Nonlinear Control Systems

1.1 Linear Time-Invariant Systems.

1.2 Nonlinear Systems.

1.3 Equilibrium Points.

1.4 First-Order Autonomous Nonlinear Systems.

1.5 Second-Order Systems: Phase-Plane Analysis.

1.6 Phase-Plane Analysis of Linear Time-Invariant Systems.

1.7 Phase-Plane Analysis of Nonlinear Systems.

1.8 Higher-Order Systems.

1.9 Examples of Nonlinear Systems.

1.10 Exercises.

Mathematical Preliminaries.

2.1 Sets.

2.2 Metric Spaces.

2.3 Vector Spaces.

2.4 Matrices.

2.5 Basic Topology.

2.6 Sequences.

2.7 Functions.

2.8 Differentiability.

2.9 Lipschitz Continuity.

2.10 Contraction Mapping.

2.11 Solution of Differential Equations.

2.12 Exercises.

Lyapunov Stability I: Autonomous Systems.

3.1 Definitions.

3.2 Positive Definite Functions.

3.3 Stability Theorems.

3.4 Examples.

3.5 Asymptotic Stability in the Large.

3.6 Positive Definite Functions Revisited.

3.7 Construction of Lyapunov Functions.

3.8 The Invariance Principle.

3.9 Region of Attraction.

3.10 Analysis of Linear Time-Invariant Systems.

3.11 Instability.

3.12 Exercises.

Lyapunov Stability II: Nonautonomous Systems.

4.1 Definitions.

4.2 Positive Definite Functions.

4.3 Stability Theorems.

4.4 Proof of the Stability Theorems.

4.5 Analysis of Linear Time-Varying Systems.

4.6 Perturbation Analysis.

4.7 Converse Theorems.

4.8 Discrete-Time Systems.

4.9 Discretization.

4.10 Stability of Discrete-Time Systems.

4.11 Exercises.

Feedback Systems.

5.1 Basic Feedback Stabilization.

5.2 Integrator Backstepping.

5.3 Backstepping: More General Cases.

5.4 Examples.

5.5 Exercises.

Input-Output Stability.

6.1 Function Spaces.

6.2 Input-Output Stability.

6.3 Linear Time-Invariant Systems.

6.4 L_p  Gains for LTI Systems.

6.5 Closed Loop Input-Output Stability.

6.6 The Small Gain Theorem.

6.7 Loop Transformations.

6.8 The Circle Criterion.

6.9 Exercises.

Input-to-State Stability.

7.1 Motivation.

7.2 Definitions.

7.3 Input-to-State Stability (ISS) Theorems.

7.4 Input-to-State Stability Revisited.

7.5 Cascade Connected Systems.

7.6 Exercises.

Passivity.

8.1 Power and Energy: Passive Systems.

8.2 Definitions.

8.3 Interconnections of Passivity Systems.

8.4 Stability of Feedback Interconnections.

8.5 Passivity of Linear Time-Invariant Systems.

8.6 Strictly Positive Real Rational Functions.

Exercises.

Dissipativity.

9.1 Dissipative Systems.

9.2 Differentiable Storage Functions.

9.3 QSR Dissipativity.

9.4 Examples.

9.5 Available Storage.

9.6 Algebraic Condition for Dissipativity.

9.7 Stability of Dissipative Systems.

9.8 Feedback Interconnections.

9.9 Nonlinear L₂ Gain.

9.10 Some Remarks about Control Design.

9.11 Nonlinear L₂-Gain Control.

9.12 Exercises.

Feedback Linearization.

10.1 Mathematical Tools.

10.2 Input-State Linearization.

10.3 Examples.

10.4 Conditions for Input-State Linearization.

10.5 Input-Output Linearization.

10.6 The Zero Dynamics.

10.7 Conditions for Input-Output Linearization.

10.8 Exercises.

Nonlinear Observers.

11.1 Observers for Linear Time-Invariant Systems.

11.2 Nonlinear Observability.

11.3 Observers with Linear Error Dynamics.

11.4 Lipschitz Systems.

11.5 Nonlinear Separation Principle.

Proofs.

Bibliography.

List of Figures.

Index.