A First Course in FUZZY and NEURAL CONTROL

3.5. COMBINING FUZZY RULES 113

Example 3.8Take the fuzzy setsAidefinedinEquation3.8andthefunctions

f 1 (x)=2+xf 2 (x)=1+x

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

2

4

(^123) x
f 1 (x)=2+x
0
1
2
3
(^123) x
f 2 (x)=1+x
Then the rules
Ri:IfxiisAithenfi(x),i=1, 2
produce the function
0
1
2
3
4
5
y
0.5 1 1.5x 2 2.5
R(x)=
A 1 (x)f 1 (x)+A 2 (x)f 2 (x)
A 1 (x)+A 2 (x)
A 1 andA 2 (dashed lines)
f 1 andf 2 (dotted lines)

3.5.5 Tsukamotomodel ......................

Given rules ìIfxisAithenyisCi,îi=1,...,n,withCiall monotonic (either
strictly increasing or strictly decreasing), the Tsukamoto model produces the
function

y=

Pn

i=1C

− 1

P i Ai(x)

n

i=1Ai(x)

The monotonicity of the fuzzy setsCiis necessary in order to compute the
inverse functionsCi−^1.