Abstract
The paper deals with the Bayesian state estimation of nonlinear stochastic dynamic systems. The focus is aimed at the stochastic integration filter, which represents the Gaussian filters with the state and measurement prediction moments calculated by the stochastic integration rule. Besides the value of the integral, the rule also provides the covariance matrix of the integral value error. In the paper an improved mean-square error of the state estimate is proposed based on utilization of the integral error covariance matrix. The improved calculation is illustrated using two numerical examples for the stochastic integration filter of the third and fifth degrees.
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Notes
- 1.
Note that even though the optimization methods and the GFs were derived by different approaches and assumptions, finally they provide formally same structure of the algorithm, which is computationally reasonable and often used in practice.
- 2.
For the sake of simplicity all PDFs will be given by their argument, if not stated otherwise, i.e., \( p(\mathbf w_k) = p_{\mathbf w_k}(\mathbf w_k) \).
- 3.
For example, the Cholesky decomposition can be used.
- 4.
Note that the Algorithm 1 computes approximate value of the integral for given vector functions \( \varvec{\gamma }\), but Algorithm 2 requires calculation of the predictive CMs (24), (25), and (27) for given matrix function. This can be resolved by column-wise stacking of matrix functions to obtain a vector function.
- 5.
Note that \( I_{2} \) denotes \( 2\times 2 \) identity matrix.
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Acknowledgements
This work was supported by the Czech Science Foundation, project no. GA 16-19999J.
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Havlík, J., Straka, O., Duník, J., Ajgl, J. (2018). Stochastic Integration Filter with Improved State Estimate Mean-Square Error Computation. In: Madani, K., Peaucelle, D., Gusikhin, O. (eds) Informatics in Control, Automation and Robotics . Lecture Notes in Electrical Engineering, vol 430. Springer, Cham. https://doi.org/10.1007/978-3-319-55011-4_21
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