Page 934 - The Mechatronics Handbook

0066_Frame_C30 Page 45 Thursday, January 10, 2002 4:45 PM

State Feedback Loop Shaping
If one selects

B 2 = B (30.246)

C 1 = M (30.247)
0 n × n
u

0 n × n
D 12 = y u (30.248)
rI n × n
u u
R = rI n × (30.249)
u n u

T
then D 12 C 1 = 0 and we have

G c = 1 T (30.250)
---B 2 X
r

where X ≥ 0 is the unique (at least) positive semi-definite solution of the CARE:

1
A X + XA + M MXB---B X = 0 (30.251)
T
T
T
–
r
The following LQFDE may be derived from the CARE:
1
1
H
[
(
[ I + G LQ jw( )] I + G LQ jw)] = I + -------G OL jw( ) H -------G OL jw( ) (30.252)
r r
where

–
(
G OL = MsI A) B (30.253)
1
–
(
G LQ = G c sI A) B (30.254)
–
1
–
Given this, the loop shaping ideas discussed earlier are applicable. A designer may use the matrix M and
the scalar ρ > 0 to shape G OL in an effort to get a desirable loop G LQ . The matrix M, specifically, may be
used to match singular values at low frequencies, high frequencies, all frequencies, etc. Assuming that
(A, B) is stabilizable and (A, M) has no imaginary modes that are unobservable, a stabilizing solution is
guaranteed to exist. Moreover, the resulting G LQ loop will possess nominal sensitivity and stability
robustness properties—a consequence of the LQFDE. The resulting control gain matrix G c may be used
within a state feedback loop, a modified state feedback loop, or within a model-based compensator.

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