2.3. H∞AND H2DESIGN METHODOLOGIES 31
extend these approaches tothe case where P(s) is replaced by Fu(P(s),∆), ∆ ∈B∆.
2.3.1 H∞Design Overview
Again, recall from Section 2.1.2, the H∞is norm of G(s)is,
kG(s)k
∞=sup
ω
σ
max[G(ω)].
The H∞norm is the induced L2to L2norm. Therefore minimizing the H∞norm of
G(s) will have the effect of minimizing the worst-caseener gy ofeover all bo unded
energy inputs at w.
Consider γ(K) to be the closed loop H∞norm achieved for a particular controllerK.In
other words,
γ(K)=kF
l
(P,K)k∞.
There is a choice of controller,K, which minimizes γ(K). This is often referred to as the
optimal valueof γand is denoted by γopt. Furthermore, there is no stabilizing controller
which satisfies,
kG(s)k∞<γ
opt.
In a particular design problem, γopt is not known a priori. Therefore the functions
calculating the H∞controller use some form of optimizationto obtain a value of γclose
to γopt.
The first approaches to the solutiono fthis problem were described by Doyle [1]. The
book by Francis[57] gives a good overview of the early version of this theory. A
significant breakthrough was achieved with the development of state-space calculation
techniques for the problem. These are discussed in the paper colloquia llyknown as
DGKF [58]. The algorithmic details are actually given by Glover and Doyle [59].