46 CHAPTER 2. OVERVIEWOF THE UNDERLYING THEORY
Fu(G(s),∆) stable for all ∆∈B∆
if and only if kµ(G11(s))k∞<1.
where
kµ(G11(s))k∞=sup
ω
µ[G
11(ω)].
The use of this notation masks the fact that µis also a function of thep erturbation
structure, ∆. The above definition of µapplies to the more general block structure given
in Section 2.2.4. We can even consider the some of the blocks to be real valued,r ather
than complex valued. The robust stability lemma is still valid; however the calculation
of µbecomes significantly more difficult.
In applying the matrix definition of µto a real-rational G11(s), it has been assumed that
∆ is a complex constant at each frequency. This arises from the assumptionthat ∆ is
linear and time-invariant. Under this assumption we can examine the combination of
system and perturbation independently at each frequency. The analysis then involves
looking for the worst case frequency. If ∆ is not time-invariant then the frequency by
frequency analysis does not apply; ∆ can be used to shift energy between frequencies
and cause instability not predicted by the above analysis.
In practice this µtest is applied by selecting a frequency grid and at each frequency
calculating µ(G11(ω)). The choice of range and resolution for this grid is a matter of
engineering judgement. If very lightly damped modes are present a fine grid may be
required in the region of those modes.
2.4.3 Robust Performance
The obvious extension to the above isto considerperformance in the presence of
perturbations ∆. For e,w∈BP or BL2robust performance is a simple extension of
robust stability.
Lemma 8 (Robust Performance)
Fu(G(s),∆) is stable and kFu(G(s),∆)k∞≤1for all ∆∈B∆