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In-Depth Information
v
t
R
1
=
(
(
),
(
))
E
v
k
k
(1.54)
e
t
R
12
=
(
(
),
(
))
E
v
k
k
(1.55)
e
t
R
2
=
E
(
e
(
k
),
(
k
))
(1.56)
It is also assumed that the initial condition x(0) is Gaussian distributed with
m
0
=
E
(
x
(
0
))
(1.57)
E
(
t
R
(
0
)
=
x
(
0
)
−
m
0
)(
x
(
0
)
−
m
0
)
(1.58)
where E(.) is the expectation operator. It is supposed that
x
ˆ
(
k
/
k
−
1
),
u
(
k
)
and
y
(
k
)
are known and the objective is to estimate
. The prediction problem can be
improved by introducing the difference between the measured and estimated outputs,
x
ˆ
(
k
+
1
/
k
)
y
)
as a feedback gain:
(
k
)
−
C
x
ˆ
(
k
/
k
−
1
x
ˆ
(
k
+
1
/
k
)
=
ˆ
x
(
k
/
k
−
1
)
+
u
(
k
)
+
K
(
k
)(
y
(
k
)
−
C
x
ˆ
(
k
/
k
−
1
))
(1.59)
The resultant prediction error is the difference between the state of the real system
and the estimated one which can be stated as the following:
ε(
k
+
1
)
=
x
(
k
+
1
)
−ˆ
x
(
k
+
1
/
k
)
(1.60)
(
)
(
)
It should be observed that as above mentioned Gaussian errors
v
k
and
e
k
are
with zero mean, it can be verified that:
ε(
k
+
1
)
=
(
−
K
(
k
)
C
)ε(
k
)
(1.61)
Thus,
)
⇒ˆ
m
0
⇒∀
ε(
0
x
(
0
)
=
k
>
0
ε(
k
)
=
0
(
ˆ
x
(
k
)
=
m
k
)
(1.62)
And if the dynamics of (
1.61
) is stable, then:
∀
x
(
0
)
lim
→∞
¯
e
(
k
)
=
0
⇒
lim
→∞
ˆ
x
(
k
)
=
m
k
(1.63)
k
k
The secondary objective is to minimize the covariance matrix which is denoted
as P(k),
t
P
(
k
)
=
E
((ε
−
ε).(ε
−
ε)
)
(1.64)
in the sense that it approaches its minimum for:
t
P
n
min
(α
(
k
)α)
∀
α
∈
(1.65)
The algorithm of Kalman filter can be summarized by the following iterative
process:
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