Example H6.9
MATLAB code for example 6.9 from the book "Regeltechniek voor het HBO"
Application of the product rule
- Date : 02/03/2022
- Revision : 1.0
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Table of Contents
Determine the radius of a circular course in the root locus
In most cases it is not easy to describe the course of a root locus outside the real axis. We like to leave that to the computer. However, with the product rule we can prove in example 6.6 the circular course of the root locus.
In figure 6.10 we transform the imaginary axis to the left so that it passes through the point 
. Poles and zeros relative to the transformed coordinate system we will indicate by the index
.
. Poles and zeros relative to the transformed coordinate system we will indicate by the index.The poles of the open system are in the original coordinate system: 
en 
,
en , and in the transformed coordinate system this equals 
and 
.
and .We take two arbitrary complex poles on the root locus in the original coordinate sytem: 
and 
,
and ,and in the transformed coordinate system we can write them as: 
and 
.
and .Because 
, we can apply formula 6.40: 
→ 
.
, we can apply formula 6.40: → .So since and are complex conjugated in the original coordinate system, we see that in the transformed coordinate system apparently applies: 
. So the root locus outside the real axis is a circle centered at the origin of the transformed coordinate system ( in the original coordinate system) and radius 
.
and are complex conjugated in the original coordinate system, we see that in the transformed coordinate system apparently applies: . So the root locus outside the real axis is a circle centered at the origin of the transformed coordinate system ( in the original coordinate system) and radius .MATLAB code for this example
% clear all variables from Workspace and close all figures.
clear variables;
close all;
% Define 's' variable
s=tf('s');
% Determine parameters.
K=1;
% Create the 2nd order proces
H=K*(s+8)/((s+2)*(s+6));
% Show the root locus plot of H
figure(101);
rlocus(H);
grid on;
