Rethinking the Mathematical Framework and Optimality of Set-Membership Filtering

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 1960s, studies on SMF have used set-based description as its mathematical framework. One important issue that overlooked is optimality SMF. In this work, we put forward a new framework using concepts uncertain variables. We establish two basic properties variables, namely, law total range (a non-stochastic version probability) and equivalent Bayes' rule. This enables us to general SMFing established optimality. Furthermore, obtain optimal under Markov condition, which shown be fundamentally Bayes filter. Note classical literature only obtained condition. When condition violated, show not gives an outer bound estimate.