20 CHAPTER 2. OVERVIEWOF THE UNDERLYING THEORY
The issue of the invertibility of (I−P11∆) 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 µanalysisapproach.
Note that Equation 2.7 indicates that we have mblocks, ∆i, in our model. For
notational purposes we will assume that each of these blocks is square. This is a ctually
without loss of generality as in all of the analysis we will do here we can square up Pby
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, (k1,...,k
m), giving the dimensions of each
∆iblock. 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 appropriatetype (complex
matrices, real matrices, or operators, for example) and have the appropriate dimensions.
In Figure 2.5 the elementsP11 and P12 are not shown partitioned with respectto the
∆i. For consistency the sum of the column dimensions of the ∆imust equal the row
dimension of P11. Now define ∆as
∆=ndiag (∆1,...,∆
m)
dim(∆i)=k
i×k
io.
It is assumed that each ∆iis norm bounded. Scaling Pallows the assumption that the
norm bound is one. If the input to ∆iis ziand the outputis vi,then
kv
i
k=k∆
i
z
i
k≤kz
i
k.
It will be convenient to denote the unit ball of ∆, the subset of ∆norm bounded by
one, by B∆. More formally
B∆ =n∆∈∆k∆k≤1
o.
Putting all of this together gives the following abbreviated representation of the
perturbedmodel,
y=Fu(P,∆)u, ∆∈B∆.(2.8)