Biomedical Engineering Reference
In-Depth Information
4.1
Kalman Filter
The Kalman filter is an iterative update scheme that minimizes the cost function
J
when the system state evolves according to a linear model,
u
n
=
M
n
u
n
−
1
,andthe
observation operator is also linear
y
i
=
H
i
u
i
+
ε
i
.
(12)
We begin the derivation with some assumptions. Suppose at time
t
n
−
1
we have the
minimizer,
u
a
n
−
1
=
, which we assume represents a Gaussian distribution
with an associated covariance matrix
P
a
n
−
1
. This assumption is motivated by the
fact that a Gaussian distribution propagates to a Gaussian distribution under a linear
model. In the absence of new observations the most likely estimate of the true
system state is the background
u
(
t
n
−
1
)
u
b
n
=
M
n
u
a
n
−
1
.
(13)
The background covariance matrix is
M
n
P
a
n
−
1
M
n
+
P
b
n
=
C
n
.
(14)
Here
C
n
is assumed to be positive definite and represents model error. For
simplicity we assume that
C
n
=
0. If
u
is a state vector and
c
an arbitrary constant,
we assume algebraically that the analysis completes the square of the cost function:
n
1
j
=
1
[
y
j
−
H
j
M
n
−
1
,
j
u
]
−
T
R
−
1
j
y
j
−
T
P
−
1
[
H
j
M
n
−
1
,
j
u
]=[
u
−
u
a
n
−
1
]
a
n
−
1
[
u
−
u
a
n
−
1
]+
c
.
(15)
When a new observation vector,
y
n
, becomes available at time
t
n
, a simple induction
argument applied to Eq. (
15
) at the new time shows how the updated analysis
minimizes the cost
T
P
−
1
b
n
T
R
−
1
n
y
n
−
y
n
−
J
[
u
(
t
)] = [
u
−
u
b
n
]
[
u
−
u
b
n
]+[
H
n
u
]
[
H
n
u
]
(16)
T
P
−
1
=[
−
u
a
n
]
[
−
u
a
n
]+
.
u
u
c
a
n
If one thinks of the covariance matrices in Eq. (
16
) as numbers, the effect of
updating the analysis is intuitive. For example suppose
P
b
n
is large compared to
R
n
.
Then the inverse
P
−
1
b
n
will be smaller than
R
−
n
. Hence, the cost function,
J
,gives
more weight to the observation and the resulting analysis will give more preference
to the observations. Figure
1
illustrates the result of this process. The analysiss
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