A First Course in FUZZY and NEURAL CONTROL

7.2. BASIC PRINCIPLES OF FUZZY-NEURAL SYSTEMS 235

Figure 7.2. Standard neuron

to the neuron. The neuron uses its transfer functionf, often a sigmoidal function
f(x)=1+^1 e−x, to compute the output

y=f(net)=f(w 1 x 1 +w 2 x 2 )

In the crisp case, we use addition and multiplication operators to perform
aggregation of the input signals (Figure 7.2). However, this is not possible in
the case of fuzzy numbers. If we employ operators like a t-norm or a t-conorm
to combine the incoming data to a neuron, we obtain what we call ahybrid
neural net.Thesemodifications lead to a fuzzy-neural architecture based on
fuzzy arithmetic operations. A hybridneural net may not use multiplication,
addition, or a sigmoidal function because the results of these operations are not
necessarily in the unit interval.

Definition 7.1Ahybrid neural netis a neural net with crisp signals and
weights and a crisp transfer function for which

ïThe signals and weightsxiandwi,bothofwhichlieintheinterval[0,1],

can be combined using a t-norm, t-conorm, or some other continuous op-

eration.

ïThe resultsp 1 andp 2 can be aggregated with a t-norm, t-conorm, or any

other continuous function from[0,1]to[0,1].

ïThe transfer functionfcan be any continuous function from[0,1]to[0,1].

A processing element of a hybrid neural net is called afuzzy neuron.

We emphasize that all inputs, outputs, and weights of a hybrid neural net
are real numbers taken from the unit interval[0,1].

Example 7.2 (AND)Figure 7.3 illustrates an AND fuzzy neuron. The signals
xiand weightswiare combined by a triangular conormSto produce the output

pi=S(wi,xi), i=1, 2

The input informationpiis aggregated by a triangular normTto produce the
output
y=AND(p 1 ,p 2 )=T(p 1 ,p 2 )=T(S(w 1 ,x 1 ),S(w 2 ,x 2 ))