A First Course in FUZZY and NEURAL CONTROL

3.9. DEFUZZIFICATION 121

where[a,b]is an interval containing the support ofC. If the support ofCis
finite, the computation is

z 0 =

Pn

Pj=1zjC(zj)

n

j=1C(zj)

The output set obtained in Example 3.6 produces the following value.

0

y0.5

2468101214 y

z 0 (1.25) = 7. 3187

The center of area defuzzification is the most widely used technique. The
defuzzified values tend to move smoothly in reaction to small changes, and it is
relatively easy to calculate.

This solution is reminiscent of statistical decision theory. If we normalize
C(∑|x)=Cx, we obtain the probability density function

C(∑|x)

R

ZC(z|x)dz

A common way to summarize a distribution is to use its center of location ñ
that is, its expected value:

z 0 =

R

RZzC(z|x)dz

ZC(z|x)dz

3.9.2 Height-centerofareamethod ................

The height-center of area defuzzification method ignores values of the fuzzy
set below some levelα, then uses the center of area method on the resulting
curve. Forα=0. 5 , the output set obtained in Example 3.6 produces the value
z 0 (1.25) = 7. 5.