Gaussian/Gaussian-mixture filters for non-linear stochastic systems with delayed states

The Gaussian mixture approximation to the probability density function of the state is more appropriate than the single Gaussian approximation. A Gaussian mixture filter (GMF) is proposed for a class of non-linear discrete-time stochastic systems with the multi-state delayed case. First, a novel non-augmented filtering framework of the constituent Gaussian filter (GF) in GMF is derived, which recursively operates by analytical computation and non-linear Gaussian integrals. The implementation of such GF is thus transformed to the computation of such non-linear integrals in the proposed framework, which is solved by applying different numerical technologies for developing various variations of the non-augmented GF, for example, GF-cubature Kalman filter (CKF) based on the cubature rule. Secondly, a non-augmented GMF is discussed by a weight sum of the above proposed GF, where each GF component is independent from the others and can be performed in a parallel manner, and its corresponding weigh is updated by using the measurements according to Bayesian formula. Naturally, a variation or implementation of such GMF based on the cubature rule is the GMF-CKF. Finally, the performance of the new filters is demonstrated by a numerical example and a vehicle suspension estimation problem.