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1 Introduction 1

1.1 Nonlinear Models and Nonlinear Phenomena 1

1.2 Examples 5

1.2.1 Pendulum Equation 5

1.2.2 Tunnel-Diode Circuit 6

1.2.3 Mass-Spring System 8

1.2.4 Negative-Resistance Oscillator 11

1.2.5 Artificial Neural Network 14

1.2.6 Adaptive Control 16

1.2.7 Common Nonlinearities 18

1.3 Exercises 24

2 Second-Order Systems 35

2.1 Qualitative Behavior of Linear Systems 37

2.2 Multiple Equilibria 46

2.3 Qualitative Behavior Near Equilibrium Points 51

2.4 Limit Cycles 54

2.5 Numerical Construction of Phase Portraits 59

2.6 Existence of Periodic Orbits 61

2.7 Bifurcation 69

2.8 Exercises 76

3 Fundamental Properties 87

3.1 Existence and Uniqueness 88

3.2 Continuous Dependence on Initial Conditions and Parameters 95

3.3 Differentiability of Solutions and Sensitivity Equations 99

3.4 Comparison Principle 102

3.5 Exercises 105

4 Lyapunov Stability 111

4.1 Autonomous Systems 112

4.2 The Invariance Principle 126

4.3 Linear Systems and Linearization 133

4.4 Comparison Functions 144

4.5 Nonautonomous Systems 147

4.6 Linear Time-Varying Systems and Linearization 156

4.7 Converse Theorems 162

4.8 Boundedness and Ultimate Boundedness 168

4.9 Input-to-State Stability 174

4.10 Exercises 181

5 Input-Output Stability 195

5.1 L Stability 195

5.2 L Stability of State Models 201

5.3 L2 Gain 209

5.4 Feedback Systems：The Small-Gain Theorem 217

5.5 Exercises 222

6 Passivity 227

6.1 Memoryless Functions 228

6.2 State Models 233

6.3 Positive Real Transfer Functions 237

6.4 L2 and Lyapunov Stability 241

6.5 Feedback Systems：Passivity Theorems 245

6.6 Exercises 259

7 Frequency Domain Analysis of Feedback Systems 263

7.1 Absolute Stability 264

7.1.1 Circle Criterion 265

7.1.2 Popov Criterion 275

7.2 The Describing Function Method 280

7.3 Exercises 296

8 Advanced Stability Analysis 303

8.1 The Center Manifold Theorem 303

8.2 Region of Attraction 312

8.3 Invariance-like Theorems 322

8.4 Stability of Periodic Solutions 329

8.5 Exercises 334

9 Stability of Perturbed Systems 339

9.1 Vanishing Perturbation 340

9.2 Nonvanishing Perturbation 346

9.3 Comparison Method 350

9.4 Continuity of Solutions on the Infinite Interval 355

9.5 Interconnected Systems 358

9.6 Slowly Varying Systems 365

9.7 Exercises 372

10 Perturbation Theory and Averaging 381

10.1 The Perturbation Method 382

10.2 Perturbation on the Infinite Interval 393

10.3 Periodic Perturbation of Autonomous Systems 397

10.4 Averaging 402

10.5 Weakly Nonlinear Second-Order Oscillators 411

10.6 General Averaging 413

10.7 Exercises 419

11 Singular Perturbations 423

11.1 The Standard Singular Perturbation Model 424

11.2 Time-Scale Properties of the Standard Model 430

11.3 Singular Perturbation on the Infinite Interval 439

11.4 Slow and Fast Manifolds 443

11.5 Stability Analysis 449

11.6 Exercises 460

12 Feedback Control 469

12.1 Control Problems 469

12.2 Stabilization via Linearization 475

12.3 Integral Control 478

12.4 Integral Control via Linearization 481

12.5 Gain Scheduling 485

12.6 Exercises 499

13 Feedback Linearization 505

13.1 Motivation 505

13.2 Input-Output Linearization 509

13.3 Full-State Linearization 521

13.4 State Feedback Control 530

13.4.1 Stabilization 530

13.4.2 Tracking 540

13.5 Exercises 544

14 Nonlinear Design Tools 551

14.1 Sliding Mode Control 552

14.1.1 Motivating Example 552

14.1.2 Stabilization 563

14.1.3 Tracking 572

14.1.4 Regulation via Integral Control 575

14.2 Lyapunov Redesign 579

14.2.1 Stabilization 579

14.2.2 Nonlinear Damping 588

14.3 Backstepping 589

14.4 Passivity-Based Control 604

14.5 High-Gain Observers 610

14.5.1 Motivating Example 612

14.5.2 Stabilization 619

14.5.3 Regulation via Integral Control 623

14.6 Exercises 625

A Mathematical Review 647

B Contraction Mapping 653

C Proofs 657

C.1 Proof of Theorems 3.1 and 3.2 657

C.2 Proof of Lemma 3.4 659

C.3 Proof of Lemma 4.1 661

C.4 Proof of Lemma 4.3 662

C.5 Proof of Lemma 4.4 662

C.6 Proof of Lemma 4.5 663

C.7 Proof of Theorem 4.16 665

C.8 Proof of Theorem 4.17 669

C.9 Proof of Theorem 4.18 675

C.10 Proof of Theorem 5.4 676

C.11 Proof of Lemma 6.1 677

C.12 Proof of Lemma 6.2 680

C.13 Proof of Lemma 7.1 684

C.14 Proof of Theorem 7.4 688

C.15 Proof of Theorems 8.1 and 8.3 690

C.16 Proof of Lemma 8.1 699

C.17 Proof of Theorem 11.1 700

C.18 Proof of Theorem 11.2 706

C.19 Proof of Theorem 12.1 708

C.20 Proof of Theorem 12.2 709

C.21 Proof of Theorem 13.1 710

C.22 Proof of Theorem 13.2 712

C.23 Proof of Theorem 14.6 713

Note and References 719

Bibliography 724

Symbols 740

Index 742