Lecture 3: Fuzzy Membership Function

Professor. Dr. Hani Hagras

Fuzzy Logic Control and Hybrid Systems

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) ax b

(c-x)/(c-b) bx c

0 otherwise

(x-a)/b-a) ax b

1 bx c

(d-x)/(d-c) cx d

0 otherwise

))/)((5.0( 2axe

• Average = a= = (x1+x2+…xn)/N

Where N is the total number of data points

• Standard deviation = θ =

x

-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

c-a

c

c+a

slope = -b/2a

• 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