Page 188 - Modern Control Systems

162 Chapter 3 State Variable Models

3.1 INTRODUCTION

In the preceding chapter, we developed and studied several useful approaches to
the analysis and design of feedback systems. The Laplace transform was used to
transform the differential equations representing the system to an algebraic
equation expressed in terms of the complex variable s. Using this algebraic equa-
tion, we were able to obtain a transfer function representation of the input-output
relationship.
The ready availability of digital computers makes it practical to consider the time-
domain formulation of the equations representing control systems. The time-domain
techniques can be used for nonlinear, time-varying, and multivariable systems.

A time-varying control system is a system in which one or more of the
parameters of the system may vary as a function of time.

For example, the mass of a missile varies as a function of time as the fuel is ex-
pended during flight. A multivariable system, as discussed in Section 2.6, is a system
with several input and output signals.
The solution of a time-domain formulation of a control system problem is facili-
tated by the availability and ease of use of digital computers. Therefore we are in-
terested in reconsidering the time-domain description of dynamic systems as they
are represented by the system differential equation. The time domain is the mathe-
matical domain that incorporates the response and description of a system in terms
of time, t.
The time-domain representation of control systems is an essential basis for modern
control theory and system optimization. In Chapter 11, we will have an opportunity
to design an optimum control system by utilizing time-domain methods. In this
chapter, we develop the time-domain representation of control systems and illus-
trate several methods for the solution of the system time response.

3.2 THE STATE VARIABLES OF A DYNAMIC SYSTEM

The time-domain analysis and design of control systems uses the concept of the
state of a system [1-3,5].

The state of a system is a set of variables whose values, together with the input
signals and the equations describing the dynamics, will provide the future state
and output of the system.

For a dynamic system, the state of a system is described in terms of a set of state
variables [*i(f)» x 2{t), • • • > x n(t)]. Th e s t a t e variables are those variables that deter-
mine the future behavior of a system when the present state of the system and the
excitation signals are known. Consider the system shown in Figure 3.1, where y\{i)