A First Course in FUZZY and NEURAL CONTROL

104 CHAPTER 3. FUZZY LOGIC FOR CONTROL

3.4 Sensitivityoffunctions........................

The modeling of fuzzy concepts through the assignment of membership func-
tions, as well as the choice of fuzzy connectives, is subjective. In specificap-
plications, choices must be made. We illustrate some possible guidelines for
making these choices when sensitivity of membership functions or of fuzzy log-
ical connectives with respect to variations in their arguments is a factor. We
look at two measures of sensitivity ñ extreme and average sensitivity.
Note that here, by sensitivity we mean sensitivity with respect to measure-
ment errors or to theflexibility in assigning degrees of membership for fuzzy
concepts. Thus, in general, a least sensitive operator among a class of operators
is preferred.

3.4.1 Extrememeasureofsensitivity ...............

Definition 3.9For a mappingf:[0,1]n→[0,1]andδ∈[0,1],let

ρf(δ)=

W

|xi−yi|≤δ

|f(x)−f(y)|

wherex=(x 1 ,x 2 ,...,xn)andy=(y 1 ,y 2 ,...,yn). The functionρf:[0,1]→[0,1]
is called anextreme measure of sensitivityoff. We say thatf isless
sensitivethangif for allδ,ρf(δ)≤ρg(δ), with strict inequality at someδ.

Example 3.4Iff(x, y)=x∧y,thenfor|x−u|≤δand|y−v|≤δ,wehave
x≤u+δandy≤v+δ.So

(x∧y)≤(u+δ)∧(v+δ)=(u∧v)+δ

and thus(x∧y)−(u∧v)≤δ. Similarly,(u∧v)≤(x∧y)+δ,sothat(u∧v)−
(x∧y)≤δ.Thus,ρ∧(δ)≤δ.
Takingx=y=δandu=v=0,wehave

(x∧y)−(u∧v)=(δ∧δ)−(0∧0) =δ

soρ∧(δ)≥δ. It follows that
ρ∧(δ)=δ
Here are some additional examples.

Function Sensitivity

f(x)=1−xρf(δ)=δ

f(x,y)=xy ρf(δ)=2δ−δ^2

f(x,y)=x+y−xy ρf(δ)=2δ−δ^2

f(x,y)=(x+y)∧ 1 ρf(δ)=2δ∧ 1

f(x,y)=x∨yρf(δ)=δ

Theorem 3.3At-norm 4 and the t-conormOthat is dual to 4 with respect
toα(x)=1−x, have the same sensitivity.