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Moreover, the prediction formula for the nonlinear case is the following:
x
ˆ
(
k
+
1
)
=
ˆ
x
(
k
/
k
−
1
)
+
(
k
)
+
K
(
k
)(
y
(
k
)
−
g
(
ˆ
x
(
k
/
k
−
1
),
u
(
k
)))
(1.74)
It must be noted that in this case, the system matrices in this depend on both
the state and input of the system in each instant. Thus, it becomes necessary the
calculation of these matrices in each iteration of the algorithm.
1.5.3 Kalman Filter for Parameters' Estimation
One of the applications of Kalman filter is the identification of parameters. Let us
suppose that a function depends on q parameters
p
1
,
p
2
...
p
q
n
m
f
:
→
(1.75)
y
=
f
(
x
1
,
x
2
,...,
x
n
,
p
1
,
p
2
...
p
q
)
=
f
(
x
,
p
)
(1.76)
The problem of identification of parameters can be explained as a problem of
estimation of systems' states.
p
(
k
+
1
)
=
p
(
k
)
(1.77)
(
+
)
=
(
(
))
+
(
)
y
k
1
f
p
k
e
k
(1.78)
of the function to be
identified, Kalman filter can be used with the following particularities. The matrix
Then, if we have a set of m samples
{
x
1
k
,
x
2
k
,...,
x
nk
,
y
k
}
will be an identity matrix in this case. It is assumed a free system without an external
input so the matrix
is null and the matrix C can be calculated as follows:
))
=
∂
f
C
(
p
(
k
p
|
p
=
p
(
k
)
(1.79)
∂
Thematrices
R
1
and
R
12
become null, while
R
2
is selected based on trial and error.
If
y
I
, which would correspond to Gaussians error
functions N(0,1), and the function is a linear one, the algorithm becomes equivalent
to the recursive minimum square one. The initial covariance state matrix is supposed
to be
P
∈
and we suppose that
R
2
=
cI
where C is a number relatively large with respect to the data of the
problem. The formulation of the algorithm becomes:
(
0
)
=
K
(
k
)
=
P
(
k
/
k
−
1
)
C
t
CP
(
k
/
k
−
1
)
C
t
+
R
2
−
1
(1.80)
p
(
k
+
1
/
k
)
=
p
(
k
/
k
−
1
)
+
K
(
k
)(
y
k
−
f
(
x
1
k
,
x
2
k
,...,
x
nk
,
p
(
k
/
k
−
1
)))
(1.81)
P
(
k
+
/
k
)
=
P
(
k
/
k
−
)
−
K
(
k
)
CP
(
k
/
k
−
)
(1.82)
1
1
1
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