Biomedical Engineering Reference
In-Depth Information
(a)
State and Parameter:
K
k;1
z
.k/
H.#
.k
p
/
u
.k/
;
u
.k/
c
D
u
.k/
p
K
k;2
z
.k/
H.#
.k
p
/
u
.k/
:
#
.k/
c
D #
.k/
p
(b)
Covariance estimate:
"
K
k;1
H.#
.k
p
/ƒ
.k/
#
:
p;
u
K
k;1
H.#
.k
p
/ƒ
.k/
p;
u
#
ƒ
.k/
augm;c
D ƒ
.k/
augm;p
K
k;2
H.#
.k
p
/ƒ
.k/
p;
u
K
k;2
H.#
.k
p
/ƒ
.k/
p;
u
#
It is worth noting that in this way we have a sort of adaptive filtering, since
the improvement of the knowledge of the parameter affects the quality of the state
estimate in a self-learning process.
As we have pointed out in the Introduction, there are several ways to perform
parameter estimation (see, e.g., [
2
,
3
]), this one is just an example. In Sect.
6.4.2
we
present an example relevant to FSI. Since EKF suffers from the computation of the
tangent operators, this can be avoided by resorting to a different extension of the
Kalman Filter, that we introduce in the next section.
Remark 6.4.
EKF can be regarded as the result of the application of one iteration of
the Gauss-Newton method for the minimization of a suitable mismatch functional,
as we have seen for the linear case. For more details, see [
43
]
6.2.5
The Unscented Kalman Filter
As pointed out above, errors associated with the linearization of EKF lead in
general to sub-optimal performances. In the unscented Kalman filter (UKF) [
45
],
the basic idea is to approximate the evolution of the nonlinear dynamic system not
by linearization but by deterministic sampling, following the so-called
unscented
transformation
(UT). The basic idea of UT is that “it is easier to approximate a
Gaussian distribution than it is to approximate an arbitrary nonlinear function or
transformation” [
44
]. For this reason, the nonlinear dynamics in UKF is statistically
approximated by mean and covariance of samples suitably selected for the state
variable to be estimated.
For instance, suppose to have a scalar Gaussian random variable
u
.k/
with mean
and variance
2
. At the first step we determine two samples of
u
.k/
,ass
1;2
D Ǚ.
If we need to approximate a nonlinear evolution
u
.k
C
1/
D f.
u
.k/
/, we compute the
samples f
i
f.s
i
/ and take
E
u
.k
C
1/
w
1
f
1
C
w
2
f
2
f;
E
f.
u
.k/
/
E
f.
u
.k/
/
2
w
1
.f
1
f/
2
C
w
2
.f
2
f/
2
;
where
w
i
are suitable weighting coefficients.
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