TABLE OF CONTENTS The most powerful tools for analysis and design of control systems operate only on linear models. It is therefore potentially very attractive when undertaking the design of a controller for a non-linear system to replace the non-linear system model by a linear approximation.
Questions that arise next are:
What is meant by linearisation?
How is it undertaken?
To what extent are designs, produced using linear approximations, valid in practice when applied to the original non-linear system?
The volume V of a sphere is given by
V = 4 ? r 3/3
where r is the radius of the sphere
Suppose r 0 = 10 then V = 4188.79
Suppose r 1 = 10.1 then V = 4315.7147
Suppose r 2 = 11 then V = 5575.27956
These are the full solutions of the non-linear equation for three different r values.
To linearise the equation we operate as follows. Let V = V 0+ ?v, r = r 0+ ?r. Then
while from earlier
Substracting the last equation from the one above yields
Linearisation consists in neglecting terms in ?r 2, ?r 3, etc., i.e.
and this result could have been obtained directly by using
To complete this little illustration, we will see how good the approximations are for two cases, keeping r 0
