Loop shaping for PID controller design based on time and frequency specifications ^⁎

In this paper it is described a new method to design PID controllers using a linear programming approach for optimizing performance subject to robustness constraints. It is based on the shaping of a reference loop gain transfer function which forms a convex region on the Nyquist diagram which contains and bounds the designed loop gain Nyquist plot. The convex region is approximated by a set of lines in order to formulate a linear optimization problem. It is also presented an optional performance specification related to the crossover frequency of the designed loop gain. The class of stable linear time-invariant single-input simple-output (SISO) systems is considered and the optimization problem is proposed, solved and analyzed.