Why Should I Care About Stochastic Hybrid Systems?

Abstract

Hybrid systems combine continuous-time dynamics with discrete modes of operation. The states of such system usually have two distinct components: one that evolves continuously, typically according to a differential equation; and another one that only changes through instantaneous jumps.

We present 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, in SHSs the rate at which transitions occur depends on both the continuous and the discrete states of the hybrid system. The combination of continuous dynamics, discrete events, and stochasticity results in a modeling framework with tremendous expressive power, making SHSs appropriate to describe the dynamics of a wide variety of systems. This observation has been the driving force behind the several recent research efforts aimed at developing tools to analyze these systems.

In this talk, we use several examples to illustrate the use of SHSs as a versatile modeling tool to describe dynamical systems that arise in distributed control and estimation, networked control systems, molecular biology, and ecology. In parallel, we will also discuss several mathematical tools that can be used to analyze such systems, including the use of the extended generator, Lyapunov-based arguments, infinite-dimensional moment dynamics, and finite-dimensional truncations.