Llinear Control Systems - Page 1-49 (2)

8/20/2019 Llinear Control Systems - Page 1-49 (2)

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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 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 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


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


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


433

9 7

Exercises

433

BIBLIOGRAPHY

437

INDEX

459


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Llinear Control Systems - Page 1-49 (2)