A First Course in FUZZY and NEURAL CONTROL

90 CHAPTER 3. FUZZY LOGIC FOR CONTROL

The values ofσdetermine either increasing or decreasing functions, while the
parametermshifts the function right or left. These same shapes can be achieved
with hyperbolic tangent functions since^12 (1 + tanhx)=1+^1 e 2 x:

0

1

-4 -2 (^2) x 4
1+tanhx
0
1
-4 -2 2 x 4
1 −tanhx
The product of two sigmoidal functions is sometimes used
0
0.2
0.4
0.6
0.8
-6 -4 -2 (^2) x 4 6
1 −tanh^2 x
All of these functions can be useful for different applications.

3.3 Combiningfuzzysets.........................

In fuzzy control theory, where we work with collections of fuzzy subsets, we
need useful ways to combine them. These ways of combining should coincide
with known methods when the sets in question are ordinary sets. In other
words, methods of combining fuzzy sets should generalize common methods for
ordinary sets. The various operators used to combine fuzzy sets are calledfuzzy
connectivesoraggregation operators.
The variety of operators for the aggregation of fuzzy sets can be confusing.
If fuzzy set theory is used as a modeling language for real situations or systems,
it is not only important that the operators satisfy certain axioms or have certain
formal qualities (such as associativity and commutativity), that are certainly of
importance, but the operators must also be appropriate models of real-system
behavior, and this can normally be proven only by empirical testing. In practice,
numerical efficiency in computations can also be an important consideration.

3.3.1 Minimum,maximum,andcomplement...........

Ordinary subsets, also known ascrispsets, of a setXare often combined or
negated viaintersection(AND),union(OR), andcomplement(NOT):

AND: A∩B={x∈X:x∈Aandx∈B}(intersection)

OR: A∪B={x∈X:x∈Aorx∈B}(union)

NOT: X−A={x∈X:x/∈A}(complement)

(3.3)