Contents

Preface     xiii

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

1.2.7 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 L 2 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 L 2 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

8.1 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     553

14.1 Sliding Mode Control ........................... 554

14.1.1 Motivating Example ....................... 554

14.1.2 Stabilization ........................... 565

14.1.3 Tracking .............................. 574

14.1.4 Regulation via Integral Control ................. 577

14.2 Lyapunov Redesign ............................ 581

14.2.1 Stabilization ............................ 581

14.2.2 Nonlinear Damping ....................... 590

14.3 Backstepping ............................... 591

14.4 Passivity-Based Control .......................... 606

14.5 High-Gain Observers ............................ 612

14.5.1 Motivating Example ....................... 614

14.5.2 Stabilization ........................... 621

14.5.3 Regulation via Integral Control ................. 625

14.6 Exercises .................................. 627

A Mathematical Review     649

B Contraction Mapping     655

C Proofs     659

C.1 Proof of Theorems 3.1 and 3.2 ...................... 659

C.2 Proof of Lemma 3.4 ............................ 661

C.3 Proof of Lemma 4.1 ............................ 663

C.4 Proof of Lemma 4.3 ............................ 664

C.5 Proof of Lemma 4.4 ............................ 664

C.6 Proof of Lemma 4.5 ............................ 665

C.7 Proof of Theorem 4.16 .......................... 667

C.8 Proof of Theorem 4.17 .......................... 671

C.9 Proof of Theorem 4.18 .......................... 677

C.10 Proof of Theorem 5.4 ........................... 678

C.11 Proof of Lemma 6.1 ............................ 679

C.12 Proof of Lemma 6.2 ............................ 682

C.13 Proof of Lemma 7.1 ............................ 686

C.14 Proof of Theorem 7.4 ........................... 690

C.15 Proof of Theorems 8.1 and 8.3 ...................... 692

C.16 Proof of Lemma 8.1 ............................ 701

C.17 Proof of Theorem 11.1 .......................... 702

C.18 Proof of Theorem 11.2 .......................... 708

C.19 Proof of Theorem 12.1 .......................... 710

C.20 Proof of Theorem 12.2 .......................... 711

C.21 Proof of Theorem 13.1 .......................... 712

C.22 Proof of Theorem 13.2 .......................... 714

C.23 Proof of Theorem 14.6 .......................... 715

Note and References     721

Bibliography     726

Symbols     742

Index     744