A First Course in FUZZY and NEURAL CONTROL

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

The derived training set consists of input-output pairs(Ai,Bi).ForaMIMO
fuzzy system, the derived training set can be written in the form

(

©

Aij

™N

j=1,

©

Bik

™M

k=1),i=1,...,n

For a MISO system, this simplifies to

(

©

Aij

™N

j=1,Bi),i=1,...,n

and for a SISO system, the derived training set can be written in the form

(A 1 ,B 1 ),...,(An,Bn)

We also need to incorporate defuzzification operations into neural networks.
These operations, when necessary, take place at thefinal level of the neural
network and are the same operations as those discussed in Chapter 3.

7.2 Basic principles of fuzzy-neural systems

Rules express the behavior of a system. They allow us to specify the output,
given the input. This property is particularly useful in that we can define
the desired output to obtain the ìbestî performance of the system being con-
trolled. For a PID controller, this gives theflexibility to specify the various
gains, namely, the proportional gain, the derivative gain and the integral gains
to achieve the desired performance.
In this section we look at how fuzzy ìIf... then... î rules express the input-
output behavior of fuzzy-neural systems. Knowledge of rules allows us to ex-
amine appropriate trainable neural network architectures. We show how the
structure of fuzzy rules can be transformed into neural networks, giving rise to
fuzzy-neural systems.
TherearetwomainapproachestoimplementfuzzyìIf...then...îrulesbya
standard-error backpropagation network ñ a methodfirst proposed by Umano
and Ezawa [75] in 1991, and a modification by Uehara and Fujise [74] in 1992
that uses afinite number ofα-level sets to represent fuzzy numbers. In 1993 Jang
[35] extended these methods toward the development of an adaptive network
fuzzy inference system (ANFIS) that has spawned numerous applications. We
will discuss ANFIS later in this chapter.
In the method proposed by Umano and Ezawa, a fuzzy set is represented by
afinite number of its membership values. Suppose we have a set of fuzzy rules
of the form
Ri:IfxisAithenyisBi

i=1,...,n,whereAiandBiare fuzzy sets. Let the interval[a 1 ,a 2 ]contain the
support of all theAi, plus the support of any other functions we might have
as input to the system. Also, let[b 1 ,b 2 ]contain the support of all theBi,plus
the support of any other functions we canobtain as outputs from the system,i
=1,...,n.