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2.4. µ ANALYSIS 47
if and only if µ(G(s))
∞
< 1,
where µ is taken with respect to an augmented structure
∆,
∆ =
diag(∆,
ˆ
∆)
∆ ∈ ∆,
ˆ
∆=C
dim(w)×dim(e)
.
The additional perturbation block,
ˆ
∆ can be thought of as a “performance block”
appended to the ∆ blocks used to model system uncertainty. This result is the major
benefit of the choice of input and output signal norms; the norm test for performance is
the same as that for stability. Robust performance is simply another µ test with one
additional full block.
The frequency domain robustness approach, outlined above, assumes that the
perturbations, ∆, are linear and time-invariant. This assumption is the most commonly
applied. Section 2.4.6 will consider a robustness analysis from a state-space point of
view. This form of analysis applies to norm bounded non-linear or time varying
perturbations. We will first look more closely at the properties of µ, particularly as they
relate to its calculation.
2.4.4 Properties of µ
The results presented here are due to Doyle [68]. Fan and Tits [69, 70] have done
extensive work on algorithms for tightening the bounds on the calculation of µ.
Packard [3] has also worked on improvement of the bounds and the extension of these
results to the repeated block cases. The most comprehensive article on the complex
singular value is that by Packard and Doyle [20]. More detail is contained in the
technical report by Doyle et al. [71].
We will look at simple bounds on µ. The upper bound results are particularly important
as they will form the basis of the design procedure provided in this software (D-K
iteration).
Defining a block structure made up of one repeated scalar, (∆ = {λI | λ ∈C})makes
the definition of µ the same as that of the spectral radius.
∆ =
λI
λ ∈C
⇒ µ(M)=ρ(M).