Hilbert Transform, Causality, PI Controller

[angryturtle]

TL;DR Summary

Help understanding why PI controller is causal.

I was told that PI controller is a causal filter, and has frequency response represented by H(w) = Ki/(iw)+ Kp.

I was also told that causal filter should satisfy this relationship H(w) = G(w) -i G_hat(w) where G_hat(w) is the Hilbert transform of G(w).

Does this mean that we cannot freely select gain Ki and Ki/w must be the Hilbert transform of Kp?

 

[Olexandrid]

Good afternoon. I learned the theory of automatic control. But It's hard for me to understand your question. Maybe you'll give more information? Example, for which object of control you'll plan to use PI-controller?

 

[berkeman]

Summary: Help understanding why PI controller is causal.

I was told that PI controller is a causal filter, and has frequency response represented by H(w) = Ki/(iw)+ Kp.

I was also told that causal filter should satisfy this relationship H(w) = G(w) -i G_hat(w) where G_hat(w) is the Hilbert transform of G(w).

Does this mean that we cannot freely select gain Ki and Ki/w must be the Hilbert transform of Kp?

If you mean a PID controller, it would seem to satisfy the criteria of being LTI and dependent only on current and past inputs, no? So how could it be non-causal? If the PID characteristics could be altered real-time by itself (like in Machine Learning), then it would no longer be time-invariant, but I think that is a different situation...

https://en.wikipedia.org/wiki/Causal_filter

https://en.wikipedia.org/wiki/PID_controller

 

[DaveE]

Summary: Help understanding why PI controller is causal.

Does this mean that we cannot freely select gain Ki and Ki/w must be the Hilbert transform of Kp?

No. You can choose any gain terms and this filter will remain causal.