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100 Introduction to Control Theory

or

2
(s +k v s+k p )Q(s)=(as+b)[s R(s)+S(s)].
2
At this stage, we may choose for example R(s)=1 and suppose for the
purpose of illustration that a=b=k p =1 and that k v =9. Then we solve for S(s)=9s
and Q(s)=s. Note that in this case, the feedback block S(s)/R(s) is improper
but this does not cause any problems in cases (such as robot control) where
velocity measurements are available.

The question addressed next is to find conditions on (2.11.1) so that a
static-output feedback controller will render the closed-loop system SPR.
We will consider the SISO case only since this is the only case encountered
in this book.

(2.11.8)

or in the frequency-domain

(2.11.9)

We present a simple frequency domain result to show the existence of K and
γ that will render the closed-loop system SPR. The result first appeared in
[Gu 1988].

THEOREM 2.11–5: Let system (2.2.11) have no common poles and zeros.
Then there exists a nonsingular K and a positive scalar γ such that the closed-
loop system (2.11.8, 2.11.9) is SPR, if and only if P(s) has no zeros in the
right half plane and if P(s) has n poles and n - 1 zeros. In fact, one such K is
given by

K=P(CB) -1

where P is any symmetric, positive-definite matrix.

Copyright © 2004 by Marcel Dekker, Inc.