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Contents
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 L1 Stability of State Models
5£®3 L2 Gain
5£®4 Feedback Systems£º 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£º 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
Note and References
Bibliography
Symbols
Index