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2.3. H∞AND H2DESIGN METHODOLOGIES 37
and the components, C1xand D12uare orthogonal. D12 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.
Toso lvethe H∞design pr oblem wedefine 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∞=ATγ−2CT
1C1−CT
2C2
−B1BT
1−A.
The following theorem gives the solution to the problem.
Theorem 3 There exists a stabilizing controller satisfying kG(s)k∞<γ if 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 aresatisfied, one such controller is,
K∞(s)=ˆ
A
∞−Z
∞
L
∞
F
∞0,
where,
F∞=−BT
2X∞