Understanding Engineering Mathematics

equations and studying the properties of the solutions. This question is a significant

project in which you are asked to repeat as much of the Chapter 15, Applications,

Question 6 as you can using Laplace transform methods.

Show that with zero initial conditions (i.e. a system that is initially at rest) the Laplace

transform ofx(t)can be written

L[x(t)]= ̃x(s)=

f(s) ̃

ms^2 +βs+α

wheref(s) ̃ is the Laplace transform of the right-hand sidef(t).

Find the solutionsx(t)for the constant forcing function

f(t)=F 0 H(t) (H(t)the unit stepfunction)

for various values ofm,β,α, considering cases of real, equal and complex roots of

the quadratic.

Consider also the sinusoidal forcing function

f(t)=f 0 cosυt

and compare your solutions with the results of Chapter 15, Applications, Question 6.

2.The sorts of equations considered in question 1 are often used to describe acontrol
systemin whichx(t)represents theresponseor output of the system to aninputf(t).
In this case the Laplace transform of the equation plays an important role in control
theory, and the connection between the Laplace transform of the input to that of the
output is called thetransfer functionof the system, denotedF(s). We write:

x(s) ̃ =F(s)f(s) ̃

So, for example the transfer function of the system described in question 1 would take

the form

F(s)=

1

ms^2 +βs+α

Thestability of the control system depends crucially on thepolesof the transfer

function – that is the roots of the denominator,ms^2 +βs+α, in this case. Control

systems can obviously be of great complexity, and a particularly important feature

is that offeedback. A control system has feedback when the output or some other

signal is fed back to the input as a means of influencing the behaviour of the control

system. A simple example is shown in Figure 17.8. Here, the inputxi(t)is modified

by subtraction of a feedbackxf(t)to formxi(t)−xf(t)which is input to the control

system which has transfer functionG(s). The feedbackxf(t)is formed by a modifying

control system that converts the outputxo(t)toxf(t)with a transfer functionF(s), thus

forming a feedback loop. Ifx ̃i(s)andx ̃o(s)are respectively the Laplace transforms of

the input and output, show that the overall transfer function of the feedback system is

G(s)

1 +F(s)G(s)