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12
Stochastic Hybrid Systems: Application to Communication Networks
 in Hybrid Systems: Computation and Control, ser. Lect. Notes in Comput. Science
, 2004
"... Abstract. We propose a model for Stochastic Hybrid Systems (SHSs) where transitions between discrete modes are triggered by stochastic events much like transitions between states of a continuoustime Markov chains. However, the rate at which transitions occur is allowed to depend both on the continu ..."
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Cited by 51 (14 self)
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Abstract. We propose a model for Stochastic Hybrid Systems (SHSs) where transitions between discrete modes are triggered by stochastic events much like transitions between states of a continuoustime Markov chains. However, the rate at which transitions occur is allowed to depend both on the continuous and the discrete states of the SHS. Based on results available for PiecewiseDeterministic Markov Process (PDPs), we provide a formula for the extended generator of the SHS, which can be used to compute expectations and the overall distribution of the state. As an application, we construct a stochastic model for onoff TCP flows that considers both the congestionavoidance and slowstart modes and takes directly into account the distribution of the number of bytes transmitted. Using the tools derived for SHSs, we model the dynamics of the moments of the sending rate by an infinite system of ODEs, which can be truncated to obtain an approximate finitedimensional model. This model shows that, for transfersize distributions reported in the literature, the standard deviation of the sending rate is much larger than its average. Moreover, the later seems to vary little with the probability of packet drop. This has significant implications for the design of congestion control mechanisms. 1
A framework for worstcase and stochastic safety verification using barrier certificates
 IEEE TRANSACTIONS ON AUTOMATIC CONTROL
, 2007
"... This paper presents a methodology for safety verification of continuous and hybrid systems in the worstcase and stochastic settings. In the worstcase setting, a function of state termed barrier certificate is used to certify that all trajectories of the system starting from a given initial set do ..."
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Cited by 28 (1 self)
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This paper presents a methodology for safety verification of continuous and hybrid systems in the worstcase and stochastic settings. In the worstcase setting, a function of state termed barrier certificate is used to certify that all trajectories of the system starting from a given initial set do not enter an unsafe region. No explicit computation of reachable sets is required in the construction of barrier certificates, which makes it possible to handle nonlinearity, uncertainty, and constraints directly within this framework. In the stochastic setting, our method computes an upper bound on the probability that a trajectory of the system reaches the unsafe set, a bound whose validity is proven by the existence of a barrier certificate. For polynomial systems, barrier certificates can be constructed using convex optimization, and hence the method is computationally tractable. Some examples are provided to illustrate the use of the method.
A model for stochastic hybrid systems with application to communication networks
 Nonlinear Analysis Special Issue on Hybrid Systems
, 2004
"... Abstract. We propose a model for Stochastic Hybrid Systems (SHSs) where transitions between discrete modes are triggered by stochastic events much like transitions between states of a continuoustime Markov chains. However, the rate at which transitions occur is allowed to depend both on the continuo ..."
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Cited by 23 (10 self)
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Abstract. We propose a model for Stochastic Hybrid Systems (SHSs) where transitions between discrete modes are triggered by stochastic events much like transitions between states of a continuoustime Markov chains. However, the rate at which transitions occur is allowed to depend both on the continuous and the discrete states of the SHS. Based on results available for PiecewiseDeterministic Markov Process (PDPs), we provide a formula for the extended generator of the SHS, which can be used to compute expectations and the overall distribution of the state. As an application, we construct a stochastic model for onoff TCP flows that considers both the congestionavoidance and slowstart modes and takes directly into account the distribution of the number of bytes transmitted. Using the tools derived for SHSs, we model the dynamics of the moments of the sending rate by an infinite system of ODEs, which can be truncated to obtain an approximate finitedimensional model. This model shows that, for transfersize distributions reported in the literature, the standard deviation of the sending rate is much larger than its average. Moreover, the later seems to vary little with the probability of packet drop. This has significant implications for the design of congestion control mechanisms.
Safety verification using barrier certificates
 In HSCC, volume 2993 of LNCS
, 2004
"... Abstract — We develop a new method for safety verification of stochastic systems based on functions of states termed barrier certificates. Given a stochastic continuous or hybrid system and sets of initial and unsafe states, our method computes an upper bound on the probability that a trajectory of ..."
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Cited by 18 (4 self)
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Abstract — We develop a new method for safety verification of stochastic systems based on functions of states termed barrier certificates. Given a stochastic continuous or hybrid system and sets of initial and unsafe states, our method computes an upper bound on the probability that a trajectory of the system reaches the unsafe set, a bound whose validity is proven by the existence of a barrier certificate. For polynomial systems, both the upper bound and its corresponding barrier certificate can be computed using convex optimization, and hence the method is computationally tractable. I.
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 8 (3 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
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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Cited by 7 (5 self)
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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.
Stochastic Hybrid Systems with Renewal Transitions
, 2009
"... We consider Stochastic Hybrid Systems (SHSs) for which the lengths of times that the system stays in each mode are independent random variables with given distributions. We propose an analysis framework based on a set of Volterra renewaltype equations, which allows us to compute any statistical mom ..."
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Cited by 4 (3 self)
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We consider Stochastic Hybrid Systems (SHSs) for which the lengths of times that the system stays in each mode are independent random variables with given distributions. We propose an analysis framework based on a set of Volterra renewaltype equations, which allows us to compute any statistical moment of the state of the SHS. Moreover, we provide necessary and sufficient conditions for various stability notions, and determine the exponential decay or increase rate at which the expected value of the energy of the system converges to zero or to infinity, respectively. The applicability of the results is illustrated in a networked control problem considering independently distributed intervals between data transmissions and delays. 1
Polynomial stochastic hybrid systems (extended version
, 2004
"... 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 2 (1 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
BMC Systems Biology BioMed Central Research article Hybrid stochastic simplifications for multiscale gene networks
, 2009
"... This is an Open Access article distributed under the terms of the Creative Commons Attribution License ..."
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This is an Open Access article distributed under the terms of the Creative Commons Attribution License
Approximations of Stochastic Hybrid Systems
"... Abstract—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 stochasti ..."
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Abstract—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. Index Terms—Approximation, bisimulation, stochastic hybrid systems, verification.