Type II compensation adds a zero (to increase phase margin [1]) and a pole (to filter high-frequency noise) to the compensation network, resulting in proportional-integral (PI) control.
Type III compensation introduces two zeros and two poles into the control loop, making it a proportional-integral-derivative (PID) control scheme.
... and if you want to go one step further, MPC [3] also exists:
Type I is proportional only.
Type II compensation adds a zero (to increase phase margin [1]) and a pole (to filter high-frequency noise) to the compensation network, resulting in proportional-integral (PI) control.
Type III compensation introduces two zeros and two poles into the control loop, making it a proportional-integral-derivative (PID) control scheme.
... and if you want to go one step further, MPC [3] also exists:
https://iaeme.com/MasterAdmin/Journal_uploads/IJARET/VOLUME_...
[1] https://en.wikipedia.org/wiki/Phase_margin
[2] https://eng.libretexts.org/Bookshelves/Industrial_and_System...
[3] https://en.wikipedia.org/wiki/Model_predictive_control