Closed-Form Approximation for the Convergence Time of pth-Order Kalman Filters

The Kalman filter is used in a myriad of applications for estimating a set of time-varying parameters in the minimum mean square error sense. When designing the filter, one of the points of most relevant interest is predicting its estimation performance. However, it is often difficult to find expressions in closed form that allow obtaining this information in a straightforward manner. As a consequence, the designer usually requires resorting to the numerical evaluation of the Bayesian Cramér-Rao bound or the implementation and assessment of the filter through Monte Carlo simulations. In this regard, this letter intends to contribute with a novel closed-form upper bound for the convergence time of a Kalman filter. To this end, the Kalman filtering problem is reformulated in batch mode, and the corresponding Fisher information matrix is analyzed. The contribution presented herein is based on a generic dynamic model and is not restricted to any specific order, in contrast to existing contributions. Simulation results are provided to illustrate the goodness of the proposed approach.