20.6.12.3. 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 Figure 20.175.

2. gauss2mf: Gaussian combination, See Figure 20.176.

3. gaussmf: Gaussian curve, See Figure 20.177.

4. gbellmf: Generalized bell-shape, See Figure 20.178.

5. pimf: -shape, See Figure 20.179.

6. psigmf: Composed of product of two Sigmoidally shaped mf, See Figure 20.180.

7. sigmf: Sigmoidally shape, See Figure 20.181.

8. smf: Spline-based curve , See Figure 20.182.

9. trapmf: Trapezoidal-shape, See Figure 20.183.

10. trimf: Triangular shape, See Figure 20.184.

11. zmf: Z-shape, See Figure 20.185.

Each member function and required parameters are described in Table 20.111.

Table 20.111 Member functions in Fuzzy logic

No.	Name	Number of Parameter	Mathematical definition
1	dsigmf	4	\(dsinmf(s,a_1,c_1,a_2,c_2)=\frac{1}{1+e^{-a_1(x-c_1)}}-\frac{1}{1+e^{-a_2(x-c_2)}}\) about arbitrary x MF Parameters a1,c1,a2 and c2
2	gauss2mf	4	\(gauss2mf(x,\sigma_1,c_1,\sigma_2,c_2)=\left\{ \begin{matrix} \text{left gaussian curve} e^{\frac{-(x-c_1)}{2\sigma{_1}^2}} \\ \text{right gaussian curve} 1-e^{\frac{-(x-c_2)}{2\sigma{_2}^2}} \end{matrix} \right\}\) about arbitrary x MF Parameters sig1 , c1, sig2, and c2 If c1 < c2, the maximum value is 1.
3	gaussmf	2	\(gaussmf(x,\sigma,c)=e^{\frac{-(x-c)}{2\sigma^2}}\) about arbitrary x MF Parameters sig and c
4	gbellmf	3	\(gbellmf(x,a,b,c)=\frac{1}{1+\left| \frac{x-c}{a} \right|^2b}\) about arbitrary x MF Parameters a, b and c
5	pimf	4	\(pimf(x,a,b,c,d)=\left\{ \begin{matrix} {\text{left spline curve} smf(x,a,b)} \\ {\text{right spline curve} zmf(x,c,d)} \end{matrix} \right\}\) about arbitrary x MF Parameters a,b,c and d
6	psigmf	4	\(psigmf(x,a_1,c_1,a_2,c_2)=\frac{1}{1+e^{-a_1(x-c_1)}} \times \frac{1}{1+e^{-a_2(x-c_2)}}\) about arbitrary x MF Parameters a1, c1,a2 and c2
7	sigmf	2	\(sigmf(x,a,c)=\frac{1}{e^{-a(x-c)}}\) about arbitrary x MF Parameters a and c
8	smf	2	\(smf(x,a,b)=\left\{ \begin{matrix} \text{if} x \leq a,0 \\ \text{if} a\leq x \leq \frac{a+b}{2}, 2\times \left( \frac{x-a}{b-a}\right)^2 \\ \text{if} \frac{a+b}{2} \leq x \leq b, 1-2\times \left( \frac{b-x}{b-a} \right)^2\end{matrix} \right\}\) about arbitrary x MF Parameters a and b
9	trapmf	4	\(trapmf(x,a,b,c,d)=\left\{ \begin{matrix} \text{if} x \leq a, 0 \\ \text{if} a \leq x \leq b, \frac{x-a}{b-a} \\ \text{if} b \leq x \leq c, 1 \\ \text{if} c \leq x \leq d, \frac{d-x}{d-c} \\ \text{if} d \leq x, 0 \end{matrix} \right\} = max \left( min \left( \frac{x-a}{b-a},1,\frac{d-x}{d-c} \right),0 \right)\) about arbitrary x MF Parameters a, b, c and d
10	trimf	3	\(trimf(x,a,b,c)=\left\{ \begin{matrix} \text{if} x \leq a,0 \\ \text{if} a\leq x \leq b, \frac{x-a}{b-a} \\ \text{if} b \leq x \leq c,\frac{c-x}{c-b} \\ \text{if} c \leq x, 0 \end{matrix} \right\}\) about arbitrary x MF Parameters a, b and c
11	zmf	2	\(zmf(x,a,b)=\left\{ \begin{matrix} \text{if} x \leq a,1 \\ \text{if} a \leq x \leq \frac{a+b}{2}, 1-2\times \left( \frac{x-a}{b-a}\right)^2 \\ \text{if} \frac{a+b}{2} \leq x \leq b, 2\times \left( \frac{b-x}{b-a} \right)^2 \\ \text{if} b \leq x, 0 \end{matrix} \right\}\) about arbitrary x MF Parameters a and b

Figure 20.175 ‘dsigmf’ Membership Function

Figure 20.176 ‘gauss2mf’ Membership Function

Figure 20.177 ‘gaussmf’ Membership Function

Figure 20.178 ‘gbellmf’ Membership Function

Figure 20.179 ‘pimf’ Membership Function

Figure 20.180 ‘psigmf’ Membership Function

Figure 20.181 ‘sigmf’ Membership Function

Figure 20.182 ‘smf’ Membership Function

Figure 20.183 ‘trapmf’ Membership Function

Figure 20.184 ‘trimf’ Membership Function

Figure 20.185 ‘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 \(y={{a}_{1}}{{x}_{1}}+{{a}_{2}}{{x}_{2}}+\cdots +{{a}_{n-1}}{{x}_{n-1}}+{{a}_{0}}\). The numbers of parameters are n. The number of inputs is n-1. The constant value is a0.