PID controllers
PID control consists of a proportional plus derivative (PD) compensator cascaded with a proportional plus integral (PI) compensator. The purpose of the PD compensator is to improve the transient response while maintaining the stability. The purpose of the PI compensator is to improve the steady state accuracy of the system without degrading the stability. The speed of response, accuracy and stability are what is needed for satisfactory response.
Proportional control (P-control):
A P-controller implements the simple control law
where