Example H6.6

MATLAB code for example 6.6 from the book "Regeltechniek voor het HBO"

Break away points and entry points on the real axis of a root locus

Date : 02/03/2022
Revision : 1.0

Table of Contents

Break away points and entry points on the real axis of a root locus Initialisation Given system and rootlocus equation Determine break away or entry points MATLAB code for this example

Initialisation

clear all variables from Workspace and close all figures.

clear variables;
close all;
 
% Define 's' variable
s=tf('s');

Given system and rootlocus equation

Suppose

in the system of figure 6.4 is:

The root locus equation of the closed system is:

→

We can say right now that (for positive

) the parts of the real axis between

and

and to the left of

belong to the root locus.

Determine break away or entry points

The place of break away points or entry points on the real axis we can determine by formula 6.23:

. →

→

The numerator must be zero:

Or:

→ (abc formula)

and

Is is clear that

must be a break away point and

an entry point. As will be proved later (example 6.9), in this case where the poles are complex conjugated the complex part of the root locus is a circle with radius

and center in

; see figure 6.8.

For the angles at which the branches of the root locus depart or arrive on the real axis the same rule applies as stated for the asymptotes in section 6.3.3: they divide

into equal parts. In the most common cases of two departing or arriving branches of the root locus this means that the angles are

and

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;