A First Course in FUZZY and NEURAL CONTROL

3.5. COMBINING FUZZY RULES 111

In fuzzy control, the number

Wn
i=1Ai(x)=Ai^1 (x^1 )∧Ai^2 (x^2 )∧∑∑∑∧Aik(xk)
is called thestrengthof the ruleRifor the inputx. The fuzzy setRi,x(y)=
Ai(x)∧Bi(y)is called thecontrol outputof the ruleRifor the inputx,and
the fuzzy setRx(y)is theaggregated control outputfor the inputx.

Example 3.6Take the fuzzy setsAiandBidefinedinEquation3.8.

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

(^246) x 8 10 12 14
B 1 andB 2
At the pointx=1. 25 ,therulesìIfxisAithenyisBi,i=1, 2 ,î produce the
fuzzy set
R 1. 25 (y)=



















° 3

4 ∧

1

8 y

¢

° if^0 ≤y≤^4

3

4 ∧

° 1

8 y

¢¢

∨

° 1

4 ∧

° 1

6 y−

2

3

¢¢

° if^4 ≤y≤^8

3

4 ∧

°

−^14 y+3

¢¢

∨

° 1

4 ∧

° 1

6 y−

2

3

¢¢

° if^8 ≤y≤^10

3

4 ∧

°

−^14 y+3

¢¢

∨

° 1

4 ∧

°

−^15 y+3

¢¢

° if^10 ≤y≤^12

1

4 ∧

°

−^15 y+3

¢¢

if 12 ≤y≤ 15

0 if otherwise

0

0.2

0.4

0.6

0.8

1

x

246810121416 y

[A 1 (1.25)∧B 1 (y)]∨[A 2 (1.25)∧B 2 (y)]

B 1 andB 2 (dotted lines)

3.5.3 Larsenmodel.........................

Given rules ìIfxisAithenyisBi,îi=1,...,n, they are combined in the
Larsen model as

R(x,y)=

_n

i=1

(Ai(x)∑Bi(y))

where∑indicates multiplication. For eachk-tuplex=(x 1 ,x 2 ,...,xk)this gives
a fuzzy set

Rx(y)=

_n

i=1

Ai(x)∑Bi(y)