2.2. MODELING UNCERTAINSYSTEMS 19
in an LFT format. The open-loop system is described by,
y=Fu(Polp,∆)u,
where
Polp =0Wa
IP
0
.
The unity gain, negative feedback configuration, illustrated in Figure 2.4 (and given in
Equation 2.5) can be described by,
y=Fu(Gclp,∆)r,
where
Gclp =−Wa(I+P0)−1Wa(I+P0)−1
(I+P0)−1P0(I+P0)−1
Figure 2.5 also shows the perturbation,∆ as blo ckstructured. Inotherwords,
∆ = diag(∆1,...,∆
m).(2.7)
This allows us to consider different perturbation blocks in a complex interconnected
system. If we interconnect two systems, each with a ∆ perturbation, then the result can
always be expressed as an LFT with a single, structured perturbation. This is a very
general formulation as wecan always rearrange the inputs and outputs of Pto make ∆
block diagonal.
The distinction between perturbations and noise in the model can be seen from bo th
Equation 2.6 and Figure 2.5. Additive noise will enter the model as a comp onentof u.
The ∆ block represents the unknown but bounded perturbations. It is possible that for
some ∆, (I−P11∆) is not invertible. This type of model can describe nominally stable
systems which can be destabilized by perturbations. Attributing unmodeled effects
purely to additive noise will not have this characteristic.