120 CHAPTER 3. FUZZY LOGIC FOR CONTROL
3.8.2 Orderrelations........................
A partial order≤on a set determines a relationRby(x, y)∈Rif and only
ifx≤y. There are many kinds of orders, and many generalizations have been
proposed for fuzzy sets. We will mention only the following.
Definition 3.16 Afuzzy order relationis a fuzzy relationSon a setXsuch
that for allx,y,z∈X
1.S(x,x)=1(Sis reflexive.)- Ifx 6 =y,thenS(x,y)∧S(y,x)=0(Sis antisymmetric.)
3.S(x,z)≥S(x,y)∧S(y,z)(Sis max-min transitive.)
A fuzzy order relation is a fuzzy linear ordering if it also satisfies- Ifx 6 =y,thenS(x,y)∨S(y,x)> 0
3.9 Defuzzification ............................
The most common methods for combining fuzzy rules produce a fuzzy set. In
control theory, a crisp output is often needed. This requires some process of
defuzzificationñ producing a number that best reflects the fuzzy set in some
sense. There are many techniques for defuzzification. We will mention only a
few of the most common ones here. We will demonstrate each of these on the
output set obtained by the Mamdani method in Example 3.6.
Loosely speaking, there are two types of defuzzification techniques ñ com-
posite moments and composite maximum. ìCompositeî reflects the fact that
the values are obtained from combining several fuzzy sets. Composite moment
techniquesusesomeaspectofthefirst moment of inertia, and composite max-
imum techniques extract a value for which the fuzzy set attains its maximum.
The center of area and height-center of area methods are of thefirst type, and
the max criterion,first of maxima, and middle of maxima methods are of the
second type.
3.9.1 Centerofareamethod....................
Thecenter of area,orcenter of gravity,orcentroidmethod computes the
center of area of the region under the curve defined by a fuzzy set and selects
thefirst component. IfCis the fuzzy set in question andCis integrable, then
the defuzzified value ofCby this method is
z 0 =Rb
RazC(z)dz
b
aC(z)dz