Modelling and control by Anthony Rossiter

# INVERSE LAPLACE

This is a section in the chapter on mathematical skills. Some is revision of pre-university mathematics and some is syllabus mostly coered in year 1 of engineering programmes. Use the left hand toolbar to access the other chapters and themes.

 It is implicit that for many engineering topics, students have core competence in some mathematical topics such as polynomials, roots, complex numbers, exponentials, logarithms and Laplace. This section 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.

## Inverse Laplace 1- introduction

First of a set of videos on inverse Laplace for engineers. Focus on pragmatic solutions rather than theory. This video sets the scene and assumptions for later videos.

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

#### Quick test question

The best way to do inverse Laplace is:
A. Use the analytical definition.
B. Use a look-up table.
C. From memory.
D. None of the above.

## Inverse Laplace 2 - partial fractions (expansion and cover-up)

Second of a set of videos on inverse Laplace for engineers. Focus on pragmatic solutions rather than theory. These resources introduce partial fractions expansions.

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

#### Quick test question

For a transfer function with only real roots, which partial fraction method is usually easiest.
A. Long expansion following by coefficient matching.
B. Long expansion following by substitution of values of 's' to find given residues.
C. The cover-up rule.
D. None of the above.

## Inverse Laplace 3 - partial fractions and cover-up rule

Third of a set of videos on inverse Laplace for engineers. Focus on pragmatic solutions rather than theory. These resources continue with the theme of partial fractions and the cover-up rule.

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

## Inverse Laplace 4 - partial fractions with quadratic factors

Fourth of a set of videos on inverse Laplace for engineers. Focus on pragmatic solutions rather than theory. This video continues with the theme of partial fractions and introduces quadratic factors.

• A talk through video is on youtube.

## Inverse Laplace 5 - solving ODEs

Fifth of a set of videos on inverse Laplace for engineers. Focus on pragmatic solutions rather than theory. These resources demonstrate the solutions of ODEs using Laplace methods.

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

## Inverse Laplace 6 - analytic solutions with MATLAB

Sixth of a set of videos on inverse Laplace for engineers. Focus on pragmatic solutions rather than theory. These resources demonstrate how the MATLAB tool can be used to find algebraic inverse Laplace solutions.

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

#### Quick test question

Students should learn tools such as MATLAB because:
A. Facilitates independent checking of their work.
B. Is a learning tool which can reinforce their understanding.
C. Increases confidence in automated approaches common in industry.
D. All of the above.

## Inverse Laplace 7 - numeric solutions with MATLAB

Seventh  of a set of videos on inverse Laplace for engineers. Focus on pragmatic solutions rather than theory. These resources demonstrate how the MATLAB tool can be used to find numeric solutions for inverse Laplace.

## Inverse Laplace 8 - partial fractions with MATLAB

Eighth of a set of videos on inverse Laplace for engineers. Focus on pragmatic solutions rather than theory. These resources demonstrate how the MATLAB tool can be used to find partial fractions.

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

## Inverse Laplace 9 - solving ODES with inverse Laplace

Gives a number of tutorial type examples illustrating how Laplace tools can be used to solve single ODEs or even ODEs with shared variable such as occur in feedback loops or series arrangements.

• Talk through video is on youtube.

## Exensive tutorial sheet

Gives a number of typical tutorial style questions for students to attempt. Outline solutions are given for som eand students encouraged to use MATLAB to test their solutions for others.

## Tutorial sheet 1 - Laplace transforms

Gives a short number of typical tutorial style questions for students to attempt. Latter part of video goes through the solutions using a look-up table and also using MATLAB.

• Talk through video is on youtube.

## Tutorial sheet 2 - inverse Laplace

Gives a short number of typical tutorial style questions for students to attempt. Latter part of video goes through the solutions using a look-up table and also using MATLAB.

• Talk through video is on youtube.

## Tutorial sheet 3 - final value theorem and dead-time

Gives a short number of typical tutorial style questions for students to attempt using the final value theorem and the impact of dead-time. Latter part of video goes through the solutions.

• Talk through video is on youtube.

## Tutorial sheet 4 - speed of response and behaviour

Gives a short number of typical tutorial style questions for students to attempt on links between a Laplace transform and the expected characteristics of the underlying signal/system. Latter part of video goes through the solutions using a look-up table and also using MATLAB.

• Talk through video is on youtube.