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 σ² ∈ [σ_min² , σ_max² ] and time constant τ ∈ [τ_min, τ_max]. The bounds on σ² 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 τ_max and a process noise variance inflated by (τ_max/τmin). We also show that the overbound is tightest by initializing the variance of the Gauss-Markov process with σ₀² = 2σ_max² /[1 + (τ_min/^τ_max)]. The new method is evaluated using covariance analysis for an example application in advanced receiver autonomous integrity monitoring (ARAIM) [2].