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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?

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?