Page 422 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 71 STATE SPACE ANALYSIS

7.38. Consider the system in Prob. 7.32.

(a) Is the system controllable?
(6) Is the system observable?
(a) From the result from Prob. 7.32 we have

Now

and by Eq. (7.120) the controllability matrix is

and IM,I = - 4 # 0. Thus, its rank is 2 and hence the system is controllable.
(b) Similarly,

and by Eq. (7.123) the observability matrix is

and IMol = 0. Thus, its rank is less than 2 and hence the system is not observable.
Note from the result from Prob. 7.32(b) that the system function H(z) has pole-zero
cancellation. If H(z) has pole-zero cancellation, then the system cannot be both control-
lable and observable.

SOLUTIONS OF STATE EQUATIONS FOR CONTINUOUS-TIME LTI SYSTEMS

7.39. Find eA' for

using the Cayley-Hamilton theorem method.

First, we find the characteristic polynomial c(A) of A.

=A'+ 5A +6 = (A + 2)(A + 3)
Thus, the eigenvalues of A are A, = - 2 and A, = - 3. Hence, by Eqs. (7.66) and (7.67) we have