Example H6.2

MATLAB code for example 6.2 from the book "Regeltechniek voor het HBO"

Start and end points of a root locus

Date : 02/03/2022
Revision : 1.0

Table of Contents

Start and end points of a root locus Root locus equation Solutions for various values of K' MATLAB code for this example Rootlocus plot

Root locus equation

In the block diagram of figure 6.4 the transfer function

equals:

and the transfer function of the total system then equals:

So the root locus equation becomes:

We often prefer to wtite this in so called "normaal vorm" (see formula 6.5):

Solutions for various values of K'

The solutions of this equation for

(starting points) are of course:

and

The solutions of this equation for

are:

but also

We will prove this last formally, but first of all we have to realize that two starting points always mean two end points too. That's why beside the solution

there must be a second one: when

, then

and for

we get

. So

here really is a solution of the root locus equation for

From the two starting points in

and

one branche of the root locus will leave and end in

, and the other one will leave and end somewhere infinitely far away from the origin. Later we will see where exactly.

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;

p1 = 2;

p2 = 9;

z1 = 5;

% Create the 2nd order proces

H=K*(s+z1)/((s+p1)*(s+p2));

% Create feedback

Hcl=feedback(H,1);

% Show stepresonse of system with feedback

figure(101);

Tsim=5*(1/p1); % Simulation time in seconds

step(Hcl, Tsim);

grid on;

Rootlocus plot

figure(102);

% Show the root locus plot of H

rlocus(H);