Example H5.8

Matlab code for example 5.8 from the book "Regeltechniek voor het HBO"

Root locus of a first order proces with a zero to the right of the imaginary axis

Date : 02/03/2022
Revision : 1.0

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Root locus of a first order proces with a zero to the right of the imaginary axis Root locus of a first order proces with a zero to the right of the imaginary axis Matlab code for this example Rootlocus plot

Root locus of a first order proces with a zero to the right of the imaginary axis

In a first order process we want to examine the influence of a zero to the right of the imaginary axis. In figure 5.11 we take:

The transfer function of the closed system equals:

The equation for the root locus is:

→

For

we get:

, the pole of the open system; stable because left of the imaginary axis and starting point of the root locus (because

For

; unstable or marginally stable.

For

; unstable and the so called end point of the root locus (because

Figure 5.14 shows the complete course of the root locus for

. We see for

that the pole is in the same place as the zero of the open system. Note: despite the fact that for

the pole position should be represented by a cross, we still take the circle symbol of the zero of the open system. Actually not correct, but usual.

From figure 5.14 we can conclude that the system remains a first order system (always only one pole). The system behaviour is determined by a variable pole and a constant zero (see formula 5.17). In terms of stability we see, that the system becomes unstable for

. See figure 3.11 for the transient in a step response for

or simulate a step response with the live script (limit the time to avoid a huge output signal)

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=3;

p1 = 4;

z1 = 1;

% Create the 2nd order proces

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

% Create feedback

Hcl=feedback(H,1);

% Show stepresonse of system with feedback

figure(101);

Tsim=10; % Simulation time in seconds

step(Hcl, Tsim);

grid on;

Rootlocus plot

figure(102);

% Show the root locus plot of H

rlocus(H);

grid on;