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O
.x;t/
Fig. 6.3
Test case 1: (
a
) spatiotemporal variation of the wave function .x;t/.(
b
) estimate
of the wave function provided by the derivative-free nonlinear Kalman Filter
@
2
@t
2
@x
fŒc
0
C C q
2
@
@x
g)
@
D
(6.41)
@
2
@t
2
C Œc
0
C C q
2
@
2
D . C 2q/.
@
@x
/
2
@x
2
By denoting f./D . C 2q/.
@
@x
/
2
and K.x;t/ D Œc
0
C C q
2
, one finally
obtains the following description for the wave-type dynamics of the membrane
@
2
@t
2
D K.x;t/
@
2
@x
2
(6.42)
C f./
The estimator was based on the Derivative-free nonlinear Kalman Filter as analyzed
in Sect.
6.5
and made use of a grid of N D 50 points, out of which n D 25 were
measurement points. At each measurement point a value of the state variable x
1
D
.x;t/ was obtained. In Fig.
6.3
the estimate of the wave function provided by the
distributed filtering algorithm is compared to the real wave function dynamics. Due
to boundary conditions set at points x D 0 and x D 10 of the x-axis wave type
dynamics .x;t/ is generated and propagates in time. The nonlinear Kalman Filter
estimates the wave-type dynamics without knowledge of the initial conditions of the
PDE. As it can be noticed the derivative-free nonlinear Kalman Filter approximates
closely the real wave function. Indicative results about the estimation obtained at
local grid points is given in Figs.
6.4
,
6.5
, and
6.6
.
The evaluation tests were repeated for different values in the parameters of
the wave function coefficient K.x;t/.InFig.
6.7
a new value of gain K.x;t/ is
considered and the estimate of the wave function provided by the distributed filtering
algorithm is compared again to the real wave function dynamics. Additional results
about the accuracy of estimation obtained in local grid points, when using different
coefficients c
0
, , and q in the computation of K.x;t/ are given in Figs.
6.8
,
6.9
,
and
6.10
. As it can be observed again, the derivative-free nonlinear Kalman Filter
resulted in very accurate estimates of the wave-type dynamics.
