2.3. HAND 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 Hdesign pr oblem wedefine two Hamiltonian matrices,
H=
2
B
1
B
T
1
B
2
B
T
2
C
T
1
C
1A
T
,
and
J=ATγ2CT
1C1CT
2C2
B1BT
1A.
The following theorem gives the solution to the problem.
Theorem 3 There exists a stabilizing controller satisfying kG(s)kif and only if
the following three conditions are satisfied:
a) Hdom(Ric) and X= Ric(H)0.
b) Jdom(Ric) and Y= Ric(J)0.
c) ρ(XY)
2
.
When these conditions aresatisfied, one such controller is,
K(s)=ˆ
A
Z
L
F
0,
where,
F=BT
2X