Estimation of dynamic systems using a method of characteristics filter

This paper deals with the technical formulation and implementation details of an algorithm that provides an estimate of the probability density function of the state of a nonlinear dynamical system from discrete time measurements. In contrast to other state estimation methods that propagate the conditional moments, the proposed filter is shown to propagate and update the full probability density function of the state. Characteristic solutions to the Liouville equation are used to propagate the exact probability density values along the flow of the dynamical system. The reconstruction of the state probability density function is then posed as a convex programming problem. An adaptive regression-tree based region-decomposition approach is used to efficiently compute expectation integrals involved in implementing the Bayes rule associated with posterior density function. Numerical examples capturing the non-Gaussian nature of the uncertainty in the duffing oscillator problem and the two body problem are used to demonstrate the efficacy of the proposed methods.