38 CHAPTER 2. OVERVIEWOF THE UNDERLYING THEORY
L∞=−Y∞CT
2
Z∞=(I−γ
−2
Y
∞
X
∞
)
−1
ˆ
A
∞=A+γ
−2
B
1
B
T
1
X
∞
+B
2
F
∞
+Z
∞
L
∞
C
2
.
Actually,the ab oveformulation can be used to parametrize all stabilizing controllers
which satisfy,kG(s)k∞<γ. This can be expressed as an LFT. All such controllers are
given by,
K∞=Fl(M∞,Q),
where,
M∞=
ˆ
A∞−Z∞L∞Z∞B2
F∞
−C2
0I
I0
,
and Qsatisfies: Q∈RH
∞
,kQk
∞<γ.NotethatifQ= 0 we get back the controller
given in Theorem 3. This controller is referred to as the central controller and it is the
controller calculated by the software.
Note also that the controller givenabove satisfies kGk∞<γ. It is not necessarily the
controller that minimizes kGk∞and is therefore referredto as a suboptimal H∞
controller. In practice this is not a problem, and mayeven be an advantage. The optimal
H∞controller has properties which may not be desirable from an implementation point
of view. One typical property is that the high frequency gain is often large. Suboptimal
central controllers seem to be less likely to exhibit this characteristic.
2.3.5 Further Notes on the H∞Design Algorithm
Now that we havecovered the problem solution we can look at the areas that might give
potential numerical problems. The above results give a means of calculating a controller
(if one exists) for a specified γvalue. As we mentioned earlier, the smallest such γis
referred to as γopt. An iterative algorithm is used to find γclose to γopt and calculate
the resulting controller. The algorithm canb e stated conceptuallyas follows: