Biomedical Engineering Reference
In-Depth Information
As to be expected, most of the analysis holding for the linear case cannot
be trivially extended to this case, since the covariance matrices associated with
the errors depend on the linearization procedure. In particular, they depend on
the set of observations so they are a random process. In addition, they are only
an approximation of the error covariance and this leads to biased state estimates
(
e
.k
c
¤ 0). Another drawback is the computational cost associated with
the tangent operators, that for problems coming from the discretization of partial
differential equations may be fairly expensive.
Nevertheless, we address the case of parameter estimation with the EKF.
E
EKF and Parameter Estimation
Let us apply EKF to (
6.29
), with
2
3
H.#/
@H
@#
u
:
A.#/
@A
@#
u
4
5
;
A
0
.
v
/ D
H
0
.
v
/ D
O
I
H
0
.
v
/ is zero. We
assume that this is the case hereafter. In this form, the parameter estimation is
performed following the EKF steps. The covariance matrix of the augmented status
will be
In many cases, H is independent of # so that the last entry in
ƒ
u
;
ƒ
u
#;
ƒ
u
#;
ƒ
#;
ƒ
augm;
where the dot can be either p in the prediction or c for the correction (or estimate).
Then,wehavethefollowingsteps.
1.
P
REDICTION
(a)
u
.k/
.#
.k
1
c
/
u
.k
1
c
; #
.k/
D #
.k
1/
D
A
p
p
c
"
Aƒ
.k
1/
#
@A
T
@#
@A
@#
u
.k
1/
ƒ
.k
1/
u
;c
A
T
C
B
C
B
T
#;c
u
.k
1/;T
C
C
(b)
ƒ
.k/
augm;p
D
c
c
ƒ
.k
1/
#;c
C
T
@A
T
@#
,CD Aƒ
.k
1/
@A
@#
u
.k
1
c
ƒ
.k
1/
where B D Aƒ
.k
1/
u
#;c
u
.k
1/;T
C
and all
c
u
#;c
#;c
the occurrences of A and its derivative are computed in #
.k
1/
.
c
2.
C
ORRECTION
Kalman gain:
"
ƒ
.k/
#
Hƒ
.k/
K
k;1
K
k;2
:
C R
k
1
p;
u
H
T
ƒ
.k/
p;
u
H
T
K
k
D
D
p;
u
#
H
T
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