Example H6.7

MATLAB code for example 6.7 from the book "Regeltechniek voor het HBO"

Application of the sum rule

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A system with open loop poles in

and

has a root locus (as for the real axis) as shown in figure 6.9.

According to formula 6.31 for every point s on the root locus must hold (note: if a zero is absent, the value 1 must always be entered for

→

Since this relation holds for any value of s on the root locus, this means that the factor for

and

on both sides of the

" sign must be equal. This determines that for the factor of

we have to demand:

→

Apparently the sum of the closed loop poles everywhere on the root locus for a certain value of

equals the sum of the open loop poles.

With this for instance we can easily determine the break away point

of the root locus on the real axis (double pole!). Because there

we get:

, so

. Of course you can check it with formula 6.23.

Note: in this case

is exactly in the middle of the root locus on the real axis. In other cases it need not be. You will always have to calculate them.

% 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+2)*(s+4));

% Show the root locus plot of H

figure(101);

rlocus(H);

grid on;