Linear, Time-varying Approximations to Nonlinear Dynamical Systems: with Applications in Control and Optimization (Lecture Notes in Control and Information Sciences, 400)

✍ Scribed by Maria Tomas-Rodriguez, Stephen P. Banks

• Publisher: Springer
• Year: 2010
• Tongue: English
• Leaves: 303
• Category: Library

✦ Synopsis

Linear, Time-varying Approximations to Nonlinear Dynamical Systems introduces a new technique for analysing and controlling nonlinear systems. This method is general and requires only very mild conditions on the system nonlinearities, setting it apart from other techniques such as those – well-known – based on differential geometry. The authors cover many aspects of nonlinear systems including stability theory, control design and extensions to distributed parameter systems.

Many of the classical and modern control design methods which can be applied to linear, time-varying systems can be extended to nonlinear systems by this technique. The implementation of the control is therefore simple and can be done with well-established classical methods. Many aspects of nonlinear systems, such as spectral theory which is important for the generalisation of frequency domain methods, can be approached by this method.

✦ Table of Contents

Title Page
Preface
Contents
Introduction to Nonlinear Systems
Overview
Existence and Uniqueness
Logistic Systems
Control of Nonlinear Systems
Vector Fields on Manifolds
Nonlinear Partial Differential Equations
Conclusions and Outline of the Book
References
Linear Approximations to Nonlinear Dynamical Systems
Introduction
Linear, Time-varying Approximations
The Lorenz Attractor
Convergence Rate
Influence of the Initial Conditions on the Convergence
Notes on Different Configurations
Comparison with the Classical Linearisation Method
Conclusions
References
The Structure and Stability of Linear, Time-varying Systems
Introduction
Existence and Uniqueness
Explicit Solutions
Stability Theory
Lyapunov Exponents and Oseledec’s Theorem
Exponential Dichotomy and the Sacker-Sell Spectrum
Conclusions
References
General Spectral Theory of Nonlinear Systems
Introduction
A Frequency-domain Theory of Nonlinear Systems
Exponential Dichotomies
Conclusions
References
Spectral Assignment in Linear, Time-varying Systems
Introduction
Pole Placement for Linear, Time-invariant Systems
Pole Placement for Linear, Time-varying Systems
Generalisation to Nonlinear Systems
Application to F-8 Crusader Aircraft
Conclusions
References
Optimal Control
Introduction
Calculus of Variations and Classical Linear Quadratic Control
Nonlinear Control Problems
Examples
The Hamilton-Jacobi-Bellman Equation, Viscosity Solutions and Optimality
Characteristics of the Hamilton-Jacobi Equation
Conclusions
References
Sliding Mode Control for Nonlinear Systems
Introduction
Sliding Mode Control for Linear Time-invariant Systems
Sliding Mode Control for Linear Time-varying Systems
Generalisation to Nonlinear Systems
Conclusions
References
Fixed Point Theory and Induction
Introduction
Fixed Point Theory
Stability of Systems
Periodic Solutions
Conclusions
References
Nonlinear Partial Differential Equations
Introduction
A Moving Boundary Problem
Solution of the Unforced System
The Control Problem
Solitons and Boundary Control
Conclusions
References
Lie Algebraic Methods
Introduction
The Lie Algebra of a Differential Equation
Lie Groups and the Solution of the System
Solvable Systems
The Killing Form and Invariant Spaces
Compact Lie Algebras
Modal Control
Conclusions
References
Global Analysis on Manifolds
Introduction
Dynamical Systems on Manifolds
Local Reconstruction of Systems
Smooth Transition Between Operating Conditions
From Local to Global
Smale Theory
Two-dimensional Manifolds
Three-dimensional Manifolds
Four-dimensional Manifolds
Conclusions
References
Summary, Conclusions and Prospects for Development
Introduction
TravellingWave Solutions in Nonlinear Lattice Differential Equations
TravellingWaves
An Approach to the Solution
A Separation Theorem for Nonlinear Systems
Conclusions
References
Appendix
Linear Algebra
Vector Spaces
Linear Dependence and Bases
Subspaces and Quotient Spaces
Eigenspaces and the Jordan Form
References
Lie Algebras
Elementary Theory
Cartan Decompositions of Semi-simple Lie Algebras
Root Systems and Classification of Simple Lie Algebras
Compact Lie Algebras
References
Differential Geometry
Differentiable Manifolds
Tangent Spaces
Vector Bundles
Exterior Algebra and de Rham Cohomology
Degree and Index
Connections and Curvature
Characteristic Classes
References
Functional Analysis
Banach and Hilbert Spaces
Examples
Theory of Operators
Spectral Theory
Distribution Theory
Sobolev Spaces
Partial Differential Equations
Semigroup Theory
The Contraction Mapping and Implicit Function Theorems
References
Index