Approach to an unknown orthogonal Beta Signal with a known alfa in single phase dq transformation
Sounds like you seek to phase_track a signal, for which you have some inphase and some quadrature information.
And you need to preserve the polarity of the information, so you can have a Negative Feedbac control system instead of a Positive Feedback system.
Show all your information_extarction maths, and the sensor input to the system.
Then think about how to preserve the needed polarity of the control output.
================================================
If you want to phase_track a sinusoid, then yes you can generate a quadrature signal using a differentiator. This will be noisy.
If you want to phase_track a non_sinusoid, you need to decide which edge to align with, and use a phase_frequency digital_phase_frequency circuit; this has 2 FlipFlips, a ORing gate, and (usually) a tiny delay needed to ensure healthy RESET pulse behavior.