How to use Final Value Theorem for Inverse Laplace transform?

Question

I do not understand how the Inverse Laplace transform is taken of G(s).

Please see my working. I understand how the final value theorem is used, but I do not know why it is relevant to find the output, and I do not understand how the time response output is found.

I know the second line of my working is probably not right, where I take the constant A out, into it's own inverse Laplace transform. If I had

A + 1/(s+a)

then I could do it, and the two inverse laplace terms would be added together. But I am not sure how to tackle this example.

Thanks

Answer 1

I think the second line you have is correct. Just, since A is a constant you can say that L^-1{A} = A. And you probably don't even need to take that step.

Then, note that G(s) = Y(s)/U(s) = Y(s)/(1/s) = sY(s)

Therefore Y(s) = G(s)/s

Going forward in the problem you have to simplify and use partial fractions, then take the inverse Laplace transform. Hope this gets you going on the problem.

I'm also not exactly clear on how that fact about the FVT helps, though.