A First Course in FUZZY and NEURAL CONTROL

3.5. COMBINING FUZZY RULES 109

whereA,A,B,ò Bòare fuzzy sets representing fuzzy concepts. The computation of
Bòcan be carried out through a basic rule of inference called thecompositional
rule of inference,namely,Bò=R◦AòwhereRis a fuzzy relation representing
the implication or fuzzy conditional proposition ìPremise 2.î This inference
scheme is sometimes described as a problem ofinterpolation.Interpolation
lies at the heart of the utility of fuzzy rule-based systems because it makes it
possible to employ a relatively small number of fuzzy rules to characterize a
complex relationship between two or more variables.
A number of formulas have been proposed for this implication, most com-
monly thecompositional conjunction

R(x,y)=A(u)∧B(v)

ThenBòis defined as

Bò(v)=

≥

R◦Aò

¥

(u)=

W

u

≥

Aò(u)∧A(u)∧B(v)

¥

(See page 118 for a discussion of max-min composition with fuzzy relations.)
We describe four inference and aggregation mechanisms ñ named after
Mamdani, Larsen, Takagi-Sugeno-Kang, and Tsukamoto ñ commonly used to
interpret a collection of rules

IfxisAithenyisBi,i=1, 2 ,...,n

Applications of these methods in control theory will be discussed in Chapter 4.
In the following examples, we will use the same four fuzzy setsA 1 ,A 2 ,B 1 ,
andB 2 to illustrate the combination of fuzzy rules:

A 1 (x)=







x if 0 ≤x≤ 1

2 −x if 1 ≤x≤ 2

0otherwise

B 1 (y)=







1

8 y if^0 ≤y≤^8

−^14 y+3 if 8 ≤y≤ 12

0otherwise

A 2 (x)=







x− 1 if 1 ≤x≤ 2

3 −x if 2 ≤x≤ 3

0otherwise

B 2 (y)=







1

6 y−

2

3 if^4 ≤y≤^10

−^15 y+3 if 10 ≤y≤ 15

0otherwise

(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

(^246) x 8 10 12 14
B 1 andB 2
First we look at theproductof fuzzy sets, as this greatly simplifies the
presentation of fuzzy rules for the Mamdani and Larsen models.