A First Course in FUZZY and NEURAL CONTROL

3.2. FUZZY SETS IN CONTROL 89

TheGaussian functions, the familiar bell-shaped curve, are of the form

A(x)=e−

(x−c)^2

2 σ^2

These are related to the well-known normal or Gaussian distributions in prob-
ability and have useful mathematical properties.

0

0.5y

Gaussiane−

x 22

0

0.5y

Gaussiane−

(x−5)^2

25

The parameterscandσdetermine the center and the shape of the curve, respec-
tively. The valuesc=0andσ=1define thestandard Gaussian membership

functione−
x 22
, centered atc=0, and with area under the curve equal to

√

2 π.
This is the Gaussian curve depicted on the left above.
A Cauchy function, orgeneralized bell curve,isgivenbyfunctionsofthe

formA(x)=1/

≥

1+

Ø

Øx−c

a

Ø

Ø^2 b

¥

. The parametercdetermines the center of the

curve, andaandbdetermine its shape.

0.2

0.4

0.6

0.8

-4 -2 0 2 x 4 6

1

.≥

1+

Ø

Øx−^1

2

Ø

Ø^2

¥

0.2

0.4

0.6

0.8

-200 (^0200) x 400
1

.≥

1+

Ø

Øx−^100

2

Ø

Ø^1 /^2

¥

0

0.2

0.4

0.6

0.8

-2-1 (^1234) x
1

.≥

1+

Ø

Øx−^1

2

Ø

Ø^200

¥

0

0.2

0.4

0.6

0.8

-1500 -1000 -500 500 x 1000 1500

1

.≥

1+

Ø

Øx−^1

200

Ø

Ø^2

¥

The S- and Z-functions aresigmoidal functionsof the form

A(x)=

1

1+e−(x−m)σ

0

y0.5

S-function1+e^1 −x+1

0

y0.5

Z-function1+e^1 x− 1