Tight Bounds for Uncertain Time-Correlated Errors With Gauss–Markov Structure in Kalman Filtering

Safety-critical navigation applications require that estimation errors be reliably quantified and bounded. This can be challenging for linear dynamic systems if the process noise or measurement errors have uncertain time correlation. In many systems (e.g., in satellite-based or inertial navigation systems), there are sources of time-correlated sensor errors that can be well modeled using Gauss–Markov processes (GMP). However, uncertainty in the GMP parameters, particularly in the correlation time constant, can cause misleading error bounds. In this article, we develop time-correlated models that ensure tight upper bounds on the estimation error variance, assuming that the actual error is a stationary first-order GMP with a variance and time constant that are only known to reside within an interval. We first use frequency-domain analysis to derive stationary GMP models in both the continuous and discrete-time domains, which outperform models previously described in the literature. Then, we derive an even tighter estimation error bound using a nonstationary GMP model, for which we determine the minimum initial variance that guarantees bounding conditions. Both models can easily be implemented in a linear estimator like the Kalman filter.