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relating the parameters of the MFs together based on the sum-normality conditions,
then projecting this matrix in each iteration of optimization to constrain the MFs to
sum normal types.
Since the derivatives of the functions are used in Kalman filtering, it is limited
to special type of MFs because of complicated and time consuming calculations. So
far only triangular types are optimized for both inputs and outputs of a Fuzzy Logic
Controller (FLC) (Simon 2002b , c ).
The rest of the chapter is organized as follows. Section 1.2 recalls the estimation
of TS fuzzy model. Section 1.3 introduces restrictions of TS identification Method.
In Sect. 1.4 the proposed non iterative approach is explained. The proposed iterative
approach by applying Extended Kalman filter is explained in Sect. 1.5 . Section 1.6
entails an example of an inverted pendulum to demonstrate the validity of the pro-
posed approach. The results show that the proposed approach is less conservative
than those based on (standard) TS model and illustrate the utility of the proposed
approach in comparison with TS model. The conclusions of the effectiveness and
validity of the proposed approach are explained in Sect. 1.7 .
1.2 Estimation of Fuzzy TS Model's Parameters
An interestingmethod of identification is presented in Takagi and Sugeno ( 1985 ). The
idea is based on estimating the nonlinear system parameters minimizing a quadratic
performance index. The method is based on the identification of functions of the
following form:
n
f
:
−→
(1.1)
y
=
f
(
x 1 ,
x 2 ,...,
x n )
(1.2)
Each IF-THEN rule R i 1 ... i n , for an n th order system can be written as follows:
S ( i 1 ... i n ) :
If x 1 is M i 1
x n is M i n
and
...
then
(1.3)
p ( i 1 ... i n )
0
p ( i 1 ... i n )
1
p ( i 1 ... i n )
2
p ( i 1 ... i n )
ˆ
=
+
x 1 +
x 2 +···+
y
x n
(1.4)
n
where the fuzzy estimation of the output is:
p ( i 1 ... i n )
0
x n
r 1
i 1
1 ... r n
+ p ( i 1 ... i n )
1
x 1 + p ( i 1 ... i n )
x 2 +···+ p ( i 1 ... i n )
1 w ( i 1 ... i n ) ( x )
n
=
i n
=
2
y
ˆ
=
r 1
i 1
1 ... r n
1 w ( i 1 ... i n ) (
x
)
=
i n
=
(1.5)
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