Synthesizing All Filtered Proportional–Integral and PID Controllers Satisfying Gain, Phase, and Sensitivity Specifications

This article presents a method for calculating all the proportional–integral/proportional–derivative controllers with a low-pass filter of a given order, which satisfy gain–phase margins and sensitivity (or complementary sensitivity) specifications. The method is explicit and requires solving a set of a polynomial and two linear equations. It is shown how to extend the method to design all the proportional–integral–derivative or cascaded controllers with a low-pass filter assuming that a minimum value of one of the controller components is given.