A First Course in FUZZY and NEURAL CONTROL

7.2. BASIC PRINCIPLES OF FUZZY-NEURAL SYSTEMS 233

Wecanworkwithafinite number of membership values by taking positive
integersM≥ 2 andN≥ 2 ,andsetting

xj = a 1 +

j− 1

N− 1

(a 2 −a 1 )

yk = b 1 +

k− 1

M− 1

(b 2 −b 1 )

for 1 ≤j≤Nand 1 ≤k≤M. This gives us a discrete version of the continuous
training set consisting of the input-output pairs

(Ai(x),Bi(y)) = ((Ai(x 1 ),...,Ai(xN)),(Bi(y 1 ),...,Bi(yM)))

fori=1,...,n, and the fuzzy-neural network turns into anN-input andM-
output standard network that can be trained by the generalized delta rule (see
page 180).

Example 7.1Assume our fuzzy rule base consists of three rules

R 1 :Ifxis small thenyis negative

R 2 :Ifxis medium thenyis about zero

R 3 :Ifxis big thenyis positive

where the membership functions of fuzzy terms are defined by

μsmall(u)=A 1 (u)=

Ω

1 − 2 u if 0 ≤u≤ 1 / 2

0otherwise

μmedium(u)=A 2 (u)=

Ω

1 − 2 |u− 1 / 2 | if 0 ≤u≤ 1

0otherwise

μbig(u)=A 3 (u)=

Ω

2 u− 1 if 1 / 2 ≤u≤ 1

0otherwise

0

0.5

1

0.2 0.4 0.6 0.8 1u

Fuzzy setssmall (dots), medium (solid),andbig(dashed)

μnegative(u)=B 1 (u)=

Ω

−u if − 1 ≤u≤ 0

0otherwise

μaboutzero(u)=B 2 (u)=

Ω

1 − 2 |u| if − 1 / 2 ≤u≤ 1 / 2

0otherwise

μpositive(u)=B 3 (u)=

Ω

u if 0 ≤u≤ 1

0otherwise