Polynomial Stochastic Hybrid Systems

Hespanha, João P.

doi:10.1007/978-3-540-31954-2_21

Cited by 32 publications

(24 citation statements)

References 11 publications

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“…SHSs have been introduced in [17] and further developed in [6,7,14,15,22]. Here, we consider a restricted form of SHSs that are closely related to (and heavily inspired by) the PiecewiseDeterministic Markov Process (PDMPs) introduced by Davis [11] and, in fact, our SHSs can be viewed as a special case of PDMPs and thus inherent many of the PDMPs properties.…”

Section: Stochastic Hybrid Systemsmentioning

confidence: 99%

“…The proposition P8 is a direct consequence of (39), which can be used to provide an explicit estimate on the state bound δ, as a function of ρ, in the definition in (14): for given , ρ ą 0, we simply need to pick µ, δ ą 0 sufficiently small so that }xptq} ą ρ ñ V`qptq, xptq˘ě µ and }x 0 } ď δ ñ V pq 0 , x 0 q µ ď .…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…In this case, for every initial condition qp0q " q 0 P Q, xp0q " x 0 P R n with }x 0 } ď δ, we have that (14) holds.…”

Section: Proof Of Theoremmentioning

confidence: 99%

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Networked Control Systems

Wang

Liu

2008

207

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This paper aims at familiarizing the reader with Stochastic Hybrid Systems (SHSs) and enabling her to use these systems to model and analyze Networked Control Systems (NCSs). Towards this goal, we introduce two different models of SHSs and a set of theoretical tools for their analysis. In parallel with the presentation of the mathematical models and results, we provide a few simple examples that illustrate the use of SHSs to models NCSs.Keywords: Network Control Systems; Hybrid Systems; Stochastic Processes; Stability; Markov Processes Networked Control SystemsThe expression Networked Control Systems (NCSs) typically refers to feedback control systems for which some of the sensors, controllers, and actuators communicate with each other using a shared communication network. The use of a multi-purpose shared network reduces installation and maintenance costs and adds flexibility, as it permits the system reconfiguration and/or expansion with minimal additional infrastructure costs. In view of this, NCSs are finding application in numerous areas that include the automotive industry, the aviation industry, robotics, process control, and building control, among others.While NCSs are attractive from the perspective of cost of deployment and maintenance, they introduce significant design challenges, because the traditional unity feedback loop that operates in continuous time or at a fixed sampling rate is not adequate when sensor data arrives from multiple sources, asynchronously, delayed, and possibly corrupted. Consequently, NCSs have been the focus of intense study in the last few years [16,25,29]. This paper is focused on two aspects of NCSs that are responsible for important challenges in analyzing and designing NCSs and that are prompting the development of new formal tools to study these systems. This paper is partially based on material presented in a plenary lecture at the 4th IFAC Workshop on Distributed Estimation and Control in Networked Systems (NECSYS'13).

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Section: Stochastic Hybrid Systemsmentioning

confidence: 99%

Section: Proof Of Theoremmentioning

confidence: 99%

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Networked Control Systems

Wang

Liu

2008

207

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“…If all quantified variables have a definite value, i.e. the current prefix is empty, we can execute the choice rule (15).…”

Section: Algorithm For Ssmtmentioning

confidence: 99%

“…In full generality, stochastic hybrid system (SHS) models can cover all such ingredients [16,6]. While such models have a vast potential of application, results related to their analysis and verification are limited, and often based on Monte-Carlo simulation [4,15]. For certain subclasses of piecewise deterministic Markov processes, of probabilistic hybrid automata, and of stochastic hybrid systems, reachability probabilities can be approximated (e.g.…”

Section: Introductionmentioning

confidence: 99%

Stochastic Satisfiability Modulo Theory: A Novel Technique for the Analysis of Probabilistic Hybrid Systems

Fränzle

Hermanns

Teige

Hybrid Systems: Computation and Control

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Abstract. The analysis of hybrid systems exhibiting probabilistic behaviour is notoriously difficult. To enable mechanised analysis of such systems, we extend the reasoning power of arithmetic satisfiability-modulotheory solving (SMT) by a comprehensive treatment of randomized (a.k.a. stochastic) quantification over discrete variables within the mixed Booleanarithmetic constraint system. This provides the technological basis for a fully symbolic analysis of probabilistic hybrid automata. Generalizing SMT-based bounded model-checking of hybrid automata [2, 11], stochastic SMT permits the direct and fully symbolic analysis of probabilistic bounded reachability problems of probabilistic hybrid automata without resorting to approximation by intermediate finite-state abstractions.

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Stochastic models for chemically reacting systems using polynomial stochastic hybrid systems

Hespanha

Singh

2005

Intl J Robust & Nonlinear

Self Cite

152

173

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Abstract. A stochastic model for chemical reactions is presented, which represents the population of various species involved in a chemical reaction as the continuous state of a polynomial Stochastic Hybrid System (pSHS). pSHSs correspond to stochastic hybrid systems with polynomial continuous vector fields, reset maps, and transition intensities. We show that for pSHSs, the dynamics of the statistical moments of its continuous states, evolves according to infinite-dimensional linear ordinary differential equations (ODEs), which can be approximated by finite-dimensional nonlinear ODEs with arbitrary precision. Based on this result, a procedure to build this types of approximation is provided.This procedure is used to construct approximate stochastic models for a variety of chemical reactions that have appeared in literature. These reactions include a simple bimolecular reaction, for which one can solve the master equation; a decaying-dimerizing reaction set which exhibits two distinct time scales; a reaction for which the chemical rate equations have a continuum of equilibrium points; and the bistable Schögl reaction. The accuracy of the approximate models is investigated by comparing with Monte Carlo simulations or the solution to the Master equation, when available.

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Polynomial Stochastic Hybrid Systems

Cited by 32 publications

References 11 publications

Networked Control Systems

Networked Control Systems

Stochastic Satisfiability Modulo Theory: A Novel Technique for the Analysis of Probabilistic Hybrid Systems

Stochastic models for chemically reacting systems using polynomial stochastic hybrid systems

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