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PPLIE
NONLINE R
CONTROL
Jean
Jacques E Slotine
WeipingLi
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Applied
Nonlinear
Control
JEAN JACQUES E SLOTlNE
assachusetts nstitute Technology
WEIPING LI
assachusetts nstitute Technologv
Prentice Hall
Englewood Cliffs New Jersey 7632
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Library Congress Cataloging·in·Publication Data
Siotine, I I E. Jean-Jacques E.
Appliednonlinear 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
629.8 312 dc20
Editorial/production supervision and
interior design: JENNIFER
WENZEL
Cover design: KAREN
STEPHENS
Manufacturing Buyer: LORI BULW[N
© 1991
by Prentice-Hall, Inc.
A Division of Simon
Schuster
Englewood Cliffs, New Jersey 07632
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 18
17 16
15
II. Tit le.
90-33365
CIP
S N
13 4 89 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
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Contents
Preface
Introduction
Why Nonlinear
ontrol
1 2 Nonlinear System Behavior
3
An Overview
of
the Book
1 4 Notes and References
Part Nonlinear Systems Analysis
Introduction
to
Part I 14
2 Phase Plane Analysis
2
Concepts
of
Phase Plane Analysis
2
Phase Portraits
2 1 2 Singular Points 2
2 1 3 Symmetly 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
4
2
3
4
7
8
23
29
3
32
36
38
38
38
vii
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viii
Fundamentals
Lyapunov Theory
4
3 1 Nonlinear Systems an d Equilibrium Points
3 2 Concepts of Stability
3 3 Linearization and Local Stabili ty
3 4 Lyapunov s Direct Method
3.4.1 Positive Definite Functions and Lyapunov Functions
3.4.2 Equilibrium Point Theorems
6
3.4.3 Invariant Set Theorems 68
3 5
System
Analysis Based
on
Lyapunov s Direct
Method
3.5.1 Lyapunov Analysis ofLinear Time Invariant Systems
3 5 2
Krasovskii s Method
8
3.5.3 The Variable Gradient Method 86
3.5.4 Physically Motivated Lyapunov Functions 88
3.5.5 Pelformance Analysis 9
3 6 Control Design Based on Lyapunov s Direct
Method
3 7
Summary
3 8 Notes and References
3 9 Exercises
58
77
4
47
53
57
76
94
96
97
105
lOS
]]6
117
2
22
22
26
4 1 Concepts of Stability for
Non Autonomous
Systems
4 2 Lyapunov Analysis of Non Autonomous Systems
4 2 1 Lyapullov S DirectMethodfor Non Autonomous Systems
4.2.2 Lyapullov Analysis
of
Linear Time Varying Systems 114
4 2 3
The Linearization Methodfor N o n u t o n o ~ o u s Systems
4 3
Instability Theorems
4 4 Existence of Lyapunov Functions
4 5 Lyapunov Like Analysis Using
Barbalat s
Lemma
4.5.1 Asymptotic Properties
of
Functions and Their Derivatives
4 5 2 Barbalat s Lemma 123
4 6 Positive
Linear
Systems
4.6.1 PR and SPR Transfer Functions 126
4.6.2 The Kalman Yakubovich Lemma 130
4.6.3 Positive Real Transfer Matrices 131
4 7
Th e
Passivity Formalism 132
4.7.1 Block Combinations 132
4.7.2 Passivity in Linear Systems 7
4 Advanced Stability Theory
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ix
4 8
Absolute Stability
142
4 9
Establishing Boundedness Signals
147
4 10
Existence and Unicity
Solutions
151
4 11
Summary
153
4 12
Notes and References
153
4 13 Exercises
154
5 Describing Function Analysis
157
5 1 Describing Function Fundamentals
158
5 1 1
An Example
of
Describing Function Analysis
158
5 1 2
Applications Domain 162
5 1 3 Basic Assumptions 164
5 1 4
Basic Definitions
165
5 1 5 Computing Describing Functions 167
5 2
Common Nonlinearities In Control Systems
169
5 3
Describing Functions
Common Nonlinearities
172
5 4
Describing Function Analysis Nonlinear Systems
179
5 4 1 The Nyquist Criterion and Its Extension 180
5 4 2 Existence
of
Limit Cycles 182
5 4 3
Stability ofLimit Cycles
184
5 4 4 Reliability ofDescribing Function Analysis
186
5 5
Summary
187
5 6 Notes and References
188
5 7 Exercises
188
Part II: Nonlinear Control Systems Design 9
ntroduction
to
Part
191
6 Feedback Linearization
2 7
6 1
Intuitive Concepts 2 8
6 1 1 Feedback Linearization nd The Canonical Form 208
6 1 2 Input State Linearization 213
6 1 3
Input Output Linearization 216
6 2
Mathematical Tools
229
6 3
Input State Linearization SISO Systems
236
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x
6.4 Input Output Linearization SISO Systems
6.5
Multi Input Systems
6.6 Summary
6.7 Notes and References
6.8 Exercises
246
266
270
27
27
7
Sliding Control
7.1 Sliding Surfaces
7 1 1 A Notational Simplification 278
7.1.2
Filippov s Construction of the Equivalent Dynamics
7 1 3 Pe/fect Peiformance
At
a Price 285
7 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
7
277
83
89
290
301
303
306
307
307
8 Adaptive Control
8
Basic Concepts in Adaptive Control
8 1 1 Why Adaptive Control? 312
8 1 2 What
s
Adaptive Control? 315
8 1 3 How To Design Adaptive Controllers? 323
8.2 Adaptive Control First Order Systems
8.3 Adaptive Control
Linear Systems With Full State Feedback
8.4 Adaptive Control
Linear Systems With Output Feedback
8 4 1 Linear Systems With Relative Degree One 4
8 4 2 Linear Systems With Higher Relative Degree 346
8.5 Adaptive Control Nonlinear Systems
8.6 Robustness Adaptive Control Systems
8.7 On Line Parameter Estimation
8 7 1 Linear Parametrization Model 359
8 7 2 Prediction Error Based Estimation Methods 364
8 7 3 The Gradient Estimator 364
8 7 4 The Standard Least Squares Estimator 370
3
312
326
335
339
350
353
358
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xi
8 7 5
Least Squares With Exponential Forgetting
374
8 7 6
Bounded Gain Forgetting
376
8 7 7 Concluding Remarks and Implementation Issues
38
8 8 Composite Adaptation
382
8 9
Summary
388
8 10
Notes and References
389
8 11
Exercises
389
9
ontrol of Multi Input Physical Systems
392
9 1 Robotics as a Prototype
393
9
Position Control
394
9 2
Trajectory Control
9
9 2
Adaptive Robot Trajectory Control
403
9 2
The Basic Algorithm 404
9 2 2
Composite Adaptive Trajectory Control
4
9 3
Putting Physics in Control
416
9 3
High Frequency Unmodeled Dynamics
4 6
9 3 2 Conservative n Dissipative Dynamics
4 8
9 3 3 Robotics as a Metaphor
4 9
9 4
Spacecraft Control
422
9 4
The Spacecraft Model
422
9 4 2 Attitude Control
425
9 5 Summary
432
9 6
Notes and References
433
9 7
Exercises
433
BIBLIOGRAPHY
437
INDEX
459
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