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 = ( b ) Similarly,
- 4 # 0. Thus, its rank is 2 and hence the system is controllable.
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 controllable 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 Thus, the eigenvalues of A are A , =
- 2 and A ,
=(
=
A
+ 2)(A + 3)
- 3. Hence, by Eqs. (7.66)and (7.67)we have