Information Technology Reference
In-Depth Information
We define
X
(
k
) as the least squares optimal estimate, i.e., the linear re-
gression of the random state vector
X
(
k
) onto the random vector of past
measurements up to time
k
:
Y
(
k
)=[
Y
(1);
...
;
Y
(
k
)]. Let
ϑ
(
k
+1) be the
innovation at time
k
+ 1. It is defined by
HA X
(
k
)
ϑ
(
k
+1)=
Y
(
k
+1)
−
−
HBu
(
k
)
.
The innovation filter recursive equation is
X
(
k
+1)=
A X
(
k
)+
Bu
(
k
)+
K
k
+1
ϑ
(
k
+1)
where the innovation gain is inferred from the computation formula of linear
regression:
K
k
+1
=Cov[
X
(
k
+1)
,
ϑ
(
k
+ 1)]Var[
ϑ
(
k
+1)]
−
1
.
X
(
k
)and
P
k
stands for the covariance matrix of the estimation error
X
(
k
)
−
P
k
+1
stands for the covariance matrix of the prediction error
A X
(
k
)
X
(
k
+1)
−
−
Bu
(
k
)
.
Let us compute covariance of the prediction error. One obtains
A X
(
k
)
X
(
k
)] +
V
(
k
+1)
.
X
(
k
+1)
−
−
Bu
(
k
)=
A
[
X
(
k
)
−
X
(
k
), the prediction error covari-
ance propagation equation is easily computed using a quadratic expansion,
Because
V
(
k
+ 1) is uncorrelated to
X
(
k
)
−
P
k
+1
=
AP
k
A
T
+
Q
.
From the definition of innovation error,
HA X
(
k
)
ϑ
(
k
+1)=
Y
(
k
+1)
−
−
HBu
(
k
)
X
(
k
)] +
V
(
k
+1)
=
H
{
A
[
X
(
k
)
−
}
+
W
(
k
+1)
.
The value of its covariance matrix is deduced in a similar way, expressed as a
function of the prediction error at time
k
Var[
ϑ
(
k
+1)]=
HP
k
+1
H
T
+
R
.
Let us compute the covariance between the state
X
(
k
+ 1) and the innovation
ϑ
(
k
+1),
HA X
(
k
)
Cov[
X
(
k
+1)
,
Y
(
k
+1)
−
−
HBu
(
k
)]
X
(
k
)]
=Cov
{
AX
(
k
)+
V
(
k
+1)
,
HA
[
X
(
k
)
−
+
HV
(
k
+1)+
W
(
k
+1)
}
X
(
k
)]
=Cov
{
AX
(
k
)
,
HA
[
X
(
k
)
−
}
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