Nonlinear Systems

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.

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

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