Posterior Linearization Filter: Principles and Implementation Using Sigma Points

Abstract

This paper is concerned with Gaussian approximations to the posterior probability density function (PDF) in the update step of Bayesian filtering with nonlinear measurements. In this setting, sigma-point approximations to the Kalman filter (KF) recursion are widely used due to their ease of implementation and relatively good performance. In the update step, these sigma-point KFs are equivalent to linearising the nonlinear measurement function by statistical linear regression (SLR) with respect to the prior PDF.In this paper, we argue that the measurement function should be linearised using SLR with respect to the posterior rather than the prior to take into account the information provided by the measurement. The resulting filter is referred to as the posterior linearisation filter (PLF). In practice, the exact PLF update is intractable but can be approximated by the iterated PLF (IPLF), which carries out iterated SLRs with respect to the best available approximation to the posterior. The IPLF can be seen as an approximate recursive Kullback-Leibler divergence minimisation procedure. We demonstrate the high performance of the IPLF in relation to other Gaussian filters in two numerical examples.

Iterative Filtering and Smoothing in Nonlinear and Non-Gaussian Systems Using Conditional Moments

“…The IEKF/IEKS iteratively linearise the system by Taylor series around the current mean of the approximate filtering/smoothing distribution [22], [23]. On the other hand, IPLF/IPLS iteratively linearise the system using SLR with respect to the current approximate filtering/smoothing density [25], [26]. The RTS smoother is given in Alg.…”

“…1. The filter is presented in Section IV-A, where emphasis is on the update as it is the only stage of the algorithm that utilises iterations [25]. The iterative smoother is presented in Section IV-B and a convergence theorem is provided in Section IV-C.…”

“…The key insight of [25] was that posterior moment approximations can be improved by re-computing the C t , d t and Σ Rt using the current approximate posterior together with SLR, while keeping the original predictive moments in the Update step of Alg. 1.…”

“…In a similar manner to [25], [26] a local convergence analysis can be carried out, resulting in Thm. 2.…”

“…This approach was recently extended on the basis that it ought to be better to perform the SLR with respect to the posterior distribution rather than the prior distribution. While this is intractable, it leads to a scheme where the SLR is iteratively computed using the current best approximation to the posterior, why it is called iterated posterior linearisation filtering/smoothing (IPLF/IPLS) [25], [26]. However, these methods assume additive Gaussian noise.…”