Skip to main content
SpringerLink
Log in

Adaptive Masreliez–Martin Fractional Embedded Cubature Kalman Filter

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

In this paper, two fractional embedded cubature Kalman filters are proposed. Based on Masreliez–Martin (M–M) method, the first filter named M–M fractional embedded cubature Kalman filter (MMFECKF) increases the robustness of estimation under the situations where the measurement noise is non-Gaussian. To deal with state estimation of fractional nonlinear discrete stochastic models with unknown measurement noise covariance, the second filter named adaptive M–M fractional embedded cubature Kalman filter (AMMFECKF) is put forward by introducing the direct covariance matching approach to the first filter. The simulations on re-entry ballistic target tracking system have demonstrated the effectiveness and accuracy of the two proposed filters. Moreover, the influences of initial measurement noise covariance and contaminated measurement noise on AMMFECKF are analyzed, with the conclusion that AMMFECKF can achieve more accurate and robust state estimation than MMFECKF.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

References

  1. R. Abdolrahman, S. Behrouz, A modified fractional-order unscented Kalman filter for nonlinear fractional-order systems. Circuits Syst. Signal Process. 37(9), 3756–3784 (2018). https://doi.org/10.1007/s00034-017-0729-9

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Almagbile, J. Wang, W. Ding, Evaluating the performances of adaptive Kalman filter methods in GPS/INS integration. J. Glob. Position. Syst. 9(1), 33–40 (2010)

    Article  Google Scholar 

  3. I. Arasaratnam, S. Haykin, Cubature Kalman Filters. IEEE Trans. Automat. Contr. 54(6), 1254–1269 (2009)

    Article  MathSciNet  Google Scholar 

  4. I. Arasaratnam, S. Haykin, T.R. Hurd, Cubature Kalman filtering for continuous-discrete systems: theory and simulations. IEEE Trans. Signal Process. 58(10), 4977–4993 (2010). https://doi.org/10.1109/TSP.2010.2056923

    Article  MathSciNet  MATH  Google Scholar 

  5. R. Caballero-Águila, A. Hermoso-Carazo, J. Linares-Pérez, Extended and unscented filtering algorithms in nonlinear fractional order systems with uncertain observations. Appl. Math. Sci. 6(30), 1471–1486 (2012)

    MathSciNet  MATH  Google Scholar 

  6. L. Chang, B. Hu, G. Chang, A. Li, Huber-based novel robust unscented Kalman filter. IET Sci. Meas. Technol. 6(6), 502–509 (2012)

    Article  Google Scholar 

  7. X. Chen, Z. Gao, R. Ma, X. Huang, Hybrid extended-unscented Kalman filters for continuous-time nonlinear fractional-order systems involving process and measurement noises. Trans. Inst. Meas. Control. 42(9), 1618–1631 (2020)

    Article  Google Scholar 

  8. M. Dalir, M. Bashour, Applications of fractional calculus. Appl. Math. Sci. 4(21), 1021–1032 (2010)

    MathSciNet  MATH  Google Scholar 

  9. M. Das, A. Dey, S. Sadhu, T.K. Ghoshal, Adaptive central difference filter for non-linear state estimation. IET Sci. Meas. Technol. 9(6), 728–733 (2015)

    Article  Google Scholar 

  10. A. Dey, M. Das, S. Sadhu, T.K. Ghoshal, Adaptive divided difference filter for parameter and state estimation of non-linear systems. IET Signal Process. 9(4), 369–376 (2015)

    Article  Google Scholar 

  11. G.A. Einicke, L.B. White, Robust extended Kalman filtering. IEEE Trans. Signal Process. 47(9), 2596–2599 (1999)

    Article  Google Scholar 

  12. Z. Gao, Cubature Kalman filters for nonlinear continuous-time fractional-order systems with uncorrelated and correlated noises. Nonlinear Dyn. 96, 1805–1817 (2019)

    Article  Google Scholar 

  13. Z. Gao, Fractional-order Kalman filters for continuous-time fractional-order systems involving colored process and measurement noises. J. Frankl. Inst. 355(2), 922–948 (2018). https://doi.org/10.1016/j.jfranklin.2017.11.037

    Article  MathSciNet  MATH  Google Scholar 

  14. Z. Gao, Reduced order Kalman filter for a continuous-time fractional-order system using fractional-order average derivative. Appl. Math. Comput. 338, 72–86 (2018)

    MathSciNet  MATH  Google Scholar 

  15. H. Jiang, Y. Cai, Adaptive fifth-degree cubature information filter for multi-sensor bearings-only tracking. Sensors 18, 3241 (2018)

    Article  Google Scholar 

  16. S. Julier, J. Uhlmann, H.F. Durrant-Whyte, A new method for the nonlinear transformation of means and covariances in filters and estimators. IEEE Trans. Autom. Contr. 45(3), 477–482 (2000)

    Article  MathSciNet  Google Scholar 

  17. R.E. Kalman, A new approach to linear filtering and prediction problems. J. Basic Eng. 82, 35–45 (1960)

    Article  MathSciNet  Google Scholar 

  18. C.D. Karlgaard, H. Schaub, Comparison of several nonlinear filters for a benchmark tracking problem, in AIAA Guidance, Navigation, and Control Conference and Exhibit Keystone, Colorado, pp. 6243–6259 (2006)

  19. C.D. Karlgaard, H. Schaub, Huber-based divided difference filtering. J. Guid. Control Dyn. 30(3), 885–892 (2007)

    Article  Google Scholar 

  20. W. Li, S. Sun, Y. Jia, J. Du, Robust unscented Kalman filter with adaptation of process and measurement noise covariances. Digit. Signal Process. 48, 93–103 (2016). https://doi.org/10.1016/j.dsp.2015.09.004

    Article  MathSciNet  Google Scholar 

  21. T. Liu, S. Cheng, Y. Wei, A. Li, Y. Wang, Fractional central difference Kalman filter with unknown prior information. Signal Process. 154, 294–303 (2019). https://doi.org/10.1016/j.sigpro.2018.08.006

    Article  Google Scholar 

  22. C.J. Masreliez, R.D. Martin, Robust Bayesian estimation for the linear model and robustifying the Kalman Filter. IEEE Trans. Autom. Control. 22(3), 361–371 (1977)

    Article  MathSciNet  Google Scholar 

  23. D. Meng, L. Miao, H. Shao, Composite embedded cubature Kalman filter. Int. J. Adapt Control Signal Process. 31(12), 1743–1753 (2017). https://doi.org/10.1002/acs.2797

    Article  MathSciNet  MATH  Google Scholar 

  24. A.H. Mohamed, K.P. Schwarz, Adaptive Kalman filtering for INS/GPS. J. Geod. 73(4), 193–203 (1999)

    Article  Google Scholar 

  25. M. Nϕrgaard, N.K. Poulsen, O. Ravn, New developments in state estimation for nonlinear systems. Automatica 36(11), 1627 (2000)

    Article  MathSciNet  Google Scholar 

  26. A. Ramezani, B. Safarinejadian, J. Zarei, Novel hybrid robust fractional interpolatory cubature Kalman filters. J. Frankl. Inst. 357(1), 704–725 (2020)

    Article  MathSciNet  Google Scholar 

  27. C. Shan, W. Zhou, Y. Yang, Z. Jiang, Multi-fading factor and updated monitoring strategy adaptive Kalman filter-based variational Bayesian. Sensors. 21, 198 (2021)

    Article  Google Scholar 

  28. Y. Shi, X. Tang, X. Feng, D. Bian, X. Zhou, Hybrid adaptive cubature Kalman filter with unknown variance of measurement noise. Sensors 18, 4335 (2018)

    Article  Google Scholar 

  29. D. Sierociuk, A. Dzielinski, Fractional Kalman filter algorithm for the states, parameters and order of fractional system estimation. Int. J. Appl. Math. Comput. Sci. 16(1), 129–140 (2006)

    MathSciNet  MATH  Google Scholar 

  30. D. Sierociuk, I. Tejado, B.M. Vinagre, Improved fractional Kalman filter and its application to estimation over lossy networks. Signal Process. 91, 542–552 (2011)

    Article  Google Scholar 

  31. V. Stojanovic, N. Nedic, Robust Kalman filtering for nonlinear multivariable stochastic systems in the presence of non-Gaussian noise. Int. J. Robust Nonlinear Control. 26(3), 445–460 (2016). https://doi.org/10.1002/rnc.3319

    Article  MathSciNet  MATH  Google Scholar 

  32. Y. Sun, Y. Wang, X. Wu, Y. Hu, Robust extended fractional Kalman filter for nonlinear fractional system with missing measurements. J. Frankl. Inst. 355(1), 361–380 (2018). https://doi.org/10.1016/j.jfranklin.2017.10.030

    Article  MathSciNet  MATH  Google Scholar 

  33. Y. Sun, X. Wu, J. Cao, Z. Wei, G. Sun, Fractional extended Kalman filtering for non-linear fractional system with Lévy noises. IET Control. Theory Appl. 11(3), 349–358 (2017). https://doi.org/10.1049/iet-cta.2016.1041

    Article  MathSciNet  Google Scholar 

  34. Y. Sunahara, K. Yamashita, An approximate method of state esitmation for nonlinear dynamical systems. Int. J. Control. 11(6), 957–972 (1969)

    Article  Google Scholar 

  35. H. Torabi, N. Pariz, A. Karimpour, A novel cubature statistically linearized Kalman filter for fractional-order nonlinear discrete-time stochastic systems. J. Vib. Control. 24(24), 5880–5897 (2018). https://doi.org/10.1177/1077546317692943

    Article  MathSciNet  Google Scholar 

  36. J.H. Yoon, D.Y. Kim, V. Shin, Window length selection in linear receding horizon filtering, in International Conference on Control, Automation and Systems (Seoul, Korea, 2008), pp. 2463–2467

  37. X.-C. Zhang, Y.-L. Teng, A new derivation of the cubature Kalman filters. Asian J. Control. 16(5), 1501–1510 (2014)

    Article  MathSciNet  Google Scholar 

  38. X. Zhang, A novel cubature Kalman filter for nonlinear state estimation, in 52nd IEEE Conference on Decision and Control, pp. 7797–7802 (2013).

  39. Y. Zhang, Y. Huang, N. Li, L. Zhao, Interpolatory cubature Kalman filters. IET Control. Theory Appl. 9(11), 1731–1739 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Funding

This work is supported by National Natural Science Foundation of China under grant numbers (62177037, 52072293) and Key Project of Department of Science and Technology of Shaanxi Province under grant number (2019GY-067).

Author information

Authors and Affiliations

  1. School of Computer Science and Engineering, Xi’an Technological University, Xi’an, 710021, China

    Jing Mu, Xiaojun Bai & Changyuan Wang

  2. Division of Natural and Applied Science, Duke Kunshan University, Kunshan, 215316, China

    Feng Tian

  3. School of Construction Machinery, Chang’an University, Xi’an, 710064, China

    Jianlian Cheng

Authors
  1. Jing Mu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Feng Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiaojun Bai
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Changyuan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Jianlian Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jing Mu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mu, J., Tian, F., Bai, X. et al. Adaptive Masreliez–Martin Fractional Embedded Cubature Kalman Filter. Circuits Syst Signal Process 41, 6051–6074 (2022). https://doi.org/10.1007/s00034-022-02060-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-022-02060-0

Keywords

Search

Navigation