Hilbert Transform, Causality, PI Controller

AI Thread Summary
The discussion centers on the causality of the PI controller, which is defined as a causal filter with a frequency response of H(w) = Ki/(iw) + Kp. It is noted that a causal filter must satisfy the relationship H(w) = G(w) - i G_hat(w), where G_hat(w) is the Hilbert transform of G(w). Participants clarify that the gain terms Ki and Kp can be chosen freely without violating causality. Additionally, the distinction between a PID controller and its potential non-causal characteristics is mentioned, emphasizing that a standard PI controller remains causal. Ultimately, the consensus is that the selection of gain terms does not impact the causality of the filter.
angryturtle
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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?
 
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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?
 
angryturtle said:
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
 
angryturtle said:
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.
 
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