A First Course in FUZZY and NEURAL CONTROL

98 CHAPTER 3. FUZZY LOGIC FOR CONTROL

Table 3.2 (a). Nilpotent t-norms,a> 0

Type t-norm L-generator/Residual

Bounded (x+y−1)∨ 0 L-Gen:x

product Res: 1 −x

Schweizer-Sklar ((xa+ya−1)∨0)

(^1) a
L-Gen:xa
Res:(1−xa)
(^1) a
Yager

≥

1 −((1−x)a+(1−y)a)

(^1) a¥
∨ 0 L-Gen: 1 −(1−x)a
Res: 1 −(1−(1−x)a)
(^1) a
Sugeno-Weber (a(x+y−1)−(a−1)xy)∨ 0 L-Gen:−loga

°

1 −x+^1 ax

¢

a 6 =1 Res: 1 −^1 −a−x 1

ax

Table 3.2 (b). Nilpotent t-conorms,a> 0

Type t-conorm L-cogenerator/Residual

Bounded sum (x+y)∧ 1 L-Cog: 1 −x

Res: 1 −x

Schweizer-Sklar 1 −(((1−x)a+ L-Cog:(1−x)a

(1−y)a−1)∨0)

(^1) a
Res: 1 −(1−(1−x)a)
(^1) a
Yager (xa+ya)
(^1) a
∧ 1 L-Cog: 1 −xa
Res:(1−xa)
(^1) a
Sugeno-Weber (x+y+(a−1)xy)∧ 1 L-Cog: 1 −loga(1−x+ax)
a 6 =1 Res: 1+(^1 a−−x1)x
A t-conorm that satisfiesx◦x=xfor allxis calledidempotent.The
maximum t-conorm is idempotent, and it is the only idempotent t-conorm.
A continuous t-conorm that satisfies x◦x>xfor allx 6 =0, 1 is called
Archimedean. All continuous t-conorms we will consider, other than max-
imum, are Archimedean.
An Archimedean t-conorm for whichx◦x=1only whenx=1is called
strict. The algebraic sum t-conorm is the prototype for a strict t-conorm.
Archimedean t-conorms that are not strict arenilpotent. The bounded sum
t-conorm is the prototype for a nilpotent t-conorm. See Tables 3.1 (b) and 3.2
(b) for examples of strict and nilpotent t-conorms.
Nonassociative AND operations Averaging two AND operations produces
a new AND operation we will denote by ì&.î The result is a nonassociative
operation that can be used to model real-life expert reasoning.
Example 3.2Take theŁukasiewicz and max-min t-norms for the two AND
operations. If we know the degrees of certainty (subjective probabilities)p 1 =