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In-Depth Information
J
=
N
x
k
w
)
2
(
y
k
−
k
=1
from
N
(1
,n
) input vectors (
x
1
,...,
x
k
,...,
x
N
)and
N
output scalars (
y
1
,...
,
y
k
,...,y
N
), or, equivalently that minimizes the half mean squared error
(MSE),
N
1
2
N
x
k
w
)
2
.
φ
N
(
w
)=
(
y
k
−
k
=1
We restrict ourselves to the classical case of a scalar output. The extension
to the case of a vector output is trivial. It is well known (see Chap. 2) that a
quadratic cost function has a single minimum, which can be derived through
the following matrix formula:
w
=
X
T
X
−
1
X
T
Y
,
where the (
N,n
) observation matrix
X
=(
x
1
;
...
;
x
k
;
...
x
N
)andthe(
N
,1)
column vector
y
=(
y
1
;
...
;
y
k
;
...
;
y
N
) are constructed from the input and the
output data. This result is available if the quadratic minimization problem is
well-posed, i.e., the matrix (
X
T
X
) is invertible.
That algorithm may be used for autoregressive model identification. That
type of model (ARX model) was introduced in the previous section.
x
(
k
+1)=
a
1
x
(
k
)+
···
+
a
p
x
(
k
−
p
+1)+
b
0
v
(
k
+1)+
c
1
u
(
k
)+
···
+
c
r
u
(
k
−
r
+1)
.
Note that there is a correlation of the input and the output. Here the
regression coe
cient vector is
w
=[
a
1
,...,a
p
,b
0
,c
1
,...,c
r
]
T
.
When an input trajectory [
u
(1)
,...,u
(
k
)
,...,u
(
N
)] and an output tra-
jectory [
x
(1
,...,x
(
k
)
,...,x
(
N
)] are available, the (1
,p
+
r
) input vectors of
the regression are constructed as follows:
x
k
=
x
(
k
);
...
;
x
(
k
−
p
+1);
u
(
k
);
...
;
u
(
k
−
r
+ 1)] for
k
varying from max(
p,r
)+1to(
N
−
1) and the
associated output is
y
k
=
x
(
k
+1).
High quality results are obtained provided that the linear model of the
estimator is relevant. This assertion is supported by the following example.
4.2.1.2 Example of Application
Let us consider the (2, 2) order ARX model,
x
(
k
+1)=
a
1
x
(
k
)+
a
2
x
(
k
−
1) +
b
0
v
(
k
+1)+
c
1
u
(
k
)+
···
+
c
2
u
(
k
−
1)
,
with the following real values for the parameters:
a
1
=1
.
2728
,
2
=
−
0
.
81
,
0
=0
.
5
,
1
=0
.
5
,
2
=
−
0
.
5
,
and where the operator-designed input trajectory (
u
k
) is a white noise.
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