Optimal and Suboptimal Importance Density Functions for Rao-Blackwellized Particle Filter

For the standard particle filter, sampling efficiency is decreasing rapidly with time, especially when the state dimension is large. Rao-Blackwellized particle filter can improve the sampling efficiency by marginalizing over one part of the state and applying particle filter to the other part with lower dimension. However, Rao-Blackwellization will lead to a non-Markovian model, and this causes the existing optimal importance density function theory for the standard particle filter is inapplicable. In this paper, in the sense of the minimum conditional variance of importance weights, we derive the corresponding optimal importance density function for Rao-Blackwellized particle filter. The concrete calculation formula for optimal importance density function is provided for two particular conditionally linear Gaussian models. When optimal importance density function cannot be computed analytically, suboptimal importance density functions are designed by Gaussian approximation methods. The effectiveness of the proposed methods are illustrated through three simulation examples including two typical target tracking models and a fourth order mixed linear/nolinear Gaussian model.