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Consider the gaussian stationary stochastic process associated to the
AR(
p
)model,
p
+1)+
b
0
V
(
k
+1)
.
Assume that the model is stable (i.e., the roots of the polynomial
P
(
z
)=
X
(
k
+1)=
a
1
X
(
k
)+
···
+
a
p
X
(
k
−
a
p
z
p
are outside the unit ball) and that the white noise (
V
k
)
is gaussian and centered. Then set
r
j
=Cov(
X
k
,X
k−j
), and take the co-
variance of the two members of the previous equation binding the variables
(
X
k−i
)
i
=0
...p−
1
. Then the classical Yule-Walker relations are obtained
⎧
⎨
1
−
a
1
z
−···−
+
a
p
r
p
............................
r
p
=
a
1
r
p−
1
+
···
+
a
p
r
0
The same relations are obeyed approximately by the empirical estimators
r
1
=
a
1
r
0
+
···
⎩
p
)
k
=
N
r
i
i
) of the covariance coe
cients
r
j
and by
the mean-squared estimators
a
i
of the regression coe
cients
a
j
(for higher
accuracy, one should consider the truncation errors that vanish as the ratio
p/N
).
The estimators
r
i
, however, are consistent, without bias and asymptot-
ically gaussian with a variance of order 1
/N
. Then, it may be proved that
the estimators
a
i
are consistent, asymptotically without bias, asymptotically
gaussian with a variance of order 1
/N
too. That result allows us to build
statistical tests in order to validate the model.
Remark.
An estimator is said to be consistent if the mean squared estimation
error goes to 0 when the sample size goes to infinity.
We have just provided here a cursory introductory outline of linear re-
gression. Actually, statisticians and control engineers have improved those
methods to a great extent. Spectral representation is a key tool of linear mod-
eling, and the transfer function of linear filters associated to ARMA models
are generally the object of identification process. Those basic techniques are
addressed in the literature (see the references) and are not within the scope of
this topic. Neural networks are a methodology that is relevant in the nonlinear
framework.
=1
/
(
N
−
k
=
i
+1
x
(
k
)
x
(
k
−
4.2.1.4 Application to a Linear System: The Harmonic Oscillator
Let us use the previous algorithm to identify the harmonic oscillator that
was described in the previous section. Suppose we know only the input tra-
jectory and the angle trajectory (oscillator position). If a hundred step data
file is available, ARX(2, 2) model-based identification gives the correct coef-
ficients with high accuracy. Note that the order of the model is 2. If we use
an ARX(2, 1) model to perform the identification, the results are significantly
corrupted. That can easily be explained: since the control is implemented on
speed increment, its order is 2.
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