2.4. µANALYSIS 55
iii)There exists a constant β∈[0,1] such that for each fixed ∆∈B∆,G(z)is stable
and for zero initial state response, esatisfies kek2≤βkwk2(robust
performance).
The frequency domain µtest is implemented by searching for the maximum value of µ
over a user specified frequency grid. Theorem 11 shows that this is equivalent to a
single, larger, µtest. There are subtle distinctions between the two tests. As we would
expect, calculation of the larger µtest is more difficult. More importantly, the result
does not scale. In the frequency domain test, if
max
ω∈[0,2π]µ∆p[Fu(Gss,eωInx)] = β,
where β>1, then weare robust with respect to perturbations up to size 1/β.Inthe
state-space test, if µ∆s(Gss)=β, where β>1, then we cannot draw anyconclusio ns
about the robust performance characteristicso fthe system. Wemust scale the inputs or
outputs and repeat the calculation until the µtest gives a result less than one.
In practice we can only calculate upper and lower bounds for both of these µtests.
Although the state-space and frequency domain µtests are equivalent, their upper
bound tests have different meanings. We will see that this difference can be used to
study the difference between linear time-invariantpertur bations and linear time-varying
(and some classes of non-linear) perturbations.
Tocla rify this issue, consider the Dscaleswhich correspond to ∆sand ∆p;
Ds=diag(D1,d
2I
2,D)
D
T
1=D
1>0,dim(D1)=nx×nx,
d2>0,dim(I2) = dim(w)×dim(w),D∈D,
D
p=diag(d2I2,D)
d
2>0,dim(I2) = dim(w)×dim(w),D∈D.
In the above Dis the set of D-scales for the perturbation structure ∆, and, for
notational simplicity,we have assumed that dim(w) = dim (e). Now, the upper bound
tests are: