Modelling and analysis of stochastic hybrid systems

Abstract

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 the modelling of transmission control protocol (TCP) algorithm for congestion control both for long-lived and on–off flows, state-estimation for networked control systems, and the stochastic modelling of chemical reactions. These examples illustrate the use of SHSs as a modelling tool. Attention is mostly focused on polynomial stochastic hybrid systems (pSHSs) that generally correspond to SHSs 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). It is shown that these ODEs can be approximated by finite-dimensional non-linear 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.