Example A.1

MATLAB code for example 1 from appendix A of the book "Regeltechniek voor het HBO"

Laplace techniques for a double pole

Date : 02/03/2022
Revision : 1.0

Table of Contents

Laplace techniques for a double pole

Given is a process, input

, output

and differential equation:

We assume that for

there is static equilibrium and x and y are constant (so

). Moreover all derivatives with respect to t are zero and the constant values of x and y we take as reference (value zero). A change of

of course will change

from

; in that case we get for the differential equation in Laplace-form:

So:

With

→

It appears to be possible to write:

(Mathematics has taught us to operate like this, but it is very simple: if you try somthing else you won't get a solution).

, and this has to be correct for all values of s

;

This must be true for every s and that means we get three equations for the factors before

and

→

and

, and by table lookup:

Due to the double pole in

the graphically supported method of section 3.4.4 cannot be applied. A practical solution would be to slightly separate the two poles in

, for instance to

and

. Now the method of section 3.4.4 can be applied again. This is also more in line with reality, because due to inaccuracies, tolerances, wear etc. Poles (and zeros too) will never be exactly equal to each other and/or in a fixed, rigid position in the s-plane..

The change of pole positions proposed here would provide the solution according to the graphical method:

Although this expression looks completely different from the previous one, its numeric values for

are found to be almost equal.