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APPLIEDNONLINEAR
CONTROL
Jean-Jacques E SlotineWeipingLi
AppliedNonlinearControl
JEAN-JACQUES E. SLOTlNEMassachusetts Institute of Technology
WEIPING LIMassachusetts Institute of Technologv
Prentice HallEnglewood 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. I. Li, Weiping.
QA402.35.S56 1991629.8'312-dc20
Editorial/production supervision andinterior design: JENNIFER WENZEL
Cover design: KAREN STEPHENSManufacturing Buyer: LORI BULW[N
© 1991 by Prentice-Hall, Inc.A Division of Simon & SchusterEnglewood Cliffs, New Jersey 07632
All rights reserved. No part of this book may bereproduced, in any form or by any means,without permission in writing from the publisher.
Printed in the United States of America
20 19 18 17 16 15
II. Title.
90-33365CIP
ISBN 0-13-040890-5
Prentice-Hall International (UK) Limited, LondonPrentice-Hall of Australia Pty. Limited, SydneyPrentice-Hall Canada Inc., TorontoPrentice-Hall Hispanoamericana, S.A., MexicoPrentice-Hall of India Private Limited, New DelhiPrentice-Hall of Japan, Inc.. TokyoSimon & Schuster Asia Pte. Ltd., SingaporeEditora Prentice-Hall do Brasil, Ltda., Rio de Janeiro
Contents
Preface
1. Introduction
1.1 Why Nonlinear Control?
1.2 Nonlinear System Behavior
1.3 An Overview of the Book
1.4 Notes and References
Part I: Nonlinear Systems AnalysisIntroduction to Part I 14
2. Phase Plane Analysis
2.1 Concepts of Phase Plane Analysis2.1.1 Phase Portraits 182.1.2 Singular Points 202.1.3 Symmetly in Phase Plane Portraits 22
2.2 Constructing Phase Portraits
2.3 Determining Time from Phase Portraits
2.4 Phase Plane Analysis of Linear Systems
2.5 Phase Plane Analysis of Nonlinear Systems
2.6 Existence of Limit Cycles
2.7 Summary
2.8 Notes and References
2.9 Exercises
xi
1
1
4
12
13
14
17
18
23
29
30
32
36
38
38
38
vii
viii
3. Fundamentals of Lyapunov Theory 40
3.1 Nonlinear Systems and Equilibrium Points
3.2 Concepts of Stability
3.3 Linearization and Local Stability
3.4 Lyapunov's Direct Method
3.4.1 Positive Definite Functions and Lyapunov Functions3.4.2 Equilibrium Point Theorems 613.4.3 Invariant Set Theorems 68
3.5 System Analysis Based on Lyapunov's Direct Method
3.5.1 Lyapunov Analysis ofLinear Time-Invariant Systems3.5.2 Krasovskii's Method 833.5.3 The Variable Gradient Method 863.5.4 Physically Motivated Lyapunov Functions 883.5.5 Pelformance Analysis 91
3.6 Control Design Based on Lyapunov's Direct Method
3.7 Summary
3.8 Notes and References
3.9 Exercises
58
77
41
47
53
57
76
94
95
9697
100
101105
lOS
]]6
117
120122
122
126
4.1 Concepts of Stability for Non-Autonomous Systems
4.2 Lyapunov Analysis of Non-Autonomous Systems
4.2.1 Lyapullov'S Direct Methodfor Non-Autonomous Systems4.2.2 Lyapullov Analysis ofLinear Time-Varying Systems 1144.2.3 * The Linearization Methodfor Non-Autono~ousSystems
4.3 * Instability Theorems
4.4 * Existence of Lyapunov Functions
4.5 Lyapunov-Like Analysis Using Barbalat's Lemma
4.5.1 Asymptotic Properties ofFunctions and Their Derivatives4.5.2 Barbalat's Lemma 123
4.6 Positive Linear Systems
4.6.1 PR and SPR Transfer Functions 1264.6.2 The Kalman-Yakubovich Lemma 1304.6.3 Positive Real Transfer Matrices 131
4.7 The Passivity Formalism 132
4.7.1 Block Combinations 1324.7.2 Passivity in Linear Systems 137
4. Advanced Stability Theory
ix
40 4.8 * Absolute Stability 142
4.9 * Establishing Boundedness of Signals 147
41 4.10 * Existence and Unicity of Solutions 15147 4.11 Summary 15353 4.12 Notes and References 15357 4.13 Exercises 154
.~
5. Describing Function Analysis 15776
5.1 Describing Function Fundamentals 1585.1.1 An Example ofDescribing Function Analysis 1585.1.2 Applications Domain 1625.1.3 Basic Assumptions 1645.1.4 Basic Definitions 165
94 5.1.5 Computing Describing Functions 16795 5.2 Common Nonlinearities In Control Systems 16996 5.3 Describing Functions of Common Nonlinearities 17297 5.4 Describing Function Analysis of Nonlinear Systems 179
5.4.1 The Nyquist Criterion and Its Extension 1805.4.2 Existence ofLimit Cycles 182
100 5.4.3 Stability ofLimit Cycles 1845.4.4 Reliability ofDescribing Function Analysis 186
101 5.5 Summary 187
105 5.6 Notes and References 188
1055.7 Exercises 188
l-t116
117 Part II: Nonlinear Control Systems Design 191
120Introduction to Part II 191
122122 6. Feedback Linearization 207
1266.1 Intuitive Concepts 208
6.1.1 Feedback Linearization And The Canonical Form 2086.1.2 Input-State Linearization 2136.1.3 Input-Output Linearization 216
1326.2 Mathematical Tools 2296.3 Input-State Linearization of SISO Systems 236
x
6.4 Input-Output Linearization of SISO Systems
6.5 * Multi-Input Systems
6.6 Summary
6.7 Notes and References
6.8 Exercises
246
266
270
271
271
7. Sliding Control
7.1 Sliding Surfaces7.1.1 A Notational Simplification 2787.1.2 * Filippo v,s Construction of the Equivalent Dynamics7.1.3 Pe/fect Peiformance - At a Price 2857.1.4 Direct 1mplementations ofSwitching Control Laws
7.2 Continuous Approximations of Switching Control Laws
7.3 The Modeling/Perfonnance Trade-Offs
7.4 * Multi-Input Systems
7.5 Summary
7.6 Notes and References
7.7 Exercises
276
277
283
289
290
301
303
306
307
307
8. Adaptive Control
8.1 Basic Concepts in Adaptive Control8.1.1 Why Adaptive Control? 3128.1.2 What 1s Adaptive Control? 3158.1.3 How To Design Adaptive Controllers? 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 Feedback8.4.1 Linear Systems With Relative Degree One 3408.4.2 Linear Systems With Higher Relative Degree 346
8.5 Adaptive Control of Nonlinear Systems
8.6 Robustness of Adaptive Control Systems
8.7 * On-Line Parameter Estimation8.7.1 Linear Parametrization Model 3598.7.2 Prediction-Error-Based Estimation Methods 3648.7.3 The Gradient Estimator 3648.7.4 The Standard Least-Squares Estimator 370
311
312
326
335
339
350
353
358
xi
246 8.7.5 Least-Squares With Exponential Forgetting 374266 8.7.6 Bounded-Gain Forgetting 376
2708.7.7 Concluding Remarks and Implementation Issues 381
2718.8 Composite Adaptation 382
2718.9 Summary 3888.10 Notes and References 3898.11 Exercises 389
276
2779. Control of Multi-Input Physical Systems 392
283 9.1 Robotics as a Prototype 3939.1.1 Position Control 394
'9 9.1.2 Trajectory Control 397
290 9.2 Adaptive Robot Trajectory Control 403301 9.2.1 The Basic Algorithm 404
303 9.2 .2 * Composite Adaptive Trajectory Control 411
306 9.3 Putting Physics in Control 416
3079.3.1 High-Frequency Unmodeled Dynamics 4169.3.2 Conservative and Dissipative Dynamics 418
307 9.3.3 Robotics as a Metaphor 419
9.4 Spacecraft Control 4229.4.1 The Spacecraft Model 422
311 9.4.2 Attitude Control 425
9.5 Summary 432312 9.6 Notes and References 433
9.7 Exercises 433
326 BIBLIOGRAPHY 437
335 INDEX 459
339
350353
358