Unformatted text preview: p urposes of t his c hapter. 13.1 S tate-Space Analysis
So far we have been describing systems in terms of e quations r elating certain
o utput t o a n i nput ( the i nput-output r elationship). T his t ype o f description is
a n external description o f a system (system viewed from t he i nput a nd o utput
t erminals). As n oted in C hapter 1, such a description may b e i nadequate in some
cases, a nd we need a systematic way of finding system's internal description. S tate
s pace analysis of systems meets this need. I n t his method, we first select a s et o f
k ey variables, called t he s tate v ariables, in t he system. T he s tate variables have
t he p roperty t hat every possible signal o r v ariable in t he s ystem a t a ny i nstant t
c an b e expressed in t erms o f the s tate variables a nd t he i nput(s) a t t hat i nstant t.
I f w e know all t he s tate variables as a function o f t, we c an determine every possible
signal or variable in t he s ystem a t a ny i nstant w ith a relatively simple relationship.
T he s ystem description in this method consists o f two parts:
2 F inding t he e quation(s) relating t he s tate variables t o t he i nput(s) ( the s tate
F inding the o utput variables in terms of t he s tate variables ( the o utput e quation). T he analysis procedure, therefore, consists of solving t he s tate e quation first,
a nd t hen solving t he o utput equation. T he s tate space description is c apable of
determining every possible system variable (or o utput) from t he knowledge of t he
i nput a nd t he initial s tate (conditions) of t he system. For this reason it is a n internal
description o f t he s ystem.
B y i ts nature, t he s tate variable analysis is eminently s uited for multiple-input,
multiple-output (MIMO) systems. I n a ddition, t he s tate-space techniques are useful
for several other reasons, including t he following: 1 . Time-varying p arameter systems a nd nonlinear systems can be characterized
effectively with state-space descriptions.
2 . S tate equations lend themselves readily t o a ccurate simulation on analog or
3 . For second-order systems (n = 2), a graphical m ethod called p hase-plane
a nalysis c an b e used on s tate e quations, whether t hey a re linear o r nonlinear. Introduction F rom t he discussion in C hapter 1, we know t hat t o d etermine a system's response(s) a t a ny i nstant t, we need t o know t he s ystem's i nputs d uring its entire
past, from - 00 t o t . I f t he i nputs a re known only for t > to, we c an still determine
t he s ystem o utput(s) for any t > to, p rovided we know certain initial conditions
in t he s ystem a t t = to. T hese initial conditions collectively a re called t he i nitial
s tate o f the s ystem ( at t = to).
The s tate o f a s ystem a t a ny i nstant to is the smallest s et o f numbers X l (to),
X2(tO), . .. , xn(tO) which is sufficient t o determine the behavior o f t he s ystem for
all time t > to when the input(s) to the system is known for t > to. T he variables
X l, X2, . .. , x n a re known as s tate v ariables.
T he i nitial conditions o f a s ystem...
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