1.21 Find the system type given an open loop transfer function

Problem: Find the system type for the following transfer functions

1. \(\frac {s+1}{s^{2}-s}\)
2. \(\frac {s+1}{s^{3}-s^{2}}\)
3. \(\frac {s+1}{s^{5}}\)

To find the system type, the transfer function is put in the form \(\frac {k\sum _{i}\left ( s-s_{i}\right ) }{s^{M}\sum _{j}\left ( s-s_{j}\right ) }\), 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 \(s^{1}\left ( s-1\right )\) and the second system has type 2 since the denominator can be written as \(s^{2}\left ( s-1\right ) \) 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