Systems and Control
Broad in scope, this text 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.
TABLE OF CONTENTS Systems and Control
Chapter 2: Analysis of Modeling Equations
To be sure, mathematics can be extended to any branch of knowledge, including economics, provided the concepts are so clearly defined as to permit accurate symbolic representation. That is only another way of saying that in some branches of discourse it is desirable to know what you are talking about.
James R. Newman
Overview
Most of dynamical systems analyzed in this book are modeled by ordinary differential equations. A main use of a mathematical model is to predict the system transient behavior. Unlike linear systems, where closed-form solutions can be written in terms of the system's eigenvalues and eigenvectors, finding analytical, exact, solutions to nonlinear differential equations can be very difficult or impossible. However, we can approximate the solutions of differential equations with difference equations whose solutions can be obtained easily using a computer. In this chapter we discuss a number of methods for solving differential equations. The first class of methods allows us to graphically determine solutions to second-order differential equations. Then, we discuss numerical techniques for solving differential equations. After that, we present two methods of linear approximation of nonlinear systems.
2.1 State-Plane Analysis
The state plane, which is a state space of two-dimensional systems, is also called the phase plane. Analysis in the state plane is applicable to linear and nonlinear systems modeled by second-order ordinary differential equations. The state-plane methods are graphical procedures for solving such equations. Using state-plane methods, one can graphically determine the transient response of a second-order dynamical system. We now...
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