In maintaining tank level, the error is volume, and the control variable is flow rate. On the z axis of drone flight control, the error is position, and the control variable affects acceleration. The dimensions of control variables are the first and second time derivative of the dimensions of error, respectively. Why are both methods effective?

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I don’t quite get what you’re asking. Controllers are designed to be broadly applicable in a variety of situations. I’m school I even did a lot of stuff abstracted from physical systems. You could (probably) control the tank with the acceleration of flow if you wanted to.

But looking at these systems, you’re controlling a first rise system with the first derivative (ie the flow shows up in the question) and a second order system with the second derivative (ie the acceleration shows up in the equation)

I’m currently writing a PhD on that topic, but I have no clue what you’re really asking.

Do you mean “why can a PID controller handle both first and second order derivative systems?”

Well that’s because you can parametrize them differently. If you pick a slow PID then you can just wait a bit longer and the control error will eventually evolve based on the control input. Higher order systems aren’t really that different from low order systems with longer time constants. In your

There are even rules to transform a second order system to a similar first order system to simplify controller design.

PID control can even work when you have a control input that is the 5th derivative of the control error. You have to be very slow and careful then, but it works.

It boils down to the desired control quality. At some point it gets so glacially slow that you’d rather switch to more complicated controller setups.