How do I get the amplitude of a variable sine function?

Question

Let's assume that we have a dynamical system

$$G(s) = \frac{Y(s)}{U(s)}$$

And we have measure the input $u(t)$

And we have measure the output $y(t)$.

Here we can see that if we increase the frequency at the input of the system, the output is going to decrease.

The question is simple: How can I find the amplitudes over time of by just knowing $y(t)$ and $u(t)$. Nothing more.

My goal is to estimate a transfer function from arbitrary frequency input and output and I going to use least square to curve fit on a transfer function.

Can I just drag a line of all peaks/tops of the sine waves?

Answer 1

Since you have $u(t)$ and $y(t)$, you can compute the Laplace transform of each and get, by your first equation,

$$ G(s) = \frac{L(y(t))}{L(u(t))}. $$

If your measurements are continuous time and perfect then this is exact. If you have noisy measurements, then this is approximate. If you have discrete measurements you need to look at Hankel matrices and realization theory.