Stability of Linear Set-Membership Filters With Respect to Initial Conditions: An Observation-Information Perspective

Abstract: The issue of filter stability with respect to (w.r.t.) the initial condition refers to the unreliable filtering process caused by improper prior information of the initial state. This paper focuses on analyzing and resolving the stability issue w.r.t. the initial condition of the classical Set-Membership Filters (SMFs) for linear time-invariance systems, which has not yet been well understood in the literature. To this end, we propose a new concept -- the Observation-Information Tower (OIT), which describes how the measurements affect the estimate in a set-intersection manner without relying on the initial condition. The proposed OIT enables a rigorous stability analysis, a new SMFing framework, as well as an efficient filtering algorithm. Specifically, based on the OIT, explicit necessary and sufficient conditions for stability w.r.t. the initial condition are provided for the classical SMFing framework. Furthermore, the OIT inspires a stability-guaranteed SMFing framework, which fully handles the stability issue w.r.t. the initial condition. Finally, with the OIT-inspired framework, we develop a fast and stable constrained zonotopic SMF, which significantly overcomes the wrapping effect.