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71
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 53 (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
Mode estimation of probabilistic hybrid systems
 In Intl. Conf. on Hybrid Systems: Computation and Control
, 2002
"... Abstract. Modelbased diagnosis and mode estimation capabilities excel at diagnosing systems whose symptoms are clearly distinguished from normal behavior. A strength of mode estimation, in particular, is its ability to track a system’s discrete dynamics as it moves between different behavioral mode ..."
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Cited by 53 (16 self)
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Abstract. Modelbased diagnosis and mode estimation capabilities excel at diagnosing systems whose symptoms are clearly distinguished from normal behavior. A strength of mode estimation, in particular, is its ability to track a system’s discrete dynamics as it moves between different behavioral modes. However, often failures bury their symptoms amongst the signal noise, until their effects become catastrophic. We introduce a hybrid mode estimation system that extracts mode estimates from subtle symptoms. First, we introduce a modeling formalism, called concurrent probabilistic hybrid automata (cPHA), that merge hidden Markov models (HMM) with continuous dynamical system models. Second, we introduce hybrid estimation as a method for tracking and diagnosing cPHA, by unifying traditional continuous state observers with HMM belief update. Finally, we introduce a novel, anytime, anyspace algorithm for computing approximate hybrid estimates. 1
Extended stochastic hybrid systems and their reachability problem
 in Hybrid Systems: Computation and Control, LNCS 2993
, 2004
"... Abstract. In this paper we generalize a model for stochastic hybrid systems. First, we prove that this model is a right Markov process and it satisfies some mathematical properties. Second, we propose a method based on the theory of Dirichlet forms to study the reachability problem associated with t ..."
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Cited by 30 (5 self)
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Abstract. In this paper we generalize a model for stochastic hybrid systems. First, we prove that this model is a right Markov process and it satisfies some mathematical properties. Second, we propose a method based on the theory of Dirichlet forms to study the reachability problem associated with these systems.
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.
General stochastic hybrid systems: Modelling and optimal control
 in Proc. 43rd IEEE Conf. Decision Control
, 2004
"... Abstract — We develop a model for General Stochastic Hybrid Systems (GSHS) which is a generalization of PiecewiseDeterministic Markov Processes (PDMP), introduced by Davis and stochastic hybrid systems proposed by Hu, Lygeros and Sastry. This model possesses certain desirable properties, as the str ..."
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Cited by 26 (2 self)
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Abstract — We develop a model for General Stochastic Hybrid Systems (GSHS) which is a generalization of PiecewiseDeterministic Markov Processes (PDMP), introduced by Davis and stochastic hybrid systems proposed by Hu, Lygeros and Sastry. This model possesses certain desirable properties, as the strong Markov property and the càdlàg property. Extending results available for PDMP, we develop the extended generator formula and the differential formula for GSHS. Then we investigate the dynamic programming for GSHS, using the differential formula. I.
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 ..."
Abstract

Cited by 24 (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.
Computational Methods for Reachability Analysis of Stochastic Hybrid
 Systems, Hybrid Systems: Computation and Control 2006 LNCS 3927
, 2006
"... Abstract. Stochastic hybrid system 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 algorithms for reachability analysis is challen ..."
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Cited by 20 (8 self)
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Abstract. Stochastic hybrid system 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 algorithms for reachability analysis is challenging because of the interaction between the discrete and continuous stochastic dynamics. In this paper, we propose a probabilistic method for reachability analysis based on discrete approximations. The contribution of the paper is twofold. First, we show that reachability can be characterized as a viscosity solution of a system of coupled HamiltonJacobiBellman equations. Second, we present a numerical method for computing the solution based on discrete approximations and we show that this solution converges to the one for the original system as the discretization becomes finer. Finally, we illustrate the approach with a navigation benchmark that has been proposed for hybrid system verification. 1
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 19 (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.
Stochastic hybrid models: An overview
 In Proceedings IFAC Conference on Analysis and Design of Hybrid Systems
, 2003
"... Abstract: An overview of Stochastic Hybrid Models developed in the literature is presented. Attention is concentrated on three classes of models: Piecewise Deterministic Markov Processes, Switching Diffusion Processes and Stochastic Hybrid Systems. The descriptive power of the three classes is compa ..."
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Cited by 17 (0 self)
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Abstract: An overview of Stochastic Hybrid Models developed in the literature is presented. Attention is concentrated on three classes of models: Piecewise Deterministic Markov Processes, Switching Diffusion Processes and Stochastic Hybrid Systems. The descriptive power of the three classes is compared and conditions under which the classes coincide are developed. The theoretical analysis is motivated by modelling problems in Air Traffic Management. Copyright, 2003, IFAC
Modeling Subtilin Production in Bacillus subtilis Using Stochastic Hybrid Systems
 Hybrid Systems: Computation and Control 2004, LNCS
, 2004
"... Abstract. The genetic network regulating the biosynthesis of subtilin in Bacillus subtilis is modeled as a stochastic hybrid system. The continuous state of the hybrid system is the concentrations of subtilin and various regulating proteins, whose productions are controlled by switches in the geneti ..."
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Cited by 16 (0 self)
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Abstract. The genetic network regulating the biosynthesis of subtilin in Bacillus subtilis is modeled as a stochastic hybrid system. The continuous state of the hybrid system is the concentrations of subtilin and various regulating proteins, whose productions are controlled by switches in the genetic network that are in turn modeled as Markov chains. Some preliminary results are given by both analysis and simulations. 1 Background of Subtilin Production In order to survive, bacteria develop a number of strategies to cope with harsh environmental conditions. One of the survival strategies employed by bacteria is the release of antibiotics to eliminate competing microbial species in the same ecosystem [15]. It is observed that the production of antibiotics in the cells is affected by not only the environmental stimuli (e.g. nutrient levels, aeration, etc.) but also the local population density of their own species [12]. Therefore, the physiological states of the cell and the external signals both contribute to the regulation of antibiotic synthesis. Our study focuses on the subtilin, an antibiotic