54 CHAPTER 2. OVERVIEWOF THE UNDERLYING THEORY
Note that the nominal system is given by,
Gnom(z)=F
u AB
1
C
1D
11 ,z−1I,
and the perturbed system is,
G(z)=F
u
(F
l
(G,∆),z−1I).
We assume that ∆ is an element of a unity norm bounded block structure, ∆ ∈B∆.
For th e µanalysiswe will define a block structure corresponding to Gss,
∆s=diag(δ1Inx,∆2,∆) δ1∈C,∆
2∈C
dim(w)×dim(e),∆∈∆.
Consider also a block structure corresponding to Fu(Gss,z−1I),
∆p=diag(∆2,∆) ∆2∈C
dim(w)×dim(e),∆∈∆.
This is identical to the ∆sstructure except that the δ1Inx block, corresponding to the
state equation, is not present. The following theorem gives the equivalence between the
standard frequency domain µtest and a state-space µtest for robust performance (first
introduced by Doyle and Packard[23]. Thenotation µ∆swill denote a µtest with
respect to the structure ∆s,andµ
∆
pis a µtest with respect to the ∆pstructure.
Theorem 11
The following conditions are equivalent.
i)µ∆s(Gss)<1(state-space µtest);
ii)ρ(A)<1and max
ω∈[0,2π]µ∆p(Fu(Gss,eωI)) <1(frequency domain µtest);