Example H5.5

Matlab code for example 5.5 from the book "Regeltechniek voor het HBO"

Investigation of the stability of an electronic circuit

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Investigation of the stability of an electronic circuit Assignment Solution Matlab code for this example

Assignment

The transfer function of an electronic circuit looks like this: . Is this system stable or unstable?

Solution

We write: . The pole and zero plot is shown in figure 5.9.
According to the pole and zero plot this system is unstable. But because the response of this system contains an undamped vibration with a constant amplitude, it is also called a marginally stable system. The poles are on the imaginary axis and have a real value of zero. We see here the mathematical representation of a harmonic oscillator circuit. Just watch a stepresponse (see e.g. example 4.4 how to do that).
Note aside: so far you have seen in all examples that if there are complex poles (or zero's) these are added complex in pairs. Mathematics can prove that for physically realizable processes this is always the case. This is due to the fact that process parameters are always real.

Matlab code for this example

% clear all variables from Workspace and close all figures.
clear variables;
close all;
 
% Define 's' variable
s=tf('s');
 
% Make the transferfunction H
H=5*s/(s^2+9);
 
% Find the poles and zeros
zpk(H)
ans = 5 s --------- (s^2 + 9) Continuous-time zero/pole/gain model.
 
% plot the pz map
figure(101);
pzmap(H);
 
% Plot the stepresponse
figure(102);
Tsim=10; % Simulate time in seconds
step(H,Tsim);
grid on;