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20 CHAPTER 2. OVERVIEW OF THE UNDERLYING THEORY
The issue of the invertibility of (I − P
11
∆) is fundamental to the study of the stability of
a system under perturbations. We will return to this question in much more detail in
Section 2.4. It forms the basis of the µ analysis approach.
Note that Equation 2.7 indicates that we have m blocks, ∆
i
, in our model. For
notational purposes we will assume that each of these blocks is square. This is actually
without loss of generality as in all of the analysis we will do here we can square up P by
adding rows or columns of zeros. This squaring up will not affect any of the analysis
results. The software actually deals with the non-square ∆ case; we must
specify the input and output dimensions of each block.
The block structure is a m-tuple of integers, (k
1
,...,k
m
), giving the dimensions of each
∆
i
block. It is convenient to define a set, denoted here by ∆, with the appropriate block
structure representing all possible ∆ blocks, consistent with that described above. By
this it is meant that each member of the set of ∆ be of the appropriate type (complex
matrices, real matrices, or operators, for example) and have the appropriate dimensions.
In Figure 2.5 the elements P
11
and P
12
are not shown partitioned with respect to the
∆
i
. For consistency the sum of the column dimensions of the ∆
i
must equal the row
dimension of P
11
. Now define ∆ as
∆ =
diag (∆
1
,...,∆
m
)
dim(∆
i
)=k
i
×k
i
.
It is assumed that each ∆
i
is norm bounded. Scaling P allows the assumption that the
norm bound is one. If the input to ∆
i
is z
i
and the output is v
i
,then
v
i
= ∆
i
z
i
≤z
i
.
It will be convenient to denote the unit ball of ∆, the subset of ∆ norm bounded by
one, by B∆. More formally
B∆ =
∆ ∈ ∆
∆≤1
.
Putting all of this together gives the following abbreviated representation of the
perturbed model,
y = F
u
(P, ∆)u, ∆ ∈ B∆. (2.8)