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Modeling and analysis of stochastic hybrid systems
 IEE Proc — Control Theory & Applications, Special Issue on Hybrid Systems 153(5
, 2007
"... The author describes a model for Stochastic Hybrid Systems (SHSs) where transitions between discrete modes are triggered by stochastic events. The rate at which these transitions occur is allowed to depend both on the continuous and the discrete states of the SHS. Several examples of SHSs arising fr ..."
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The author describes a model for Stochastic Hybrid Systems (SHSs) where transitions between discrete modes are triggered by stochastic events. The rate at which these transitions occur is allowed to depend both on the continuous and the discrete states of the SHS. Several examples of SHSs arising from a varied pool of application areas are discussed. These include modeling of the Transmission Control Protocol’s (TCP) algorithm for congestion control both for longlived and onoff flows; stateestimation for networked control systems; and the stochastic modeling of chemical reactions. These examples illustrate the use of SHSs as a modeling tool. Attention is mostly focused on polynomial stochastic hybrid systems (pSHSs) that generally correspond to stochastic hybrid systems with polynomial continuous vector fields, reset maps, and transition intensities. For pSHSs, the dynamics of the statistical moments of the continuous states evolve according to infinitedimensional linear ordinary differential equations (ODEs). We show that these ODEs can be approximated by finitedimensional nonlinear ODEs with arbitrary precision. Based on this result, a procedure to build this type of approximations for certain classes of pSHSs is provided. This procedure is applied to several examples and the accuracy of the results obtained is evaluated through comparisons with Monte Carlo simulations. I.
Optimal Control of Stochastic Hybrid Systems Based on Locally Consistent Markov Decision Processes
, 2005
"... This paper applies a known approach for approximating controlled stochastic diffusion to hybrid systems. Stochastic hybrid systems are approximated by locally consistent Markov decision processes that preserve local mean and covariance. A randomized switching policy is introduced for approximating ..."
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Cited by 16 (7 self)
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This paper applies a known approach for approximating controlled stochastic diffusion to hybrid systems. Stochastic hybrid systems are approximated by locally consistent Markov decision processes that preserve local mean and covariance. A randomized switching policy is introduced for approximating the dynamics on the switching boundaries. The validity of the approximation is shown by solving the optimal control problem of minimizing a cost until a target set is reached using dynamic programming. It is shown that using the randomized switching policy, the solution obtained based on the discrete approximation converges to the solution of the original problem.
Computational Methods for Verification of Stochastic Hybrid Systems
 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS  PART A
, 2008
"... Stochastic hybrid system (SHS) models can be used to analyze and design complex embedded systems that operate in the presence of uncertainty and variability. Verification of reachability properties for such systems is a critical problem. Developing sound computational methods for verification is ch ..."
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Cited by 14 (5 self)
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Stochastic hybrid system (SHS) models can be used to analyze and design complex embedded systems that operate in the presence of uncertainty and variability. Verification of reachability properties for such systems is a critical problem. Developing sound computational methods for verification is challenging because of the interaction between the discrete and the continuous stochastic dynamics. In this paper, we propose a probabilistic method for verification of SHSs based on discrete approximations focusing on reachability and safety problems. We show that reachability and safety can be characterized as a viscosity solution of a system of coupled Hamilton–Jacobi–Bellman equations. We present a numerical algorithm for computing the solution based on discrete approximations that are derived using finitedifference methods. An advantage of the method is that the solution converges to the one for the original system as the discretization becomes finer. We also prove that the algorithm is polynomial in the number of states of the discrete approximation. Finally, we illustrate the approach with two benchmarks: a navigation and a room heater example, which have been proposed for hybrid system verification.
R.: Specification and analysis of distributed objectbased stochastic hybrid systems
 In: HSCC
, 2006
"... Abstract. In practice, many stochastic hybrid systems are not autonomous: they are objects that communicate with other objects by exchanging messages through an asynchronous medium such as a network. Issues such as: how to compositionally specify distributed objectbased stochastic hybrid systems ..."
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Abstract. In practice, many stochastic hybrid systems are not autonomous: they are objects that communicate with other objects by exchanging messages through an asynchronous medium such as a network. Issues such as: how to compositionally specify distributed objectbased stochastic hybrid systems (OBSHS), how to formally model them, and how to verify their properties seem therefore quite important. This paper addresses these issues by: (i) defining a mathematical model for such systems that can be naturally regarded as a generalized stochastic hybrid system (GSHS) in the sense of [7]; (ii) proposing a formal OBSHS specification language in which system transitions are specified in a modular way by probabilistic rewrite rules; and (iii) showing how these systems can be subjected to statistical model checking analysis to verify their probabilistic temporal logic properties. 1
Polynomial stochastic hybrid systems
 In: Hybrid Systems : Computation and Control (HSCC) 2005
, 2005
"... Abstract. This paper deals with polynomial stochastic hybrid systems (pSHSs), which generally correspond to stochastic hybrid systems with polynomial continuous vector fields, reset maps, and transition intensities. For pSHSs, the dynamics of the statistical moments of the continuous states evolve a ..."
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Cited by 12 (4 self)
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Abstract. This paper deals with polynomial stochastic hybrid systems (pSHSs), which generally correspond to stochastic hybrid systems with polynomial continuous vector fields, reset maps, and transition intensities. For pSHSs, the dynamics of the statistical moments of the continuous states evolve according to infinitedimensional linear ordinary differential equations (ODEs). We show that these ODEs can be approximated by finitedimensional nonlinear ODEs with arbitrary precision. Based on this result, we provide a procedure to build this type of approximations for certain classes of pSHSs. We apply this procedure for several examples of pSHSs and evaluate the accuracy of the results obtained through comparisons with Monte Carlo simulations. These examples include: the modeling of TCP congestion control both for longlived and onoff flows; stateestimation for networked control systems; and the stochastic modeling of chemical reactions. 1
Approximations of Stochastic Hybrid Systems
, 2009
"... This paper develops a notion of approximation for a class of stochastic hybrid systems that includes, as special cases, both jump linear stochastic systems and linear stochastic hybrid automata. Our approximation framework is based on the recently developed notion of the socalled stochastic simulat ..."
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Cited by 12 (0 self)
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This paper develops a notion of approximation for a class of stochastic hybrid systems that includes, as special cases, both jump linear stochastic systems and linear stochastic hybrid automata. Our approximation framework is based on the recently developed notion of the socalled stochastic simulation functions. These Lyapunovlike functions can be used to rigorously quantify the distance or error between a system and its approximate abstraction. For the class of jump linear stochastic systems and linear stochastic hybrid automata, we show that the computation of stochastic simulation functions can be cast as a tractable linear matrix inequality problem. This enables us to compute the modeling error incurred by abstracting some of the continuous dynamics, or by neglecting the influence of stochastic noise, or even the influence of stochastic discrete jumps.
Cairano,“Optimal Control of Discrete Hybrid Stochastic Automata
 Hybrid Systems: Computation and Control, number 3414
, 2005
"... Abstract. This paper focuses on hybrid systems whose discrete state transitions depend on both deterministic and stochastic events. For such systems, after introducing a suitable hybrid model called Discrete Hybrid Stochastic Automaton (DHSA), different finitetime optimal control approaches are exa ..."
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Cited by 11 (1 self)
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Abstract. This paper focuses on hybrid systems whose discrete state transitions depend on both deterministic and stochastic events. For such systems, after introducing a suitable hybrid model called Discrete Hybrid Stochastic Automaton (DHSA), different finitetime optimal control approaches are examined: (1) Stochastic Hybrid Optimal Control (SHOC), which “optimistically ” determines the trajectory providing the best trade off between the tracking performance and the probability that stochastic events realize as expected, under specified chance constraints; (2) Robust Hybrid Optimal Control (RHOC) which, in addition, less optimistically, ensures that the system remains within a specified safety region for all possible realizations of stochastic events. Sufficient conditions for the asymptotic convergence of the state vector are given for recedinghorizon implementations of the above schemes. The proposed approaches are exemplified on a simple benchmark problem in production system management. 1
Adjointbased optimal control of the expected exit time for stochastic hybrid systems
 Hybrid Systems: Computation and Control, 8th Int. Workshop (HSCC 2005), volume 3414 of Lecture Notes in Computer Science
, 2005
"... Abstract. In this paper, we study the problem of controlling the expected exit time from a region for a class of stochastic hybrid systems. That is, we find the least costly feedback control for a stochastic hybrid system that can keep its state inside a prescribed region for at least an expected am ..."
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Cited by 8 (4 self)
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Abstract. In this paper, we study the problem of controlling the expected exit time from a region for a class of stochastic hybrid systems. That is, we find the least costly feedback control for a stochastic hybrid system that can keep its state inside a prescribed region for at least an expected amount of time. The stochastic hybrid systems considered are quite general: the continuous dynamics are governed by stochastic differential equations, and the discrete mode evolves according to a continuous time Markov chain. Instead of adopting the usual HamiltonJacobi viewpoint, we study the problem directly by formulating it as a PDE constrained optimization problem, and propose a solution using adjointbased gradient descent methods. Numerical results of the proposed approach are presented for several representative examples, and, for the simple case, compared with analytical results. 1
Stochastic equilibria of AIMD communication networks
 SIAM JOURNAL OF MATRIX ANALYSIS
, 2006
"... In this paper tools are developed to analyse a recently proposed random matrix model of communication networks that employ additiveincrease multiplicativedecrease (AIMD) congestion control algorithms. We investigate properties of the Markov process describing the evolution of the window sizes of ..."
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Cited by 7 (4 self)
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In this paper tools are developed to analyse a recently proposed random matrix model of communication networks that employ additiveincrease multiplicativedecrease (AIMD) congestion control algorithms. We investigate properties of the Markov process describing the evolution of the window sizes of network users. Using paracontractivity properties of the matrices involved in the model, it is shown that the process has a unique invariant probability, and the support of this probability is characterized. Based on these results we obtain a weak law of large numbers for the average distribution of resources between the users of a network. This shows that under reasonable assumptions such networks have a welldefined stochastic equilibrium. ns2 simulation results are discussed to validate the obtained formulae. (The simulation program ns2, or network simulator, is an industry standard for the simulation of Internet dynamics.)