# Hilbert Transform, Causality, PI Controller

• angryturtle
In summary: However, if you want to satisfy the relationship H(w) = G(w) -i G_hat(w), then Ki/w must be the Hilbert transform of Kp.
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?

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.

berkeman

## 1. What is the Hilbert Transform?

The Hilbert Transform is a mathematical operation that converts a time-domain signal into its corresponding frequency-domain representation. It is commonly used in signal processing and analysis to extract the instantaneous amplitude and phase of a signal.

## 2. How is causality related to the Hilbert Transform?

Causality refers to the principle that the output of a system should only depend on its past inputs. The Hilbert Transform is a causal operation, meaning that the output at any given time depends only on the input values up to that time. This property is important in many applications, such as control systems and filtering.

## 3. What is a PI controller?

A PI controller is a type of feedback control system that uses both proportional and integral control actions to regulate a system's output. It is commonly used in engineering and control systems to improve stability and performance.

## 4. How does the Hilbert Transform relate to PI controllers?

The Hilbert Transform can be used in the design and analysis of PI controllers. It can help to identify the frequency response of a system and determine the appropriate parameters for the controller to achieve desired performance.

## 5. Can the Hilbert Transform be used for non-linear systems?

Yes, the Hilbert Transform can be applied to non-linear systems. However, it may not always provide accurate results due to the non-linear nature of the system. In such cases, alternative methods may need to be used for analysis and control.

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