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TheKalman filter can be conceptualized as two distinct phases: predict and update.
The predict phase uses the estimated state from the previous time-step to produce
an estimation of the state at the current time-step. This predicted state is also known
as the a priori estimated state because, although it is an estimate of the state at
the current time-step, it does not include observation information from the current
time-step. In the update phase, the current a priori prediction is combinedwith current
observation information to refine the estimate state. This improved estimate is termed
the a posteriori estimate state.
Following the previous, the EKF can be solved by iterative application of the
following set of equations (Grewal and Andrews
2001
):
Predict:
x
ˆ
(
k
|
k
−
1
)
=
(
k
)
ˆ
x
(
k
−
1
|
k
−
1
)
+
(
k
)
u
(
k
)
(2.10)
T
P
(
k
|
k
−
1
)
=
(
k
)
P
(
k
−
1
|
k
−
1
)
(
k
)
+
R
v
(2.11)
Update:
R
ve
C
R
e
−
1
C
T
C
T
K
(
k
)
=
(
k
)
P
(
k
|
k
−
1
)
(
k
)
+
(
k
)
P
(
k
|
k
−
1
)
(
k
)
+
(2.12)
)
y
)
x
ˆ
(
k
|
k
)
= ˆ
x
(
k
|
k
−
1
)
+
K
(
k
(
k
)
− ˆ
y
(
k
(2.13)
C
R
ve
T
T
P
(
k
|
k
)
=
(
k
)
P
(
k
|
k
−
1
)
(
k
)
+
R
v
−
K
(
k
)
(
k
)
P
(
k
|
k
−
1
)
(
k
)
+
,
(2.14)
where
represent respectively the estimate of
x
predicted (a
priori), and updated estimation (a posteriori) given observations up to, and including
at time
k
.
P
x
ˆ
(
k
|
k
−
1
)
and
x
ˆ
(
k
|
k
)
(
k
|
k
−
1
)
is the predicted (a priori) estimate covariance, and
P
(
k
|
k
)
is
the updated (a posteriori) estimate covariance matrix.
K
)
are the estimated outputs, and
R
v
,
R
ve
and
R
e
are the noise covariance matrices,
estimated from the hope operator,
E
(
k
)
is the Kalman gain,
y
ˆ
(
k
(
·
)
:
E
v
v
T
R
v
=
(
k
)
(
k
)
(2.15)
E
v
e
T
R
ve
=
(
)
(
)
k
k
(2.16)
E
e
e
T
R
e
=
(
k
)
(
k
)
.
(2.17)
The iterative process starts with an initial estimate of state vector
ˆ
x
(
0
)
=
m
0
=
E
(
x
(
0
))
and an initial value of the covariance matrix
P
(
0
)
=
E
((
x
(
0
)
−
m
0
)(
x
(
0
)
T
−
m
0
)
)
, being known
x
(
0
|−
1
)
,
u
(
0
)
and
y
(
0
)
. Then it is evolving online with
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