Reach Probability Estimation of Rare Events in Stochastic Hybrid Systems

Abstract

This thesis conducts a series of interrelated research studies on reach probability estimation of rare events for stochastic hybrid systems. Chapter 1 explains that the motivation for these studies stems from the need to assess safety and capacity of a design for a future Air Traffic Management (ATM) concept of operations (ConOps). The safety/capacity of an ATM ConOps can be expressed in terms of the amount of traffic that can be handled in such a way that the probability of rare events remains sufficiently low. Chapter 1 also explains that the dynamic and stochastic behaviours in an ATM ConOps design can be captured by a General Stochastic Hybrid System (GSHS) model, and that the rare events to be studied can be defined as events that the state of a GSHS model reaches an unsafe set. In ATM safety studies, an unsafe set often considered is the closed subset in the GSHS state space where the physical shapes of two aircraft overlap. The state of a GSHS model consists of two components: i) a Euclidean valued component, and ii) a discrete valued component. The evolution of these two components influence each other; therefore a GSHS model can capture various types of dynamic and stochastic behaviours, including Brownian motion and spontaneous jumps. In contrast to forced jumps, that happen when the GSHS state reaches a boundary in the hybrid state space, spontaneous jumps occur according to a Poisson point process. A mathematically important property of GSHS, is that a GSHS execution satisfies the strong Markov property...