Modelling and control by Anthony Rossiter

# SYSTEMS BEHAVIOURS AND LINKS TO LAPLACE

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.

## Behaviour 1 - LHP and RHP

This set of videos considers the links between pole positions in a transfer function and expected system behaviour. The first video focuses on stability and concepts of LHP and RHP and uses first order models for clarity.

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

#### Quick test question

Poles correspond to convergent behaviour if
A. The real part of the pole is positive.
B. The real part of the pole is negative.
C. The imaginary part of the pole is zero.
D. None of the above.

## Behaviour 2 - Signal or system

This set of videos considers the links between pole positions in a transfer function and expected system behaviour. These resources focus on links and possible confusion between signal dynamics and system dynamics.

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

#### Quick test question

Which of the following is true?
A. As a pole moves further in the LHP, the corresponding response converges slower.
B. As a pole moves further in the RHP, the corresponding response diverges slower.
C. As a pole moves nearer to the origin but in the LHP, the corresponding response converges faster.
D. None of the above.

## Behaviour 3 - Speed of response

This set of videos considers the links between pole positions in a transfer function and expected system behaviour. These resources focus on speed of response and for systems with real poles.

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

## Behaviour 4 - Quadratic factors

This set of videos considers the links between pole positions in a transfer function and expected system behaviour. These resources give a summary of what might be considered desirable pole positions and includes both real and complex pole positions.

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

## Behaviour 5 - tutorial sheet

Gives a number of tutorial questions on links between pole positions and expected behaviour. These questions include a discussion of system gain which is covered in the first two resources in the chapter on offset of the feedback section.