45
then the pair
F
0
6
-BH
A
>
0
L
(3-18)
is stabilizable. If in addition, the word "stabilizable" in (B.l) is
replaced by "controllable" then the pair of (3-18) is controllable.
The next theorem shows that when and Kg are selected to
stabilize (3-17) the pair (A, Kg) is observable. This is the precise
condition needed for Theorem 2.1 and the final condition required for
our discussion.
Theorem 3.4: If all eigenvalues of A are in the closed right half-plane
and the system LCT described by (3-17) is asymptotically stable then
the pair (A, Kg) is observable.
Proof: We use contradiction. Suppose that the system LCT is
asymptotically stable but (A, Kg) is not observable. This implies that
there exists a vector v such that
XI-A
k2
v = 0
(3-19)
for some x which is an eigenvalue of A. Consequently, we can write
"xi F + GKX GKg
~ o'
BH XI-A
V
(3-20)