Information Technology Reference
In-Depth Information
a
b
50
50
40
40
30
30
20
20
10
10
0
0
−10
−10
−20
−20
−30
−30
−40
−40
−50
−50
0
5
10
15
20
0
5
10
15
20
t
t
Fig. 6.10
Test case 2: estimate (
dashed green line
) and real value (
red continuous line
)(
a
)of
.x
1
5;t/ at grid point 15 (
b
)of.x
20
;t/at grid point 20
6.6.2
Assessment of the Filter's Performance
6.6.2.1
Stability of the Filtering Method
The stability and convergence conditions for the proposed filtering algorithm are
those of the standard Kalman Filter, which means that the filter converges if the
system's model in its linearized canonical form satisfies obvervability (detectability)
criteria. About the method for solution of a PDE through its decomposition into
a set of local ODEs, numerical stability conditions have been provided and these
are related to the step of spatio-temporal discretization of the initial PDE [
139
].
The method converges and remains numerically stable if the discretization steps
are taken to be sufficiently small. For the basic wave-type PDE
@
2
y
@t
2
D K
@
2
y
@x
2
the
convergence condition is 2K
Dt
Dx
2
1 where
Dt
and
Dx
are the discretization steps in
space and time [
139
].
6.6.2.2
Implementation Stages and Advantages of the Filtering Method
The first stage of the filter's design is to analyze the PDE that describes the
system's dynamics into a set of ordinary differential equations. Actually, at each
iteration of the algorithm in time, a numerical solution of the PDE is computed at
aspatialgridofN points and at each point of the grid the system's dynamics is
described by a nonlinear ODE. At a second stage differential flatness theory and
a diffeomorphism (change of coordinates) is used to transform the local nonlinear
ODEs into linear ODEs which in turn can take the form of a linear state space
equation in the canonical (Brunovsky) form. The Derivative-free nonlinear Kalman
Filter consists of the standard Kalman Filter recursion on the linearized equivalent
model of the valve and on computation of state and disturbance estimates using
