Riemannian Trust-Region based Adaptive Kalman filter with unknown noise Covariance matrices.

The problem of adaptive Kalman filtering for a discrete observable linear time-varying system with unknown noise covariance matrices is addressed in this paper. The measurement difference autocovariance method is used to formulate a linear least squares cost function containing the measurements and the process and measurement noise covariance matrices. Subsequently, a Riemannian trust-region optimization approach is designed to minimize the least squares cost function and ensure symmetry and positive definiteness for the estimates of the noise covariance matrices. The noise covariance matrix estimates, under sufficient excitation of the system, are shown to converge to their unknown true values. Saliently, the exponential stability and convergence guarantees for the proposed adaptive Kalman filter to the optimal Kalman filter with known noise covariance matrices is shown to be achieved under the relatively mild assumptions of uniform observability and uniform controllability. Numerical simulations on a linear time-varying system demonstrate the effectiveness of the proposed adaptive filtering algorithm.