  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.
