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2.2. MODELING UNCERTAIN SYSTEMS 19
in an LFT format. The open-loop system is described by,
y = F
u
(P
olp
,∆)u,
where
P
olp
=
0 W
a
IP
0
.
The unity gain, negative feedback configuration, illustrated in Figure 2.4 (and given in
Equation 2.5) can be described by,
y = F
u
(G
clp
,∆)r,
where
G
clp
=
−W
a
(I + P
0
)
−1
W
a
(I + P
0
)
−1
(I + P
0
)
−1
P
0
(I + P
0
)
−1
Figure 2.5 also shows the perturbation, ∆ as block structured. In otherwords,
∆ = 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 we can always rearrange the inputs and outputs of P to make ∆
block diagonal.
The distinction between perturbations and noise in the model can be seen from both
Equation 2.6 and Figure 2.5. Additive noise will enter the model as a component of u.
The ∆ block represents the unknown but bounded perturbations. It is possible that for
some ∆, (I − P
11
∆) 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.