MATHEMATICS AND COMPUTING FOR CONTROL

LAPLACE TRANSFORMS

Module Topics

1. Definition

2. Laplace transform of standard functions

3. Inverse Laplace transforms

4. Solution of second order linear ordinary differential equations with constant coefficients

Work Scheme based on James(Advanced Modern Engineering Mathematics(4th Edition)

a) Laplace transforms prove a very useful tool in analysing engineering systems and an

introduction to the technique can be found in chapter 5 of James. Turn to pg.346 and read the

introductory section 5.1.

b) Study section 5.2.1 on pg.348 which includes the definition of a Laplace transform, formula

(5.2). Note that the transform variable s is in general complex, not real.

c) Laplace transform pairs are usually quoted from tables. You should know how the simplest

results are derived, however, so work through Examples 5.1 and 5.2. Note that the transforms

exist only for particular values of 𝑠. Read through Examples 5.3 and 5.4 which supply additional

standard results. The table of Laplace transforms on your Formula Sheet is from James.

d) Turn to pg.355 and section 5.2.4 on the properties of the Laplace transform. You must

understand and be able to use the shaded results in this section, but the proofs are less

important.

#### Practice with the Matlab comments. ####

***Using the Formula Sheet do Exercises 3(a),(c),(g) on pg.364***

e) Study section 5.2.7 on the inverse Laplace transform, denoted by 𝐿−1, up to equation (5.12). If

the transform F(s) appears in the tables the corresponding function f(t) can easily be written

down. Study Examples 5.14 and 5.15, which are very straightforward. The linearity property at

the end of the section is important in applications, since it extends the range of transforms

which can be easily inverted.

f) Study section 5.2.8 including Examples 5.16 and 5.17. The use of partial fractions is crucial in

many situations.

In Example 5.16 the original denominator is a product of two linear factors. The split into partial

fractions is then relatively simple, the associated constants can be calculated and the function

f(t) is easily written down using the standard results in the table.

Example 5.17 has a more complicated denominator which must be expressed as the sum of

three fractions. Again the constants must be found by the standard procedure, before using the

linearity property and the table to carry out the inversion. As you know from earlier work, when

a quotient is split into a linear combination of fractions the determination of the constants can

be a lengthy process, and it this stage of the solution procedure which takes the most time.

***Do Exercises 4(a),(c),(f),(h) on pg .369***

g) In this module only the basic ideas of Laplace transforms have been presented. The method is

widely used in many Engineering applications.

Exercise

1. Obtain the Laplace transform of the following functions:

(i) 1𝑠3

+ 1𝑠+1

(ii) 1𝑠2−2𝑠+5

(iii) 2𝑠2(𝑠2+1)

2. Obtain the inverse Laplace transform of the following functions:

(i) 1𝑠3

+ 1𝑠+1

(ii) 1𝑠2−2𝑠+5

(iii) 2𝑠2(𝑠2+1)

Book: James Glyn, “Advanced Modern Engineering Mathematics”, 4th Edition