On Hybrid State Estimation for Stochastic Hybrid Systems

This paper considers the state estimation problem for the general continuous-time Stochastic Hybrid System (SHS) which has various applications. Defined on the hybrid state space, the SHS has the interacting discrete dynamics and continuous dynamics subject to various uncertainties. The hybrid state estimation problem is to estimate both the continuous state and the discrete state of the SHS with the information given by a continuous-time observation process. In this paper, the hybrid state estimation problem is mathematically formulated and the corresponding filtering equations that are stochastic partial differential equations are derived to describe the evolution of the hybrid state estimates conditioned on the observation history. A numerical algorithm based on a finite-difference approach is proposed to solve the filtering equations. A Markov Chain (MC) is constructed on the descretized hybrid state space to approximate the infinitesimal generator of the SHS and then hybrid state estimation for the SHS is reduced to estimating the state of the MC. It is proved that the state estimation results of the MC converge to the solution to the filtering equations as the constructed MC converges to the SHS. An illustrative example of aircraft tracking is used to demonstrate the performance of the proposed algorithm.