Markov Set-Chains as Abstractions of Stochastic Hybrid Systems

  • Alessandro Abate
  • Alessandro D’Innocenzo
  • Maria D. Di Benedetto
  • Shankar S. Sastry
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4981)

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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Alessandro Abate
    • 1
  • Alessandro D’Innocenzo
    • 2
  • Maria D. Di Benedetto
    • 2
  • Shankar S. Sastry
    • 3
  1. 1.Department of Aeronautics and AstronauticsStanford UniversityUSA
  2. 2.Department of Electrical Engineering and Computer Science, Center of Excellence DEWSUniversity of L’AquilaItaly
  3. 3.Department of Electrical Engineering and Computer SciencesUniversity of CaliforniaBerkeleyUSA

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