Inverse Kalman Filtering for Systems with Correlated Noises

This paper focuses on two inverse problems of the Kalman filter in which the process and measurement noises are correlated. The unknown covariance matrix in a stochastic system is reconstructed from observations of its posterior beliefs. For the standard inverse Kalman filtering problem, a novel duality-based formulation is proposed, where a well-defined inverse optimal control (IOC) problem is solved instead. Identifiability of the underlying model is proved, and a least squares estimator is designed that is statistically consistent. The time-invariant case using the steady-state Kalman gain is further studied. Since this inverse problem is ill-posed, a canonical class of covariance matrices is constructed, which can be uniquely identified from the dataset with asymptotic convergence. Finally, the performances of the proposed methods are illustrated by numerical examples.