Page 905 - The Mechatronics Handbook

0066_Frame_C30 Page 16 Thursday, January 10, 2002 4:43 PM

where the control gain matrix G c ∈ R n × n is given by
u
1
T
T
–
G c = R [ B 2 X + D 12 C 1 ] (30.79)
X = Ric(H con ) ≥ 0 is the unique (at least) positive semi-definite solution of the CARE:
( AB 2 R D 12 C 1 ) X + X A B 2 R D 12 C 1 ) + C 1 1 D 12 R D 12 )C 1 – XB 2 R B 2 X = 0 (30.80)
(
T
(
1
T
–
–
T
–
1
T
1
T
T
T
1
–
–
–
–
and the filter gain matrix H f ∈ R n × n y is given by
T
T
H f = [ YC 2 + B 1 D 21 ]Θ – 1 (30.81)
Y = Ric(H fil ) ≥ 0 is the unique (at least) positive semi-definite solution of the FARE:
T
1
(
1
T
1
T
T
T
T
( AB 1 D 12 Θ C 2)Y + Y A B 1 D 21 Θ C 2 ) + B 1 ID 21 Θ D 21 )B 1 – YC 2 Θ C 2 Y = 0 (30.82)
(
1
–
–
–
–
–
–
–
Moreover, the minimum norm is given by
1/2
(
T wz K opt ) 2 = M c B 1 2 2 + R G c M f L 2 2 (30.83)
L L
T
T
(
(
= trace B 1 XB 1 ) + trace RG c YG c ) (30.84)
where
–
M c = [ AB 2 G c , I n × , C 1 – D 12 G c ] (30.85)
n
–
M f = [ AH f C 2 , I n × , B 1 – H f D 21 ] (30.86)
n
Finally, the closed loop poles are the eigenvalues of A − B 2 G c and A − H f C 2 .
2
Comment 30.11 (Computing Optimal HH Controller in MATLAB)
2
The following MATLAB command sequence may be used to compute the optimal H controller K opt and
the resulting closed loop transfer function matrix T wz :
tss_g = mksys(a, [b1 b2], [c1; c2], [0∗ones(nz, nw) d12; d21 0∗ones(ny,
nu),‘tss’)
[ss_k ss_twz] = h21qg(tss_g, ‘schur’)
[a_k, b_k, d_k] = branch(ss_k, ‘a,b,c,d’)
The “mksys” command packs the two-port state space data for the generalized plant G into a column
vector data structure (called a tree) possessing the “tss” (two-port state space) variable designation. All
dimension information is encoded into the column vector. The “h21qg” command computes the optimal
2
H controller K opt and the associated closed loop system from the exogenous signals w to the regulated
signals z. An eigenvalue-eigenvector method is the default method used to solve the two relevant algebraic
Riccati equations. A Schur method—based on Schur’s unitary transformation of a matrix to upper
triangular form—may be used by including the “schur” option. The results are stored in the tree vectors
ss_k and ss_twz, respectively. The ‘branch’ command is then used to retrieve the state space represen-
tation for K opt from the tree vector ss_k.
Comment 30.12 (Relationship to LQG, Stability Robustness Margins)
2
Theorem 30.1 shows that the optimal H output feedback controller is identical in structure to that found
in classical LQG problems. While certain LQR, KBF, and LQG/LTR problem formulations do result in
feedback loops possessing stability robustness margins, LQG controllers need not possess margins [3].
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