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Explain fuzzy membership functions

written 19 days ago by | • modified 19 days ago |

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written 19 days ago by |

Following are the different fuzzy membership functions:

*1. Triangular membership function:*

(triangle(x:a,b,c) = \left{ 0 \ x<=a \ \frac{x-a}{b-a} \ a<=x<=b\ \frac{c-x}{c-b} \ b<=x<=c\ 0 \ c<=x\right})

Eg:

$triangle(x:20,60,80)$

In design only triangular membership function is used.

*2. Trapezoidal membership function:*

(trapezoid(x:a,b,c,d) = \left{ 0 \ x<=a \ \frac{x-a}{b-a} \ a<=x<=b \ 1 \ b<=x<=c\ \frac{d-x}{d-c} \ c<=x<=d\ 0 \ d<=x\right})

$trapezoid(x:10,20,60,95)$

*3. Gaussian membership function:*

$Gaussian(x:c,\sigma)\\ =e^{-\frac{1}{2}(\frac{x-c}{\sigma})^2}$

Eg:

$gaussian(x:50,20)$

*4. Generalized bell membership function:*

$Bell(x:a,b,c)\\=\frac{1}{1+| \frac{x-c}{a}|^{2b}}$

Eg:

$Bell(x:20,4,80)$

*5. Sigmoidal membership function:*

$sig(x:a,c)=\frac{1}{1+exp[-a(x-c)]}$

Eg:

$sig(x:1,-5)$

In this case 'c' is also a slope control parameter but it is connected with 'a'

*6. A left-right membership function:*

(LR(x:c,\alpha,\beta)\ =\left{F_L (\frac{c-x}{\alpha}),x<=c\ F_R(\frac{x-c}{\beta}),x<=c\right})

Eg:

$LR(x:65,60,10)$

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