Example H7.1

MATLAB code for example 7.1 from the book "Regeltechniek voor het HBO"

Characteristic times and overshoot in a step response of a second order process

Date : 02/03/2022
Revision : 1.0

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Characteristic times and overshoot in a step response of a second order process Assignment Solution

Assignment

Of a second order process with input

and output

the differential equation is:

What is the indication time, response time, peak time and overshoot in a step response?

Solution

Usually we note the transfer function directly from the differential equation, but maybe it would be good to discuss the Laplace transformation of this differential equation once more (and for the last time in this book). The following applies:

According to the mathematical rules this becomes:

In control theory the input of a process/system (often a step) we always want to be applied to the process in rest/balance/equilibrium/static state. Otherwise you don't know which part of the response is caused by the input and which part by the transient of a previous input. In other words: we always assume that

and

are constant; they no longer change due to previous inputs. This state of the process is starting point at a new input and the numerical value zero is associated with it. We consider it as reference level. So beside the fact that

, this also means that the first and higher derivatives of

and

with respect to t are also zero. So, in control theory our differential equation becomes in Laplace notation:

and the transfer function is:

→

rad/s

and from

we get

second.

From the well known formula

we find:

For the indication time

we must solve the equation:

→

For the response time

we must solve the equation:

→