Nonlinear Systems
Second Edition
Hassan K. Khalil
Prentice Hall
The second edition is different from the first one in three aspects:
1. The material of the book has been reorganized into analysis vs
design chapters. The first nine chapters deal with nonlinear analysis
tools that apply to general nonlinear systems. Chapter 10 presents
analysis tools for feedback systems. The last three chapters deal with
control design for nonlinear systems.
2. There is an expanded coverage of some material as well as coverage of
new topics not included in the first edition. The new topics include:
Passivity approach to analysis of feedback systems.
Gain scheduling
Backstepping
Sliding mode control
The expanded coverage includes:
Input-output stability (including L2 gain)
Design via linearization (including output feedback and
integral control)
Feedback linearization (including input-output linearization
and differential geometric approach)
3. New exercises have been added, bringing the total number of exercises
to over 400, many of which are multiple-part exercises. A solution manual
is available from the publisher for instructors.
The complete table of contents is given below.
1 Introduction 1
1.1 Examples 5
1.1.1 Pendulum Equation 5
1.1.2 Tunnel Diode Circuit 7
1.1.3 Mass-Spring System 8
1.1.4 Negative-Resistance Oscillator 11
1.1.5 Artificial Neural Network 14
1.1.6 Adaptive Control 16
1.2 Second-Order Systems 19
1.2.1 Qualitative Behavior of Linear Systems 23
1.2.2 Multiple Equilibria 33
1.2.3 Qualitative Behavior Near Equilibrium Points 36
1.2.4 Limit Cycles 41
1.2.5 Numerical Construction of Phase Portraits 46
1.3 Exercises 47
2 Fundamental Properties 57
2.1 Mathematical Preliminaries 58
2.1.1 Euclidean Space 58
2.1.2 Mean Value and Implicit Function Theorems 62
2.1.3 Gronwall-Bellman Inequality 63
2.1.4 Contraction Mapping 64
2.2 Existence and Uniqueness 67
2.3 Continuous Dependence on Initial Conditions and Parameters 78
2.4 Differentiability of Solutions and Sensitivity Equations 81
2.5 Comparison Principle 84
2.6 Exercises 88
3 Lyapunov Stability 97
3.1 Autonomous Systems 98
3.2 The Invariance Principle 113
3.3 Linear Systems and Linearization 120
3.4 Nonautonomous Systems 132
3.5 Linear Time-Varying Systems and Linearization 143
3.6 Converse Theorems 148
3.7 Exercises 154
4 Advanced Stability Analysis 167
4.1 The Center Manifold Theorem 167
4.2 Region of Attraction 177
4.3 Invariance Theorems 191
4.4 Exercises 196
5 Stability of Perturbed Systems 203
5.1 Vanishing Perturbation 204
5.2 Nonvanishing Perturbation 211
5.3 Input-to-State Stability 217
5.4 Comparison Method 222
5.5 Continuity of Solutions on the Infinite Interval 228
5.6 Interconnected Systems 230
5.7 Slowly Varying Systems 239
5.8 Exercises 250
6 Input-Output Stability 261
6.1 L Stability 262
6.2 L Stability of State Models 269
6.3 Input-to-Output Stability 275
6.4 L_2 Gain 276
6.5 Exercises 284
7 Periodic Orbits 289
7.1 Second-Order Systems 289
7.2 Stability of Periodic Solutions 299
7.3 Exercises 309
8 Perturbation Theory and Averaging 313
8.1 The Perturbation Method 314
8.2 Perturbation on the Infinite Interval 326
8.3 Averaging 330
8.4 Weakly Nonlinear Second-Order Oscillators 339
8.5 General Averaging 342
8.6 Exercises 347
9 Singular Perturbations 351
9.1 The Standard Singular Perturbation Model 352
9.2 Time-Scale Properties of the Standard Model 358
9.3 Slow and Fast Manifolds 365
9.4 Stability Analysis 372
9.5 Singular Perturbation on the Infinite Interval 384
9.6 Exercises 388
10 Analysis of Feedback Systems 399
10.1 Absolute Stability 400
10.1.1 Circle Criterion 407
10.1.2 Popov Criterion 419
10.1.3 Simultaneous Lyapunov Functions 423
10.2 Small-Gain Theorem 430
10.3 Passivity Approach 436
10.4 The Describing Function Method 450
10.5 Exercises 468
11 Feedback Control 479
11.1 Control Problems 479
11.2 Design via Linearization 485
11.2.1 Stabilization 485
11.2.2 Regulation via Integral Control 488
11.3 Gain Scheduling 493
11.4 Exercises 509
12 Exact Feedback Linearization 519
12.1 Input-State Linearization 519
12.2 Input-Output Linearization 531
12.3 State Feedback Control 545
12.3.1 Stabilization 545
12.3.2 Tracking 552
12.3.3 Regulation via Integral Control 554
12.4 Differential Geometric Approach 558
12.4.1 Differential Geometric Tools 558
12.4.2 Input-Output Linearization 564
12.4.3 Input-State Linearization 567
12.5 Exercises 570
13 Lyapunov-Based Design 577
13.1 Lyapunov Redesign 578
13.1.1 Robust Stabilization 578
13.1.2 Nonlinear Damping 586
13.2 Backstepping 588
13.3 Sliding Mode Control 601
13.4 Adaptive Control 617
13.4.1 Model Reference Controller 618
13.4.2 Model Reference Adaptive Control 622
13.5 Exercises 640
Appendix A: Proofs 651
Notes and References 705
Bibliography 711
Symbols 725
Index 727