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54 CHAPTER 2. OVERVIEW OF THE UNDERLYING THEORY
Note that the nominal system is given by,
G
nom
(z)=F
u
AB
1
C
1
D
11
,z
−1
I
,
and the perturbed system is,
G(z)=F
u
(F
l
(G, ∆),z
−1
I).
We assume that ∆ is an element of a unity norm bounded block structure, ∆ ∈ B∆.
For the µ analysis we will define a block structure corresponding to G
ss
,
∆
s
=
diag(δ
1
I
nx
, ∆
2
, ∆)
δ
1
∈C,∆
2
∈C
dim(w)×dim(e)
, ∆ ∈ ∆
.
Consider also a block structure corresponding to F
u
(G
ss
,z
−1
I),
∆
p
=
diag(∆
2
, ∆)
∆
2
∈C
dim(w)×dim(e)
, ∆ ∈ ∆
.
This is identical to the ∆
s
structure except that the δ
1
I
nx
block, corresponding to the
state equation, is not present. The following theorem gives the equivalence between the
standard frequency domain µ test and a state-space µ test for robust performance (first
introduced by Doyle and Packard [23]. The notation µ
∆
s
will denote a µ test with
respect to the structure ∆
s
,andµ
∆
p
is a µ test with respect to the ∆
p
structure.
Theorem 11
The following conditions are equivalent.
i) µ
∆
s
(G
ss
) < 1 (state-space µ test);
ii) ρ(A) < 1 and max
ω∈[0,2π]
µ
∆
p
(F
u
(G
ss
, e
ω
I)) < 1 (frequency domain µ test);