2.2. MODELING UNCERTAINSYSTEMS 25
=Cz−1(I−z−1A)−1B+D
=Fu(Pss,z−1I),
where Pss is the real valued matrix,
Pss =AB
CD
,
and the scalar ×identity,z−1I, has dimension equal to the statedimension 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,
∆=δInxδ∈C
,
where nx is the state dimension. The set of models,
Fu(Pss,∆),∆∈B∆,
is equivalentto P(z),|z|≥1. Thishints 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 themodel 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 eng ine
speed, v(rpm). We model the relationship as,
y=e
−T
d
s0.9
1+T
c
s+0.1
1+su,