If p < q the pair (A, B) is not controllable If m < q the pair (A, K2) is not observable
Proof: The proofs to properties (1) and (2) are similar so we only prove (1). This will be accomplished by showing the existence of a row vector v' such that
v'EXI - A , B] = 0
(2-50)
for some x which is an eigenvalue of
From the structure of A, it is C and hence there is a row vector w'
A.
apparent that x is an eigenvalue of such that
w'[XI - C] = 0
Now define a matrix Q . Rqxqr to be the following
Q= 0
* o . Wm]
It can be readily seen that
Q [xI - A] = 0
Now let D . RqxP be the matrix product of Q and B. That is
(2-51)
(2-52)
(2-53)