# In the context of transfer functions, what is the relationship between the terms “proper”, “causal”, and “realizable”?

I am thinking about these terms in the context of linear control.

A transfer function is proper if the degree of the numerator is not greater than the degree of the denominator. I've read often that improper transfer functions are "not causal". I also often see the word "unrealizable" used often in this context.

If a control transfer function I've designed is improper, does that mean it is "causal" and/or "unrealizable"? What is the difference between these terms? What do they mean in practice?

• This question is related. – Matt L. Mar 2 '17 at 15:52
• Thank you for the link. I buy that proper systems (and only proper systems) are causal. I suppose I am interested in whether applying the label "causal" gives us any other useful information about the system (beyond the definition of causality). "Realizability" seemed like an obvious candidate which is why I mentioned it. – Max Mar 2 '17 at 15:56
• An improper transfer-function can be realizable, and casual. E.g., what is the relative degree of a PID control law? – Arnfinn Mar 3 '17 at 9:08
• But real PD control laws don't actually have infinite high-frequency gain, right? It seems like any time you would actually try to implement (for example) a derivative term of the form $K(T_ds+1)$, there's really a high-frequency pole that we just don't account for in the model, i.e., $G_c = K(T_ds+1)/(\tau s+1)$. It's just that the high frequency pole at $1/\tau$ is negligible so we don't include it in the model, but it really is there and it makes the transfer function proper. Similar to how lumped parameter models break down at high frequencies. Is this the case? – Max Mar 3 '17 at 13:44
• Well, yes, in practice it is impossible to implement a differentiator, and it is also a bad idea to try to approximate one to a high degree in most cases, but mathematically speaking it is still realizable and causal. – Arnfinn Mar 4 '17 at 5:38