Systems and Control

Description

Systems and Control presents modeling, analysis, and control of dynamical systems. Introducing students to the basics of dynamical system theory and supplying them with the tools necessary for control system design, it emphasizes design and demonstrates how dynamical system theory fits into practical applications. Classical methods and the techniques of postmodern control engineering are presented in a unified fashion, demonstrating how the current tools of a control engineer can supplement more classical tools.
Broad in scope, Systems and Control shows the multidisciplinary role of dynamics and control; presents neural networks, fuzzy systems, and genetic algorithms; and provides a self-contained introduction to chaotic systems. The text employs Lyapunov's stability theory as a unifying medium for different types of dynamical systems, using it--with its variants--to analyze dynamical system models. Specifically, optimal, fuzzy, sliding mode, and chaotic controllers are all constructed with the aid of the Lyapunov method and its extensions. In addition, a class of neural networks is also analyzed using Lyapunov's method.
Ideal for advanced undergraduate and beginning graduate courses in systems and control, this text can also be used for introductory courses in nonlinear systems and modern automatic control. It requires working knowledge of basic differential equations and elements of linear algebra; a review of the necessary mathematical techniques and terminology is provided.

Table of Contents

Each chapter ends with Notes and Exercises
Preface
1. Dynamical Systems and Modeling
1.1. What Is a System?
1.2. Open-Loop Versus Closed-Loop
1.3. Axiomatic Definition of a Dynamical System
1.4. Mathematical Modeling
1.5. Review of Work and Energy Concepts
1.6. The Lagrange Equations of Motion
1.7. Modeling Examples
1.7.1. Centrifugal Governor
1.7.2. Ground Vehicle
1.7.3. Permanent Magnet Stepper Motor
1.7.4. Stick Balancer
1.7.5. Population Dynamics
2. Analysis of Modeling Equations
2.1. State Plane Analysis
2.1.1. Examples of Phase Portraits
2.1.2. The Method of Isoclines
2.2. Numerical Techniques
2.2.1. The Method of Taylor Series
2.2.2. Euler's Methods
2.2.3. Predictor-Corrector Method
2.2.4. Runge's Method
2.2.5. Runge-Kutta Method
2.3. Principles of Linearization
2.4. Linearizing Differential Equations
2.5. Describing Function Method
2.5.1. Scalar Product of Functions
2.5.2. Fourier Series
2.5.3. Describing Function in the Analysis of Nonlinear Systems
3. Linear Systems
3.1. Reachability and Controllability
3.2. Observability and Constructability
3.3. Companion Forms
3.3.1. Controller Form
3.3.2. Observer Form
3.4. Linear State-Feedback Control
3.5. State Estimators
3.5.1. Full-Order Estimator
3.5.2. Reduced-Order Estimator
3.6. Combined Controller-Estimator Compensator
4. Stability
4.1. Informal Introduction to Stability
4.2. Basic Definitions of Stability
4.3. Stability of Linear Systems
4.4. Evaluating Quadratic Indices
4.5. Discrete-Time Lyapunov Equation
4.6. Constructing Robust Linear Controllers
4.7. Hurwitz and Routh Stability Criteria
4.8. Stability of Nonlinear Systems
4.9. Lyapunov's Indirect Method
4.10. Discontinuous Robust Controllers
4.11. Uniform Ultimate Boundedness
4.12. Lyapunov-Like Analysis
4.13. LaSalle's Invariance Principle
5. Optimal Control
5.1. Performance Indices
5.2. A Glimpse at the Calculus of Variations
5.2.1. Variation and Its Properties
5.2.2. Euler-Lagrange Equation
5.3. Linear Quadratic Regulator
5.3.1. Algebraic Riccati Equation (ARE)
5.3.2. Solving the ARE Using the Eigenvector Method
5.3.3. Optimal Systems with Prescribed Poles
5.3.4. Optimal Saturating Controllers
5.3.5. Linear Quadratic Regulator for Discrete Systems on an Infinite Time Interval
5.4. Dynamic Programming
5.4.1. Discrete-Time Systems
5.4.2. Discrete Linear Quadratic Regulator Problem
5.4.3. Continuous Minimum Time Regulator Problem
5.4.4. The Hamilton-Jacobi-Bellman Equation
5.5. Pontryagin's Minimum Principle
5.5.1. Optimal Control With Constraints on Inputs
5.5.2. A Two-Point Boundary-Value Problem
6. Sliding Modes
6.1. Simple Variable Structure Systems
6.2. Sliding Mode
6.3. A Simple Sliding Mode Controller
6.4. Sliding in Multi-Input Systems
6.5. Sliding Mode and System Zeros
6.6. Nonideal Sliding Mode
6.7. Sliding Surface Design
6.8. State Estimation of Uncertain Systems
6.8.1. Discontinuous Estimators
6.8.2. Boundary Layer Estimators
6.9. Sliding Modes in Solving Optimization Problems
6.9.1. Optimization Problem Statement
6.9.2. Penalty Function Method
6.9.3. Dynamical Gradient Circuit Analysis
7. Vector Field Methods
7.1. A Nonlinear Plant Model
7.2. Controller Form
7.3. Linearizing State-Feedback Control
7.4. Observer Form
7.5. Asymptotic State Estimator
7.6. Combined Controller-Estimator Compensator
8. Fuzzy Systems
8.1. Motivation and Basic Definitions
8.2. Fuzzy Arithmetic and Fuzzy Relations
8.2.1. Interval Arithmetic
8.2.2. Manipulating Fuzzy Numbers
8.2.3. Fuzzy Relations
8.2.4. Composition of Fuzzy Relations
8.3. Standard Additive Model
8.4. Fuzzy Logic Control
8.5. Stabilization Using Fuzzy Models
8.5.1. Fuzzy Modeling
8.5.2. Constructing a Fuzzy Design Model Using a Nonlinear Model
8.5.3. Stabilizability of Fuzzy Models
8.5.4. A Lyapunov-Based Stabilizer
8.6. Stability of Discrete Fuzzy Models
8.7. Fuzzy Estimator
8.7.1. The Comparison Method for Linear Systems
8.7.2. Stability Analysis of the Closed-Loop System
8.8. Adaptive Fuzzy Control
8.8.1. Plant Model and Control Objective
8.8.2. Background Results
8.8.3. Controllers
8.8.4. Examples
9. Neural Networks
9.1. Threshold Logic Unit
9.2. Identification Using Adaptive Linear Element
9.3. Backpropagation
9.4. Neural Fuzzy Identifier
9.5. Radial-Basis Function (RBF) Networks
9.5.1. Interpolation Using RBF Networks
9.5.2. Identification of a Single-Input, Single-State System
9.5.3. Learning Algorithm for the RBF Identifier
9.5.4. Growing RBF Network
9.5.5. Identification of Multivariable Systems
9.6. A Self-Organizing Network
9.7. Hopfield Neural Network
9.7.1. Hopfield Neural Network Modeling and Analysis
9.7.2. Analog-to-Digital Converter
9.8. Hopfield Network Stability Analysis
9.8.1. Hopfield Network Model Analysis
9.8.2. Single Neuron Stability Analysis
9.8.3. Stability Analysis of the Network
9.9. Brain-State-in-a-Box (BSB) Models
9.9.1. Associative Memories
9.9.2. Analysis of BSB Models
9.9.3. Synthesis of Neural Associative Memory
9.9.4. Learning
9.9.5. Forgetting
10. Genetic and Evolutionary Algorithms
10.1. Genetics as an Inspiration for an Optimization Approach
10.2. Implementing a Canonical Genetic Algorithm
10.3. Analysis of the Canonical Genetic Algorithm
10.4. Simple Evolutionary Algorithm (EA)
10.5. Evolutionary Fuzzy Logic Controllers
10.5.1. Vehicle Model and Control Objective
10.5.2. Case 1: EA Tunes Fuzzy Rules
10.5.3. Case 2: EA Tunes Fuzzy Membership Functions
10.5.4. Case 3: EA Tunes Fuzzy Rules and Membership Functions
11. Chaotic Systems and Fractals
11.1. Chaotic Systems Are Dynamical Systems with Wild Behavior
11.2. Chaotic Behavior of the Logistic Equation
11.2.1. The Logistic Equation---An Example From Ecology
11.2.2. Stability Analysis of the Logistic Map
11.2.3. Period Doubling to Chaos
11.2.4. The Feigenbaum Numbers
11.3. Fractals
11.3.1. The Mandelbrot Set
11.3.2. Julia Sets
11.3.3. Iterated Function Systems
11.4. Lyapunov Exponents
11.5. Discretization Chaos
11.6. Controlling Chaotic Systems
11.6.1. Ingredients of Chaotic Control Algorithm
11.6.2. Chaotic Control Algorithm
Appendix: Math Review
A.1. Notation and Methods of Proof
A.2. Vectors
A.3. Matrices and Determinants
A.4. Quadratic Forms
A.5. The Kronecker Product
A.6. Upper and Lower Bounds
A.7. Sequences
A.8. Functions
A.9. Linear Operators
A.10. Vector Spaces
A.11. Least Squares
A.12. Contraction Maps
A.13. First-Order Differential Equation
A.14. Integral and Differential Inequalities
A.14.1. The Bellman-Gronwall Lemma
A.14.2. A Comparison Theorem
A.15. Solving the State Equations
A.15.1. Solution of Uncontrolled System
A.15.2. Solution of Controlled System
A.16. Curves and Surfaces
A.17. Vector Fields and Curve Integrals
A.18. Matrix Calculus Formulas
Notes
Exercises
Bibliography
Index