Control Theory -- Systems where the controller also changes with time

[mad mathematician]

Usually we look at a system of ODEs of the form:
$$\dot{x}=f(x,u,t)$$
$$y=g(x,u,t)$$

Why not look at systems where the controller also changes with time, i,e functions of terms $\dot{u}$?

I took quite a handful of Control Theory courses and yet as of yet never seen one incorporating this derivative.

Perhaps it's impractical, my pure side of me doesn't really care though...

 

[DaveE]

I assume you are referring to LTI systems? Because you can construct all kinds of weird #$@! otherwise.

In the very sloppiest, hand wavy way (because it's late here and it's been a long time since I did this stuff for real), you will end up with a state variable that represents the effect of $\dot u$, if it matters to the output $y$.

Consider the "D" part of a PID controller:
https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=9933758

Maybe try a simple example, like an RC HPF (in the EE context)? This stuff is often clearer when you work through a simple case.