You need to do the PID control on a frame that rotates with the AC waveform. In other words, you need to find a way to measure your output signal and transform it to components either in phase or out of phase with the target sinusoid. This is commonly known as a d-q (direct-quadrature) transform.
Once you have these values, which should not oscillate at the AC frequency, you can use PID control. Then, you have to convert back to the oscillating frame to generate your inverter pulses.
The following paper might have way too much detail, but it's the best I found:
- Sebastian
