To add two points to the question: In the context of a known disturbance $d(t)$ in a discrete control loop, what is the $\Delta t$ at which the control loop has to be executed in order to function?

If you know the system output for the case of no controller for an ordinary control loop (see the pictures). Lets say you want to have a constant system output of 380. Is there a general rule how fast a discrete control loop has two operate in order to achieve this?

If there is no general rule for an arbitray controller, we can restrict the question to an ordinary PID-Controller.

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