1.31 Draw the root locus from the open loop transfer function

Problem: Given $L(s)$, the open loop transfer function, draw the root locus. Let

$$L(s)=\frac{s^2+2s+4}{s(s+4)(s+6)(s^2+1.4s+1)}$$

Root locus is the locus of the closed loop dominant pole as the gain $k$ is varied from zero to infinity.

Mathematica

Remove["Global`*"] sys = TransferFunctionModel[ k*(s^2+2 s+4)/ (s(s+4)(s+6)(s^2+1.4s+1)),s]; RootLocusPlot[sys,{k,0,100}, ImageSize->300, GridLines->Automatic, GridLinesStyle->Dashed, Frame->True, AspectRatio -> 1]

 

Matlab

clear all; close all; s=tf('s'); sys=(s^2+2*s+4)/... (s*(s+4)*(s+6)*(s^2+1.4*s+1)); rlocus(sys,0:100) set(gcf,'Position',[10,10,420,420]);