The Weighted Kalman Filter

Abstract

This paper proposes a new version of the Kalman filter, referred to as weighted Kalman filter (WKF). In the WKF some recent results on the weighted linearization of nonlinear systems are exploited to incorporate modifications in the equations of the extended Kalman filter (EKF). More specifically, the computation of the Jacobian matrices at the current mean of the estimated state is replaced by the multiple integral over the state space of the Jacobian matrix functions multiplied by a weighting function. Similar modifications are introduced in the equations used to account for the available nonlinear model and compute the so-called a priori state and output estimates. The weighting function is chosen to be a multivariable Gaussian function where the generalized variance is selected as proportional to the current covariance matrix of the state estimate. An illustrative example is used to describe the step-by-step derivation of the WKF equations and compare its performance against the EKF in terms of convergence properties and estimation error performance.