2.5. µSYNTHESIS AND D-KITERATION 61
dynamic system. This requires fitting an approximation to the upper bound D-scale in
the iteration. We will now look at this issue more closely.
The D-Kiteration procedure is illustrated schematically in Figure 2.12. It can b e
summarized as follows:
i) Initialize procedure with K0(s): H∞(or other) controllerfor P(s).
ii) Calculate resulting closed loop: Fl(P(s),K(s)).
iii) Calculate Dscales for µupper bound:
inf
D(ω)∈Dσmax[D(ω)Fl(P(s),K(s))D(ω)−1].
iv) Approximate frequency data, D(ω), by ˆ
D(s)∈RH
∞
,withˆ
D(ω)≈D(ω).
v)DesignH
∞controller for ˆ
D(s)P(s)ˆ
D−1(s).
vi) Gotostepii).
Wehave used the notation D(ω) to emphasize that the Dscale arises from frequency by
frequency µanalyses of G(ω)=F
l
(P(ω),K(ω)) and is therefore a function of ω.Note
that it is NOT the frequency response of some transfer function and therefore we do
NOT use the notation D(ω).
The µanalysis of the closed loop system is unaffected by the D-scales. However the H∞
design problem is strongly affected by scaling. The procedure aims at finding at Dsuch
that the upper bound for the closed loop system is a close approximation to µfor the
closed loop system. There are several details about this pro cedure that will now be
clarified.
At each frequency,a scaling ma trix, D(ω), can be found such that
σmax(D(ω)G(ω)D(ω)−1) is a close upper bound to µ(G(ω)) (Figure 2.12c). The D
scale is block diagonal and the block corresponding to the eand wsignals can be chosen
to be the identity. The part of Dcorresponding to the zsignal commutes with ∆ and
cancels out the part of D−1corresponding to the vsignal. To illustrate this, consider
the Dscale that might result from a block structure with only mfull blocks. At each