repeated roots in residue fn, how to read/ display??

my code is as follows

clc ;

clear;

n1=[5 20 6];

d1=[1 2 1 0];

tf1=tf(n1,d1)

[r,p,k]= residue(n1,d1)

% r values are the numeratior of the tf

% p values are the poles,

%NOTE

%THEY ARE oposite

%VALUE AKA p = -5 is ACT (s+5)

Id really like to be able to get some repeated root values from this but i dont know what to do code wise to get there..

Btw this is from the tf bellow

( 5s^2 + 20s + 6 ) / ( s^3+ 2s^2 + s )

this should yeild 6/s - 1/(s+1) + 9(s+1)^2

but i get

r =

-1

9

6

p =

-1

-1

0

which is great, but doesnt show that the (s+1) of the 9/(s+1) should be ^2 ... can i edit my code some how to show that i needs to be squared?

Answer 1

[Mahesh Taparia]

Hi,

It is possible to know if the fraction contains multiple roots. As per your given code, ‘r’ contains the residue, ‘p’ contains the poles and ‘k’ contains the direct terms of your transfer function. If ‘p’ contains any duplicate element entry, then it means there are multiple roots at the same location. As per your case, p= [-1 –1 0] it means there are two roots at –1.

For more information on residue function, you can refer to the documentation link given below:

https://www.mathworks.com/help/matlab/ref/residue.html

[Aidan Macdougall]

yes, however the second root repeated root (-1) is to the power of two when you do partial fraction expansion, this isnt clear from reading from the p array, and so my question is how do i know form this code that my final result should be -1/(s+1) +9/(s+1)^2 +6/s as you can see the bit i have underlined is to the second power... is there some display i can use to for this type of result rather than just knowing that that is the form?

[Mahesh Taparia]

Hi,

If multiple roots are present at same location, then residue are arranged as per the pole multiplicity. In your case, '-1' residue corresponds to (s+1), '9' residue is for (s+1)^2. In general, for a pole of multiplicity m, you can refer to the following link for residue arrangement:

https://www.mathworks.com/help/matlab/ref/residue.html#bupof9u-5

Hope it will help.

[Aidan Macdougall]

Okay, i understand from this that it is purely a matter of knowing that the repeated values in the p array are going to be of second or higher power, this is helpful thank you both, i was hopeful i could get it to display this simply but ill just to add a note to those lines, regarless thanks for your time :)