Optimal Set-Membership Smoothing

TL;DR

This paper introduces an optimal set-membership smoothing framework for non-stochastic Hidden Markov Models using uncertain variables, providing new algorithms for linear and nonlinear systems with numerical validation.

Contribution

It establishes the first optimal SMSing framework for non-stochastic HMMs and develops two algorithms for linear and nonlinear systems with set-membership constraints.

Findings

The optimal SMSing framework reveals fundamental principles of set-membership smoothing.

The linear system algorithm provides a closed-form solution.

Numerical simulations confirm the effectiveness of the proposed methods.

Abstract

This article studies the Set-Membership Smoothing (SMSing) problem for non-stochastic Hidden Markov Models. By adopting the mathematical concept of uncertain variables, an optimal SMSing framework is established for the first time. This optimal framework reveals the principles of SMSing and the relationship between set-membership filtering and smoothing. Based on the design principles, we put forward two SMSing algorithms: one for linear systems with zonotopic constrained uncertainties, where the solution is given in a closed form, and the other for a class of nonlinear systems. Numerical simulations corroborate the effectiveness of our theoretical results.