That equation (I am not aware of) could be helpful. What I am suggesting is keep your controls as variables and change then till you get a lock and then keep them
If you want to follow the diagram you posted that is ok. C1 corresponds to scale factor on input and so keep it variable and start with the above equation value.
C2 is a scale factor on input to integrator and there is no scale factor on feedback term. You can try that and keep C2 also variable.
My IIR suggests a further scale factor on feedback term equal to (1-alpha) producing a typical IIR with unity dc gain. Your case does not care about filter dc gain and it may well work or get out of control. I suggest you add scale factor on feedback term and set to 1 if it turned out to be irrelevant.
Thus the IIR cutoff is under control of these variables and these variables will eventually be fixed in design by trial and error.
I don't know what is PI (proportional Integrator?).Yes we are talking about same thing but I am conceptually splitting the single system into two.