Page 420 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 71 STATE SPACE ANALYSIS

Rewriting Eq. (7.125) as a matrix equation, we get

Thus, to find a unique solution for q[O], the coefficient matrix of Eq. (7.126) must be
nonsingular; that is, the matrix

M, =

must have rank N.

7.35. Consider the system in Prob. 7.7

(a) Is the system controllable?
(b) Is the system observable?
(c) Find the system function H(z).
(a) From the result from Prob. 7.7 we have

Now

and by Eq. (7.120) the controllability matrix is

and IM,l = - 1 # 0. Thus, its rank is 2 and hence the system is controllable.
(b) Similarly,

and by Eq. (7.123) the observability matrix is

and (MoI = - & # 0. Thus, its rank is 2 and hence the system is observable.