Closed-Form Approximation for the Steady-State Performance of Second-Order Kalman Filters

The Kalman filter is adopted in a myriad of applications for providing the minimum mean square error estimation of time-varying parameters in a simple and systematic manner. However, determining the Kalman filter performance is not so straightforward, particularly when process noise is present. In that case, one must often resort to numerical evaluations of the recursive Bayesian Cramer-Rao bound, or alternatively to implement the filter and assess the performance through Montecarlo simulations. This letter is intended to circumvent this limitation. It proposes a closed-form approximation for the steady-state performance of a Kalman filter based on a second-order dynamic model, while at the same time providing a novel closed-form upper bound for the convergence time. These two results are obtained by reformulating the Kalman filter in batch mode and analyzing the inner structure of the Bayesian information matrix. Simulation results are provided to illustrate the goodness of the proposed approach.