A First Course in FUZZY and NEURAL CONTROL

108 CHAPTER 3. FUZZY LOGIC FOR CONTROL

so the triangular functionAwith end points(a,0)and(b,0)andhighpoint
(c,α)has average sensitivity

S(A)=

1

(b−a)

√Z

c

a

μ

α

c−a

∂ 2

dx+

Zb

c

μ

α

c−b

∂ 2

dx

!

=

1

(b−a)

μ

α^2

c−a

+

α^2

b−c

∂

=

α

(c−a)

α

(b−c)

which is the product of the absolute values of the slopes of the sides of the
triangle.
For the sigmoidal membership functionf(x)=1+e^1 −x+1 on the interval
[− 5 ,10], the average sensitivity is

S(f)=

1

(10−(−5))

Z 10

− 5

μ

d

dx

1

1+e−x+1

∂ 2

dx

=

1

15

Z 10

− 5

√

e−x+1

(1 +e−x+1)^2

! 2

dx

=0.233 33

and for the Gaussian membership functionf(x)=e−
x 22
on the interval[− 5 ,5],
the average sensitivity is

S(f)=

1

(5−(−5))

Z 5

− 5

μ

d

dx

e−

x^2

2

∂ 2

dx

=

1

10

Z 5

− 5

≥

−xe−

(^12) x 2 ¥^2
dx
=8.862 3◊ 10 −^2

3.5 Combiningfuzzyrules ........................

The rules used in a rule-based system are generally expressed in a form such as
ìIfxisAthenyisB,î whereAandBare fuzzy sets,xis in the domain ofA,
andyis in the domain ofB. This sounds like an implication, such as ìAimplies
B.î There are many generalizations of the classical logical implication operation
to fuzzy sets, but most inference mechanisms used in fuzzy logic control systems
are not, strictly speaking, generalizations of classical implication.
The reasoning applied in fuzzy logic is often described in terms of agener-
alized modus ponens

Premise 1 xisAò

Premise 2 IfxisAthenyisB

Conclusion yisBò