Markov Set-Chains as Abstractions of Stochastic Hybrid Systems

  • Authors:

  • Alessandro Abate;Alessandro D'Innocenzo;Maria D. Benedetto;Shankar S. Sastry

  • Affiliations:

  • Department of Aeronautics and Astronautics, Stanford University, USA;Department of Electrical Engineering and Computer Science, Center of Excellence DEWS, University of L'Aquila, Italy;Department of Electrical Engineering and Computer Science, Center of Excellence DEWS, University of L'Aquila, Italy;Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, USA

  • Venue:
  • HSCC '08 Proceedings of the 11th international workshop on Hybrid Systems: Computation and Control
  • Year:
  • 2008

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Abstract

The objective of this study is to introduce an abstraction procedure that applies to a general class of dynamical systems, that is to discrete-time stochastic hybrid systems (dt-SHS). The procedure abstracts the original dt-SHS into a Markov set-chain (MSC) in two steps. First, a Markov chain (MC) is obtained by partitioning the hybrid state space, according to a controllable parameter, into non-overlapping domains and computing transition probabilities for these domains according to the dynamics of the dt-SHS. Second, explicit error bounds for the abstraction that depend on the above parameter are derived, and are associated to the computed transition probabilities of the MC, thus obtaining a MSC. We show that one can arbitrarily increase the accuracy of the abstraction by tuning the controllable parameter, albeit at an increase of the cardinality of the MSC. Resorting to a number of results from the MSC literature allows the analysis of the dynamics of the original dt-SHS. In the present work, the asymptotic behavior of the dt-SHS dynamics is assessed within the abstracted framework.