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respect the system, obtaining a solution that minimizes both estimation error and its
covariance matrix for the linearization obtained at each instant.
2.2 Application of the EKF to Fuzzy Modeling
A so interesting application of EKF is the adaptive identification of parameters in
nonlinear systems, which allows the online obtaining of the adaptive parameters set
of a discrete nonlinear model with noise presence and in a pseudo-optimal way (is
optimal in linear systems). The identification of a TS fuzzy model can be seen as the
obtaining of parameters of a nonlinear model, so the Kalman filter can be applied
using the extended algorithm for estimating these parameters.
First it is necessary to raise the problem of estimation by EKF. For this, a system
whose states depend directly on the parameters to be estimated (Simon
2002
)must
be build, and then apply recursively from (
2.10
)to(
2.14
).
Let
p
the set of
outputs of this fuzzy system, the system represented in (
2.18
) and the diagram shown
in Fig.
2.1
, allows to obtain these parameters using the EKF.
(
k
)
be the set of adaptive parameters of a fuzzy system, and
y
(
k
)
)
=
p
(
k
+
1
p
(
k
)
y
(
k
)
=
h
(
x
(
k
),
p
(
k
))
+
e
(
k
).
(2.18)
is a constant matrix that relates the parameters between themselves, for exam-
ple, to meet the requirements of the Standard Fuzzy Partition (SFP) (Xiu and Ren
2005
) frequently used in control applications (Al-Hadithi et al.
2007
; Xiu and Wang
2007
). If all parameters can be adjusted freely,
)
is the uncertainty of the measurement of the output system and is represented by a
white noise, whose covariance is
R
e
.
Thus, the first thing to do is the calculation of Jacobian matrices of the system
using (
2.7
), (
2.8
) and (
2.9
). Applying these expressions on (
2.18
):
will be the identity matrix.
e
(
k
(
p
(
k
))
=
(2.19)
(
p
(
k
))
=
0
(2.20)
and
p
=
p
(
k
)
∂
h
C
(
p
(
k
))
=
(2.21)
∂
p
being
the current estimation of the parameters vector of the TS fuzzy model.
Note that, given formulation exposed in Sect.
2.1.1
, the estimation problem of the
TS fuzzy model (
2.1
) is to determine the values of the adaptive parameters of both
p
ˆ
(
k
)
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