2.4. µANALYSIS 57
The gap between the state-space (or constant D) upper bound and the frequency
domain upper bound is more significant. In the state-space upp er bound, a single D
scale is selected. This gives robust performance for all ∆ satisfying, kvk≤kzkfor all
e∈L
2
. This can be satisfied for linear time-varying perturbations or non-linear cone
bounded perturbations. The formal result is g ivenin the following theorem (given in
Packardand Doyle [20]).
Theorem 12
If there exists Ds∈D
ssuch that
σmax[DsGssD−1
s]=β<1,
then there exists constants,c1≥c2>0such that for all perturbation sequences,
{∆(k)}∞
k=0 with ∆(k)∈∆,σmax[∆(k)] <1/β, the time varying uncertain system,
x(k+1)
e(k)=F
l
(G
ss,∆(k))x(k)
w(k),
is zero-input, exponentiallystable, and furthermore if {w(k)}∞
k=0 ∈l2,then
c
2
(1−β2)kxk2
2+kek2
2≤β2kwk2
2+c1kx(0)k2.
In particular,
kek2
2≤β2kwk2
2+c1kx(0)k2.
The user now has a choice of robust performance tests to apply. The most appropriate
depends on the assumed nature of the perturbations. If the state-space upper b ound test
is used, the class of allowable perturbations is now very muchlarger and includes
perturbations with arbitrarily fast time variation. If the actual uncertainty were best
modeled by a linear time-invariantp erturbation then the state-space µtest could be
conservative. The frequency domain upper bound is probably the most commonly used
test. Even though the uncertainties in a true physical system will not be linear, this
assumption givessuitable analysis results in a wide range of practical examples.