IEEE/CAA Journal of Automatica Sinica
Citation:  Q. B. Ge, X. M. Hu, Y. Y. Li, H. L. He, and Z. H. Song, “A novel adaptive Kalman filter based on credibility measure,” IEEE/CAA J. Autom. Sinica, vol. 10, no. 1, pp. 103–120, Jan. 2023. doi: 10.1109/JAS.2023.123012 
It is quite often that the theoretic model used in the Kalman filtering may not be sufficiently accurate for practical applications, due to the fact that the covariances of noises are not exactly known. Our previous work reveals that in such scenario the filter calculated mean square errors (FMSE) and the true mean square errors (TMSE) become inconsistent, while FMSE and TMSE are consistent in the Kalman filter with accurate models. This can lead to low credibility of state estimation regardless of using Kalman filters or adaptive Kalman filters. Obviously, it is important to study the inconsistency issue since it is vital to understand the quantitative influence induced by the inaccurate models. Aiming at this, the concept of credibility is adopted to discuss the inconsistency problem in this paper. In order to formulate the degree of the credibility, a trust factor is constructed based on the FMSE and the TMSE. However, the trust factor can not be directly computed since the TMSE cannot be found for practical applications. Based on the definition of trust factor, the estimation of the trust factor is successfully modified to online estimation of the TMSE. More importantly, a necessary and sufficient condition is found, which turns out to be the basis for better design of Kalman filters with high performance. Accordingly, beyond trust factor estimation with SageHusa technique (TFESHT), three novel trust factor estimation methods, which are directly numerical solving method (TFEDNS), the particle swarm optimization method (PSO) and expectation maximizationparticle swarm optimization method (EMPSO) are proposed. The analysis and simulation results both show that the proposed TFEDNS is better than the TFESHT for the case of single unknown noise covariance. Meanwhile, the proposed EMPSO performs completely better than the EM and PSO on the estimation of the credibility degree and state when both noise covariances should be estimated online.
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