Non-Gaussian filter based on the Method of Characteristics for Nonlinear Dynamical Systems

This paper deals with the development of a non-Gaussian filter for nonlinear systems with discrete time measurements. Specifically, for systems with no process noise, the evolution of the state probability density function is governed by the Liouville equation. In general, solving the Liouville equation is computationally challenging. To this end, we leverage the method of characteristics to propagate probabilities along the characteristics solutions of the Liouville equation. Further, a convex optimization procedure is proposed to reconstruct the state probability density function from these characteristic solutions. Numerical examples of capturing the non-Gaussian nature of the uncertainty in the duffing oscillator and the two body problem are illustrated.