7.3. BASIC PRINCIPLES OF NEURAL-FUZZY SYSTEMS 237
represent a fuzzy system in terms of a neural network is to utilize the learning
capability of neural networks to improve performance, such as adaptation of
fuzzy systems. Thus, the training algorithm in the modified neural networks
should be examined.
7.3.1 Adaptive network fuzzy inference systems
To illustrate the use of neural networksfor fuzzy inference, we present some
successful adaptive neural network fuzzy inference systems, along with training
algorithms known as ANFIS. These structures, also known as adaptive neuro-
fuzzy inference systems or adaptive network fuzzy inference systems, were pro-
posed by Jang [35]. It should be noted that similar structures were also proposed
independently by Lin and Lee [40] and Wang and Mendel [77]. These structures
are useful for control and for many other applications.
Tofix the ideas, consider the problem of graphically representing the way
fuzzy control is achieved in the Sugeno-Takagi model. For a simple example,
consider a fuzzy rule base consisting of only two rules:
R 1 :Ifx 1 isA 1 andx 2 isB 1 theny=f 1 (x)
R 2 :Ifx 1 isA 2 andx 2 isB 2 theny=f 2 (x)whereAiandBiare fuzzy sets and
f 1 (x)=z 11 x 1 +z 12 x 2 +z 13
f 2 (x)=z 21 x 1 +z 22 x 2 +z 23Recall that when numerical inputx=(x 1 ,x 2 )is presented, the inference mech-
anism will produce the numerical output
y∗=
A 1 (x 1 )B 1 (x 2 )f 1 (x)+A 2 (x 1 )B 2 (x 2 )f 2 (x)
A 1 (x 1 )B 1 (x 2 )+A 2 (x 1 )B 2 (x 2 )A fuzzy-neural network for implementing the above is shown in Figure 7.5. The
observed inputx=(x 1 ,x 2 )is presented to Layer 1 by Input Layer 0 .The
output of Layer 1 is
(O 11 ,O 12 ,O 13 ,O 14 )=(A 1 (x 1 ),A 2 (x 1 ),B 1 (x 2 ),B 2 (x 2 ))where the membership functionsAi,Bi,i=1, 2 , are specified in some paramet-
ric way from a family of membership functions, such as triangular or Gaussian.
Layer 2 consists of fuzzy neurons with an aggregation operator being some
t-norm. We use the product t-norm in this example, in view of the way that
product is used in Sugeno-Takagiís inference procedure. The output of Layer 2
is
(O 21 ,O 22 )=(A 1 (x 1 )B 1 (x 2 ),A 2 (x 1 )B 2 (x 2 ))
Layer 3 is a normalizer. The output of Layer 3 is
(O 31 ,O 32 )=μ
O 21
O 21 +O 22,
O 22
O 21 +O 22
∂
=
≥
A 1 (x 1 )B 1 (x 2 )
A 1 (x 1 )B 1 (x 2 )+A 2 (x 1 )B 2 (x 2 ),A 2 (x 1 )B 2 (x 2 )
A 1 (x 1 )B 1 (x 2 )+A 2 (x 1 )B 2 (x 2 )