A First Course in FUZZY and NEURAL CONTROL

112 CHAPTER 3. FUZZY LOGIC FOR CONTROL

Note that for the 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)

Example 3.7Take the fuzzy setsAiandBidefined in Equation 3.8.

0

0.2

0.4

0.6

0.8

1

y

0.5 1 1.5x 2 2.5 3

A 1 andA 2

0

0.2

0.4

0.6

0.8

1

y

2 4 6 x 8 10 12 14

B 1 andB 2

At the pointx=1. 25 ,therulesìIfxisAithenyisBi,îi=1, 2 ,produce the
fuzzy set

0

0.2

0.4

0.6

0.8

1

x

(^2468) y 10 12 14 16
R 1. 25 (y)=(A 1 (1.25)∑B 1 (y))∨(A 2 (1.25)∑B 2 (y))
B 1 andB 2 (dotted lines)

3.5.4 Takagi-Sugeno-Kang(TSK)model .............

For the TSK model, rules are given in the form

Ri:Ifx 1 isAi 1 andx 2 isAi 2 and ... andxkisAik

thenfi(x 1 ,x 2 ,...,xk),i=1, 2 ,...,n

or
Ri:IfxiisAithenfi(x),i=1, 2 ,...,n

wheref 1 ,f 2 ,...,fnare functionsX=X 1 ◊X 2 ◊∑∑∑◊Xk→RandAi=

Vk
j=1Aij.
These rules are combined to get a function

R(x)=

A 1 (x)f 1 (x)+A 2 (x)f 2 (x)+∑∑∑+An(x)fn(x)

A 1 (x)+A 2 (x)+∑∑∑+An(x)

Thus, this model produces a real-valued function.