Global Positioning System Reference
In-Depth Information
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w
k
is the system process noise and is distributed as
w
k
∼
N(
o
,
Q
w
k
)
. The notation
(
−
)
indicates the predicted value. Thus,
x
k
(
−
=
Φ
k
−
1
x
k
−
1
+
)
w
k
(4.390)
x
k
(
)
is the predicted parameter vector at epoch
k
, based on the estimated parameter
x
k
−
1
(
−
)
from the previous epoch and the dynamic model. The solution that generated
x
k
−
1
also generated the respective cofactor matrix
Q
k
−
1
. The observation equations
fo
r epoch
k
are given in the familiar form
−
v
k
=
A
k
x
k
+
k
(4.391)
w
ith
v
k
∼
k
)
.
The first step in arriving at the Kalman filter formulation is to apply variance-
co
variance propagation to (4.389) to predict the parameter cofactor matrix at the next
epoch,
N(
o
,
Q
[16
Lin
—
5.7
——
Lon
PgE
k
Q
k
(
−
)
=
Φ
k
−
1
Q
k
−
1
Φ
+
Q
w
k
(4.392)
−
1
Expression (4.392) assumes that the random variables
k
and
w
k
are uncorrelated. The
various observation sets
k
are also uncorrelated, as implied by (4.88). The second
step involves updating the predicted parameters
x
k
(
−
)
, based on the observations
k
.
Following the sequential least-squares formulation (4.140) to (4.144), we obtain
Q
)
A
k
−
1
T
k
=
k
+
A
k
Q
k
(
−
(4.393)
[16
K
k
A
k
x
k
(
k
x
k
=
x
k
(
−
)
−
−
)
+
(4.394)
Q
k
=
[
I
−
K
k
A
k
]
Q
k
(
−
)
(4.395)
v
T
Pv
k
−
1
+
A
k
x
k
(
+
k
T
T
k
A
k
x
k
(
+
k
v
T
Pv
k
=
−
)
−
)
(4.396)
w
here the matrix
)
A
k
T
k
K
k
=
Q
k
(
−
(4.397)
is called the Kalman gain matrix.
If the parameter
x
k
+
1
depends only on the past (previous) solution
x
k
, we speak of
a first-order Markov process. If noise
w
k
has a normal distribution, we talk about a
first-order Gauss-Markov process,
x
k
+
1
=
ϕx
k
+
w
k
(4.398)
with
w
k
∼
n(
0
,q
w
k
)
. In many applications a useful choice for
ϕ
is
e
−
T/
τ
ϕ
=
(4.399)
