Biomedical Engineering Reference
In-Depth Information
.;1/.Set
u
.1/
Assume also that the initial data
u
.1/
G
D . Then, the Kalman
p
filter formulas read
u
.k/
p
D
u
.k
1
c
;
.k/;2
D
.k
1/;2
c
p
.k/;2
p
.k/;2
K
k
D
C 1
p
.k/;2
.k/;2
1
.k/;2
p
.k/;2
p
.k/;2
u
.k/
c
D
u
.k/
p
.
z
.k/
u
.k
p
/
u
.k/
p
z
.k/
C
D
C
C 1
C 1
C 1
p
p
p
.k/;2
.k
1/;2
p
.k/;2
c
.k
1/;2
.k/;2
c
D
C 1
D
:
C 1
p
c
We have therefore
C
z
.1/
2
1
2
;
u
.1/
u
.1/
p
D ;
.1/
p
D
u
.2
p
:
D 1; K
1
D
D
c
Notice that the prediction at k D 2 is just the sample average of the “past” and the
new data. Similarly we obtain at a generic step k
C
P
j
D
1
z
.j/
k C 1
u
k
C
1
p
D
u
c
D
:
Actually, we have the arithmetic average of the available data at t
k
, that is somehow
intuitively expected. Moreover, we have the recursive formula
.k/;2
p
.k/;2
.k
C
1/;2
p
; with
.1/
p
D
D 1:
C 1
p
1
k
: Consequently we have that
By induction one can check that
.k/;2
D
p
.k
p
D 0, i.e., the prediction is asymptotically exact; similarly, the estimate
is asymptotically exact;
2.
the ARE
2
1.
lim
k
!1
D
2
=.1 C
2
/ has only one solution, that is 0;
3.
the Kalman filter is asymptotically stable, whereas the dynamic system is not
asymptotically stable.
This example provides the case of an asymptotically stable estimator even when
the dynamical system is not stable. As we have pointed out, the “reverse” situation
(system is stable, estimator is unstable) is not possible: when the system is stable,
the predictor is automatically stable.
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