If I have a linear dynamical system (assume continuous time for the time being) I can create the transfer function, let's say: $$\frac{1}{(s+a_1)(s+a_2)}$$ and the pole-zero map (this one is for e.g. $-2\pm i$), then I can construct the output signal with the given modes for a given input.
Now, I would like to do the opposite, I have an unknown system, and by having the input and output data I want to work out the system transfer function (at least the dominant modes). Kind of doing a Fourier transform, but I'm also interested in the damping, not just the frequency components.
I was thinking on applying the Laplace transform $$F(s)=\int _{0}^{\infty }e^{-st}f(t)\,dt$$ then finding the singularities of $F(s)$ would give the answer (i.e. the same pole-zero map, that is shown above.
However due to region of convergence (ROC) issues of the Laplace integral (it is only valid for $\Re(s)>-2$), it doesn't give a sensible result. See plot.
So my question is, why is this? Also, why can the signal (or system) be analysed on the whole s-plane (for finding the poles for example), when the ROC is much smaller? For example if $a_1=-1$, $a_2=-2$, then the ROC is $\Re(s)>-1$, but we still work with the pole at $-2$.
Thanks