Results 1 - 10 of 36

A Toolbox of Hamilton-Jacobi Solvers for Analysis of Nondeterministic Continuous and Hybrid Systems

by Ian M. Mitchell, Jeremy A. Templeton - In HSCC 2005, LNCS 3414 , 2005
"... Submitted to HSCC 2005. Please do not redistribute Abstract. Hamilton-Jacobi partial differential equations have many applications in the analysis of nondeterministic continuous and hybrid systems. Unfortunately, analytic solutions are seldom available and numerical approximation requires a great de ..."
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Submitted to HSCC 2005. Please do not redistribute Abstract. Hamilton-Jacobi partial differential equations have many applications in the analysis of nondeterministic continuous and hybrid systems. Unfortunately, analytic solutions are seldom available and numerical approximation requires a great deal of programming infrastructure. In this paper we describe the first publicly available toolbox for approximating the solution of such equations, and discuss three examples of how these equations can be used in systems analysis: cost to go, stochastic differential games, and stochastic hybrid systems. For each example we briefly summarize the relevant theory, describe the toolbox implementation, and provide results. 1
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Modeling and analysis of stochastic hybrid systems

by João P. Hespanha - 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 ..."
Abstract - Cited by 17 (7 self) - Add to MetaCart
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 long-lived and on-off flows; state-estimation 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 infinite-dimensional linear ordinary differential equations (ODEs). We show that these ODEs can be approximated by finite-dimensional 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.
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Stability analysis of deterministic and stochastic switched systems via a comparison principle and multiple lyapunov functions

by Debasish Chatterjee, Daniel Liberzon - SIAM Journal on Control and Optimization , 2004
"... Abstract. This paper presents a general framework for analyzing stability of nonlinear switched systems, by combining the method of multiple Lyapunov functions with a suitably adapted comparison principle in the context of stability in terms of two measures. For deterministic switched systems, this ..."
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Abstract. This paper presents a general framework for analyzing stability of nonlinear switched systems, by combining the method of multiple Lyapunov functions with a suitably adapted comparison principle in the context of stability in terms of two measures. For deterministic switched systems, this leads to a unification of representative existing results and an improvement upon the current scope of the method of multiple Lyapunov functions. For switched systems perturbed by white noise, we develop new results which may be viewed as natural stochastic counterparts of the deterministic ones. In particular, we study stability of deterministic and stochastic switched systems under average dwell-time switching.

Polynomial stochastic hybrid systems

by João Pedro Hespanha - 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 ..."
Abstract - Cited by 12 (4 self) - Add to MetaCart
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 infinite-dimensional linear ordinary differential equations (ODEs). We show that these ODEs can be approximated by finite-dimensional 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 long-lived and on-off flows; state-estimation for networked control systems; and the stochastic modeling of chemical reactions. 1
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Stochastic Hybrid Systems with Renewal Transitions

by Duarte Antunes, João P. Hespanha, Carlos Silvestre , 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 renewal-type equations, which allows us to compute any statistical mom ..."
Abstract - Cited by 11 (5 self) - Add to MetaCart
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 renewal-type 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
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Generalized fokker-planck equation for piecewise-diffusion processes with boundary hitting resets

by J. Bect, H. Baili, G. Fleury - in: Proceedings of the International Symposium on Mathematical Theory of Networks and Systems (MTNS 2006), 2006, p. Paper WeA05.6
"... Abstract. This paper is concerned with the generalized Fokker-Planck equa-tion for a class of stochastic hybrid processes, where diffusion and instanta-neous jumps at the boundary are allowed. The state of the process after a jump is defined by a deterministic reset map. We establish a partial diffe ..."
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Abstract. This paper is concerned with the generalized Fokker-Planck equa-tion for a class of stochastic hybrid processes, where diffusion and instanta-neous jumps at the boundary are allowed. The state of the process after a jump is defined by a deterministic reset map. We establish a partial differen-tial equation for the probability density function, which is a generalisation of the usual Fokker-Planck equation for diffusion processes. The result involves a non-local boundary condition, which accounts for the jumping behaviour of the process, and an absorbing boundary condition on the non-characteristic part of the boundary. Two applications are given, with numerical results obtained by finite volume discretization. 1.
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A unifying formulation of the Fokker-Planck-Kolmogorov equation for general stochastic hybrid systems

by Julien Bect - In Proceedings of the 17th IFAC World Congress , 2008
"... A general formulation of the Fokker-Planck-Kolmogorov (FPK) equation for stochastic hybrid systems is presented, within the framework of Generalized Stochastic Hybrid Systems (GSHS). The FPK equation describes the time evolution of the probability law of the hybrid state. Our derivation is based on ..."
Abstract - Cited by 6 (0 self) - Add to MetaCart
A general formulation of the Fokker-Planck-Kolmogorov (FPK) equation for stochastic hybrid systems is presented, within the framework of Generalized Stochastic Hybrid Systems (GSHS). The FPK equation describes the time evolution of the probability law of the hybrid state. Our derivation is based on the concept of mean jump intensity, which is related to both the usual stochastic intensity (in the case of spontaneous jumps) and the notion of probability current (in the case of forced jumps). This work unifies all previously known instances of the FPK equation for stochastic hybrid systems, and provides GSHS practitioners with a tool to derive the correct evolution equation for the probability law of the state in any given example. Key words: Stochastic hybrid systems, Stochastic system with jumps, Markov processes, Fokker-Planck equation 1.

Stability analysis and stabilization of randomly switched systems

by Debasish Chatterjee, Daniel Liberzon - IN PROCEEDINGS OF THE 45TH IEEE CONFERENCE ON DECISION AND CONTROL , 2006
"... This article is concerned with stability analysis and stabilization of randomly switched systems with control inputs. The switching signal is modeled as a jump stochastic process independent of the system state; it selects, at each instant of time, the active subsystem from among a family of determi ..."
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This article is concerned with stability analysis and stabilization of randomly switched systems with control inputs. The switching signal is modeled as a jump stochastic process independent of the system state; it selects, at each instant of time, the active subsystem from among a family of deterministic systems. Three different types of switching signals are considered: the first is a jump stochastic process that satisfies a statistically slow switching condition; the second and the third are jump stochastic processes with independent identically distributed values at jump times together with exponential and uniform holding times, respectively. For each of the three cases we first establish sufficient conditions for stochastic stability of the switched system, when the subsystems do not possess control inputs, and are not all stable. Thereafter we design feedback controllers by employing our analysis results such that the switched control system is stable in closed loop, when subsystems are affine in control. Multiple Lyapunov functions and Sontag’s universal formulae for feedback stabilization of nonlinear systems constitute the primary tools for analysis and control design.
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Modeling Communication Networks With Hybrid Systems

by Junsoo Lee, Stephan Bohacek, João P. Hespanha, Senior Member, Katia Obraczka
"... Abstract—This paper introduces a general hybrid systems framework to model the flow of traffic in communication networks. The proposed models use averaging to continuously approximate discrete variables such as congestion window and queue size. Because averaging occurs over short time intervals, dis ..."
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Abstract—This paper introduces a general hybrid systems framework to model the flow of traffic in communication networks. The proposed models use averaging to continuously approximate discrete variables such as congestion window and queue size. Because averaging occurs over short time intervals, discrete events such as the occurrence of a drop and the consequent reaction by congestion control can still be captured. This modeling framework, thus, fills a gap between purely packet-level and fluid-based models, faithfully capturing the dynamics of transient phenomena and yet providing significant flexibility in modeling various congestion control mechanisms, different queueing policies, multicast transmission, etc. The modeling framework is validated by comparing simulations of the hybrid models against packet-level simulations. It is shown that the probability density functions produced by thens-2 network simulator match closely those obtained with hybrid models. Moreover, a complexity analysis supports the observation that in networks with large per-flow bandwidths, simulations using hybrid models require significantly less computational resources thanns-2 simulations. Tools developed to automate the generation and simulation of hybrid systems models are also presented. Their use is showcased in a study, which simulates TCP flows with different roundtrip times over the Abilene backbone. Index Terms—Congestion control, data communication networks, hybrid systems, simulation, TCP, UDP. I.
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On stability of randomly switched nonlinear systems

by Debasish Chatterjee, Daniel Liberzon - IEEE Transactions on Automatic Control
"... Abstract. This article is concerned with stability analysis and stabilization of randomly switched nonlinear systems. These systems may be regarded as piecewise deterministic stochastic systems: the discrete switches are triggered by a stochastic process which is independent of the state of the syst ..."
Abstract - Cited by 3 (1 self) - Add to MetaCart
Abstract. This article is concerned with stability analysis and stabilization of randomly switched nonlinear systems. These systems may be regarded as piecewise deterministic stochastic systems: the discrete switches are triggered by a stochastic process which is independent of the state of the system, and between two consecutive switching instants the dynamics are deterministic. Our results provide sufficient conditions for almost sure global asymptotic stability using Lyapunov-based methods when individual subsystems are stable and a certain “slow switching” condition holds. This slow switching condition takes the form of an asymptotic upper bound on the probability mass function of the number of switches that occur between the initial and current time instants. This condition is shown to hold for switching signals coming from the states of finite-dimensional continuous-time Markov chains; our results therefore hold for Markov jump systems in particular. For systems with control inputs we provide explicit control schemes for feedback stabilization using the universal formula for stabilization of nonlinear systems.
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