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Stochastic Integration Filter with Improved State Estimate Mean-Square Error Computation

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Informatics in Control, Automation and Robotics

Part of the book series: Lecture Notes in Electrical Engineering ((LNEE,volume 430))

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. 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. 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. 3.

    For example, the Cholesky decomposition can be used.

  4. 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. 5.

    Note that \( I_{2} \) denotes \( 2\times 2 \) identity matrix.

References

  1. Anderson, B.D.O., Moore, J.B.: Optimal Filtering. Prentice Hall, New Jersey (1979)

    MATH  Google Scholar 

  2. Arasaratnam, I., Haykin, S.: Cubature Kalman filters. IEEE Trans. Autom. Control 54(6), 1254–1269 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  3. Arasaratnam, I., Haykin, S., Elliott, R.J.: Discrete-time nonlinear filtering algorithms using Gauss-Hermite quadrature. Proc. IEEE 95(5), 953–977 (2007)

    Article  Google Scholar 

  4. Bhaumik, S., et al.: Cubature quadrature Kalman filter. IET Signal Process. 7(7), 533–541 (2013)

    Article  MathSciNet  Google Scholar 

  5. Doucet, A., De Freitas, N., Gordon, N.: An introduction to sequential Monte Carlo methods. Sequential Monte Carlo Methods in Practice. Springer, Berlin (2001)

    Chapter  Google Scholar 

  6. Duník, J., Straka, O., Šimandl, M.: Stochastic integration filter. IEEE Trans. Autom. Control 58(6), 1561–1566 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  7. Genz, A., Monahan, J.: Stochastic integration rules for infinite regions. SIAM J. Sci. Comput. 19(2), 426–439 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. Havlík, J., Straka, O., Duník, J., Ajgl, J.: On nonlinearity measuring aspects of stochastic integration filter. In: Proceedings of the 13th International Conference on Informatics in Control, Automation and Robotics. Lisbon, Portugal (2016)

    Google Scholar 

  9. Ito, K., Xiong, K.: Gaussian filters for nonlinear filtering problems. IEEE Trans. Autom. Control 45(5), 910–927 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  10. Jia, B., Xin, M., Cheng, Y.: Sparse-grid quadrature nonlinear filtering. Automatica 48(2), 327–341 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  11. Julier, S.J., Uhlmann, J.K.: Unscented filtering and nonlinear estimation. IEEE Rev. 92(3), 401–421 (2004)

    Article  Google Scholar 

  12. Kramer, S.C., Sorenson, H.W.: Recursive Bayesian estimation using piece-wise constant approximations. Automatica 24(6), 789–801 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  13. Nørgaard, M., Poulsen, N.K., Ravn, O.: New developments in state estimation for nonlinear systems. Automatica 36(11), 1627–1638 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ristic, B., Arulampalam, S., Gordon, N.: Beyond the Kalman filter: Particle filters for tracking applications. Artech House (2004)

    Google Scholar 

  15. Sorenson, H.W.: On the development of practical nonlinear filters. Inf. Sci. 7, 230–270 (1974)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by the Czech Science Foundation, project no. GA 16-19999J.

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Authors and Affiliations

  1. European Centre of Excellence - New Technologies for Information Society, Department of Cybernetics, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 8, Plzeň, Czech Republic

    Jindřich Havlík, Ondřej Straka, Jindřich Duník & Jiří Ajgl

Authors
  1. Jindřich Havlík
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  2. Ondřej Straka
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  3. Jindřich Duník
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  4. Jiří Ajgl
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Corresponding author

Correspondence to Jindřich Havlík .

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Editors and Affiliations

  1. Images, Signals, and Intelligent Systems Laboratory, University Paris-EST Créteil (UPEC), Créteil, France

    Kurosh Madani

  2. MAC Team, LAAS-CNRS, Toulouse Cedex 4, France

    Dimitri Peaucelle

  3. Ford Research and Advanced Engineering, Dearborn, Michigan, USA

    Oleg Gusikhin

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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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  • DOI: https://doi.org/10.1007/978-3-319-55011-4_21

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-55010-7

  • Online ISBN: 978-3-319-55011-4

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