86 CHAPTER 3. FUZZY LOGIC FOR CONTROL
If a strong wind is blowing right to left, then aim to the right of the goalposts.The use of basic rules of this form is the basic idea behind fuzzy control. Lin-
guistic variables such as fast, slow, large, medium, and small are translated into
fuzzy sets; mathematical versions of ìIf...then...î rules are formed by combining
these fuzzy sets.
3.2 Fuzzysetsincontrol
It is easy to express rules in words, but the linguistic notion of fuzziness as
illustrated in the rules above needs to be represented in a mathematical way
in order to make use of this notion in control theory. How can we do that?
The mathematical modeling of fuzzy concepts wasfirst presented by Professor
LotfiZadeh in 1965 to describe, mathematically, classes of objects that do not
have precisely defined criteria of membership. His contention is that meaning
in natural language is a matter of degree.
Zadeh gave the examples, ìthe class ofall beautiful womenî and ìthe class
of all tall men.î The notion of ìtallî can be depicted by a graph such as Figure
3.1, where thex-axis represents height in centimeters, and they-axis represents
the degree, on a scale of 0 to 1, of the tallness attributed to that height. Of
00.20.40.60.81y(^50100) x 150 200 250
Figure 3.1. ìTallnessî of height in centimeters
course, the scaling of this function depends on the context, but the shape is
descriptive in a fairly general setting.
Before defining a fuzzy subset mathematically, wefirstlookatordinary
subsets in a special way that will allow us to expand the notion of subset to
that of fuzzy subset. An ordinary subsetAof a setXcan be identified with a
functionX→{ 0 , 1 }fromXto the 2-element set{ 0 , 1 },namely
A(x)=
Ω
1 ifx∈A
0 ifx/∈AThis function is called thecharacteristic functionorindicator functionof
the fuzzy setA.IfAis the set of real numbers equal to or bigger than 10 and less
than or equal to 40 ,thesetAwouldbedepictedasinFigure3.2.Incontrast,
elements of fuzzy subsets ofXcan have varying degrees of membership from 0
to 1.