A First Course in FUZZY and NEURAL CONTROL

234 CHAPTER 7. FUZZY-NEURAL AND NEURAL-FUZZY CONTROL

-1^01

u

Fuzzy sets negative(dots),about zero (solid), and positive (dashed)

The fuzzy training set derived from this rule base can be written in the form

{(small, negative), (medium, about zero), (big, positive)}

Let[0,1]contain the support of all the fuzzy sets we might have as input to the
system. Also, let[− 1 ,1]contain the support of all the fuzzy sets we can obtain
as outputs from the system. LetM=N=5and

xj =(j−1)/ 4

yk = −1+(k−1)2/4=−1+(k−1)/2=− 3 /2+k/ 2

for 1 ≤j≤ 5 and 1 ≤k≤ 5. Substituting foriandj,weget

x =(x 1 ,x 2 ,x 3 ,x 4 ,x 5 )=(0, 0. 25 , 0. 5 , 0. 75 ,1)

y =(y 1 ,y 2 ,y 3 ,y 4 ,y 5 )=(− 1 ,− 0. 5 , 0 , 0. 5 ,1)

A discrete version of the continuous training set consists of three input-output
pairs for use in a standard backpropagation network

(A 1 (x),B 1 (y)) =

°°

10. 5000

¢

,

°

10. 5000

¢¢

(A 2 (x),B 2 (y)) =

°°

00. 510. 51

¢

,

°

00100

¢¢

(A 3 (x),B 3 (y)) =

°°

0000. 51

¢

,

°

0000. 51

¢¢

To demonstrate the distinction between a regular neural network and a fuzzy-
neural network, we briefly consider the crisp case of a neural network that was
discussed in detail in Chapter 5. Consider a simple neural net as in Figure
7.2. All signals and weights are real numbers. The two input neurons do not
change the input signal, so their output is the same as their input. The signal
xiinteracts with the weightwito produce the product

pi=wixi,i=1, 2

The input informationpiis aggregated by addition to produce the input

net=p 1 +p 2 =w 1 x 1 +w 2 x 2