2.3. H∞AND H2DESIGN METHODOLOGIES 41
framework. We again assume the simplifying assumptions usedin Section 2.3.4 The H2
design solution is obtained (at least conceptually) from the H∞design procedure by
setting γ=∞and using the resulting central controller. It is interesting to compare the
H∞solution, given above, and the H2solution given below.
Define two Hamiltonians, H2and J2,by,
H
2=A−B
2
B
T
2
−C
T
1C
1−A
T,
and
J2=AT−CT
2C2
−B1BT
1−A.
The sign definiteness of the off-diagonal blocks guarantees that H2∈dom(Ric),
J2∈dom(Ric) and X2= Ric(H2)≥0andY
2= Ric(J2)≥0. The following theorem
gives the required result.
Theorem 4 The unique H2optimal controller is given by,
K2(s)=ˆ
A
2−L
2
F
20,
where,
F2=−BT
2X2
L2=−Y2CT
2
ˆ
A2=A+B2F2+L2C2.
Weco mmentedabove that the controller, K2, is (conceptually) obtained by choosing
γ=∞in the H∞design procedure. This does not mean that kGk∞=∞; it simply
means that we can makeno a priori prediction about kGk∞for this controller. K2
minimizes kGk2and yields a finite kGk∞. As such, it is often useful for determining an