A First Course in FUZZY and NEURAL CONTROL

268 CHAPTER 8. APPLICATIONS

Table 8.5. Linguistic variables and linguistic

values for fuzzy ìIf... then... î rules

Linguistic Variable Linguistic Values

Area {small, large}

Solidity {small, medium, large}

Edif {small, large}

Formulation of membership functions Even though the training set used
to develop the ìIf...then...î rules is a set of number pairs, the membership
functions normally used to implement these rules have a universe of discourse
Xconsisting of the real lineR. These membership functions are chosen from
known parameterized families of functions such as the generalized bell functions.
A generalized bell membership function, commonly referred to as bell MF,
is characterized by three parameters namelya,b,c.

bell(x;a,b,c)=

1

1+

Ø

Øx−c

a

Ø

Ø^2 b

A desired, generalized bell membership function can be obtained with the proper
selection of the parametersa,b,c. The parametersaandcrepresent the width
and the center of the bell function, andbrepresents the slopes at the crossover
points. Figure 8.13 illustrates the parameters that describe the generalized bell
function.

0.0

0.5

1.0

c-a 2 a c c+a

Slope = -b/2a

MF

X

Figure 8.13. Parameter description of a generalized bell function

Various other membership functions such as triangular, trapezoidal, Gaussian,
andsigmoidalcanbeusedintheformulationofmembershipfunctions. See
Section 3.2 for details. The triangular and trapezoidal membership functions,
due to their simplicity and computational efficiency, are used extensively in the
formulation of membership functions for many applications. However, these
membership functions consist of line segments and are not smooth at the corner
specified by the parameters. The Gaussian,the generalized bell function, and