Fuzzy Membership Functions

 

Inputs and outputs of Mamdani type and Inputs of Sugeno type can be chosen from 11 member functions.

 

1.  dsigmf: Composed of different between two spline curves, See Fig 1.

2.  gauss2mf: Gaussian combination, See Fig 2.

3.  gaussmf: Gaussian curve, See Fig 3.

4.  gbellmf: Generalized bell-shape, See Fig 4.

5.  pimf: -shape, See Fig 5.

6.  psigmf: Composed of product of two Sigmoidally shaped mf, See Fig 6.

7.  sigmf: Sigmoidally shape, See Fig 7.

8.  smf: Spline-based curve , See Fig 8.

9.  trapmf: Trapezoidal-shape, See Fig 9.

10.      trimf: Triangular shape, See Fig 10.

11.      zmf: Z-shape, See Fig 11.

 

Each member function and required parameters are described in Table 1.

 

No.

Name

Number of Parameter

Mathematical definition

1

dsigmf

4

about arbitrary x

MF Parameters a1,c1,a2 and c2

2

gauss2mf

4

about arbitrary x

MF Parameters sig1 , c1, sig2, and c2

If c1 < c2, the maximum value is 1.

3

gaussmf

2

about arbitrary x

MF Parameters sig and c

4

gbellmf

3

about arbitrary x

MF Parameters a, b and c

5

pimf

4

about arbitrary x

MF Parameters a,b,c and d

6

psigmf

4

about arbitrary x

MF Parameters a1, c1,a2 and c2

7

sigmf

2

about arbitrary x

MF Parameters a and c

8

smf

2

about arbitrary x

MF Parameters a and b

9

trapmf

4

about arbitrary x

MF Parameters a, b, c and d

10

trimf

3

about arbitrary x

MF Parameters a, b and c

11

zmf

2

about arbitrary x

MF Parameters a and b

Table 1  Member functions in Fuzzy logic

 

img6.gif

Figure 1  'dsigmf' Membership Function

 

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Figure 2  'gauss2mf' Membership Function

 

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Figure 3  'gaussmf' Membership Function

 

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Figure 4  'gbellmf' Membership Function

 

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Figure 5  'pimf' Membership Function

 

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Figure 6  'psigmf' Membership Function

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Figure 7  'sigmf' Membership Function

 

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Figure 8  'smf' Membership Function

 

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Figure 9  'trapmf' Membership Function

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Figure 10  'trimf' Membership Function

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Figure 11  'zmf' Membership Function

 

Output member functions of Sugeno type is two. Actually this member functions (MFs) is related Fuzzy rule of Sugeno type. The parameters of this MFs are consists of a value of linear first order polynomial equation including input numbers. Therefore, the number of parameters is the number of inputs plus 1.

 

     constant: one of a linear type that just has zero coefficient of first order value.

     linear: for example ' '. The numbers of parameters are n. The number of inputs is n-1. The constant value is a0.