48 CHAPTER 2. OVERVIEWOF THE UNDERLYING THEORY
For the other extreme consider a single full block (∆={∆|∆∈C
n×n
}); the definition
of µis now the same as that for the maximum singular value,
∆={∆|∆∈C
n×n
}⇒µ(M)=σ
max(M).
Observe that every possible block structure, ∆, contains {λI |λ∈C}asa perturbation;
and every possible block structure, ∆, is contained in Cn×n. These particular block
structures are the boundary cases. This meansthat the resulting µtests act as bounds
on µfor any block structure, ∆. This gives the following bounds.
ρ(M)≤µ(M)≤σmax(M).
The above bounds can be arbitrarily conservativebut can be improved by using the
following transformations. Define the set
D=ndiag(D1,...,D
q,d
1I
1,...,d
mI
m,)
D
j=D
∗
j>0,
dim(Ii)=k
i
,d
i
∈R,d
i
>0
o
.(2.15)
This is actually the set of invertible matrices that commute with all ∆ ∈∆. This allows
us to say that for all D∈Dand for all ∆ ∈∆,
D−1∆D=∆.
Packard[3] shows that the restriction that dibe positive real is without loss of
generality. Weca n actually takeone of these blocks to be one (or the identity).
Now define Qas the set of unitary matrices contained in ∆:
Q=nQ∈∆Q∗Q=Io.(2.16)
The sets Dand Qcan be used to tighten the bounds on µin the following way (refer to
Doyle [68]).
max
Q∈Qρ(QM )≤µ(M)≤inf
D∈Dσmax(DM D−1).(2.17)