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including a weighting of the norm of P vector We propose to describe the problem
as an optimum estimation of the state of the system.
p
(
k
+
1
)
=
p
(
k
)
(1.89)
y
k
δ
f
(
x
1
k
,
x
2
k
,...,
x
nk
,
p
(
k
))
=
+
e
(
k
)
(1.90)
p
0
δ
p
(
k
)
where
is a relatively small positive number in order not to disturb the principal
objective of the problem. It should be noted that the matrix C of the algorithm vary
as follows:
δ
∂
f
∂
p
|
p
=
p
(
k
)
C
(
ˆ
p
(
k
))
=
(1.91)
δ
I
It should be noticed that the function f is a linear one with respect to the parameters
and therefore it can be calculated in a direct form as follows:
w
(
i
1
...
i
n
)
(
∂
f
x
k
)
|
p
=
p
(
k
)
=
r
1
i
1
1
...
r
n
(1.92)
p
(
i
1
...
i
n
)
0
1
w
(
i
1
...
i
n
)
(
x
k
)
∂
=
i
n
=
and
w
(
i
1
...
i
n
)
(
∂
f
x
k
)
|
p
=
p
(
k
)
=
r
1
i
1
=
1
...
r
n
x
jk
for
j
=
1
...
n
(1.93)
p
(
i
1
...
i
n
)
j
i
n
=
1
w
(
i
1
...
i
n
)
(
x
k
)
∂
Therefore, the Jacobian coincides with the row k of the matrix defined in Sect.
1.2
.
x
nk
(1.94)
∂
∂
β
(
1
...
1
)
k
β
(
1
...
1
)
k
x
1
k
...β
(
1
...
1
)
k
x
nk
...β
(
r
1
...
r
n
)
k
...β
(
r
1
...
r
n
)
k
p
|
p
=
p
(
k
)
=
X
k
=
And thus, the problem can be formulated as an estimation of the state of the linear
system
p
(
k
+
1
)
=
p
(
k
)
(1.95)
y
k
δ
=
C
(
ˆ
p
(
k
)).
p
(
k
)
+
e
(
k
)
(1.96)
p
0
where
X
k
δ
C
(
ˆ
p
(
k
))
=
(1.97)
I
The prediction formula in this case becomes:
y
k
δ
p
ˆ
(
k
+
1
/
k
)
=ˆ
p
(
k
/
k
−
1
)
+
K
(
k
)
−
C
(
p
(
k
)).
p
(
k
)
(1.98)
p
0
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