On hybrid systems and closed-loop MPC systems

The following five classes of hybrid systems were recently proved to be equivalent: linear complementarity (LC) systems, extended linear complementarity (ELC) systems, mixed logical dynamical (MLD) systems, piecewise affine (PWA) systems, and maxmin-plus-scaling (MMPS) systems. Some of the equivalences were obtained under additional assumptions, such as boundedness of system variables. In this paper, for linear or hybrid plants in closed-loop with a model predictive control (MPC) controller based on a linear model and fulfilling linear constraints on input and state variables, we provide a simple and direct proof that the closed-loop system (cl-MPC) is a subclass of any of the former five classes of hybrid systems. This result opens the use of tools developed for hybrid systems (such as stability, robust stability, and safety analysis tools) to study closed-loop properties of MPC.