plot the impulse and step responses of a system from its transfer function

1.3 plot the impulse and step responses of a system from its transfer function

Problem: Find the impulse and step responses for the continuous time system deﬁned by the transfer function \[ H\left ( s\right ) =\frac {1}{s^{2}+0.2s+1}\] and display these on the same plot up to some \(t\) value.

Side note: It was easier to see the analytical form of the responses in Mathematica and Maple so it is given below the plot.

Mathematica

Clear["Global`*"]; 
SetDirectory[NotebookDirectory[]] 
sys = TransferFunctionModel 
        [1/(s^2 + 2/10 s + 1), s]; 
 
yStep = Assuming[t > 0, 
  Simplify@OutputResponse[ 
             sys, UnitStep[t], t]]

{1 - E^(-t/10) Cos[(3 Sqrt[11] t)/10] 
 - ( 
E^(-t/10) Sin[(3 Sqrt[11]t)/10])/ 
                       (3 Sqrt[11])}

yImpulse = Simplify@ 
  OutputResponse[sys,DiracDelta[t],t]

{(10 E^(-t/10) HeavisideTheta[t] 
  Sin[(3 Sqrt[11] t)/10])/(3 Sqrt[11])}

p = Grid[{ 
  {Plot[Evaluate@{yStep, yImpulse}, 
   {t, 0, 50}, 
  PlotRange -> {{0, 50}, {-0.8, 2.0}}, 
  Frame -> True, 
  FrameLabel -> {{"y(t)", None}, 
     {t, "step and impulse reponse"}}, 
  GridLines -> Automatic, 
  GridLinesStyle -> Dashed, 
  ImageSize -> {300, 300}, 
  PlotStyle -> {Red, Black}, 
  AspectRatio -> 1] 
  }, 
  {Row[{"step response=", Re@yStep}]}, 
  {Row[{"impulse response=",Re@yImpulse}]} 
  }, Alignment -> Left, Frame -> True]

pict

Matlab

clear all; close all; 
t   = 0:0.05:50; 
s   = tf('s'); 
sys = 1/(s^2+0.2*s+1); 
y   = step(sys,t); 
plot(t,y,'-r') 
hold on 
y  = impulse(sys,t); 
plot(t,y,'-k') 
title('Step and Impulse responses'); 
xlabel('t'); 
ylabel('y(t)'); 
xlim([0 50]); 
ylim([-0.8 2]); 
legend('step','impulse'); 
grid on; 
set(gcf,'Position',[10,10,310,310]);

pict

Maple

Using Maple DynamicSystems

restart: 
alias(DS=DynamicSystems): 
sys:=DS:-TransferFunction(1/(s^2+0.2*s+1)): 
p1:=DS:-ResponsePlot(sys, DS:-Step(), 
   duration=50,color=red,legend="step"): 
p2:=DS:-ImpulseResponsePlot(sys, 
                   50, 
                   legend="impulse"): 
plots:-display([p1,p2],axes=boxed, 
   title=`step and impulse reponses`);

pict

Using Laplace transform method:

with(inttrans): 
with(plottools): 
H:=1/(s^2+0.2*s+1); 
input:=laplace(Heaviside(t),t,s): 
yStep:=invlaplace(input*H,s,t); 
input:=laplace(Dirac(t),t,s): 
yImpulse:=invlaplace(input*H,s,t); 
plot([yStep,yImpulse],t=0.001..50, 
     title=`step response`, 
     labels=[`time`,`y(t)`], 
     color=[red,black], 
     legend=["step","impulse"]);

pict

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