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The corrector equation (covariance of the smoothed estimate) is
P
nn
(
)
=
[
IK
()
G
]
P
nn
1
(
)
(9.133)
Finally, the predictor equation is
x
n
(
+
1
n
)
=
Φ
x
nn
(
)
(9.134)
αβγ
9.9.1. The Singer
-Kalman Filter
The Singer
1
filter is a special case of the Kalman where the filter is gov-
erned by a specified target dynamic model whose acceleration is a random pro-
cess with autocorrelation function given by
t
1
τ
m
------
E
ßß
()
ßß
σ
2
{
(
t
+
)
}
=
e
(9.135)
1
where is the correlation time of the acceleration due to target maneuvering
or atmospheric turbulence. The correlation time may vary from as low as
10 seconds for aggressive maneuvering to as large as 60 seconds for lazy
maneuvering cases.
τ
m
τ
m
Singer defined the random target acceleration model by a first order Markov
process given by
ßß
ßß
() 1 ρ
2
(
n
+
1
)
=
ρ
m
+
σ
m
w
()
(9.136)
where is a zero mean, Gaussian random variable with unity variance,
is the maneuver standard deviation, and the maneuvering correlation coef-
ficient
w
()
σ
m
ρ
m
is given by
T
τ
m
-----
ρ
m
=
e
(9.137)
The continuous time domain system that corresponds to these conditions is the
same as the Wiener-Kolmogorov whitening filter which is defined by the dif-
ferential equation
d
v
()
=
β
m
v
()
+
w
()
(9.138)
d
t
1. Singer, R. A., Estimating Optimal Tracking Filter Performance for Manned Maneu-
vering Targets,
IEEE Transaction on Aerospace and Electronics, AES-5
, July, 1970,
pp. 473-483.
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