Global Positioning System Reference
In-Depth Information
The Kalman filter for estimating x produces an estimate of x , incorporating the
current measurement y ( t n ) and the estimate of x just prior to the measurement,
denoted as
xt . Since x is a constant, there is no difference between
$
()
xt n + 1
$
(
)
, the
estimate just after incorporating the previous measurement, and
xt . This is not
true for more general system models, as we shall see when we discuss real-world
applications in this section. The measurement update equation is given by:
$
()
[
]
() () ()()
()
$
+
$
$
xt
=
xt
+
kt
yt
xt
(9.2)
n
n
n
n
n
xt is corrected by addition of new informa-
tion contained in the measurement. The estimation error
$
In this equation, the prior estimate
()
~ ()
xt +
is given by
~
()
()
$
xt
+
=−
x
xt
+
n
n
xt
Due to the simple measurement model and the zero mean nature of
ε m ( t n ),
$
()
is effectively an estimate of measurement y ( t n ) [i.e.,
xt
$
()
=
yt
$
()
]. The difference
n
n
sequence yt
is called the innovation process and contains the new infor-
mation obtained by the measurements. The parameter k ( t n ) in (9.2) is the Kalman
gain and contains the statistical parameters required to form the combination of the
prior estimate and new data resulting in a minimum error variance estimate. The
quality of the estimate is characterized by the error variance, but since the estimate
is unbiased, the error variance equals the estimate variance
()
yt
$
()
n
n
t 2 , and its value is
different before and after updating the estimate by incorporating the measurement.
The Kalman gain is computed as follows:
σ $ ()
x
n
()
()
2
σ
t
$
n
()
x
kt
=
(9.3)
n
2
σ
t
+
σ
2
$
x
n
m
σ 2
Note that if the measurement is less accurate (i.e.,
is large), the weighting
σ 2 appears in the denominator. After the
measurement update, the error covariance is reduced according to
given to the new data is reduced because
()
[
]
()
()
2
2
σ
t
+
=−
1
k t
σ
t
(9.4)
$
n
n
$
n
x
x
If we further assume that the measurement noise variance is constant, then it is easy
to show that the error variance is given by
and thus asymptotically
approaches zero as more data is obtained. This property makes the estimate a con-
sistent estimate. Thus, (9.2) to (9.4) provide a data processing scheme that recur-
sively combines our previous state estimate with new measurement data in a way
that is statistically optimal. A block diagram of the basic filter structure is shown in
Figure 9.6.
In real-world applications, the state vector contains several components that are
not constant but evolve dynamically, such as position and velocity. Also, the system
state model includes plant noise, which expresses modeling errors as well as actual
noise and system disturbances. Since, in general, the system state varies dynamically
between measurements, the estimate just after the measurement update
σ
2
()
t
+
=
σ
2
/
n
$
x
n
m
xt +
$
()
must
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