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

Kurzfassung

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]. [1] Tupysev, V. A., Stepanov, O. A., Loparev, A. V., and Litvinenko, Y. A. “Guaranteed Estimation in the Problems of Navigation Information Processing.” Proceedings of the 18th IEEE International Conference on Control Applications, St. Petersburg, Russia, 2009, pp. 1672 – 1677. [2] Working Group C. “ARAIM Technical Subgroup. Milestone 3.0 Report.” Technical report, EU-US Cooperation on Satellite Navigation, 2016.