A First Course in FUZZY and NEURAL CONTROL

88 CHAPTER 3. FUZZY LOGIC FOR CONTROL

Trianglesandtrapezoids, which are piecewise-linear functions, are often
used in applications. Graphical representations and operations with these fuzzy
sets are very simple. Also, they can be constructed easily on the basis of little
information.

0

0.5

1

Triangular

0

0.5

Trapezoidal

The triangular functionAwith endpoints(a,0)and(b,0),andhighpoint(c,α)
is defined by

A(x)=















α

μ

x−a

c−a

∂

ifa≤x≤c

α

μ

x−b

c−b

∂

ifc≤x≤b

0 otherwise

(3.1)

The trapezoidal functionBwith endpoints(a,0)and(b,0), and high points
(c,α)and(d,α)is defined by

A(x)=



















α

μ

x−a

c−a

∂

ifa≤x≤c

α ifc≤x≤d

α

μ

x−b

d−b

∂

ifd≤x≤b

0 otherwise

(3.2)

When the domain isfinite, these fuzzy sets can be depicted as follows:

0

0.5

Triangular

0

0.5

Trapezoidal

Several fuzzy sets are often depicted in the same plot, as follows:

0

0.5

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