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2.2. MODELING UNCERTAIN SYSTEMS 25
= Cz
−1
(I −z
−1
A)
−1
B +D
= F
u
(P
ss
,z
−1
I),
where P
ss
is the real valued matrix,
P
ss
=
AB
CD
,
and the scalar × identity, z
−1
I, has dimension equal to the state dimension of P(z).
This is now in the form of an LFT model with a single scalar × identity element in the
upper loop.
One possible use of this is suggested by the following. Define,
∆ =
δI
nx
δ ∈C
,
where nx is the state dimension. The set of models,
F
u
(P
ss
, ∆), ∆ ∈ B∆,
is equivalent to P(z), |z|≥1. This hints at using this formulation for a stability analysis
of P(z). This is investigated further in Section 2.4.6.
In the analyses discussed in Section 2.4 we will concentrate on the assumption that ∆ is
complex valued at each frequency. For some models we may wish to restrict ∆ further.
The most obvious restriction is that some (or all) of the ∆ blocks are real valued. This is
applicable to the modeling of systems with uncertain, real-valued, parameters. Such
models can arise from mathematical system models with unknown parameters.
Consider, for example a very simplified model of the combustion characteristics of an
automotive engine. This is a simplified version of the model given by Hamburg and
Shulman [21]. The system input to be considered is the air/fuel ratio at the carburettor.
The output is equivalent to the air/fuel ratio after combustion. This is measured by an
oxygen sensor in the exhaust. Naturally, this model is a strong function of the engine
speed, v (rpm). We model the relationship as,
y =e
−T
d
s
0.9
1+T
c
s
+
0.1
1+s
u,