# What is the best function of the following shape?

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How can I construct this shape? what is the best function of it?

It is more like sigmoid function + straight line

But I cant get how can I represent it by one equation and How to control its limits!

The Y limits should be from 0 to 1

( I want to use it as a fuzzy memebership function)

Is there any recommendation?

##### 2 Comments

Sam Chak
on 7 Sep 2022

### Accepted Answer

Sam Chak
on 7 Sep 2022

Edited: Sam Chak
on 12 Sep 2022

Hi @M

The original code was accidentally highlighted and deleted due to sensitive touchpad. It was constructed based on @Torsten's idea on the Arctangent function but I replaced it with the Logistic function as shown in this link:

Since you don't want anything to do with the Fuzzy Logic Toolbox, this new solution, a Quintic function (not in the toolbox, so you can probably claim novelty in the Fuzzy MF paper), is tailor-made for you so that you adjust the position of the cutoff point (end point of blue line) easily.

Example #1: The Quintic function, a 5th-order polynomial, is commonly used in the trajectory planning of a mobile robot. So I modified the function for a custom fuzzy membership function.

I have not tested everything. But this is the basic concept for you to develop. Hope it helps.

% Quintic function-based MF (blue line as main)

% Start point (xA, yA); End point (xB, yB);

xf = 1;

x = linspace(0, xf, xf*1e4 + 1);

yA = 0;

yB = 1;

xA = 0.1;

xB = 0.9;

y = (yA + (10*((x - xA)/(xB - xA)).^3 - 15*((x - xA)/(xB - xA)).^4 + 6*((x - xA)/(xB - xA)).^5)*(yB - yA)).*((0 <= (x - xA)) & ((x - xA) < (xB - xA)));

plot(x, y, -(x-xf), y), grid on, ylim([0 1.2])

legend('MF_1', 'MF_2', 'location', 'north')

xlabel('Universe of Discourse, \it{x}'), ylabel('\mu(\it{x})')

xf = 1;

x = linspace(0, xf, xf*1e4 + 1);

xA = 0.1; % Left bound of the blue S-curve

xB = 0.9; % Right bound of the blue S-curve

a = 2*9.2; % (logistic growth rate) adjust the steepness of S-curve

c = 0.5; % shift the center of S-curve to x = c

y = sigmf(x, [a c]).*((0 <= (x - xA)) & ((x - xA) < (xB - xA)));

plot(x, y, -(x-xf), y), grid on, ylim([0 1.2])

legend('MF_1', 'MF_2', 'location', 'north')

xlabel('Universe of Discourse, \it{x}'), ylabel('\mu(\it{x})')

##### 7 Comments

Sam Chak
on 12 Sep 2022

@M, Theoretically, the polynomial function offers more flexibility in adjusting the shape of the S-curve in anyway you like as long as it is monotonically increasing/decreasing, and if you know how to adjust the coefficients of the terms.

For simplicity, you may want to choose the Logistic function (aka sigmf), because you just need adjust one parameter, the logistic growth rate (a) to shape the steepness of the S-curve.

I have added a second example in the edited Answer.

Bruno Luong
on 12 Sep 2022

((0 <= (x - xA)) & ((x - xA) < (xB - xA)))

or equivalently

((x >= xA) & (x < xB))

### More Answers (4)

Bruno Luong
on 7 Sep 2022

Try this:

xmin = 1;

xmax = 3;

pp = struct('form','pp',...

'breaks',[-1e-100 0 1 inf],...

'coefs',[0 0 0 0 0 0;6,-15,10,0,0,0;0 0 0 0 0 1],...

'pieces',3,...

'order',6,...

'dim',1);

sfun = @(x) ppval(pp,(x-xmin)/(xmax-xmin));

ezplot(sfun,xmin-1,xmax+1)

##### 13 Comments

Walter Roberson
on 12 Sep 2022

In analysis, a "function" is any mapping of values to other values. It might or might not be "one to one" or "onto". There is no requirement that the function maps all possible inputs in the domain.

In analysis, functions over infinite sets are treated like the limit of finite mappings. For example, x->x^2, x in Z, is treated as-if it were the set of mappings {..., -2 -> 4, -1 -> 1, 0 -> 0, 1 -> 1, 2 -> 2, ...} rather than as being something fundamentally different. There does not have to be a formula for a function: the complete list of mappings defines the function; formulae can make it significantly more compact to express though. The mappings are the fundamental property of the function; formulae are conveniences to express mappings.

Bruno Luong
on 12 Sep 2022

Edited: Bruno Luong
on 12 Sep 2022

"There is no requirement that the function maps all possible inputs in the domain."

Wrong to my book, all elements of input set (domain) must be mapped (defined) by the function.

Walter Roberson
on 6 Sep 2022

Looks like tansig

##### 2 Comments

Image Analyst
on 12 Sep 2022

James Tursa
on 6 Sep 2022

Edited: James Tursa
on 6 Sep 2022

##### 3 Comments

Image Analyst
on 12 Sep 2022

@M there are a variety of equations that can do that. See

and just pick one.

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