Example H6.5

Parts of the real axis belonging to the root locus

• Date : 02/03/2022
• Revision : 1.0

Which parts of the real axis belong to the root locus?

Suppose in the system of figure 6.4 is:

.

→

or: .

There are four starting points and thus four ending points. Three poles will end in the zeros (poles in that case!) and one in infinity. Also indicated is the direction in which the poles move from the starting points with increasing . Note that figure 6.7 only shows that part of the root locus that belongs to the real axis. The two branches of the root locus that leave from 0 and have no end points until now (we will discuss that later). In this example we also see that the zeros (and also poles if they were there) that are not on the real axis, have no influence on the course of the root locus on the real axis. That is because the contribution to the total phase of two complex conjugate poles or zeros with respect to a point s on the real axis is zero: just investigate that in figure 6.6 for two complex conjugates to a point s on the real axis; the contribution to the phase is zero in that case, so there is no contribution to the condition according to formula 6.7.

A closer look at the real axis makes it clear, that parts of it alternately belong to the root locus. This applies in general. From the far right of the real axis we always start (with positive !) with a part that does not belong to the root locus. When we move to the left along the real axis, we see that as soon as we pass a pole or zero point, that part of the axis does belong to the root locus. After passing the next pole or zero that part again does not belong to the root locus, and then again and again not, etc. When passing a double pole or zero nothing changes. If you want, this can be seen by moving the double poles (or zeros) an infinitesimal distance apart. An infinitesimal small part of the real axis then will change, but after passing the left pole (or zero) of the double pole (ore zero) it changes again. So, overall nothing changes when passing a double pole or zero. In the same way it can be seen that a triple pole (or zero) means that what applied to the right, changes on the left.

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+4)*(s^2+10*s+34))/(s*(s+2)^2*(s+6));

 

% Note that because of the ^2 there are two poles in s=-2;

ans = (s+4) (s^2 + 10s + 34) ---------------------- s (s+6) (s+2)^2 Continuous-time zero/pole/gain model.

 

% Show the root locus plot of H

figure(101);

rlocus(H);

grid on;