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available, and thus a two state filter model can be used to replace the three state
model. In this case,
T
3
T
2
⁄
3
⁄
2
2σ
2
τ
m
C
=
(9.153)
T
2
⁄
2
T
1
T
01
Φ
=
(9.154)
αβγ
9.9.2. Relationship between Kalman and
Filters
The relationship between the Kalman filter and the
αβγ
Φ
filters can be easily
obtained by using the appropriate state transition matrix
, and gain vector
K
corresponding to the
αβγ
in Eq. (9.127). Thus,
k
1
()
xnn
(
)
xnn
1
(
)
ß
ß
=
+
[
x
0
()
xnn
1
(
)
]
k
2
()
(9.155)
(
nn
)
(
nn
1
)
ßß
ßß
(
nn
)
(
nn
1
)
k
3
()
with (see
Fig. 9.21
)
T
2
2
T
ß
s
ßß
s
-----
xnn
1
(
)
=
x
s
(
n
1
)
+
(
n
1
)
+
(
n
1
)
(9.156)
ß
nn
ß
s
T
ßß
s
(
1
)
=
(
n
1
)
+
(
n
1
)
(9.157)
ßß
nn
ßß
s
(
1
)
=
(
n
1
)
(9.158)
Comparing the previous three equations with the
αβγ
filter equations yields
α
β
---
γ
T
2
k
1
k
2
k
3
=
(9.159)
-----
Additionally, the covariance matrix elements are related to the gain coeffi-
cients by
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