A First Course in FUZZY and NEURAL CONTROL

3.2. FUZZY SETS IN CONTROL 87

0

0.2

0.4

0.6

0.8

10 20 30 40 50

Figure 3.2.A=[10,40]

Definition 3.1Afuzzy subsetAof a setXis a functionA:X→[0,1]from
Xto the unit interval[0,1].

The value ofA(x)is thought of as the degree of membership ofxinA.This
function is sometimes called themembership functionofA.Inthespecial
case that the membership function takes on only the values 0 and 1 ,Ais called
anordinaryorcrisp subsetofXand its membership function coincides with
its characteristic function. The setX is sometimes called theuniverse of
discourse. A fuzzy subset is often referred to simply as afuzzy set.
Other notation in common use for the membership function of the fuzzy
subsetAofX includesμA:X→[0,1]and sometimes

R

x∈XμA(x)/xor, if
the domain is discrete,

P

x∈XμA(x)/x. These more complicated notations do
not convey additional information, and we do not make a notational distinction
between a fuzzy subsetAand its membership functionA:X→[0,1].
Thesupportof a functionA:X→[0,1]is the set

supp(A)={x∈X|A(x) 6 =0}

For most purposes, it is not critical whetherXis the support ofAor some set
larger than the support. Thus, given a collection of fuzzy subsets of the real
numbers, for example, we may assume that they share the same universe of
discourse by takingXto be the union of the supports of the fuzzy sets. If two
fuzzy sets have the same support and take the same values on their support, we
generally need not distinguish between them.
AfuzzysetA:X→[0,1]isnormalif there is anx∈Xsuch thatA(x)=1.
AfuzzysetA:R→[0,1]isconvexif givenx≤y≤z,itmustbetruethat
f(y)≥f(x)∧f(z). A fuzzy setA:R→[0,1]withfinite support that is both
normal and convex is often called afuzzy number. Most fuzzy sets used in
control are both normal and convex.
Since data is generally numerical, the universe of discourse is most often
an interval of real numbers, or in practice, afinite set of real numbers. The
shape of a membership function depends on the notion the set is intended to
describe and on the particular application involved. The membership functions
most commonly used in control theory are triangular, trapezoidal, Gaussian,
and sigmoidal Z- and S-functions, as depicted in the followingfigures.