A First Course in FUZZY and NEURAL CONTROL

106 CHAPTER 3. FUZZY LOGIC FOR CONTROL

membership function:

ρA(δ)=

_

|x−y|≤δ



























Ø

Ø

Øαxc−−ya

Ø

Ø

Ø ifa≤x,y≤c

Ø

Ø

Øαx(c−b)(−c−ya()(c−ca−)+b)ac−bc

Ø

Ø

Ø ifa≤x≤c≤y≤b

Ø

Ø

Øαx(c−a()c−−ya(c)(−cb−)+b)ac−bc

Ø

Ø

Ø ifa≤y≤c≤x≤b

Ø

Ø

Øαxc−−yb

Ø

Ø

Ø ifc≤x,y≤b

0 otherwise



























≤















α

c−aδ ifa≤x≤c,a≤y≤c

α(b−a)

(c−a)(b−c)δ ifa≤x≤c≤y≤bora≤y≤c≤x≤b

α

b−cδ ifc≤x≤b,c≤y≤b

0 otherwise















Note thatc−αa andb−αcare the absolute values of the slopes of the triangular
function at the left and right ofc, respectively.

For the Gaussian functionsA(x)=e−

(x−c)^2

2 σ^2 , the sensitivity function is

ρA(δ)=

W

|x−y|≤δ

Ø

Ø

Ø

Øe

−(x−c)

2

2 σ^2 −e−

(y−c)^2

2 σ^2

Ø

Ø

Ø

Ø

and for the sigmoidal functions of the formf(x)=1+e−(^1 x−m)σ, the sensitivity
function is

ρf(δ)=

W

|x−y|≤δ

Ø

Ø

Ø

Ø

1

1+e−(x−m)σ

−

1

1+e−(y−m)σ

Ø

Ø

Ø

Ø

3.4.2 Averagesensitivity......................

An alternative to the measure above, of extreme sensitivity of fuzzy logical
connectives and membership functions, is a measure ofaverage sensitivity.
Letf:[a,b]→R. One measure of the sensitivity of differentiable functionsf
at a point in[a,b]is the squaref^0 (x)^2 of its derivative at that point. Its average
sensitivity is the average over all points in[a, b]off^0 (x)^2 , namely, the quantity

1

b−a

Zb

a

f^0 (x)^2 dx

Iffis a function of two variables, sayf:[a, b]^2 →R, the average sensitivity of
fis

S(f)=

1

(b−a)^2

Zb

a

Zb

a

√μ

∂

∂x

f(x,y)

∂ 2

+

μ

∂

∂y

f(x,y)

∂ 2!

dxdy