Stochastic Hybrid Models: An Overview

Pola, Giordano; Bujorianu, Manuela L.; Lygeros, John; Benedetto, Maria

doi:10.1016/s1474-6670(17)36405-4

Cited by 83 publications

(61 citation statements)

References 2 publications

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“…The transition rates of the Markov process can depend on the state but transitions do not generate jumps on the continuous state (i.e., no resets). The reader is referred to [42] for a comparison of the models in [14,18,21]. The SHSs considered here can be viewed as special cases of general jump-diffusion processes [24].…”

Section: Introductionmentioning

confidence: 99%

Stochastic Hybrid Systems: Application to Communication Networks

Hespanha

2004

Lecture Notes in Computer Science

134

122

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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 continuous-time 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 Piecewise-Deterministic 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 on-off TCP flows that considers both the congestion-avoidance and slow-start 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 finite-dimensional model. This model shows that, for transfer-size 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.

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Section: Introductionmentioning

confidence: 99%

Stochastic Hybrid Systems: Application to Communication Networks

Hespanha

2004

Lecture Notes in Computer Science

134

122

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“…whereμ := µ (3) evolves according toμ (3) = 3µ (2) RT T 2 − 7pµ (4) 8 . In this case, (17) has a unique solution ϕ, resulting in a truncated system given by (18) and…”

Section: Examples Of Truncationsmentioning

confidence: 99%

“…However, (21) does not satisfy condition 2 in Lemma 1 because the different discrete modes do not appear decoupled:μ (3) ss depends on µ (0) off , andμ (3) ca depends on µ (4) ss . For the purpose of determining ϕ, we ignore the cross coupling terms and approximate (21) bẏ…”

Section: Examples Of Truncationsmentioning

confidence: 99%

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Polynomial Stochastic Hybrid Systems

Hespanha

2005

Lecture Notes in Computer Science

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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 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.

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“…The theory of hybrid dynamical systems [1], [3] is therefore a natural framework for power systems modeling and control, since it allows to simultaneously capture both kinds of behaviour. Although the conventional formulation is deterministic, there have been many attempts to introduce randomness in the theory, in order to cope with the inherent uncertainty in many practical problems (see [4] and the references therein for a survey).…”

Section: Introductionmentioning

confidence: 99%

On the Fokker-Planck Equation for Stochastic Hybrid Systems: Application to a Wind Turbine Model

Bect

Phulpin

Baili

et al. 2006

2006 International Conference on Probabilistic Methods Applied to Power Systems

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Abstract-This paper presents some recent results concerning a class of continuous-time Markov processes called "stochastic hybrid systems"1 . These processes describe the evolution of a multidimensional hybrid-state dynamical system subject to Gaussian white noise inputs. After a brief recall of the formalism, we state the generalized Fokker-Planck equation, which is a partial differential equation satisfied by the probability density function of the system. As an illustration, we consider a variablespeed wind turbine, with a switching controller that combines stall regulation and pitch control. For a given value of the mean wind speed, the stationary distribution of the state variables is computed numerically. This truly dynamical analysis of the system yields a complete probabilistic characterization of the uncertain power output, which is much more accurate than the usual static analysis.

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Stochastic Hybrid Models: An Overview

Cited by 83 publications

References 2 publications

Stochastic Hybrid Systems: Application to Communication Networks

Stochastic Hybrid Systems: Application to Communication Networks

Polynomial Stochastic Hybrid Systems

On the Fokker-Planck Equation for Stochastic Hybrid Systems: Application to a Wind Turbine Model

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