60 CHAPTER 2. OVERVIEWOF THE UNDERLYING THEORY
Recall that this is an upper bound for the µproblem of interest, implying that,
µ[Fl(P(s),K(s)] ≤1,
as required. However the upper bound may be conservative, meaning that in order to
guarantee thatµ[Fl(P(s),K(s)]≤1, we have had to back off on the performance and/or
the stability margins.
With the appropriate choiceof Dscalingsthe upper bound will be much closer to µ.In
otherwords thereexists Dsuch that, kDFl(P(s),K(s))Dk∞isa close upper bound to
µ[Fl(P(s),K(s)]. The µ-synthesis problem can be replaced with the following
approximation (based on the upper bound):
inf
D∈D
K(s)stabilizing
kDFl(P(s),K(s))D−1k∞.(2.18)
The reader is referred to Doyle [1] for details of this problem.
If this is considered as an optimization of two variables, Dand K(s), the problem is
convex in eachof the variables separately, but not jointlyconvex. Doyle[2] gives an
example where this method reaches a local nonglobal minimum.
D-Kiteration involves iterating between using D∈Dand K(s) to solve Equation 2.18.
There are several practicalissues to b e addressedin doingthis and we discuss those in
the next section.
2.5.2 The D-KIteration Algorithm
The objective is to design a controller which minimizes the upper bound to µfor the
closed loop system;
inf
D∈D
K(s)stabilizing
kDFl(P(s),K(s))D−1k∞.
The major problem in doing this is that the D-scale that results from the µcalculation
is in the form of frequency by frequency data and the D-scale required above must be a