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2.4. µ ANALYSIS 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 upper bound, a single D
scale is selected. This gives robust performance for all ∆ satisfying, v≤zfor all
e ∈L
2
. This can be satisfied for linear time-varying perturbations or non-linear cone
bounded perturbations. The formal result is given in the following theorem (given in
Packard and Doyle [20]).
Theorem 12
If there exists D
s
∈D
s
such that
σ
max
[D
s
G
ss
D
−1
s
]=β<1,
then there exists constants, c
1
≥ c
2
> 0 such that for all perturbation sequences,
{∆(k)}
∞
k=0
with ∆(k) ∈ ∆, σ
max
[∆(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, exponentially stable, and furthermore if {w(k)}
∞
k=0
∈ l
2
,then
c
2
(1 − β
2
) x
2
2
+ e
2
2
≤ β
2
w
2
2
+ c
1
x(0)
2
.
In particular,
e
2
2
≤ β
2
w
2
2
+ c
1
x(0)
2
.
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 bound test
is used, the class of allowable perturbations is now very much larger and includes
perturbations with arbitrarily fast time variation. If the actual uncertainty were best
modeled by a linear time-invariant perturbation then the state-space µ test 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 gives suitable analysis results in a wide range of practical examples.