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Note that the evolution of the covariance matrix does not depend on the
measurements. That remark is of practical importance for real-time applica-
tions of the Kalman filter (on-board navigation devices, for instance) because
the computation of the Kalman gain sequence may be performed once and for
all, from the model equations and the initial covariance error.
4.4.2.2 Properties of the Kalman Filter
The consequences of the previous paragraph are very important. Some of
them may be extended to more general models. Let us summarize the main
properties of the Kalman filter:
•
When we compare the results of the computation of the innovation gain in
the variational framework and in the probabilistic framework, the Kalman
filter appears to be an optimal innovation filter in the sense of the varia-
tional framework. Penalties are time-dependent, in matrix form, and may
be pre-computed. They are interpreted as the covariance of the predic-
tion error (penalizing the model uncertainty) and the covariance of the
measurement error (penalizing the measurement uncertainty).
•
It was shown that the Kalman filter is unconditionally stable and is a
consistent estimate of the state: the dynamics of the error converges to-
wards an optimal steady regime even when the dynamical system itself is
unstable (for details and proofs see [Anderson 1979; Haykin 1996]).
•
The innovation sequence is the result of successive linear regressions.
Therefore, it is uncorrelated and independent in the gaussian model.
Whitening of innovation is an optimality characteristics, which may be
computationally observed and tested.
4.4.2.3 Kalman Filtering for a Time-Varying Linear System
Kalman filtering may be applied in a straightforward way for linear nonsta-
tionary models. Let the state equation be
X
(
k
+1)=
A
(
k
)
X
(
k
)+
B
(
k
)
u
(
k
)+
V
(
k
+1)
and the measurement equation
Y
(
k
)=
H
(
k
)
X
(
k
)+
W
(
k
)
,
where the state noise sequence
V
(
k
) and the measurement noise sequence
W
(
k
) have time-varying covariance matrices
Q
(
k
)and
R
(
k
). The filter equa-
tion is
X
(
k
+1)=
A
(
k
)
X
(
k
)+
B
(
k
)
u
(
k
)+
K
k
+1
ϑ
(
k
+1)
,
with
H
(
k
+1)
A
(
k
)
X
(
k
)
ϑ
(
k
+1)=
Y
(
k
+1)
−
−
H
(
k
+1)
B
(
k
)
u
(
k
)
.
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