Markov Set-Chains

Hartfiel, D. J.

doi:10.1007/bfb0094586

Cited by 84 publications

(93 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We also recall some useful results from [8], which contains a compendium of literature on the subject. Definition 3 (Transition Set).…”

Section: Endmentioning

confidence: 99%

“…More formally, for large values of k should be accounted for. However, it is possible to compute tight (see [8]) upper and lower bounding matrices…”

Section: Endmentioning

confidence: 99%

“…This will become clear when studying the asymptotic properties of the MSC, and is related to the regularity properties of the matrices that build up the MSC [8]. The exact value of T ([Π]) can be hard to compute, but it can be upper bounded as follows:…”

Section: Endmentioning

confidence: 99%

“…The above matrix norm is taken from [8] and is a modification of the induced 1-norm. The following notion connects to Definition 5:…”

Section: Endmentioning

confidence: 99%

“…Namely, if T ∞ < 1 we can arbitrarily increase the accuracy δ of our abstraction until, by equation (8), λ * k δ ≤ ε (1 − T ∞ ). When this happens the algorithm terminates, and we compute the steady state π ∞ , or compute [π ∞ ] using the upper and lower bounding matrices L ∞ , H ∞ , as described in section 3.…”

Section: Steady State Computation Using the Msc Abstractionmentioning

confidence: 99%

See 4 more Smart Citations

Markov Set-Chains as Abstractions of Stochastic Hybrid Systems

Abate

D’Innocenzo

Benedetto

et al. 2008

Hybrid Systems: Computation and Control

View full text Add to dashboard Cite

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.

show abstract

“…We also recall some useful results from [8], which contains a compendium of literature on the subject. Definition 3 (Transition Set).…”

Section: Endmentioning

confidence: 99%

“…More formally, for large values of k should be accounted for. However, it is possible to compute tight (see [8]) upper and lower bounding matrices…”

Section: Endmentioning

confidence: 99%

Section: Endmentioning

confidence: 99%

“…The above matrix norm is taken from [8] and is a modification of the induced 1-norm. The following notion connects to Definition 5:…”

Section: Endmentioning

confidence: 99%

Section: Steady State Computation Using the Msc Abstractionmentioning

confidence: 99%

See 3 more Smart Citations