Is there a strategy for discrete control of a system with dynamics near sample rate?

Question

I'm trying to control a system where the controller sample rate is physically fixed and the plant has significant dynamics on the same order as the sample rate. I understand that one would prefer to have the sample rate considerably faster than the plant dynamics, but the physics of this system are such that this is inherently impossible. I can get some control of the system by hand-tuning a PID, but it seems considerably sub-optimal.

Is there a strategy for developing a controller like this?

Edit: I should add that the plant is this situation is a relatively complex LTI plant with reverberations caused by pure time delays. The delay cycles are a bit slower than the Nyquist frequency. The plant is stable.

Answer 1

The best strategy depends on your control objectives and whether or not the fast poles are first or second order. Here is my advice:

(1) If the fast poles are all first order, the response possesses an asymptotic form where they die off and stop contributing, so unless you need something special out of the transient response, you should be fine with ignoring them and controlling only the dominant poles. The same goes for complex poles with a fast decaying envelope. The discretized plant will basically smooth the fast poles and end up looking no different than the discretized version of the the dominant pole approximation of the analog plant. Implementing a digital controller for this plant will succeed in controlling the asymptotic response of the analog plant at a practical sampling rate, but (obviously) can't shape the transient response.

(2) If the envelope is slow but the frequency of oscillations is large, this is bad no matter what. If your fastest possible sampling rate still aliases these oscillations, you need to include some sort of analog controller specifically to damp these poles. Although you mention a PID controller, this is much better done with either a simple P controller or a lead-lag compensator, which you can design using the typical Nyquist plot/root locus method. These sort of controllers can be mechanized from op-amps just as a PID can be, so it should not be a problem to implement an analog version. One thing you may need to do in this case is separate your plant into fast and slow parts and create a cascade control where the inner loop only feeds back to the fast part.

(3) You say

I can get some control of the system by hand-tuning a PID, but it seems considerably sub-optimal.

This may just be semantics, but optimality is rarely a practical requirement--the main reason optimal control is rarely if ever practically applied. One thing I might suggest is figuring out exactly what practical control requirements you have to meet. This will be essential for coming up with a good controller architecture regardless of whether or not there are bumps like this.