We present a general framework for the formal specification and algorithmic analysis of hybrid systems. A hybrid system consists of a discrete program with an analog environment. We model hybrid systems as nite automata equipped with variables that evolve continuously with time according to dynamical laws. For verification purposes, we restrict ourselves to linear hybrid systems, where all variables follow piecewise-linear trajectories. We provide decidability and undecidability results for classes of linear hybrid systems, and we show that standard program-analysis techniques can be adapted to linear hybrid systems. In particular, we consider symbolic model-checking and minimization procedures that are based on the reachability analysis of an infinite state space. The procedures iteratively compute state sets that are definable as unions of convex polyhedra in multidimensional real space. We also present approximation techniques for dealing with systems for which the iterative procedures do not converge.

. The analysis, verification, and control of hybrid automata with finite bisimulations can be reduced to finite-state problems. We advocate a time-abstract, phasebased methodology for checking if a given hybrid automaton has a finite bisimulation. First, we factor the automaton into two components, a boolean automaton with a discrete dynamics on the finite state space B m and a euclidean automaton with a continuous dynamics on the infinite state space R n . Second, we investigate the phase portrait of the euclidean component. In this fashion, we obtain new decidability results for hybrid systems as well as new, uniform proofs of known decidability results. For example, we prove that if two hybrid automata have finite bisimulations, and both can be calibrated to a common time scale, then their product also has a finite bisimulation. 1 Introduction A hybrid automaton [2] is a mathematical model for a digital program that interacts with an analog environment. Hybrid automata are usef...

Rectangular hybrid automata model digital control programs of analog plant environments. We study rectangular hybrid automata where the plant state evolves continuously in real-numbered time, and the controller samples the plant state and changes the control state discretely, only at the integer points in time. We prove that rectangular hybrid automata have nite bisimilarity quotients when all control transitions happen at integer times, even if the constraints on the derivatives of the variables vary between control states. This is in contrast with the conventional model where control transitions may happen at any real time, and already the reachability problem is undecidable. Based on the nite bisimilarity quotients, we give an exponential algorithm for the symbolic sampling-controller synthesis of rectangular automata. We show our algorithm to be optimal by proving the problem to be EXPTIME-hard. We also show that rectangular automata form a maximal class of systems for which the sampling-controller synthesis problem can be solved algorithmically.

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