Frequency parameterization of H∞ PID controllers via relay feedback: A graphical approach

This paper focuses on a graphical approach to determine the region of proportional-integral-derivative (PID) controllers in the parameter space for which the closed-loop system is internally stable and the H∞ optimization criteria are satisfied for a class of single-input single-output arbitrary order plant with or without dead time. Unlike conventional methods which the analytical models, such as transfer functions and state space models, are needed, the design information of the proposed approach is only the frequency response data, which are directly calculated from a single relay test for stable plants, or extracted from the closed-loop system frequency response data by dividing out the known stabilizing compensator for unstable plants using relay feedback methods. It is shown that the problem to be solved can be translated into simultaneous stabilization of the closed-loop characteristic function and a family of characteristic functions. Based on the technique of D-decomposition, the analytical boundaries of root invariant regions are derived and the admissible H∞ region in the parameter space is the intersection of the admissible sets, and it can be drawn and identified immediately, not to be computed mathematically. A practical algorithm of determining the H∞ region is proposed and two examples are used to illustrate the proposed method.