Iterated Extended Kalman Smoother-Based Variable Splitting for $L_1$-Regularized State Estimation

Abstract

In this paper, we propose a new framework for solving state estimation problems with an additional sparsitypromoting L1-regularizer term. We first formulate such problems as minimization of the sum of linear or nonlinear quadratic error terms and an extra regularizer, and then present novel algorithms which solve the linear and nonlinear cases. The methods are based on a combination of the iterated extended Kalman smoother and variable splitting techniques such as alternating direction method of multipliers (ADMM). We present a general algorithmic framework for variable splitting methods, where the iterative steps involving minimization of the nonlinear quadratic terms can be computed efficiently by iterated smoothing. Due to the use of state estimation algorithms, the proposed framework has a low per-iteration time complexity, which makes it suitable for solving a large-scale or high-dimensional state estimation problem. We also provide convergence results for the proposed algorithms. The experiments show the promising performance and speed-ups provided by the methods.

Augmented Sigma-Point Lagrangian Splitting Method for Sparse Nonlinear State Estimation

“…The overall architecture is shown in Figure 1. Compared to our previous work [3], the main difference here is better capturing the higher order…”

“…In many applications, the fundamental problem is to estimate the original states from degraded measurements, based upon prior knowledge on the nonlinear stochastic dynamics [1], [2]. This problem is of central importance, for example, in target tracking, biological processes, tomographic imaging reconstruction, and automatic music transcription [2], [3]. Generally, nonlinear Gaussian filtering and smoothing methods, for instance, Taylor series expansion based methods [4] or sigma-point based methods [2] can be used to estimate the state by analytical or statistical linearization of the nonlinear functions.…”

“…Promoting sparsity in state estimation is difficult but very much desired [3], [5]. Typically, the state is estimated by performing penalized least squares along with a sparsity-inducing regularized function [5].…”

“…However, when the datasets are extremely large, consisting of hundreds of millions or billions of states in a dynamic system, solving such problems by conventional splitting methods becomes challenging. In our recent paper, we have proposed a general variable splitting framework for L 1 -regularized state estimation, which is based on the combination of iterated extended Kalman smoother (IEKS) and alternating direction method of multipliers (ADMM) [3]. However, IEKS requires the Jacobians of the model functions, and may lead to an imprecise estimation result when the system is strongly nonlinear.…”

A Novel Smooth Variable Structure Smoother for Robust Estimation

“…The state and parameter estimation of dynamic systems plays a key role in various application fields. In practice, not only the real-time state estimation of the system is required, but also an estimation result with higher precision through the smoother by using additional measurements made after the time of the estimated state vector is necessary [1][2][3][4]. For example, in a radar system, a smoother is used to obtain higher precision track trajectory for the battlefield situation estimation, and to make more accurate estimates for the aircraft performance.…”