New smoothed-state estimation for correlated process and measurement noises

关键词:  New Kalman smoother · Correlated noises · Fixed interval · Linear time-varying system · SPIKE algorithm

Key words:  New Kalman smoother · Correlated noises · Fixed interval · Linear time-varying system · SPIKE algorithm

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.

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.