On the construction of invariant sets for piecewise affine systems using the transition graph

The present paper introduces an algorithmic construction of the maximal invariant set for a PWA (piecewise affine) system. The classical analysis of this type of dynamical systems is based on the construction of a Lyapunov function which lead subsequently to the invariant set description by means of the Lyapunov function level sets. As an alternative, expansive/contractive schemes exploit the global one-step forward/backward evolution of the system dynamics in order to obtain the invariant set as a fixed point of the set iterates. Both approaches address the problem of finding an invariant set from a global point of view, and therefore result in very demanding computations. The conditions under which the resulting invariant sets are finitely determined are not clear. The approach proposed in this paper is different in the sense that each polyhedral region is treated separately considering only the infinite-time endogenous (by the same region of the state space) transitions, providing (at least from a local point of view) clear conditions on the finite determinedness. In order to address with the global behavior, a transition graph between local piecewise descriptions is used.