This study proposes a geometric procedure for robust controller tuning under parametric uncertainty, based on root-contour analysis of the closed-loop control system. For a fixed candidate controller tuning, the set of possible pole locations induced by the admissible variations of the control plant parameters is constructed. Robust admissibility is formulated as a geometric set-inclusion problem, requiring this set to remain inside a prescribed dynamic performance region in the complex s-plane. A distinction is introduced between nominal admissibility, robust stability, and robust admissibility, showing that stability over the entire uncertainty set is not sufficient to guarantee the desired dynamic performance. To quantify the root contours, several indices are defined, including the dispersion along the real and imaginary axes, the maximum pole displacement with respect to the nominal pole locations, and the geometric margin to the boundary of the performance region. The procedure is applied to the selection and verification of PI controller tunings for an uncertain single-input–single-output (SISO) control system and is further validated through examples with different structures of parametric uncertainty, including a system with a single uncertain parameter and a PID-controlled system with several uncertain control plant parameters. The results show that root-contour analysis can distinguish tunings that are only robustly stable from tunings that preserve the prescribed dynamic performance over the entire uncertainty set. Thus, the method can be used as a practical tool for the diagnosis, comparison, and selection of controller tunings under parametric uncertainty.
Vesela Karlova-Sergieva (Sat,) studied this question.