The square-root unscented Kalman filter for state and parameter-estimation

Abstract

Over the last 20-30 years, the extended Kalman filter (EKF) has become the algorithm of choice in numerous nonlinear estimation and machine learning applications. These include estimating the state of a nonlinear dynamic system as well estimating parameters for nonlinear system identification (e.g., learning the weights of a neural network). The EKF applies the standard linear Kalman filter methodology to a linearization of the true nonlinear system. This approach is sub-optimal, and can easily lead to divergence. Julier et al. [1] proposed the unscented Kalman filter (UKF) as a derivative-free alternative to the extended Kalman filter in the framework of state-estimation. This was extended to parameter-estimation by Wan and van der Merwe [2, 3]. The UKF consistently outperforms the EKF in terms of prediction and estimation error, at an equal computational complexity of O(L3)1 for general state-space problems. When the EKF is applied to parameter-estimation, the special form of the state-space equations allows for an O(L2) implementation. This paper introduces the square-root unscented Kalman filter (SR-UKF) which is also O(L3) for general state-estimation and O(L2) for parameter estimation (note the original formulation of the UKF for parameter-estimation was O(L3)). In addition, the square-root forms have the added benefit of numerical stability and guaranteed positive semi-definiteness of the state covariances.