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!
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 = ≥
 −
 ≤ <
−
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
 − 
= −  
ï£ ï£¸ï£¯  
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
=
−
+