State estimation for Stochastic Hybrid Systems based on deterministic Dirac mixture approximation

In this paper, we consider state estimation for Stochastic Hybrid Systems (SHS). These are systems that possess both continuous-valued and discrete-valued dynamics. For SHS with nonlinear hybrid dynamics and/or non-Gaussian disturbances, state estimation can be implemented as an Interacting Multiple Model (IMM) particle filter. However, a disadvantage of particle filtering is the computational load caused by the large number of particles required for a sufficiently good estimation. We address this issue by first expressing the probability density that describes the state of the SHS as a collection of densities of the continuous-valued state only conditioned on the discrete-valued state. Then, we deterministically approximate these individual densities with Dirac mixtures. The employed approximation method places the particles so that a so called modified Cramér-von Mises distance between the true and the approximated density is minimized. Deterministic approximation requires far less particles than the stochastic sampling used by particle filters. To avoid particle degeneration that can occur when a density is multiplied with the likelihood, the filter uses progressive density correction. The presented filter is demonstrated in a numerical maneuvering target tracking example.