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

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 (eg, 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. (1997), 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 et al., (2000). The UKF consistently outperforms the EKF in terms of prediction and estimation error, at an equal computational complexity of (OL³)^l 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