On the stability of quadratic forms based model predictive control of constrained PWA systems

In this paper we investigate the stability of discrete-time PWA systems in closed-loop with quadratic cost based model predictive controllers (MPC) and we derive a priori sufficient conditions for Lyapunov asymptotic stability. We prove that Lyapunov stability can be achieved for the closed-loop system even though the considered Lyapunov function and the system dynamics may he discontinuous. The stabilization conditions are derived using a terminal cost and constraint set method. An S-procedure technique is employed to reduce conservativeness of the stabilization conditions and a linear matrix inequalities set-up is developed in order to calculate the terminal cost. A new algorithm for computing piecewise polyhedral positively invariant sets for PWA systems is also presented. In this manner, the on-line optimization problem associated with MPC leads to a mixed integer quadratic programming problem, which can be solved by standard optimization tools.