In electronics equipment, a unit named phase-locked loop (PLL) is used.
Simply speaking, it adjusts the phase $p_r$ of a reference signal like $r(t)=sin(f_r*t+p_r)$ with constant frequency $f_r$ to a given input signal $x(t)$ by a feedback control loop acting on $p_r$ so that $\int f(t)\ r(t)\ dt$ gets maximised, minimised or absolutely minimised.
This control still works if some noise is added to the input signal or the input signal is moderately distorted to an non-sine periodically signal.
If $\int f(t)\ r(t)\ dt$ is used as a negative feedback itself, it is absolutely minimised and $p_r$ locks to an about 90 degreed shifted phase of a clean sine input signal (empirical knowledge from software implementation).
The same feedback however, can also be used to control the reference frequency $f_r$ for a given constant phase $p_r=const.$, if starting conditions are in close range.
My question now is, if it is possible to control both parameters $f_r$ and $p_r$ from the same integral term (empirically it seems to work sometimes on well tuned feedback parameters), why it is possible to control two parameters from a seemingly one dimensional term, and what the conditions have to be met for stable operation.
Phase is a time shift between a Reference vs. Feedback in periodic signals. To measure this correctly, the frequency has to be (almost) the same.
If by some reason the frequencies have a slight difference, the first thing that happens is a change in phase shift: one of the signals is running at a higher frequency than the other, so the controller has to adjust above the exact frequency to return the shift in phase back to zero. So when you are controlling the phase shift, in reality you are moving the frequency slightly over the exact value to return the shift to zero and then to stop the shift, put the both Reference and Feedback frequencies exactly the same when both signals are at a desired phase shift.