Towards a mathematical theory of PID control

The well-known and nearly hundred years old “proportional-integral- derivative” (PID) controller is a linear feedback control method that adjusts the input signal by using the regulation error of the systems only. Because of its simple structure, good robustness and high reliability, it is by far the most widely used control method in automatic control. For example, it has been reported that 95 percent control loops are of PID type in process control. However, one of the most challenging tasks for the implementation of the PID controller is how to design the three parameters of the controller.