Stochastic hybrid systems with renewal transitions

Singh, AbhyudaiStochastic dynamics of several systems can be modeled via piecewise deterministic time evolution of the state, interspersed by random discrete events. Within this general class of systems we consider time-triggered stochastic hybrid systems (TTSHS), where the state evolves continuously according to a linear dynamical system. Discrete events occur based on an underlying renewal process (timer), and the intervals between successive events follow an arbitrary continuous probability density function. Moreover, whenever the event occurs, the state is reset based on a linear affine transformation that allows for inclusion of state-dependent and independent noise terms. ☐ Traditional analysis of stochastic hybrid systems (SHS) relies heavily on various Monte Carlo simulation techniques, which come at a significant computational cost. Since one is often interested in computing only the lower-order moments of the state variables, much time and effort can be saved by directly computing these statistical moments without having to run Monte Carlo simulations. Unfortunately, moment calculations in SHS can be non-trivial due to the problem of unclosed dynamics: the time evolution of lower-order moments of the state space depends on higher-order moments. In such cases, moments are usually approximated by employing closure schemes, that close the system of differential equations by approximating higher-order moments as nonlinear functions of lower-order moments. ☐ The key contribution of this thesis is to develop novel methods for different classes of TTSHS for obtaining exact analytical expressions for the steady-state moments, along with derivation of necessary and sufficient conditions for the stability of statistical moments. The method developed here is applied to a wide range of problems from systems biology to nano sensors. Moreover, we used TTSHS as an efficient tool to design a controller in the presence of disturbance, noise, and random discrete events in a system. Finally, for SHS models which cannot be solved analytically, we develop a new moment closure approach to approximate their moments with low approximation error.University of Delaware, Department of Electrical and Computer EngineeringPh.D