A Combined Approach to Determine Robustly Stabilizing Parameter Spaces

Guaranteeing the stability of dynamical systems is one of the core problems of control engineering. The complexity of the problem increases significantly when uncertain parameters take place. Most of the studies in the literature focus on the stability of a given uncertain polynomial family and omit the effect of free controller parameters. In this study, a combined approach is proposed in order to determine the robustly stabilizing controller parameter spaces of a given parametric uncertain system. The proposed approach is composed of two main steps. In the first step, currently existing theorems for robust stability (Kharitonov, Edge, 16-plants, etc.) are used to determine the polynomials that should be stable for robust stability. In the second step, a Lyapunov Equation based generic stability mapping approach is presented in order to determine the bounds of free parameters that guarantee the stability of predetermined polynomials. In this way, it is easily possible to determine the exact bounds of controller parameters that achieve robust stability. The presented stability mapping approach is independent of the controller type and the number of free controller parameters. Two benchmark case studies are included in order to verify the effectiveness and correctness of the derived theoretical results.