Electronic – Brushless DC motor model for model-based design

If the nonlinear relationship holds for a wide variety of conditions, you could create a mapping function that removes the nonlinearity. The only issue would be if the function is non-monotonic (ie has a dip). Of course, you could always just apply the PID and experiment with parameter values. Unless the system is VERY badly behaved, you will probably find a result you like.

TL;DR Stop worrying and just try it

Brushless DC means exactly that - a DC motor with brushless commutation. The controller's job is to switch the phases in and out at the correct rotor angles (not frequency) just like the commutator in a brushed motor. The motor spins at whatever speed it wants to, depending on supply voltage and load.

If the motor has sensors then the controller can be very simple, since it just has to read the sensors and turn on the appropriate phases depending on rotor angle. It has no direct control over commutation frequency, but it can 'control' motor speed indirectly by varying the effective supply voltage (using either a regulator or PWM).

Sensorless controllers have a harder job because they must monitor the back-emf waveform for zero crossings. At startup there is no back-emf so the ESC cannot detect the rotor's position. To get the rotor spinning it pulses the phases at low power like a stepper motor, gradually increasing speed until it gets a strong enough back-emf to switch into synchronous operation.

During this startup period only, the sensorless ESC controls motor speed by varying commutation frequency. However since it is basically dragging the rotor up to speed, any sudden change in load can cause it to loose sync. Also the motor may start in reverse, then it has to stop and try again. This may result in the rotor jumping back and forth a few times until the ESC sees a good back-emf.