### 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 `