Stochastic Linear Hybrid Systems: Modeling, Estimation, and Application in Air Traffic Control

Abstract

We propose a discrete-time model for a stochastic hybrid system (SHS) in which the continuous state evolution is described by stochastic difference equations and the discrete state evolution is governed by stochastic guards (or conditions that contain random parameters). The stochastic guards model system's uncertainties that directly affect the discrete state evolution and thus incorporate uncertainties that are different from those due to the stochastic evolution of the continuous state. A special class of the SHS family, called stochastic linear hybrid system (SLHS), is then presented. In the SLHS, the continuous state evolution is described by a set of stochastic linear difference equations, and the stochastic guards are a set of polyhedral partitions of a real vector space (which is a union of the continuous state space and the random parameter space). Some examples of practical systems that can be modeled as the SLHS are presented. Utilizing the properties of the SLHS, we propose a corresponding hybrid estimation algorithm. The algorithm approximates the conditional probability density functions (pdfs) of the continuous state by Gaussian pdfs. Both the conditional continuous state pdf and the discrete state pdf are then propagated analytically using a Bayesian approach. The SLHS model and the proposed hybrid estimation algorithm are used in an aircraft tracking application in air traffic control. It is illustrated that, in this application, a more accurate modeling of the discrete state transitions with the stochastic guards results in better hybrid estimation accuracy compared with other models/algorithms that assume deterministic guards or constant transition probabilities.