%0 Book Section
%A Rubagotti, Matteo
%A Trimboli, Sergio
%A Bernardini, Daniele
%A Bemporad, Alberto
%B Proceedings of the 18th IFAC World Congress
%D 2011
%F eprints:740
%I IFAC
%K Hybrid systems stability
%T Stability and invariance analysis of approximate explicit MPC based on PWA Lyapunov functions
%U http://eprints.imtlucca.it/740/
%V 18
%X For piecewise affine (PWA) systems whose dynamics are only defined in a bounded and possibly non-invariant set X, this paper proposes a numerical approach to analyze the stability of the origin and to find a region of attraction. The approach relies on introducing fake dynamics outside X and on synthesizing a piecewise affine and possibly discontinuous Lyapunov function on a larger bounded set containing X by solving a linear program. The existence of a solution proves that the origin is an asymptotically stable equilibrium of the original PWA system and determines a region of attraction contained in X. The procedure is particularly useful in practical applications for analyzing a posteriori the stability properties of approximate explicit model predictive control laws defined over a bounded set X of states, and to determine whether, for a given set of initial states, the closed-loop system evolves within the domain X where the control law is defined.