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phenomena that are not modeled with the required accuracy through prior
knowledge. Since a larger amount of prior knowledge is used in the design of a
semiphysical model than in the design of a black-box model, a smaller amount
of experimental data is required to estimate its parameters reliably.
2.8.1.2 Design and Training of a Dynamic Semiphysical Model
Design Principle
The design of a semi-physical model requires the availability of a knowledge-
based model, which is usually in the form of a set of coupled, possibly
nonlinear, differential, partial differential, and algebraic, equations. We as-
sume that model to be in standard state-space form,
d
x
d
t
=
f
[
x
(
t
)
,
u
(
t
)]
y
(
t
)=
g
[
x
(
t
)]
,
where
x
is the vector of state variables,
y
is the vector of outputs,
u
is the
vector of control inputs, and where
f
and
g
are known vector functions. That
model may be unsatisfactory for various reasons: functions
f
and
g
(or some
of their components) may be too inaccurate for the purpose that the model
should serve, or they may involve parameters that are not estimated accu-
rately, etc. In a black-box model, neural networks are used to approximate
functions
f
and
g
; they are trained from experimental data. In a semiphysi-
cal neural model, those functions that are not known accurately enough are
implemented as neural models, whereas those functions, which are known re-
liably, are either kept under their analytic form, or implemented as a neural
network with fixed parameters and nonlinearities.
In general, the design of a semiphysical neural model is performed in three
steps:
•
Step 1: construction of a discrete-time semiphysical model that is de-
rived, by an appropriate discretization scheme (discussed below) from the
knowledge-based model.
•
Step 2: training of the semiphysical model, or of specific parts thereof, from
results obtained by numerical integration of the knowledge-based model;
that step is generally necessary in order to obtain appropriate initial values
of the parameters, to be used in step 3.
•
Step 3: training of the semiphysical neural model from experimental data.
That design strategy is exemplified in the next section.
An Illustrative Example
A knowledge-based model is described by the following equations:
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