Consequently, why is it not allowed to produce physically some controllers for processes that are described by a transfer function that is an improper function?
A simple example is the driven harmonic oscillator. So the equation in this case is $mx'' +Ax'+k=f(t)$, where $x$ is the variable that stands for the position of the mass, $m$ is the mass, $A$ is a certain constant for the friction and $f$ is a force that pull the mass. Omitting some passages we get a transfer function (in the frequency domain) that is: $$W(s)=\frac {1/m}{s^2+2\alpha s+w_0^2}$$ where $\alpha = \frac Am$ and $w_0 = \sqrt\frac km$. So the harmonic function is: $$W(jw) = \frac {1/m}{(w_0^2-w^2)+2\alpha jw}$$ Now the module of the function is $M=\frac {1/m}{\sqrt{(w_0^2-w^2)^2+4\alpha^2 w^2}}$ and $\lim_{x\to+\infty} M = 0$. So the process has a visible response just in "low" values of the frequency of the force ($w$). The professor added that in some controlling problems of some process we could obtain a transfer function of the controller that is an improper transfer function, and that a controller of this type is impossible to produce physically. I'd like just some kind of intuitive explaination for this.
