Nonlinear Systems

Title page
Preface
1 Introduction
1.1 Nonlinear Models and Nonlinear Phenomena
1.2 Examples
1.2.1 Pendulum Equation
1.2.2 Tunnel-Diode Circuit
1.2.3 Mass-Spring System
1.2.4 Negative-Resistance Oscillator
1.2.5 Artificial Neural Network
1.2.6 Adaptive Control
1.2.7 Common Nonlinearities
1.3 Exercises
2 Second-Order Systems
2.1 Qualitative Behavior of Linear Systems
2.2 Multiple Equilibria
2.3 Qualitative Behavior Near Equilibrium Points
2.4 Limit Cycles
2.5 Numerical Construction of Phase Portraits
2.6 Existence of Periodic Orbits
2.7 Bifurcation
2.8 Exercises
3 Fundamental Properties
3.1 Existence and Uniqueness
3.2 Continuous Dependence on Initial Conditions and Parameters
3.3 Differentiability of Solutions and Sensitivity Equations
3.4 Comparison Principle
3.5 Exercises
4 Lyapunov Stability
4.1 Autonomous Systems
4.2 The Invariance Principle
4.3 Linear Systems and Linearization
4.4 Comparison Functions
4.5 Nonautonomous Systems
4.6 Linear Time-Varying Systems and Linearization
4.7 Converse Theorems
4.8 Boundedness and Ultimate Boundedness
4.9 Input-to-State Stability
4.10 Exercises
5 Input-Output Stability
5.1 L Stability
5.2 L Stability of State Models
5.3 L₂ Gain
5.4 Feedback Systems: The Small-Gain Theorem
5.5 Exercises
6 Passivity
6.1 Memoryless Functions
6.2 State Models
6.3 Positive Real Transfer Functions
6.4 L₂ and Lyapunov Stability
6.5 Feedback Systems: Passivity Theorems
6.6 Exercises
7 Frequency Domain Analysis of Feedback Systems
7.1 Absolute stability
7.1.1 Circle Criterion
7.1.2 Popov Criterion
7.2 The Describing Function Method
7.3 Exercises
8 Advanced Stability Analysis
8.1 The Center Manifold Theorem
8.2 Region of Attraction
8.3 Invariance-like Theorems
8.4 Stability of Periodic Solutions
8.5 Exercises
9 Stability of Perturbed Systems
9.1 Vanishing Perturbation
9.2 Nonvanishing Perturbation
9.3 Comparison Method
9.4 Continuity of Solutions on the Infinite Interval
9.5 Interconnected Systems
9.6 Slowly Varying Systems
9.7 Exercises
10 Perturbation Theory and Averaging
10.1 The Perturbation Method
10.2 Perturbation on the Infinite Interval
10.3 Periodic Perturbation of Autonomous Systems
10.4 Averaging
10.5 Weakly Nonlinear Second-Order Oscillators
10.6 General Averaging
10.7 Exercises
11 Singular Perturbations
11.1 The Standard Singular Perturbation Model
11.2 Time-scale Properties of the Standard Model
11.3 Singular Perturbation on the Infinite Interval
11.4 Slow and Fast Manifolds
11.5 Stability Analysis
11.6 Exercises
12 Feedback control
12.1 Control Problems
12.2 Stabilization via Linearization
12.3 Integral Control
12.4 Integral Control via Linearization
12.5 Gain Scheduling
12.6 Exercises
13 Feedback Linearization
13.1 Motivation
13.2 Input-Output Linearization
13.3 Full-State Linearization
13.4 State Feedback Control
13.4.1 Stabilization
13.4.2 Tracking
13.5 Exercises
14 Nonlinear Design Tools
14.1 Sliding Mode Control
14.1.1 Motivating Example
14.1.2 Stabilization
14.1.3 Tracking
14.1.4 Regulation via Integral Control
14.2 Lyapunov Redesign
14.2.1 Stabilization
14.2.2 Nonlinear Damping
14.3 Backstepping
14.4 Passivity-Based Control
14.5 High-Gain Observers
14.5.1 Motivating Example
14.5.2 Stabilization
14.5.3 Regulation via Integral Control
14.6 Exercises
A Mathematical Review
B Contraction Mapping
C Proofs
C.1 Proof of Theorems 3.1 and 3.2
C.2 Proof of Lemma 3.4
C.3 Proof of Lemma 4.1
C.4 Proof of Lemma 4.3
C.5 Proof of Lemma 4.4
C.6 Proof of Lemma 4.5
C.7 Proof of Theorem 4.16
C.8 Proof of Theorem 4.17
C.9 Proof of Theorem 4.18
C.10 Proof of Theorem 5.4
C.11 Proof of Lemma 6.1
C.12 Proof of Lemma 6.2
C.13 Proof of Lemma 7.1
C.14 Proof of Theorem 7.4
C.15 Proof of Theorems 8.1 and 8.3
C.16 Proof of Lemma 8.1
C.17 Proof of Theorem 11.1
C.18 Proof of Theorem 11.2
C.19 Proof of Theorem 12.1
C.20 Proof of Theorem 12.2
C.21 Proof of Theorem 13.1
C.22 Proof of Theorem 13.2
C.23 Proof of Theorem 14.6
Note and References
Bibliography
Symbols
Index