Find the system type given an open loop transfer function

1.21 Find the system type given an open loop transfer function

Problem: Find the system type for the following transfer functions

$\frac{s + 1}{s^{2} - s}$
$\frac{s + 1}{s^{3} - s^{2}}$
$\frac{s + 1}{s^{5}}$

To ﬁnd the system type, the transfer function is put in the form $\frac{k \sum_{i} (s - s_{i})}{s^{M} \sum_{j} (s - s_{j})}$ , then the system type is the exponent $M$ . Hence it can be seen that the ﬁrst system above has type one since the denominator can be written as $s^{1} (s - 1)$ and the second system has type 2 since the denominator can be written as $s^{2} (s - 1)$ and the third system has type 5. The following computation determines the type

Mathematica

Clear["Global`*"]; 
p=TransferFunctionPoles[TransferFunctionModel[ 
    (s+1)/(s^2-s),s]]; 
Count[Flatten[p],0]

Out[171]= 1

p=TransferFunctionPoles[TransferFunctionModel[ 
    (s+1)/( s^3-s^2),s]]; 
Count[Flatten[p],0]

Out[173]= 2

p=TransferFunctionPoles[ 
   TransferFunctionModel[(s+1)/ s^5 ,s]]; 
Count[Flatten[p],0]

Out[175]= 5

Matlab

clear all; 
s=tf('s'); 
[~,p,~]=zpkdata((s+1)/(s^2-s)); 
length(find(p{:}==0))

ans = 
     1

[~,p,~]=zpkdata((s+1)/(s^3-s^2)); 
length(find(p{:}==0))

ans = 
     2

[~,p,~]=zpkdata((s+1)/s^5); 
length(find(p{:}==0))

ans = 
     5

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