$y(0)$ terms in the Laplace transform

Question

When taking the Laplace transform (in my case, for building a transfer function) of a signal $y(n)$ the substitution below is often made directly:

$$\mathscr{L} \big\{ y^{(n)}(t) \big\} = s^n \mathscr{L} \big\{ y(t) \big\}$$

But this ignores some $y(0)$ terms, for example if we take the Laplace transform $$\mathscr{L} \big\{ x(t) \big\} = \int_0^\infty x(t) e^{-st} \ dt$$

of $y'(t)$ then we get:

$$-y(0) + s\mathscr{L} \big\{ y(t) \big\}$$

and with higher-derivaties, these terms get even more complex and start including $s$. So I guess we are making some assumptions about $y(n)$ so that $y(0)=0$? If so what are these and why are they valid? (i.e. often we are filtering signals that do not start at $0$).

Answer 1

Linear systems, by definition, obey the the superposition principle. As such, depending on what you're doing, the initial conditions can be irrelevant.

If you want to know what the overall response is, then including the initial conditions is required because they make up part of the overall response. For example, currently I'm solving the diffusion equation and include initial conditions when using the Laplace transform to solve the differential equations so that I can see how things evolve. As you say, the initial conditions will contribute a component to the transient response, and possibly even the steady state response.

In contrast, if you want to know how the system will respond to a specific input, the initial conditions are irrelevant because you can just consider each input independently thanks to superposition, so for simplicity set the initial conditions to zero. For example, whenever I am designing filters or controllers, I ignore all initial conditions and just consider different types of inputs/disturbances independently (stability is not affected by initial conditions).

You can initialise digital filters to minimise the transient response due to the initial conditions, which relates to my first previous paragraph of when there is a concern regarding the overall response, and you're right, if you don't initialise the filter with some other state, the assumption is that the input into the filter has been zero for all time prior to you beginning to feed in your signal (this is certainly the behaviour of Matlab's filter function).