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Fu-Xi Chen1,2,Li-Hui Geng1,et al.[en_title][J].Control Theory and Technology,2025,23(2):176~192.[Copy]
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New smoothed-state estimation for correlated process and measurement noises
Fu-XiChen1,2,Li-HuiGeng1,2,BrettNinness3,Yong-LiZhang1,2
0
(1 School of Automation and Electrical Engineering, Tianjin University of Technology and Education, Tianjin, 300222, China 2 Tianjin Key Laboratory of Information Sensing and Intelligent Control, Tianjin, 300222, China;3 Faculty of Engineering and Built Environment, The University of Newcastle, Callaghan, 2308, NSW, Australia)
摘要:
This paper addresses the computational problem of fixed-interval smoothing state estimation in linear time-varying Gaussian stochastic systems. A new fixed-interval Kalman smoothing algorithm is proposed, and the corresponding form of the smoother is derived. The method is able to accommodate situations where process and measurement noises are correlated, a limitation often encountered in conventional approaches. The Kalman smoothing problem discussed in this paper can be reformulated as an equivalent constrained optimization problem, where the solution corresponds to a set of linear equations defined by a specific co-efficient matrix. Through multiple permutations, the co-efficient matrix of linear equations is transformed into a block tridiagonal form, and then both sides of the linear system are multiplied by the inverse of the co-efficient matrix. This approach is based on the transformation of linear systems described in the SPIKE algorithm and is particularly well-suited for large-scale sparse block tridiagonal matrix structures. It enables efficient, parallel, and flexible solutions while maintaining a certain degree of block diagonal dominance. Compared to directly solving block tridiagonal co-efficient matrices, this method demonstrates appreciable advantages in terms of numerical stability and computational efficiency. Consequently, the new smoothing algorithm yields a new smoother that features fewer constraints and broader applicability than traditional methods. The estimates, such as smoothed state, covariance, and cross-covariance, are essential for fields, such as system identification, navigation, guidance, and control. Finally, the effectiveness of the proposed smoothing algorithm and smoother is validated through numerical simulations.
关键词:  New Kalman smoother · Correlated noises · Fixed interval · Linear time-varying system · SPIKE algorithm
DOI:https://doi.org/10.1007/s11768-025-00252-y
基金项目:
New smoothed-state estimation for correlated process and measurement noises
Fu-Xi Chen1,2,Li-Hui Geng1,2,Brett Ninness3,Yong-Li Zhang1,2
(1 School of Automation and Electrical Engineering, Tianjin University of Technology and Education, Tianjin, 300222, China 2 Tianjin Key Laboratory of Information Sensing and Intelligent Control, Tianjin, 300222, China;3 Faculty of Engineering and Built Environment, The University of Newcastle, Callaghan, 2308, NSW, Australia)
Abstract:
This paper addresses the computational problem of fixed-interval smoothing state estimation in linear time-varying Gaussian stochastic systems. A new fixed-interval Kalman smoothing algorithm is proposed, and the corresponding form of the smoother is derived. The method is able to accommodate situations where process and measurement noises are correlated, a limitation often encountered in conventional approaches. The Kalman smoothing problem discussed in this paper can be reformulated as an equivalent constrained optimization problem, where the solution corresponds to a set of linear equations defined by a specific co-efficient matrix. Through multiple permutations, the co-efficient matrix of linear equations is transformed into a block tridiagonal form, and then both sides of the linear system are multiplied by the inverse of the co-efficient matrix. This approach is based on the transformation of linear systems described in the SPIKE algorithm and is particularly well-suited for large-scale sparse block tridiagonal matrix structures. It enables efficient, parallel, and flexible solutions while maintaining a certain degree of block diagonal dominance. Compared to directly solving block tridiagonal co-efficient matrices, this method demonstrates appreciable advantages in terms of numerical stability and computational efficiency. Consequently, the new smoothing algorithm yields a new smoother that features fewer constraints and broader applicability than traditional methods. The estimates, such as smoothed state, covariance, and cross-covariance, are essential for fields, such as system identification, navigation, guidance, and control. Finally, the effectiveness of the proposed smoothing algorithm and smoother is validated through numerical simulations.
Key words:  New Kalman smoother · Correlated noises · Fixed interval · Linear time-varying system · SPIKE algorithm