Page 380 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 71 STATE SPACE ANALYSIS

vector :

Then Eqs. (7.4~) and (7.46) can be rewritten compactly as

where

Equations ( 7.6~) and (7.66) are called an N-dimensional state space representation (or
state equations) of the system, and the N x N matrix A is termed the system matriu. The
solution of Eqs. (7.6~) and (7.66) for a given initial state is discussed in Sec. 7.5.

B. Similarity Transformation:
As mentioned before, the choice of state variables is not unique and there are infinitely
many choices of the state variables for any given system. Let T be any N X N nonsingular
matrix (App. A) and define a new state vector

v[n] = m[n] (7.7)
where q[n] is the old state vector which satisfies Eqs. (7.6~) and (7.66). Since T is
nonsingular, that is, T-I exists, and we have

q[n] = T-'v[n] (7.8)
Now
+
v[n + 11 = Tq[n + 11 = T(~q[n] bx[n])
= TAq[n] + Tbx[n] = TAT-'v[n] + Tbx[n] (7.9~)

Thus, if we let

A
b=Tb ;=(q-' i=d
then Eqs. (7.9~) and (7.9b) become

v[n + 11 = R[n]+ bx~nl