Robustness margin for piecewise affine explicit control law

Classical robustness margin i.e., gain margin and phase margin, considers the gain variation and phase variation of the model for which the stability of the closed loop is preserved. In this paper, an attempt to find the same kind of margin for a piecewise affine (PWA) controller is done. This type of controller obtained for example via explicit model predictive control (MPC) is defined over a convex region of the state space X. Starting from the invariance property of the closed loop obtained involving a discrete dynamic model and PWA controller in a convex region of the state space, we calculate the two robustness margin preserving this invariance property. The first one will be denoted as gain margin corresponding to the variation of the gain of the model guaranteeing the invariance. The second one, denoted, the robustness margin against first order neglected dynamics will correspond to the slowest first order neglected dynamic allowed in the system preserving the invariance property.