Re: PID controller with second derivative?

Scott L wrote:
In my math doodlings, I wrote the second derivative as:

(e[n] + e[n-2] - 2*e[n-1]) / dt^2

I.e., the typical finite difference approximation (obviously not
centered). Are you saying that's too naive in the presence of noise?

Yes, but forget about noise for now. The real enemy is quantizing. I see application every day where the feed back resolution is only 0.0001m but even if the feedback resolution is .00001m there is still problems. Assume the sample time is .001s and the resolution is .00001m then the best resolution on the acceleration is 10m/s^2 or a little over 1g. That isn't really usable for motion control. If you try to use the 0.00001m resolution you will be severely limited in how high the double derivative gain can be increased without the control output going wild. Low pass output filter will help only a little but not enough.

Fortunately, most system can be modeled as having just one or two poles so a normal PID will do well enough until you start to push it to point where higher order un-modeled poles start to affect control. Then you must start identifying these higher frequency poles too and add higher order derivative gains the the controller so these poles can be placed somewhere safe. In the process you will need feedback to that can measure these gains directly or you need some sort of filter or observer that can estimate the higher order derivatives better than you can measure them directly.

When noise is added the problem is even worse.

Peter Nachtwey




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