Nonlinear Control Systems : Analysis and Design

A thorough yet highly readable introduction to the complex world of nonlinear science

Today’s technological advances have brought about a growing demand for better performance. This, coupled with the availability of low-cost computing power, has led control engineers to face problems of increasingly higher complexity. Consequently, the linear approximations once used to analyze these problems are giving way to more accurate and realistic nonlinear models, forcing both students and industry practitioners to abandon the peripheral view of the linear world and immerse themselves in the reality of nonlinear science. Nonlinear Control Systems: Analysis and Design addresses the need for an up-to-date yet readable pedagogical presentation of this difficult subject. Assuming no previous background on the subject, the author takes readers from the very basics to some of the most current research topics being addressed today.

Highlights of the text include:

• Complete yet concise coverage of both the Lyapunov and Input-Output stability theories
• An introduction to the popular backstepping approach to nonlinear control design
• Thorough discussion of the concept of input-to-state stability
• Coverage of the fundamentals of feedback linearization and related results
• Detailed coverage of the fundamentals of dissipative systems theory and its application in the so-called L2gain control problem
• In-depth discussion of nonlinear observers, a very important problem not commonly covered in introductory textbooks

The author’s friendly, accessible treatment of even highly complex topics makes this text an invaluable resource for students and professionals alike.

Table of Contents

Introduction.

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.