A First Course in FUZZY and NEURAL CONTROL

114 CHAPTER 3. FUZZY LOGIC FOR CONTROL

Example 3.9Take the two fuzzy setsA 1 andA 2 of Equation 3.8, and the
fuzzy setsC 1 andC 2 and their inverses.

C 1 (y)=y/ 2

C 2 (y)=

√

y

C 1 −^1 (z)=2z

C 2 −^1 (z)=z^2

The rules ìIfxisAithenyisCi,îi=1, 2 , produce the function

R(x)=

P 2

i=1C

− 1

P i (Ai(x))

2

i=1Ai(x)

=







2 if 0 ≤x≤ 1

5 − 4 x+x^2 if 1 ≤x≤ 2

3 −x if 2 ≤x≤ 3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

y

0.5 1 1.5x 2 2.5 3

R(x)=

P 2

i=1C

− 1

P i (Ai(x))

2

i=1Ai(x)

A 1 andA 2 (dashed lines)

C 1 andC 2 (dotted lines)

3.6 Truthtablesforfuzzylogic .....................

In classical two-valued logic, truth tables can be used to distinguish between
expressions. The following tables define the operations of∨,∧,and^0 on the
truth values{ 0 , 1 },where 0 is interpreted as ìfalseî and 1 is interpreted as
ìtrue.î

∨ 01

0 01

1 11

∧ 01

0 00

1 01

0

0 1

1 0

Truth tables, in a slightly different form, can be used to determine equivalence
of logical expressions. Take two variablesx,yin classical two-valued logic, let
p=xandq=(x∧y)∨(x∧y^0 ), and compare their values: