@incollection{eprints499,
       publisher = {Springer-Verlag},
         journal = {Hybrid Systems: Computation and Control},
          author = {Alberto Bemporad and Fabio Danilo Torrisi and Manfred Morari},
       booktitle = {Hybrid Systems: Computation and Control},
          editor = {Nancy Lynch and Bruce Krogh},
          volume = {1790},
           pages = {45--58},
           title = {Optimization-based verification and stability characterization of piecewise affine and hybrid systems},
          series = {Lecture Notes in Computer Science},
            year = {2000},
             url = {http://eprints.imtlucca.it/499/},
        abstract = {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. }
}