9780130673893

This book is written is such a way that the level of mathematical sophistication builds up from chapter to chapter. It has been reorganized into four parts: basic analysis, analysis of feedback systems, advanced analysis, and nonlinear feedback control.Updated content includes subjects which have proven useful in nonlinear control design in recent yearsnew in the 3rd edition are: expanded treatment of passivity and passivity-based control; integral control, high-gain feedback, recursive methods, optimal stabilizing control, control Lyapunov functions, and observers.For use as a self-study or reference guide by engineers and applied mathematicians.

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Preface

xiii

Introduction

1

(34)

Nonlinear Models and Nonlinear Phenomena

1

(4)

Examples

5

(19)

Pendulum Equation

5

(1)

Tunnel-Diode Circuit

6

(2)

Mass-Spring System

8

(3)

Negative-Resistance Oscillator

11

(3)

Artificial Neural Network

14

(2)

Adaptive Control

16

(2)

Common Nonlinearities

18

(6)

Exercises

24

(11)

Second-Order Systems

35

(52)

Qualitative Behavior of Linear Systems

37

(9)

Multiple Equilibria

46

(5)

Qualitative Behavior Near Equilibrium Points

51

(3)

Limit Cycles

54

(5)

Numerical Construction of Phase Portraits

59

(2)

Existence of Periodic Orbits

61

(8)

Bifurcation

69

(7)

Exercises

76

(11)

Fundamental Properties

87

(24)

Existence and Uniqueness

88

(7)

Continuous Dependence on Initial Conditions and Parameters

95

(4)

Differentiability of Solutions and Sensitivity Equations

99

(3)

Comparison Principle

102

(3)

Exercises

105

(6)

Lyapunov Stability

111

(84)

Autonomous Systems

112

(14)

The Invariance Principle

126

(7)

Linear Systems and Linearization

133

(11)

Comparison Functions

144

(3)

Nonautonomous Systems

147

(9)

Linear Time-Varying Systems and Linearization

156

(6)

Converse Theorems

162

(6)

Boundedness and Ultimate Boundedness

168

(6)

Input-to-State Stability

174

(7)

Exercises

181

(14)

Input-Output Stability

195

(32)

L Stability

195

(6)

L Stability of State Models

201

(8)

L2 Gain

209

(8)

Feedback Systems: The Small-Gain Theorem

217

(5)

Exercises

222

(5)

Passivity

227

(36)

Memoryless Functions

228

(5)

State Models

233

(4)

Positive Real Transfer Functions

237

(4)

L2 and Lyapunov Stability

241

(4)

Feedback Systems: Passivity Theorems

245

(14)

Exercises

259

(4)

Frequency Domain Analysis of Feedback Systems

263

(40)

Absolute Stability

264

(16)

Circle Criterion

265

(10)

Popov Criterion

275

(5)

The Describing Function Method

280

(16)

Exercises

296

(7)

Advanced Stability Analysis

303

(36)

The Center Manifold Theorem

303

(9)

Region of Attraction

312

(10)

Invariance-like Theorems

322

(7)

Stability of Periodic Solutions

329

(5)

Exercises

334

(5)

Stability of Perturbed Systems

339

(42)

Vanishing Perturbation

340

(6)

Nonvanishing Perturbation

346

(4)

Comparison Method

350

(5)

Continuity of Solutions on the Infinite Interval

355

(3)

Interconnected Systems

358

(7)

Slowly Varying Systems

365

(7)

Exercises

372

(9)

Perturbation Theory and Averaging

381

(42)

The Perturbation Method

382

(11)

Perturbation on the Infinite Interval

393

(4)

Periodic Perturbation of Autonomous Systems

397

(5)

Averaging

402

(9)

Weakly Nonlinear Second-Order Oscillators

411

(2)

General Averaging

413

(6)

Exercises

419

(4)

Singular Perturbations

423

(46)

The Standard Singular Perturbation Model

424

(6)

Time-Scale Properties of the Standard Model

430

(9)

Singular Perturbation on the Infinite Interval

439

(4)

Slow and Fast Manifolds

443

(6)

Stability Analysis

449

(11)

Exercises

460

(9)

Feedback Control

469

(36)

Control Problems

469

(6)

Stabilization via Linearization

475

(3)

Integral Control

478

(3)

Integral Control via Linearization

481

(4)

Gain Scheduling

485

(14)

Exercises

499

(6)

Feedback Linearization

505

(46)

Motivation

505

(4)

Input-Output Linearization

509

(12)

Full-State Linearization

521

(9)

State Feedback Control

530

(14)

Stabilization

530

(10)

Tracking

540

(4)

Exercises

544

(7)

Nonlinear Design Tools

551

(96)

Sliding Mode Control

552

(27)

Motivating Example

552

(11)

Stabilization

563

(9)

Tracking

572

(3)

Regulation via Integral Control

575

(4)

Lyapunov Redesign

579

(10)

Stabilization

579

(9)

Nonlinear Damping

588

(1)

Backstepping

589

(15)

Passivity-Based Control

604

(6)

High-Gain Observers

610

(15)

Motivating Example

612

(7)

Stabilization

619

(4)

Regulation via Integral Control

623

(2)

Exercises

625

(22)

A Mathematical Review

647

(6)

B Contraction Mapping

653

(4)

C Proofs

657

(62)

Proof of Theorems 3.1 and 3.2

657

(2)

Proof of Lemma 3.4

659

(2)

Proof of Lemma 4.1

661

(1)

Proof of Lemma 4.3

662

(1)

Proof of Lemma 4.4

662

(1)

Proof of Lemma 4.5

663

(2)

Proof of Theorem 4.16

665

(4)

Proof of Theorem 4.17

669

(6)

Proof of Theorem 4.18

675

(1)

Proof of Theorem 5.4

676

(1)

Proof of Lemma 6.1

677

(3)

Proof of Lemma 6.2

680

(4)

Proof of Lemma 7.1

684

(4)

Proof of Theorem 7.4

688

(2)

Proof of Theorems 8.1 and 8.3

690

(9)

Proof of Lemma 8.1

699

(1)

Proof of Theorem 11.1

700

(6)

Proof of Theorem 11.2

706

(2)

Proof of Theorem 12.1

708

(1)

Proof of Theorem 12.2

709

(1)

Proof of Theorem 13.1

710

(2)

Proof of Theorem 13.2

712

(1)

Proof of Theorem 14.6

713

(6)

Note and References

719

(5)

Bibliography

724

(16)

Symbols

740

(2)

Index

742

What is included with this book?

The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

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This text is intended for a first-year graduate-level course on nonlinear systems or control. It may also be used for self study or reference by engineers and applied mathematicians. It is an outgrowth of my experience teaching the nonlinear systems course at Michigan State University, East Lansing. Students taking this course have had background in electrical engineering, mechanical engineering, or applied mathematics. The prerequisite for the course is a graduate-level course in linear systems, taught at the level of the texts by Antsaklis and Michel 9, Chen 35, Kailath 94), or Rugh 158. The linear systems prerequisite allowed me not to worry about introducing the concept of "state" and to refer freely to "transfer functions," "state transition matrices," and other linear system concepts. The mathematical background is the usual level of calculus, differential equations, and matrix theory that any graduate student in engineering or mathematics would have. In the Appendix, I have collected a few mathematical facts that are used throughout the book. I have written the text in such a way that the level of mathematical sophistication increases as we advance from chapter to chapter. This is why the second chapter is written in an elementary context. Actually, this chapter could be taught at senior, or even junior, level courses without difficulty. This is also the reason I have split the treatment of Lyapunov stability into two parts. In Sections 4.1 through 4.3, I introduce the essence of Lyapunov stability for autonomous systems where I do not have to worry about technicalities such as uniformity, classkfunctions, etc. In Sections 4.4 through 4.6, I present Lyapunov stability in a more general setup that accommodates nonautonomous systems and allows for a deeper look into advanced aspects of the stability theory. The level of mathematical sophistication at the end of Chapter 4 is the level to which I like to bring the students, so that they can comfortably read the rest of the text. There is yet a higher level of mathematical sophistication that is assumed in writing the proofs in the Appendix. These proofs are not intended for classroom use. They are included to make the text on one hand, self contained, and, on the other, to respond to the need or desire of some students to read such proofs, such as students continuing on to conduct Ph.D. research into nonlinear systems or control theory. Those students can continue to read the Appendix in a self-study manner. This third edition has been written with the following goals in mind: To make the book (especially the early chapters) more accessible to first-year graduate students. As an example of the changes made toward that end, note the change in Chapter 3: All the material on mathematical background, the contraction mapping theorem, and the proof of the existence and uniqueness theorem have been moved to the Appendix. Several parts of the books have been rewritten to improve readability. To reorganize the book in such a way that makes it easier to structure nonlinear systems or control courses around it. In the new organization, the book has four parts, as shown in the flow chart. A course on nonlinear systems analysis will cover material from Parts 1, 2, and 3, while a course on nonlinear control will cover material from Parts 1, 2, and 4. To update the material of the book to include topics or results that have proven to be useful in nonlinear control design in recent years. New to the third addition are the: expanded treatment of passivity and passivity-based control, integral control, sliding mode control, and high-gain observers. Moreover, bifurcation is introduced in the context of second-order systems. On the technical side, the reader will find Kurzweil's converse Lyapunov theorem, nonlocal results in Chapters 10 and 11, and new results on integral; control and gain scheduling. To update the exercises. More

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Nonlinear Systems

Nonlinear Systems

0130673897

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