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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.

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Exponential fall-off in frequency domain is equivalent to smoothness in time domain. Since physical processes usually happen smoothly, they act as a low-pass filter.

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Physical systems are either proper or strict proper. This means, they either show low-pass properties or a 'constant' one (imagine the two end-points of a bar that can move in one direction and can't rotate: every movement, you apply to one end, will immediately be applied at the other end).

If a system's transfer function was improper, your system can have non-continuous behavior even if your input is conintuous. For example, if $x$ is your variable denoting some coordinate, it could change value instantaneously even if the input is continuous. This is a property physical system usually don't have. Also this behavior would require an infinite amount of energy.

From a control theory point of view, an improper system has a differentiator. So if your input is not smooth, your system has to have some points, at which it changes state in zero time.

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