There are plenty of compensator types you can think of when attempting to compensate a given plant. First off, you need the plant dynamic response. This is what is called the control-to-output transfer function. From this response, you can infer what compensation strategy is needed to fulfill your goals. Basically, at least in power electronics, there are three compensator types: type 1, type 2 and type 3. They can be built around an active amplifier like an op amp, an OTA and a shunt regulator (TL431) for instance. The below picture shows you what they can do in terms of dynamic response.
You select the type by knowing the amount of "phase boost" you need to meet the phase margin criteria. The phase boost is the amount of positive phase lead you need to compensate the lag incurred by the power stage at the selected crossover frequency \$f_c\$.
A type 1 is a simple integrator. It features a pole at the origin and lags the phase by 270°. There is no boost. A type 2 combines a pole at the origin and a pole-zero pair. By adjusting the distance between the pole and the zero, you adjust the boost up to 90° in theory. Finally, a type 3 adds another pole/zero pair to the original type 2 and lets you boost the phase up to 180°. I have a complete seminar on the subject of compensator that you can download here and a book you could consider for closing the loop is this one. Good luck with your project!