Introduction to Modeling and Simulation

3System Fundamentals

3.1 Introduction

In the previous chapter we discussed continuous‐time second‐order systems. In this chapter we consider continuous‐time systems of any order. We discuss existence and uniqueness of solutions, linearization about hyperbolic equilibrium points, and the matrix exponential, which allows a compact representation of solutions of linear systems of any order. We then present a result, known as the Hartman–Grobman theorem, which relates the response of a continuous‐time nonlinear system in a neighborhood of a hyperbolic fixed point to the corresponding linear approximation about the fixed point. We also discuss singular perturbations, which are used to model systems with very large or very small parameters.

3.2 Existence and Uniqueness of Solution

In this section we consider the question of existence and uniqueness of solutions for an th‐order autonomous system of the form

where is a vector field on the state space . For an th‐order autonomous linear ...