A First Course in FUZZY and NEURAL CONTROL

7.3. BASIC PRINCIPLES OF NEURAL-FUZZY SYSTEMS 239

It is important to keep in mind that when we develop a set of prototype fuzzy
rules, we are in fact representing possibly nonlinear input-output relationships.
The effectiveness of fuzzy models representing nonlinear input-output relation-
ships depends on the membership functions involved. Thus, the tuning of mem-
bership functions is an important issue in fuzzy modeling. This tuning task
canbeviewedasanoptimizationproblem;neuralnetworksoffer a possibility
to solve this problem.
In order to train a fuzzy-neural network, we need a set of training data in
the form of input-output tuples, and a specification of the rules, including a
preliminary definition of the corresponding membership functions. A standard
approach is to assume a certain shape for the membership functions so that the
membership functions depend on parameters that can be learned by a neural
network.
We describe one method for learning the membership functions of the an-
tecedent and consequent parts of fuzzy ìIf...then...î rules. Suppose an un-
known function, or control law, to be realized by a fuzzy inference system is
known only through the training set
©°
x^1 ,y^1

¢

,...,

°

xK,yK

¢™

wherexk=

°

xk 1 ,...,xkn

¢

∈Rnandyk∈R. To model the unknown function, we
use fuzzy ìIf...then...î rulesRi,i=1,...,m, of the following type

Ri:Ifxk 1 isA^1 iand ... andxknisAni theny=

Pn

j=1z

j

ix

k

j+zi

whereAjiare fuzzy membership functions andzijare real numbers.
LetOkbe the output from the fuzzy system corresponding to the inputxk.
Suppose the fuzzy AND of each rule is implemented by the product, so that the
antecedent of theithrule is given by

αki=

Qn

j=1

Aji

°

xkj

¢

We could also use other t-norms for modeling the logical connective AND. We
compute the output of the system as

Ok=

Pm

i=1α

k

i

≥P

n

j=1z

j

ix

k

j+z

0

i

¥

Pm

i=1α

k

i

=

Pm

i=1

≥Q

n

j=1A

j

i

°

xkj

¢¥≥Pn

j=1z

j

ix

k

j+z

0

i

¥

Pm

i=1

Qn

j=1A

j

i

°

xkj

¢

and define the measure of error for thekthtraining pattern as

Ek=

1

2

°

Ok−yk

¢ 2

whereOkis the computed output from the fuzzy system corresponding to the
input patternxk,andykis the desired output,k=1,...,K.Standardneural
network learning methods are used to learnzij,j=0, 1 ,...,nin the consequent
part of the fuzzy ruleRi.