%0 Book Section
%A Bemporad, Alberto
%A Torrisi, Fabio Danilo
%A Morari, Manfred
%B Hybrid Systems: Computation and Control
%D 2000
%E Lynch, Nancy
%E Krogh, Bruce
%F eprints:499
%I Springer-Verlag
%P 45-58
%S Lecture Notes in Computer Science
%T Optimization-based verification and stability characterization of piecewise affine and hybrid systems
%U http://eprints.imtlucca.it/499/
%V 1790
%X In this paper, we formulate the problem of characterizing the stability of a piecewise affine (PWA) system as a verification problem. The basic idea is to take the whole IR n as the set of initial conditions, and check that all the trajectories go to the origin. More precisely, we test for semi-global stability by restricting the set of initial conditions to an (arbitrarily large) bounded set X(0), and label as “asymptotically stable in T steps” the trajectories that enter an invariant set around the origin within a finite time T, or as “unstable in T steps” the trajectories which enter a set X inst of (very large) states. Subsets of X(0) leading to none of the two previous cases are labeled as “non-classifiable in T steps”. The domain of asymptotical stability in T steps is a subset of the domain of attraction of an equilibrium point, and has the practical meaning of collecting the initial conditions from which the settling time to a specified set around the origin is smaller than T. In addition, it can be computed algorithmically in finite time. Such an algorithm requires the computation of reach sets, in a similar fashion as what has been proposed for verification of hybrid systems. In this paper we present a substantial extension of the verification algorithm presented in [6] for stability characterization of PWA systems, based on linear and mixed-integer linear programming. As a result, given a set of initial conditions we are able to determine its partition into subsets of trajectories which are asymptotically stable, or unstable, or non-classifiable in T steps.