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Fig. 6.2
Grid points for
measuring .x;t/
Measurement update
:
K.k/ D P
.k/C
d
ŒC
d
P
.k/C
d
C R
1
y.k/ Dy
.k/ C K.k/Œ
z
.k/ C
d
y
.k/
P.k/D P
.k/ K.k/C
d
P
.k/
(6.39)
Time update
:
P
.k C 1/ D A
d
.k/P.k/A
d
.k/ C Q.k/
y
.k C 1/ D A
d
.k/ y.k/C B
d
.k/
u
.k/
(6.40)
Therefore, by taking measurements of .x;t/ at time instant t at a small number of
measuring points, as shown in Fig.
6.2
, j D 1; ;n
1
it is possible to estimate the
complete state vector, i.e. to get values of in a mesh of points that covers efficiently
the variations of .x;t/. By processing a sequence of output measurements of the
system, one can obtain local estimates of the state vector y. The measuring points
can vary in time provided that the observability criterion for the state-space model
of the PDE holds.
Remark.
The proposed derivative-free nonlinear Kalman Filter is of improved pre-
cision because unlike other nonlinear filtering schemes, e.g. the Extended Kalman
Filter it does not introduce cumulative numerical errors due to approximative
linearization of the system's dynamics. Besides it is computationally more efficient
(faster) because it does not require to calculate Jacobian matrices and partial
derivatives.
6.6
Simulation Tests
6.6.1
Evaluation Experiments
The proposed filtering scheme was tested in estimation of the dynamics of a
wave equation of the form of Eq. (
6.24
) under unknown boundary and initial
conditions. It has been shown that it is possible to obtain voltage measurements from
dendritic trees using, for example, laser-based scanning techniques [
69
,
84
,
202
].
In the simulation experiments the following wave-type dynamics in the neuron's
membrane was considered
