Page 417 - Schaum's Outline of Theory and Problems of Signals and Systems

404 STATE SPACE ANALYSIS [CHAP. 7

Now from the result from Prob. 7.24 we have

and cAnq[0]= [-4 $1

= f + '(9)"
6 3

,.An- I -k b= [-) ;I

Thus,

1 1 1 " "-1 ] , n+l-k 1 I n
I + ( [ ( j ](i) +&)

which is the same result obtained in Prob. 4.38(b).

732. Consider the discrete-time LTI system shown in Fig. 7-19.

(a) Is the system asymptotically stable?
(b) Find the system function H(z).
(c) Is the system BIB0 stable?
(a) From Fig. 7-19 and choosing the state variables q,[n] and q,[n] as shown, we obtain
9,[n + 11 = tclz[n1 +x[nl

92[n + 11 = -$?l[n] +292[n]
Y ~ I
=q,[nl-9*[nl
In matrix form
q[n + 1] = Aq[n] + bx[n]
Y [ ~ I = cq[nl