smf

S-shaped membership function

Syntax

```y = smf(x,[a b])
```

Description

This spline-based curve is a mapping on the vector `x`, and is named because of its S-shape. The parameters a and b locate the extremes of the sloped portion of the curve, as given by:

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

Examples

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S-Shaped Membership Function

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