Example H6.12
MATLAB code for example 6.12 from the book "Regeltechniek voor het HBO"
Determination of the root locus manually and with MATLAB
- Date : 02/03/2022
- Revision : 1.0
Copyright (c) 2022, Studieboeken Specialist Permission is granted to copy, modify and redistribute this file, provided that this header message is retained.
Table of Contents
Assignment
A process has three time constants: 1.0, 0.25 and 0.1. The static gain 
of the process is 10. This system is provided with negative unit feed back and in the forward path of the loop a variable gain 
is included.
of the process is 10. This system is provided with negative unit feed back and in the forward path of the loop a variable gain is included. Draw the root locus for 
as accurate as possible. Determine also the value of K for which this system becomes unstable and 
in that case.
as accurate as possible. Determine also the value of K for which this system becomes unstable and in that case.Solution
The block diagram of this system will be as shown in figure 6.14.
The static gain of the process (
) indeed equals 
and 
.
) indeed equals and .The root locus equation is:
, or in "normaal vorm":
.The system has three poles (
) and no zeros (
).
) and no zeros (). The characteristics of the root locus are:
- there are 3 branches of the root locus, starting in
, 
and 
- there are 3 asymptotes (because →
) - the asymptotes intersect on the real axis in
- the angles
with the real axis are (formula 6.13): 
→ 
, 
and 
(the latter is correct for angles!) - The parts of the real axis that belong to the root locus are (alternating and starting to the right with "not"): to the right of not, between and
and not, and to the left of yes - the break away and/or return points on the real axis follow from:
→ 
→ the numerator must be zero: 
→ 
and 
. The point 
is acceptable and is a break away point. The point 
is between and and therefore not acceptable (with negative 
it would be a break away point). - An alternative to determine the break away and/or return points is to solve the equation:
. That provides the same solutions. (You can do it yourself).
With all this data the root locus can be drawn (figure 6.15). But with the following MATLAB live script it would be much easier:
% clear all variables from Workspace and close all figures.
clear variables;
close all;
% Define 's' variable
s=tf('s');
H=400/((s+1)*(s+4)*(s+10));
figure(101);
rlocus(H)
The system becomes unstable as soon as the root locus passes the imaginary axis in 
and 
.
and . In that case, with the aid of the sum rule (
and 
), we can determine the value of the third pole 
.
and ), we can determine the value of the third pole . Substituting this value of 
in the root locus equation gives us 
.
in the root locus equation gives us . But for this value of 
the points/poles 
and 
are solutions of the root locus equation too, so 
.
the points/poles and are solutions of the root locus equation too, so . The oscillation frequency that arises at these points we can find by substituting 
(or 
) in the root locus equation for 
:
that arises at these points we can find by substituting (or ) in the root locus equation for : 
→
54
, or:
; these are two equations in one: 
and 
.
and .The mathematical solutions are: 
and 
. We only can accept 
rad/s.
and . We only can accept rad/s.