Alternative framework of the Gaussian filter for non-linear systems with synchronously correlated noises

To conveniently deduce the Gaussian filter (GF), it is based on the assumption that the process and measurement noises are independent. However, it does not always satisfy this assumption in practice. In this study, a novel framework of GF for non-linear system in which the process and measurement noises are correlated at the same time is proposed. The novel framework of GF takes the type of synchronous correlation into account. It is inspired by the fact that the probability of process noises conditioned on measurement noises has much more correlation information than its original probability in the situation of measurements data received, and exploits the conditional Gaussian distributions to improve the GF accuracy. In addition, the issue about multi-dimensional integrals in the implementation of novel framework of GF is solved by the spherical-radial cubature rule. The cubature Kalman filter with synchronously correlated noises (CKF-SCN) is derived. The results of simulations on a non-linear example show better performance of the CKF-SCN in contrast to the extended Kalman filter with correlation noises, the unscented Kalman filter with correlation noises, the Gaussian approximation recursive filter, and the standard cubature Kalman filter.