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APPLIED NONLINEAR CONTROL
Jean-Jacques E Slotine WeipingLi
Applied Nonlinear Control
JEAN-JACQUES E. SLOTlNE Massachusetts Institute of Technology
WEIPING LI Massachusetts Institute of Technologv
Prentice Hall Englewood Cliffs, New Jersey 07632
Library of Congress Cataloging·in·Publication Data Siotine, I.-I. E. (Jean-Jacques E.) Applied nonlinear control/Jean-Jacques E. Siotine, Weiping Li p.
em.
Includes bibliographical references. ISBN 0-13-040890-5 1, Nonlinear control theory. QA402.35.S56 1991 629.8'312-dc20
I. Li, Weiping.
Editorial/production supervision and interior design: JENNIFER WENZEL Cover design: KAREN STEPHENS Manufacturing Buyer: LORI BULW[N
All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher.
Printed in the United States of America 20
19
ISBN
18
17
16
15
0-13-040890-5
Prentice-Hall International (UK) Limited, London Prentice-Hall of Australia Pty. Limited, Sydney Prentice-Hall Canada Inc., Toronto Prentice-Hall Hispanoamericana, S.A., Mexico Prentice-Hall of India Private Limited, New Delhi Prentice-Hall of Japan, Inc.. Tokyo Simon & Schuster Asia Pte. Ltd., Singapore Editora Prentice-Hall do Brasil, Ltda., Rio de Janeiro
II. Title. 90-33365 CIP
Contents
Preface
xi
1
1. Introduction 1.1 1.2 1.3 1.4
1 4 12 13
Why Nonlinear Control? Nonlinear System Behavior An Overview of the Book Notes and References
14
Part I: Nonlinear Systems Analysis Introduction to Part I
Constructing Phase Portraits Determining Time from Phase Portraits Phase Plane Analysis of Linear Systems Phase Plane Analysis of Nonlinear Systems Existence of Limit Cycles Summary Notes and References Exercises
18
22
23 29 30 32 36 38 38 38 vii
viii
3. Fundamentals of Lyapunov Theory
40
3.1 Nonlinear Systems and Equilibrium Points
41 47
3.2 Concepts of Stability 3.3 Linearization and Local Stability
53
3.4 Lyapunov's Direct Method
57
3.4.1 Positive Definite Functions and Lyapunov Functions 3.4.2 Equilibrium Point Theorems 61 3.4.3 Invariant Set Theorems 68
58
76
3.5 System Analysis Based on Lyapunov's Direct Method 3.5.1 Lyapunov Analysis of Linear Time-Invariant Systems 3.5.2 Krasovskii's Method 83 3.5.3 The Variable Gradient Method 86 3.5.4 Physically Motivated Lyapunov Functions 88 91 3.5.5 Pelformance Analysis
77
3.6 Control Design Based on Lyapunov's Direct Method
94 95
3.7 Summary 3.8 Notes and References
96
3.9 Exercises
97
100
4. Advanced Stability Theory
101
4.1 Concepts of Stability for Non-Autonomous Systems 4.2 Lyapunov Analysis of Non-Autonomous Systems
105
4.2.1 Lyapullov'S Direct Methodfor Non-Autonomous Systems lOS 114 4.2.2 Lyapullov Analysis ofLinear Time- Varying Systems ]]6 4.2.3 * The Linearization Methodfor Non-Autono~ousSystems
4.3 * Instability Theorems
117
4.4 * Existence of Lyapunov Functions 4.5 Lyapunov-Like Analysis Using Barbalat's Lemma
120 122
4.5.1 Asymptotic Properties of Functions and Their Derivatives 123
122
4.5.2 Barbalat's Lemma
126
4.6 Positive Linear Systems 4.6.1 PR and SPR Transfer Functions 4.6.2 The Kalman-Yakubovich Lemma 4.6.3 Positive Real Transfer Matrices
126 130 131
4.7 The Passivity Formalism 4.7.1 Block Combinations 132 4.7.2 Passivity in Linear Systems
132 137
ix
40 41 47 53 57
4.8 * Absolute Stability 4.9 * Establishing Boundedness of Signals 4.10 * Existence and Unicity of Solutions 4.11 Summary 4.12 Notes and References 4.13 Exercises
142 147 151 153 153 154
.~
5. Describing Function Analysis
157
76 5.1 Describing Function Fundamentals
94 95 96 97
100 101 105 105
5.1.1 5.1.2 5.1.3 5.1.4 5.1.5
An Example ofDescribing Function Analysis Applications Domain 162 Basic Assumptions 164 Basic Definitions 165 Computing Describing Functions 167
158 158
169 172 179
5.2 Common Nonlinearities In Control Systems 5.3 Describing Functions of Common Nonlinearities 5.4 Describing Function Analysis of Nonlinear Systems 5.4.1 5.4.2 5.4.3 5.4.4
The Nyquist Criterion and Its Extension 180 Existence ofLimit Cycles 182 Stability ofLimit Cycles 184 186 Reliability of Describing Function Analysis
187 188 188
5.5 Summary 5.6 Notes and References 5.7 Exercises
l-t
116
117 120 122 122
Part II: Nonlinear Control Systems Design Introduction to Part II
191
191
207
6. Feedback Linearization 126
132
6.1 Intuitive Concepts 6.1.1 Feedback Linearization And The Canonical Form 213 6.1.2 Input-State Linearization 216 6.1.3 Input-Output Linearization
6.2 Mathematical Tools 6.3 Input-State Linearization of SISO Systems
208 208
229 236
x 6.4 Input-Output Linearization of SISO Systems
246
6.5 * Multi-Input Systems 6.6 Summary 6.7 Notes and References 6.8 Exercises
266 270 271 271
7. Sliding Control
276
7.1 Sliding Surfaces
277
7.1.1 A Notational Simplification 278 7.1.2 * Filippo v,s Construction of the Equivalent Dynamics 285 7.1.3 Pe/fect Peiformance - At a Price 7.1.4 Direct 1mplementations ofSwitching Control Laws
283 289
7.2 Continuous Approximations of Switching Control Laws 7.3 The Modeling/Perfonnance Trade-Offs
290 301
7.4 * Multi-Input Systems 7.5 Summary 7.6 Notes and References 7.7 Exercises
303 306 307 307
8. Adaptive Control
311
8.1 Basic Concepts in Adaptive Control 8.1.1 Why Adaptive Control? 312 8.1.2 What 1s Adaptive Control? 315 8.1.3 How To Design Adaptive Controllers?
312
323
8.2 Adaptive Control of First-Order Systems 8.3 Adaptive Control of Linear Systems With Full State Feedback 8.4 Adaptive Control of Linear Systems With Output Feedback 8.4.1 Linear Systems With Relative Degree One 8.4.2 Linear Systems With Higher Relative Degree
340 346
8.5 Adaptive Control of Nonlinear Systems 8.6 Robustness of Adaptive Control Systems 8.7
