Estimating parameters to damped harmonic oscillator model early in process

Question

I am observing a process that I assume fits the model of a damped harmonic oscillator. The solution to the diff eq is $P(t) = A + B e^{-\alpha t} + C e^{-\beta t} \cos (\omega t - \phi)$, where $A, B, \alpha, C, \beta, \omega, \phi$ are parameters determined by experimental data. Given some complete data that fit the model, it is easy to estimate those parameters. If I am observing live data, and I believe that the process is beginning to behave like the d.h.o, can I estimate the solution parameters from just the first few data points? If so, how?

Answer 1

You have $7$ parameter to estimate, hence you should have at least $7$ data-points. However, for efficient result you should have much more. Regarding the method, note that your model is non-linear w.r.t. the parameters, hence you should check the non-linear regression here.

Answer 2

In theory seven points will give enough data. If the measurements are perfect, they will. I suspect the solution process will be badly conditioned because certain pairs of parameters, particularly $B,C$ and $\alpha, \beta$ have almost the same effect. It will be a nonlinear problem that can probably only be solved numerically.