# smf

S-shaped membership function

## Description

This function computes fuzzy membership values using a spline-based S-shaped membership function. You can also compute this membership function using a fismf object. For more information, see fismf Object.

This membership function is related to the zmf and pimf membership functions.

example

y = smf(x,params) returns fuzzy membership values computed using the spline-based S-shaped membership function given by:

$f\left(x;a,b\right)=\left\{\begin{array}{ll}0,\hfill & x\le a\hfill \\ 2{\left(\frac{x-a}{b-a}\right)}^{2},\hfill & a\le x\le \frac{a+b}{2}\hfill \\ \text{1}-\text{2}{\left(\frac{x-b}{b-a}\right)}^{\text{2}},\hfill & \frac{a+b}{2}\le x\le b\hfill \\ 1,\hfill & x\ge b\hfill \end{array}\right\}$

To specify the a and b parameters, use params.

Membership values are computed for each input value in x.

## Examples

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x = 0:0.1:10;
y = smf(x,[1 8]);
plot(x,y)
xlabel('smf, P = [1 8]')
ylim([-0.05 1.05])

## Input Arguments

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Input values for which to compute membership values, specified as a scalar or vector.

Membership function parameters, specified as the vector [a b]. Parameter a defines the foot of the membership function, and b defines its shoulder.

## Output Arguments

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Membership value returned as a scalar or a vector. The dimensions of y match the dimensions of x. Each element of y is the membership value computed for the corresponding element of x.

## Alternative Functionality

### fismf Object

You can create and evaluate a fismf object that implements the smf membership function.

mf = fismf("smf",P);
Y = evalmf(mf,X);

Here, X, P, and Y correspond to the x, params, and y arguments of smf, respectively.