What Does 's' Represent in the Laplace Transform of Control Theory?

[Rob K]

Good day to you people,

I have just started learning control theory at Uni, as part of my course, and I have to admit it is quite difficult to grasp.

I am starting from the basics, and I am having difficulty understanding what 's' is supposed to represent as regards the Laplace transform.

for example:

L{f(t)} = F(s)

Now to me these variables represent the following:

L - as in Laplace transform of a
f - function of
t - time

is equal to

F - Laplace function of
s - ??

I am slightly confusing it with Dynamics and Statics I think, where 's' refers to displacement. And so I am naming that as something I am aware of.
My guessing is that the 's' is referring to the s-plane as a complex plane, but still that is a difficult concept to grasp.
Can anyone give me some sort of analogy or point me at a resource that might be able to help a student understand this concept and it's relationship to the poles of a system.

Sorry for giving such a vague post, but I don't know how to narrow this down any more at the moment.

Kind regards

Rob

 

[Antiphon]

S is the complex frequency.

The imaginary part is the usual frequency. In fact, you often see S=jw when there is no real part. w=2pi * frequency (Hz)

The real part represents an imaginary frequency which is also called exponential damping.

f(t) = e^(-St).

If S is pure imaginary you have time harmonic signals (sinusoids). If its only real you have exponentially decaying (or growing) signals. If S is complex then you have a ringing bell- an exponentially decaying sinusoid.

It's a very general way to exite a linear system, a little more general than Fourier analysis where S is pure imaginary.

 

[Rob K]

Brilliant, after an evening of studying the intuition of complex numbers, this now makes a little bit of sense to me.

Thank you very much antiphon.

I a may be back for a little more, but this is good for me to be going on with.

Kind regards

Rob K