Page 386 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 71 STATE SPACE ANALYSIS

Rearranging Eq. (7.36a), we have
(21 - A)Q(z) = zq(0) + bX(z) (7.37)

Premultiplying both sides of Eq. (7.37) by (zI - A)-' yields
+
Q ) = 1 - A)-~Z~(O) (21 - A)-'~x(z) (7.38)
Hence, taking the inverse unilateral z-transform of Eq. (7.38), we get

q[n] = ~;'((ZI - A)-'z)~(o) + ~~'((~1 A)-'b~(z)) (7.39)
-
Substituting Eq. (7.39) into Eq. (7.2261, we get
y[n]=~~;~((z1-~)~'z)~(0)+~~;'((~1-~)~'b~(z))+dr[n] (7.40)

A comparison of Eq. (7.39) with Eq. (7.23) shows that

D. System Function H(z):
In Sec. 4.6 the system function H(z) of a discrete-time LTI system is defined by
H(z) = Y(z)/X(z) with zero initial conditions. Thus, setting q[O] = 0 in Eq. (7.38), we have

Q(Z) = (21 - A)-'~x(z)
The substitution of Eq. (7.42) into Eq. (7.36b) yields

Y(z) = [c(z1- A)-'b + d]x(z)

Thus,

E. Stability:
From Eqs. (7.25) and (7.29) or (7.34) we see that if the magnitudes of all eigenvalues
A, of the system matrix A are less than unity, that is,

bkl < 1 all k (7.45)
then the system is said to be asymptotically stable; that is, if, undriven, its state tends to
zero from any finite initial state q,. It can be shown that if all eigenvalues of A are distinct
and satisfy the condition (7.45), then the system is also BIB0 stable.