Explicit robustness margins for discrete-time linear systems with PWA control

The paper considers the class of discrete-time linear dynamics in closed loop with a piecewise affine control law defined over a bounded region of the state space χ. Given a polytopic uncertainty on the nominal model of the system, the objective is to describe the largest subset of this parametric uncertainty for which the positive invariance of the set χ is guaranteed with respect to the closed loop dynamics. This subset describes in fact a robust invariance margin which represents in general terms an important element in the study of the robust stability of any piecewise affine dynamics. It is shown that the necessary and sufficient conditions for set inclusion lead to convex formulations of the robustness margin. A series of remarks on the structure of the numerical problems are discussed and a numerical example is provided for illustration.