A First Course in FUZZY and NEURAL CONTROL

110 CHAPTER 3. FUZZY LOGIC FOR CONTROL

3.5.1 Productsoffuzzysets ....................

The (Cartesian)product of ordinary setsX 1 ,X 2 ,...,Xnis the set ofn-tuples

X 1 ◊X 2 ◊∑∑∑◊Xn=

Yn

i=1

Xi={(x 1 ,x 2 ,...,xn)|xi∈Xi}

Theproduct of fuzzy setsAi:Xi→[0,1],i=1,...,n, is the fuzzy set

A:

Yn

i=1

Xi→[0,1]

defined by
A(x 1 ,x 2 ,...,xn)=A 1 (x 1 )∧∑∑∑∧An(xn)
Given rules

Ri:IfAi 1 andAi 2 and ... andAikthenBi,i=1, 2 ,...,n

whereAij:Xj→[0,1]andBi:Y→[0,1], interpreting ìandî as minimum, we
can represent these rules as

Ri:IfAithenBi,i=1, 2 ,...,n

whereAi=

Qn

j=1Aij:

Qn
j=1Xj →[0,1]; so in a situation like this, we can
always assume that we have just one fuzzy setAifor eachi, with domain
X=

Qk
j=1Xjan ordinary product of sets. All of the models we describe allow
this simplification. A ruleRiis said tofireatxifAi(x) 6 =0,inotherwords,
ifxis in the support ofAi.

3.5.2 Mamdanimodel .......................

Given rules ìIfxisAithenyisBi,îi=1,...,nwherex=(x 1 ,x 2 ,...,xk),they
are combined in the Mamdani model as

R(x,y)=

_n

i=1

(Ai(x)∧Bi(y))

For eachk-tuplex=(x 1 ,x 2 ,...,xk)this gives a fuzzy setRxdefined by

Rx(y)=

_n

i=1

Ai(x)∧Bi(y)

Note that for the expanded set of rules

Ri:IfAi 1 andAi 2 and ... andAikthenBi,i=1, 2 ,...,n

this looks like

Rx(y)=R(x 1 ,x 2 ,...,xk,y)=

_n

i=1

(Ai 1 (x 1 )∧Ai 2 (x 2 )∧∑∑∑∧Aik(xk)∧Bi(y))