Part of Lecture series on EE646, Fuzzy Theory & Applications delivered by me during First Semester of M.Tech. Instrumentation & Control, 2012 Z H College of Engg. & Technology, Aligarh Muslim University, Aligarh Reference Books: 1. T. J. Ross, "Fuzzy Logic with Engineering Applications", 2/e, John Wiley & Sons,England, 2004. 2. Lee, K. H., "First Course on Fuzzy Theory & Applications", Springer-Verlag,Berlin, Heidelberg, 2005. 3. D. Driankov, H. Hellendoorn, M. Reinfrank, "An Introduction to Fuzzy Control", Narosa, 2012. Please comment and feel free to ask anything related. Thanks!

1. EE-646
Lecture-4
Types of Membership Functions

2. Basic Definitions
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3. Symmetric MF
A fuzzy set is symmetric if its membership
function (MF) is symmetric about a certain
point x = c and we write
( ) ( );A Ac x c x x Xµ µ+= − ∀ ∈
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4. Decreasing MF (Open Left)
A fuzzy set is open left or decreasing if
membership values continuously decrease
(from 1) as we increase x
lim ( ) 1 & lim ( ) 0A A
x x
x xµ µ
→−∞ →+∞
= =
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5. Increasing MF (Open Right)
A fuzzy set is open right or increasing if
membership values continuously increase (up
to 1) as we increase x
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lim ( ) 0 & lim ( ) 1A A
x x
x xµ µ
→−∞ →+∞
= =

6. Approximating MF (Closed Fuzzy Set)
A fuzzy set is closed if values on both ends
decrease to zero i.e.
lim ( ) lim ( ) 0A A
x x
x xµ µ
→−∞ →+∞
= =
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7. Types of Membership Functions
• Increasing Type (Γ-function, S-function)
• Decreasing Type (L or Z-function)
• Approximation Type (Triangular function,
Trapezoidal function, Gaussian Function, Bell
function)
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8. Γ- Function
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( )
0,
: , 1,
,
x
x x
x
x
α
α β β
α
α β
β α

 <

Γ = ≥
 −
 ≤ <
−

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10. S - Function
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( )
2
2
0,
2 ,
: , ,
1 2 ,
1,
x
x
x
S x
x
x
x
α
α
α β
γ α
α β γ
α
β γ
γ α
γ
<

 −
≤ <  −  
= 
  −
− ≤ <  − 
 ≥

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By symmetry, we can reduce the no. of parameters and
we can take
2
α γ
β
+
=
µ (x)
x

12. L or Z - Function
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( )
1,
: , ,
0,
x
x
L x x
x
α
α
α β α β
β α
β
 <

−
= ≤ <
−
 ≥

13. µ(x)
x
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14. Triangular Function
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( )
( ) ( )
( ) ( )
0 ,
/ ,
or : , ,
/ ,
0 ,
x
x x
x
x x
x
α
α β α α β
α β γ
γ γ β β γ
γ
≤

− − < ≤
∆ Λ =
− − < ≤
 >

15. µ (x)
x
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16. Trapezoidal or Π-Function
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( )
( ) ( )
( ) ( )
0 ,
/ ,
: , , , 1 ,
/ ,
0 ,
x
x x
x x
x x
x
α
α β α α β
α β γ δ β γ
δ δ γ γ δ
δ
≤

− − < ≤

Π= < ≤
 − − < ≤

 >

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µ(x)
x
α β γ δ

18. Gaussian Function
• Also known as normalized distribution
function. It is defined as
• It can be used as inc, dec or approx. type
function by controlling only two parameters
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( )
2
1
Gaussian : , exp
2
c
c
x x
x x σ
σ
 − 
= −  
   

19. Gaussian Function
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xc
x
µ(x)

20. Generalized Bell Function
• Crossover points are c ± a
• BW is 2a
• Flat on top
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( ) 2
1
Bell : , ,
1
b
x a b c
x c
a
=
−
+

21. Generalized Bell Function
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22. Effect of Change in parameters
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23. Sigmoidal Function
• Used extensively in ANN theory
• Please see yourself
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24. Today’s Task
• Find out the MATLAB commands for these
functions and generate some sample
functions
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