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Bounding Sequential Estimation Errors Due to Gauss-Markov Noise with Uncertain Parameters

This paper describes the derivation and implementation of a new method to overbound Kalman filter (KF) based estimate error distributions in the presence of time-correlated measurement and process noise. The method is specific to problems where each input noise component is first-order Gauss-Markov with a distinct variance σ ∈ [σmin 2 , σmax 2 ] and time constant τ ∈ [τmin, τmax]. The bounds on σ and τ are known. Reference [1] derives an overbound for the continuous-time KF, and we extend the result to the more common case of sampled-data systems with discrete-time measurements. We prove that the KF covariance matrix overbounds the estimate error distribution when Gauss-Markov processes are defined using a time constant τmax and a process noise variance inflated by (τmax τmin ⁄ ). We also show that the overbound is tightest by initializing the variance of the Gauss-Markov process with σ0 2 = 2σmax 2 [1 + (τmin τmax ⁄ )] ⁄ . The new method is evaluated using covariance analysis for an example application in advanced receiver autonomous integrity monitoring (ARAIM) [2]. The author's affiliation with The MITRE Corporation is provided for identification purposes only and is not intended to convey or imply MITRE's concurrence with, or support for, the positions, opinions, or viewpoints expressed by the author.