Page 924 - The Mechatronics Handbook

0066_Frame_C30 Page 35 Thursday, January 10, 2002 4:44 PM

Design Plant Singular Values
50

0
Singular Values (dB)

−50

−100
−1 0 1 2
10 10 10 10
Frequency (rad/sec)
FIGURE 30.10 PUMA 560 robotic manipulator design plant singular values.

FIGURE 30.11 Augmented system used for designing target loop.
−l
The matrix L L matches the singular values of G FOL = C(sI − A) L at low frequencies. The matrix
L H matches the singular values at high frequencies. Together, L L and L H match the singular values
−l
of G FOL = C(sI − A) L at all frequencies (see Fig. 30.12). Why is this?
This selection for L L and L H results in

I
1
–
(
G FOL = C p sI A p ) L H + ( – – 1  (30.210)
–
s 
- L L
C p sI A p ) B p
I
= C p sI A p ) – 1 L H +  (30.211)
(
–
s 
- L L
B p
I
(
= C p sI A p ) – 1 – ( A p ) B p L L +  (30.212)
1
–
–
s 
B p
- L L
I
(
= C p sI A p ) [ sI A p ] – – 1  (30.213)
(
1
–
–
-
–
s 
A p ) B p L L
I
(
= C p – – 1  (30.214)
-
s 
A p ) B p L L
= I - (30.215)
s
The resulting gain crossover frequency in Fig. 30.12 is 1 rad/s, as expected.
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