A First Course in FUZZY and NEURAL CONTROL

4.2. MAIN APPROACHES TO FUZZY CONTROL 141

Note that the premise is in the same form as for the Mamdani and Larsen
methods, but the consequent is of a different form.
In the Takagi-Sugeno method eachfjis a linear function

fj(x 1 ,...,xn)=a 0 j+

Xn

i=1

αijxi

Other forms in common use are quadratic

fj(x 1 ,...,xn)=a 0 j+

Xn

i=1

αijx^2 i

and trigonometric

fj(x 1 ,...,xn)=exp

√n

X

i=1

αijsinxi

!

The choice of the consequentsfj(x)depends upon the particular application.

Example 4.1The Takagi-Sugeno rules provide a means to interpolate between
piecewise linear mappings. For the partial mapping

-6

-4

-2

0

2

4

6

8

10

-3 -2 -1 (^1) x 2 3
y=

Ω

3+3xforx≤− 1
−1+4xforx≥ 1
take two fuzzy sets

A 1 (x)=







1 if x≤− 1

1 −x

2 if −^1 ≤x≤^1

0 if 1 ≤x

and

A 2 (x)=







0 if x≤− 1

1+x

2 if −^1 ≤x≤^1

1 if 1 ≤x