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To apply the Kalman filter algorithm, the nonlinear evolution model is
replaced by its linearization around the current estimate
X
(
k
), and the non-
linear measurement model is replaced by its linearization around the predicted
state
f
[
X
(
k
)
,
u
(
k
)] in order to compute the covariance propagation.
Thus,
A
(
k
) stands for the gradient of
f
with respect to
x
at the point
[
X
(
k
)
,
u
(
k
)], and
H
(
k
+ 1) stands for the gradient of
h
at the point
f
[
X
(
k
),
u
(
k
)].
The filter equation is determined by the predictor-corrector scheme,
X
(
k
+1)=
f
[
X
(
k
)
,
u
(
k
)] +
K
k
+1
ϑ
(
k
+1)
,
with
f
[
X
(
k
)
,
u
(
k
)]
ϑ
(
k
+1)=
Y
(
k
)
−
h
{
}
.
The iteration of covariance update is (using convenient linearizations, see
[Anderson 1979])
P
k
+1
=
A
(
k
)
P
k
A
(
k
)
T
+
Q
(
k
+1)
K
k
+1
=
P
k
+1
H
(
k
+1)
T
[
H
(
k
+1)
P
k
+1
H
(
k
+1)
T
+
R
(
k
+1)]
−
1
P
k
+1
=[
I
K
k
+1
H
(
k
+1)][
A
(
k
)
P
k
A
(
k
)
T
+
Q
(
k
+1)]
−
K
k
+1
H
(
k
+1)]
T
+
K
k
+1
R
(
k
+1)
K
T
×
[
I
−
k
+1
.
The computation of the gain is subject here to an approximation. There-
fore, no optimality property can be provided any longer. If the approximation
is valid, that algorithm can provide a sub-optimal solution (the quality of
the solution is near optimal). The stability of the linearized Kalman filter is
much more di
cult to prove than for time-varying linear Kalman filter. More-
over, the gain computation must be performed on-line. That is very hard for
real-time applications and on-board computers. For that type of applications,
tracking a reference trajectory, and computing a filter for a linearization of the
model along that trajectory, are usually preferred. In that case, the algorithm
of the previous section is used. Nevertheless, the extended Kalman filter is
currently used, especially for identification problems. In the following section,
we address that issue, using a state extension.
4.4.3.2 Using Extended Kalman Filter for Parametric
Identification
Consider the following model for an observed controlled dynamical system
X
(
k
+1)=
A
(
θ
)
X
(
k
)+
B
(
k
)
u
(
k
)+
V
(
k
+1)
Y
(
k
)=
H
(
θ
)
X
(
k
)+
W
(
k
)
,
where the model depends on an unknown parameter
θ
. One has to estimate
the unknown parameter. Depending on the application,
θ
may be constant
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