A First Course in FUZZY and NEURAL CONTROL

264 CHAPTER 8. APPLICATIONS

in Figure 8.12 (b), where nodes of the same layer have similar functions. In the
discussion, the termOl,idenotes the output of theithnode in layerl.

Layer 1: Every nodeiin this layer is an adaptive node with a node function

O 1 ,i=Ai(x), fori =1, 2

O 1 ,i=Bi− 2 (x), fori =3, 4 (8.4)

wherex(ory)istheinputtonodeiandAi(orBi− 2 ) is a linguistic label

(such as ìsmallî or ìlargeî) associated with the node. In other words,

O 1 ,iis the membership grade of a fuzzy set

A=(A 1 ,A 2 ,B 1 or B 2 )

and it specifies the degree to which the given inputx(ory)satisfies a

quantifierA. Here the membership function forAcan be an appropriately

parameterized membership function such as the generalized bell function

A(x)=

1

1+

Ø

Ø

Øx−aici

Ø

Ø

Ø

2 b

where{ai,bi,ci}is the parameter set. As the values of these parameters

change, the bell-shaped function varies accordingly, thus exhibiting various

forms of membership functions for fuzzy setA. Parameters in this layer

are generally referred to aspremise parameters.

Layer 2: Every node in this layer is afixed node labeledπ, whose output is

the product of all the incoming signals:

O 2 ,i=wi =Ai(x)Bi(y), fori =1, 2. (8.5)

Each node output represents thefiring strength of a rule. In general, any

other t-norm operators, all of which perform a fuzzy AND, can be used

as the node function in this layer.

Layer 3: Every node in this layer is afixed node labeledN.Theithnode

calculates the ratio of theithruleísfiring strength to the sum of all ruleís

firing strengths:

O 3 ,i=wbi =

wi

w 1 +w 2

fori =1, 2 (8.6)

The outputs of this layer are referred to asnormalizedfiring strengths.

Layer 4: Every nodeiin this layer is an adaptive node with a node function

O 4 ,i=wbifi =wbi(ai 0 +ai 1 x+ai 2 y) fori =1, 2 (8.7)

wherewbiis the normalizedfiring strength from layer 3 and

©

ai 0 ,ai 1 ,ai 2

™

is the parameter set of this node. Parameters in this layer are referred to

asconsequent parameters.