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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 25 (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
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.
M.: Approximate abstractions of stochastic hybrid systems
 IEEE Transactions on Automatic Control
"... Abstract—We present a constructive procedure for obtaining a finite approximate abstraction of a discretetime stochastic hybrid system. The procedure consists of a partition of the state space of the system and depends on a controllable parameter. Given proper continuity assumptions on the model, t ..."
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Cited by 11 (3 self)
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Abstract—We present a constructive procedure for obtaining a finite approximate abstraction of a discretetime stochastic hybrid system. The procedure consists of a partition of the state space of the system and depends on a controllable parameter. Given proper continuity assumptions on the model, the approximation errors introduced by the abstraction procedure are explicitly computed and it is shown that they can be tuned through the parameter of the partition. The abstraction is interpreted as a Markov setChain. We show that the enforcement of certain ergodic properties on the stochastic hybrid model implies the existence of a finite abstraction with finite error in time over the concrete model, and allows introducing a finitetime algorithm that computes the abstraction. Index Terms—Stochastic Hybrid Systems, Markov Chains. I. INTRODUCTION AND RELATED WORK The study of complex, heterogeneous, and probabilistic models
Safety Analysis of Sugar Cataract Development Using Stochastic Hybrid Systems
 HYBRID SYSTEMS: COMPUTATION AND CONTROL 2007 LNCS 4416
, 2007
"... Modeling and analysis of biochemical systems are critical problems because they can provide new insights into systems which can not be easily tested with real experiments. One such biochemical process is the formation of sugar cataracts in the lens of an eye. Analyzing the sugar cataract developmen ..."
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Cited by 6 (5 self)
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Modeling and analysis of biochemical systems are critical problems because they can provide new insights into systems which can not be easily tested with real experiments. One such biochemical process is the formation of sugar cataracts in the lens of an eye. Analyzing the sugar cataract development process is a challenging problem due to the highlycoupled chemical reactions that are involved. In this paper we model sugar cataract development as a stochastic hybrid system. Based on this model, we present a probabilistic verification method for computing the probability of sugar cataract formation for different chemical concentrations. Our analysis can potentially provide useful insights into the complicated dynamics of the process and assist in focusing experiments on specific regions of concentrations. The verification method employs dynamic programming based on a discretization of the state space and therefore suffers from the curse of dimensionality. To verify the sugar cataract development process we have developed a parallel dynamic programming implementation that can handle large systems. Although scalability is a limiting factor, this work demonstrates that the technique is feasible for realistic biochemical systems.
Modelling and analysis of the sugar cataract development process using stochastic hybrid systems
 IET SYSTEMS BIOLOGY
, 2008
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Approximation Metrics based on Probabilistic Bisimulations for General StateSpace Markov Processes: a Survey
 HAS 2011
, 2011
"... This article provides a survey of approximation metrics for stochastic processes. We deal with Markovian processes in discrete time evolving on general state spaces, namely on domains with infinite cardinality and endowed with proper measurability and metric structures. The focus of this work is to ..."
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Cited by 3 (0 self)
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This article provides a survey of approximation metrics for stochastic processes. We deal with Markovian processes in discrete time evolving on general state spaces, namely on domains with infinite cardinality and endowed with proper measurability and metric structures. The focus of this work is to discuss approximation metrics between two such processes, based on the notion of probabilistic bisimulation: in particular we investigate metrics characterized by an approximate variant of this notion. We suggests that metrics between two processes can be introduced essentially in two distinct ways: the first employs the probabilistic conditional kernels underlying the two stochastic processes under study, and leverages notions derived from algebra, logic, or category theory; whereas the second looks at distances between trajectories of the two processes, and is based on the dynamical properties of the two processes (either their syntax, via the notion of bisimulation function; or their semantics, via sampling techniques). The survey moreover covers the problem of constructing formal approximations of stochastic processes according to the introduced metrics.
K.: Verification of biochemical processes using stochastic hybrid systems
 In: Intelligent Control
, 2007
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Optimal Eventtriggered Control under Costly Observations
"... Abstract — Digital control design is commonly constrained to timetriggered control systems with equidistant sampling intervals. The emergence of more and more complex and distributed systems urges the development of advanced triggering schemes that utilize computational and communication resources ..."
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Abstract — Digital control design is commonly constrained to timetriggered control systems with equidistant sampling intervals. The emergence of more and more complex and distributed systems urges the development of advanced triggering schemes that utilize computational and communication resources efficiently. This paper considers a linear stochastic continuoustime setting, where the design objective is to find an eventtriggered controller that optimally meets the tradeoff between control performance and resource utilization. This is reflected by imposing a cost penalty on updating the controller by current observations that is added to a quadratic control cost. It is shown that the underlying optimization problem results in an eventtriggered controller, where the controller is updated, when the estimation error of the controller exceeds a apriori determined threshold. The controller design is related to linear quadratic Gaussian regulation and to optimal stopping time problems. Contrary to the initial problem, these can be solved by standard methods of stochastic optimal control. Numerical examples underline the effectiveness compared to optimal timetriggered controllers.
Stochastic reachability as an exit problem
 17th Mediterranean Conference on Control and Automation, 2009
"... Abstract — For stochastic hybrid systems, safety verification methods are very little supported mainly because of complexity and difficulty of the associated mathematical problems. The key of the methods that succeeded in solving various instances of this problem is to prove the equivalence of these ..."
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Abstract — For stochastic hybrid systems, safety verification methods are very little supported mainly because of complexity and difficulty of the associated mathematical problems. The key of the methods that succeeded in solving various instances of this problem is to prove the equivalence of these instances with known problems. In this paper, we apply the same pattern to the most general model of stochastic hybrid systems. Stochastic reachability problem can be treated as an exit problem for a suitable class of Markov processes. The solutions of this problem can be characterised using Hamilton Jacobi theory. I.
Reachability Analysis of a Biodiesel Production System Using Stochastic Hybrid Systems
"... Abstract — Modeling and analysis of chemical reactions are critical problems because they can provide new insights into the complex interactions between systems of reactions and chemicals. One such set of chemical reactions defines the creation of biodiesel from soybean oil and methanol. Modeling an ..."
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Abstract — Modeling and analysis of chemical reactions are critical problems because they can provide new insights into the complex interactions between systems of reactions and chemicals. One such set of chemical reactions defines the creation of biodiesel from soybean oil and methanol. Modeling and analyzing the biodiesel creation process is a challenging problem due to the highlycoupled chemical reactions that are involved. In this paper we model a biodiesel production system as a stochastic hybrid system, and we present a probabilistic verification method for reachability analysis. Our analysis can potentially provide useful insights into the complicated dynamics of the chemicals and assist in focusing experiments and tuning the production system for efficiency. The verification method employs dynamic programming based on a discretization of the state space and therefore suffers from the curse of dimensionality. To verify the biodiesel system model we have developed a parallel dynamic programming implementation that can handle large systems. Although scalability is a limiting factor, this work demonstrates that the technique is feasible for realistic biochemical systems. I.