On Kullback-Leibler's information and discrete-time uncertain nonlinear systems

The idea of Kullback-Leibler's information is applied to discrete-time nonlinear state estimation problems when there are model uncertainties. The paper first points out that consistent statistical properties between residuals and the assumed noise model only means a goodness of fit between the data and the model while the Kullback-Leibler's information covers both the goodness of fit and a measure to model reliability. Conditions that assure the mean Kullback-Leibler's information (MKLI) has a unique minimum point for nonlinear systems are derived. Under these conditions, the maximum likelihood function is shown to be equivalent to the MKLI. Finally, the generalized mean Kullback-Leibler's information is defined and applied to select important parameters including the covariance matrix of process noise for an extended Kalman filter.