Modified Unscented Kalman Filter Considering Maximum Point of Probability Density Function

The unscented Kalman filter (UKF) has demonstrated its effectiveness for state estimation in highly nonlinear systems over the past two decades. However, the UKF's assumption of a single Gaussian probability density function (PDF) adversely affects its filtering accuracy. To address this limitation, this article proposes a variant of the UKF, referred to as MaxUKF, which enhances estimation accuracy by incorporating the maximum point of the PDF, denoting the point getting maximum value of the PDF. First, two mathematical theorems are presented, providing direct ways to accurately predict the maximum point of the PDF of the state after nonlinear transformation. Subsequently, a Gaussian sum PDF is constructed to accurately match the mean, covariance, and maximum point of the PDF after nonlinear transformation. Then, the MaxUKF algorithm is presented in a manner analogous to the UKF, following a two-step process: time-update step and measurement-update step, during which the non-Gaussian PDF of the state is represented by the constructed Gaussian sum PDF. Finally, the effectiveness of the proposed MaxUKF is verified through a series of numerical simulations. The results demonstrate that the MaxUKF algorithm offers higher estimation accuracy compared to the conventional UKF while requiring similar computational time.