Modelling and control by Anthony Rossiter

# FIRST ORDER RESPONSES

This is the first chapter in the theme on modelling and simulation of linear models. Use the left hand toolbar to access the other chapters and themes. It is implicit in several of these chapters that students have core competence in some mathematical topics such as polynomials, roots, complex numbers, exponentials and Laplace. More information on these can be found in the Mathematics theme on the left hand toobar.

This chapter contains the following topics. Under each topic there are hardcopy (pdf) notes, a video talk through of key derivations with example problems and also a tutorial sheet for users to test themselves.

## First order responses 1 - no input

An introduction to the behaviour of systems modelled with a 1st order ODE. Considers response with non-zero initial condition and no input.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

#### Quick test question

First order responses have the following dominant characteristic:
A. A constant multiplied by time, that is 'y=at+b'.
B. A constant divided by time, that is 'y=a/t + b'.
C. An exponential term.
D. None of the above.

## First order responses 2 - step input

An introduction to the behaviour of systems modelled with a 1st order ODE. Second video in series. Considers response with zero initial condition and a step input.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

#### Quick test question

The step response of a stable 1st order model, converges how quickly?
C. Depends upon the size of the step input.
D. None of the above.

## First order responses 3 - step input and non zero initial condition

An introduction to the behaviour of systems modelled with a 1st order ODE. Third video in series. Considers response with non-zero initial condition and a step input.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

## First order responses 4 - sketching

An introduction to the behaviour of systems modelled with a 1st order ODE. Fourth video in series. Considers how to sketch responses response with zero initial condition and a step input.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

## First order responses 5 - tutorials

An introduction to the behaviour of systems modelled with a 1st order ODE. Fifth  video in series. largely comprises tutorial questions for students to test themselves.

## First order responses 6 - using MATLAB

An introduction to the behaviour of systems modelled with a 1st order ODE. Sixth video in series. Illustrates how the MATLAB tool might be used to produce solutions and sketches for 1st order models.

• A summary of key MATLAB commands and screen dumps is given in the pdf file.
• A talk through video is on youtube.

## First order responses 7 - tutorial on transfer functions

An introduction to the behaviour of systems modelled with a 1st order ODE. Seventh video in series. This video is constructed like a tutorial, so viewers have questions to try before looking at example solutions. The focus here is on the development of transfer function models.

• A talk through video is on youtube.

## First order responses 8 - tutorial on responses with Laplace

An introduction to the behaviour of systems modelled with a 1st order ODE. Eighth video in series. This video is constructed like a tutorial, so viewers have questions to try before looking at example solutions. A number of simple questions asking students to solve for the response of 1st order systems using Laplace methods.

• A talk through video is on youtube.

## First order responses 9 - tutorial on response dependence on parameters

An introduction to the behaviour of systems modelled with a 1st order ODE. Ninth video in series. This video is constructed like a tutorial, so viewers have questions to try before looking at example solutions. These questions invite viewers to explore how changing individual parameters effects the system response characteristics.

• A summary of key facts and derivations is given in the pdf file.
• Talk through video is on youtube.

## First order responses 10 - tutorial on system design

An introduction to the behaviour of systems modelled with a 1st order ODE. Tenth video in series. This video is constructed like a tutorial, so viewers have questions to try before looking at example solutions. The focus is on going from response specifications such as desired settling times and gains to the selection of appropriate system parameters.

• Talk through video is on youtube.

## First order responses 11 - typical exam style questions

This video gives some step by step examples of how to go about answering typical examination questions beginning from core knowledge and concepts.

• Talk through video is on youtube.  