Page 912 - The Mechatronics Handbook

0066_Frame_C30 Page 23 Thursday, January 10, 2002 4:44 PM

Given this, the compensator transfer function is given by

−1
(
K opt = – G c sI A + B 2 G c + H f C 2 ) H f (30.130)
–
(
= – 61 + 1 + 1/r) (30.131)
------------------------------------------
s ++ 1 + 1/r
6
For small ρ (cheap control), this yields

(
K opt ≈ – 61/ r) (30.132)
------------------------
s + 1/ r
Open Loop Transfer Function. The associated open loop transfer function is given by

(
PK opt = – C 2 sI A) B 2 G c sI A + B 2 G c + H f C 2 ) H f (30.133)
(
−1
−1
–
–
61 +
1 +
1/r)
(
–
1
= ---------- ------------------------------------------ (30.134)
s 1 s ++ 1 + 1/r
–
6
For small ρ (cheap control), this becomes
61/ r)
(
1
–
PK opt ≈ ---------- ------------------------ (30.135)
s 1 s + 1/ r
–
Loop Transfer Recovery (LTR). From this, we see that as control weighting parameter r approaches zero
(cheap control), the open loop transfer function approaches the KBF open loop transfer function G KF ; i.e.,
–
6
lim G 22 K opt = ---------- (30.136)
–
r → 0 + s 1
= G KF (30.137)
This shows that as r approaches zero (cheap control), the actual open loop transfer function PK opt
approaches the target open loop transfer function G KF . The above procedure of recovering a target open
loop transfer function (with desirable closed loop properties) using an LQG controller is called LQG
with loop transfer recovery or LQG/LTR.
Selection of Far Away Closed Loop Regulator Pole. For small r, the closed loop system is stable with
closed loop poles at s = −5 and s ≈ 1/ r– . A good selection for ρ might be r = 1/2500. This results in
a fast closed loop pole at s ≈ −50 and makes the closed loop filter pole at s = −5 the dominant closed
loop pole, as required.
2
Stability Robustness Margins. It is well known that H and LQG designs need not possess good stability
robustness margins. In fact, they can be arbitrarily bad [3]. LQG/LTR designs for minimum phase plants
(such as ours: P = 1/(s − 1) have guaranteed stability robustness margins. LQG/LTR designs provide
margins that approach those associated with LQR and KBF designs; i.e., infinite upward gain margin, at
least 6 dB downward gain margin, and at least ±60° phase margin. Our final LQG/LTR design

50
6
–
PK opt = ---------- ------------- (30.138)
s 1 s + 50
–
©2002 CRC Press LLC