State-Variable Representation - Control of Color Imaging Systems: Analysis and Design - page 176

4 State-Variable

Representation

4.1 INTRODUCTION
Modem control system theory is based on the state-space formulation of an under-
lying system. It provides a uni

ed approach for system modeling and design that is
applicable to both linear and nonlinear systems, time-varying and time-invariant
systems as well as single-input single-output (SISO) and multiple-input multiple-
output (MIMO) systems [1]. In this chapter, we present system modeling in state
space, solution of state equations, and controllability and observability of linear time-
invariant (LTI) systems.

4.2 CONCEPT OF STATES
The state-space approach is a uni ed approach for representation of both continuous-

and discrete-time dynamical systems. It covers a broad range of systems from
nonlinear to linear, time-invariant to time-varying systems. It can also be used in
modeling stochastic dynamic systems. In general terms, the states of a dynamic
system is the minimum number of variables called state variables such that know-
ledge of these states at any given time together with the input at that time and future
time uniquely determines the behavior of the system past that time [2].

4.3 STATE-SPACE REPRESENTATION OF CONTINUOUS-TIME
SYSTEMS

4.3.1 D EFINITION OF S TATE
The state of a continuous-time dynamic system is the minimum number of variables
called state variables such that the knowledge of these variables at time t ¼ t 0
together with the input for t t 0 uniquely determines the behavior of the system
for t t 0 .IfN variables are needed, then these N variables are considered as
components of an N-dimensional vector x called the state vector. The N-dimensional
space whose coordinates are the states of the system is called state space. The state of
a system at time t is a point in the N-dimensional state space [2,3].

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