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Fig. 2.2 EKF(c) algorithm.
Adaptation of consequents of
a TS fuzzy model
1: p c (
0
|−
1
)=
0
2: P c (
0
|−
1
)=
I
α
=
..
3: for k
0
k end do
Calculate P c (
k
|
k
1
)
by (2.11)
4:
Estimate y
(
k
)
using the fuzzy model
5:
Calculate C c (
k
)
by (2.22)
6:
Get K c (
k
)
by (2.12)
7:
Update p c (
k
|
k
)
by (2.10) and (2.13)
8:
Update P c (
|
)
k
k
by (2.14)
9:
10: end for k
x
b
∂μ Tri [
a
,
b
,
c
] (
x
)
if b
<
x
<
c
=
2
in other case .
(2.35)
(
c
b
)
c
0
It is possible to suppose that the derivatives in a , b or c are the same as the
derivative in a point infinitesimally close to the right, to the left, or the average of
these derivatives.
2.3 Algorithms for the Parametric Adaptation
of a TS Fuzzy Model Based on EKF
In this section, two algorithms that allow to adjust the antecedents and consequents of
a TS fuzzy model based on the Kalman filter are presented. Before executing adjust-
ment algorithms, the fuzzy model must be initialized. If there is no prior information
about the plant, all consequents can be initialized to 0, and the antecedents can be
initialized using uniform partitioning, or it could be initialized by other procedures
Benmakrouha ( 1997 ). However, if data are available for the system to be modeled,
these data can be used to obtain a better initial model, for example, by applying a
clustering algorithm (Bezdek and Dunn 1975 ; Bezdek 1981 ;Chiu 1994 ; Dunn 1973 ;
Gustafson and Kessel 1979 ; Kim et al. 1997 ; Wang and Mendel 1992b ), even an
offline modeling algorithm (Andújar et al. 2006 ; Jang 1993 ; Jang and Sun 1995 ;
Wang and Mendel 1992a ).
Once it has an initial model, the EKF algorithm can be used for online fuzzy
modeling of the system. The algorithm shown in Fig. 2.2 can be employed to adjust
the consequents of a TS fuzzy model. k is the discrete time,
p c is the adaptable set of
parameters of the consequents, P c , C p and K c arematrices of the EKF, I is the identity
matrix and
˜
is a positive integer which represents the certainty that the algorithm
should give to the initial parameters. For example,
α
α
will be higher if the consequents
 
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