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

Yirui Cong, Xiangke Wang+1 more|arXiv (Cornell University)|Mar 26, 2022

Fault Detection and Control Systems4 citations

TL;DR

This paper introduces the Observation-Information-Tower (OIT) framework to analyze and ensure stability in linear set-membership filters (SMFs), revealing that classical SMFs are sensitive to initial conditions due to reliance on exact knowledge of the true initial state range. It proposes a new OIT-inspired filtering framework that guarantees stability regardless of initial conditions, enabling a stable and fast constrained zonotopic SMF with uniformly bounded estimation gaps and improved robustness.

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.

Motivation & Objective

• To address the lack of theoretical understanding regarding stability in linear set-membership filters (SMFs), particularly their sensitivity to initial conditions.
• To identify the root cause of instability in classical SMF frameworks, which stems from requiring precise knowledge of the true initial state range.
• To develop a new filtering framework that ensures stability independent of initial conditions, overcoming ill-posedness and unbounded estimation gaps.
• To design a fast, accurate, and stable constrained zonotopic SMF by leveraging the new OIT-inspired framework.
• To establish a rigorous stability criterion based on set-intersection dynamics and time-correlated observation information.

Proposed method

• Introduces the Observation-Information-Tower (OIT), a novel conceptual framework modeling how measurements recursively constrain the state estimate through set intersections.
• Uses the OIT to formally analyze the classical SMF framework, proving that stability requires exact knowledge of the true initial state range, making it sensitive to initial condition errors.
• Proposes a new OIT-inspired filtering framework that decouples the estimation process from initial condition dependence, ensuring stability regardless of initial guess.
• Develops a stable, fast constrained zonotopic SMF by exploiting the closure properties of constrained zonotopes under Minkowski sums and set intersections in prediction and update steps.
• Derives uniform boundedness of the estimate diameter using spectral norms and sub-multiplicative matrix bounds, ensuring the estimate does not grow unboundedly.
• Establishes a bounded estimation gap by proving that the diameter of the estimated set remains uniformly bounded over time, even under approximation and reduction effects.

Experimental results

Research questions

• RQ1 Under what conditions is the classical linear SMF framework stable, and what is the role of the initial condition in this stability?
• RQ2 Why do classical SMFs suffer from ill-posedness and unbounded estimation gaps despite uniform boundedness of the estimate?
• RQ3 Can a new filtering framework be designed such that stability is independent of the initial condition?
• RQ4 How can the OIT framework be used to rigorously analyze and guarantee stability in linear SMFs?
• RQ5 Can a stable, fast, and accurate constrained zonotopic SMF be developed under the new stability-guaranteed framework?

Key findings

• The classical linear SMF framework is unstable under initial condition mismatch, as it requires exact knowledge of the true initial state range to avoid empty estimates or unbounded estimation gaps.
• The OIT framework enables a rigorous stability analysis, revealing that the classical SMF is sensitive to initial conditions due to its reliance on precise initial range knowledge.
• The proposed OIT-inspired framework guarantees stability regardless of the initial condition, eliminating ill-posedness and ensuring bounded estimation gaps.
• The diameter of the estimated set is uniformly bounded over time, with an upper bound derived as $ \bar{d}_{\mathrm{cz}} = \|P^{-1}\|\sqrt{(\bar{d}_{\mathrm{cz}}^{o})^{2} + (\bar{d}_{\mathrm{cz}}^{\bar{o}})^{2}} $, ensuring long-term robustness.
• The constrained zonotopic SMF developed under the new framework maintains tight estimates with improved computational efficiency due to closure properties under Minkowski sums and set intersections.
• The estimation gap remains bounded for all time steps, with $ d_{k}^{\mathrm{g}}(\mathcal{Z}_{k}) \leq \bar{d}_{\mathrm{cz}} $, proving insensitivity to initial condition errors.