Page 918 - The Mechatronics Handbook

0066_Frame_C30 Page 29 Thursday, January 10, 2002 4:44 PM

is a consequence of the structure of model-based compensators and has nothing to do with
optimal control and filtering problems.
• Assuming that a limiting condition lim rG c = WC holds for some invertible matrix W,
r→ 0 + −l
loop transfer recovery of the target loop L o = C(sI − A) H f may be proven as follows: For
small r we have

G c ≈ WC (30.163)
---------
r
which gives yields the following:

PK opt = PG c sI A + BG c + H f C) H f (30.164)
(
−1
–

WC
WC
≈ P --------- sI A + B ---------   −1 H f (30.165)
–

r r
WC
WC
(
(
≈ P --------- sI A) −1 I + B --------- sI A) −1 −1 H f (30.166)
–
–
r r
(
(
WC sI A) −1 WC sI A) −1 −1
–
–
≈ P --------------------------------- I + B --------------------------------- H f (30.167)
r r
(
(
WC sI A) −1 −1 WC sI A) −1
–
–
≈ PI + ---------------------------------B ---------------------------------H f (30.168)
r r
(
−1
–
WP
≈ P --------- −1 W --------------------------------- (30.169)
CsI A) H f
r r
−1
(
≈ CsI A) H f = L o (30.170)
–
The central idea (underneath the algebra) is that as r goes to zero, the C feedback path
within the compensator K opt = [A − BG c − H f C, G c , H f ] is broken (see Fig.30.7 ) and the nice
properties that hold at the so-called innovations v (e.g., open loop transfer function matrix
−l
at v is L o = C(sI − A) H f ) in Fig. 30.7 get transferred to the error signal e (compensator input,
or plant output) within the feedback loop.
• Loop Transfer Recovery at Plant Input (LTRI).
−1
1. Design of Target Loop L i . The first step is to design a target loop L i = G C (sI − Α) B with desirable
closed loop properties (e.g., stability, sensitivity, complementary sensitivity, stability robustness
margins, etc). This may be done using any method! (Any method you feel comfortable enough
with.)
One procedure that results in good properties at the plant input is based on LQR methods.
The idea is to select the design (shaping) matrix M so that the singular values of G OL = M(sI −
−l
Α) B look nice; e.g., large minimum singular value at low frequencies, small maximum singular
value at high frequencies, singular values cross 0 dB with slopes of −20 dB/dec, etc.
We then solve the CARE with A, B, M, R = rI n × n u , using r > 0 to adjust the bandwidth
u
−l
of our target loop L i = G LQR G c (sI − A) B. A smaller (larger) r results in a larger (smaller)
bandwidth.
Guidelines for Shaping of Target Loop L i = G LQ .
• The so-called LQ frequency domain equality (LQFDE) guides our loop shaping:
1
1
[ I + G LQ jw)] I + G LQ jw)] = I + -------G OL jw( ) H -------G OL jw( ) (30.171)
[
(
(
H
r r
©2002 CRC Press LLC