a neighborhood of the origin with a desired rate of convergence. In addition, each control input is subject to an upper bound on its magnitude or norm. This research is an outgrowth of that presented in (Corless and Leitmann, 1993; and Leitmann, 1979). (Corless and Leitmann, 1993) considers the same desired closed loop behavior as that considered here; however there the uncertain terms in the system description are required to satisfy a linear growth condition. That restriction is removed here; this is useful in application to robot control problems where the linear growth condition is usually not satisfied. Also, Corless (1993) considers a single bound on the Euclidean norm of the full control input vector. Here, as is usually the case in practice, one has individual bounds on the magnitude of each control input. This results in a control design in which controllers are designed separately for each control component; this was the approach taken in (Leitmann, 1979) for problems without control constraints.
176 S o m e n o t a t i o n . The symbol IRn denotes the set of real n-vectors, i.e., the set of all ordered n-tuples z = [ xl
z2
...
x,~ IT
where the superscript 'T' denotes transpose. The notation usual Euclidean norm of x, i.e.,
Ilzll
denotes the
Ilxll :-- (z~ + z~ + . . . + x2,) ½ If M is a real m × n matrix, the notation by the Euclidean norm, i.e.,
IlMll := sup { [[Mz[[ LIft----f-
IIMII denotes
the norm of M induced
: ~ e n~~, • ¢ o
}
.
This quantity is explicitly given by T
1
IIMI1 = Amax(M M)~
where, for a matrix N with real eigenvalues, Amin(N) and /~max(N) denote its minimum and maximum eigenvalues, respectively. If V : IRn -+ I1%is continuously differentiable, then for any x E IR'~,
DV(x) A [ OV =
~(x)
OV ...
o~(~)
] .
This is sometimes written as -OV ~ (x) or v v T ( z ) .
9.2
Problem
We consider
Statement
uncertain, systems described
by
l
~(t) =/(~(t)) + ~ [ B , ~,(t) + nF/(t, x(t), u/(t))],
(9.1)
i=1
where t E IR is the time variable, x(t) E lR~ is the state and u/(t) E lRm', i = 1, 2 , . . . , I , are control inputs. The continuous function f and the constant matrices B/, i = 1, 2 , . . . , l, are known; they define the nominal system
x(t) : / ( ~ ( t ) )
+ B~(t),
(9.2)
where ix U~
Ul U2
Ul
B:=[B1
B2 ...
B~].
177
All the uncertainty and time-dependence in the system is represented by the terms AFi which are assumed to be continuous functions. Each control input ui is subject to a hard constraint of the form
Iludt)/I _< f~
(9.3)
where the bound fii > 0 is prescribed. We shall consider the control input ui to be generated by a memoryless state feedback controller Pi, i.e.,
ui(t) = pi(x(t)).
(9.4)
The resulting closed loop system is described by x(t) = F(t, x(t))
(9.5)
with l
F(t, x) := f(x) + Z [B, p,(x) + AFt(t, ~, p,(x))].
(9.6)
i=1
For any scalar r > 0, the ball of radius r is defined by U(r) := {x e ~
: Ilxll < r}.
Consider any scalar a > 0 and any set .4 C IR'~ containing a neighborhood of the origin.
D e f i n i t i o n 9.1 System (9.5) is uniformly exponentially convergent to B(r) with rate c~ and region of attraction .4 if there exists a scalar/3 > 0 such that
the following hold. (i) Existence of solutions. For each to E I R and xo E .4, there exists a solution x ( . ) : [to,t1) ~ ]1%n of (9.5) with to < tl and x(to) = Xo. (ii) Indefinite extension of solutions. Every solution x(. ) : [to, tl) --+ IR'~ of (9.5), with x(t0) E .4, has an extension 37(. ) : [to, oo) -+ ]1%'~, i.e., 2(t) = x(t) for alI t E [to, tl) and 2(. ) is a solution of (9.5). (iii) Uniform exponential convergence of solutions. is any solution of (9.5) with x(t0) ~ .4, then
IIx(t)It <_ r +/9ttx(to)lt exp[-c~(t - to)],
If x( . ) : [to, oo) --~ IR '~
Vt _> to.
Remark. If system (9.5) satisfies the requirements of the above definition with r = 0, then it is uniformly exponentially stable with region of attraction .4. When r is nonzero it can be regarded as a measure of the departure of the system behavior from that of exponential stability.
178 Definition 9.2 System (9.5) is globally uniformly exponentially convergent to B(r) if it is exponentially convergent with .A = ]Rn as a region of attraction. D e f i n i t i o n 9.3 A set ,4 is invariant for (9.5) if every solution x ( . ) which starts in M remains in M, i.e., x(to) • ,4 implies x(t) • A for all t >_to. The problem we wish to consider is as follows. PROBLEM STATEMENT. Consider a system described by (9.1), subject to control constraints (9.3), and let ~ > 0 and r > 0 be specified scalars. Find memoryless state feedback controllers pi, i = 1, 2 , . . . , l, which render the closed loop system (9.5) uniformly exponentially convergent to B(r) with rate ~.
9.3
A Lyapunov Exponential
Sufficient Condition Convergence
for
The following theorem provides a sufficient condition which is useful in guaranteeing that a given system is uniformly exponentially convergent. T h e o r e m 9.4 Consider a system described by (9.5) with F(.) continuous and suppose there exists a continuous function V(.) : ]Rn -+ IR with the following properties. (i) There are scalars q > 1, wl > 0 and w2 > 0 such that
~1 tixllq _< V(x) < ~2 Ilxliq for all x E IRn . (ii) There are scalars V and V__, with 0 ~_ V_.< V ~_ co, such that whenever
v_ < v(x) < v , V is continuously differentiable and n V ( x ) F ( t , x) ~
-q~[V(x) -
for all t E IR. Then system (9.5) has the following properties. (a) The set A = Ix • ~" :
is invariant for (9.5).
v(~) < v }
V_]
179
(b) System (9.5) is uniformly exponentially convergent to B(r) with rate a and region of attraction ~4 where r = (V_/~I) llq,
and
= (,~2/~,1)1/q.
Example
9.5
Consider a scalar s y s t e m ~(t) = - x ( t ) + e(t)
where e(t) is due to an unknown b o u n d e d disturbance with known m a g n i t u d e b o u n d p, i.e., le(t)l < p. Considering V(x) = Ixl, v is continuously differentiable on the set of all nonzero real numbers. Hypothesis (i) is assured with q = 1 and Wl = a~2 = 1. Also,
O V ( x ) [ - x + e(t,)] <_ - ( I x l -
p);
hence, hypothesis (ii) is assured with V = p, V = oe, and a = 1. Hence all solutions of this system satisfy
Ix(t)l < p + Ixol e x p [ - ( t - to)] for all t _> to. We refer to a function V which satisfies the hypotheses of the above theor e m as a Lyapunov function tbr (9.5). Note that, if V is infinity, the convergence is global. Also, _.V = 0 yields uniform exponential stability.
9.3.1
Proof
of Theorem
9.4
Existence of solutions follows from the continuity of F; see Hale (1980). We now d e m o n s t r a t e the invariance of A. Consider any initial time to E Ill and state x0 C IR ~' and let x(.) : [to, tl) --+ IPt'~ be any solution of (9.5) with x(to) = xo. Hypothesis (ii) implies t h a t whenever
v_ < v ( x ( t ) ) < v one has
av(x(t)) dt
--
DV(x(t))F(t, x(t))
<
-q ~[v(x(t))
<
0.
- v__]
180 o
To demonstrate the invariance of.4, we show that if V(xo) < V, then V(z(t)) < V for to <_ t < tl. Suppose, on the contrary that V(zo) < V, but, there exists a time t4 > tQ such that Y(z(t4)) > V. Since z(.) and V are continuous, V(z(.)) is also continuous. Hence, there exists two times t2 and t3 such that y(~(t3)) = v and V < V(z(t)) < V
for t2 <_ t < t3.
So, t3 is the first time x(t) leaves .4. Since V(z(.)) is continuous on the closed interval [t2, t3] and differentiable on the open interval (t2,t3), it follows from the Mean Value Theorem that there is a time t. E (t2,tz) such that
V(z(ta)) - V(z(t2))
dV(x(t.))
ta - t2
dt
Since V < V(x(t,)) < V, it follows from hypothesis (ii) that ~ thus
Y
= <
< 0;
v(x(t3)) < v(x(t2)) V
and we obtain a contradiction. Hence .,4 must be invariant. Since .4 is invariant and bounded, all solutions which originate in .4 are bounded. Since .4 is open, this guarantees that all solutions can be continued indefinitely over the interval [to, oo); see Hale (1980). Using hypothesis (ii) and arguments similar to those used in demonstrating the invariance of .4, one can also show that
V(z(to)) < V__ =¢,
V(z(t)) < V
for t > to.
(9.7)
In other words, the set {x ~ n~" : V(x) < V} is also invariant. We now prove exponential convergence for x0 E .4. Consider first the case in which V(zo) < V. In this case, as we have already shown, V(z(t)) < V for all t > 0. Utilizing hypothesis (i), we obtain that for all t > to, It~(011
<
[v(~(t))/~x] 1/q
<_ (K/~I) I/q =
r;
hence, ttx(t)ll < r +
~]lxott exp[-a(t
-
to)].
Consider now the case in which V < V(xo) < V . Defining
181
~(t) :=
v(,(t))
- _v,
? := v - v__,
(9.8)
we have r/(to) > 0 and from the invariance of .A it follows that r/(t) < 0 for all t _> to. Since i1 = D V ( x ) F ( t , x ) , it follows from hypothesis (ii) that 0 < rl(t) < 0
~
O(t) _< - q c~](t).
(9.9)
V~re now show t h a t
rl(t) ~ o(to) exp[--qa(t -- to)]
(9.~o)
for t _> to. Let ts = inf{t E [t0,eo): rl(t ) < 0 }. Considering r E [to, ts), we have 0 < rl(r) < ~/; hence, utilizing (9.9) we obtain exp(qar)/t(r) + qa exp(qar) rl(r ) < O;
i,e,~ d d--'~(exp(qar)rl(r)) <- O. Considering any t E [to, ts) and integrating over [to, t] yields (9.10). If t5 ~ ~ , rl(ts) < 0 and it follows from (9.7) that (9.10) holds also for t E Its, ~ ) . Using (9.10) it now follows that for all t _> to.
V(x(t))
<_ V__÷[V(xo) - V__]exp[-qa(t - to)] <
v + V(~o) e x p [ - q ~ ( t - to)];
hence Ilx(t)ll
<
[V(x(t))/~l] 1/q
< < <
[Y_/~I + (V(xo)/~l) exp[-q~(t to)]] '/q (V__l~)~l q + ( V ( x o ) l ~ ) ~lq exp[-~(t - to)] ( Z / ~ ) l l q + (~2/~1)~/q11~oll exp[-~(t -to)], -
i.e.,
IIx(t)ll < ~"÷ #llxoll exp[-~(t - to)].
182
9.4 Exponential Stabilization of the Nominal System The following assumption guarantees that the nominal system (9.2) can be globally stabilized with rate a > 0. A s s u m p t i o n 9.6 There exist positive-definite symmetric matrices P and Q and a scalar cr > 0 which satisfy 2 x T p f ( x ) < --2e~xTpx -- xTQx + ~rxTpBBTpx
(9.11)
for all x E lRn • L i n e a r n o m i n a l s y s t e m . The above assumption is always satisfied when the nominal system is linear and controllable. To see this, consider a nominal system described by = Ax + Bu where the matrix pair (A, B) is controllable. For each positive-definite symmetric matrix Q and each ~r, a > 0, there exists a positive-definite symmetric matrix P which satisfies the Riccati equation P(A + aI) + (A + aI)T p - - a P B B T p + Q = O. Pre- and postmultiplying this equation by x T and x, respectively, yields 2xT p A x _= --2axT p x -- xTQx + qxT p B B T p x . Section 9.9 contains an example of a nonlinear nominal system which satisfies the above assumption. L i n e a r stabilizing controller for t h e n o m i n a l s y s t e m . We now demonstrate that when Assumption 9.6 holds, the linear controller given by O"
T
ul = - - ~ B i Px globally stabilizes the nominal system (9.2). To achieve this we show that the corresponding closed loop system, ~(t) = f ( x ( t ) ) - 2 B B T px(t) satisfies the hypotheses of Theorem 9.4 with V(x) = x T p x , V = 0, and ~ = cx~, Since )~min(P) llxll = < x r p x < ~max(P)Ilxll 2 for all x E IR'~, hypothesis (i) is satisfied with q = 2 and
1~33
wl = ~min(P) > 0,
cz2 = Ama×(P) > 0.
Also, utilizing (9.11), we obtain that for all x E IR~, ff
DV(x)[f(x)--~BB
T
Px]
(9"
=
2xTp[f(x)--~BB
<
--20~xTpx-- x T Q x
T
Px]
< Hence hypothesis (ii) is satisfied with Lz = 0 and V = ec. The theorem now guarantees that the above closed loop system is globally uniformly exponentially stable with rate of convergence a.
9.5 A s s u m p t i o n s on U n c e r t a i n t y The following assumption, which is sometimes referred to as a matching condition, is common in the literature on control of uncertain systems. To achieve an arbitrary rate of convergence for the closed loop system, this assumption is necessary for some classes of uncertain systems; see (Swei and Corless, 1991).
Assumption 9.7 For each i = 1 , 2 , . . . , l , there is a function ei(.) such that AFi = Biei.
(9.t2)
Assumption 9.8 _bbr each i = 1 , 2 , . . . , l, there exist scalars kol, k~i, with k2i < 1,
(9.13)
and a continuous non-decreasing function kli such that ltei(t, x, ui)ll <_ koi + lqi(llxlI)ttxll + k2iltuilt
(9.14)
for alt t E IR, x E IR '~, u E IR mi.
Assumption 9.9 For each i = 1, 2 , . . . , l, Pi > k0i/(1 - k2i).
(9.15)
If this last assumption is not satisfied, an uncertain term ei can completely cancel the effect of the corresponding control input ui; see (Corless and Leitmann, 1993).
Remark. Section 9.9, (Lee and Leitmann, 1994) and (Leitmann and Lee, to appear) contain practical examples of nonlinear systems satisfying the above assum~)tions.
184
9.6
Unconstrained Convergence
Controllers
and Global
In this section we ignore control constraints (9.3) and present controllers which yield exponential convergence which is global. In what follows, a positivedefinite symmetric matrix P which, along with a positive-definite matrix Q and a positive scalar or, guarantees satisfaction of Assumption 9.6 will be referred to as a Lyapunov matrix. Neglecting the control constraints, the proposed controllers consist of a bounded part and an unbounded part; namely, choosing any scalar c > 0,
ui = pi(x) = -Pi s a t ( c - l B T p x ) -- ^/i(]lxll)B TPx,
(9.16)
where P is a Lyapunov matrix, Pi := (1-k~i)-lk0~,
(9.17)
7i is any continuous function which satisfies
(llxll)
_>
--
2(1-k2i)
+
2p(1-k2i)'
(9.18)
with :-- )~min (Q),
and the saturation function sat is given by sat(y) :=
y
]]yll-ly
if if
[Iyll _< 1, [[YI[> 1.
(9.19)
Consider an uncertain system described by (9.1), satisfying assumptions 9.6-9.9 and subject to control given by (9.16). Then the resulting closed loop system (9.5) is globally uniformly exponentially convergent to B(re) with rate a, where
T h e o r e m 9.10
re :--
[¢ko/aAmin(P)]½,
l ko :-" E koi.
(9.20)
i=1
Proof. The proof consists of demonstrating that the function V(-), given by V(x) = xTPx, satisfies the hypotheses of Theorem 9.4. In Section 9.4 we demonstrated that V satisfies hypothesis (i) with q = 2 and Wl -- Amin(P),
082 "- Amax(P) •
Utilizing (9.6)-(9.19), one obtains (the detailed calculations are given in the Appendix) that for all t E IR and x E lRn
185
DV(x)F(t,x)
=
2xT p F ( t , x )
<
- 2 a V ( x ) + 2ek0;
hence hypothesis (ii) is satisfied with V = eko/a and V = oo. Applying Theorem 9.4 yields the desired result.
Remark. It follows from (9.20) that the radius r~ of the ball of convergence can be made arbitrary small, but non-zero, by choosing e > 0 sufficiently small. Convergence to zero, i.e. exponential stability, requires k0 = 0. 9.6.1
Exponential
Stabilization
If k0 = 0, the proposed controllers are given by
p~(x)
:
-~(llxfl)BJPx.
In view of (9.20) exponential stability is achieved and we have the following corollary. C o r o l l a r y 9.11 Suppose the hypotheses of Theorem 9.10 are satisfied and ko = O. Then the closed loop system (9.5) is globally uniformly exponentially stable with rate c~.
9.7
Constrained
Controllers
and
Main
Result
Taking control constraints (9.3) into account, the proposed controllers are simply saturating versions of those presented in the previous section. They are given by
ui = p~(x) =
-p~sat(e-lBTPx) -
~ sat (~;-a'y~(llxlI)B~Px),
(9.21)
where 7i, Pi, and P and the saturation function 'sat' are as defined in Section 9.6, and fii := fii - Pi. (9.22)
The positive real scalar c is chosen sufficiently small to satisfy < e* := ac*2/ko,
(9.23)
with e* := min{ci : i = 1 , 2 , . . . , t }
(9.24)
where ci > 0 satisfies
~(~e~)c~ <_ and
(9.25)
186 T
1
hi := ,~rn~x(Bi PBi) ~,
,~ :-- )~min(P) -½
Remark. For constant 7_i, condition (9.25) simply reduces to ci < ..~i..... - 7___i~i " Before introducing the main result, consider any real scalar c >_ 0 and define the Lyapunov ellipsoid
E(c) := {x • ~R'~ : xr P~, < d ) . T h e o r e m 9.12 Consider an uncertain system described by (9.1), satisfying Assumptions 9.6-9.9 and subject to bounded control given by (9.21). Then the resulting closed loop system (9. 5) is uniformly exponentially convergent to B(re) with rate v~ and region of attraction Yl = g(c*).
Proof. First note that the constrained controller (9.21) always has the same direction as a corresponding unconstrained controller (9.16). Hence, whenever
~_,(llxll)
llBT Pxlt <_ k ,
(9.26)
the control given by the constrained controller (9.21) is the same as that given by an unconstrained controller (9.16) with
~'~(11~11) >_ ~'__.:(11~11)• Since
IIB, T P~ft 2 -< I1"11~
)~max(BiTpB') x T P x = A ? * T P x '
_< )~min(P)-IxT p x = )~xT Px,
and kli is a nondecreasing function, it follows that inequality (9.26) is satisfied whenever ' T Px)~1 < ~i. _7 i ( ~ ( x T p x ) ~))q(x (9.27) Since kli is a nondecreasing function, it follows from (9.251 that (9.27) is satisfied w h e n e v e r x T p x ~_ c i2 . Thus, whenever x T p x < e* , inequality (9.26) holds, and hence the control given by the constrained controller is the same as that given by an unconstrained controller. Proceeding now as in the proof of Theorem 9.10, one shows that the hypotheses of Theorem 9.4 hold with V(x) = x T p x , V = eko/a, and V = c*~. The requirement (9.23) on e guarantees that V < V. The next corollary yields exponential convergence with a ball as a region of attaction.
187 C o r o l l a r y 9.13 Suppose the hypotheses of Theorem 9.I0 are satisfied. Then the closed loop system (9.5) is uniformly exponentially convergent to B(re) with rate a and region of attraction .d = B(r*), where * ~
r*
t n \
1
Proof. Since x T p x <_ Am~x(P)[fxtt2, it follows that whenever Ilxtt ~ r*, one has x T p x <_c'2; hence B(r*) C g(c*). The result now follows from Theorem 9.12. 9.7.1
Special
C a s e : kli = 0
When the uncertainty in the system is due mainly to persistently acting disturbances (see example is Section 9.9) then the following condition, which is a stronger version of Assumption 9.8, may be satisfied: For each i = 1, 2 , . . . ,l, there exist scalars koi, k2i, with k2i < i,
(9.28)
Ilei(t, x, ud]l _ k0g+ k2~lluill
(9.29)
such that for a l l t EIR, x E I R n , u C I R m~. In this case we have kli = 0,
for i = 1 , 2 , . . . , l .
If one examines the proof of Theorem 9.10 closely, one will see that in this case, Q need only be positive semi-definite; in other words inequality (9.11) can be replaced with 2xT p f ( x ) <_ --2ctxT p x 4- c~xTp B B T Px. (9.30) Also, the expression for 7i simplifies to 7i = ~/2(1 - k~i).
9.8
Relaxation
of Assumptions
Assumptions 9.6-9.9 guarantee the following condition. C o n d i t i o n 9.14 There exist a positive-definite matrix symmetric P, matrices Bi, continuous functions ~/i and nonnegative scalars Pi, i = 1, 2,..., rn, a non-negative scalar ko, and a positive scalar ~ such that when control is given by (9.16) the resulting closed loop system (9.5) .satisfies
xT pF(t, x) <_ --o~xT p x -k ~ko.
188 A careful examination of the proof of the main results of the chapter shows that we only need the above condition to be satisfied. In this case, constraint (9.25) on ci is replaced with
i( c )ci < There is a large body of literature dealing with uncertain systems of the form x(t) = A(5(t))x(t) + B(5(t))u(t). (9.31) where the uncertain parameter vector 5(t) lies in some known compact set A and the matrix-valued functions A(-) and B(-) are continuous; see, e.g., (Barmish et al, 1987; Corless, 1993b; Petersen and Hollot, 1986; and Rotea and Khargonekar, 1989). In the literature on robust stabilization of these systems using quadratic Lyapunov functions, one commonly encounters a stabilizing controller of the form
u = -TBTpx, where B is a "nominal value" of/~(5), 7 is some positive real number, and P is a positive-definite symmetric matrix with the property that the closed loop system k(t) = [A(5(t)) - 7B(5(t))BTp]x(t) (9.32) satisfies P[A(5) - 7[~(5)BT P] + [A(5) - 7B(5)BT p]T p < 0 for all (f E A. Since A is compact and fi,(-) and B(.) are continuous, the above condition guarantees the existence of a positive scalar a such that P[A(5) - 7B(5)BT p] + [.A(5) - 7B(5)BT p]T p < -aP. Letting
F(t, x) = [A(5(t)) - 7B(5(t))BTP]x, one obtains
xT pF(t, x) ~_ --axT px. Thus, condition 9.14 is assured with 7/ = 7, pi = 0, k0 = 0, and Bi consists of the columns of B corresponding to ul. Hence the closed loop system (9.32) is globally uniformly exponentially stable with convergence rate a. Also one can apply the results of the Section 9.7 to compute a region of attraction when the control is subject to constraints.
189
9.9 S t a b i l i z a t i o n Motion
of Spacecraft Rotational
Here we consider the rotational motion of a spacecraft subject to three bounded control torques. We also suppose the spacecraft is subject to three bounded disturbance torques. We wish to consider the problem of designing controllers which stabilize the rotational motion of the spacecraft in the sense that the angular velocity is driven close to zero. Let 81,82, b3 be a triad of mutually orthogonal unit vectors fixed in the spacecraft and parallel to the spacecraft principal axes at the mass center. The control and disturbance torques are Utbl, u252, u3b 3 and elbl, e2b2, e353, respectively. Let the angular velocity of the spacecraft relative to an inertial reference frame be
zig1 + z 2 4 + z383 • Then the rotational motion of the spacecraft is described by xl
=
I2-- I3
x2z3
ul I"-1
+
(I3-I1"]
~
+
u2
= \ \ / ~
+
~
-I1- ,
e2 +
~,
where I1,I2, I3 > 0 are the principal moments of inertia of the spacecraft about its mass center. The condition x = 0 corresponds to no rotation of the spacecraft. We consider the control inputs to be constrained by
l~(t)l, I~(t)i, lu~(t)l<_1 and the unknown disturbance inputs to satisfy
i~(t)I, te2(t)l, 1~3(t)t < p < 1 with known bound p. This system is described by (9.1) with
f(z)=
B1
=
,
0
Ia ] XlX2 and
B2 =
o]
B3 ~
[i]
190 A F i ( t , x, ui) = B i e i ( t ) •
Clearly Assumptions 9.7-9.8 hold with kl/=O,
koi=p,
i=1,2,3
k21=O,
and this system falls into the special case discussed in Section 9.7.1. To demonstrate satisfaction of Assumption 9.6 we show that inequality (9.30) holds with
P=
[100] 0 0
h 0
0 /3
.
Since 2xTpf(x)
= O,
BTpx
= x
and
zWPx <
IMIIxll 2,
where I M := m a x { I 1 , / 2 , / 3 } ,
Im := min{I1, I2,/3},
it should be clear that, for any c~ > 0, inequality (9.30) is satisfied with ~r = 20lIM.
Consider any specified rate of convergence c~. Choosing any 7 >_ a i M
(9.33)
= 7/(1 - p),
(9.34a)
< c* = ( 1 - p ) 2 I ~ / 3 p I ~ a ,
(9.34b)
and letting
0<
c
the constrained controller is simply given by ui = - p s a t ( e - l x i )
( l - p ) sat(~xi).
-
(9.35)
These controllers guarantee exponential convergence with rate c~ to the ball of radius r, = (3p~/~I~)~
for all initial states in the ball of radius 1
3
r* = (1 - p ) I ~ / r ~ . R e m a r k . It can readily seen that any controller of the form (9.35) with any ~, ~ > 0, yields a closed loop system for which all solutions are bounded. To see this consider any ~ satisfying
191
1.2 ...... 1.1i
~0
/
0.8
0.7 ~// 0t
1~0
Figure 9.1
1=5 (see)
~'s
30
time
Response of nominal uncontrolled system
0 < a < max
{ ~(1 - p)
?~
(i~- pp)2Im } /-~
,
.
Then ~' and c satisfy (9.34a) and (9.34b), respectively. This implies that the closed loop system is exponentially convergent to B(rc) with region of attraction B(r*); hence all solutions originating within are bounded. Now note that r* can be chosen arbitrarily large by considering a > 0 sufficiently small.
B(r*)
9.9.1
Numerical
Simulation
Results
For numerical simulations we chose the following system parameters: I i = 3 k g m 2,
/ 2 = 4 k g m 2,
/3=5kgm 2
and initial condition, x(0)=[1
1
1]T.
Figure 9.1 illustrates the x3-response of the nominal uncontrolled system, i.e., u~(t) = e~(t) - 0. We considered the disturbance bound, p = 0.5 Nm. Figure 9.2 illustrates the x3-response of the uncontrolled system with the following constant disturbances: ei(t) = 0 . 5 N m , i = 1,2,3. It seems the response is unbounded. For controller parameters we chose c=0.02,
~=1.
16 14
1°
xt
t
J I
C
i
s
Figure 9.2
10
t's
time (see)
2'0
2'5
30
Response of disturbed uncontrolled system
Figure 9.3 demonstrates the effectiveness of the controller. The control history for u3 is illustrated in Figure 9.4.
Acknowledgements. This research was supported by the National Science Foundation under Grant MSS-90-57079.
9.10
Appendix
Consider l
xT pF(t, x) -- xT pf(x) + Z [xT PB'pi(x) + xT pBiei(t' x, pi(x))] . (9.38) i=l
For each i = 1 , 2 , . . . , l ,
xr l~s~e~ (t, x, p~(z) ) ~< IIB~TP~II Ile,(t, ~,P,(~))II
5 ItB, Tp~II (ko, ÷ kl,(ll~tl)It~11 + Since the direction of Pi (x) is opposite to that of
k2i
BiTpx, we have
x TPBipi(x) = -[[Bi TPxl[ I[pi(x)ll; also
IIp,(~)ll = p, ]lsat (~-~B*TPx)I[ + ~,(ll~ll)lIB, TPx[[. Hence.
ttp,(x)tl).
193
0.9 0.8 0.7
x 0,4 O,a 0,2 0,1
5
time (se¢)
Figure 9,3
£i(t,x)
Response of disturbed controlled system
xTpBipi(x)+xTpBiei(t,x,pi(x)) k0, liB, T Pxtt- (1 - k=,)p, llBFPxlt tlsat (~-1B~ ex)ll kli(J[xJl ) ]]BiTpx[I IJxJl- (1 - k2i)Ti([[xH)I[BiTpx[I 2 . (9.37)
:= <
+
Since # = Amin(Q) > o, one has
kli([[x[[) [[BiTpx[[ Ilxll < [(l/#)kli(llxll) 2 H.BiTpxll 2 + (/.t/l) HxH2] /2. Also, for M1 y C ]Rm~
tl~tl tlsat (~-~y)ll > tiyll -
~.
Substitution into (9.37) yields ~,(t, x)
<
ko, [[BJP~II -
- (1 - k~)p,
[(1 - k~,)'~,(l[~ll) -
[ll'FPxl[- ~] Zk~,(ll~ll)~/2~,] I1,, ~'pxH 2
+ (,/2z)ilxlI ~
<
ko,¢- (~/2)IIBJPxII = + (~/2z)llxll
Recalling (9.11) and noting that l
PBBT p
= E PBiBiT p'
xTQx
> Amin(Q) llx[[ 2,
i=1
one obtains
~
.
(9.38)
194
-0.4
-0.5
~'-0.6 =lI
~
-0.7
#
2
-0.8
-0.9
1'o F i g u r e 9.4
;~
time (sec)
~o
&
Control history for u3
l
2xTpf(x)
~
-2c~x T Px + E o" liB, T PxH ~ - ~ Hxll2 .
(9.39)
i----1
Substituting (9.38) and (9.39)into (9.36) yields 2xT p F ( t , x) ~_ --2axT p x + 2ek0.
References Barmish, B. R., Corless, M., Leitmann, G., 1987, A new class of stabilizing controllers for uncertain dynamical systems, SIAM Journal of Control and Optimization, 21,246-255; 1983, reprinted in Dorato, P., editor, Robust control, IEEE Press, New York. Blanchini, F., 1991, Constrained control for uncertain linear systems, Journal of Optimization Theory and Applications, 71,465-484. Corless, M., 1990, Guaranteed rates of exponential convergence for uncertain systems, Journal of Optimization Theory and Applications, 64, 481-494. Corless, M., 1993a, Control of uncertain nonlinear systems, ASME Journal of Dynamic Systems, Measurement, and Control, 115~ 362-372. Corless, M., 1993b, Robust stability analysis and controller design with quadratic Lyapunov functions, in Zinober, A. S. I., editor, Variable structure and Lyapunov control, Springer-Verlag, London. Corless, M., Leitmann, G., 1988, Controller design for uncertain systems via Lyapunov functions, Proceedings American Control Conference, Atlanta, Georgia.
195 Corless, M., Leitmann, G., 1990, Deterministic control of uncertain systems: A Lyapunov theory approach, in Zinober, A. S. I., editor, Deterministic control of uncertain systems, Peter Peregrinus Ltd., London, 220-251. Corless, M., Leitmann, G., 1993, Bounded controllers for robust exponential convergence, Journal of Optimization Theory and Applications, 76, 1-12. Corless, M., Leitmann, G., 1994, Componentwise bounded controllers for robust exponential convergence, Proceedings IEEE Workshop on Robust Control via Variable Structure and Lyapunov Techniques, Benevento, Italy. Dolphus, R. M., Schmitendorf, W. E., 1991, Stability analysis for a class of linear controllers under control constraints, Proceedings IEEE Conference on Decision and Control, Brighton, England, 1, 77-80. Franchek, M. A., Niu, W., 1993, Performance limitations imposed by actuator constraints, ASME Winter Annual Meeting, New Orleans. Gutman, P.-O., Hagander, P., 1985, A new design of constrained controllers for linear systems, IEEE IYansactions on Automatic Control, 30, 22-33. Hale, J., 1980, Ordinary differential equations, Krieger. Hached, M., Madani-Esfahani, S. M., Zak, S. H., 1990, On the stability and estimation of ultimate boundedness of nonlinear/uncertain dynamic systems with bounded controllers, Proceedings American Control Conference, San Diego, California, 2, 1180-1185. Lee, C. S., Leitmann, G., 1994, A bounded harvest strategy for an ecological system in the presence of uncertain disturbances, Proceedings of the international workshop on intelligent systems and innovative computations, Tokyo, Japan. Leitmann, G., 1979, Guaranteed asymptotic stability for some linear systems with bounded uncertainties, ASME Journal of Dynamic Systems, Measurements and Control, 101,212-216. Leitmann, G., 1990, Deterministic control of uncertain systems via a constructive use of Lyapunov stability theory, in Sebastian, H. J., Tammer, K., editors, System Modeling and Optimization, Lecture Notes in Control and Information Sciences, Springer-Verlag, Berlin, Germany, 143, 38-55. Leitmann, G., 1993, On one approach to the control of uncertain systems, ASME Journal of Dynamic Systems, Measurements and Control, 115,373380. Leitmann, G., Lee, C. S., to appear, Stabilization of an uncertain competing species system, Computers and Mathematics with Applications. Madani-Esfahani, S. M., Hui, S., Zak, S. H., 1988, On the estimation of sliding domains and stability regions of variable structure control systems with bounded controllers, Proceedings of the 26th Allerton Conference on Communication, Control, and Computing, Monticello, Illinois, 518-527. Madani-Esfahani, S. M., Zak, S. H., 1987, Variable structure control of dynamical systems with bounded controllers, Proceedings American Control Conference, Minneapolis, Minnesota, 1, 90-95. Petersen, I. R., Hollot, C. V., 1986, A Riccati equation approach to the stabilization of uncertain linear systems, Automatica, 22, 397-411.
196 Rotea, M. A., Khargonekar, P. P., 1989, Stabilization of uncertain systems with norm bounded uncertainty - a control Lyapunov function approach, SIAM Journal on Control and Optimization, 27, 1462-1476. Soldatos, A. G., Corless, M., 1991, Stabilizing uncertain systems with bounded control, Dynamics and Control, 1,227-238. Swei, S. M., Cortess, M., 1991, On the necessity of the matching condition in robust stabilization, Proceedings IEEE Conference on Decision and Control, 3. 2611-2614.
10. Quadratic Stabilization of
Uncertain Linear Systems Francesco A m a t o , Franco Garofalo, Luigi Glielmo and Alfredo Pironti 10.1 Introduction The concept of quadratic stability dates back to the late Seventies with the germinal papers by Leitmann (1979) and Barmish (1983). An uncertain linear system is said to be quadratically stable if there exists a quadratic time-invariant Lyapunov function whose derivative along the trajectories of the system is negative definite for all possible values of the uncertain parameters. In this case uniform asymptotic stability of the origin of the state space is guaranteed with respect to all possible time behaviours of the parameters within their bounding set. In the same way, considered a linear plant described by uncertain matrices, it makes sense to consider the problem of finding a linear feedback controller which quadratically stabilizes the overall closed loop system. Regarding the state feedback case, a sufficient condition guaranteeing quadratic stabilizability via linear control is the so-called Matching Assumption (see Barmish, 1983). However, since this structural requirement is too restrictive, the effort of the researchers has been devoted to understand when it is possible to state conditions which are both necessary and sufficient. In Khargonekar et al (1990) it is shown that when the dependence on parameters is in the so-called norm bounded one-block form, the stabilizing controller exists if and only if a suitable algebraic Riccati equation has a positive definite solution, while in Bernussou et al (1989) a system depending affinely on parameters is proved to be quadratically stabilizable if and only if a certain convex optimization problem involving linear matrix inequalities (LMIs) admits a solution. The output feedback case is much more complicated to deal with. There is no equivalent of the results holding in the state feedback case; however, as shown recently in Becket et al (1993), the use of parameter dependent dynamic controllers allows to design a quadratically stabilizing controller using a sort of separation principle, which again converts the original problem to the solution of an LMI based problem. In this chapter we consider the case in which the system and input matrices depend on parameters in the form of ratio of multiaffine polynomials. First, in Sections 10.2 and 10.3, we show that, in this case, it is still possible to solve the quadratic stabilization problem via convex optimization algorithms. Then we deal with the general case of non-multiaffine dependence. In Section 10.4 we introduce a procedure presented in Amato et al (1995) to cover the image of
198 a non-multiaffine matrix valued function by the image of a function depending multiaffinely on parameters; via this procedure we can substitute the original non-multiaffine system matrices with multiaffine ones and apply the previous algorithms. Some examples illustrate the theory.
10.2
Quadratic Stabilization via State Feedback of a Class of Uncertain
Linear
Systems Consider an uncertain system in the form 2(t) = A(p)x(t), wherez(t) E l R ' , p = ( p l p~ ... ters, R is a hyperrectangle, i.e.
tEIR +,
pET~cIR n",
(10.1)
Pnp )T is the vector of uncertain parame-
u := ~1,~1] × ~2,~2] × . , . ×
~_~,P~,],
(lO.2)
and A(. ) is continuous. We denote by p(i), i - 1 , . . . , 2n~, the vertices of T~. Definition 10.1 The uncertain system (10.1) is said to be quadratically stable if there exists a positive definite matrix P E ]Rn×n such that for all p E £(p) := - [ A T (p)P + PA(p)] > 0.
(10.3)
Quadratic stability guarantees uniform asymptotic stability of the linear time-varying system it(t) = A ( p ( t ) ) z ( t ) ,
t E IR+
(10.4)
for all Lebesgue measurable functions p : IR + --+ Tl, t ~-~ p(t). In the same way, given an uncertain linear plant in the form x(t) = A(p)x(t) + B(p)u(t),
(10.5)
with u(t) ~ ~'~, and A(.) and B(. ) continuous, we can state the following definition. Definition 10.2 The uncertain system (10.5) is said to be quadratically stabilizable via linear state feedback control if there exists a matrix K E IRm×n such that the unforced system, obtained from (10.5) letting u = K x , z(t) = (A(p) + B ( p ) K ) z ( t ) is quadratically stable.
199
We recall that quadratic stabilizability via dynamic, time-varying state feedback linear control implies quadratic stabilizability via memoryless, timeinvariant, linear state feedback control (see Petersen, 1988). Hence we do not lose any generality in Definition 10.2. On the contrary, as shown by Petersen (1985), quadratic stabilizability without any other specification does not imply quadratic stabilizability via linear control, hence this specification in Definition 10.2 is mandatory. From Definition 10.2 follows that the uncertain system (10.5) is quadratically stabilizable via linear control if and only if there exist a positive definite P E IR'~x'~ and a matrix K E IRmxn such that - [(A(p) + B ( p ) h ' ) T P + P(A(p) + B(p)K)] > 0,
Vp e T~.
(10.6)
As shown in Bernussou et al (1989), letting S = p-1,
L = K P -~
(10.7)
it is readily seen that system (10.5) is quadratically stabilizable if and only if the following problem admits a solution. P r o b l e m 10.3 Find a symmetric matrix S E IR~x'~ and a matrix L E IRmxn such that
i) s > o , . ii) £K(P) := -- [SAT(p) + A ( p ) S + LTBT(p) + B(p)L] > O,
Vp e Tt.
Now we focus our attention on particular classes of matrix valued functions. D e f i n i t i o n 10.4 The matrix-valued mapping M : IR ~ -+ IR hxk, p E Tt ~+ M(p), is said to be multiafIine in p if it is aJ]fine in each variable pi, that is 1
M(p) =
. . . . p',,7.
(10.S)
i l ,...,i,~1, =O
In the sequel we shall need the following well known results. F a c t 10.5 A set of matrices is positive definite (i.e. each matrix in the set is positive definite) if and only if its convex hull is positive definite. When the convex hull is a polytope, the set of matrices is positive definite if and only if the vertices are. F a c t 10.6 (Zadeh and Desoer, i963). Let M : IR~ -+ lR hxk, p E Tt ~-+ M(p) be multiaJ~ne; then
co({M(p),pele})=co({M(p(,)),i=l,
(10.9)
200
where 'co ( . ) ' denotes the operation of taking the convex hull of the argument. A corollary of Facts 10.5 and 10.6 is the following lemma. L e m m a 10.7 (Garofalo et al, 1993). Assume that F : IRn~ --+ IRh×h is the ratio of multiaffine polynomials in p, that is
1 E Fili2,,,inpPl I'" •PnpZ~P NF(p) ._ i, .....i.,=o F(p)--" dF(p) .-1
E
(10.10)
p':;
il,...,inp =0
where fili2...i,p E IR and dF(p) ~ 0 for all p ET¢. Then F(p) > 0 for all p E 7~ if and only if F(p(i)) > 0 for all i = 1,2,...,2 n~. Proof.
We must show that the condition F(p(i)) > 0,
i = 1,..., 2np
(10.11)
implies the condition F(p) > 0,
Vp E 7~.
(10.12)
Since dR(p) • 0 in T~, we have that dF(.) does not change sign in 7~. Assume that dF(p) > 0, Vp E n ; (10.13) this implies that dF(p(o ) > 0 for i = 1,..., 2np. This fact, together with (10.11), implies that NF(p(,)) > 01 i = 1,..., 2np . (10.14) Now, since NF(. ) is multiaffine, using Facts 10.5 and 10.6, we can conclude that (10.14) implies NF(p) > 0, Vp E 7~. (10.15) Condition (10.15) together with (10.13) guarantees (10.12). When dF(p) < 0 in 7~ the proof is analogous. We remark that (10.10) is the largest class of symmetric matrix functions for which it is possible to state vertices results as in Lemma 10.7. Using Lemma 10.7, we can readily prove the following theorem. T h e o r e m 10.8 Assume that A and B are ratio of multiaffine polynomials in p with denominators nonzero in T¢. Then system (10.5) is quadratically stabilizable via linear control if and only if there exist S > 0 and a matrix L E IRm×n such that for alli = 1,...,2n~ £K(P(O) > 0.
(10.16)
201 Proof. The hypothesis ensures that /:K (") is the ratio of multiaffine polynomials in p; therefore, by virtue of Lemma 10.7, it is positive definite in T¢ if and only if condition (10.16) holds.
Note that the application of Theorem 10.8 requires to check that the denominators of A(. ) and B(- ) are nonzero in T~. Since these denominators are multiaffine functions of p, this check can be done in a simple way by using an obvious corollary of Facts 10.5 and 10.6. C o r o l l a r y 10.9 Let us consider a multiajCfine scalar function f : IR nv -+ ]R. Then f ( p ) # 0 for all p E Tt if and only if either f(p(~)) > 0 for all i = 1 , 2 , . . . , 2 n p or f(P(i)) < 0 for all i = 1 , 2 , . . . , 2 n~. Remark. Theorem 10.8 includes as particular case Theorem 1 in Bernussou et al (1989), where the affine dependence on parameters was considered.
By virtue of Theorem 10.8, when A and B depend muttiaffinely on parameters, Problem 10.3 is equivalent to the following. P r o b l e m 10.10 Find a symmetric matrix S E ]RÈx~ and a matrix L E IRm×'~ such that
i) s > 0 ; i = 1 , . . . , 2 ~p .
ii) £K(P(i)) > O,
Problem 10.10 is a feasibility problem involving Linear Matrix Inequalities (LMIs), and it can be solved by one of the algorithms in Gahinet et al (1995). E x a m p l e 10.11
Let
A(p)
=
B(p)
=
0 0)
1 a32(p) (1
-1 1
,
1 2) 'r
(10.17a) (10.17b)
where all(P) aa2(p)
= =
--1 +pip2 1.79 + 0.84p2 -- 0.01p3 -4- 0.21p4 + 0.09p2p3 + 0.01pap4
with p E 7"/:= [1, 2.22] x [0, 0.8] × [0, 1]2. The LMI Toolbox routine f e a s p solves Problem 10.10, and the resulting K is: K = (-24.8003
29.3263
-7.2079).
(10.18)
202 When the dependence on parameters of the system matrices in equation (10.5) is more "complex" than the ratio of multiaffine polynomials, constraint (ii) in Problem 10.3 is no longer reducible to a finite number of convex constraints as in (ii) of Problem 10.10, with the obvious consequence that the LMIs formulation cannot be utilized.
10.3 Quadratic Stabilization via O u t p u t Feedback In this section we consider a system in the form
y
= =
A(p)x + B(p)u, C(p)z+D(p)u.
(10.19a) (10.195)
Since in this case the full state is not available for feedback, we need to search for dynamic output feedback controllers. To this aim we state the following definition.
Definition 10.12 System (10.19) is said to be quadratically stabilizable via parameter dependent output feedback linear control if there exists a dynamic compensator K(s,p) such that the feedback connection of system (10.19) and K(s, p) yields a quadratically stable closed loop system. Note that in the above definition the controller is allowed to depend on the uncertain parameters; therefore we implicitly assume to use a gain scheduled control scheme. The next result is due to Becker et al (1993). T h e o r e m 10.13 System (10.19) is quadratically stabilizable if there exist positive definite matrices P and Q and matrices K and N such that, for all p E T~, - [(A(p) + B ( p ) K ) ~ P + P ( A ( p )
+ B(p)K)]
-[(A(p)+NC(p))TQ+Q(A(p)+NC(p))]
>
0,
(10.20a)
>
0.
(10.20b)
In this case a quadratically stabilizing controller has the following structure ~¢
=
g(p)xc + B(p)u + g[C(p)x¢ + D(p)u - y] ,
(10.21a)
u
--
Kxc.
(10.21b)
Note that (10.21) has the classical state feedback-state observer structure. Moreover Theorem 10.13 shows that we can design separately the feedback and the observer part of the controller, as in the classical Separation Principle.
203 Now, following again Bernussou and Geromel (1989), we let S := p - l ,
L := K P - t ,
W := Q N ,
(10.22)
so that conditions (10.20) become
f--g(P)
:=
-- [SAT(p) + A(p)S + LT BT(p) + B(p)L] > 0,
£N(P)
:=
-- [AT (p)Q + QA(p) + CT (p)W T + WC(p)] > 0 (10.235)
(10.23a)
If the dependence of system matrices on parameters is the ratio of multiaffine polynomials, using again Theorem 10.8 we can conclude that it is necessary and sufficient to check conditions (10.23) at the vertices of 7~. In this case we can derive the following result. C o r o l l a r y 10.14 Assume A(. ), B(. ), C(. ) and D(. ) are ratio of multia]fine polynomials. Then system (10.19) is quadratically stabilizabte if there exist
i) a positive definite matrix S and a matrix L such that the following 2~ constraints are satisfied f-.K(P(i)) :> 0, i = 1,...,2riP;
(10.24)
ii) a positive definite matrix Q and a matrix W such that the following 2~ constraints are satisfied g~N(P(i)) > O, i = 1 , . . . , 2 rip.
(10.25)
E x a m p l e 10.15 Consider system (10.19) with A(. ) and B ( - ) given in (10.17), and C=(1 1 0) . (10.26) Using the LMI Toolbox, a matrix gain N solving (10.25) is given by
: (1 ::41
\ 40.03 ]
Therefore a quadratically stabilizing controller is given by (10.21), with K and N given by (10.18) and (10.27) respectively.
10.4 A G e n e r a l P r o c e d u r e In this section we treat the case in which the system matrices in (10.5) and (10.19) depend non-multiaffinely on parameters. The class of functions considered is specified in the next assumption. Note that we consider, for the
204 sake of simplicity, the state feedback case. By virtue of Theorem 10.13 the procedure is immediately generalizable to the output feedback case. A s s u m p t i o n 10.16 The matrix valued functions A ( . ) and B ( . ) in (10.5) can be written in the following way
A(p)
=
E
• ' i~(p) , Ail,i2 ..... i~a]t(P)a~(P)'"au
(10.28a)
Bil,i2 .....i, bi11(P)b~2(P)'"b~"(P),
(10.28b)
I1 ~$21-,.llv
B(p)
=
E I1 lt2~---~Itl
where is E {0, 1}, Ai~,i~.....i~ 6 IRn×", Bi~,i~.....i, E IR "×m and the scalar functions aj and bj are continuous. For j = 1 , . . . , v, if as is not multiaj~ne it is assumed to be convex or concave; the same holds for the functions bs. Note that Assumption 10.16 includes as a particular case the multiaffine dependence but not the dependence (10.10). We could further generalize the structure in Assumption t0.16 considering the ratio of non-multiaffine functions; however this is not done in this chapter to avoid cumbersome notation.
Remark. It is interesting to note that many functions are convex or concave only if the parameters on which they depend do not change sign. This fact does not cause any loss of generality in practical applications because in many situations the parameters have a physical significance and hence are inherently positive. In any case we can always split, prior to the application of the algorithm, the original hyperrectangle in smaller hyperrectangles, each contained only in one orthant of 113.np .
10.4.1 An Algorithm to Cover the Image of a Non-Multiaffine Function by the Image of a Multiaffine Function Consider a matrix valued function
F(p) = E F,~,,~ .....i. f~' (P)f~:(P)"" f ~ ( P )
(10.29)
satisfying Assumption 10.16. The next algorithm (Amato et al, 1995) constructs a multiaffine function whose image covers that of F(- ).
Algorithm 10.17 Step 1 For j -- 1 , . . . , v , if fj(p) in (10.29) is not multiaffine, appl._y the procedure detailed below to construct a pair of aJ:fine functions f_j and fd such that
L(p) _
Vp ~ n .
(10.30)
205 We show how to find these bounding functions in case of convex functions (the procedure is similar for the concave case). Let p* E 7~ and 7rj(p*) the gradient offj(p) in p*. The following inequality holds (see Rockafellar, 1970, Theorem 25.1)
fj(p) _> fj(p*) +
pen.
(i0.3i)
Then set
f_j (p) = fj (p*) + 7rj (p*)(p - p*).
(10.32)
Now consider the images of the vertices of 7~ under fj, namely fjt := fj(P(t)) ,..., fj2~p := fj(P(~'~p)), and order them with decreasing magnitude, say fjhl >_ ... >_ fjh~p. Simple considerations from linear algebra show that there exists an integer r E {np q- 1 , . . . , 2n,} such that the hyperplane hi(P) in the space IR~'+1, connecting the points (PYh~),fiat),..., (PT(h,), fjh~), is univocally determined. By virtue of the convexity of fj we have that, for all p E 7~, fj (p) <_ hj (p); hence set 7 (p) = hi(p). (10.33)
Step 2 Construct the matrix function ~(p, 5) obtained from (10.29) substituting for fj (p), j = 1 , . . . , u, the multiaffine function ( 1 - S j ) f j ( p ) + Sj-f j(p)
[
fj(p)
-
if fj is not multiaNne, if fj is multiaNne,
(10.34) where the f_j's and the f j ' s are affine functions constructed according to Step 1 and [0, 1] if fj is not multiattine (10.35) 5j C Ij := {0} if fj is multiamne. Let 7) := I1 x h x --- x I,; hence g~(p,5) is defined over 7~ x 7). It is simple to recognize that
× 7)) n F ( n ) . Observe that ~(p,5) is an a n n e function of the 5i's and a polynomial function of the pi's; moreover since in (10.29) there are at most ~ products, the maximum exponent of each Pi cannot exceed p.
Step 3 Replace each Pia~ , ai > 1, with a multiaNne function as in (10.34); obviously in this case the bounding functions (which can be easily computed by applying Step 1) will depend only on pi. The number of replacements, say 7, may vary between zero (when ~ is already multiaNne, that is ~ does not contain powers of p l , . . . , P n p greater than 1) and nv(u - 1). In this way we introduce a new vector of fictitious variables e ranging in the hyperrectangle £ defined in the same way as 7) and define the new function
206 ¢(p, ~, e); in Theorem 3 of Amato moreover
et al (1993) it is shown that ~ is multiaffine,
F(n).
2
Remark. Note that 7~ x 7) x g is a hyperrectangle with 2~+u+~ vertices, where # is the number of non-muItiaffine functions in the expression (10.29) of F and, as said, 7 is the number of replacements in Step 3 of Algorithm 10.17. 10.4.2
Solution
of Problem
10.3 in the
Non-Multiafflne
Case The solution of Problem 10.3 when A ( . ) and B ( . ) satisfy Assumption 10.16 is composed of two steps:
Step a) Apply Algorithm 10.17 to construct two multiaffine functions
~A(P,~A,eA) : T~ X ~)A X gA
~
I~nxn ,
such that
¢A(7~,Z)A,gA) D_ A(Tt),
(10.36)
~B(~r~,Z)B,gB)
(10.37)
~
B(Tt).
Let us define ~ := (~T ~T) T e ]R t~A'I'#B and e := (e T eT) T e ]RYA+"/B; in the same way we define :D : = ~)A X "I)B and g := CA x EB; note that :D and £ have 2 uA+'B and 27A+7s vertices respectively. Finally let £2 := 7~ x 2) x g and 2"~ = 2np+ÈA+~s+TA+7B the number of its vertices. In order to simplify the notation, we will consider ~A and ~B functions of the whole vector w =
( pT
¢~T ET ) T"
Step b) Define • ~:K (w) := s~T(w) + ~A(W)S +
LTq~T(w) +q~B(w)L.
(10.38)
Then solve Problem 10.10 with ~zK replacing ~ g . In this case, the existence of a solution of the problem is no longer equivalent to the quadratic stabilizability of the system under consideration. This is obviously due to the fact that the immersion of the images of A and B introduces conservatism. In the next section we shall show how to reduce this conservatism.
207 E x a m p l e 10.18 We present an example to clarify how the algorithm works when the system depends non-multiaffinely on parameters. Consider again syst e m (10.17) with all(P)
=
--l+exp(pl)P2,
a32(p)
=
2 +p2sin(p2)
with p E [0, .8] 2. Note that B does not depend on parameters, so that ¢B(W) = B; moreover, without loss of generality we can replace an(p) with
where/5 = (/51 P2 ) and/51 • [exp(0), exp(0.8)]. Now we have to compute g'A(a,'). T h e function A ( - ) can be written in the standard form (10.28) by letting a1(/5) =/51 ;
a2(/5) = P2;
a3(/5) = sin(p2).
(10.40)
Step 1 In this case the non-multiaffine functions is aa. Using Step 1 of Algorithm 10.17, a choice of the bounding functions is a_a(p ) = 0.84p~ ;
g3(P) = 0.05 + 0.88p2 •
(10.41)
Step 2 Substituting a T to a3 we obtain
all(f) ~A (/5, 6) =
o o
0 0) 1
~32 (/5, 6)
-1 1
,
with ¢32(/5, 5) = 2 + 0.05p253 + 0.84p~ + 0.04p~53.
(10.42)
Step 3 T h e only non-multiaffine function in (10.42) is 9(P) = P~; as bounding functions we choose g_(p) = - 0 . 2 5 +p~ and g(p) = P2. Substituting 9__and g in (10.42) we obtain that
~A(f, 6, c ) =
(11tt 0 0) 0 0
1 ¢32(f, 6,~)
--1 1
,
where ¢32(/5, g, e) = 1.79 + 0.84p~ - 0.015a + 0.2tc + 0.09p253 -4- 0.0153~.
(10.43)
Note t h a t ¢A is muttiaffine, moreover @A : D --+ IR 3×3 where := 7~ x / ) x g = [1, 2.22] x [0, 0.8] x [0, 1] x [0, 1].
(10.44)
We have reduced this example to Example 10.11 with p3 = 5a, and P4 Therefore a quadratically stabilizing K is given by (10.18).
:
E.
208
10.4.3 A Refinement Procedure For a given S and L, recalling that OzK(12) ~ ~ K ( ~ ) and that 4~z~,. is multiaffine, application of Facts (10.5) and (10.6) enables to conclude that co (£K(7~)) C_ co ({OZK (w(i)), i = 1 , . . . , 2"~ }) ,
(10.45)
where, as usual, we denote the vertices of 12 by w(i). In the next theorem we show that, if we split the hyperrectangle T~ in smaller hyperrectangles and consider the union of the images of these subhyperrectangles, the right hand side in (10.45) converges to the left hand side when the splitting is made finer and finer. We say that T = { ' ~ 1 , ~T~2, • . . , ~r~l} is a rectangular covering of the hyperrectangle T~ if Ulr=lT~r = 7~. We define T as the set of all rectangular coverings of 7~. Now let C(IRn) the set composed of all closed sets in lR'~. We equip this set with the Hausdorff metric
dH(S1,S2) := m a x { maxd(xl,S2), maxd(x2, S1)} kxiES1
x2ES2
(10.46)
with $1, $2 e C(lRn) and where d(x, S) denotes the usual Euclidean distance of the point x from the set S. Let {Th}he~, 7h E T, a sequence of rectangular coverings, set up as follows. T1 = {7~}; 72 is constructed by splitting 7~ into 2nv hyperrectangles by lines parallel to the coordinate axes; 7~ is obtained applying the same procedure to each hyperrectangle in T2 and so on; note that 7h is composed of 2 (h-1)nv hyperrectangles; the partition is set up in such a way that the Lebesgue measure of each hyperrectangle goes to zero when h --~ +oc. We denote by T~h~ the r-th hyperrectangle in 7~. T h e o r e m 10.19
(Amato et al, 1995) Consider the sequence {7~}he~- Then
2(h-D-p
lim
h--+~
U
i = 1,..., 2"-}
) = LK(n)
r=l
in the metric space (C(IR'), dH), where ~ : is the function constructed on T~hr according to Algorithm 10.17 and w~ is the i-th vertex of 12hr := T~hr x 7:) x g. Based on Theorem 10.19 we propose an algorithm for the solution of Problem 10.3 when the system matrices are non-multiaffine and satisfy Assumption 10.16. Given an ordered set $, Sj denotes the j-th element of $ and card(S) the number of elements of S.
209
A l g o r i t h m 10.20 Step 1 (Initialization) Let S = {T~}, h = 0.
Step 2 Let h e- h + 1; for j = 1 , . . . , c a r d ( S ) := mh repeat the followi.ng procedure: apply Algorithm I0.17 to A(- ) and B ( - ) restricted to Sj and solve the following problem. P r o b l e m 10.21 Sufficient condition Find S > 0 and L E IR'nx'~ such that
~Ei~.(Wj(i))
> 0, i = 1 , . . . , 2 '~r2 , j = 1 , . . . , m h ,
where a~j(i) is the i-th vertex of £2j := Sj x 7) x g. If Problem 10.21 admits a solution then s t o p , the system is quadratically stabilizable and K = LS -1 is a compensator which quadratically stabilizes the system; otherwise test the necessary condition described below. P r o b l e m 10.22 Necessary condition Find S > 0 and L E IRmx~ such that
£K(Pj(i)) > _ d , i =
1,...,2np,j=
1,...,mh,
where Pj(i) is the i-th vertex of Sj. If this problem does not admit solution then s t o p , the system is not quadratically stabilizable; otherwise goto Step 3.
Step 3 Split each element belonging to S into 2"~ hyperrectangles, put them into S and go to Step 2. When Algorithm 10.20 does not converge in a finite number of steps, the solutions of Problem 10.22 return a pair of sequences of positive definite matrices {S N} and {LN), where S g and L N are the solution of Problem 10.22 at iteration N. T h e o r e m 10.23 (Amato et al, 1996) If system (10.1) is quadratically stabilizable then Algorithm 10.20 converges in a finite number of iterations; i] system (10. I) is not quadratically stabilizable either Algorithm 10.20 converges in a finite number of iterations or at least one of the sequences {S N } and {L N } is unbounded.
Proof. If system (10.5) is quadratically stabilizable there exist S > 0 and L E i~rn ×n such t h a t / 2 g (p) > 0 for all p E TO. Following continuity arguments there
210
exists an open set O D 7~ such that system (10.5) is quadratically stabilizable in ~9. Obviously/:K(O) D/:K(T~); therefore by virtue of Theorem 10.19 there will exist an index s such that 2(g-- 1)r~lo
first
r=l
At iteration s Problem 10.21 admits a solution and therefore Algorithm 10.20 stops. Suppose now that system (10.5) is not quadratically stabilizable and that the algorithm does not terminate in a finite number of steps. Assume, on the contrary, that the sequences {S N} and {LN} are bounded. From these sequences we can extract a pair of subsequenees {S yt }, {L N~} which converg_.e to and T respectively. The way Problem 10.22 is stated ensures that S and L are a solution of Problem 10.3. Hence, either {S N} or {L N } must be unbounded. The weak point of the procedure presented in this section relies in the fact that the number of points to test (which corresponds to the number of constraints to satisfy in Problems 10.21 and 10.22) grows exponentially with the number of parameters. However, as shown in Amato et al (1995), the covering performed by Algorithm 10.17 is very tight. This, in many practical situations, makes Algorithm 10.20 converge in a few iterations, before the computational burden becomes too hard.
10.5 C o n c l u s i o n s In this chapter we have considered linear systems subject to parametric uncertainties. The quadratic stabilization problem can be solved in an elegant way, via convex optimization procedures, if the dependence of the system matrices on parameters is the ratio of multiaffine polynomials. In the other cases we can use an algorithm which, under mild assumptions, transforms a given nonlinear matrix function into a multiaffine one. Via this algorithm and at the price of some conservatism, we can solve the quadratic stabilization problem referred to a fictitous system depending multiaffinely on parameters. An algorithm to reduce such conservatism has also been presented.
References Amato, F., 1994, Stability analysis for linear systems depending on uncertain time-varying parameters, PhD Thesis [in Italian], Dipartimento di Informatica e Sistemistica, Napoli. Amato, F., Garofalo, F., Glielmo, L., 1993, Polytopic coverings and robust stability analysis of linear systems, in Zinober, A. S. I., editor, Variable Structure and Lyapunov Control, Springer Vertag, London.
211 Amato, F., Garofalo, F., Glielmo, L., Pironti, A., 1995, Robust and quadratic stability via Polytopic set covering, Int. J. Robust Nonlinear Control, 5,745756. Amato, F., Corless, M., Mattei, M., Setola, R., 1996, A muttivariable stability margin in the presence of time-varying, bounded rate gains, Submitted for publication. Barmish, B. R., 1983, Stabilization of uncertain systems via linear control, IEEE Transactions on Automatic Control, 28,848-850. Becket, G., Packard, A., Philbrick, D., Balas, G., 1993, Control of parametrically dependent linear systems: A single quadratic Lyapunov approach, Proceedings American Control Conference, San Francisco. Bernussou, J., Peres, P. L. D., Geromel, J. C., 1989, A linear programming oriented procedure for quadratic stabilization of uncertain systems, Systems and Control Letters, 13, 65-72. Boyd, S., E1 Ghaoui L., Feron E., Batakrishnan V., 1994, Linear Matrix Inequalities in System and Control Theory, SIAM Press. Corless, M., 1993, Robust stability analysis and controller design with quadratic Lyapunov functions, in Zinober, A. S. I., editor, Variable Structure and Lyapunov 6~ntrol, Lectures Notes in Control and Information Sciences, SpringerVerlag, London. Garofalo, F., Celentano, G., Glielmo, L., 1993, Stability robustness of interval matrices via Lyapunov quadratic forms, IEEE Transactions on Automatic Control, 38, 281-284. Gahinet, P., Nenirowski, A., Laub, A. J., Chilali, M., 1995, LMI Control Toolbox, The Mathworks. Geromel, J. C., Peres, P. L. D., Bernussou, J., 1991, On a convex parameter space method for linear control design of uncertain systems, SIAM Journal on Control and Optimization, 29,381-402. Leitmann, G., 1979, Guaranteed asymptotic stability for some linear systems with bounded uncertainties, ASME Journal of Dynamic Systems, Measurements and Control, 101,212--216. Khargonekar, P. P., Petersen, I. R., Zhou, K., 1990, Robust stabilization of uncertain linear systems: quadratic stabilizability and 7/~ control theory, IEEE Transactions on Automatic Control, 35, 356--361. Petersen, I. R., 1987, A stabilization algorithm for a class of uncertain linear systems, Systems and Control Letters, 8,351-357. Petersen, I. R., 1985, Quadratic stabilizability of uncertain linear systems: existence of a nonlinear stabilizing control does not imply existence of a linear stabilizing control, IEEE Transactions on Automatic Control, 30,291-293. Petersen, I. R., 1988, Quadratic stabitizability of uncertain linear systems containing both constant and time-varying uncertain parameters, Journal oJ Optimization Theory and Applications, 57,439-461. Rockafetlar, R. T., 1970, Convex Analysis, Princeton University Press, Princeton. Zadeh, L., Desoer, C. A., 1963, Linear System Theory, McGraw-Hill, New York.
11. P i e c e w i s e - l i n e a r F u n c t i o n s in Robust Control Franco B l a n c h i n i a n d S t e f a n o M i a n i 11.1 I n t r o d u c t i o n In the last twenty years the control of uncertain systems has been one of the most intensively investigated problems in the automatic control theory. This subject is strongly motivated by the control engineering practice when the control designer has often to cope with inaccurate modeling, disturbances and neglected nonlinearities which render the classical control design techniques often not applicable. Moreover the subject of robust control analysis and design has played a fundamental role in deep understanding basic phenomena occurring in system dynamics. As it is well known, several different approaches have been followed to obtain good robustness analysis tools and robust design methods. Investigating or even mentioning all of them is far out of the scope of this chapter whose main purpose is to describe some recent developments in the area of robust control via Lyapunov techniques. We rather refer to good tutorial papers (see Dorato et al, 1993; Siljak, 1989) while limiting ourselves to mention some connections with other areas whenever they arise. The intensive analysis and synthesis of systems with parametric uncertainties via Lyapunov techniques started essentially twenty years ago with the pioneer works by Leitmann (1979a, 1979b) and Gutman and Leitman (1975). These results involving Lyapunov functions where natural developments of previous results concerning dynamic game theory. Since then a lot of researchers have been involved in the subject of Lyapunov based robust control systems. In particular we recall the works by Barmish (1985), where necessary and sufficient conditions for quadratic stabilizability via state feedback have been given, Petersen and Hollot (1986), who proposed a Riccati equation approach, and Rotea and Khargonekar (1989), who showed that it' the uncertainties are unstructured then robust stabilizability implies robust stabilizability via linear state feedback. The problem of the robustness analysis via quadratic Lyapunov functions received at the same time a tot of attention (see Patel and Toda, 1980; Yedavalli and Liang, 1986; Sezer and Siljak, 1989; Garofalo et al, 1993) as well. We stress that the list provided here is far to be complete and the reader is referred to the tutorial work by Corless (1993). It turns out that the class of Lyapunov functions that have been mainly considered are the quadratic ones. The quadratic functions have their force in the fact that they can be managed via the well known linear Lyapunov equation. However it can be shown, by very simple counterexamples, that the results obtained via quadratic Lyapunov functions can be conservative with respect
214 to the results obtained via wider classes of Lyapunov functions (Zelentsowsky, 1994; Olas, 1991). For instance in the paper by Zelentsowsky (1994) it has been given an example in which the stability margin has been improved of the 50% passing from quadratic to homogeneous polynomial functions. The question therefore arises if there are other classes of functions which turn out to be less conservative and thus may be more suitably used for analysis and synthesis. This question has a positive answer and the candidate functions we consider here are the piecewise linear (polyhedral) functions. By polyhedral function we mean a function which is positively homogeneous and whose level surfaces are the boundary of polytopes (thus having the same shape). Polyhedral functions for the robustness analysis problem have been considered by (Barabanov, 1988; Molchanov and Pyatnitskii, 1986; Gerasimov and Yunger, 1993; Brayton and Tong, 1980; Michael et al, 1984; Ohta et al, 1993; Bhaya and Mota, 1994). In recent contributions by Blanchini (1993, 1995) it has been proved that these function are universal in the sense that if a system is stabilizable via static state feedback via a Lyapunov function, then it can be stabilized via a polyhedral Lyapunov function and a piecewise linear state feedback control (i.e. a control which is linear in some subsets of a proper partition of the state-space). The polyhedral Lyapunov functions and the associated polyhedral invariant sets where previously used also for the control of systems with constraints by (Benzaoiua and Burgat, 1989; Tan and Gilbert, 1991; Castelan and Hennet, 1992; G u t m a n and Cwikel, 1986; Vassilaki and Bitsoris, 1989; and Sznaier, 1993). In this chapter we provide a summary of recent results concerning robust analysis and synthesis for uncertain systems via piecewise linear (polyhedral) Lyapunov functions. In Section 11.2 we recall some basic properties of the polyhedral functions and their use in the solution of robust analysis problems. In particular we express some conditions in terms of matrix equalities and inequalities which assure that a certain polyhedral function is a Lyapunov function for a linear uncertain system. In Section 11.3 we show how to construct a polyhedral Lyapunov function. In Section 11.4 several applications to the analysis of robustness for systems with time-varying parametric uncertainties will be presented, such as the determination of the "best" transient estimate and the determination of the largest domain of attraction under state a n d / o r control constraints. Then we consider the problem of robust synthesis in Section 11.5. We recall the basic result of polyhedral Lyapunov functions known as their universality property that means that if the robust stabilization via state feedback can be achieved via a Lyapunov function then it can be achieved via a polyhedral Lyapunov function associated to a proper piecewise linear controller. We provide equations which state necessary and sufficient conditions for a polyhedral function to be a Lyapunov function. Finally we describe a procedure for the generation of polyhedral Lyapunov functions and we analyze several (nonlinear) controllers which can be associated to this kind of Lyapunov functions. We describe an on-line optimization based control, a piecewise linear variable structure controller and a bang-bang control. Several numerical exam-
215 pies will be provided to illustrate the described methods. Finally the directions for further research on the subject will be pointed out.
11.2 11.2.1
Preliminary
results and notations
Notations
In this chapter, we use the following symbols. Unless differently specified, calligraphic letters ,4, B , . . . , denote sets, and capital letters A, B , . . . , denote matrices. Given a set S, we denote by 0 8 its boundary, by int{$} its interior, by conv{S} its convex hull. If S is a polytope, we denote by vert{S} the set of its vertices. The scaled set A8 is defined as AS = {x := Ay, y E 8}. We call C-set a convex and compact set having the origin as an interior point.. We define the infinity-norm of a vector x as Ilxll~ = supi Ix~l. The inequality A _< B between matrices or vectors has to be intended componentwise. The acronym L P means linear programming. Assume that f : IR --+ IR is a continuous function. We define D + f ( t ) as the upper right Dini derivative of f
7?+ f ( t ) := lira sup f ( t + r) - f ( t ) "t-+O+
(11.1)
T
Given a continuous function !P : IR~ --+ 1t{, and a real constant k, we define the (possibly empty) open set N[~,k] := t x ~ n ~ : ~(x) < k}, and its closure
~[~, k] := {~ ~ ~
: ~(~)
< k}
Definition 11.1
We say that ~ : IR ~ -+ IR, is a gauge function if, for every x, y E IR ~, it fulfills the following properties
~(~) > o, k~(Ax) = A~(x),
for every A >_ 0,
~ ( , + y) < e(~) + ~(y). It follows from the definition that a gauge function is convex and #(0) = 0. Moreover, any norm is a 0-symmetric (in the sense that # ( x ) = ~ ( - x ) ) gauge function and any 0-symmetric gauge function is a norm. If g7 is a gauge function, then 2q[#, k] is a C-set for all k > 0. Conversely any C-set S induces a gauge function, known as the Minkowski functional of S, which is defined as ~(x) := inf{p > 0 : x E p 8 } . So a C-set $ can be thought as the unit ball S = ~T[~, 1] of a gauge function k~ and x E S iff k~(x) < 1. Clearly a 0-symmetric (i.e. such that
216
x E S ~ - x E S) C-set induces a norm. A polyhedral C-set :P in IR'~, can be represented in the standard face representation
P={xEIRn:Fix~_I, i= 1,...,s}, or synthetically P = {x E m " :
F x < i},
(11.2)
where Fi is the i-th row of F and i is the vector defined as i:=[11
...
1] T .
Alternatively, we can represent the set 7) in terms of its vertices {zl, i = 1,...,r}
t)
= {X =
fi
OtiXi,
i=1
fi
O~i = 1, ai >__0,
i = 1,...,r},
i=1
or synthetically, denoting by X = Ix1 x ~ . . . xr] the matrix whose j-th column is x j, P={x=Xa, iTa=l, a_>0}. (11.3) We call polyhedral function the Minkowski functional of a polyhedral C-set. This function has the expression ~P(x) - max Fix
(11.4)
1
or, in a dual form ~(x)=min{tt:x=Xa,
iTa=#, ak0}.
(11.5)
Finally in the sequel we will denote by Z(x) the set of indices in which the maximum in (11.4) is achieved, i.e. Z(x) = {i: Fix = ~P(x)}. 11.2.2
Systems
under
consideration
and
(11.6) main
definitions
In this section we start by describing the class of systems under consideration, then we introduce some basic concepts which are used in the sequel. We consider the class of linear uncertain systems of the form
&(t) = A(w(t))x(t) + B(w(t))u(t) + Ed(t),
(11.7)
x(t + 1) = A(w(t))x(t) + B(w(t))u(t) + Ed(t).
(11.8)
or of the form
where x E IRn is the state, u E IRq is the control and d E IRm is a disturbance. We assume that A and B are polytopes of matrices
217
p
p
A(w) =
B(w) =
w B,
(tl.9)
1, w i > 0 } .
(11,10)
i=1
with
i=1
p
wE14;={w:
Ewi= i----1
In the continuous-time case we work under the following assumption: Assumption
11.2 d(.) and w(.) in (11.7) are piecewise-continuous functions.
The above assumption is not strictly necessary but simplifies m a t h e m a t i c a l developments, We assume that the following constraints for both control and disturbances input
uEU,
dED,
are given, where U and 7) are polyhedral sets containing the origin in their interior. W'e assume that U m a y be possibly equal to IRq while 7) is always assumed to be compact. Moreover, we will consider a "target" C-set 2d for the state which, depending on the specific problem, will be a constraint set or a neighbourhood or the origin where the state has to be driven or just the unit ball of a relevant norm. In this chapter we are interested in a state-feedback control of the form u(t) = ¢(x(t)), (11.11) where ~ is to be determined in order to assure proper design specifications. Depending on the context, one of the design specifications listed below can be pursued. GRS
G l o b a l R o b u s t S t a b i l i t y . Assume E = 0 and 34 = IRq. Then for all initial conditions x(0) E ]R~, and all w(t) E W, the closed loop state evolution is such that IIx(t)II < tlx(0)l[¢(t), t ~ 0, where ¢ ( t ) : IR + --+ IR + is a strictly decreasing continuous function such that limt-+~ ¢(t) = 0.
UUB
U n i f o r m U l t i m a t e B o u n d e d n e s s . Assume that the target C-set A' is assigned. Then for all x(0) E IRn, w(t) E W and all d(t) E 7), there exists T(x(O)) such that for t >_ T(x(O)), x(t) E 2d.
C C S C o n s t r a i n e d C o n t r o l S t a b i l i t y . Assume E = 0 and that the state constraint set X, containing the origin as an interior point, as well as an initial condition set ;)do C X, are assigned. Then for all x(0) E ;to and for all w(t) E W, the conditions x(t) E X and u(t) = ~(x(t)) E U hold for t _> 0, and x(t) -+ 0 as t -+ oc. CDR
C o n s t r a i n e d D i s t u r b a n c e R e j e c t i o n . Assume that the constraint C-set X as well as an initial condition set X0 are assigned. Then for every x(O) E Xo, for all w(t) E )4; and all d(t) E /), the conditions x(t) E X and u(t) = q~(x(t)) E 3/ hold for t ~_ O.
218 In the next section we will associate these different goals with different concepts of Lyapunov functions. Remark. Concerning the problem of determining a control assuring the CCS and CDR specifications, we will actually provide (up to an arbitrarily close approximation) the largest set of initial condition 7) C 2( such that the required conditions hold. Thus the specification will be met if X0 C 7). 11.2.3
Lyapunov
functions
and
invariant
sets
In this section we consider some basic properties of the polyhedral Lyapunov functions. Then in the next sections we will show some applications. We choose candidate Lyapunov functions in the class of locally Lipschitz functions. Since no differentiability is required, to manage the continuous-time case, we have to invoke the generalized Dini derivative. Given the system (11.7) define the expression D + ~ ( x , u , w , d ) := l i m s u p ~ ' ( x + h[A(w)x + B(w)u + E d ] ) - ~ ( x ) h-+0+ h
(11.12)
If u(t) is a continuous function then, denoting by x(t) a trajectory corresponding to u(t) and w(t) and setting x = x(t), u = u(t), w -- w(t +) and d = d(t+), we have that the Lyapunov (generalized) derivative (11.1) is given by l ) + ~ ( x ( t ) ) = lim sup ~ ( x ( t + h)) - ~(x(t)) = n + ~ ( x , u, w, d). h-+O+ h We remark that if we assume u, w and d only to be measurable instead of piecewise continuous, the inequality above holds for almost all t. In the discretetime case we obviously define as counterpart of the Lyapunov derivative the one-step Lyapunov difference (we use the same notations for obvious analogy reasons) D+gJ(x, u, w, d) = g* (A(w)x + B(w)u + Ed) - g* (x). It is known that a basic property of a Lyapunov function is that of having the derivative (or the Lyapunov difference) upper bounded by a negative number D + (x, ~(x), w, d) < -~ (11.13) in a suitable region of the state space. D e f i n i t i o n 11.3 Given a C-set X and the system (11.7) (or (11.8)) with the continuous control u = ¢ ( x ) a locally Lipschitz function ~ : IRn -+ IR + such that
o] = {o}, (i.e. positive everywhere except in the origin where it is equal to zero) is said to be
219 L F O a Lyapunov Function Outside the set X if there exists ~ > 0 such that c
x,
and there exists/3 > 0 such that for each x q~ H[~,~] condition (11.i3) holds. L F I a Lyapunov Function Inside the set ,t2 if there exists ~ > 0 such that
¢'[~, ~] C X, and for each 0 < ~' < ~ there exists/3 > 0 such that for each x E ¢~[gt ~] and x ~ N'[gt,~ '] condition (11.13) holds. G L F a 91obal Lyapunov function if for each ~t > 0 there exists/3 > 0 such that for each x ~ Af[gt,~'] the condition (11.13) holds, The three concepts above are related to the following definitions. D e f i n i t i o n 11.4 A C-set S is said ,k-contractive (0 < 'k < 1), for the system (11.8) if for all x E $ there exists a control u-(x) E U such that
A(w)x + Z(w)u(x)
+ E d E AS f o r all
e W
and d
V . For
=
1 we
say that S is positively invariant. D e f i n i t i o n 11.5 A C-set $ is said/3-contractive, ~ > O, for the system (11.7) if for all x E 05 there exists a control u(x) E bl such that D + ( x , u ( x ) , w , d ) <_ -/3 for all w E 14] and d E D. For/3 = 0 we say that $ is positively invariant. These two definitions allow for a geometrical interpretation of the concept of Lyapunov function: ifO is a LFO then the sets ff[~', #] (# _> 0) are contractive for p >_ ~; i f ~ is a LFI then the sets ff[~, p] are contractive for # <_ ~; and ifgr is a GLF then the sets 3J'[~, #] are contractive for all #. Note that in the discretetime case the decreasing condition (11.13) is given in terms of/~ instead of 'k, but it is easily seen that there is no difference in stating the condition (11.13) in terms of 'k. Thus the design specifications previously defined are related to the definitions of Lyapunov functions in the sense that the existence of a LFO or GLF implies the existence of a control which assures UUB and GRS respectively. Moreover the existence of a LFI assures a solution to the CCS provided that the initial condition set X0 C ff[~P, ~], in view of the invariance of ff[g~, ~]. Finally the existence of an invariant set included in X is a necessary and sufficient condition for the existence of a control assuring CDR. We recall that invariant sets have received a great consideration in the past for solving the CDR problem via state feedback (Bertsekas and Rhodes, 1989; Glover and Schweppe, 1971). Very recently this set-theoretic approach to the disturbance rejection has been reconsidered by Shamma (1994) and Blanchini and Sznaier (1995a), as an alternative approach to the l 1 theory (Dahleh and Pearson, 1987).
220 Now let us consider the class of the polyhedral Lyapunov functions. It has recently been shown that these functions have the following fundamental property. F a c t 11.6 (Blanchini, 1995) Assume that there exists a continuous control u = Ol(x) and a locally Lipschitz function ~1 which is a Lyapunov function outside the C-set X (or inside the C-set X or globally) for the system (11.7) or (11.8). Then there exist a piecewise linear 1 globally Lipschitz feedback control ~ and a polyhedral Lyapunov function ~t outside the C-set X (or inside the C-set X or global).
This property is named universality property of the polyhedral Lyapunov functions. As a particular case, when no control is considered (i.e. B = 0) the previous result holds for robust performance and stability analysis. The properties of the polyhedral functions in the robustness analysis of linear systems with time varying uncertain parameters had been already pointed out by (Molchanov and Pyatnitskii, 1986; Barabanov, 1988; Gerasimov and Yunger, 1993; Brayton and Tong, 1980; Michael et al, 1984). Analysis and synthesis of dynamic systems via Lyapunov functions are usually performed with the basic intention of assuring the Lyapunov derivative to be negative and the lack of differentiability proper of the polyhedral functions might seem to introduce some difficulties, since the Lyapunov derivative is not continuous. Nevertheless, we have that the polyhedral Lyapunov functions, when associated to systems of the considered form, have very nice extreme point properties which allow to check condition (11.13) only on a finite number of points in which a linear programming problem must be solved. To explain this fact we first consider the condition (11.13) for fixed x, u, w and d. Let X and F be the vertex and the plane representation of the unit ball 2~f[~, 1] of a polyhedral function ~ and recall that Z(x) is the set of indices i corresponding to the planes which satisfy Fix = ff'(x). Denoting by y = A(w)x + B(w)u + Ed we have that in the continuous-time case D+(x, u, w, d) = limsup ~t(x + hy) - ~ ( x ) = max Fiy, h-.o+ h iEl(x) while in the discrete-time case, we have that D+(x,u,w,d)=gt(y)-~(x)=
max F i y -
i=1,...,~
max Fix,
i=l,...,s
thus the quantity D+(x, u, w, d) is readily computable in both continuous and discrete-time case. The following properties are true for polyhedral Lyapunov functions (Blanchini, 1995). 11.7 Let ~ a polyhedral function and assume It = ]Rq (unbounded control}. Then there exists a control • such that ~t is a Lyapunov function for (11.7)
Fact
1We mean that the state space can be partitioned into a finite numberof polyhedral cones with vertices in the origin and in each of them the control is linear (see Section 11.5).
221
or (11.8) outside the C-set f/'[~, 1] 2 if and only if for all x E vert{~¢'[~, 1]} there exists u(x) assuring the condition (11.13) for all w E vert{W} and d E vert{D}. F a c t 11.8 Let ~ a and U = IR q. Then function for (11.7) exists u(x) assuring
polyhedral function and assume E = 0 (noise-free system) there exists a control q5 such that ~ is a global Lyapunov or (11.8) if and only if for all x E vert{2~/'[~t, 1]} there the condition (11.13) for all w E vert{W}.
F a c t 11.9 Let ~ a polyhedral function and assume bt compact (bounded control) and E = O. Then there exists a control • such that qs is a Lyapunov function for (11.7) or (11.8) inside the C-set 2('[gt, 1] if and only if for all x E vert{ff[gt, 1]} there exists u(x) ELt assuring the condition (11.13) for all w E vert{W}. The results above can be stated in terms of equalities and inequalities as follows. For the simple exposition consider the case E = 0 and unbounded control, then we briefly show how to extend the result. T h e o r e m 11.10 Let the function ~ be the Minkowski functional of the polyhedral C-set P = f/'[~, 1]. Let X be the n × r matrix whose columns are the vertices of 7). There exists a control function u = O(x) such that ~ is a global Lyapunov function for the discrete-time system (11.8) if and only if there exists a set ofp nonnegative square r x r matrices {Hi, i = 1 , . . . , p}, a q x r matrix U and a non-negative ~ < 1 such that
A i X + BiU = X H i ,
Hi >_ 0,
(11.14)
and Hii < hi,
i= 1,...,p.
(11.15)
Proof. As mentioned above there exists O(x) such that gs is a Lyapunov function if and only if for some 0 _< A < 1 and for each vertex xj o f P = .M[~, 1] there exists uj such that Aixj + Biuj E ~7), for all i = 1 , . . . , p . This means that there exist h (i) >_ 0, j = 1 , . . . , r such that Aixj + Biuj = E ~ = I xJ h(i), and ~ kr= l h (i) k -< )~" Define the k-j entry of Hi as [Hi]kj = h~i) and U = [Ul, u 2 , . . . , u~ and the proof follows immediately. In the continuous-time case, we have the following result. Define A,i-matrix a real matrix M (see Castelan and Hennet, 1992; Vassilaki and Bitsoris, 1989) whose non-diagonal elements are all non-negative, i.e. such that
Mij >_ O,
if i • j.
2Being ~ a gauge function there is no restriction in setting ~ = 1.
222 T h e o r e m 11.11 Let the function ~P be the Minkowski functional of the polyhedral C-set P = )~f[~P,1]. Let X be the n × r matrix whose columns are the vertices of P . There exists a control function u = q~(x) such that ~v is a global Lyapunov function for the system (11.7) if and only if there exists a set of p square r x r M-matrices {Mi, i = 1 , . . . , p } , a q x r matrix U and/3 ~ 0 such that A i X + B~U = X M i , (11.16) and Mii _~ -/31,
i = 1,...,p.
(11.17)
Proof. From Blanchini (1995) there exists a control • such that ~ is a Lyapunov function for (11.7) if and only if there exists 0 < A < 1 and ? such that for all r < ~ and each vertex xj of 7) =/V[~, 1] there exists uj such that [I + rAi]xj + r B i u j E /kP. Moreover,/? and ~ are related as )~ = 1 - r ~ (see Fact 11.14 in the next section). From Theorem 11.10 we have that there exist Hi, for i = 1 , . . . , r , such that [ I + r A i ] X + r B i U = X H i , and H i i _~ hi, say A i X + B i U = X - ~ -L. Now, Mi = H,-Ir are M-matrices. Set ~ = I~_..Aand necessity is proved. To prove sufficiency we proceed in the same way, if there exist M-matrices Mi, take Hi = rMi + Mi and A = 1 - r/3. If v • 0 is small enough, we have Hi > 0 and 0 ~_ ~ < 1. By applying the result in Blanchini (1995) sufficiency follows. The columns of the matrix U in the theorems above represent the control vectors which assure condition (11.13) on the vertices of ~r[~, 1]. If the vertex matrix X of the set X is assigned, checking the existence of a feasible solution to the above set of linear equalities and inequalities requires just to solve a linear programming problem of minimizing A or maximizing ~. Thus determining a control which renders contractive an assigned polyhedral set for the closed-loop system is solvable via LP. As shown by Blanchini (1991a) and by Sznaier (1993), in this case we can force the control to be linear just adding the constraints U = K X assuming the feedback gain K as new unknown. As a particular case these conditions can be used to check if a certain polyhedral function is a Lyapunov function for the uncertain system x(t + 1) = A ( w ( t ) ) x ( t ) or k(t) = A ( w ( t ) ) x ( t ) . Extensions of the theorems above are straightforward. If for instance U is compact and we ask if there exists a control such that ~P is a Lyapunov function inside the set 2~f[~P,1] then, we have to add to the theorems above the condition that the columns of U are in the set U. If for instance U = {u : Ilullo¢ _~ 1}, this is equivalent to the matrix inequalities - U + < U ~_ U + where U + has all the entries equal to 1. Also the extension to the case in which E is non zero is easily managed. Consider the polyhedral set P* = { x :
x+EdEP,
VdED}
and assume that the vertex representation for T~* is X*. Then, in the discretetime case, the equation (11.14) in theorem (11.10) must be replaced by
223
AIX + B~U = X ' H i . A similar result can be found for the continuous-time case. Although equations (11.16) and (11.14) allow for a compact characterization of a polyhedral Lyapunov function via its unit ball, it seems very difficult that they could be directly useful to generate a Lyapunov function because, if X is not fixed, these equations are bilinear, and it m a y be very hard to find a solution in terms of U, Mi (or Hi) and X. In the next section we will show an iterative procedure to synthesize polyhedral Lyapunov functions. We have stated that the existence of a control which renders a certain polyhedral function a Lyapunov function for the closed-loop system is equivalent to the existence for each vertex of P = ~/'[~P, 1] of a proper control vector u assuring the condition (11.13) for all d E 7) and w E W. This fact is due to the linearity of equations (11.7), (11.8). For instance in the discrete-time case, given x on the boundary of A/[~, 1] it can be written as convex combination of some vertices as x = X a , a > 0, i T a = 1. It is simple to verify that u = U a assures (11.13). The same reasoning easily extends by linearity to states outside :P (if no control constraints are present) and inside P (in the disturbance-free case). The determination of the control action u to associate to each state x m a y be done in principle by computing the "control at vertices" and proceeding as proposed by G u t m a n and Cwikel (1986) for systems without uncertainties. Furthermore, given x a control u(x) can be found as a solution of the following m i n - m a x problem min
max
uEbl d E V , w E W
D +(x,u,w,d),
and in Section 11.5 we will show that this is an LP problem. However, the question of the continuity of the controller arises. In the paper by Blanchini (1991b), it is shown that by taking the m i n i m u m Euclidean norm in the set of solutions of this LP problem we obtain a control which is Lipschitz in x. However the resulting control is somewhat complicated, while for practical implementation simple control laws are desired. In Section 11.5 we will go back to this problem and we will illustrate several alternative control laws.
11.3 Synthesis of polyhedral Lyapunov functions We now focus our attention on the construction of polyhedral functions for discrete and continuous systems. First we furnish a procedure to compute f f [ ~ , 1] for discrete time systems of the form (11.8), which is based on the notion of controllability regions. Then the continuous-time case will be managed by introducing the Euler Approximating System. We first consider the case U = IRq. Given a polyhedral C-sets X and :D, the unit ball P of a polyhedral Lyapunov function outside X can be constructed starting from X as follows.
224 A l g o r i t h m 11.12 C o n s t r u c t i o n o f a p o l y h e d r a l L y a p u n o v f u n c t i o n L e t X = {z E I R n : F z < i} = { z = Z a , i T a = 1, a > 0} be the assigned polyhedral C-set. Fix ~ : 0 <_ ~ < 1, and a tolerance ¢ > 0 such that ~ q- c < 1. Set k = O. Set X (°) = X . Consider the set Q(k) = {(x,u) : F(k)[Aix + Biu] < )~[ - A (k), i = 1 , . . . ,p} C ]Rn+q, where the vector A (k) has components A} k) = maxdez~ F(k) Ed; (ii) Compute the projection of Q(k) on IR'~ : 79(k) = {:e : 3u E / / :
Q(k)};
(x, u) E
(iii) Compute the polyhedron X(k+l) = X(k) N 79(k) ; (iv} If 0 E int{X(k+l)}, that is X (k+l) is a C-set, continue, otherwise stop (the procedure has failed with the assigned )~); set X (k+l)={z:
F(k+l)z<_i}={x=X(k+t)a,
iTch=l, a>O}
(v) If X (k+l) is (3~ + e)-contractive then set 79 - X (k+l) and stop, else set k = k + l and go to (i). The above procedure deserves some comments. For an efficient implementation both vertex and plane representation of the initial set 2( and the sets X (k) are considered. Step ii: The first element of the pair (z, u) is a state which is mapped by the control u in X (k) for all d E :D and w E W. Thus the resulting set is the controllability set to 2((k). The projection operation can be performed via Fourier-Motzkin elimination method (see for instance the work by Keerthi and Gilbert (1987) for details). Step iii: X(k+l) is the subset of all states in 2((k) which can be remapped in X (k) itself. Finding a minimal representation for this set is the most delicate and time consuming step of the whole procedure. To perform this step we set = X(k) and we consider the plane description of 79(k) = {G(k)z _< ~} and we intersect the set X with all the half-spaces///(k) = { x : G~k)z < 1} which generate 79(k), say P~ := A~N//~ k), for i = 1 , 2 , . . . The intersection of the region 5( described in terms of vertex matrix X and plane matrix F with the hyperplane f x < 1 is computed as follows: • Compute Xext,Zint and Xpla, the subset of vertices of X (the vertex matrix of X) which satisfy f z > 1, f x < 1 and respectively f x = 1 (within a certain numerical tolerance). • If Zext m. 0 then stop: the result of the intersection is ,~. * If Xint = ~ then stop: the result of the intersection is an empty set.
225
If Next ¢ ~ then discard all the planes of X which do not contain internal vertices, store those which contain internal and external vertices in a matrix Fn~w and store the remaining ones in F. Determine the set of all the points, which can be obtained by solving the linear equation constituted by considering f x = 1 and n - 1 planes of /~new, and store them in a matrix Xpos. Remove from Xpos the points which do not satisfy all the constraints induced by Fnew. • The result of the intersection is the polyhedron whose vertex and plane representation is given by 2 = [Xpos IXint t~'~'pla] and/~ = [fT iF~w tFT]T.
Step v: To check the contractivity of the set we must have that for all the vertices xi of X (k+l) the optimal values of the LP problems min~ s.t. F(k+l)Bu - ~i < - F ( k + l ) A x i - A (k+l) is less than or equal to ,~ + e. We remark that, in view of the high number of inequalities involved, it is normally easier to solve the dual problem. If the procedure stops successfully, the final set P is the largest A-contractive set in X (up to the tolerance c). Let ¢r the Minkowski functional of P (i.e. N[~, 1] = P). I f U = IRq, then ~ is a LFO the set X and if E = 0, then ~ is a GLF. If the procedure is run with a compact U, and E = 0, then ~r, is a LFI the set X. To extend the procedure to the case in which the control is constrained, then it is enough to take in Step i the set Q(k) = {(x, u) C Q(k) : u E U}, instead of Q(k). The described procedure can be used to manage the continuous-time case as follows. D e f i n i t i o n 11.13 Given r > O, the Euler Approximating Systems of (1I. 7) is the following discrete-time system
x(t + 1) = [I + rA(w(t))]x(t) + rB(w(t))u(t) + rEd(t).
(11.18)
The following fact holds (Blanchini, 1995). F a c t 11.14 Assume that g] is a polyhedral Lyapunov function for (11.18) (outside, inside a set X or global), with a coefficient of contraction )~ < 1. Then it is a Lyapunov function for (11.7) with a coefficient of attraction/3 = (1 )~)r -1. Conversely if ~ is a polyhedral Lyapunov function for (11.7) (outside, inside a set X or global), with a coefficient of contraction/3 > O, then for all /3' < [3 there exists r > 0 such that ~ is a Lyapunov function for (I1.18) with a coefficient of attraction ~1 = 1 - r/3' < 1.
226 This property assures the possibility of finding a polyhedral Lyapunov function by applying Procedure 11.12 to (11.18), with a sufficiently small v and A = 1 - r f l < 1. If the procedure fails we can reduce r a n d / o r fl to increase A and restart. As a final remark we have that as A --+ I, with A < 1, (or fl, r -+ 0 in the continuous-time case), the evaluated set approaches the largest invariant set. This means that we can solve the problems of determining controllers which assure CCS and CDR by providing an approximation of the largest set of initiM condition within an arbitrarily small tolerance.
11.4 R o b u s t a n a l y s i s via p i e c e w i s e - l i n e a r functions As a particular case of the problem of determining a Lyapunov function and a stabilizing control we can check whether or not a certain system is stable and which is the speed of convergence to the origin. It is known that a Lyapunov function offers a qualitative description of the system transient in terms of estimation of the convergence speed to the origin and of the domain of attraction. In this section we investigate these two features. 11.4.1
Transient
analysis
Consider now the system without control input (continuous or discrete-time)
~(t) = A(w(l))x(t) + Ed(t), x(t q- 1) = A(w(t))x(t) + Ed(t). An important problem for systems of this kind is that of checking their stability/instability (i.e. checking whether or not the state converges asymptoticMly to zero when d -- 0). Moreover, to have a more detailed description of the system behaviour, we are interested in obtaining a quantitative measure of the system stability/instability. Furthermore, when the system is stable we wish to know the smallest neighbourood of the origin to which the state converges under the action of d. Both these problems can be solved via polyhedral Lyapunov functions. The existence of a polyhedral Lyapunov function k~ for the system outside a certain C-set 2' implies that
< max{h'
1} (or
< max{e-S'
1}),
while a global Lyapunov function for the case E = 0 implies the condition
~(z(t)) _< ~t~(z(O))
(or ~(z(t)) _< e -~'~(z(O))).
However the information above may be not satisfactory for the fact that the gauge function ~(-) may be inappropriate as a measure of the distance
from the origin. In practice one may desire to measure the distance from the origin using a given norm. Suppose that tl' ]l is an assigned polyhedral norm (for instance a weighted infinity-norm) and take for the moment being E = 0; then we call transient estimate a pair of constants (A, C) (resp. (fl, D)) such that, for all t > 0, we have
Ilx(t)tl
cIMo))ll
(or IMt)ll
DIM0))IIe-
) •
(11.19)
The constant C (resp. D) is clearly greater or equal to one, being a unit if and only if ]1' t[ is a Lyapunov function. It is clear that to estimate the transient one should take ,~ as small as possible (resp. fl as large as possible), and the smallest constant C (resp. D). Consider the infimum ~ of )~ (the supremum ¢) of fl) such that an inequality of the form (11.19) holds, and let C' ( or /)) be the smallest constant (if it exists) such that (C, ~) (or (J), fl)) are transient estimate, then (C', ~) (or (D,/~) ) is called the best transient estimate. If we assume that the polytope of matrices A(w) has no common invariant subspaces, 3 a best transient estimate exists. The assumption of the non-existence of common invariant subspaces is not restrictive in stability analysis (see Barabanov (1988) for details). We have the following result (Blanehini and Miani, 1995a). F a c t 11.15 Assume that [[. I[ is an assigned polyhedral norm. Then, given > ~, the number C()~) defined as the smallest constant C such that (~, C) is a transient estimate, is given by
[
]_1
supllxlt
xE'P
,
where ;P = J~[~, 1] is the largest A-contractive set included in the unit ball X = frill" I[,1].
Figure 11.1 provides a geometric interpretation of C()~), which is the ratio between the dimension of A' (the big square) and the small square inside 7). Computing C(,~) is extremely simple if II' II is a polyhedral norm, indeed C(A) =
max k~p(x).
xEvertX
The application of this result jointly to Procedure 11.12 leads to the determination of the best transient estimate. Let ]]. ]I be an assigned polyhedral norm whose unit ball is X. A l g o r i t h m 11.16
Computation
of the best transient estimate.
(i) Fix a tolerance e > O, set )~- = O, ~+ - +oo; (ii) choose A in the interval ()~-, ~+) and compute the largest A-contractive set P~ in X; SA subspace Z which is invariant for A(w) for all w E YV.
228
(iii) if Px is not a C-set, set A- = A and, if A+ - A- > e, go to Step ii; (iv) if P~ is a C-set set A+ = A and, ira + - A- > e, go to Step ii; (v) compute C(A +) as described above.
Note that in the procedure above we may find A > t and clearly in this case the system is unstable. N o t e that the definition of contractive set is still valid even if A > 1. The above procedure terminates by providing bounds of the form A- < ~ < A+ within a tolerance e. The continuous-time case can be managed by applying Procedure 11.16 to the corresponding EAS, i.e. to the system x(t + 1) = [I + rA(w)]x(t). We recall that if a set P is A-contractive for the EAS, then it is fl-contractive for the system z(t) = A(w(t))x(t), where/~ = (1 - A ) r - 1 . Moreover, i f P is fl-contractive, then for each j3~ < fl there exists v such that P is ~3/-contractive with/~1 = (1 - A~)r -1. Thus it turns out that the best transient estimate for a continuous-time system can be approximated arbitrarily closely by taking r sufficiently small and running Procedure 11.16 for the EAS. Suppose now E ¢ 0 and [[d[[oo < 1. Given a certain norm I[' [[, a transient estimate may be defined as a triple of numbers (C, A, p) (or (D,/~, p)) such that
IIx(t)ll < max{CAtllx(O)ll,p}
(or IIx(t)ll <
max{De-a~llx(O)ll,P}),
and the procedure above can be extended in a straightforward way to find proper constants C, A, p (resp. D, fl, p) by iterating on the two parameters A and p.
Example 11.17 Computation o f a t r a n s i e n t e s t i m a t e We report an example to illustrate the features of the proposed method. Consider the following continuous-time uncertain system whose state matrix is
where [w(t)[ < #. For this system a quadratic function which proves stability exists if and only if the amplitude of # is less then the value/~@ = x/~/2 (which can be computed as the inverse of the 7/°~ norm of the continuous-time transfer function W ( s ) = H ( s I - A 0 ) - I E , see the papers by Khargonekar et al (1990) and by Petersen et al (1993), whereas, by using polyhedral functions, we can prove stability for each p < 1, which is about a 15% improvement. The matrix A ( w ) has no proper invariant subspaces, hence the results here presented apply. For this system we first evaluated the maximal allowable level for the uncertainty and then we considered a variable bound # on the uncertainty amplitude (i.e. we considered the system above when [w(t)[ < p) and we computed the best transient estimate (D(/~), ~(#)) of the system, with respect to the infinity norm, as a function of # with # varying from 0 up to the polyhedral
stability margin. The approximations of the contractive regions for the continuous time system have been obtained by computing the best transient estimate for the EAS, with r = .1. The obtained results are reported in Table 11.1. The accuracy used in the determination of the coefficient ~+ is 10 -3, that is the real value ~(#) lies in the interval [/~+ - 10 -3, ~+]. The number of rows n~ of the matrix F corresponding to ~+ is also reported to give an idea of the complexity of the method. Note that there is no relationship between the number of rows of F and the contractivity. In practice the value of n~ depends on the difference between ~+ and the actual value ~(#). tn Figure t1.1 we report the largest invariant set computed for ¢~ = 0.001.
, 0.00 0.20 0.40 0.60 0.80 0.866 0.90 0.999
Z+
o(Z +)
0.460 0.386 0.312 0.234 0.149 0.115 0.096 0.001
1.9970 2.0391 2.0913 2.1506 2.2324 2.2685 2.2958 2.0029
n~ 114'" 80 82 86 100 114 132 58
Table 11.1
11.4.2
Determination
of the
largest
domain
of attraction
As stated before, another feature of a Lyapunov function is to provide a domain of attraction of the state to the origin. Here we show how to solve the problem of finding the largest domain of attraction inside a C-set X. Applications and basic properties of the largest domain of attraction have been described by Tan and Gilbert (1991) (for linear time invariant systems) and Blanchini (1992) for systems with additive disturbances. We show here that the largest domain of attraction, when the system is affected by both parametric and additive uncertainties, can be derived as a particular case of what is reported in the works by Btanchini (1994, 1995), just setting B = 0 and A = 1. This extension is shown to have further applications to the robust stability analysis of systems with both parameter and operator uncertainties as shown by Blanchini and Sznaier (19955). Assume that a certain polyhedral set A' is given. Then, given the system x(t + 1) = A(w(t))x(t) + Ed(t), by applying the Procedure 11.16 with )~ = 1 we derive the largest invariant set included in the set X. It can be easily shown that the following property holds. Assume that the set Xp = {x : IIFxll < p} is assigned (for simplicity we take it symmetric), where p is a positive parameter, and assume that 7? = {d : Ildll~ < 1}. For x(0) = 0 define the l ~ to 1~ induced
230 I 0.8 0.6 0.4 0.2 o -0.2 -o.4 -o.6 -o.8 -I -I
-0.S
F i g u r e 11.1
0
0.5
The final set when/3 -- 0.001
norm of the system (A(w(t)), E, F) (Dahleh and Kammash, 1993; Dahleh and Pearson, 1987) as =
sup
sup Ily(t)ll
w(t)e~V t>0 d(t)e2)
where y(t) = Fx(t) is the system output. Then the following property holds.
Assume that there exists a vector if: E )/V such that the pairs (A(~:), E) and (A(~:), F) are reachable and observable respectively. Then the two following statements are equivalent
F a c t 11.18
a l The largest invariant set included in Xp is nonempty and compact; a2 The system is internally stable and p <_ p.
Moreover for # < p the largest invariant set is a polyhedral C-set. The above property can be applied to determine via set computation whether p
Example 11.19 Computation o f the largest domain o f attraction We now report an example to show how the notion of maximal domain of attraction can be used to analyze some robustness properties. Consider the system having the following state space representation:
231
ic(t) = (w(t)A1 + (l - w(t))A2)x(t) + Ed(t), with 0 <
w(t) <_ 1,
A i .=
[01 1
3
mi
mi
[] E=
0
m l = 5 - 5, rn2 = 5 + 5 and IId(t)tI~ ~ "/. For this system we consider a constraint on the maximal oscillation (i.e. we require the zero state evolution to belong to the region Ixll _< k) and we focus our attention on the determination of:
• The maximal allowable disturbance amplitude 7 for a certain value of 5 * The maximal value of 5 for a prescribed value of 7 which guarantee the state not to exceed the imposed constraint starting from zero initial conditions. The results of such an inquiry can be extrapolated by the two dimensional array reported in Table 11.2, obtained using the EAS for r = 0.1 where k(7 , 5), the minimal output level guaranteed is calculated for different values 7 and 5 (the value of k has been determined by excess on the last decimal digit). Conversely k and the associated set computed for the couple of values (5, 7) can be seen as the limit output value and limit bounding set to which the state will converge in its evolution, starting from an arbitrary initial condition, when the parameter variation is less equal 5 and the disturbance amplitude does not exceed 3'. We report in Figure 11.2 the maximal domain of attraction for the couple of values 5 = 1, 7 = .25.
5/7 4.00 3.00 2.00 1.00 Table 11.2
1 138.9 110.7 90.45 74.02
.75 101.6 83.00 67.84 54.91
.50 67.72 55.41 45.23 37.01
.25 33.86 27.67 22.62 18.52
The output level for different values of 5 and 7
11.5 Robustly stabilizing control laws We now investigate the structure of the controllers which can be associated to a polyhedral Lyapunov function. Henceforth we will assume that a proper Lyapunov function k~ (inside, outside the set X or global) is computed and
232 20
15
10
-;..v
F i g u r e 11.2
-15
-10
-5
0
5
10,
15
The maximal domain of attraction for cf = 1, 7 = .25
that P = A~f[O,1]. We assume that the matrices F and X of the plane description (11.2) and the vertex description (11.3) of 7~ are known as well as the matrix U of the controls "at vertices" guaranteeing condition (11.14) and (11.16) respectively in the discrete and continuous time case. We first reconsider an LP based controller then we describe a linear-variable structure continuous one (piecewise-linear). Finally we show a (discontinuous) piecewise-constant control.
11.5.1
Linear
programming
control
The contractivity condition ~ ( z ( t + 1)) _< R~(z(t)) can be expressed as the existence of u(x) such t h a t
F[A(w)x + B(w)u(x) + Ed] < hi ~(x), for all w E )4; and d E 7?. This condition suitably rephrased can be used to find the control which maximizes the contraction. The control can be chosen in the set U(z) of the solutions of the following on-line LP min )~ > 0 uE//
F[Akx + Bku(z)]
< AiO(z)-.4,
being .4 the vector whose components are
k = 1,...,p,
233 Ai = maxFiEd,
i = 1,. . .,s.
dED
It can be shown that if we take the smallest Euclidean norm vector in U(x) the selection function u(x) is Lipschitz in x. Thus this fact allows, in principle, to apply this control to continuous-time systems via EAS. A modified version of this controller holds in the single-input case as shown in Blanchini et al (1995), where the case E = 0 is considered. In this case A is fixed as the guaranteed contractivity factor and the set U(x) is the nonempty intervM of all control inputs which guarantee the contraction ~'(x(t)) _< A~'(x(t)). So, finding the smallest absolute vMue control is straightforward.
11.5.2 Piecewise-linear robust stabilizing control We consider now a linear variable-structure control introduced by G u t m a n and Cwikel (1986) and reconsidered by Blanchini (1994, 1995) and Blanchini and Sznaier (1995a). The unit ball P of the given polyhedral Lyapunov function ~" can be divided into sectors as follows. Denote by X (h) the matrix (a submatrix of X) x(h)
r (h) (h) " . • Z(h)], k~J
=tXklXk2
X kj (h) E vert{P},
j=
i,
...,n,
derived by selecting n distinct vertices of P. Consider the simplex
~i:~h - -
x(h)a kj 3•,
X =
aj < -- 1 , a j > -- 0 j=l
j=l
}
"
The vertices forming the matrices X (h) can be selected in such a way that :Ph has a non-empty interior; if k ~ h, Ph ~ P k has empty interior and is an m-dimensional face (with m < n) of both Ph and Pk; and Uk Pk = P . In this way the set Ph is associated to the simplicial cone
Oh=
(h) . x=2_.,xk~j,
7j_>0
,
j=l
having the origin as its vertex. Note that [-Jh Ch = IR~. For each point x E gh the unique vector of the coefficients 7j associated to x is given by @h) = [ x ( h ) ] - l x (X(h) is invertible because int{gh} is non-empty). For each matrix X (h) consider the matrix U (h) formed by the control vectors "~*kj (h) , j = 1, ' " ., n, (which are columns of U) associated to the vertices x kj (h) (which are columns of X). Now set
K (h) = U(h)[X(h)] -1, and consider the following control law for x E Ch, u = O(x) = K ( h ) x .
(11.20)
234
This control is such that the closed loop system satisfies the decreasing condition (11.13) for x E ~o if E = 0, for x ~ ~o if u is unbounded, and everywhere if both conditions hold. The proposed partition of the polyhedral C-set into simplicial sectors can be obtained using a procedure proposed in the book by G r u m b a u m (1967). An heuristic but efficient technique to find the partition has been proposed by Miani (1993). Once the simplicial partition has been derived, to apply the control we need to be able to determine online in which sector the state is contained to apply the proper gain K (h). This problem can be solved by considering an auxiliary simplicial polytope that is a polytope whose faces contain exactly n vertices. As shown by G r u m b a u m (1967), there exists a polytope ~5 whose vertices $j = 7jxj, 7j > 1 are those of :P scaled by proper factors 7i, such that each proper face of ~ is simplicial (i.e. 7~ is a simplicial polytope), in the sense that if ~ = {x e IRn : F x < i ) , then the hyperplane represented by fi'hx = 1 contains exactly n vertices of ~O. Clearly each of the simplicial sectors associated to 75 uniquely corresponds to a simplicial sector of the original polytope 7) and it is associated to the same cone Ch. Then, to discover which cone Ch contains the state we can consider the auxiliary polytope 9 . Then, i f / ~ is the plane representation of ~b, the sector index is identified by h(x) = arg max {Fjz}. j=l,...,s
The practical implementation of the control can run into computational problems due to the number of simplicial sectors ~h involved in the decomposition of :P. So, to reduce complexity, in the next section we describe a variablestructure controller proposed by Blanchini and Miani (1995b). 11.5.3
Bang-bang control for single-input continuous-time systems
Henceforth we assume a single control input, say q = 1. The compensator we are presenting is of particular interest in the cases in which the control is saturated or quantized. Therefore we refer to the CCS problem and we assume E = 0. Then we briefly mention possible extensions. Assume that the control is constrained as u = {u : u - < u < u + ) ,
where the interval H contains the origin as an interior point. The proposed controller assures convergence of the state to the origin for all the initial states included in the unit ball of a polyhedral Lyapunov function. The control requires the division of the polyhedral domain into sectors, but in this case the sectors are not simplicial (i.e. can be associated to more than n vertices). In contrast with the compensator previously proposed, which is linear in each sector, this new controller has the property of being constant in each sector. As a consequence, it introduces sliding surfaces in the state space. Assume that a polyhedral domain of attraction of the form (11.2) is produced by the Procedure 11.12. We recall that Procedure 11.12 can be arranged in order to take
235
into account control constraints. Then the system state can be driven asymptotically to the origin using only control values assumed on the e x t r e m a of ht and the origin. As a first step, the state space must be divided into cones defined as the set
Ci = {x: Fix > IS)x,j • i}, each of them being the smallest cone centered in the origin and containing the i-th face of P , i.e. that associated to the constraint fix _< 1. The fact that the sectors involved are not necessarily simplicial is fundamental because it strongly reduces the number of cones involved thus reducing the complexity of the compensator. Consider the sector Pi = Ci N P given by the convex hull of the origin and one of the i-th face of P. In each of these sectors (we call these sectors of P proper in contrast with the simplicial sectors required by the previous control) the control assumes the value in H which assures the m a x i m u m rate of convergence with respect to the metric induced by the polyhedral function associated to P. Given the state x, such a value of the control is obtained by solving the following m i n - m a x problem r / = min m a x D+#(x, u, w). u~Lt w E W
(11.21)
We recall that Z(x) = {i: Fix = at(x)}, and thus D+C'(z, u, w) = m a x Fi[A(w)x + B(w)u]. iez(~)
(1t.22)
If x is in the interior of the cone Ci we have Z(x) = {i} and therefore the following proposition is immediate. 11.20 Let x E int{Ci}. The problem in (11.21) has a solution u which is reached on the vertices of U and it is constant for all x E int{Ci}. The minimizing solution and the minimum value are obtained as the solution
Proposition
of r/i =
min max{FiBku}. ue{u-,u+} k
Note that if in particular x E int{Pi} by construction we have that rJi < -~(z). If the values corresponding to u - and u + of the m a x i m u m in the expression above are equal we choose u = 0. The proposition above tells us how to associate to the sector i a unique control u(i) E { u - , O, u+}. There is an indeterminacy of the control on the points on the intersection of sectors. In this case, we can arbitrarily assign to the control any of the values a m o n g those assumed in one of the concurrent sectors. The feedback control, which is of the bang-bang type, is derived by performing on-line the following two steps. 11.21 B a n g - b a n g c o n t r o l Let x be the current state and let F be the plane description of ~.
Algorithm
236 "-" O. Else for the current value of the state x c o m the index i = argmaxj{Fjx}. 5 is the index of the cone containing the state)
(i) If x = 0 set u = ~BB(X) pute
(ii) Apply the control u = 6'BB(X) := u(i),
(11.23)
the value associated to the i-th sector by means of Proposition 11.20. We have the following theorem. T h e o r e m 11.22 Assume that the polyhedral function ~t is a Lyapunov function
by means of Theorem 11.11. Then for x E 7) the optimal value of the min-max problem (11.21) is such that min max D+ ~(x, u, w) < -t3 ~(x). uE~ wEW
Moreover, the control qSBB assures a rate of convergence to the origin t3 (measured with respect to ~ ) for every initial condition in the unit ball 7) = N[~, 1]. Proof.
If x E 7) is in the interior of a sector 7)i, then the condition D + # ( x , ~ s s ( x ) , w) _~ -13#(x),
is assured. Consider now the case in which x(t) is not in the interior of a sector, say Z(x) contains more than one element. The set Z(x) identifies a linear manifold
£ = {z E IR~ : Fikz = ~(x),ik e Z(x)}. Using the theory of the equivalent control (Utkin, 1974), we have that f o r / : to be a sliding surface for the system (otherwise we fall into the previous case), there must exist an equivalent control value Ueq such that
U-- ~ Ueq ~_~U+~ and such that the equivalent derivative k E £. Thus at the time t, in view of (11.22) we have
Fix = Fi[A(w(t+))x(t +) + B(w(t+))Ueq] - ~,
for all i E Z(x).
By assumption, there exists u E U such that D+ffs(x,u,w(t+)) <_ -13~(x). Now the proof proceeds exactly as in Blanchini and Miani (1995b), showing that if we assume D+k~(x, ueq, W) ~ --13~(x), we fall in a contradiction and hence the convergence is assured for the equivalent motion of the system. We extend the basic idea that leads to a control of the form ¢)BB (x), to consider the case in which there are no control constraints and the state is not
237
necessarily confined in the set P. Assume that u E IR and x G lR ~. In this case, the problem (11.21) may have not a minimum since u is not bounded. To consider this case we generalize the control as
t / = p(x) eBb(x), where p(x) is a suitable weighting function having the property that p({x), ~ _> 0 is a non-decreasing function of { for all x. One way of solving the problem is that of considering a fictitious constraint on the control variable of the form It/-, u+], where u - and u + are the smallest and the largest value of the control associated to the vertices of 5° (say the largest and the smallest entry of U). Now, if we take the weighting function p(x) = e ( x ) we have, by Theorem 11.22, that the decreasing condition (t1.13) is satisfied for every x on the boundary of P (as in this case we have ~(x) = 1). By linearity it is immediate to check that, the control ~u assures the condition with the same/3 on the state ~x for any ~ _> 0. Thus, in absence of control constraints the control u = p(X)~)BB(X) assures global conveTyenee to the origin (i.e. for all initial conditions). This weighting function has not only the property of assuring global convergence but it has the advantage of reducing the control effort as the state approaches the origin. In practice this fact is fimdamental in view of the digital implementation of the control because the inter-sampling ripples are strongly reduced while the speed of convergence/3 is preserved. A further choice of the weighting function is particularily useful when the control is quantized. Assume for brevity that u - = - u + = t/ma× and the control can assume values only on the discrete set f2 formed by taking 2N + 1 equispaced points in the interval [u-, u +] J~ : {U : k~,
-N
<~ k < N,
N ~ : Umax}.
In this case, denoting by [#] the smallest integer greater than/2 _> 0, a proper weighting function is -
N
The control with this weighting function assures convergence to the origin inside the set T'. This is easy to see because, since condition (11.13) is true for x E ~ o , by linearity we get the same condition for every x E :P. Moreover, if the control is not constrained, an interval [-urea×, Urea×] can be just fictitiously chosen as described above and the resulting control assures global convergence. Finally, in the case in which E ~£ 0, it is easy to see that the quantized control above solves the UUB problem of driving the state into P.
Remark. From the sliding motion theory, it is well known that for practical implementation the control needs an hysteresis. This can be easily done as follows. Suppose that the trajectory passes from 8j to $i. Then the control is kept equal to u(j) even in $i until O(x) >_ ~+fjx, where e is a small tolerance.
238 E x a m p l e 11.23 B a n g - b a n g c o n t r o l o f a D C m o t o r To illustrate the details of the implementation, we consider the stabilization problem for a DCelectric m o t o r governing a rigid arm. A third order model is considered, the state variables xl, x2, x3 are respectively the current, the angular speed, and the difference between the motor and the reference angle. The system matrices are
A =
[o 0] 5 0
0 1
A 0
,
B =
,
(11.24)
0
where c~ = 484, t3 = 21, 5 = 2636, 7 = 295. The coefficient A = 550 cos(00) is a t e r m which depends on the nominal linearization angle O0. Since the control is to be designed for all possible linearization points, we consider it as an uncertain t e r m subject to Iz~l _< 550. We formed the (EAS) of the system for v = 0.001 and we computed a contractive set applying Procedure 11.12 with = 0.99 starting from the initial set Ixll < 4, Ix21 < 40, lx3t < 2. We assume that lu[ < 20. The resulting polyhedral set P is bounded by 26 planes, it is 0-symmetric and can be represented in the form {x : [Fx[ _< i}, where F is reported below. Then the function ~ ( x ) is the norm kP(x) = [[FxI[~.
1
0.000E + 0
0.000E + 0
0.000E + 1 0.000E + 1
2,500E -
2.500E - 2 0.000E + 0
0.000E + 0 5.000E - 1
6.665E2 0,000E + 0
2,525E2 5.050E - 4
-1.389E2 5.050E - 1
6.665E2 1.347E - 3
2.525E2 1.020E - 3
1.389E5.105E -
2 1
2.706E3 4.057E - 3 5.380E - 3
1.538E3 2.050E3 2.548E - 3
5.135E5.140E5.120E -
1 1 1
6.655E - 3 7.860E - 3
3.026E - 3 3.477E - 3
5.075E 5.010E -
1 1
-8.990E
- 3
-3.896E
- 3
-4.926E-
1
The matrix F.
By dividing 7) into simplicial sectors we obtained 92 sectors, in each of t h e m the control is linear. The number of sectors is still reasonable for this application. The control eBB does not require the partition into simplicial sectors, so 26 sectors only (i.e. those associated to the rows of the m a t r i x F ) have to be considered. In each of these sectors the control assumes values equal to -4-20 or 0 at the origin. If in a sector the two values u = 20 and u = - 2 0 give the same derivative the zero control does the same and it has been considered in order to reduce the control effort. In Figure 11.3 it is shown the transient from the angle of ~r/4 to 3~r/4 with the bang-bang control (the horizontal line and the curve which converges to it are the reference angle signal and angle values, the oscillating one is the motors'). In order to reduce the control effort, the following function p has been considered
239
P(*)-
2o
"
Note that with this function, the control assumes integer values, and it is comm o n to find suppliers which are only able to furnish such kind of control. In Figure 11.4 the transient corresponding to this control is shown. Note that virtually the convergence of the angle is the same while the current fluctuation is strongly reduced. 35--
,
,
,
3
25
F i g u r e 11.3
xt and x3 with
¢BB(X)
3
25
f 2
15
f ......... o'i
o12
Figure
11,4
o'~
oi.
o15
oi~
xl a n d x~ w i t h
o17
o18
o19"
p(X)¢BB(X)
240
11.6 C o n c l u s i o n s In this contribution we have described basic properties of piecewise-tinear (polyhedral) Lyapunov functions in the control of systems with parametric uncertainties and unknown but bounded disturbances. We have shown some numerical procedures for their synthesis and we have presented some applications. It is important to remark that using this kind of functions may require a nontrivial computational effort due to the complexity of their representation. This problem is important, but it may be considered not very crucial if we take into account the modern computer technology. There are still open problems that may be of interest for future research. For instance, so far the considered compensators are of the state feedback type, while in many applications only output feedback is available. Another important problem is to reduce the complexity of the compensators which can be associated to Lyapunov functions. A further important problem is to find numerical efficient algorithms for the generation of Lyapunov functions of the polyhedral type.
References Barabanov, N. E., 1988, Lyapunov indicator of discrete inclusion, parts I, II, III, Automation and Remote Control, 49(2), 152-157; (4), 283-287; (5), 558-566. Barmish, B. R., 1985, Necessary and sufficient conditions for quadratic stabilization of uncertain systems, Journal of Optimization Theory and Applications, 46, 398-408. Benzaouia, A., Burgat, C., 1989, Existence of non-symmetrical Lyapunov functions for linear systems, International Journal of Systems Science, 20, 597607. Bertsekas, D. P., Rhodes, I. B., 1971, On the minmax teachability of target set and target tubes, Automatica, 7, 233-247. Bhaya, A., Mota, F. C., 1994, Equivalence of stability concepts for discrete time-varying systems, International Journal of Robust and Nonlinear Control, 4(6), 725-740. Blanchini, F., 1991a, Constrained control for uncertain linear systems, Journal of Optimization Theory and Applications, 71(3), 465-484. Blanchini, F., 1991b, Ultimate boundedness control for discrete-time uncertain system via set-induced Lyapunov functions, Proceedings IEEE Conference on Decision and Control, 1755-1760. Blanchini, F., 1992, Constrained control for systems with unknown disturbances, in Leondes, C. T., editor, Control and Dynamic Systems, 51, Academic Press; see also: Constrained control for uncertain linear systems, 1990, Proceedings IEEE Conference on Decision and Control, 3464-3468.
241
Blanchini, F., 1993, Robust control for uncertain linear systems via polyhedral Lyapunov functions, Proceedings IEEE Conference on Decision and Control, 2532-2533, San Antonio, Texas. Blanchini, F., 1994, Ultimate boundedness control for discrete-time uncertain system via set-induced Lyapunov functions, IEEE Transactions on Automatic Control, 39(2), 428-433. Btanchini, F., 1995, Nonquadratic Lyapunov function for robust control, Automatica, 31(3), 451-461. Blanchini, F., Miani, S., 1995a, On tile transient estimate for linear systems with time-varying uncertain parameters, IEEE Transactions on Circuits and Systems, Part I, to appear. Blanchini, F., Miani, S., 1995b, Constrained stabilization of continuous-time linear systems, Systems and Control Letters, to appear. Blanchini, F., Sznaier, M., 1995a, Persistent disturbance rejection via static state feedback, IEEE Transactions on Automatic Control, 40(6). Blanchini, F., Sznaier, M., 1995b, Necessary and sufficient conditions for robust performance of systems with mixed time-varying gains and structured uncertainty, Proceedings American Control Conference, Seattle, 2391. Blanchini, F., Mesquine, F., Miani, S., 1995, Constrained stabilization with assigned initial condition set, International Journal of Control, 62(3), 601617. Brayton, R. K., Tong, C. H., 1980, Constructive stability and asymptotic stability of dynamicM systems, IEEE Transactions on Circuits and Systems, 27(11), 1121-1130. Castelan, E. B., Hennet, J. C., 1992, Eigenstrueture assignment for state constrained linear continuous-time systems, Automatica, 28(3), 605-611. Corless, M., 1993, Robust analysis and design via quadratic Lyapunov functions, Proceedings IFAC World Conference, Sidney, 18-23. Dahleh, M. A., Kammash, M. H., 1993, Controller design for plants with structured uncertainty, Automatica, 29(1), 37-56. Dahleh, M. A., J. B. Pearson, 1987,/l-optimal feedback controllers for MIMO discrete-time systems, IEEE Transactions on Automatic Control, 32, 314322. Dorato, P., Tempo, R., Muscato, G., 1993, Bibliography on robust control, Automatiea, 29(1), 201-214. Garofalo, F., Celentano, G., Glielmo, L., 1993, Stability robustness of interval matrices via Lyapunov quadratic forms, IEEE Transactions on Automatic Control, 38(2). Gerasimov, O. I., Yunger, I. B., 1993, Matrix criterion for absolute stability of pulsed automatic control systems, Automation and Remote Control, 54(2), 290-294. Glover, J. D., Schweppe, F. C., 1971, Control of linear dynamic systems with set constrained disturbances, IEEE Transactions on Automatic Control, 16, 411-423. Grumbaum, B., 1967, Convex polytopes, John Wiley & Sons, New York.
242 Gutman, P. O., Cwikel, M., 1986, Admissible sets and feedback control for discrete-time linear systems with bounded control and states, IEEE Transactions on Automatic Control, 31(4), 373-376. Gutman, P. O., Leitman, G., 1975, On a class of linear differential games, Journal of Optimization Theory and Applications, 17(5/6). Khargonekar, P., Petersen, I. R., Zhou, K., 1990, Robust stabilization of uncertain systems and Hoo optimal control, IEEE Transactions on Automatic Control, 35, 356-361. Keerthi, S. S., Gilbert, E. G., 1987, Computation of minimum-time feedback contol laws for discrete-time systems with state-control constraints, IEEE Transactions on Automatic Control, 32(5), 432-435. Lasserre, J. B., 1993, Reachable, controllable sets and stabilizing control of constrained systems, Automatica, 29(2), 531-536. Leitmann, G., 1979a, Guaranteed asymptotic stability for a class of uncertain linear dynamical systems, Journal of Optimization Theory and Applications, 27(1).
Leitmann, G., 1979b, Guaranteed asymptotic stability for some linear systems with bounded uncertainties, Transaction of the ASME, 101,212-216. Miani, S., 1993, Tecniche di sintesi del controllo per sistemi con vincoli su stati e ingressi, Engineering Degree Thesis, Padova, Italy. Michael, A. N., Nam, B. H., VittM, V., 1984, Computer generated Lyapunov functions for interconnected systems: improved results with applications to power systems, IEEE Transactions on Circuits and Systems, 31(2). Molchanov, A. P., Pyatnitskii, E. S., 1986, Lyapunov functions specifying necessary and sufficient conditions of absolute stability of nonlinear nonstationary control system, Automation and Remote Control, parts I, II, III, 47(3), 344-354; (4), 443-451; (5), 620-630. Ohta, Y., Imanishi, H., Gong, L., Haneda, H., 1993, Computer generated Lyapunov functions for a class of nonlinear systems, IEEE Transactions on Circuits and Systems, 40(5). Olas, A., 1991, On robustness of systems with structured uncertainties, Proc of the IV Workshop on Control Mechanics, California. Patel, R. V., Toda, M., 1980, Quantitative measure of robustness for multivariable systems, Proceedings American Control Conference, San Francisco. Petersen, I. R., Hollot, C., 1986, A Riccati equation approach to the stabilization of uncertain systems, Automatica, 22,397-411. Petersen, I. R., McFarlane, D. C., Rotea, M. A., 1993, Optimal guaranteed cost control of discrete-time uncertain linear systems, Proceedings IFAC World Conference, Sidney, 407-410. Rotea, M. A., Khargonekar, P. P., 1989, Stabilization of uncertain systems with norm-bounded uncertainty: a Lyapunov function approach, SIAM Journal on Control and Optimization, 27, 1462-1476. Sezer, M. E., Siljak, D. D., 1989, A note on robust stability bound, IEEE Transactions on Automatic Control, 34(11). Siljak, D. D., 1989, Parameter space methods for robust control design: a guided tour, IEEE Transactions on Automatic Control, 34(7), 674-688.
243 Shamma, J. S., 1994, Nonlinear state feedback for 11 optimal control, Systems and Control Letters, 21, 40-41. Sznaier, M., 1993, A set-induced norm approach to the robust control of constrained linear systems, S I A M Journal on Control and Optimization, 31(3), 733-746. Tan, K., Gilbert, E., 1991, Linear systems with state and control constraints: the theory and the applications of the maximal output admissible sets, I E E E Transactions on Automatic Control, 36(9), 1008-1020. Utkin, V. I., 1974, Sliding modes and their applications in variable structure systems, Moskow, Mir Publisher. Vassitaki, M., Bitsoris, 1989, Constrained regulation of linear continuous-time dynamical systems, Systems and Control Letters, 13,247-252. Yedavalli, R. K., Liang, Z., 1986, Reduced conservatism is stability robustness bounds by state transformations, I E E E Transactions on Automatic Control, 31. Zelentsowsky, A. L., 1994, Nonquadratic Lyapunov functions for robust stabitity analysis of linear uncertain systems, I E E E Transactions on Automatic Control, 39(1), 135-138.
12. A Lie-B icklund A p p r o a c h to D y n a m i c Feedback Equivalence and F l a t n e s s Michel Fliess, Jean Lfivine, P h i l i p p e M a r t i n and Pierre R o u c h o n 12.1 I n t r o d u c t i o n (Differentially) fiat nonlinear systems were introduced in 1992 (Ftiess et al, 1992a, 1992b, 1995a; Martin, 1992) by means of differential algebraic methods (Fliess and Glad, 1993). Their potential importance is due to numerous applications, such as the motion planning of several non-holonomic mechanical systems (Rouchon et al, 1993; Fliess et al, 1995a) or the control of aircraft (Martin, 1992; Martin et al, 1994), chemical reactors (Rothfuss et al, 1995) and electrical motors (Martin and Rouchon, 1996). The differential-algebraic definitions of flatness and system equivalence by endogenous feedback have been recently extended to the differential geometric case in (Fliess et al, 1993, 1994). A slightly different approach is proposed in (Pomet, 1993). The purpose of this chapter is, on one hand, to make an elementary presentation of this differential geometric standpoint and, on the other hand, to establish some connections with the classical notions of linear and nonlinear control theory. The need to describe control systems as geometric objects, such as manifolds and vector fields, with infinitely many components, made up of a finite number of independent variables and all their time derivatives of any order, is first explained and motivated on five examples including a simplified model of VTOL aircraft (Hauser et al, 1992; Martin, 1992; Martin et al, 1994), and an inverted pendulum model (Martin, t992; Ftiess et al, 1995a). To be more specific, by analogy with the differential-algebraic case (see (Fliess et aI, 1995a)), the notion of differential flatness is based on the concept of system equivalence by endogenous feedback, since, by definition, a flat system is equivalent, in this sense, to a linear controllable one. Because general dynamic feedbacks are not invertible, we are confronted with the problem of defining a true equivalence relation between systems having different dimensions. Our claim is that in the differential geometric context, the right equivalence concept, which copies and extends the one of the differential algebraic theory, must be stated in the context of the geometry of infinite jet bundles. Moreover, as will be seen on the five introductory examples, it naturally fits with important physical considerations, and confirms the fact that the classical geometric state space approach, where the state lives in a fixed finite dimensional manifold, and where
246 the allowed transformations (change of coordinates and feedback) must leave this manifold invariant, is too restrictive for our purpose. Connections with hierarchical control (Kokotovid and Khalil, 1986) and backstepping (Krstid et al, 1995) becomes also natural in this setting.
12.2 Five introductory 12.2.1
The
VTOL
examples
example
The Vertical Take Off and Landing (VTOL) aircraft model (Hauser et al, 1992) is the model of a mechanical system with two controls, whose evolution is restricted in the vertical plane, described by the following equations: Y: =
ulsinO-eu2cosO,
fl
ulcosO+eu2sinO-1,
=
(12.1a) (12.1b)
=
(12.1c)
with e a small parameter, (x, y) E IR 2 the position of the center of gravity and 0 E S 1 the roll angle. The control variables ul and u2 are the vertical acceleration and the torque, respectively, applied to the center of gravity. The acceleration of gravity is here normalized to 1. 12.2.2
The
VTOL
with
a model
of the
actuators
We now add to (12.1) a model of the actuators, namely differential equations that describe the way the acceleration and torque are produced.
=
u2,
(12.2a) (12.2b) (t2.2c)
Ul
:
21 (z, x, y, y, 0, 0, Ul, •2, Vl, v2),
(12.2d)
iL2
-.~ ~/2(X, x, y, ~I, O, O, Ul, ~t2, Vl, V2).
(12.2e)
~l
=
- u l sin0 + su2 cos0,
=
UlCOSO+eu2sinO-1,
The smooth functions 7~, i = 1, 2, which need not be specified here, are such that the mapping (vl, v~) ~-+ 7(x, ~?, y, ~), 0, 0, ul, us, vl, v2) is invertible for every (x, k, y, ~), 0, 0, ul, u2) where the vector function 7 stands for the vector According to this invertibility, and since the variables used to describe (12.2) can be expressed in terms of the variables of the VTOL system and a finite number of their derivatives (endogenous dynamic extension), it seems natural to call this new extended system equivalent to (12.1), though its dimension is different. Clearly, to every integral curve C of the system (12.1), defined by
247
t ~-~ C(t) = (x(t), [e(t), y(t), y(t), O(t), O(t), ul(t), u2(t)) for t in an open interval I of IR, there corresponds an integral curve g of (12.2), defined, on the interval I, by
t ~ ~(t) = (~(t), ~(t), ~(t), b(t), ~(t), ~(t), ~ ( t ) , ~ ( t ) , v~(t), ~(t)) where:
~(t) ~(t) ~(t) ~(t)
= = = =
~(t), y(t), 0(t), ~(t),
~(t) ~(t) ~ ~(t) ~2(t)
= = = =
x(t), 9(t), ~(t) , ~(t)
(12.3)
and
( vl(t) ) v2(t)
= 7
-1
.
.
.
.
(x[t),x(t),y(t),y(t),O(t),O(t),ul(t),itt(t),u2(t),it2(t)).
(12.4) We have thus constructed a mapping ¢ sending the integral curves g on tile integral curves C. ¢ preserves tangent vectors since to the tangent vector ~dg ( t ) there corresponds the tangent vector dE(t). Conversely, it is immediate to check that • is invertible: every point of the integral curve g can be uniquely expressed as a function of g. This new (inverse) mapping also preserves tangent vectors. It results that the trajectories of (12.1) and (12.2) are two different descriptions of the same object. We thus have the right to declare that (12.1) and (12.2) are equivalent. Since the above trajectories live in manifolds with different dimensions, and since their state representations are both strongly accessible (see, for example, (Isidori, 1989; Nijmeijer and van tier Sehaft, 1990)), this equivalence relation is indeed an unusual one. Moreover, it may involve derivatives of the input coordinates (see (12.4)). The above invertible mapping ¢ is called a LieB~cklund isomorphism and the associated equivalence relation, a Lie-B~cklund equivalence (see Section 12.3.2 below). Note that, since the model of the actuators can be realized from (12.1) by the invertible dynamic feedback
it1
-= 71(x,x,y,y,O,0, ul,u2, Vl,V2),
(12.5a)
~2 = ~.~(~,~,, y, ij, o, o, ~, ~2, v~, v2),
(12.5b)
the above equivalence relation can be interpreted as a dynamic feedback equivalence. We will see later on that the VTOL is flat, or, in other words, equivalent to a linear system. Therefore, system (12.2) is also flat by the transitivity of the equivalence relation. It is interesting to remark that this equivalence concept is strongly related to Cartan's absolute equivalence (Cartan, 1914), as noted by Shadwick (1990) (see also (Sluis, 1992)). It is also strongly related to the notions of hierarchical
248
control (Kokotovi6 et al, 1986), or backstepping (Krsti~ et al, 1995), that consist in shortening the integrators by invertibility or high-gain considerations. It gives a precise meaning to simplifications that are useful to clarify the structure of a complex system. Other examples of the same nature are presented below. Remark also that if, in place of a model of actuators of the first order, we had a higher order dynamic extension, using only the system variables and a finite number of their time derivatives (endogenous dynamic feedback), the same result would still hold true, with the Lie-Bgcklund isomorphism involving higher order derivatives of ul and u2. Therefore, it may be more convenient to define such mappings as ranging the manifold with coordinates
ij, o,& ul, u2, ul, into itself. This implies in turn that all the notions of systems, manifolds, vector fields, etc., may also be expressed in this setting. This is precisely what we have in mind. 12.2.3
The
inverted
pendulum
example
We consider an inverted pendulum in the (x, y)-plane (see (Martin, 1992; Fliess et al, 1995a)) given by: -- wl, ~) ---- w2, 80
--Wl COS 0 -{- (W 2 "[- 1) sin 0,
:
(12.6a) (12.6b) (12.6c)
where e is here interpreted as the inertia of the pendulum and where wl (resp. w2) is the horizontal (resp. vertical) component of the acceleration. Clearly, the same remarks as with the previous example can be done. To every trajectory of (12.1) with its associated tangent vector field, there corresponds a unique trajectory of (12.6) with its associated tangent vector field, and conversely, by the formulae: wl
=
ulsinO-eu2cosO,
w2
=
ulcosO+eu2sinO-1,
(12.7a) (12.7b)
and ul u2
-=
w l s i n 0 + ( w ~ + l ) cos0, 1
- ( - w t cos0 + (w2 + 1) sin 0), g
(12.8a) (12.8b)
x, y, 0 being the same in both systems. Note that, since the dimension is preserved, the mapping • is a diffeomorphism: our notion of equivalence coincides here with the classical equivalence by diffeomorphism and static feedback (see, for example, (Isidori, 1989; Nijmeijer and van der Schaft, 1990)).
249 12.2.4
An
implicit
model
of pendulum
We also consider the following variant of (12.6), obtained by eliminating the control variables wl and w2:
(1:2.9)
e0 = 55c o s 0 + (~) + 1) sin 0.
This is a single implicit differential equation with 3 unknown functions (x, y, 0). Such an implicit differential equation is thus called underdetermined. The number of degrees of freedom is precisely the number of independent inputs: by posing 2 = wl and ~) = w2, the system becomes determined and the equivalence with the inverted pendulum is clear. Therefore, beside the fact that the control variables are not specified, it contains exactly the same information as (12.6). This shows in particular that our notion of equivalence is intrinsic in the sense that it does not depend on the choice of input, state and other variables to describe the system (compare with (Willems, 1991)). 12.2.5
The
Huygens
oscillation
center
Finally, we consider the following implicit differential system (see (Fliess et al, 1992b; Martin, 1992)) with 4 unknown functions and 2 equations:
~(¢ -
( ( _ ut)2 + (¢_u~)2
u~) - (~ - ~1)(¢ + 1)
= =
~2,
(12.10)
o.
It can be checked that all the variables of this system, namely ~, ~, 4', ¢, ~, ~:, t'l, v2, can be expressed, in an invertible way, in terms of the variables of (12.6) or (12.1). We have: = x + ~sin0, ¢ = y+seos0, (12.11) //1
"--
X,
u2
=
y,
and conversely, x
=
¢-¢-
4"
(12.12a)
y
=
¢-e
~+ 1 ~((~')~ + (~ + 1) 2,
(12.12b)
0
=
arctan .. ~'
wl
-
~/-~ ~-eV/((~.)~ + ( ¢ + 1) ~
4+1'
d2(
)
(12.12c)
,
(12.12d)
250
w2
-
~
V((~.)2+(~+1) 2
C-
•
(12.12e)
Note that the coordinates (~,~) are known as the Huygens oscillation center (Whittaker, 1937). Comparing this example with the second one, we find that, here, the control variables are vl and v2. Remind that, by (12.11), we have i)1 = Wl and /)2 = w2, where (wl,w2) is the input of the pendulum model (12.6). Thus, (12.10) may be seen as a reduced model where we have removed the double integrators of the VTOL or the inverted pendulum. 12.2.6
Conclusion
To summarize, the first three examples are given by state representations with different dimensions, the first and the third example being described by 6 variables, whereas the second is 8-dimensional, and the last two examples are not given in classical state variable representation. However, they are all equivalent as far as we accept to deal with coordinates and a finite, but a priori unprescribed, number of their derivatives with respect to time. This is why, in the first example, we must consider the time t and coordinates of the form
(x,
y, vy, e, v.,
where vx, vy and vo correspond to the velocity of x, y and 0 respectively, and where u!f ) corresponds to the derivative of order p of uj with respect to time. The manifold associated to (12.1) is thus the Cartesian product: J
~x~x~x~xS
1 x~x~
° ° x ] R °°.
A simple introduction to the basic properties of such infinite dimensional differential manifolds is provided in the next section.
12.3 I n f i n i t e - d i m e n s i o n a l differential g e o m e t r y 12.3.1
Infinite
number
of coordinates
12.3.1.1 Explicit state variable representations = f ( x , u)
Consider the dynamics
(12.13)
where f is smooth on an open subset X × U C IRn x IRm. f is in fact an infinite collection of vector fields parametrized by u. More precisely, to define an integral curve of (12.13), we must specify not only the initial condition x0 at time t -- 0, but also the smooth time function t ~-~ u(t). This
251 infinite-dimensional dependence on the input u is not we!l-adapted when considering dynamic feedback. According to the examples of the previous section, we develop a slightly different standpoint where the integral curves of (12.13) are described in a more compact way as smooth functions t ~+ (x(t), u(t)), parametrized by initial conditions only. More precisely, we consider initial conditions in the form of the infinite sequence ~0 = (x0, u0, u 0 , . . . , u ~ ' ) , . - - ) , where the derivatives of u of any order at time t = 0 are noted u (~), with p _> 0. We are therefore led to complete the original coordinates (x, u) by the infinite sequence of coordinates ~ = (x, u, h , . . . , u ( ' ) , . . . ) E X x U x I R ~ , where we have denoted by IR~ = IR"~ x IR"~ x - •., the product of a countably infinite number of copies of IR"~. In this context, a smooth function is a function smoothly depending on a finite (but arbitrary) number of coordinates. Then, if we prolong the original vector field f as F(~) = (f(x, u), u, ~ , . . . ) , equation (12.13) reads = F((),
(12.14)
with ~(0) = ~0. Therefore, (12.14) defines a vector field, in the classical sense, on the infinite dimensional manifold X x U x IR~. The same conclusion is reached in another way, by considering the next Lie derivative formula: take a smooth function h, i.e. smoothly depending on x, u and a finite number r of derivatives of u. The time derivative of h along a trajectory of (12.13) is given by
dh Oh Oh. Oh u(~+l) dt - ~x f + -~u u + " " + cgu(r)
(12.15)
at every point (x(t),u(t), h ( t ) , . . . , u(r)(t),...). Note that, though h depends only on derivatives of u up to order r, the coordinate u (~+1) is required, which is another motivation to consider the coordinates made up with the whole sequence of derivatives of u. This formula may be interpreted as the Lie derivative of h with respect to the infinite-dimensional vector field
(X, tt, U (1), U(2),...) ~ F(X, U, U (1), U(2),...) : (f(x, u), u (1), u(2),...)
(12.16)
or, with notations easily understood from the Lie derivative formula (12.15), 0 = f(x,
~-~ u(,+:) +
0 '
(12.17)
#=0 where u = u (°). Note that each component of F is a smooth function, i.e. depends smoothly on a finite number of coordinates. Therefore, to the controlled system (12.13), where f is a family of vector fields parametrized by u, we substitute the following system definition with a "true" vector field on an infinite dimensional manifold.
252
D e f i n i t i o n 12.1 A classical system is a pair (X × U × ] R ~ , F ) where F is a smooth vector field on X × U × l R ~ .
Remark. To be rigorous we must say something of the underlying topology and differentiable structure of lRm ~ to be able to speak of smooth objects. This topology is the Frdchet topology (see (Zharinov, 1992)), which makes things look as if we were working on the product of k copies of IRm for a "large enough" k. For our purpose it is enough to know that a basis of the open sets of this topology consists of infinite products U0 × UI × -- • of open sets of IR"~, and that a function is smooth if it depends on a finite but arbitrary number of variables and is smooth in the usual sense. In the same way a mapping : IR~ -4 IR~ is smooth if all of its components are smooth functions. Notice also that IRm ~ equipped with the Frdchet topology has otherwise pretty bad properties: very useful theorems such as the implicit function theorem, Frobenius theorem or the straightening out theorem (Abraham et al, 1988) do no longer hold true. This is only because IRm ~ is a really big space: indeed the Frdchet topology on the product of k copies of IRm for any finite k coincides with the usual Euclidean topology. Remark. We saw above how a "classical" control system of the form (12.13) fits into our definition. There is nevertheless an important difference: we loose the notion of state dimension (see the VTOL examples with and without a model of the actuators as another illustration of this aspect). Indeed = f ( x , u),
(x, u) E X x U C IR n × ]R m
(12.18)
and
= =
f ( x , u), v,
(12.19a) (12.19b)
oo now have the same description (X × U x IRm, F), with
F ( x , u, u (1), u(2),...) -- ( f ( x , u), u (1), u(2), . . .), in our formalism: indeed t ~4 (x(t), u(t)) is a trajectory of (12.18) if and only if t ~-~ (x(t), u(t), i~(t)) is a trajectory of (12.19). This situation is not surprising since the state dimension is of course not preserved by dynamic feedback. On the other hand we will see that there is still a notion of input dimension. E x a m p l e 12.2 The trivial system (]R~, Fro), with coordinates (y, y(1), y(2),...) and vector field
Fm(y, y(1), y(2),...) __ (y(1) y(~), y(3),...) or, with the differential operator notation (y _-- y(0))
F m ( y , y ( 1 ) y(2)
") = Z Y ( V + I ) v>0
0
253
describes any system made up of m independent chains of integrators of arbitrary lengths, and in particular the direct transfer y = u.
12.3.1.2 The general case As suggested by the implicit representations of the pendulum and oscillation center presented in the first section, it may be interesting to generalize the above considerations to systems without reference to the particular state variable representation we are working with. We consider a pair (2vI, F) where ~4 is a smooth n:lanifotd--possibly of infinite dimension--and F is a smooth vector field on A/I. This pair does not generally define a system since F may not depend on a finite number of independent input channels, as in the classical case. This notion is precised in the next definition, where the notions of submersion and fiber are straightforward extensions of their finite-dimensional counterparts (see, e.g., (Abraham et al, 1988)). D e f i n i t i o n 12.3 We say that the pair (A/l, F) is a system if, and only if, there exists a smooth submersion 7r to the trivial system (IRm ~ , Fro) with global coordinates ~ = (u, it, il, . . .), such that every fiber ( = 1r-1(~) is finite dimensional with locally constant dimension for every ~. In what follows, we always work with submersions for which the number m of independent input channels is maximal. Locally, a control system looks like an open subset of tR" x IR~ with coordinates ~ = (~, u, ~) and vector field
= g((,
_
0
+ >0Z
O
(t2.20)
where all the components of g depend only on a finite number of coordinates. h trajectory of the system is a mapping t ~4 ~(t) = (((t),~(t)) such that =
Note that our definition of system does not distinguish between state and input variables since they are both deduced from the submersion lr which is assumed to exist, but not fixed a priori. E x a m p l e 12.4 For the classical system (X x U x l R ~ , F ) with local coordinates = (x,3) = (x,u, it, 5 , . . . ) and
= f(x,
0 Ou(~)
+ ~>0
the submersion 7r is just the projection (x, 3) ~+ ~ from X × U x IR~ to the ~ .~ (v+l) o~-7~" o trivial system (IR,~, F,~) with F,~ = vz-,~>_0 E x a m p l e 12.5 In the coordinates ( x , v ~ , y , vy,O, ve,u~U),u~'),# > 0), the vector field associated to the system (12.1) is given by:
254
F
=
0 v = ~ x + ( - s i n e ul +
0
u2)
+ (cos O ul + e sin O u2 - 1) ~
0
0
+
0
0
vo -~ + u20vo .
(12.21)
Similar expressions can be easily obtained for the vector fields corresponding to (12.2) or (12.6), which correspond to a different choice of submersion. The vector field associated to (12.9), in the coordinates (0, vo, z, x(1), x(2),..., y, y(1), y(2),...) is
ve -~-~+0-~l(x(2)c°sO+(Y(2 ) + 1) sin 0)
c9 oo oo ~v8 +~__0 x ("+1) Oz(~)O+ E Y(~+l)~=0
Oy(")'c3 (12.22)
As seen in the first section, the integral curves of (12.21) and (12.22) are transformed into each other by a smooth mapping and the respective tangent vectors are transformed accordingly. Therefore, the same transformation should change the vector field (12.21) into (12.22). The underlying equivalence relation will be precised in the next section.
Remark. Our definition of a system is adapted from the notion of diffiety introduced in (Vinogradov, 1984, 1994) to deal with systems of (partial) differential equations. By definition a diffiety is a pair (M, CTAd) where A~ is smooth manifold--possibly of infinite dimension--and CTY~4 is a Caftan distribution on ~4, namely an involutive finite-dimensional distribution on 2~4. Remind that involutive means that the Lie bracket of any two vector fields of CTJ~4 is itself in CTA~. The dimension of CTM is equal to the number of independent variables. As we are only working with systems with lumped parameters, hence governed by ordinary differential equations, we consider diffieties with one-dimensional Caftan distributions (see (Fliess et al, 1996) for more details). Until now, we have chosen to single out a particular vector field rather than work with the distribution it spans. The difference is simply explained in terms of time scaling: the distribution spanned by the vector field F, noted span (F), is made of vector fields of the form F7 ~ 7.F where 7 is a smooth function on .h4. Therefore, changing F into F~ can be interpreted as the time-scaling dr 1 -in a neighborhood of a point where 7 ¢ 0. Note that this time change
dt
7
may depend on a finite number of components of ~. Indeed, ~
= 7(~)F(()
d~ becomes ~ = F(~). Though not always necessary, it is often useful to introduce an additional coordinate, corresponding to time, to deal with such scalings and
255 the original manifold is thus replaced by IR x 3,t with coordinates (t, ~). The 0 vector field F is, in this case, replaced by ~-~ + F. The above calculations show that span(~--~ + s p a n ( F ) ) = s p a n ( s p a n ( ~ ) - t - F ) .
Remark. It is easy to see that the manifold Ad is finite-dimensional only when there is no input, i.e., to describe a determined system of differential equations--as many equations as variables. In the presence of inputs, the system becomes underdetermined--more variables than equations--which accounts for the infinite dimension. Remark. In place of the distribution spanned by our vector field F, we could have worked with eodistributions. On the manifold M with coordinates (t, x, u, u(1),.. ,), we can indeed define a 1-form w as afinite linear combination of {dr, dxi, du~~) I i = 1 , . . . , n; j = 1 , . . . , m; p >_ 0}. The finiteness requirement is implied by the fact that, by definition, a smooth function, and therefore its differential, depends only upon a finite number of variables. The codistribution orthogonal to the vector field F is spanned by the 1-forms ca such that (F, ca} = 0. If F is defined by (12.17), its orthogonal codistribution is spanned by the infinite set of Cartan 1-forms
{dxi-fi(x,u)dt,
d u ~ U ) - u (j U + l ) d t l i = l , ,
...,
n; j = 1, ... , m; p > 0}._
A comparable approach may be found in (Nieuwstadt et al, 1994). 12.3.2
Changes of coordinates mappings
and Lie-B~icklund
In this section we define an equivalence relation formalizing the idea that two systems are "equivalent" if there is an invertible transformation exchanging their trajectories. As we will see later, the relevance of this rather natural equivalence notion lies in the fact that it admits an interpretation in terms of dynamic feedback. Consider two systems (Ad, F) and (H, G) and a smooth mapping ¢ : ~4 -4 N" (remember that by definition every component of a smooth mapping depends only on a finite number of coordinates). If t ~-~ ~(t) is a trajectory of (JM, F), i.e., vt,
=
the composed mapping t ~-~ •(t) = ¢(~(t)) satisfies the chain rule
256
(~(t)).F(~(t)).
~(t) = ~ - ( ~ ( t ) ) . ~ ( t ) =
We insist that the above expressions involve only finite sums even if the matrices 0O and vectors have infinite sizes: indeed a row of ~ - contains only a finite number of non zero terms because a component of • depends only on a finite number of coordinates. Now if the vector fields F and G are O-related, i.e., =
00 u~
then 4(t) =
= G(¢(t)),
which means that t ~-~ #(t) = O(~(t)) is a trajectory of (A;, G). If moreover • has a smooth inverse ~ then obviously F, G are also ~Vrelated, and there is a one-to-one correspondence between the trajectories of the two systems. Such an invertible • relating F and G corresponds the a more general notion of Lie.Biicklund isomorphisms. D e f i n i t i o n 12.6 We say that • is a Lie-Bgcktund isomorphism if 4) is a smooth mapping from M to Af, preserving the distributions span(F) and span(G), namely such that its tangent mapping TO satisfies TO (span(F)) C span(G), and if it has a smooth inverse ~ from Af to M with T~(span(G)) C span(F). Clearly, • is Lie-Bgcklund if, and only if, it relates F to an element ~G of the distribution span (G). A Lie-Bgcklund isomorphism such that ~ _= 1, which, in some sense, preserves the time, will be called an endogenous transformation. D e f i n i t i o n 12.7 Two systems ( M , F) and (iV, G) are differentially equivalent, or shortly equivMent, at (p, q) E M x A/" if there exists an endogenous transformation from a neighborhood of p to a neighborhood of q. (.tel, F) and (X, G) are equivalent if they are equivalent at every pair of points (p, q) of a dense open subset of M × N . This (differential geometric) notion of equivalence corresponds to the notion of equivalence in the differential algebraic setting where the ground field is a field of constant (Fliess et al, 1992b, 1995a). D e f i n i t i o n 12.8 Two systems (A~, F) and (Af , G) are orbitally equivalent at (p, q) E .&4 x Af if there exists a Lie-Biicklund isomorphism from a neighborhood of p to a neighborhood of q. ( M , F) and (.IV',G) are orbitally equivalent if they are orbitally equivalent at every pair of points (p, q) of a dense open subset of.L4 × Af.
257
Orbital equivalence means that there exists a one to one correspondence between the curves on 2¢[ tangent to span (F) and the curves on Af tangent to span (G). As opposed to the definition of differential equivalence, curve parametrization is not necessarily preserved. Notice that when 34 and N" have the same finite dimension, the systems are necessarily equivalent by the straightening out theorem. This is no longer true in infinite dimension. E x a m p l e 12.9 Consider the two systems (X x U x IR~, F ) and (Y x V x lR~, G) respectively describing the dynamics
= =
f(x,u), g(y,v),
(x,u)~XxUc~R (y,~)ey×vc~×~
(12.23)
~ x l a "~, ~.
(12.24)
The vector fields F , G are defined by
F(x,u,u(1),...) G(y, v, ?)(1),...)
=
(f(z, u), u (1), u(~),...),
~_~
(g(y, v), v(~), v(2),...).
If the systems are equivalent the endogenous transformation • takes the form
~(x, ~, ~(~),...) = (~(x, ~), ~(~, ~), ~(x, ~),...) = (y, ~, ~,...). Here we have used the short-hand notation ~ = (u, u(1),..., u(k)), where k is some finite but otherwise arbitrary integer. Hence ~ is completely specified by the mappings ~ and a, i.e, by the expression of y, v in terms of x,~. Similarly, the inverse ~P of ~ takes the form
e(y, v, vl,...) = (¢(y, ~), Z(y, v), f)(~, v),...) = (x, u, u(1),...). As ¢ and kP are inverse mappings we have ¢(~(x,~),~(x,~)) ~(~(x,~),~(~,~))
= =
x, u,
and vice versa. Moreover F and G ~-related implies
/(¢(y,~),~(y,~))
=
.g(y,v)+~
k i=O
for some large enough k. In other words, every time
t ~ (~(t), u(t)) is a trajectory of (12.23), then
0 ¢ v(i+l)
,
(12.25)
258
(y(t), v(t)) =
)
is a trajectory of (12.24), and vice versa. The adaptation of the above interpretation to Lie-BKcklund isomorphisms is easily done by locally expressing the time scalings as functions of a common time scale. An important property of endogenous transformations and Lie-B~icklund isomorphisms is that they preserve the input dimension: T h e o r e m 12.10 Consider two systems (.A4, F) and (2v~, G). If they are LieB(icklund equivalent they have the same number of independent input channels. Proof. By definition, A/l (resp. Af) may be locally identified to X × U x IR~ (resp. Y × V x IR~). Consider the mapping • u o f ~ on X x U x (IR'~) ~,
• ,:XxUx(IRm+k)" (x,u,u(1),...,u(k+"))
--+ Y x V x ( ~ t ' ) " ~ (y,v,v(1),...,v(")),
i.e., the first p + 2 blocks of components of ~; k is just a fixed "large enough" integer. Because ~ is invertible, ~ is a surjection for all p. Hence the dimension of the source is greater than the dimension of the target, V#,
dim(x) + m ( k + # + 1) _> dim(y) + s(# + 1),
which implies m _> s. Using the same idea on ~P leads to s _> m.
12.3.3 F l a t n e s s Recall that a trivial system is a system (IR s , s), where F~(y, y(1), y(2) (y(1), y(2), y(3),...), with y E IR ~.
.)
D e f i n i t i o n 12.11 The control system (.A4, F) is said to be differentially flat (or shortly flat) at p, if, and only if, it is equivalent to a trivial system in a neighborhood of p. It is said differentially flat if it is differentially flat at every p of an open dense subset of 2~4. The set y --- {yj ] j -- 1 , . . . , s} is called a flat or linearizing output of 2¢1. D e f i n i t i o n 12.12 The control system (A/l, F) is said to be orbitally flat at a point p if, and only if, it is orbitally equivalent at p to a trivial system. Orbital flatness means orbital flatness at every point of an open dense subset of A/[. We immediately deduce from Theorem 12.10 the following result. C o r o l l a r y 12.13 Consider a fiat system (orbitally or differentially). The number of components of a flat output is equal to the number of independent input channels.
259 E x a m p l e ( V T O L a i r c r a f t ( c o n t i n u e d ) ) As shown in (Martin et al, 1995), this time-invariant system is differentially flat. The flat output is the Huygens oscillation center:
(12.26)
(Yl,Y2) = (x + esinO, y + e cosO).
" (")~j~,>_oj 1 ,% Consider the manifold M with coordinates ( x , v x , y , vy,O, vo,~ru(~) and vector field F defined by (12.21). Let us also consider the trivial system
(n~7, F2). The mapping • :
y,
o, vo,
" (") ,Y2(")')t,>_o u2 )~,>o) ~ tYl
where y~0) = y(1) = y~2) = y(3)
y~0) = x + e s i n 0
y~l) ----vx -I- e cos 0 vo y~2) = sinO(ul -- e(vo) 2) ~(a) _ d t -- ~t ( s i n O ( u l - e(vo)2)) ,(4) _ d ~ (sinO(ul ~1 - dt 2
Y + a cos 0 vy - ~ sin 0 vo - 1 + cos O(ul - c(vo) 2) d
= ~ ( ~(cos0(~ o),: -
y~4)
e(vo)2))
= ~
(cos0(~ -
))
E(VO)2))
is an isomorphic endogenous transformation such that F and F2 are J-related. The inverse mapping is based on the relations
x
--
y~0) csin
arctan
(12.27a)
1-y12) ] ] ' Y
0
y~0)
=
arctan
(arctan ( l y ( 2oz )
2 1 -~2))
.
(12.275) (12.27c)
As noted in (Fliess et al, 1995a), differential flatness means that the state and input may be completely recovered from the flat output without integrating the system differential equations. The consequences on the solution to the motion planning problem as well as for trajectory stabilization are immediately understood. The reader may refer once more to (Fliess et al, 1995a) and the bibliography therein for an extensive collection of examples illustrating the various applications of flatness.
260
12.4
Interpretation feedback
of equivalence
in terms
of
Consider the two systems (X × U × iRm ~ , F ) and (Y × V × iR~, G) respectively describing the dynamics
=
f(x,u),
(x,u) E X × U C I R n
~) =
g(y, v),
(y, v) E Y × V C iR~ × iR~.
×iRa,
(12.28) (12.29)
The vector fields F, G are defined by
F(~, ~, ~(~),...) = G(y, ~, v(*),...) =
(f(~, ~), ~(~), ~(~),...), (g(y,v), ~(~),,(=),...).
Note that the general case can be reduced to the above case as follows. We have seen that a system may be locally described by a manifold X × IRm ~ with coordinates (~, ~) and a vector field of the form (12.20). Since g contains only a finite number of derivatives of u, let us denote by r the highest order of derivation and v = u (r). Then setting x = (if, u, ~ , . . . , u(rl)), we easily see that the vector field in these new coordinates is expressed as a classical vector field. We thus only sketch the results in the classical case since they easily extend to the general case by the same remark. In order to avoid some technicalities related to the intrinsic definition of a general dynamic feedback, I the next result, though valid for general systems, is only stated in the classical setting. If systems (12.28) and (12.29) are equivalent, we are going to show that it is possible to go from (12.28) to (12.29) and vice versa by a dynamic feedback
u = ~(x, z, ~), =
a(x,z,w),
(12.30a) (12.30b)
with z E Z C IRq. Of course we cannot hope to go from one dynamics to the other without changing the state dimension. But this is in some sense the only thing we lose: T h e o r e m 12.14 Assume the systems (X x U x IR~, F) and (Y × V × IR~, G)
are equivalent. Then, there exists a dynamic feedback such that the closed-loop system (12.28)-(12.30) is diffeomorphic to (12.29) prolonged by sufficiently many integrators. Here "(12.29) prolonged by sufficiently many integrators" means 1A general dynamic feedback is a Lie-B~icklundcorrespondence (see (Fliess et at, 1994)).
261 9
=
g(y,~),
~) = ,)(i) =
v(i), v(2),
7)(u) =
(12.31)
w
for # large enough.
Proof. Remind that it suffices to prove the result in the classical case. We follow the proof in (Martin, 1992): let ~ = (y, v, v(1),..., v (~)) and w = v ('+1) . Using the notations of Example 12.9, we see that, for # large enough, ¢ depends only on ~ and/3 only on (~, w), i.e., the endogenous transformation ~ takes the form ~ ( 9 , ~ , ~(~), . . .) = ( ¢ ( 9 ) , z ( ~ , ~ ) , ) ( 9 , ~ ) , . . .),
and equation (12.25) now reads f(g)(~),/3(~, w)) = 0@~).~(~, w),
(12.32)
where ~ = (g, vO),...,v(U)). Let ~ = (~,, t)b) be a splitting of the components of~ such that the mapping ~ K(9) = (¢(9), 9b)
is invertible (such a splitting exists because ~, being a block of components of the invertible mapping ~, is full rank). Apply now the dynamic feedback u
=
~(K-l(x,~),w),
where gb stands for the part of .~ corresponding to Yb, to get the closed-loop dynamics = ] ( ~ , z, ~) =
~b(K -1 (x, z), ~)
Using (12.32), we have
](K(~), w) = : -
(f(g?(~))'~b(t), 3(tJ'w)) ) w ) ( O0~y(t))O 0 ) i .~(~,W)
OK
~ (9)~(9, w),
which means (12.33) and (12.31) are diffeomorphic.
"
262 Notice that it is not a priori obvious that an equivalence relation based on dynamic feedback transformation is interesting from a control point of view. Indeed, consider the scalar integrator k = u acted upon by the dynamic feedback k = v, u = v. Though the feedback is, according to the usual terminology (Nijmeijer and van der Schaft, 1990), invertible, the closed-loop system is not controllable, and it is not possible to restore controllability by feedback. This means that the class of feedbacks associated to our equivalence notion by Theorem 12.14 is smaller than the class of invertible dynamic feedbacks. In fact the feedback build in the proof has the a true "reversibility" property: there exists another feedback which leads back to the original system extended by integrators. Such a feedback is called endogenous (Martin, 1992, 1994) because the new z variables it contains can be expressed as functions of the state and (finitely many) derivatives of the input. E x a m p l e ( V T O L a i r c r a f t ( e n d ) ) According to (12.27), the state of (12.1), i.e., (x, v,, y, vu, O, vo) is a function of the linearizing output (Yt, Y2) and is derivative up to order a = 3. Thus, according to the method presented here above for computing the dynamic feedback, there exists an endogenous dynamic feedback leading to the following closed-loop system (a + 1 = 4):
A possible linearizing dynamic feedback is as follows (q = 2): d
=
¢2 =
w l s i n O + w 2 c o s O + (ve)2¢1,
ul
¢t
=
2, Wl cos 0 - w2 sin 0 - 2voG
U2
¢1
It can be obtained by applying the previous constructive method with ¢ = (ul - e(v0) 2, ~il - 2evou2), or by considering classical inversion methods via dynamic feedback on (12.1) with output (yl, y2) defined in (12.26). Notice that this dynamic feedback is not a simple prolongation where time derivatives of the control (Ul,U2) are added to the state. For this system, one can prove, using the characterization in (Jakubczyk and Respondek, 1980) that, for any #1 >_ 0 and ,2 >_ 0, the prolonged system (12.1) with u~m) = vl and u~m) = v2, is not linearizable via static feedback. This point constitutes one of the major difficulty for finite characterization of flat system. 12.4.1
Trivial residual
dynamics
Dynamics (12.13) with output y = ( y l , . . . , Y,~) is a square left and right inputoutput invertible system, where, moreover, any component of u or x may be recovered from y without integrating any differentiM equation: we will say that it possesses a trivial residual dynamics. The residual dynamics is slightly different from the more usual zero dynamics (Isidori, 1989; Nijmeijer van der Schaft,
263
1990): we are, for instance, not necessarily working at an equilibrium point. It yields the
Proposition
12.15 (12.13) is differentially fiat at a point if, and only if, it is possible to find an output y = ( Y l , . . . , Ym), which smoothly depends around that point on x l , . . . , x ~ , u l , . . . , u m and on a finite number of derivatives of the uj 's, such that the resulting square system is left and right input-output invertible, with a trivial residual dynamics.
It is possible to consider this proposition as definition of a flat system. With such a definition the underlying equivalence relation would be hidden and the geometry of the problem is lost: flat system are systems equivalent to linear controllable ones; two flat systems are equivalent if, and only if, they have the same number of inputs.
12.5 Controllability and first integrals 12.5.1
On the
strong
accessibility
property
Assume, for the sake of simplicity, that in (12.13) the input variables appear linearly
i: = Ao(x) + ~
Aj(x) uj.
(12.34)
j=l
Ao, A 1 , . . . , Am are smooth functions from IR'~ to IR~, which are considered as vector fields. 2 If the strong accessibility distribution 3 is not full rank, i.e., of dimension strictly less than n on some open subset of IRn, then (12.34) may locally be decomposed into ~1
=
A~(xI),
~
=
A2tx 1: x2~? -t- V"m 0'~ Z-,j=l
A2(xl ~ x2~/ Uj,
~*3 \
where (x 1, z 2) is another local set of coordinates. Then, as well known (see, e.g., (Reinhard, 1982)), there exist for the uncontrolled ditferential equation ie1 = A~(xt), non-constant first integrals, i.e., smooth functions h(t, x 1) of t and x 1, such that T dh_ = 0 . Assume, for the converse, the existence of a smooth function
I(t,x,u,,
,urn,
,u(m )
2 O u r restriction to s y s t e m s of f o r m (12.34) is d u e to t h e fact t h a t t h e y are very well covered in recent b o o k s (Isidori, 1989; Nijmeijer v a n d e r Schaft, 1990). "3Recall t h a t t h e accessibility d i s t r i b u t i o n is s p a n n e d by t h e v e c t o r fields A0, A1 ,.. •, A m . T h e s t r o n g accessibility d i s t r i b u t i o n is t h e s m a l l e s t s u b - d i s t r i b u t l o n A, which c o n t a i n s A1, . . . . Am, a n d s u c h t h a t , for a n y X E A, t h e Lie bracket [X,A0] also b e l o n g s to A.
264 such that ~ydl= 0. Then ~T,dlwhich contains 0u---~,j°t. (a+l), can be identically zero J
for any value of uJa+l) if, and only if,
Ol -= 0. Thus, I must be independent
of uJ~) and, by induction, of any component of u and their derivatives of any order. Then dI OI m d-Y = - ~ + LA°I + Z ujLAJ I" j=l
Thus, as above, Laj I -- 0 for j = 1,..., m. The commutativity between ~ and the LA, 's, i.e., [ o , LA,] = 0, yields d2 I 021 L OI m dr---~ - ~Ot + A o - ~ + L2Aot + Z ujL[Ao,Aj] • j-~ l
As above, L[Ao,Aj]I -- 0 for j =- 1,..., m. By induction, L x h - O, for any vector field X E A. It implies that the strong accessibility distribution A cannot be of full rank if a non-trivial first integral exists, i.e., which is non-constant. We have proved the
Proposition 12.16
The strong accessibility distribution A in (12.34) is full rank if, and only if, there does not exist any non.trivial first integral. 12.5.2
A general
definition
of controllability
Take a system (AJ, F) as in Definition 12.1, consider the Cartesian product IR × M which corresponds to the system (IR x M _' 0t a -P F). Consider the set {I [ L ~ + F I = 0} of smooth local real-valued functions on IR x AJ, such that their Lie-derivatives with respect to any Cartan field ~ + F is identically zero. A function I satisfying the above proposition is called a first integral. The preceding proposition leads to the
Definition 12.17 The control system A4 is said to be controllable at a point p E Ad if, and only if, there exists an open neighborhood 0 of p such that the set of first integrals on IR × 0 is trivial, i.e., equal to IR. This definition is not only independent, like in (Fliess et al, 1995a), of any distinction between the system variables, but also of any time-scaling. Notice furthermore that any system, which is orbitally equivalent to a controllable one, is again controllable.
1 2 . 5 . 3 C o n t r o l l a b i l i t y o f fiat s y s t e m s . (~j)
Consider the trivial system (IR~, Fro), with global coordinates lYj
]J =
1,..., m; vj > 0} and a smooth function I(t, y l , . . . , y ~ l ) , . . . , y m , . . . , y(a")).
265 Using the vector field d
0
d-/= o-7 +Fm over the manifold IR x lRm ~ with global coordinates {t, yJui) }j = 1 , . . . , m; ~,j >_ 0}, we have dI
Oh
dt Since dI
0, the coefficients
(~,j+l)
+ j=l E vj=O E 0z
OI
Oy?
of the highest order derivatives y~j+l)
must be identically zero. A straightforward induction argument shows that oI =- 0 for all the uj's. Thus I can only depend on t. But Ol = 0 shows that I is a constant (see also (Zharinov, 1992; Tsujishita, 1989)). T h e o r e m 12.18
The trivial system ( I R ~ , Fro) is controllable at any point.
The next corollary is obvious: C o r o l l a r y 12.19 around that point.
12.5.4
A linear
A n y orbitally flat system around a point is controllable
controllable
dynamics
is f l a t
Consider the linear dynamics = Ax + Bu,
(t2.35)
where x = (Zl,...,Zn)T~ U : ( U l , . . . , U m ) T, A e IR~x~, B E IR~x'~. For simplicity's sake, m < n and the matrix B is of rank m. Assume, following the above corollary, that (12.35) is controllable, i.e., rank(B, A B , . . . , A ~ - I B ) = n. By an appropriate static state feedback change of coordinates in the state space, (12.35) may be written in the Brunovsky (Kailath, 1980) canonical form y~UJ) = vj,
j=
1,...,m,
where * the uj's are the controllability indices; • x = P ( Y l , . . . , Y ~ ~'~-1), . .. ,.y m. ,.
, y(m""-l))T, where P e IR'*×~ is invert-
ible; • the new input variables v = (vl, • . . , vm) T are related to u by u = F x + G v .
266 Thus, any component of x and u may expressed as a linear function of the yj's and of a finite number of their derivatives. On the other hand, any yj is a linear function of the component of x. We have proved that (12.35) is differentially equivalent to the trivial system (]Rm ~ , Fro). The next result follows immediately T h e o r e m 12.20 The linear dynamics (12.35) is differentially flat if, and only if, it is controllable.
12.6 C o n c l u s i o n s This infinite geometry, as well as differential algebra, is useful to prove important results such as the fact that any dynamic feedback linearizable system is differentially flat (Fliess et al, 1995b).
References Abraham, R., Marsden, J. E., Ratiu, T., 1988, Manifolds, Tensor Analysis, and Applications, Springer-Verlag, Berlin, second edition. Caftan, E., 1914, Sur t'~quivalence absolue de certains syst~mes d'~quations diff~rentielles et sur certaines familles de courves, Bull. Soe. Math. France, 42, 12-48; also in Oeuvres Completes, part II, 2, pp. 1133-1168, CNRS, Paris, 1984. Fliess, M., Glad, S. T., 1993, An algebraic approach to linear and nonlinear control, in Trentelman, H. J., Willems, J. C., editors, Essays on Control: Perspectives in the Theory and its Applications, Birkh~user, Boston, 223267. Fliess, M., L~vine, J., Martin, Ph., Rouchon, Ph., 1992a, On differentially fiat nonlinear systems, Proceedings IFAC Symposium NOLCOS'92, Bordeaux, 408-412. Fliess, M., L~vine, J., Martin, Ph., Rouchon, P., 1992b, Sur les syst~mes non lin~aires diff~rentiellement plats, C. R. Acad. Sci. Paris, 1-315, 619-624. Fliess, M., L~vine, J., Martin, Ph., Rouchon, P., 1993, Lin~arisation par bouclage dynamique et transformations de Lie-B~cklund, C. R. Acad. Sci. Paris, 1-317, 981-986. Fliess, M., L~vine, J., Martin, Ph., Rouchon, P., 1994, Nonlinear control and Lie-B~izklund transformations: Towards a new differential geometric standpoint, Proceedings IEEE Conference on Decision and Control, Lake Buena Vista, Florida, 981-986. Fliess, M., L~vine, J., Martin, Ph., Rouchon, P., 1995a, Flatness and defect of nonlinear systems: introductory theory and examples, International Journal of Control, 61(6), 1327-1361.
267 Fliess, M., L~vine, J., Martin, Ph., Ollivier, F., Rouchon, P., 1995b, Flatness and dynamic feedback linearizability: two approaches, in Proceedings European Control Conference, Rome. Fliess, M., L~vine, J., Martin, Ph., Rouchon, P., 1996, Deux applications de la g~om~trie locale des diffi~t~s, Annales de l'Institut Henri Poincard, Physique Thdorique. Hauser, J., Sastry, S., Meyer, G., 1992, Nonlinear control design for slightly nonminimum phase systems: Application to V/STOL aircraft, Automatiea, 28(4), 665-679. Isidori, A., 1989, Nonlinear Control Systems, Springer-Verlag, New York, second edition. Jakubczyk, B., Respondek, W., 1980, On linearization of control systems, Bull. Acad. Pol. Sci. Ser. Sci. Math., 28, 517-522. Kailath, T., 1980, Linear Systems, Prentice-Hall, Englewood Cliffs, NJ. Kokotovid, P. V., Khalil, H. K., 1986, Singular Perturbations in Systems and Control, IEEE Press, New York. Kokotovi~, P. V., Khalil, H. K., O'Reilty, J., 1986, Singular Perturbation Methods in Control: Analysis and Design, Academic Press, London. Krsti5, M., Kanellakopoulos, I., KokotoviS, P. V., 1995, Nonlinear and Adaptive Control Design, J. Wiley, New-York. Martin, Ph., 1992, Contribution 5 l'dtude des syst~mes diff~rentietlement plats, PhD thesis, l~cole des Mines de Paris. Martin, Ph., 1994, Endogenous feedbacks and equivalence, in Systems and Networks: Mathematical Theory and Applications, Akademie Verlag, Berlin, 343346. Martin, Ph., Devasia, S., Paden, B., t994, A different look at output tracking: control of a VTOL aircraft, P~veeedings IEEE Conference on Decision and Control, 2376-2381. Martin, Ph., Devasia, S., Paden, B., 1995, A different look at output tracking: control of a VTOL aircraft, Automatica, 32, 101-108. Martin, Ph., Rouchon, P., 1996, Flatness and sampling control of induction motors, Proceedings IFA C World Conference, San-Francisco. van Nieuwstadt, M., Rathinam, M., Murray, R. M., 1994, Differential flatness and absolute equivalence, Proceedings IEEE Conference on Decision and Control, Lake Buena Vista, Florida, 326-332. Nijmeijer, H., van der Schaft, A. J., 1990, Nonlinear Dynamical Control Systems, Springer, New York. Pomet, J. B., 1993, A differential geometric setting for dynamic equivalence and dynamic linearization, Proceeding of Workshop on Geometry in Nonlinear Control, Banach Center Publications, Warsaw. Reinhard, H., 1982, Equations diffdrentielles, Gauthier-Villars, Paris. RothfuB, R., Rudo]oh, J., Zeitz, M., to appear, Flatness based control of a nonlinear chemical reactor model, Automatica, to appear. Rouchon, P,, Fliess, M., Lgvine, J., Martin, Ph., 1993, Flatness, motion planning and trailer systems, Proceedings IEEE Conference on Decision and Control, San Antonio. 2700-2705.
268 Shadwick, W. F., 1990, Absolute equivalence and dynamic feedback linearization, Systems and Control Letters, 15, 35-39. Sluis, W. M., 1992, Absolute Equivalence and its Application to Control Theory, PhD thesis, University of Waterloo, Ontario. Tsujishita, T., 1989, Formal geometry of systems of differential equations, Sugaku Expos., 5, 25-73. Vinogradov, A. M., 1984, Local symmetries and conservation laws, Acta Appl. Math., 2, 21-78. Vinogradov, A. M., 1994, From symmetries of partial differential equations towards secondary ("quantized") calculus, Journal Geometry Physics, 14, 146-194. Whittaker, E. T., 1937, A Treatise on the Analytical Dynamics of Particules and Rigid Bodies, Cambridge University Press, Cambridge, fourth edition. Willems, J. C., 1991, Paradigms and puzzles in the theory of dynamical systems, IEEE Transactions on Automatic Control, 3(6), 259-294. Zharinov, V. V., 1992, Geometrical Aspect of Partial Differential Equations, World Scientific, Singapore.
13. A s y m p t o t i c Stability and Periodic M o t i o n s of Selector-Linear Differential Inclusions Lev B. R a p o p o r t 13.1 P r o b l e m S t a t e m e n t We start with the classic problem of absolute stability of the control system i¢ = A x + b~,
c~ = cTx,
x, b, c E IR"
(13.1)
with the input ~, the output (r and the "sector-type" nonlinear time-varying feedback = ~(t, ~r), 0 < ~(t, ~r)(r << kc~2. (13.2) The matrix A is supposed to be stable n × n real matrix, T denotes the matrix transposition. The absolute stability problem (13.1), (13.2) is known to be equivalent to the robust stability of the linear system ic = A z + u ( t ) B x ,
B = bcr ,
(13.3)
with the time-varying uncertainty u(.) from the class U(k) of Lebesgue measurable functions subjected to the condition 0 < u(.) _< k.
(13.4)
Many methods were used to obtain the sufficient conditions for this problem. The first one consists of constructing a quadratic Lyapunov function. The second one involves a certain Popov-type frequency condition. Connection between two approaches were established by Kalman-Yakubovich-Popov lemma. These approaches gave rise to active research on quadratic stability of differential inclusions. However the quadratic stability is known to be only sufficient for absolute stability of (13.1), (13.2). There are systems (13.1), (13.2) which are absolutely stable but not quadratically stable. Another approach is based on variational technique (Pyatnitskiy, 1971). In the planar case (n = 2) this approach leads to the convenient necessary and sufficient conditions for absolute stability. It was shown in (Pyatnitskiy, 1971) that the class U(k) in the absolute stability analysis of the system (13.3) may be replaced by the class of the piecewise-constant periodic functions that take the values 0 and k only and have two switching points at the period. It turns out that the system (13.3) is absolutely stable in the class U(k) if and only if the condition det(exp(rA) exp(~A2) + I) ¢ 0
(A2 = A + ~bc T)
(13.5)
270 holds for any r > 0, y > 0 and 0 < ~ < k. Here I is an identity matrix. This condition is less conservative than quadratic stability since it is necessary and sufficient for absolute stability. Mention should be made of papers (Colonius and Sieveking, 1989) and (Colonius and Kliemann, 1993) where the role of periodic control is also pointed out. Our main goal is to extent the criterion listed above to the more general cases. This chapter contains some results in this direction. In particular, the above criterion is proved to be valid for the case n = 3.
13.2 Main Results Before to continue we will make some generalization in order to present the part of results for the more general problem statement. We will consider a problem of the asymptotic stability of the dynamic system described by the differential inclusion
e F(x),
F ( x ) = { B x : B E .,4 = c o ( { A l , . . . , Aq})},
(13.6)
x E IRn, co(.) denotes a convex hull. The convex polyhedron A of real n × n matrices is defined by vertices Ai, i = 1 , 2 , . . . , q. It is supposed that Ai ~ c o ( { A 1 , . . . , Ai-1, Ai+l . . . . , Aq}) for every i = 1 , . . . , q. The problem (13.3), (13.4) is reduced to (13.6) with the notation A1 = A, A2 = A + kbc T, q = 2. By the solution of (13.6) we mean an absolutely continuous vector function x(t) satisfying the condition ~(t) E F(x(t)) at almost all t. The zero solution x = 0 of (13.6) is said to be asymptotically stable if: (i) for every e > 0 there exists 5 > 0 such that for every solution x(t) of (13.6) the inequality [x(t)[ < e holds for every t > 0 if only Ix(0)[ < 5; (ii) there exists A > 0 such that limt-,¢¢ x(t) = 0 for every solution x(t) of (13.6) satisfying the inequality Ix(0)[ < A. Here I " I denotes any one of the equivalent norms on IR'~. Asymptotically stable inclusions are known to be exponentially stable in the large. Let F be the region of those sets { A 1 , . . . , Aq} for which the inclusion (13.6) is asymptotically stable. It is easily seen that /~ is an open cone. Now with a view to describe F we will restrict ourselves to the investigation of its boundary b n d ( F ) = c l ( F ) \ F , where cl(-) is a closure. Denote also by int(.) an interior of a set. If 0 E F(x) for some x ~ 0 then obviously { A 1 , . . . , A q } ~ F and the problem has a trivial solution. So we suppose that
~(1
o
F(x)
vx # o.
(13.7)
It is supposed also that matrices A1, • •., Aq have no a common nontrivial invariant subspace. This assumption when applied to the initial problem (13.1), (13.2) holds if the pair {A, b} is controllable and the pair {A, c} is observable. The absence of the common invariant subspace for the matrices A i , . . . , A q guarantees an absence of the invariant subspace of the inclusion (13.6) (Sussmann, 1976). Let X(xo, T) be the attainable set of the inclusion (13.6) from the point Xo at the time T. Let ),(t) be the q-dimensional measurable vector-function and q
A = {A(-) : 0 < Ai(t) < 1, E A i ( t )
= t}.
(13.8)
i=1
Then every solution of the inclusion (13.6) is also the solution of the linear time-varying system q
= Z
i(t).4 x
(13.9)
i=l
for some ~(.) C A. Conversely, every solution of the above system is also the solution of (13.6). For every y E IR~ we define ma×
~eX(y,T) T h e o r e m 13.1
Ix1.
Let { A 1 , . . . , A q } E bnd(F). Then:
(i) the function v(y) defined as v(y) = lim supr(y, T), T-~oo
is convex, positive for y 7£ O, sati4es the condition v(c~y) = [~]v(y) and takes the finite values only; (ii) for every initial point y • 0 there exists such a solution x*(.) of (13.6) that x*(0) = y and v(x*(t)) =
max v(x) = v(y) xEX(y,t)
(13.10)
holds for all t >_O. Here at almost all t > 0 the equality Ov(x*(t)) _ O, Oz
z = x*(t)
(13.11)
holds, Ov/az is the directional derivative. The full proof of this theorem is in (Pyatnitskiy and Rapoport, t996). Let as describe only the scheme. If it is supposed that limsuPT_.~ r(y, T) is unbounded for some y = yo then this upper limit is unbounded for any other y.
272 Otherwise the set of vectors y for which the upper limit is bounded would form the linear invariant subspace of the inclusion (13.6). That contradicts the absence of a common invariant subspace for matrices A1,. •., Aq. The unboundedness of the value limsupT.+c~ r(y, T) for every y results in existence of the exponentially growing solution of (13.6) for every initial point. Consequently for all sets {A~°),...,A (°)} close enough to {A1,...,Aq} the inclusion E F(°)(x), F(°)(x) = { B x : B E A (°) = co({A~°),...,A~°)})} also possesses exponentially growing solutions. That contradicts {A1,... ,Aq} E bud(F). Thus, the function v(y) takes finite values only. The convexity of v(y) is proved on the base of the linear dependence of solutions to the system (13.9) on the initial conditions. The proof of the statement (13.10) is based on semigroup property of solutions of (13.9) (or (13.6), which is the same). Let S be the surface {x : v(x) = 1} that is homeomorphic to the sphere S n-1. The statement (ii) of the Theorem 13.1 means that for every y E S there exists such a solution x*(.) that begins with y and belongs to S. Consider now the extreme problem
v(x) -+ max,
x E X(y,T),
(13.12)
for T > 0 and y E S. Let A E IRq and A(x,p) be the set of those A that solves the following linear programming problem: max )~
EAipTAix:
0 < )~i < 1,
i=1
Ai = 1
.
(13.13)
i=1
Having in mind the maximum principle consider the following differential inclusion: q
=
E AiAix,
(13.14a)
i=1 q
[9 =
- E AiATp'
A E Z(x,p).
(13.14b)
i=1
Let z(t) = (x(t),p(t)) be the solution of the inclusion (13.13), (13.14). It follows from the Theorem 13.1 that every solution of (13.6) that satisfies the condition x*(t) E S for all t _> 0 is also the solution of (13.12) for all T > 0 simultaneously. The following is the consequence of the maximum principle and Theorem 13.1: T h e o r e m 13.2
Let {A1,...,Aq} E bud(F). Then for every solution x*(t)
of (13.6) satisfying the condition x*(t) E S for all t >_0 and T > 0 one can find such an absolutely continuous function p* (t ) that (x* (t ) , p* (t ) ) satisfies (13.13), (I3.14) with the boundary conditions x(O) = y,
p(T) E Ov(x(T)),
(Or(.) is a subdifferential (Rockafellar, 1970)).
(13.15)
273
Let qsx(t, r) be the fundamental matrix of the systems be the fundamental matrices of the system
(13.9)
and gk(t, v)
q
[) = - E )~i(t)AT p"
(13.16)
i----1
Let # ~ ( r , r ) = I and # x ( r , v ) : I. Obviously, # ~ ( t , r ) = ( # ~ ( t , r ) T ) -1. Let C(x) be the cone tangent to the set {x • IR'~ : v(x) <_ 1} at the point x. T h e o r e m 13.3 Let {AI,..., Aq} • bnd(F). Let x* (t) be the solution of (13.6)
satisfying the condition x*(t) • S for all t > 0 and A*(.) be the corresponding vector-function from, (13.9). Then for any t > 0 and r • [0,t) the following inclusions hold 4)~.(t,r)C(x*(r)) cOv(x'(t))
C C(x*(t)), C_ #~.(t,r)cgv(x*(r)).
(13,17a) (13.17b)
Let p* (t) be the function from Theorem 13.2. Then for all t >_0 one has p*(t) •
o~(x*(t)). Proof. Let y E int(C(x*(r))). Then for e > 0 sufficiently small and xl = x*(r) +cy there follows v(xl) < 1. Since v(x*(t))= l a n d V(#x.(t,T)Xl)< V(Xl) < 1, then z = ¢&.(t,r)xl - x*(t) • C(x*(t)). But on the other hand, z : e#x. (t, r)y. Consequently, ~ . (t, T)int(C(x*(v))) C_ C(x*(t)). The above inclusion holds also at the closure of the left-hand side. The assertion (13.17a) is proved. Let p(t) : ~A.(t, r ) p ( r ) be the solution of (13.16). Since ~ . (t, r) = ( ~ . (t, r)T) -1, one has
p(t)T~ *(t) : p ( ~ ) % * ( ~ ) Then for arbitrary x 6 IR~ the following equality holds
p(t)r(~.
(t, ~)~ - ~'(t))
= :
( ~ . (t, ~ ) r ) - l p ( ~ ) ) T ( ~ . (t, . ) ( x - ~" (~))) p(~)r(x_ x,(~)).
It follows from (13.10) that v(O~. (t, r)x) <_ v(x). Consequently, from the condition v(#~. (t, r)x) - 1 > p(t)T(qs~. (t, r)x -- x*(t)) it follows
v(x) - 1 _ p(~)r(x - ~*(~)). Taking into account the equalities v(x* (t)) = v(x* (v)) = 1 one obtains that if p(t) 6 Or(x* (t)), then p(r) E Ov(x*(r)). This proves the inclusion (13.17b). It follows from Theorem 13.2 that p*(T) C av(x*(T)). Then (13.17b)implies p* (t) E Ov(x* (t)) for all t 6 [0, T). Since T can be as large as we please, we arrive at the last assertion of the theorem.
274 Recall that the convex function v(z) is differentiable at almost all points z (Rockafellar, 1970). By the uniformity of v(x) it is differentiable at almost all points of the set S. L e m m a 13.4 Let the conditions of the Theorem I3.3 are met. l f z*(t) starts with the point y = x*(0) in which the function v(x) is differentiable, then for
any t > 0 the function v(z) remains to be differentiable at z* (t). Actually, it follows from (13.17b) that if 0v(z*(0)) consists of the Proof. unique vector grad v(y), then also Ov(x*(t)) consists of the unique vector grad v(x* (t)). Let now AI(z) be the set of those ~ that solves the following optimization problem: max
Oz
:z = ~
)~ipT Aix,
0 < )q < 1,
i=1
~
)~, = 1
i=1
Given the convex function v(x) the directional derivative Ov(z)/Oz is the concave function with respect to the variable z. Consequently in general case the set Al(z) is nonconvex. In particular, if the function v(z) is differentiable at the point z then av(x) = {grad v(z)}, the above optimization problem is the linear programming problem and the set A1 (z) is convex. Consider now the following differential inclusion:
Jce fi(x),
F(x) =
z :z = ~
)~iAiz,
A e co(Al(x))
.
(13.18)
i=1
Thus for every set { A 1 , . . . , A q } E bnd(F) there exists such a function v(x) that is invariant in the sense of the extreme property (13.10). The s e t / ~ ( x ) is convex and bounded for any x E IR~ and multi-valued mapping x --+ F ( x ) is upper semicontinuous. Then for every initial data xo there exists an absolutely continuous solution x(t) of the inclusion (13.18) that satisfies the condition x(O) = zo. The following theorem is a direct consequence of Theorem 13.1 and construction (13.18): 13.5 Let { A 1 , . . . , A a} E bnd(F). Then for every Zo ~ 0 there exists such a solution z(t) of the inclusion (13.18) that z(O) = Zo and for every t >_ 0 the condition v(x(t)) = v ( x ( 0 ) ) holds. Theorem
Given the solution z(.) of (13.18) the curve in lRn described by the vector z(t) as t runs from 0 to c~ is said to be the trajectory of z(.). 13.2.1
The
Two-Dimensional
Case
Consider now the case n = 2. In this section we generalize the result in (Pyatnitskiy, 1971) to the case of the inclusions (13.6). In this case S is the boundary
275 of the convex region on the plane IR 2. Thus S is a closed curve t h a t is disposed s y m m e t r i c a l l y a b o u t the origin, x = 0. By T h e o r e m 13.1 there exists such a closed t r a j e c t o r y x*(t) of (13.6) t h a t coincides with S and is disposed s y m m e t rically around the zero, i.e.
x * ( T / 2 + t ) = -x*(t),
t > O,
(13.19)
for some T / 2 > 0. T h u s x* (t) is T-periodic. Lemma
13.6
Let n = 2. Then v(x) is continuously differentiable.
Proof. Since the convex differentiable function is continuously differentiable then it is sufficient to prove t h a t Ov(x) : {grad v(x)} for all x E IR 2. By the uniformity of v(x) this p r o p e r t y should be proven for x E S only. Suppose t h a t Or(2,) # {grad v(2)} for some 2 E 5:. T h e n
= co{>, p=},
(13.20)
where Pl #/~P2 for any # E IR. A cone tangent to the set {x E IR 2 : v(x) < 1} at the point ~ is
c(})
co{z1,
=
where
pT zl
--
o,
pT
< O,
p~r z2
----
0,
P2r Zl < 0.
(13.21)
B u t S is the t r a j e c t o r y of the periodic solution x* (t) and so x* (r) = 2 E S for some r > 0. T h e n there exists such a sequence {ti} t h a t ti < 7" and ti -+ r as i --+ oc. Define qi = x*(ti) - x*(r). Since qi E C(2) then qi = aiz~ + (1 - c~i)z2 for some ~i E [0, 1]. C o m b i n e d with (13.21) it gives
pTqi <_ 0,
p2Tqi <_ O,
for any i. Since
qi = -
f-
min{p~qi, pT2qi} < 0
(13.22)
w(O) dO,
(where w(O) E F(x*(O)) for almost all 0 E [ti,r]) then taking into account (13.22) one can obtain t h a t for every i there exists 0i E [ti T] such t h a t
pTw(Oi) > O,
p~w(Oi) >_ O,
max{pTw(Oi), pTw(Oi)} > O,
where w(Oi) E F(x*(Oi)). Directing i to e~ and selecting the subsequenee if it is necessary one can obtain w(0i) -+ ~ E F ( ~ ) = F(x*(r)) by u p p e r semicontinuity of F(.) where max{pTtb, p~Ttb} >_ 0.
276
Since pT~b > 0, pTt~ _> 0 and Pl is linearly independent of P2 then it follows from pT~b = 0, pT~ = 0 that ~ = 0. Combined with ~b • F ( 2 ) that contradicts (13.7). Consequently
max{pT ,
> 0.
(13.23)
It follows from (13.23) that Ov(2)/Off~ > 0, the contradiction with (13.11). The lemma is proved. Taking into account Lemma 13.6 we can rewrite (13.11) as
([c*(t))T grad v(z*(t)) = 0.
(13.24)
Let x* (t) be differentiable at some t. Then }* (t) = ~i=l q AiAix*(t). It follows from (13.10) that (Aix*(t))Tgrad v(x*(t)) <_0 for all i. Consequently, ifAio > 0 for some io then it follows from (13.24) that (Aio x* (t)) Tgrad v(x* (t)) = 0. Let W be the set of those ¢ for which the equality (Aix*(t))Tgrad v(x*(t)) = 0 holds for at least two indices i and j for which
mi • aAj
(13.25)
for any real scalar a. Consequently, if x* (t) ~ W then either
~*(t) =
or
x*(t)=
Aix*(t) t3i(t)Aiz*(t)
(13.26a) (13.265)
for some i = 1 , . . . , q. Let us define
Wij = i x : (Aix)Tgrad v(x) = O, (Ajx)Tgrad v(x) = 0} for i = 1 , . . . , q - i, j > i satisfying (13.25). Some of the the zero point only. Then
sets
(13.27)
Wij can contain
q
w c [_J [J
(13.2s)
i=l j > i
Consider the set W/j for some j > i. It is necessary for x • Wij that Aix = aAjx for some real a. Note that matrices Ai all are nonsingular due to (13.7). The matrix (Aj)-IAi either has no real eigenvalues or both of them are real. In the last case there are either two different one-dimensional invariant subspaces (if eigenvalues are different) or only one due to (13.25). Consequently Wij consists of either only zero point or intersection of not more than two lines spanned on real eigenvectors of (Aj)-IAi.
L e m m a 13.7 Let { A I , . . . , Aq} • bnd(F), n = 2 and condition (13.25) holds. Then for every i = 1 , . . . , q - 1 and j > i the set Wij consists of not more than two lines intersecting at the point O. Now summing up (13.27), (13.28) and the assertion of Lemma 13.7 we obtain that the T-periodic solution x* (t) has not more than 2q(q- 1) intersections with the set W at the period and not more than
277
N = q ( q - 1) at the half-period. Taking into account (13.19) and (13.26a), (13.26b) we obtain that the periodic solution x*(t) may be represented as a product of not more than N matrix exponentials for t E [0, T]. T h e o r e m 13.8 Let n = 2 and { A I , . . . , A q } E bnd(F). Then there exist the set of N integer numbers { j l , . . . , j N } from the interval [1, q] each and the set of N non-negative numbers ri such that the following condition holds N
det(I- I exp (Aj~ri) + I) = O. i=1
T h e o r e m 13.9 Let n = 2. Then it is necessary and suJ:ficient for { A 1 , . . . , Aq} E F that for every integer numbers { j l , . . . , J N } from the interval [1, q], every non-negative numbers vi and p > 1, the function N
O(T1,..., VN) = det(I"I exp (Aj,~'i) + I) i=1
does not change sign. Note that the above theorem is reduced to the criterion (13.5) for the particular case of the problem (13.3), (13.4).
13.2.2 The Three-Dimensional
Case
Consider now the case n = 3 and suppose that {A1,...,Aq} E bnd(F). Now S is the surface in IP~3 that is homeomorphic to the sphere S "~. Note that in the odd-dimensional case under conditions (13.7) there exist points x E S at which the function v(x) is nondifferentiable (Rapoport, 1993a) in contrast to the case n = 2. We investigate the flow that generates the inclusion (13.18) on the surface S when { A 1 , . . . , Aq} E bnd(F). The main result is based on a generalization of the Poincar4-Bendixson theorem. Let xo E S and x(.) be such a solution of the inclusion (13.18) that satisfies the condition x(t) E S, t >_ O, in accordance with Theorem 13.5. We call the trajectory x(.) to be closed if for some t2 > tl one has x(tt) = x(t2). Setting T = t~ - tl we get T-periodic function ~(t) = x(t + tl)
for t E [0, T].
This function is absolutely continuous and for almost all t satisfies (13.18). Hence if (13.18) has a closed trajectory then there exists a periodic solution of (13.18) (and (13.6) too).
278 Suppose that the trajectory x(.) is not a closed one and let £2z be the w-limit set of x(.). It is known that f2x _C S is weakly invariant for the inclusion (13.18) in the following sense. For any Xo E S there exists such a solution x*(.) of (13.18) that x*(0) = Xo and x*(t) E S for all t > 0. T h e o r e m 13.10 Let n = 3, condition (13.7) holds and { A 1 , . . . , A q } E bnd(F). Let x(.) be the solution of (13.18) lying on the surface S (Theorem 13.5) and its trajectory is not closed. Then there exists a closed w-limit trajectory x*(') of x(.). The proof of this theorem follows a scheme based on the construction of a "transversal section" and is a generalization of the Poincar~-Bendixson theorem for ordinary differential equations on the plane. By this theorem t2x either is a closed trajectory or consists of trajectories connecting the equilibria point. This scheme generalized to the case of the differential inclusions on the surface S (homeomorphic to the two-dimensional sphere S 2) is similar to those given in (Pyatnitskiy and Rapoport, 1991a). So we omit the whole proof. Note that the case when £2x consists of trajectories connecting the equilibria is eliminated by the condition (13.7). Thus every trajectory of (13.18) that lies on the surface S either is closed or has the closed w-limit trajectory. But for every closed trajectory of (13.18) one has a T-periodic solution of (13.18) for some T. Hereinafter we shall consider the particular case (13.3), (13.4) for the simplicity and generalize the criterion (13.5) for the case n = 3. The proof presented here is more simple than those given in (Rapoport, 1993b). It is convenient to rewright the corresponding inclusion (13.6) in the form
5: E F(x),
F ( x ) = { B x : B = Ao + ,~bsWx, --1 < )~ < 1}
(13.29)
w i t h s = ½ k c , Ao = A + b s T, b, s E lR 3. We assume that the pair {A, b} is controllable and the pair {A, c} is observable. These conditions imply the controllability of the pair {Ao, b} and observability of the pair {Ao, s} and the absence of the common invariant subspace of the matrices A1 = A = Ao - bsT and A2 = Ao + bs T Let Mr Ix] be the averaging operator:
M,[x] =
i/or
-;
x(t) dr.
L e m m a 13.11 Let x*(t) be the T-periodic solution 4 the inclusion (13.18) that exists in view of Theorem 13.10. Then
MT [x*] = 0.
(13.30)
Proof. Since x* (t) is also the solution of the inclusion (13.29) then there exists such a Lebesgue-measurable function u(t) e [-1, 1] that x*(t) is the solution of the following system of differential equations:
(13.31)
de* = (Ao + u(t)bsT)z *. Then applying Mr['] to the both sides of (13.31) we obtain
-~ (x'(~)
- x*(0)) = AoMT[~*] + bM,[~sTx*].
T
Note that x* (t) is bounded since x*(t) E S and consequently
0 = AoM~[x*] + bM~[usr~*].
(13.32)
Let ((t) = u(t)~(t), e* = M~[z*], ~ = M~[u], e = M ~ [ ~ r x *] = ~ r e . and = M~[(]. It follows from the inequalities
t~1 _< l~l lal
and u E [-1, 1] that
-t~t <_ ~ < lel. If & = 0 then ~ = 0 and it follows from (13.32) that Ao2* = 0. The condition (13.7) gives 2* = 0. If 8 ¢ 0 then = X e [-1, l].
(13.33)
The conditions (13.32) and (13.33) give
Ao~* + AbsTx * = 0 and again it follows from (13.7) that 2" = 0. Now (13.30) is the consequence of periodicity of x*(t). The lemma is proved. Turn now to the inclusion (13.13), (13.14). It takes the form
2 = Aox + AbsTx, where
A = A = A E
1 -1 [--1,1]
]) = - A o p - AsbTp, for for for
(13.34)
pTb xTs )" 0, pTbxrs
It follows from the controllability of the pair {Ao, b} and the observability of the pair {Ao, s} that the "switching function" (p(t)Tb)(x(t) T s) has only finite number of zeros on the interval [0, T] for every solution x(t), p(t) of (13.34). Consequently the periodic solution x(t) that exists in view of Theorem 13.10 and satisfies Theorem 13.2 is formed by the product of the finite number of the matrix exponentials exp(DT) where D = Ao + Abs T and A = +1. The problem now is to evaluate the number of the switching points on the period T.
Let x*(t) be the solution of the inclusion (13.18) that lies on the surface S, the function v(x) be differentiabIe at x* (0) = Xo and x* (t), p* (t) be the correspondent solution of (13.33) with the initial conditions x*(0) = xo, p*(0) = grad v(xo). Then
L e m m a 13.12
280 M ~ [p*] = 0.
(13.35)
Proof. The proof repeats the proof of Lemma 13.11 after taking into account the equality p*(t) --- grad v(x*(t)) that follows from Lemma 13.4.
Denote S+
=
{x E S : xTs > 0},
(13.36a)
S-
=
{x E S : x T s < 0},
(t3.36b)
S°
=
{x E S : xTs = 0},
(13.36c)
and Z+
:
{x E S :pTb > 0 for every p E Ov(x)},
(13.37a)
~-
:
{x E S : pTb < 0 for every p e Ov(x)},
Z°
:
{x e S : p T b : 0 for some p e 0v(x)}.
(13.375) (13.37c)
(a) Sets S +, S - , Z+ and Z -
L e m m a 13.13 subsets of S;
S ° = bnd(S +) -" bnd(S-), Z '° = bnd(Z: +) = bnd(,U-),
(b) S - = - S + , (c) z - = - z + ,
are open linearly connected
cl(S +) n cl(S-) = s°; n =
This result is geometrically obvious. Its proof is the same as the proof of the similar assertions in (Rapoport, 1993b). Thus S ° (or ,U°) is the simple closed curve separating S + from S - (or ~U+ from ~U-). Consider the curves S ° and Z:°. Denote ~
=
{x E S° : sT Aox > 0},
(13.38a)
W~
=
{x E S ° : sT Aox < 0},
(13.38b)
W~
=
{x E S ° : sTAox = 0},
(13.38c)
and W+
=
{x E ,U°: bTATp > 0 for p E av(x)},
(13.39a)
W~
=
{x E Z ° : bTATp < 0 for p E Or(x)},
(13.39b)
W°
=
{ x E Z ° : there exists p E Ov(x) : bTA~p = 0}. (13.39c)
In view of the observability of the pair {A, s} the vectors s, A T s and (AoT)2s are linearly independent. Consequently W~ from (13.38c) consists of the two vectors: W~ = {z} U {-z}, where z(AT)2s > 0. (13.40) The sets Ws+ and W s = - W s+ are the segments located symmetrically and glued together at the two points z and - z .
281 Also the following holds
W~ =
Z
u
(-Z).
(13.41.)
The sets W + and W~ = - W + are the segments located symmetrically and glued together at the two segments Z and - Z . L e m m a 13.14 Let x*(t) be the periodic solution from Theorem 13.10. The trajectory x* (t) intersects the curve S ° at the two points. The first one belongs to Ws+ and the second one belongs to W ~ .
Proof. First we show that the trajectory x*(t) has intersection with S °. This trajectory would otherwise belong to S + (or S - ) and consequently
lf0T sTx*(t) dt > 0(< 0) -~ that contradicts (13.30). Thus x*(t) E S for some t. It follows from Theorem 13.2 that there exists such a function p*(t) that x*(t), p*(t)satisfies (13.34). Since
d(sTx*(t))/dt = d2(sTx*(t))/dt 2 =
sTAox*(t), sT(Ao)2X*(t) for x*(t) E S °
then taking into account (13.38) and (13.40) we get:
d(sTx * (t)) dt
d(~ ~ ~" (t)) dt
d(sTx*(t)) > 0 for x*(t) E W +, dt
(13.42)
d(sTx*(t)) < 0 for x*(t) 6 W ~ , dt
(13.43)
- 0 - 0
and and
d ~(STx. (t)) dt 2
d 2 (sT x" (t)) dt 2
> 0 for x*(t) = z,
(13.44)
< 0 for x*(t) = - z .
(13.45)
Since
s° = w+ u ~ 7 u w ~ then there are the following possibilities for periodic trajectory x* (t) to intersect S ° : (a) x* (t) cuts S ° on the segment V/~ following direction from S - to S+; (b) x*(t) cuts S ° on the segment ~4~ following direction from S + to S - ; (c) x*(t) touches S ° at the point z or - z . Thus W + and W s are segments of "transversal section" and T-periodic trajectory z*(t) has the unique intersection with W + and the unique intersection with W s (Andronov et al, 1959). The lemma is proved.
282 L e m m a 13.15 Let x*(t) be the periodic solution from Theorem 13.10 which is the w-limit for the trajectory, starting from the point at which the function v(x) is differentiable. The trajectory x*(t) intersects the curve Z ° at the two points. The first one belongs to W + and the second one belongs to W ~ . Proof. The proof is based on (13.35) and is almost the same as the proof of Lemma 13.14. We say the solution x(.) of (13.29) satisfying the conditions x(t) • S for t > 0 and x(O) = xo to be extendable on the surface S for negative time if there exists the solution y(.) of (13.29) defined for t • [-c, co], c > 0, and satisfying the conditions y(t) • S for t • I-c, 0) and y(t) = x(t) for t ~ 0. All solutions of (13.29) starting with Xo • S \ S ° \ ~ ° are extendable for t < 0 since the right-hand side of (13.29) has unique value (Ao 4" bsT)x in some neighborhood of Xo. It is easy to conclude that all solutions, starting with the points lying on the segments W+, Ws-, W + or W~ are also extendable for t < 0 since every solution in the neighborhood of such point cuts the segment transversally. This conclusion does not hold for points that belong to Z or - Z .
Lemma 13.16 Let xo • Ws+UWs UW+ UW . Then every solution 4(13.29) lying on S and starting with xo is extendable on S for negative time. Proof. For every 5 > 0, whatever small it will be there exists the point y in the &neighbourhood of Xo in which the function v(x) is differentiable. The unique solution of (13.34) satisfying the condition p(0) = grad v(y) passes through this point, cuts one of the segments W +, W s , W + or W~ and belongs to S for t E [-2e, 2c] for some e > 0. Directing 5 to 0 and taking into account the compactness of the set of solutions of (13.34) on the closed interval of time, we get that there exists the solution of (13.34) that passes through xo and belongs to S for t E I-e, c]. We will say the T-periodic solution x* (t) to be T/2-antiperiodic if it satisfies the condition
x*(T/2) =
-x'(0).
(13.46)
L e m m a 13.17 Let the assumption of Theorem 13.10 are met. Then there exists a T/2-antiperiodic solution of the inclusion (13.29) for some T > O. Proof. The existence of the periodic solution of (13.29) follows from Theorem 13.10. Then z(t) = - y ( t ) is also the periodic solution of (13.29). Let Ly and Lz be the trajectories of solutions y(.) and z(.) respectively. If those trajectories intersect then y(tl) = z(t2) for some tl and t2. Consequently y(tl) = - y ( t 2 ) and the function
283
x*(t)
= =
y(t+tl) -x*(t + T/2)
tE[O,T/2], T / 2 = ( t 2 - t l ) for t
for
is T-periodic and satisfies (13.46). Suppose that the trajectories Ly and Lz do not intersect and let R (°) be an annular domain bounded on the surface S by the curves Ly and Lz. If the domain R (°) contains entirely the trajectory of the periodic solution different from y(.) and z(.) then we denote this periodic solution by y(1)(.) and z(I)(.) = _y(1)(.). By L(yU and L! I) we denote the correspondent closed trajectories. As before, if L (1), L! 1) have the common points then we have found the antiperiodic solution. Otherwise we obtain new annular domain R (1). Continuing this way, we obtain either an antiperiodic solution at some step or a sequence of annular domains R (i) C R (i-1). In the last case it follows from Lemmas 13.14, 13.15 that there exists such 2r that the period of every periodic solution is bounded by 2P. Then there exists nonempty intersection R ~ = fiR(i). It follows from the compactness of the set of solutions y(i)(.) and z(i)(-) = - y ( ~ ) ( . ) o n the interval [0,~ ~] that there exists the subsequence y(ij)(.) that converges uniformly on [0, T] to the periodic solution y~(.). Then if the trajectories L~' and L~ = - L ~ of the periodic solutions y~ (.) and z ~ (.) = - y ~ (-) have no common points then they form the boundaries of the annular domain R ~°. Let Ry C int(R ~) be the set of those points x E R ~ for which the function v(x) is differentiable at x and w-limit set of the unique solution of (13.29) starting with x and lying on S is L~. The analogous sense has Rz = - R v. Subsets/~y and R~ both are open. The proof is just the same as in (Andronov et al, 1959, chapter 6). Since R ~ does not contain w-limit sets different from Ly and Lz~ by the construction, then R ~ = cl(Ry) U cl(Rz).
(13.47)
The common boundary Ro = cl(Ry) N cl(Rz) is formed by those points xo from which at least two solutions of (13.29) that lies on S start. The first solution "revolves" around L~ while the second one "revolves" around Lz~ . The function v(x) is nondifferentiable at Xo due to Lemma 13.4. Let Xo E Ro. Then there exists Pl,P2 E Ov(xo) such that the solution xl(.) of (13.34) corresponding the initial conditions xo, Pl "revolves" around L~, the solution x2(-) of (13.34) corresponding the initial conditions xo, P2 "revolves" around L~', and all the solutions x(.) of (13.34) corresponding the initial conditions Xo, Po where po E int(co({pl, P2})) do not belong to S for t > 0. Let xt(t) =fi x~(t) for t > r. Then there exists such a point ~ that belong to the "sector" formed by the vertex x 1(r) and by the trajectories xl (t) and x2 (t) for t >_ r and such that none of the solutions of (13.29) starting with ~? is extendable on S for negative time. But for Xo E int(R ~) \ 5;° \ Z ° all solutions are extendable for t < 0. Consequently, by Lemma 13.16 either -- =t=z, or
(13.48)
284
fi ± Z .
(13.49)
But by Lemmas 13.14, 13.15 the periodic trajectories y~(.) and z~(.) intersect segments W+, W [ , t4~, W~ once a period. Consequently, the vectors ±z and the segments ± Z do not belong to int(R~). The contradiction with k E int(R ~) and (13.48) or (13.49) proves the lemma. Summing up Lemmas 13.14, 13.15 and 13.17 we arrive at the following: T h e o r e m 13.18 Let n = 3 and {AI,A2} E bnd(F) where AI - A and A2 = A + k b c T. Then there exist r > 0, ~ > 0 such that the following condition holds det(exp(vA) exp(~A2) + I) = 0. T h e o r e m 13.19 Let n = 3. Then it is necessary and sufficient for {A1, A2} 6 F that for any numbers v > O, ,7 > O, p > t the function
O(r, ~, #) = det(exp(rA) exp(r/A2) + pI) does not change sign.
13.2.3
The n-Dimensional
Case
The additional assumption is an existence of the convex pointed invariant cone for the inclusion (13.6). For this case an existence of the periodic solution of (13.6) is proven in (Pyatnitskiy and Rapoport, 1991b, 1996) under condition {A1,...,Aq} ~ F. Moreover {A1,...,Aq} may be as close to the boundary bnd(F) as we please. See also (Colonius and Kliemann, 1993) for the closely related considerations. The periodic motion is also formed by the finite number of the matrix exponentials but the upper bound for this number is unknown. Acknowledgements. This work is supported by the Russian Foundation of Fundamental Researches, Grant No. 94-01-00485
References Andronov, A. A., Vitt, A. A., Khaikin, S. Ye., 1959, Theory of Oscillations [in Russian], Moscow. Alexandrov, V. V., Zhermolenko, V. N., 1972, On absolute stability of second order systems [in Russian], Vestnik Mosk. Univ., set. mathem, i mechan., 5. Brayton, R. K., Tong, C. H., 1979, Stability of dynamical systems: a constructive approach, I E E E Transactions on Circuits and Systems, 26, 224-234. Colonius, F., Sieveking, M., 1989, Asymptotic properties of optimal solutions in planar discounted control problems, S l A M Journal on Control and Optimization, 27, 608-630.
285 Colonius, F., Ktiemann, W., 1993, Linear control semigroups acting on projective space, Journal of Dynamics and Differential Equations, 5, 495-527. Pyatnitskiy, E. S., 1971, Criterion for the absolute stability of second-order nonlinear control systems with one nonlinear nonstationary element, Automation and Remote Control, 32, 1-11. Pyatnitskiy, E. S., Rapoport, L. B., 1991a, Existence of periodic motions and test for absolute stability of nonlinear nonstationary systems in the threedimensional case, Automation and Remote Control, 52,648-658. Pyatnistkiy, E. S., Rapoport, L. B., 1991b, Periodic motions and test for absolute stability of nonlinear nonstationary systems, Automation and Remote Control, 52, 1379-1387. Pyatnitskiy, E. S., Rapoport, L. B., 1996, Criteria of asymptotic stability of differential inclusions and periodic motions of time-varying nonlinear control systems, IEEE Transactzons on Circuits and Systems, to appear. Rapoport, L. B., 1993a, Existence of nonsmooth invariant functions on the boundary of absolute stability domain of nonlinear nonstationary systems, Automation and Remote Control, 54, 448-453. Rapoport, L. B., 1993b, Antiperiodic motions and an algebraic criterion for the absolute stability of nonlinear time-varying systems in tlle three-dimensional case, Automation and Remote Control, 54, 1063-1075. Rockafellar, R. T., 1970, Convex Analysis, Princeton University Press, Princeton. Sussmann, H. J., 1976, Minimal realization and canonical forms for bilinear systems, Journal of Franklin Institute, 301,593-604.
14. Structured Dissipativity and A b s o l u t e Stability of Nonlinear Systems A n d r e y V. S a v k i n and Ian R. P e t e r s e n 14.1 I n t r o d u c t i o n An important problem in the area of robust control theory concerns the stability of uncertain systems containing structured uncertainty. Much research on this problem can be traced to the work of Safonov (1982) and Doyle (1982). tn both of these chapters, the structured nature of the uncertainty is exploited by introducing (frequency dependent) scaling parameters. Furthermore, it was shown in (Doyle, 1982) that for a class of uncertain systems containing three or fewer uncertainty blocks, the stability condition so obtained is both necessary and sufficient for robust stability. In recent years, a number of similar results have appeared for various classes of uncertain systems containing an arbitrary number of uncertainty blocks; e.g., see (Khammash and Pearson, 1993; Shamma, 1992; Megretski, 1993). In each of these results, the uncertainty structure is exploited by introducing a set of frequency independent scaling parameters. The key idea is to use an extension of the S-procedure Theorem (also known as Finster's Theorem); e.g., see (Yakubovich, 1971) and (Uhlig, 1979). Such a result enables the problem of robust stability of an uncertain system containing structured uncertainty to be converted into a problem of robust stability of an uncertain system containing unstructured uncertainty. This problem is then solved using existing methods such as the small gain theorem. In all of the above mentioned results, the underlying uncertain system is required to be linear. One of the main contributions of this chapter is to show that these ideas can be extended into the realm of nonlinear systems. That is, we show using a nonlinear version of the S-procedure result of Megretski and Treil (1993) that the problem of robust stability for a nonlinear uncertain system with structured uncertainty is equivalent to the existence of a set of real scaling parameters such that a corresponding nonlinear uncertain system with unstructured uncertainty is robustly stable. In order to obtain this result, we introduce a new class of nonlinear uncertain systems in which the uncertainty is described by a certain integral constraint. This uncertainty description extends to nonlinear uncertain systems the integral quadratic constraint uncertainty description introduced in (Yakubovich, 1973, 1988b, 1988a); see also (Savkin and Petersen, 1994, 1995b). Also, we introduce a new definition of absolute stability for nonlinear uncertain systems. This definition extends to nonlinear uncertain systems the notion of absolute stability introduced in (Yakubovich,
288 1973, 1988a); see also (Savkin and Petersen, 1994, 1995b). A feature of our uncertainty model and corresponding definition of absolute stability is that it is closely related to the notion of dissipativity which arises in the modern theory of nonlinear systems; e.g., see (Willems, 1972; Hill and Moylan, 1976, 1977, 1980; Byrnes et al, 1991). Thus, a major contribution of the chapter is the establishment of a connection between the areas of robust stability of nonlinear uncertain systems with structured uncertainty, the absolute stability of nonlinear uncertain systems, and the notion of dissipativity for nonlinear systems. The definition of dissipativity given in (Willems, 1972), concerns a given nonlinear system with an associated "supply function." This supply function represents the rate of "generalized energy" flow into the system. For a nonlinear system to be dissipative, it is required that there exists a suitable "storage function" for the system. This storage function must be found such that, over any given intervM of time, the change in the storage function is less than the integrM of the supply function. In this definition, the storage function acts as a measure of the generalized energy stored in the system. Thus, a dissipative system is one in which generalized energy is continually dissipated. The advantage of this general definition is that it includes such concepts as passivity and finite gain stability as special cases. The study of dissipativity is particularly useful when investigating the stability of the feedback interconnection of two nonlinear systems; e.g., see (Hill and Moylan, 1977). Such a situation would arise when a given nonlinear system is subject to a single uncertain nonlinearity. In particular, for the case of a linear uncertain system containing a single uncertain nonlinearity, this leads to the standard smM1 gain condition for robust stability. In this chapter, we introduce a new notion of "structured dissipativity". This definition, concerns a given nonlinear system and an associated collection of supply functions. In this case, we require the existence of a storage function such that over any interval of time, the system is dissipative with that storage function and at least one of the given supply functions. One of the results of this chapter shows that there is a direct connection between this notion of structured dissipativity and the absolute stability of an uncertain nonlinear system containing a number of uncertMn nonlinearities. It is because of this connection that the term structured dissipativity is used. The first main result of the chapter shows that a nonlinear system has the structured dissipativity property if and only if it is dissipative with a single supply function which is a positive linear combination of the given supply functions. From this, the property of structured dissipativity is also shown to be equivalent to the satisfaction of a certain parameter dependent integral inequality. This connection is similar to the nonlinear version of the KalmanYakubovich-Popov Lemma given in (Hill and Moylan, 1980). The second main result of the chapter concerns a given nonlinear uncertain system containing multiple structured uncertain nonlinearities each subject to an integral uncertainty constraint. These integral uncertainty constraints generalize the integral quadratic uncertainty constraints considered in (Yakubovich,
289 1973, 1988b, 1988a); see also (Savkin and Petersen, 1994, 1995b, 1995a) to the case of nonlinear uncertain systems. The result shows that this nonlinear uncertain system is absolutely stable if and only if the corresponding nonlinear system with associated supply functions (which define the integral uncertainty constraints) has tile structured dissipativity property. This result also leads to a sufficient condition for the stability of an interconnected nonlinear system. The remainder of the chapter proceeds as follows: In Section 14.2, we introduce the class of uncertain systems under consideration and define the notion of structured dissipativity. In Section 14.3, we present a technical result which extends the S-procedure theorem of Megretski and Treil (1993) to the case of nonlinear systems. This theorem is used in the proof of our main results. In Section 14.4, we present a result which shows that the notion of structured dissipativity is equivalent to the standard notion of dissipativity with a parameter dependent supply function and that this is in turn equivalent to the satisfaction of a parameter dependent integral inequality. In Section 14.5, we first introduce a class of nonlinear uncertain systems to be considered. Also defined are the integral uncertainty constraints on the structured uncertainty entering into this system and the corresponding notion of absolute stability for such a nonlinear uncertain system. The main result of this section shows that the nonlinear uncertain system is absolutely stable if and only if there exist positive scaling parameters such that a corresponding dissipativity condition is satisfied.
14.2
Definitions
Consider the nonlinear system
x(t) = #(x(t),~(t))
(14.1)
where x(t) E IRn is the state and ~(t) C IR"~ is the input. The set S of all admissible inputs consists of all locally integrable vector functions from IFI. to IRm. Associated with the system (14.1) is the following set of functions, called supply rates,
...,
(14.2)
Also, we will consider the following integral functionals:
fl (x(.), /2 (,(.), ¢(.))
/k
= fo = ff
wl (x(t), ~(t)) dr; w2(x(t), ~(t)) dt;
fo ~ wk(x(t), ~(t)) dt.
(14.3)
290 Assumptions. We assume the system (14.1) and associated set of functions (14,2) satisfy the following assumptions:
A s s u m p t i o n 14.1
The function g(., .) is continuous.
A s s u m p t i o n 14.2
The inequalities
w~(x,O) < O, w~.(z,O) < O, . . . ,
wk(x, 0) < 0
are satisfied for all x E IR~ •
A s s u m p t i o n 14.3 For all {x(.), ~(.)} E L2[0, cx)), the corresponding quantities ft (x(.),~(.)), f2 (x(.),~(.)), ..., fk (x(.),~(.)) are finite. A s s u m p t i o n 14.4
For any xo E ]R~ there exists a time T > 0 and an input ~o(') E ~ such that for the solution x(.) to the system (14.1) with initial condition x(O) = 0 and the input ~o('), we have x(T) = xo.
A s s u m p t i o n 14.5
Given any e > 0 there exists a constant ~i > 0 such that the following condition holds: For any input function ~0(') E L2[0, oo) and any x0 E {x0 ~ IRP : IIx011 _< ~}, let xl(t) denote the corresponding solution to (14.1) with initial condition x(O) = xo and let x2(t) denote the corresponding solution to (14.1) with initial condition x(O) = O. Then both xl(t) and x2(t) are defined on [0, oo) and both functions belong to L~[0, c~). Furthermore,
If, (xl(.),~o(.)) - Y, (x2(.),~o(-)) 1 < lots=
1,2,...,k.
Here II, II denotes the standard Euclidean norm, L2[0, cc) denotes the Hilbert space of square integrable vector valued functions defined on [0, c~). Note, Assumption 14.4 is a controllability type assumption on the system (14.1) and Assumption 14.5 is a stability type assumption. Consider the system (14.1) with associated function w(x(t),~(t)). For this system, we have the following standard definition of dissipativity; e.g., see (Willems, 1972; Hill and Moylan, 1976, 1977, 1980; Byrnes et al, 1991). D e f i n i t i o n 14.6 A system (14.1) with supply rate w(x(t),~(t)) is said to be dissipative if there exists a non.negative function V(xo) : IRn -+ IR called a storage function, such that V(O) = 0 and the following condition holds: Given any input ~(.) E .~ and any corresponding solution to the equation (14.1) with an interval of existence [0, t.) (that is, t. is the upper time limit for which the solution exists), then
v(x(to)) - v(x(o)) _< for all to E [0, t.).
fo'° ~(~(t), ~(t))dt
(14.4)
291 We now introduce a new definition of structured dissipativity. This definition extends the above definition to the case in which there are a number of storage functions associated with the system (14.1).
Definition 14.7 A system (14.1) with supply rates (14.2) is said to have the structured dissipativity property if there exists a non-negative storage function V(xo) : IRn ---+IR, such that V(O) = 0 and the following condition holds: Given any input ~(.) 6 S and any corresponding solution to the equation (1~.I) with an interval of existence [0,t.), then v(.(to))
v(.(0)) <
-
-
-
max
s=l,2,,,.k
if°
w.(,:(t),g(t))dt
}
(14.5)
for all to E [0, t,). Let 5 > 0 be a given constant. We introduce the following new functions associated with the system (14.1)
w~(x(t), ~(t))
=
wl(x(t),~(t)) -a(l[x(t)l[ 2 + I]g(x(t),~(t))ll 2 + ]l~(t)]12);
wa2(x(t),~(t))
=
w2(x(t),~(t)) -5(tl~e(t)ll 2 + Itg(x(t),~(t))tt ~ + tt~(t)l12);
~g(~(t),~(t))
=
wk(x(t), ~(t)) -a(llx(t)tt 2 + flg(x(*), ~(t))Ll2 + ll~(t)II2).
(14.6) Definition 14.8 The system (14.1) with supply rates (14.2) is said to have the strict structured dissipativity property if there exists a constant ~ > 0 such that the system (14.1) with supply rates (14.6) has the structured dissipativity property.
14.3
S-Procedure
for Nonlinear
Systems
In this section, we present a result which extends the so called "S-Procedure" of Megretsky and Treil (1993). The result of Megretsky and Treil is related to Finsler's Theorem concerning pairs of quadratic forms; e.g., see (Uhlig, 1979) and also (Yakubovich, 1971). The main result of Megretsky and Treil (1993) applies to a collection of integral quadratic forms defined over the space of solutions to a stable linear time-invariant system. In this section, we extend this result to a more general set of integral functionals defined over the space of solutions to a stable nonlinear time-invariant system. A slightly modified version of this result appeared in (Savkin and Petersen, 1995b) together with a partial proof. For the sake of completeness, we include the complete proof here. N o t a t i o n 14.9 For the system (14.1) and flmctions (14.2) satisfying Assumptions 14.1, 14.3 and 14.5, we define the set $2 C L2[0, oo) as follows: ~2 is the
292 set of pairs {x(.), ((.)} such that ((.) E L2[0, oo) and x(.) is the corresponding solution to (14.1) with initial condition z(0) = 0. T h e o r e m 14.10 Consider the system (14.1) and associated functionals (14.3) and suppose Assumptions 14.1, 14.3 and 14.5 are satisfied. I f f l (x(.),~(.)) >_ 0 for all pairs {z(.),~(-)} E / 2 such that f2 (z(.),((-)) < 0 , . . . , f k (z(-),~(-)) < 0, k then there exist constants rl > 0, r2 _> 0 , . . . , vk _> 0 such that ~-~=1 r~ = 1 and r~fl(x(.),~('))+r~f2(x('),~('))4-'''+7"kfk(X('),(('))>_O
for aU pairs
(14.7)
e
In order to prove this theorem, we will use the following preliminary Convex Analysis result. However, we first introduce some notation; e.g., see (Rockafellar, 1970). N o t a t i o n 14.11 Given sets S C IR'~ and T C IR~ and a constant A EtR, then S4-T:={x4-y:xES, yET},AS:={Ax:zES},andcone(S) :={ax:xE S, a _> 0} (the conic hull of the set S). Also, cl(S) denotes the closure of the set S. L e m m a 14.12
Consider a set M C IRk with the property that a 4- b E cl(M)
for all a, b E ~1. If xl > O for all vectors [ xl x2 . . . xk IT E M such that x2 < 0 , . . . , xk < O, then there exist constants 7-1 _> 0,7"2 _> 0 , . . . , vk > 0 such that
E sk= l Ts : 1
and
Tlxl + r2x~ + .... 4- rkxk > 0 for all[ xl Proof.
x2
""
zk IT E M.
In order to establish this result, we first establish the following claims:
Claim 1 Given any two sets S C IRn and T C IRn and any scalars a E ]R and E IR, then ael(S) + f c l ( T ) C cl(~S + f T ) .
To establish this claim, let a x + fly E a el(S) + / 3 cl(T) be is, there exist sequences {xi}i=l C S and {Yi}i=l C T such that yi -4 y. Hence, axi + flYi -4 a x + fly. However, axi + flyi E a S + Thus, we must have a z + fly E cl(aS + f T ) . This completes the claim. Claim 2
The set M has the property that cl(M) + cl(M) C cl(M).
given. That xi -4 x and fiT for all i. proof of the
293 To establish this claim, first note that it follows from Claim 1 that cl(M) + cl(M) C cl(M + M). However, since a + b E cl(M) for all a, b E M, we must have M + M C cl(M) and hence el(M + M) C cl(M). Combining these two facts, it follows that cl(M) + cl(M) C cl(M). Given any set S C IR~ then
Claim 3
cone(cl(S)) C cl(cone(S)).
To establish this claim, let x E cone(cl(S)) be given. We can write x = a y oo where ~ _> 0 and y E el(S). Hence, there exists a sequence {Yi}i=I C S such that Yi --+ y. Therefore ceyl -+ a y = x. However, ayl E cone(S) for all i. Thus, x E cl(cone(S)). This completes the proof of the claim. The set cl(cone(M)) is convex.
Claim 4
To establish this claim, let A E [0, t] be given and consider two points x l , x 2 E cone(M). We can write xl = a l z l and x2 = c~2z2 where a l _> 0, a2 _> 0, zl E M, and z2 E M. Hence zl E cl(M) and z2 E cl(M). It follows from Claim 2 that n q + rez2 E cl(M) for all positive integers n and re. Now consider a sequence of rational numbers mi
i=1
such that n i
AOt' 1
mi
(1 -- A)a2"
Since, ni and rni are positive integers for all i, it follows that nizl + miz2 E cl(M)
for all i. Therefore,
[
•
(1 - A)a2 L n i z l
+ z2
\ mi
~
-
(1 - A)ot2
-rei
J
(nizl + miz2) E cone(cl(M)) /
for
all
i. Hence, using Claim 3, it follows that ( 1 - ) ~ ) a 2 ~ ' Z l
cl(cone(M)) for all i. However, ( 1 - )~)°*2 ( rei ni zl +
(
-+
\ (1-A)c~2 \ ( 1 - ;~)~2 z l + z2)
=
)~cqzl + (1 - A)~2z2
=
Axl + ( 1 - A)x2.
\
+ z2)
E
294
Thus, since cl(cone(M)) is a closed set, it follows that AXl + (1 - A)x2 E el(cone(M)). Moreover, Xl e c o n e ( M ) and x2 E c o n e ( M ) were chosen arbitrarily and hence, it follows that Acone(M) + (1 - A)cone(M) C cl(cone(M)). Therefore, cl(A, c o n e ( M ) + (t - A)cone(M)) C cl(cone(M)). Now using Claim 1, we conclude A cl(cone(M)) + (1 - A)cl(cone(M)) C cl(cone(M)). Thus, el(cone(M)) is a convex set. Using the above claim, we are now in a position to complete the proof of the lemma. Indeed suppose xl > 0 for all vectors [ xl x2 .. . xk ]T E M such that x2 < 0 , . . . , xk < 0. Also, let the convex cone Q c IRk be defined by Q := {r = [ r l , r 2 , . . . , r k ] T : rl < O, r2 < O , . . . , r k < 0}. It follows that the intersection of the sets Q and M is empty. Also, since Q is a cone, it is straightforward to verify that Q N cone(M) = 0. Furthermore, the fact that Q is an open set implies Q N el(cone(M)) = 0. Thus, using the Separating Hyperplane Theorem (Theorem 11.3 of Rockafellar, 1970), it follows that there exists a hyperplane separating the convex cones Q and el(cone(M)) properly. Moreover using Theorem 11.7 of (Rockafellar, 1970), it follows that the hyperplane can be chosen to pass through the origin. Thus there exists a non-zero vector b E IRk such that x'b < 0 (14.8) for all x E cl(cone(M)) and x'b > 0
(14.9)
for all x E Q. Now ~vrite b = [ rl r2 . . . rk IT, It follows from (14.9) that 7"/ > 0 for all i. Also, since b ¢ 0, we can take coefficients r l , r 2 , . . . , r k such tha~ ~ sk= l vs = 1. Moreover, it follows from (14.8) that r l x l + rux2 + . . . + vkxk > 0
for all [ xt
x2
...
Xk ]T E M. This completes the proof of the lemma.
We are now in a position to prove Theorem 14.10. Proof of Theorem 14.10
Suppose that
f~ (x(.),~(.)) _> o
(14.1o)
for all pairs {x(.),~(.)} e f2 such that f2 (x(.),~(.)) < 0 , . . . , f k (z(.),~(.)) < 0. Also, let the set M C 1Rk be defined by
295
[
: {x(.),,~(.)} _
..
~ r~ j " ]T
It follows from (14.10) that Xl > 0 for all vectors [ Xl x2 " xa EM such that x2 < 0 , . . . , x k < 0. Now let {x~(.),(~(.)} E f2 and {Xb('),~b(')} E ~2 be given. Since xa(.) C L2[0, oc), there exists a sequence {T,i}i=t C IR such that T/ -+ oc and x~(Ti) -+ OO 0. Now consider the corresponding sequence {xi(), ~,(')}i=I C D, where
~i(t) --
{ ~(t)
t < ~;
~b(t- Ti) t > Ti.
We will establish that f~(xi(.),~i(.)) --+ f ~ ( x a ( . ) , ( ~ ( . ) ) + f,(xb('),~b(')) for s = 1 , 2 , . . . , k . Indeed, let s E { 1 , 2 , . . . , k } be given and fix i. Now suppose 2~(.) is the solution to (14.1) with input ¢(-) = {b(') and the initial condition x(0) = x~(T~). It follows from the time invariance of the system (14.1) that x~(t) -
~ ( t - ~ ) . Hence,
L(~('),~i('))
=
fO ~
~ (~(t), ¢~(t)) dt
,~0T~w~(x~(t),~(t)) dt + ~
Ti
w~ (xi(t), ~b(t - rS ) ) dt
fOTi w, (~o(0, ¢o(t)) dt + f~o w~ (2~(t - T/), ~b(t -- 7}))dt T, ~i ~0Ti ~ (~o(t), ~o(t)) dt + fo ~ ~(~b(t), ~b(t)) dt
fOTi w~ (~Jt), ~o(t)) dt + f~ (~(t), ~b(t)). Using the fact that xa(!l)) --+ 0, Assumption 14.5 implies f~(~ib(t), (b(t)) --+ f~(Xb('),(b(')) aS i -+ oo. Also, it is obvious that f?'w~(xa(t),(~(t))dt f8 (xb(.), (b(')). Hence,
fs(xi('),~i(')) -+ fs(Xa('),~a(')) + fs(Xb('),~b(')). From the above, it follows that the set M has the property that a + b E cl(M) for all a, b E M. Hence, we can apply Lemma 14.12 to conclude that k there exist constants 7t >_ 0, r; >_ 0, . . . , rk >_ 0 such that ~ s = l ~-~ = 1 and
rlxl + r~x2 + . . . + rkxt: >_ 0 for all [ xl x2 ... x~ iT E M. T h a t is condition (14.7) is satisfied. This comoletes the t)roof of the theorem.
296
14.4
The
Main
Result
The following theorem establishes a connection between dissipativity and structured dissipativity of nonlinear systems. T h e o r e m 14.13 Consider the nonlinear system (14.1) with supply rates (14.2) and suppose that Assumptions 14.1-14.5 are satisfied. Then the following statements are equivalent:
(i) The system (14.1) with supply rates (14.2) has the structured disswativity property. k
(ii) There exist constants rl > 0, r2 > 0 , . . . , vk _> 0 such that ~ s = l v8 = 1 and the following condition holds: Given any input ~(.) E ~ and the corresponding solution to the equation (14.1) with initial condition x(O) = 0 defined on an existence interval [0,t.), then fo t° w~(x(t),~(t)) dt ~_ 0
(14.11)
for all to E [0,t.), where wr(., .) is defined by k
wT(x(t),~(t)) := E
vsws(x(t),~(t)).
(14.12)
k
(iii) There exist constants 7"1 k O, v2 >_ 0 , . . . , rk >_ 0 such that ~8=1 vs : 1 and the system (14.1) is dissipative with supply rate (14.12). Proof. (i) ~ (ii) following claim.
In order to prove this statement, we first establish the
Claim 1 Consider the set £2 defined in Notation 14.9. If the system (14.1) with supply rates (14.2) has the structured dissipativity property then there exist constants rl > 0 , . . . , rk ~ 0 such that ~-~=1 vs = 1 and condition (14.7) holds for all pairs {x(.),~(.))} E £2. In order to establish this claim, we first prove that fl (x(-), ~(-)) >_ 0 for all pairs {x(.),~(.)} E £2 such that <
< o.
(14.13)
Indeed, if condition (14.13) holds for some pair {x0('),~0(')} e 12, then there exists a time T > 0 such that
~to
dt < 0 , . . . ,
~towk(xo(t),~o(t))dt < 0
(14.14)
297 for all to > T. We now consider the corresponding storage function V(-). If ~o(O) = o then V ( * o ( O ) ) = 0 and, since V ( ~ o ( t o ) ) > o we have V ( ~ o ( t o ) ) V(x0(0)) > 0. Furthermore, this and condition (14.5) imply that max
s-----l,2,...k
{/7 ws(xo(t),~o(t))dt } > O.
Theretbre, if conditions (14.14) hold, then
L
to Wl(Xo(t),~o(t)) dt >_ 0
for all to > T. Hence, k(x0('),~0(')) > 0. Now using Theorem 14.10, it follows directly that there exist constants rl > 0, r2 > 0 , . . . , rk _> 0 such that ~ =kl r~ = 1 and condition (14.7) is satisfied. This completes the proof of the claim.
Claim $ If there exist constants rl > 0, r2 ___ 0 , . . . , r k _> 0 such that ~ , =k ~ rs = 1 and condition (14.7) holds for all pairs {x(.),{(.)} C r2 then condition (14.1t), (1.4.12) holds for all to > 0 and all pairs {x(.),~(.)} such that {(') C ~ and x(.) is the corresponding solution to (14.1) with initial condition x(0) = 0. Indeed, consider an input gto(') C ! such that ~to(t) = 0 for all t _> to and the corresponding solution Xto(.) to the system (14.1) with initial condition x(0) = 0. Since {to(') E L2[0, oe), Assumption 14.5 implies that Xto (.) E L2[0, oo). Then, using Assumption 14.3, it follows that to
k
~o Wr(Xto(t),~to(t))dt
=
k
Er, fs(xto(.),~to(.))-Ers s=l
s=l
foo
ko
ws(Xto(t),O)dt
k
:> E r~fs(Xto(.),~to(.)). From this, condition (14.11) directly using condition (14.7). This completes the proof of the claim and this part of the theorem. (ii) ~ (iii) Suppose that condition (ii) holds and introduce the function V(xo) : 1Rn -+ IR defined by
V(xo) :=
sup ~(0)=~o, d.)e~,to>_0
{L- oWr(X(t),{(t)) dt } ,
(14.15)
where w~ (., .) is the function defined in (14.12). We will prove using a standard approach from the theory of dissipative systems, that the system (1.4.1) is dissipative with supply rate w~(., .) and storage function V(.); e.g., see (Hilt and Moylan, 1976, 1977, 1980). Indeed, the condition V(0) = 0 follows from the definition of V(.) by using inequality (14.1t). Furthermore, Assumption 14.2 implies that V(x0) > 0 for all x0. We now use condition (14.11) to prove that
298
V(xo) < oz for all x0. Indeed, given any x0 E IRn, it follows from Assumption 14.4 that there exists a time T > 0 and a function (0(') E E such that for the system (14.1) with initial condition x(0) = 0 and input ~0('), we have x(T) = xo. Also, let ~1(') E ~ be given and define ~(.) E ~ by
{ ~0(t) for 0 < t < T ; ~(t) :=
~l(t)
for T ~ t < c~.
Now let x(.) be the solution to the system (14.1) with initial condition x(0) = 0 and input ~(-). Then given any time tl ~ T, it follows from (14.11) that
< ~otl w~(x(t),~(t))dt =
/o
w~(x(t),~o(t)) dt+
I?
~ (~(t), ~1 (t)) dt.
Hence, -
/?
w,(x(t),~l(t))dt
<_
/o
w,(x(t),~o(t))dt
< oz.
(14.16)
We now let to = tl - T _> 0, 5:(t) = x ( t - T ) , ~l(t) = ~l(t - T ) , and use the time invariance of the system (14.1) to conclude that the (14.16) can be re-written as -
-
//
Wr(5:(t),~l(t)) dt<
f0
w,(x(t),~o(t))dt < oz.
(14.17)
Furthermore, we have x(T) = xo and hence 5:(0) = x0. Thus, given any to _> 0 and any ~1(') E ~ , then ~(.), the corresponding solution to the system (14.1) with initial condition 5:(0) = x0 and input ~1(') satisfies inequality (14.17). Hence, V(xo) < oz for all x0. To complete the proof of this part of the theorem, it remains only to establish that the dissipativity condition (14.4) is satisfied. Indeed, given any ~(') E ~ and any T > to, then if x(t) is the corresponding solution to (14.1) with initial condition x(O), it follows from (14.15) that
v(x(0)) > -
~oT wr(x(t),~(t))dt
= - ~ o t ° W , ( x ( t ) , ~ ( t ) ) d t - f t ; ~r (.(t), ¢(t)) dt. Hence, -
I;
w~(x(t),~(t))dt- V(z(0)) <
/:o
wr(x(t),~(t))dt.
(14.18)
Now define (1 (t) := ~(t - to) for t E [to, T]. Using the time invariance of the system (14.1), it follows that
-
~tT wT(x(t),~(t))dt o
=
[T-to dO
w,(2(t),~l(t))dt.
where 2(.) is the solution to (14.1) with initial condition 2(0) = x(to) and input ~:(-). Substituting this into (t4.18), it follows that
0.,o
wr(2(t),(:(t)) dt - V(x(O)) <_
/;o
w,(x(t),~(t)) dt
However, this holds for all ~:(-) E • and for all T > to. Thus using (14.15), we can conclude that the following dissipation inequality is satisfied:
V ( x ( t o ) ) - V(x(O)) <_
£°
w~(x(t), ~(t))dt.
(iii) ~ (i) If condition (iii) holds then there exists a function V(xo) > 0 such that V(0) = 0 and k v(x(to))
-
v(x(0)) <
ft0
,. [ 3----1
JU
Ws (x(t)~ ~(t)) dt
(14.19)
tbr all solutions to the system (14.1) and all to _> 0. However, since rs _> 0 and k ~ , = : r~ = 1, inequality (14.19) implies
V(x(to))-V(x(O))
<_ EkT , f0to w~(x(t),~(t))dt s=l < -
-
max s=l,2,...,k
{f
}
w,(x(t),~(t)) dt .
That is, condition (14.5) is satisfied. This completes the proof of the theorem.
14.5
Absolute
Stability
In this section we first consider an uncertain nonlinear system defined by the system (14.1) and associated functions (14.2). In this case, ~(.) is interpreted as a structured uncertainty input.
System Uncertainty The uncertainty in the system (14.1) is described by an equation of the form: ~(t) = ¢(t, x(.)lto)
(14.20)
where the following Integral Constraint is satisfied; see also (Yakubovich, 1973, 1988b, 1988a), and (Petersen, 1994; Savkin and Petersen, 1995b, 1995a).
300 D e f i n i t i o n 14.14 I n t e g r a l Uncertainty Constraint An uncertainty of the fo~n (14.20) is an admissible uncertainty for the system (14.1} if the following condition holds: Given any solution to equations (14.1), (14.20} with an interval of existence [0, t.), then there exist a sequence {ti}i~=l and constants dl > O, d2 > 0,...,dr, > 0 such that ti -+ t., ti > 0 and
fo
*'
(14.21)
dt < d,
for all i and for s = 1 , 2 , . . . , k . Note that t. and ti may be equal to infinity. Remark. Given an admissible uncertainty of the form (14.20), then for every solution to the equations (14.1), (14.20), there will be corresponding parameters ds >_ 0 such that condition (14.21) is satisfied. The value of the parameters ds is a 'measure of mismatch' between the resulting uncertainty input ~(.) and the following Homogeneous Integral Constraint: j l t' w,(x(t),~(t)) dt < 0
Vi.
Note that the uncertainty description given above allows for ~(t) to depend dynamically on x(t). Indeed, in this case, one could interpret the 'measure of mismatch' mentioned above as being due to a non-zero initial condition on the uncertainty dynamics. Also, it is clear that the uncertain system (14.1), (14.20) allows for structured uncertainty satisfying norm bound conditions. In this case, the uncertain system would be described by the state equations k
~(t) = A(x(t)) q- E
B'(x(t))As(t)Cs(x(t));
IIA,(t)tt < 1,
where As(t ) are the uncertainty matrices and tl" I1denotes the standard induced matrix norm. To verify that such uncertainty is admissible for the uncertain system (14.1), (14.20), let ~s(t) := As(t)Cs(x(t)) where llAs(t)ll _< 1 for all t > 0 and ~(t) := [ ~'l(t) (2(t) ... ~'k(t) ]T. Then ~(-) satisfies condition (14.21) with w,(z(t),~(t)) = tt~,(t)tl 2 -[tC~(z(t))[[ 2, with any ti and with do = 0. We now introduce a corresponding notion of absolute stability and for the uncertain system (14.1), (14.20). This definition extends to the case of nonlinear uncertain systems, the standard definition of absolute stability; e.g., see (Yakubovich, 1973, 1988b, 1988a; Savkin and Petersen, 1994, 1995b). D e f i n i t i o n 14.15 Consider the uncertain system (14.1), (14.20). This uncertain system is said to be absolutely stable if there exists a non-negative function W(.) : R n --+ IR and a constant c > 0 such that the following conditions hold: (i) w ( o ) = o.
301
(ii) For any initial condition x(O) = xo and any uncertainty input ~(.) L2[O, oc), the system (14.1) has a unique solution which is defined on
[o,~). (iii) Given an admissible uncertainty for the uncertain system (14.1), (14.20),
then any solution to the equations (I4.1), (14.20) satisfies [x(.), x(-), ~(-)] C L2[O, oc) (hence, t, = oo) and k
fo
¢C(Hx(t)H 2 + Ilk(t)lt 2 + ]l~(t)]l 2) dt <_ W(x(O)) + c E
d~.
(14.22)
S-.~ I
Remark. In the above definition, condition (14.22) requires that the L~ norm of any solution to (14.1) and (14.20) be bounded in terms of the norm of the initial condition and the 'measure of mismatch' d for this solution.
N o t e It follows from the above definition that for any admissible uncertainty, an absolutely stable uncertain system (14.1), (14.20) has the property that x(t) -+ 0 as t -4 oz. Indeed, using the fact that x(.) E L2[0, oc) and x(.) E L2[0, oc), it now follows that x(t) -4 0 as t -4 c~. T h e o r e m 14.16 Consider the nonlinear uncertain system (14.1), (14.20) with associated functions (14.2) and suppose that Assumptions 14.1-14.5 are satisfied. Then the following statements are equivalent: (i) The uncertain system (14.1), (14.20) is absolutely stable. (ii) There exist constants 5 > 0, rl > 0, r2 >_ O , . . . , r k > 0 such that ~ s k= l r~ = 1 and the following condition holds: Given any input ~(.) E ~, let x(.) be the corresponding solution to the equation (I4.1) with initial condition x(0) = 0. Furthermore, suppose x(.) is defined on an existence interval [O,t,), Then
/i
to w , ( x ( t ) , ~(t)) dt >_ 5
/o
(llx(t)H 2 + ll2(t)H 2 + II~(t)ll 2) dt
(14.23)
for all to C [O,t.), where w,(., .) is defined by equation (14.12). (iii) The system (14.1) with supply rate functions (14.2) has the strict dissipativity property. (iv) There exist constants 5 > O, rl >_ O, v2 >_ O , . . . , r k > 0 such that ~ =k1 v8 --- 1 and the system (14.1) is dissipative with supply rate w~(x(t),~(t)) k
:=
E
r~ws(x(t),~(t)) - 5(llx(t)ll 2 + llg(x(t),~(t))II 2 + II~(t)i12).
(14.24)
302
Proof. (i) ~ (ii) Let 5 := ~ where c is the constant from Definition 14.15. Consider functionals defined by f~ (z(.),~(.))
=
fo °° w~(x(t),~(t))dt;
f~ (x(.),~(.))
=
/5
f~ (x(-),~(.))
=
f0 ° w~k(x(t),~(t)) dt
w~(x(t),~(t)) dt;
(14.25)
where the functions w~(.,-) are as defined in (14.6). Also, let $2 be the set defined in Notation 14.9. We will prove that if f~(x(-),~(.)) < 0 , . . . , fk~(X('),~(')) < 0 for some pair {x(-),~(')} E /2, then f~(x(.),¢(.)) > 0. Indeed if f~(x(.),~(.)) < 0, we have
f0
w,(x(t),~(t)) d t < 5
f0 (llx(t)ll ~
+ Ilg(x(t), ~(t))ll 2 + II~(t)ll ~) dt
for s - 1, 2 , . . . , k. Hence, there exist constants dl, d2,..., dk such that d. < 5
fo °° (ll~(t)ll
~ + IIg(~(t),,~(t))ll 2 + II~(t)ll 2) dt
(14.26)
for s = 1, 2 , . . . , k and the pair {x(.), ~(.)} satisfies the integral constraints (14.21) with ti = ~z. However since x(0) = 0, Definition 14.15 implies that
£
k
~(llx(t)ll 2 + IIg(x(t), 5(t))ll 2 + I[¢(t)ll 2) d¢< c e d e .
(14.27)
Furthermore, since 5 = ~ , inequality (14.27) contradicts inequality (14.26). Hence, for any pair {x(-),~(.)} E / 2 such that
f~(x(.),¢(.)) < 0,..., f~(x(.),~(.)) < 0, then f~(z(-), ~(.)) > 0. Therefore, Theorem 14.10 implies that there exist conk stants vl > 0, 7-2 > 0 , . . . , rk >_ 0 such that ~ s = l v~ = 1 and k
vsfff(x(-),~(-)) ~ 0
(14.28)
for all pairs {x(-),~(-)} E ~2. Furthermore, according to Claim 2 in the proof of Theorem 14.13, condition (14.28) implies condition (14.23). This completes the proof of this part of the theorem.
(ii) ~ (iii), (iii) =~ (iv) These implications are immediate consequences of Theorem 14.13 and Definition 14.7.
303
(iv) ~ (i) Condition (iv) implies that there exist constants 5 > 0, rl _> 0 , . . . , rk > 0 and a non-negative function V(.) : IRn --+ IR such that V(0) = 0 and V ( z ( t o ) ) - V(x(O)) <_
£°
w~(x(.),((.)) dt
(14.29)
for all solutions z(.) to the system (14.1), where the function -) is defined as in equation (14.24). Therefore, if an input ~(.) satisfies the integral constraint (14.21), it follows from (14.29) that
(ll~(t)li 2 + Ilx(t)tl 2 + It~(t)ll 2) dt <_ (V(z(0)) + ~
,r~d~)
(14.30)
s..~l
for all i. However, it follows from Definition 14.14 that ti -+ t, where t, is the upper limit of the existence interval of the solution {x(.),~(.)}. Hence, condition (14.30) implies that t. = oe and the inequality (14.22) holds with W ( x o ) = ½V(xo) and with
1
C ~-~ ~
max s----1,2,...,k
This completes the proof of the theorem. Remark. For the case of linear uncertain systems with a single uncertainty and a quadratic storage function (Lyapunov function), the above theorem reduces to a well known result in the absolute stability literature; e.g., see (Yakubovich, 1973, 1988a). Furthermore in this case, condition (ii) is equivalent to the circle criterion. For the case of linear uncertain systems with structured uncertainty, the above theorem reduces to the result of Megretsky and Treil (1993). Now consider the nonlinear interconnected system
~:
~l(t)
~(t)
= =
g(z(t),~l(t),~2(t),...,~k(t)), zt(x(t),d:(t)),
~(t)
=
z2(x(t), x(t)),
1
yk(t)
=
zk(x(t),x(t));
~l(t) ~l(t)
= =
gl(xl(t),yl(t)), hl(xl(t),xl(t));
£'2
~(t) ~2(t)
=
g2(x2(t),y2(t)),
=
h~(~(t), x~(t));
Zk:
~k(t) ~k(t)
= =
gk(xk(t), yk(t)), hk(xk(t), xk(~)),
Z1
(14.31)
304 where x(t) E IRn,xl(t) E IR"l,x2(t) E lRn2,...,xk(t) E IR'~k are the state vectors; ~l(t) E IRml,~(t) E lRm~,...,~k(t) E lR'~k,yl(t) E IRql,y2(t) E IRq2,...,yk(t) E IRq~ are the input-output vectors. A block diagram of this interconnected system is shown in Figure 14.1.
~ ~
q
Yk
4
L ~k
Z
Figure 14.1
Assumptions. tions:
Interconnected nonlinear system
We assume the system (14.31) satisfies the following assump-
A s s u m p t i o n 14.17
There exists a constant.co > 0 such that
llzs(x(t), ~(t))ll ~ < co(llx(t)lt ~ + II~(t)lt ~) for s = 1 , 2 , . . . , k .
A s s u m p t i o n 14.18 Suppose that ys(') E L2[0, c¢) is a given input of the subsystem Zs and {x,(.), G (')} is any corresponding solution. Then the condition ~,(.) e L2[0, c¢) implies that {x(.), z(.)} e L2[0, c¢).
305 Note, Assumption 14.18 is an observabitity type assumption on each of the subsystems ~s. Definition 14.19 The system (14.31) is said to be globally asymptotically stable if all of its solutions {x(t), xl(t), x2(t),..., xk(t)) are defined on [0, oc) and tend to 0 as t -+ oz. Associated with the system (14.31), we will consider the following functions
wl(xl(t),~l(t)),
w2(x~(t),~2(t)),
...,
wk(xk(t),~k(t)).
(I4.32)
Now we are in a position to present a stability criterion for the nonlinear interconnected system (14.31). Consider the system (L{.31) with associated functions T h e o r e m 14.20 (14.32) and suppose that the subsystem Z satisfies Assumptions i~.1-1~.5 and the subsystems ~1, ~ 2 , - - ' , ~ k satisfy Assumptions 14.17 and 14.18. Also, suppose that the subsystem ~ with supply rates (14.32) has the strict dissipativity property and the subsystem ~s with supply rate -ws(x~(t),~s(t)) is dissipative for s = 1, 2 , . . . , k. Then the system (1~.31) is globally asymptotically stable.
The dissipativity of the subsystem )Js implies that there exists a Pro@ function V~(x~0) _> 0 such that to
_< -
L
dr.
Hence, any solution of the subsystem Zs satisfies the integral constraint
L
ti Ws(Xs(t),~s(l))dt < ds
(14.33)
with any ti _> 0 and with d~ = ~%(x8(0)). Therefore, according to Theorem 14.16, the strict structured dissipativity of the system Z implies that the system Z with inputs ~1('),~2('),...,~k(') satisfying the integral con: straints (14.33) is absolutely stable. Hence using Definition 14.19, it follows that {x(-), k('),~i(-),~2('),...,~k(-)} E L2[0, ec). This and Assumption 14.17 imply that {Yl('), Y2('),---, Yk(')} E L2[0, ec). Furthermore, according to Assumption 14.18, if {Ys('),~(')} E L2[0, ec) then {x~(.),~?~(.)} E L2[0, oc). Using {x(.),Xl(.),x2(.),...,xk(-)} E L2[0, eo) and {2(-), 21(-),x~(.),...,xk(.)} e L2[0, ec), it now follows that {x(t), xt(t), x2(t),..., xk(t)} -+ 0 as t --+ ee. This completes the proof of the theorem. Acknowledgements. Council.
This work was supported by the Australian Research
306
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