Is a proportional controller a static system?

Question

If we try to describe a generic dynamic system with a mathematical model, the result is a set of differential equations. Anyway we could say that a static system is a special case of a dynamic system. Indeed if we consider a differential equation with no derivatives, the mathematical model describes a static system. If we consider a proportional controller (with a certain gain K), could we say that the controller itself is a static system? Indeed according to what I have written above, if there are no derivatives in the expression of the differential equation, thus the system is static and the transfer function is only a constant of proportionality. Is it right or not?

Answer 1

Yes, this is correct. The equation of a standard proportional controller for some control error $e$ is

$$ u = h(e)= K e $$

so if $e$ changes, $u$ changes instantaneously as well (no dynamics involved) and the transfer function is as you said just the constant $K$.

However, keep in mind that the transfer function can be constant but the system can still be non-static.