Geoscience Reference
In-Depth Information
The practical implementation of identification by output error is
straightforward, providing the tools necessary for simulation and
optimization are available. Therefore we will not elaborate further on the
subject. There are also complete tools for identification, in particular the
system identification toolbox from Matlab
®
. They generally provide an
indication of the accuracy of the identified parameters, in the form of a
variance matrix of estimation errors, which can only be correct if the
optimality tests are positive, especially residual whiteness. If this is not the
case, the structure of the model needs to be reviewed until satisfactory
results are achieved. Although it is not optimal in terms of noise and
effective disturbances, we will nevertheless remain with OE because of its
robustness. Fortunately, this does not prevent the calculation of variance of
estimation errors. Indeed, this does not pose any particular problems, but we
describe it below due to its significance.
6.3. Estimating the error variance
We specify the dependence of the observed output
y
of a process in relation
to vector
of parameters, by writing
, under the following matrix
θ
y
=
f
(
θ
)
+
v
form:
⎡⎤⎡ ⎤⎡⎤
⎢⎥⎢ ⎥⎢⎥
⎢⎥⎢ ⎥⎢⎥
⎢⎥⎢ ⎥⎢⎥
=
y
f
()
()
θ
θ
v
1
1
1
y
f
v
θ
⎡
⎤
2
2
2
1
⎢
⎥
#
#
#
θ
⎢
2
⎥
+
, where
is the vector of the parameters and
⎢⎥⎢ ⎥⎢⎥
⎢⎥⎢ ⎥⎢⎥
⎢⎥⎢ ⎥⎢⎥
⎢⎥⎢ ⎥⎢⎥
⎢⎥⎢ ⎥⎢⎥
θ
=
y
f
()
θ
v
⎢
#
⎥
t
t
t
⎢
⎥
θ
#
#
#
⎣
⎦
n
y
f
()
θ
v
⎣⎦⎣ ⎦⎣⎦
N
N
N
where:
{}
to the
-
y
is the series of observed outputs, from the initial instant
t
=
1
t
final one
N
;
-
{
}
t
f
is the series of the outputs which would result from the
simulation of the exact model driven by the observed inputs;
-
(
θ
)
{}
t
v
represents the noise and disturbance on the output, including those
coming from input errors. This signal is assumed to be time invariant,
ergodic and centered.
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