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

Steven Langel, Omar García Crespillo, Mathieu Joerger

Abstract: 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 sigma^2 in [sigma^2_min, sigma^2_max] and time constant tau in [tau_min, tau_max]. The bounds on sigma^2 and tau 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 tau_max and a process noise variance inflated by (tau_max/tau_min). We also show that the overbound is tightest by initializing the variance of the Gauss-Markov process with sigma^2_0 = 2sigma^2_max/ [1 + (tau_min/tau_max)]. The new method is evaluated using covariance analysis for an example application in advanced receiver autonomous integrity monitoring (ARAIM) [2].
Published in: Proceedings of the 32nd International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2019)
September 16 - 20, 2019
Hyatt Regency Miami
Miami, Florida
Pages: 3079 - 3098
Cite this article: Langel, Steven, Crespillo, Omar García, Joerger, Mathieu, "Bounding Sequential Estimation Errors Due to Gauss-Markov Noise with Uncertain Parameters," Proceedings of the 32nd International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2019), Miami, Florida, September 2019, pp. 3079-3098.
https://doi.org/10.33012/2019.17014
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