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Cross-posted from statistics stackexchange:

Say we have a permanent-magnet DC motor that roughly obeys the system equation $\ddot{x}(t) = \alpha \dot{x}(t) + \beta u(t) + \gamma$, where $x(t)$ is the displacement of the rotor, and $u(t)$ the applied voltage, at time $t$.

Say we wish to determine the values of $\alpha$, $\beta$, and $\gamma$ experimentally. If we can only directly measure $x$, not $\dot{x}$ or $\ddot{x}$, how should we go about estimating these parameters from a set of time-series measurements of $u$ and $x$?

One naive approach is to compute the derivatives through some central finite difference scheme, and then perform an OLS regression - but it is unobvious how the derivative calculation interacts with the regression. Additionally, I have found in practice that this suffers from a significant amount of regression dilution if the test is allowed to run too long at steady-state (the derivatives vanish here, and so all that's left is the noise).

Is there any more "complete" method for analyzing systems like this that handles the differentiation as part of the construction of the regression model? Is there a good theory of correlations between derivatives of time-series data?

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  • $\begingroup$ Have you tried this approach to estimate $\dot{x}$ and $\ddot{x}$? $\endgroup$
    – user140541
    Commented Jan 18, 2022 at 16:07
  • $\begingroup$ I have not! It looks interesting, and I'll give it a try, but I wonder: is local polynomial fitting a valid solution for differential equations whose solutions are exponential in shape? $\endgroup$
    – user3716267
    Commented Jan 18, 2022 at 16:12
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    $\begingroup$ You can try to take logs and recover the derivatives of $x(t)$ from the estimated derivatives of $\ln(x(t))$ $\endgroup$
    – user140541
    Commented Jan 18, 2022 at 16:16
  • $\begingroup$ Are you familiar with Kalman filtering? $\endgroup$
    – Jose Avilez
    Commented Jan 18, 2022 at 17:22
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    $\begingroup$ @user3716267 Great! At first sight, it looks to me that one could write a state-equation for $(x,u)$ with observation errors, and formulate the estimation of $(\alpha , \beta, \gamma )$ as a Kalman filtering problem. Naturally, one would have to work out the details, but, as far as I know, the Kalman framework is sufficiently flexible for estimation problems like these. In fact, the ARIMAX answer given to you can be formulated in a very general state-space time series formulation, after which one could use Kalman-like tools for estimation. $\endgroup$
    – Jose Avilez
    Commented Jan 18, 2022 at 17:33

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