Example H6.8
MATLAB code for example 6.8 from the book "Regeltechniek voor het HBO"
Another usefull application of the sum rule
- Date : 02/03/2022
- Revision : 1.0
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Table of Contents
Given system and root locus equation
Suppose 
in the system of figure 6.4 is:
in the system of figure 6.4 is: 
.The pole and zero equation is: 
.
.Finding the K value for instabilty
Because of the three poles and no zeros we know, that there are three asymtotes to the three end points infinitely far away. The asymptotes intersect the real axis in a point q at angles of 60, 180 and −60 degrees (formula 6.13). The asymptotes at the angles of 
cause, that for a certain value of 
two conjugated complex poles are placed on the imaginary axis in the points 
. Then the third pole can be found easily by the sum rule (see formula 6.35: 
):
cause, that for a certain value of two conjugated complex poles are placed on the imaginary axis in the points . Then the third pole can be found easily by the sum rule (see formula 6.35: ):
→ 
→ 
.Inserting this value in the pole and zero equation gives 
, the value at which the system becomes unstable.
, the value at which the system becomes unstable.Note aside
It might be worth noting that in all transer functions we have seen until now, the highest degree of s in the numerator was less than, or at most equal to, the highest degree of s in the denominator. In physical systems this is always the case. Otherwise the s-notation of for instance a step response would contain at least a term 
(and even a term 
etc.) because the degree of the numerator is ≥ the degree of the denominator. So you can divide the numerator by the denominator at least once, until the degree of the denominator is greater than that of the nominator. In the t-domain a Laplace constant stands for a delta function. That would mean that a finite-amplitude input would generate an infinite-amplitude output (the δ-function). Impossible in the physical systeems we look at.
(and even a term etc.) because the degree of the numerator is ≥ the degree of the denominator. So you can divide the numerator by the denominator at least once, until the degree of the denominator is greater than that of the nominator. In the t-domain a Laplace constant stands for a delta function. That would mean that a finite-amplitude input would generate an infinite-amplitude output (the δ-function). Impossible in the physical systeems we look at.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+1)*(s+4)*(s+10));
% Show the root locus plot of H
figure(101);
rlocus(H);
grid on;
