2.3. H∞AND H2DESIGN METHODOLOGIES 39
a) Choose γ≥γopt
b) Form H∞and J∞
c) Check that H∞∈dom(Ric) and J∞∈dom(Ric).
d) Calculate X∞= Ric(H∞)andY
∞= Ric(J∞)
e) Check that X∞≥0andY
∞≥0
f) Check that ρ(X∞Y∞)<γ
2
g) Reduce γandgotostepb).
The value of γcan be reduced until one of the checks at steps c), e) or f) fails. In this
case, γ<γ
opt and we use the X∞and Y∞of the lowest successful γcalculation to
generate the controller. In the Xµsoftwarea bisection searchover γis used to find a γ
close to γopt. If step a) is not sa tisfied, the routine exits immediately and tells the user
to select a higher initial choice for γ.
As part of the check thatH∞∈do m(Ric), (and J∞∈dom(Ric)) the real part of the
eigenvaluesis calculated. The software uses a toleranceto determine whether or not to
consider these zero. The default tolerance works well in most cases; the user can adjust
it if necessary.
In practice determining that X∞(and Y∞) is positive definite involves checkingthat,
min
iRe{λi(X∞)}≥−.
Again, is a preset tolerance which can be adjusted by the user if necessary.
The third check is that,
ρ(γ−2X∞Y∞)<1.
Fortunately this is a relatively well conditioned test.
The software displaysthe critica lvariables relating to each of these tests. The minimum
real part of the eigenvaluesof H∞(and J∞) is displayed. Similarly the minimum