Page 432 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 71 STATE SPACE ANALYSIS

Now from Eq. (7.65), for x(t ) = 0 and to = 0,

However, by the Cayley-Hamilton theorem eA' can be expressed as

Substituting Eq. (7.136) into Eq. (7.135), we get

in view of Eq. (7.134). Thus, qo is indistinguishable from the zero state and hence the system is
not observable. Therefore, if the system is to be observable, then Mo must have rank N.

7.53. Consider the system in Prob. 7.50.
(a) Is the system controllable?
(6) Is the system observable?
(a) From the result from Prob. 7.50 we have

Now

and by Eq. (7.128) the controllability matrix is
M, = [b Ab] = [-1 -11

and IM,I = 0. Thus, it has a rank less than 2 and hence the system is not controllable.
(b) Similarly,

and by Eq. (7.133) the observability matrix is

and JMJ = - 2 # 0. Thus, its rank is 2 and hence the system is observable.
Note from the result from Prob. 7.50(b) that the system function H(s) has pole-zero
cancellation. As in the discrete-time case, if H(s) has pole-zero cancellation, then the
system cannot be both controllable and observable.

7.54. Consider the system shown in Fig. 7-22.

(a) Is the system controllable?
(b) Is the system observable?
(c) Find the system function H(s).