Constrained state estimation in particle filters

Dynamical systems are often required to satisfy certain constraints arising from basic physical laws, mathematical properties or geometric considerations. Incorporating constraints improves the performance of state estimation and increases the accuracy compared to unconstrained estimation. Particle filters (PF) have gained popularity within the signal processing community, thanks to their asymptotically optimal estimation for nonlinear and non-Gaussian state-space models. However, their constrained formulation has emerged only very recently; and the developments to incorporate state constraints in particle filters have mainly relied on constraining all particles of the PF. This approach is termed Pointwise or Particle Density Truncation (PDT). In this paper, we show that PDT constrains the posterior density of the state rather than the conditional mean estimate, which leads to more stringent and possibly completely different or even irrelevant conditions than the original constraints. Subsequently, we introduce an alternative novel solution to constrained particle filtering, which enforces the constraints on the conditional mean without further restricting the state posterior density. The proposed approach is termed Mean Density truncation (MDT) and is compared to PDT and projection methods for a severely nonlinear model.