Markov Set-Chains as Abstractions of Stochastic Hybrid Systems

  • Alessandro Abate
  • Alessandro D’Innocenzo
  • Maria D. Di Benedetto
  • Shankar S. Sastry
Conference paper

DOI: 10.1007/978-3-540-78929-1_1

Volume 4981 of the book series Lecture Notes in Computer Science (LNCS)
Cite this paper as:
Abate A., D’Innocenzo A., Di Benedetto M.D., Sastry S.S. (2008) Markov Set-Chains as Abstractions of Stochastic Hybrid Systems. In: Egerstedt M., Mishra B. (eds) Hybrid Systems: Computation and Control. HSCC 2008. Lecture Notes in Computer Science, vol 4981. Springer, Berlin, Heidelberg

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