11.8 Alternative Kalman equations
Having shown that the covariance matrix can be updated via the previous equation it is possible to
formulate an alternative Kalman gain, as follows;
K
k
=
P
0
k
H
T
,
HP
0
k
H
T
+
R
,
1
(11.53)
Inserting
P
k
P
,
1
k
and
R
R
,
1
;
K
k
=
P
k
P
,
1
k
P
0
k
H
T
R
,
1
R
,
HP
0
k
H
T
+
R
,
1
=
P
k
P
,
1
k
P
0
k
H
T
R
,
1
,
HP
0
k
H
T
R
,
1
+
I
,
1
=
P
k
,
I
+
H
T
R
,
1
HP
0
k
H
T
R
,
1
,
HP
0
k
H
T
R
,
1
+
I
,
1
=
P
k
H
T
R
,
1
,
I
+
H
T
R
,
1
HP
0
k
,
I
+
HP
0
k
H
T
R
,
1
,
1
=
P
k
H
T
R
,
1
(11.54)
Replacing
P
k
with the inverse of equation 11.50;
K
k
=
HR
,
1
H
T
+
P
0,
1
k
,
1
H
T
R
,
1
(11.55)
Which is the same as the gain calculated from the chi-square equations, conrming that the gains are
indeed equivalent.
Although an alternative recursive algorithm has been developed the ob jectivewas to demonstrate the
relationship between the Kalman lter and the chi-square statistic, showing how the Kalman lter em-
bodies this statistic. The diagram of gure
??
shows how the alternative set of lter equations maybe
used to implement a Kalman lter. This form of the lter may be attractive due to the simplied gain
calculation and some authors have been able to use this form of the lter in a distributed implementa-
tion [
?
]. However in this form the lter requires two matrix inversions which can be a computational
burden, particularly when large matrices are involved. Thus the preferred implementation here is that
given in gure 11.5.
11.9 Conclusions
This tutorial has shown how the Kalman lter may be derived from the desire to minimise the mean
squared error of a signal prediction. Several points in the derivation have been emphasised;
The minimisation of the mean squared error is shown to be applicable when the expected errors
on the signal are distribution as a Gaussian. Under such conditions the minimisation of the mean
squared error between the data and the data prediction leads to the developmentofa
maximum
likelihood statistic
It has been shown how the Kalman lter can be thought of in terms of a chi-squared minimiser
by deriving an alternative form of the Kalman lter which highlights its statistical constructs
including the processes of error propagation and data combination. This derivation leads to a
common, alternative set of lter equations.
In summary, although the Kalman lter is optimal in the mean-squared error sense, it is limited, prac-
tically by the quality and accuracy of the model which is embedded within it. However, without an
appropriate model the lter is unable to perform the task for which it is designed. The following sections
describe a new statistical approach to the solution of this problem.
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