Set-valued nonlinear estimation using the Galerkin approximation
In some estimation problems there is not enough information in the data or system model to warrant a unique estimate of the system state. Set valued estimators are able to convey the information that is available from the data and contextual considerations without overstating what can actually be known about the system state. In this work, a set-valued state estimator for nonlinear dynamic systems is presented. The estimator uses the Galerkin approximation to solve Kolmogorov's equation for the diffusion of a continuous-time continuous-state nonlinear system, as well as for implementing discrete time updates of noisy measurements. This filtering of the state is accomplished for a convex set of distributions simultaneously, and closed-form representations of the set of resulting means and covariances is provided at any desired time instant.