Page 910 - The Mechatronics Handbook

0066_Frame_C30 Page 21 Thursday, January 10, 2002 4:44 PM

Since

D 12 C 1 = 0 (30.105)
T
B 1 D T = 0 (30.106)
21
the imaginary axis rank conditions involving (A, B 2 , C l , D 12 ) and (A, B 1 , C 2 , D 21 ) in Assumption 30.2
become equivalent to (A,C l ) having no imaginary unobservable modes and (A, B 1 ) having no imaginary
uncontrollable modes. These are clearly satisfied since A = 1 has no imaginary modes. Given this, it
2
follows that all of the H output feedback problem assumptions in Assumption 30.2 are satisfied.
Plant. Finally, we note that the so-called plant (or missile) transfer function P = G 22 is given by

−1
(
–
P = G 22 = C 2 sI A) B 2 (30.107)
1
= ---------- (30.108)
–
s 1
G 22 is unstable with a right half plane pole at s = 1. G 22 is also minimum phase (i.e., no zeros in Res > 0).
T
Filter Gain Matrix H f . Since B 1 D = 0 , the associated FARE is given by
21
T
---Y =
1
–
AY + YA + B 1 B 1 – YC 2 Θ C 2 Y = Y + Y + 1 – 1 2 0 (30.109)
T
T
m
or
2
–
Y – 2mY m = 0 (30.110)
Application of the quadratic formula and selecting the positive (stabilizing) root yields:
2
Y = m + m + m (30.111)
This yields the following filter gain matrix:

1
T
H f = YC 2 Θ – 1 = 1 + 1 + --- (30.112)
µ
We now select m to achieve the given dominant pole specification:

--- =
AH f C 2 = 11 – 1 + 1 – 5 (30.113)
–
–
m
This yields
1
m = ----- (30.114)
24
The associated KBF open loop transfer function is given by

−1
G KF = – C 2 sI A) H f (30.115)
(
–
6
–
= ---------- (30.116)
–
s 1
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