Graphics Programs Reference
In-Depth Information
9.8. Fixed-Gain Tracking Filters
This class of filters (or estimators) is also known as ÐFixed-CoefficientÑ fil-
ters. The most common examples of this class of filters are the and
filters and their variations. The and trackers are one-dimensional sec-
ond and third order filters, respectively. They are equivalent to special cases of
the one-dimensional Kalman filter. The general structure of this class of esti-
mators is similar to that of the Kalman filter.
αβ
αβγ
αβ
αβγ
The standard filter provides smoothed and predicted data for target
position, velocity (Doppler), and acceleration. It is a polynomial predictor/cor-
rector linear recursive filter. This filter can reconstruct position, velocity, and
constant acceleration based on position measurements. The filter can also
provide a smoothed (corrected) estimate of the present position which can be
used in guidance and fire control operations.
αβγ
αβγ
Notation:
For the purpose of the discussion presented in the remainder of this chapter,
the following notation is adopted: represents the estimate during the
sampling interval, using all data up to and including the
xnm
(
)
nth
mth
sampling
interval;
y
n
is the
nth
measured value; and
e
n
is the
nth
residual (error).
The fixed-gain filter equation is given by
x
nn
(
)
Φ
x
n
1
=
(
n
1
)
+
K
y
n
[
G
Φ
x
n
1
(
n
1
)
]
(9.65)
Since the transition matrix assists in predicting the next state,
x
n
(
+
1
n
)
=
Φ
x
nn
(
)
(9.66)
Substituting Eq. (9.66) into Eq. (9.65) yields
x
nn
(
)
=
x
nn
1
(
)
+
K
y
n
[
Gx
n n
(
1
)
]
(9.67)
The term enclosed within the brackets on the right hand side of Eq. (9.67) is
often called the residual (error) which is the difference between the measured
input and predicted output. Eq. (9.67) means that the estimate of
x
()
is the
sum of the prediction and the weighted residual. The term
Gx
n n
(
1
)
repre-
sents the prediction state. In the case of the
αβγ
estimator,
G
is the row vector
given by
G
=
(9.68)
1
00…
and the gain matrix
K
is given by
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