Approximate timed abstractions of hybrid automata

Given a hybrid automaton and a desired precision, we aim at constructing an approximate abstraction by means of a timed automaton, whose discrete state trajectories approximate the discrete state trajectories of the original system, with the desired precision on switching times. We show that using the Euclidian metric on reals it is not always possible to construct a timed automaton that is close to a hybrid automaton with finite precision. For this reason, we motivate and introduce relative metrics on reachability time, external language and simulation relation to quantify the precision of our abstraction. Our main result is to propose a novel algorithm to construct a timed automaton that is an approximate timed abstraction of a hybrid automaton with desired precision, and study its convergence properties. For an extended version of this paper refer to [D´Innocenzo et al., 2007].