Synthesis of H∞ PID controllers

Considers the problem of synthesizing proportional-integral-derivative (PID) controllers for which the closed-loop system is internally stable and the H_∞-norm of a related transfer function is less than a prescribed level for a given single-input single-output plant. It is shown that the problem to be solved can be translated into simultaneous stabilization of the closed-loop characteristic polynomial and a family of complex polynomials. It calls for a generalization of the Hermite-Biehler theorem applicable to complex polynomials. Then a linear programming characterization of all admissible H_∞ PID controllers for a given plant is obtained