Rethinking the Mathematical Framework and Optimality of Set-Membership Filtering

• 成果类型: Article
• 署名作者: Cong, Yirui; Wang, Xiangke; Zhou, Xiangyun
• 署名单位: National University of Defense Technology - China; Australian National University
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2021.3082508
• 发表日期: 2022
• 页码: 2544-2551
• 关键词: Linear systems; Ellipsoids; probability distribution; Markov processes; Hidden Markov models; Nonlinear systems; Kalman filters; Bayes' rule for uncertain variables; law of total range; optimality; set-membership filtering (SMF); uncertain variables

摘要

Set-membership filter (SMF) has been extensively studied for state estimation in the presence of bounded noises with unknown statistics. Since it was first introduced in the 1960s, the studies on SMF have used the set-based description as its mathematical framework. One important issue that has been overlooked is the optimality of SMF. In this article, we put forward a new mathematical framework for SMF using concepts of uncertain variables. We first establish two basic properties of uncertain variables, namely, the law of total range (a nonstochastic version of the law of total probability) and the equivalent Bayes' rule. This enables us to put forward a general SMFing framework with established optimality. Furthermore, we obtain the optimal SMF under a nonstochastic Markov condition, which is shown to be fundamentally equivalent to the Bayes filter. Note that the classical SMF in the literature is only equivalent to the optimal SMF we obtained under the nonstochastic Markov condition. When this condition is violated, we show that the classical SMF is not optimal and it only gives an outer bound on the optimal estimate.