Robust PID Controller Design by Linear Programming

A linear programming approach is proposed to tune fixed-order linearly parameterized controllers for stable LTI plants. The method is based on the shaping of the open-loop transfer function in the Nyquist diagram. A lower bound on the crossover frequency and a new linear stability margin which guarantees lower bounds for the classical robustness margins are defined. Two optimization problems are proposed and solved by linear programming. In the first one the robustness margin is maximized for a given lower bound on the crossover frequency, whereas in the second one the integrated error is minimized with constraints on the new stability margin. The method can directly consider multi-model as well as frequency-domain uncertainties. An application to a high-precision double-axis positioning system illustrates the effectiveness of the proposed approach.