On the design of robust static output feedback controllers via robust stabilizability functions

A key problem in control systems consists of designing robust stabilizing controllers for systems with parametric uncertainties, in particular output feedback controllers that, without requiring to measure the uncertainty, ensure robust stability of the closed-loop system. This paper aims to establish the existence of such a controller and, more generally, to determine a robust stabilizing controller that minimize a chosen cost. This problem is considered in this paper for systems whose coefficients are polynomial functions of an uncertain vector constrained into a semialgebraic set. The admissible controllers are those in a given hyper-rectangle for which the closed-loop system is well-posed. First, the class of robust stabilizability functions is introduced, i.e. the class of functions that, when evaluated for an admissible controller, are positive if and only if the controllers robustly stabilizes the system. Second, the approximation of a robust stabilizability function with a controller-dependent lower bound is proposed through a convex program exploiting a technique developed in the estimation of the robust domain of attraction. Third, the derivation of a robust stabilizing controller from the found controller-dependent lower bound is addressed through a second convex program that provides an upper bound of the optimal cost.