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68
Stochastic Analysis of Gene Regulatory Networks Using Moment Closure
, 2007
"... Random fluctuations in gene regulatory networks are inevitable due to the probabilistic nature of chemical reactions and the small populations of proteins, mRNAs present inside cells. These fluctuations are usually reported in terms of the first and second order statistical moments of the protein po ..."
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Cited by 7 (4 self)
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Random fluctuations in gene regulatory networks are inevitable due to the probabilistic nature of chemical reactions and the small populations of proteins, mRNAs present inside cells. These fluctuations are usually reported in terms of the first and second order statistical moments of the protein populations. If the birthdeath rates of the mRNAs or the proteins are nonlinear, then the dynamics of these moments generally do not form a closed system of differential equations, in the sense that their timederivatives depends on moments of order higher than two. Recent work has developed techniques to obtain the two lowestorder moments by closing their dynamics, which involves approximating the higher order moments as nonlinear functions of the two lowest ones. This paper uses these moment closure techniques to quantify noise in several gene regulatory networks. In gene expression mechanisms in which a protein inhibits its own transcription, the resulting negative feedback reduces stochastic variations in the protein populations. Often the protein itself is not active and combines with itself to form an active multimer, which them inhibits the transcription. We demonstrate that this more sophisticated form of negative feedback (using multimerization) is more effective in suppressing noise. We also consider a twogene cascade activation network in which the protein expressed by one gene activates another gene to express a second protein. Analysis shows that the stochastic fluctuations in the population of the activated protein increases with the degree of multimerization in the activating protein.
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.
New Insights on Stochastic Reachability
"... Abstract—In this paper, we give new characterizations of the stochastic reachability problem for stochastic hybrid systems in the language of different theories that can be employed in studying stochastic processes (Markov processes, potential theory, optimal control). These characterizations are fu ..."
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Abstract—In this paper, we give new characterizations of the stochastic reachability problem for stochastic hybrid systems in the language of different theories that can be employed in studying stochastic processes (Markov processes, potential theory, optimal control). These characterizations are further used to obtain the probabilities involved in the context of stochastic reachability as viscosity solutions of some variational inequalities.
Probabilistic reachability for stochastic hybrid systems: Theory, computations, and applications
, 2007
"... Copyright c © 2007 by Alessandro Abate Probabilistic Reachability for Stochastic Hybrid Systems: ..."
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Cited by 4 (0 self)
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Copyright c © 2007 by Alessandro Abate Probabilistic Reachability for Stochastic Hybrid Systems:
Discrete Time Stochastic Hybrid Dynamical Games: Verification & Controller Synthesis
"... This paper presents a framework for analyzing probabilistic safety and reachability problems for discrete time stochastic hybrid systems in scenarios where system dynamics are affected by rational competing agents. In particular, we consider a zerosum game formulation of the probabilistic reachavo ..."
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Cited by 4 (1 self)
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This paper presents a framework for analyzing probabilistic safety and reachability problems for discrete time stochastic hybrid systems in scenarios where system dynamics are affected by rational competing agents. In particular, we consider a zerosum game formulation of the probabilistic reachavoid problem, in which the control objective is to maximize the probability of reaching a desired subset of the hybrid state space, while avoiding an unsafe set, subject to the worstcase behavior of a rational adversary. Theoretical results are provided on a dynamic programming algorithm for computing the maximal reachavoid probability under the worstcase adversary strategy, as well as the existence of a maxmin control policy which achieves this probability. The modeling framework and computational algorithm are demonstrated using an example derived from a robust motion planning application.
Robust stability of the new general 2D model of a class of continuousdiscrete linear systems,
 Bull. Pol. Acad. Sci. Techn.,
, 2010
"... Abstract. The problems of asymptotic stability and robust stability of the new general 2D model of scalar linear dynamic continuousdiscrete systems, standard and positive, are considered. Simple analytic conditions for asymptotic stability and for robust stability are given. These conditions are e ..."
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Cited by 4 (2 self)
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Abstract. The problems of asymptotic stability and robust stability of the new general 2D model of scalar linear dynamic continuousdiscrete systems, standard and positive, are considered. Simple analytic conditions for asymptotic stability and for robust stability are given. These conditions are expressed in terms of coefficients of the model. The considerations are illustrated by numerical examples. The methods proposed can be generalized to scalar FornasiniMarchesini and Roesser models of 2D continuousdiscrete systems.
Communication Logic Design and Analysis for Networked Control Systems
"... Summary. This chapter addresses the control of spatially distributed processes via communication networks with a fixed delay. A distributed architecture is utilized in which multiple local controllers coordinate their efforts through a data network that allows information exchange. We focus our work ..."
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Summary. This chapter addresses the control of spatially distributed processes via communication networks with a fixed delay. A distributed architecture is utilized in which multiple local controllers coordinate their efforts through a data network that allows information exchange. We focus our work on linear time invariant processes disturbed by Gaussian white noise and propose several logics to determine when the local controllers should communicate. Necessary conditions are given under which these logics guarantee boundedness and the tradeoff is investigated between the amount of information exchanged and the performance achieved. The theoretical results are validated through Monte Carlo simulations. The resulting closedloop systems evolve according to stochastic differential equations with resets triggered by stochastic counters. This type of stochastic hybrid system is interesting on its own. 1
STABILIZING RANDOMLY SWITCHED SYSTEMS
, 2008
"... This article is concerned with stability analysis and stabilization of randomly switched systems under a class of switching signals. The switching signal is modeled as a jump stochastic (not necessarily Markovian) process independent of the system state; it selects, at each instant of time, the ac ..."
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This article is concerned with stability analysis and stabilization of randomly switched systems under a class of switching signals. The switching signal is modeled as a jump stochastic (not necessarily Markovian) process independent of the system state; it selects, at each instant of time, the active subsystem from a family of systems. Sufficient conditions for stochastic stability (almost sure, in the mean, and in probability) of the switched system are established when the subsystems do not possess control inputs, and not every subsystem is required to be stable. These conditions are employed to design stabilizing feedback controllers when the subsystems are affine in control. The analysis is carried out with the aid of multiple Lyapunovlike functions, and the analysis results together with universal formulae for feedback stabilization of nonlinear systems constitute our primary tools for control design.
COMPUTER METHODS FOR STABILITY ANALYSIS OF THE ROESSER TYPE MODEL OF 2D CONTINUOUS–DISCRETE LINEAR SYSTEMS
"... Asymptotic stability of models of 2D continuousdiscrete linear systems is considered. Computer methods for investigation of the asymptotic stability of the Roesser type model are given. The methods require computation of eigenvalueloci of complex matrices or evaluation of complex functions. The ef ..."
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Asymptotic stability of models of 2D continuousdiscrete linear systems is considered. Computer methods for investigation of the asymptotic stability of the Roesser type model are given. The methods require computation of eigenvalueloci of complex matrices or evaluation of complex functions. The effectiveness of the stability tests is demonstrated on numerical examples.
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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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