Bounding Sequential Estimation Errors Due to Gauss-Markov Noise with Uncertain Parameters
Steven E. Langel, Omar García Crespillo, Mathieu Joerger · Proceedings of the Satellite Division's International Technical Meeting (Online)/Proceedings of the Satellite Division's International Technical Meeting (CD-ROM) · 2019
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].