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Fig. 2.51.
process response to two input sequences: (
a
) training sequence, (
b
)test
sequence
d
x
1
(
t
)
d
t
=(
x
1
(
t
)+2
x
2
(
t
))
2
+
u
(
t
)
d
x
2
(
t
)
d
t
=8
.
32
x
1
(
t
)
y
(
t
)=
x
2
(
t
)
.
The state variables
x
1
and
x
2
are measurable. Figure 2.51 shows the process
response to two sequences of input steps; throughout this section, the left-
hand input and output sequences will be used as the training set, and the
right-hand ones as the test set. The results obtained by numerical integration
of the knowledge-based model are in poor agreement with experimental mea-
surements of the output, as shown on Fig. 2.52. The mean square modeling
error on the test set is equal to 0.17, which is much larger than the noise
standard deviation of 0.01.
Experts of the process are reasonably confident that the first state equation
is valid, but there are serious doubts about the second equation because
•
The parameter 8.32 may be inaccurate.
•
The linear dependence is controversial.
•
It is even conjectured that the right-hand side of the second equation might
depend on
x
2
.
Therefore, in order to build a more accurate model, it may be advantageous
to use a semiphysical model. Actually, three different models, of increasing
complexity, may be designed in order to meet the above three criticisms. We
describe below the design of those models and the results thus obtained.
As mentioned above, the first step of the procedure consists in creating a
discrete-time model from the knowledge-based model. Since data is gathered
with a sampling period
T
, the latter is a natural candidate for being the
discretization step of the equations. The simplest discretization method is
Euler's explicit method, whereby the derivative d
f
(
t
)
/
d
t
is approximated as
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