Polyhedral Flows in Hybrid Automata

A hybrid automaton is a mathematical model for hybrid systems, which combines, in a single formalism, automaton transitions for capturing discrete updates with differential constraints for capturing continuous ows. Formal veriication of hybrid au-tomata relies on symbolic xpoint computation procedures that manipulate sets of states. These procedures can be implemented using boolean combinations of linear constraints over system variables, equivalently, using polyhedra, for the subclass of linear hybrid automata. In a linear hybrid automaton, the ow at each control mode is given by a rate polytope that constrains the allowed rst derivatives. The key property of such a ow is that, given a state-set described by a polyhedron, the set of states that can be reached as time elapses, is also a polyhedron. We call such a ow a polyhedral ow. In this paper, we study if we can generalize the syntax of linear hybrid au-tomata for describing ows without sacriicing the polyhedral property. In particular, we consider ows described by origin-dependent rate polytopes, in which the allowed rates depend, not only on the current control mode, but also on the speciic state at which the mode was entered. We establish that ows described by origin-dependent rate polytopes, in some special cases, are polyhedral.