2.4. µANALYSIS 45
Consider the case where the model has only one full ∆ block (m=1andq=0in
Equation 2.9). This is often referred to as unstructured, and the well known result (refer
to Zames [67] and Doyle and Stein [5]) is given in the following lemma.
Lemma 6 (Robust Stability,Unstructured)
Fu(G(s),∆) is stable for all ∆,k∆k∞≤1,
if and only if kG11(s)k∞<1.
A generalization of the above is required in order to handle Fu(G(s),∆) models with
more than one full ∆ block. The positive real valued function µcanbedefinedona
complex valued matrix M,by
det(I−M∆) 6=0 forall∆∈B∆,if and only if µ(M)<1.
Note that µscales linearly. In other words, for all α∈R,
µ(αM)=|α|µ(M).
In practice the test is normalized to one with the scaling being absorbed into the
interconnection structure. An alternative definition of µis the following.
µ(M)=
0ifno∆∈∆solvesdet(I+M∆) = 0
otherwise
min
∆∈∆nβ∃∆,k∆k≤β, suchthat det(I+M∆) = 0o−1
Note that µis defined as the inverse of the size of the smallest destabilizing
perturbation. This immediately gives the following lemma.
Lemma 7 (Robust Stability,Structured)