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1.33 Draw root locus for a discrete system

Problem: Given the open loop for a continuous time system as

\[ sys=\frac {5s+1}{s^{2}+2s+3}\]

convert to discrete time using a sampling rate \(0.5\) second, and display the root locus for the discrete system.

Mathematica 

Remove["Global`*"]; 
sys=TransferFunctionModel[ 
       k(5s+1)/(s^2+2s+3),s]; 
 
samplePeriod=0.5; (*sec*) 
sysd=ToDiscreteTimeModel[sys,samplePeriod,z]

pict 

p1 = RootLocusPlot[sysd,{k,0,100}, 
       ImageSize->300, 
       GridLines->Automatic, 
       GridLinesStyle->Dashed, 
       Frame->True, 
       AspectRatio->1, 
       PlotPoints->200, 
       PlotStyle->PointSize[0.01]]; 
 
p2=ParametricPlot[{Sin[t],Cos[t]},{t,0,2Pi}, 
       AspectRatio->1, 
       GridLines->Automatic, 
       GridLinesStyle->Dashed]; 
 
Show[p2,p1]

pict

 

Matlab 

clear all; close all; 
s    = tf('s'); 
sys  = (5*s+1)/(s^2+2*s+3); 
Ts   = 0.5; %sample time 
sysd = c2d(sys,Ts,'zoh'); 
 
rlocus(sysd) 
grid

pict

 

Maple 

restart; 
alias(DS=DynamicSystems): 
sys := DS:-TransferFunction((5*s+1)/(s^2+2*s+3)); 
T   :=1/2;  #will fail if 0.5 is used !! 
sys := DS:-ToDiscrete(sys, T, 'method'='zoh'); 
DS:-RootLocusPlot(sys,-1..1,gridlines);

pict

 

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