# Lecture 2: Fuzzy Membership Functions and Fuzzy Numbers

```Fuzzy Logic Control and Hybrid Systems
Lecture 3: Fuzzy Membership Function
Professor. Dr. Hani Hagras
Membership Functions
• Determining or finding input/output membership
functions is the first step of the fuzzy logic control
process where a fuzzy algorithm categorises the
information entering a system and assigns values that
represent the degree of membership in those
categories.
• Input membership functions themselves can take any
form the designer of the system requires triangles,
trapezoids, bell curves or any other shape as long as
those shapes accurately represent the distribution of
information within the system, and as long as a region
of transition exists between adjacent membership
functions.
Membership Functions
•In Rule Based applications of Fuzzy Logic, membership
functions are associated with terms that appear in the
antecedents or consequents of rules
Shapes for Membership
(x-a)/b-a) ax b
1
bx c
(d-x)/(d-c) cx d
0 otherwise
(x-a)/b-a) ax b
(c-x)/(c-b) bx c
0 otherwise
e(0.5(( x  a) /  ) )
2
• Average = a= x = (x1+x2+…xn)/N
Where N is the total number of data points
• Standard deviation = θ =
-3
a
3
Generalised Bell Membership Function
A generalised bell membership, can be specified by
three parameters {a, b, c}
bell (x: a, b, c) =
1/ (1+ (x-c/a)2b)
Membership
slope = -b/2a
c-a
c
c+a
• Due to their simple formulas and computational
efficiency, both triangular membership functions and
trapezoidal membership functions have been used
extensively, especially in real-time implementation
• However since the membership functions are
composed of straight-line segments, they are not
smooth at the switching points specified by the
parameters.
• The Guassian and the Bell membership function
provides smooth and non-linear functions that can be
used by the learning systems like Neural Networks.
Fuzzy Terminology
Support
Core
 -Cut
Height
Modifiers
```