It's really my go-to reference for any practical control theory application.
[1]: https://www.wescottdesign.com/articles/pid/pidWithoutAPhd.pd...
1: https://www.wescottdesign.com/articles/Sampling/sampling.pdf
I've heard someone say that PID is the deepest you can get without in control systems without advanced mathematics and I think hey might be right.
If you know linear algebra, that opens up a variety of state space techniques. If you know a little optimization, that opens up optimal control, including the incredibly common LQR strategy. This stuff is typically accessible for an advanced undergrad with a few of the right prereqs.
But control theory is a deep field. You can find papers that use differential geometry, topology, information theory, etc. for designing control systems.
It was fascinating, but I just couldn’t grok it. Whatever weaknesses I had in the relevant maths, that class just seemed to amplify it x10. I still wish I could have absorbed more of the knowledge
And this in turn overlaps with machine learning techniques since a problem generalized in this way can subsequently be adapted into a trained system that approximates the same result as the brute force answer.
Ideally, you can use an analytic solution, of course, and that's what you really want to use to get a result that is both fast and precise, but to do that, the "advanced mathematics" are necessary to define the problem appropriately.
It's true that there is a lot of surprisingly deep math in control theory, but even learning just a little bit of the theory can be greatly illuminating.
The trouble with PID is that it is typically treated as a black box with three magic knobs that you twiddle until your system works. This can be very frustrating.
By contrast, by applying a little bit of theory, you will be able to calculate the PID gains and have your controller work the first time.