Page 914 - The Mechatronics Handbook

0066_Frame_C30 Page 25 Thursday, January 10, 2002 4:44 PM

2
2
Check on H Output Feedback Assumptions. We now make sure that all of the H output feedback
problem assumptions in Assumption 30.2 are satisfied.
• Plant P = G 22 Assumptions. Since the plant P = G 22 = [ A, B, C ] is stabilizable and detectable, it
follows that (A, B 2 = B, C 2 = C) is stabilizable and detectable.
• Regulator Assumptions. Since

D 12 = 0 n × n u
y
rIn × n
u u
has full column rank, it follows that the control weighting matrix R = D 12 D 12 = rI n × n > 0 is
T
u u
nonsingular.
T
Since D 12 C 1 = 0, it follows that the imaginary axis (column) rank condition involving (A, B 2 ,
– 1 T −1 T
C 1 , D 12 ) in Assumption 30.2 is equivalent to (AB 2 R D 12 C 1 ,(ID 12 R D 12 )C 1 )– – = (A,C 1 )
having no unobservable imaginary modes. Since (A, M) is either detectable or has no imaginary
unobservable modes, it follows that

A,C 1 = M
0 n × n
u
has no unobservable imaginary modes. The associated Hamiltonian H con will, therefore, yield a
Riccati solution and control gain matrix G c such that A − BG c is stable.
[ mI n × ]
• Filter Assumptions. Since D 21 = 0 n × n u y n y has full row rank, it follows that the measurement
y
weighting matrix Θ = D 21 D T 21 = mI n × > 0 is nonsingular.
n
y
y
T
Since B 1 D 21 = 0 , it follows that the imaginary axis (row) rank condition involving (A, B 1 , C 2 ,
D 21 ) in Assumption 30.2 is equivalent to (AB 1 D 21 Θ C 2 ,B 1 (ID 21 Θ D 21 ))–– T −1 −1 = ( A,B 1 ) having
no uncontrollable imaginary modes. Since (A, L) is either stabilizable or has no uncontrollable
imaginary modes, it follows that (A,B 1 = [L 0 n × ]) has no uncontrollable imaginary modes.
n
y
The associated Hamiltonian H fil will therefore yield a Riccati solution and filter gain matrix H f
such that A − H f C is stable.
2
Given the above, it follows that all of the H output feedback problem assumptions in Assumption 30.2
are satisfied.
Control Gain Matrix. It follows that the control gain matrix G c is given by
G c = R B X (30.145)
−1
T
where X ≥ 0 is the unique (at least) positive semi-definite solution of the CARE:

T
−1
T
A X + XA + C CXBR BX = 0 (30.146)
–
Moreover, A − BG is stable.
Filter Gain Matrix. It follows that the filter gain matrix H f is given by
H f = YC Θ −1 (30.147)
T

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