Filter
Top Products
2.3. H
∞
AND H
2
DESIGN METHODOLOGIES 37
and the components, C
1
x and D
12
u are orthogonal. D
12
is also assumed to be
normalized. This essentially means that there is no cross-weighting between the state
and input penalties. Assumption (iv) is the dual of this; the input and unknown input
(disturbance and noise) affect the measurement, y, orthogonally, with the weight on the
unknown input being unity.
To solve the H
∞
design problem we define two Hamiltonian matrices,
H
∞
=
Aγ
−2
B
1
B
T
1
−B
2
B
T
2
−C
T
1
C
1
−A
T
,
and
J
∞
=
A
T
γ
−2
C
T
1
C
1
− C
T
2
C
2
−B
1
B
T
1
−A
.
The following theorem gives the solution to the problem.
Theorem 3 There exists a stabilizing controller satisfying G(s)
∞
<γif and only if
the following three conditions are satisfied:
a) H
∞
∈ dom(Ric) and X
∞
= Ric(H
∞
) ≥ 0.
b) J
∞
∈ dom(Ric) and Y
∞
= Ric(J
∞
) ≥ 0.
c) ρ(X
∞
Y
∞
) <γ
2
.
When these conditions are satisfied, one such controller is,
K
∞
(s)=
ˆ
A
∞
−Z
∞
L
∞
F
∞
0
,
where,
F
∞
= −B
T
2
X
∞