Robust Static Output Feedback Controllers via Robust Stabilizability Functions

This technical note addresses the design of robust static output feedback controllers that minimize a polynomial cost and robustly stabilize a system with polynomial dependence on an uncertain vector constrained in a semialgebraic set. The admissible controllers are those in a given hyper-rectangle for which the system is well-posed. First, the class of robust stabilizability functions is introduced, i.e., the functions of the controller that are positive whenever the controller robustly stabilizes the system. Second, the approximation of a robust stabilizability function with a controller-dependent lower bound is proposed through a sums-of-squares (SOS) program exploiting a technique developed in the estimation of the domain of attraction. Third, the derivation of a robust stabilizing controller from the found controller-dependent lower bound is addressed through a second SOS program that provides an upper bound of the optimal cost. The proposed method is asymptotically non-conservative under mild assumptions.