Nonlinear Systems: Stability, Dynamics And Control

✍ Scribed by Guanrong Chen

• Publisher: WSPC
• Year: 2024
• Tongue: English
• Leaves: 235
• Category: Library

✦ Synopsis

The topic of nonlinear systems is fundamental to the study of systems engineering. So extensive investigations have been carried out by both the nonlinear control and nonlinear dynamics communities, but the focus can be different — on controllers design and dynamics analysis, respectively. The last two decades have witnessed the gradual merging of control theory and dynamics analysis, but not yet to the extent of controlling nonlinear dynamics such as bifurcations and chaos. This monograph is an attempt to fill that gap while presenting a rather comprehensive coverage of the fundamental nonlinear systems theory in a self-contained and approachable manner. This introductory treatise is written for self-study and, in particular, as an elementary textbook that can be taught in a one-semester course to advanced undergraduates or entrance level graduates with curricula focusing on nonlinear systems, both on control theory and dynamics analysis.

✦ Table of Contents

Contents
Preface
1. Nonlinear Systems: Preliminaries
1.1 A Typical Nonlinear Dynamical Model
1.2 Autonomous Systems and Map Iterations
1.3 Dynamical Analysis on Phase Planes
1.3.1 Phase Plane of a Planar System
1.3.2 Analysis on Phase Planes
1.4 Qualitative Behaviors of Dynamical Systems
1.4.1 Qualitative Analysis of Linear Dynamics
1.4.2 Qualitative Analysis of Nonlinear Dynamics
2. Stabilities of Nonlinear Systems (I)
2.1 Lyapunov Stabilities
2.2 Lyapunov Stability Theorems
2.3 LaSalle Invariance Principle
2.4 Some Instability Theorems
2.5 Construction of Lyapunov Functions
2.6 Stability Regions: Basins of Attraction
3. Stabilities of Nonlinear Systems (II)
3.1 Linear Stability of Nonlinear Systems
3.2 Linear Stability of Nonlinear Systems with Periodic Linearity
3.3 Comparison Principles
3.3.1 Comparison Principle on the Plane
3.3.1.1 Comparison of zero points
3.3.1.2 Comparison of functions
3.3.2 Comparison Principle in Higher-Dimensional Spaces
3.4 Orbital Stability
3.5 Structural Stability
3.6 Total Stability: Stability under Persistent Disturbances
4. Stabilities of Nonlinear Systems (III)
4.1 Lur’e Systems Formulated in the Frequency Domain
4.2 Absolute Stability and Frequency-Domain Criteria
4.2.1 Background and Motivation
4.2.2 SISO Lur’e Systems
4.2.3 MIMO Lur’e Systems
4.3 Harmonic Balance Approximation and Describing Function
4.4 BIBO Stability
4.4.1 Small Gain Theorem
4.4.2 Relation between BIBO and Lyapunov Stabilities
4.4.3 Contraction Mapping Theorem
5. Nonlinear Dynamics: Bifurcations and Chaos
5.1 Typical Bifurcations
5.2 Period-Doubling Bifurcation
5.3 Hopf Bifurcations in 2-Dimensional Systems
5.3.1 Hyperbolic Systems and Normal Form Theorem
5.3.2 Decoupled 2-Dimensional Systems
5.3.3 Hopf Bifurcation of 2-Dimensional Systems
5.4 Poincaré Maps
5.5 Strange Attractors and Chaos
5.5.1 Chaotic Lorenz System
5.5.2 Poincaré–Bendixson Theorem
5.5.3 Some Other Chaotic Systems
5.5.4 Characterizations of Chaos
5.5.4.1 Lyaponov Exponent: Continuous-Time Systems
5.5.4.2 Lyaponov Exponent: Discrete-Time Systems
5.6 Chaos in Discrete-Time Systems
6. Nonlinear Systems Control
6.1 Feedback Control of Nonlinear Systems
6.1.1 Engineering Perspectives on Controllers Design
6.1.2 A General Approach to Controllers Design
6.2 Feedback Controllers for Nonlinear Systems
6.2.1 Linear Controllers for Nonlinear Systems
6.2.2 Nonlinear Controllers for Nonlinear Systems
6.2.3 General Ideas for Nonlinear Controllers Design
6.3 More about Nonlinear Controllers Design
6.3.1 An Illustrative Example
6.3.2 Adaptive Controllers
6.3.2.1 Observer-Based Adaptive Control
6.3.2.2 Adaptive Control of Uncertain Linear Systems
6.3.2.3 Adaptive Control of Uncertain Nonlinear Systems
6.3.3 Lyapunov Redesign of Nonlinear Controllers
6.3.4 Sliding-Mode Control
6.3.4.1 Sliding-Mode Surface Design
6.3.4.2 Sliding-Mode Controller Design
6.3.4.3 Chattering and its Attenuation
6.4 Controlling Bifurcations and Chaos
6.4.1 Controlling Bifurcations
6.4.1.1 Controlling Discrete-Time Nonlinear Systems
6.4.1.2 State-Feedback Control of Bifurcations
6.4.1.3 Controlling Period-Doubling Bifurcation
6.4.1.4 Controlling Hopf Bifurcation
6.4.1.5 More about Bifurcation Control
6.4.2 Controlling Chaos
6.4.2.1 Chaotification Problem Description
6.4.2.2 A General Chaotification Algorithm
6.4.2.3 A Modified Chaotification Algorithm
6.4.2.4 An Illustrative Example
Bibliography
Index