Example H6.10

MATLAB code for example 6.10 from the book "Regeltechniek voor het HBO"

Determination of the root locus gain graphically or through basic insight

Date : 02/03/2022
Revision : 1.0

Table of Contents

Determination of the root locus gain graphically or through basic insight Static accuracy in a given point on the rootlocus Solution 1 Solution 2 MATLAB code for this example

Static accuracy in a given point on the rootlocus

Suppose the transfer function of

in figure 6.4 equals:

The root locus of the feed back systeem is shown in figure 6.11.

The pole position

(and also the complex conjugated

) corresponds to a damping factor

(see section 4.4).

What is the static accuracy in that point?

Solution 1

In figure 6.11 we can use formula 6.46 to calculate the root locus gain :

For the static gain

of the loop we can use formula 6.48:

So the static error

in a step response will be (see the text preceding formula 6.47):

Solution 2

Instead of using all those formulas there is another way, simply based on the basic knowledge we have got:

First of all we have to remember the "final value theorem" of Laplace:

For the step response

of a system with transfer function

and input

, so

, this means:

So if we we want to know the static behaviour of a system (this only makes sense with a constant input of course, a step), we only have to substitute

, or for further calculations in all partial transfer functions of the system for further calculations.

In our case this means for

The only thing left for us to do is determing

from the root locus equation in

(or

, that doesn't matter):

From the root locus equation:

→

So a step 1 generates in the static state in output of only

, and that is

too little for what we would ideally like. So the static error

is 32%, or the static accuracy is 68%.

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=34;

% Create the 2nd order proces

H=K/((s+2)*(s+8));

% Show the root locus plot of H

figure(101);

rlocus(H);

grid on;

% Show stepresponse and static error

Hcl=feedback(H,1);

figure(102);

step(Hcl);

grid on;