Abstract
In this chapter we introduce the automata framework CPDP, which stands for Communicating Piecewise Deterministic Markov Processes. CPDP is developed for compositional modelling and analysis for a class of stochastic hybrid systems. We define a parallel composition operator, denoted as \(\left\vert {P \atop A} \right\vert\), for CPDPs, which can be used to interconnect component-CPDPs, to form the composite system (which consists of all components, interacting with each other). We show that the result of composing CPDPs with \(\left\vert {P \atop A} \right\vert\) is again a CPDP (i.e., the class of CPDPs is closed under \(\left\vert {P \atop A} \right\vert\)). Under certain conditions, the evolution of the state of a CPDP can be modelled as a stochastic process. We show that for these CPDPs, this stochastic process can always be modelled as a PDP (Piecewise Deterministic Markov Process) and we present an algorithm that finds the corresponding PDP of a CPDP. After that, we present an extended CPDP framework called value-passing CPDP. This framework provides richer interaction possibilities, where components can communicate information about their continuous states to each other. We give an Air Tra.c Management example, modelled as a value-passing CPDP and we show that according to the algorithm, this CPDP behavior can be modelled as a PDP. Finally, we define bisimulation relations for CPDPs. We prove that bisimilar CPDPs exhibit equal stochastic behavior. Bisimulation can be used as a state reduction technique by substituting a CPDP (or a CPDP component) by a bisimulation-equivalent CPDP (or CPDP component) with a smaller state space. This can be done because we know that such a substitution will not change the stochastic behavior.
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Strubbe, S., van der Schaft, A. Communicating Piecewise Deterministic Markov Processes. In: Blom, H.A.P., Lygeros, J. (eds) Stochastic Hybrid Systems. Lecture Notes in Control and Information Science, vol 337. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11587392_3
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DOI: https://doi.org/10.1007/11587392_3
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-33466-8
Online ISBN: 978-3-540-33467-5
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