H∞ robust design of PID controllers for arbitrary-order LTI systems with time delay

For a given arbitrary-order LTI (linear time-invariant) plant with time delay, we propose a directly parametric design method of the H∞ PID controller in an analytical manner. The design problem of the PID controllers satisfying the H∞ norm requirement is first cast into simultaneous stabilization problem of a family of complex quasipolynomials and the characteristic equation. Then, the linear programming characterization of the PID controllers that can ensure the stability of the complex quasipolynomials are developed on the basis of the extended Hermite-Biehler Theorem.Finally, The admissible set of the H∞ PID controllers is presented by combining such the results with the stabilizing set of the PID controllers. The results reveal that the set of the integral and derivative gains is a union of convex sets for a fixed proportional gain. The proposed scheme works without any approximation and enable one to find a set of the PID parameters satisfying the H∞ norm requirement conveniently.