How to calculate the poles of the fractional transfer function

This is my fractional transfer fucnction, I want to draw the poles of the transfer function. I tried to use the FOMCON toolbox, but there is no pzmap function.How should I do?

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clear;clc;

s=fotf('s');

lamda1=0.8;

lamda2=0.8;

w0=300;

alpha1=2*w0;

alpha2=w0^2;

beta1=2*w0;

beta2=w0^2;

k=60;

r=1;

h1=s^(lamda1+1)+alpha1*s^(lamda1)+alpha2;

h2=s^(lamda2+1)+beta1*s^(lamda2)+beta2;

m1=h1*h2-h2*alpha2-h1*beta2+beta2*(s^(lamda1+1)+alpha2);

m2=h1*h2*k+alpha2*h2*s+alpha1*beta2*s^(lamda1+1);

G=k*h1*h2/(r*m1*s+m2);

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where G is the transfer function.

[Umar]

Hi @ Sicong Wen,

Addressing your query, please use the function, I saved it as “tran.m” and code below, I did use function “pzmap“ without using toolbox and it did plot results,at the moment I can’t upload the plot results since daily upload is limited to 10 plots. Hope, you understand.

%function

% Define the fotf function to create a fractional order transfer function function F = fotf(s) F = s;

%script code

>> % Define the transfer function 's' with generic data

s = tf('s');

lambda1 = 1;

lambda2 = 1;

w0 = 300;

alpha1 = 2 * w0;

alpha2 = w0^2;

beta1 = 2 * w0;

beta2 = w0^2;

k = 60;

r = 1;

% Define the transfer function components

h1 = s^(lambda1 + 1) + alpha1 * s^lambda1 + alpha2;

h2 = s^(lambda2 + 1) + beta1 * s^lambda2 + beta2;

m1 = h1 * h2 - h2 * alpha2 - h1 * beta2 + beta2 * (s^(lambda1 + 1) + alpha2);

m2 = h1 * h2 * k + alpha2 * h2 * s + alpha1 * beta2 * s^lambda1;

% Calculate the transfer function G

G = k * h1 * h2 / (r * m1 * s + m2);

% Draw the poles of the transfer function

figure;

pzmap(G);

title('Poles of the Transfer Function');

For more guidance on pzmap function, please refer to

https://www.mathworks.com/help/control/ref/lti.pzmap.html

Please let me know if you have any further questions.

Answer 1

[Sandeep Mishra]

Hi Sicong Wen,

To plot the poles and zeros of a fractional transfer function, you can use ‘pzmap’ function in MATLAB. It takes continuous or discrete-time dynamic system model as input.

To plot the data, you need to convert the fractional-order transfer function to state-space model and use ‘pzmap’ function. This can be done by using the following code-snippet in MATLAB:

% Convert a commensurate-order fractional-order transfer function to state-space form 
[A,B,C,D,q] = tf2ss(G); 

% Creating a state-space model object 
state_space_model = ss(A, B, C, D) 

% Plotting poles and zeroes using pzmap 
pzmap(state_space_model) 

Please refer to the below documentations for more information.

1. 'tf2ss' function (FOMCOM Toolbox): https://www.mathworks.com/matlabcentral/fileexchange/66323-fomcon-toolbox-for-matlab
2. 'pzmap' (Pole-zero plot of dynamic system): https://www.mathworks.com/help/control/ref/lti.pzmap.html
3. 'ss' (State-space model): https://www.mathworks.com/help/control/ref/ss.html

I hope this helps.