Skip to main content
SpringerLink
Log in

Conditionally Minimax Nonlinear Filter and Unscented Kalman Filter: Empirical Analysis and Comparison

  • Stochastic Systems
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

We present the results of the analysis and comparison of the properties of two concepts in state filtering problems for nonlinear stochastic dynamic observation systems with discrete time: sigma-point Kalman filter based on a discrete approximation of continuous distributions and conditionally minimax nonlinear filter that implements the conditionally optimal filtering method based on simulation modeling. A brief discussion of the structure and properties of the estimates and justifications of the corresponding algorithms is accompanied by a significant number of model examples illustrating both positive applications and limitations of the efficiency for the estimation procedures. The simplicity and clarity of the considered examples (scalar autonomous regressions in the state equation and linear observations) allow us to objectively characterize the considered estimation methods. We propose a new modification of the nonlinear filter that combines the ideas of both considered approaches.

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

Similar content being viewed by others

References

  1. Julier, S.J., Uhlmann, J.K., and Durrant-Whyte, H.F., A New Approach for Filtering Nonlinear Systems, Proc. IEEE Am. Control Conf. (ACC’95), 1995, pp. 1628–1632.

    Google Scholar 

  2. Menegaz, H.M.T., Ishihara, J.Y., Borges, G.A., and Vargas, A.N., A Systematization of the Unscented Kalman Filter Theory, IEEE Trans. Autom. Control, 2015, vol. 60, no. 10, pp. 2583–2598.

    Article  MathSciNet  MATH  Google Scholar 

  3. Julier, S.J., The Scaled Unscented Transformation, Proc. IEEE Am. Control Conf. (ACC’02), 2002, pp. 4555–4559.

    Google Scholar 

  4. Xu, L., Ma, K., and Fan, B., Unscented Kalman Filtering for Nonlinear State Estimation with Correlated Noises and Missing Measurements, Int. J. Control Autom. Syst, 2018, vol. 16, no. 3, pp. 1011–1020.

    Article  Google Scholar 

  5. Li, L. and Xia, Y., Stochastic Stability of the Unscented Kalman Filter with Intermittent Observations, Automatica, 2012, vol. 48, no. 5, pp. 978–981.

    Article  MathSciNet  MATH  Google Scholar 

  6. Lee, D., Vukovich, G., and Lee, R., Robust Unscented Kalman Filter for Nanosat Attitude Estimation, Int. J. Control Autom. Syst., 2017, vol. 15, no. 53, pp. 2161–2173.

    Article  Google Scholar 

  7. Zhao, Y., Gao, S. S., Zhang, J., and Sun, Q. N., Robust Predictive Augmented Unscented Kalman Filter, Int. J. Control Autom. Syst., 2014, vol. 12, no. 5, pp. 996–1004.

    Article  Google Scholar 

  8. Scardua, L.A. and da Cruz, J.J., Complete Offline Tuning of the Unscented Kalman Filter, Automatica, 2017, vol. 80, pp. 54–61.

    Article  MathSciNet  MATH  Google Scholar 

  9. Straka, O., Dunik, J., and Simandl, M., Unscented Kalman Filter with Advanced Adaptation of Scaling Parameter, Automatica, 2014, vol. 50, no. 10, pp. 2657–2664.

    Article  MathSciNet  MATH  Google Scholar 

  10. Dunik, J., Simandl, M., and Straka, O., Unscented Kalman Filter: Aspects and Adaptive Setting of Scaling Parameter, IEEE Trans. Autom. Control, 2012, vol. 57, no. 9, pp. 2411–2416.

    Article  MathSciNet  MATH  Google Scholar 

  11. Biswas, S.K., Qiao, L., and Dempster, A.G., A Novel a Priori State Computation Strategy for the Unscented Kalman Filter to Improve Computational Efficiency IEEE Trans. Autom. Control, 2017, vol. 62, no. 4, pp. 1852–1864.

    Article  MathSciNet  MATH  Google Scholar 

  12. Sarkka, S., On Unscented Kalman Filtering for State Estimation of Continuous-Time Nonlinear Systems, Trans. Autom. Control, 2007, vol. 52, no. 9, pp. 1631–1641.

    Article  MathSciNet  MATH  Google Scholar 

  13. Li, X., Liu, A., Yu, C, and Su, F., Widely Linear Quaternion Unscented Kalman Filter for Quaternion-Valued Feedforward Neural Network, IEEE Signal Process. Lett, 2017, vol. 24, no. 9, pp. 1418–1422.

    Google Scholar 

  14. Bhotto, M.Z.A. and Bajic, I.V., Constant Modulus Blind Adaptive Beamforming Based on Unscented Kalman Filtering, IEEE Signal Process. Lett., 2015, vol. 22, no. 4, pp. 474–478.

    Article  Google Scholar 

  15. Li, L. and Xia, Y., Unscented Kalman Filter over Unreliable Communication Networks with Markovian Packet Dropouts, Trans. Autom. Control, 2013, vol. 58, no. 12, pp. 3224–3230.

    Article  Google Scholar 

  16. Wu, P., Li, X. and Bo, Y., Iterated Square Root Unscented Kalman Filter for Maneuvering Target Tracking Using TDOA Measurements, Int. J. Control Autom. Syst., 2013, vol. 11, no. 4, pp. 761–767.

    Article  Google Scholar 

  17. Jochmann, G., Kerner, S., Tasse, S., and Urbann, O., Efficient Multi-Hypotheses Unscented Kalman Filtering for Robust Localization, Led. Notes Comput. Sci. (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 7416 LNCS, 2012, pp. 222–233.

    Google Scholar 

  18. Leven, W.F. and Lanterman, A.D., Unscented Kalman Filters for Multiple Target Tracking with Symmetric Measurement Equations, Trans. Autom. Control, 2009, vol. 54, no. 2, pp. 370–375.

    Article  MathSciNet  MATH  Google Scholar 

  19. Pugachev, V.S., Recurrent Estimation of Variables and Parameters in Stochastic Systems Defined by Difference Equations, Dokl. Math., 1978, vol. 243, no. 5, pp. 1131–1133.

    Google Scholar 

  20. Pugachev, V.S., Estimation of Variables and Parameters in Discrete-Time Nonlinear Systems, Autom. Remote Control, 1979, vol. 40, no. 4, pp. 39–50.

    MathSciNet  MATH  Google Scholar 

  21. Pankov, A.R., Recurrent Conditionally Minimax Filtering of Processes In Nonlinear Difference Stochastic Systems, J. Comput. Syst. Sci. Int., 1993, vol. 31, no. 4, pp. 54–60.

    MathSciNet  MATH  Google Scholar 

  22. Pankov, A.R. and Bosov A.V., Conditionally Minimax Algorithm for Nonlinear System State Estimation, IEEE Trans. Autom. Control, 1994, vol. 39, no. 8, pp. 1617–1620.

    Article  MathSciNet  MATH  Google Scholar 

  23. Borisov, A.V., Bosov, A.V., Kibzun, A.I., Miller, G.B., and Semenikhin, K.V., The Conditionally Minimax Nonlinear Filtering Method and Modern Approaches to State Estimation in Nonlinear Stochastic Systems, Autom. Remote Control, 2018, vol. 79, no. 1, pp. 1–11.

    Article  MathSciNet  MATH  Google Scholar 

  24. Wan, E.A. and Van der Merwe, R., The Unscented Kalman Filter, in Kalman Filtering and Neural Networks, Haykin, S., Ed., New York: Wiley, 2001, pp. 221–280.

    Chapter  Google Scholar 

  25. Shiryaev, A.N., Veroyatnost’ (Probability), Moscow: Nauka, 1989.

    MATH  Google Scholar 

  26. Bhattacharya, R.N. and Lee, C., Ergodicity of Nonlinear First Order Autoregressive Models, J. Theor. Prohah., 1995, vol. 8, no. 1, pp. 207–219.

    Article  MathSciNet  MATH  Google Scholar 

  27. May, R.M., Simple Mathematical Models with Very Complicated Dynamics, Nature, 1976, vol. 261, pp. 459–467.

    Article  MATH  Google Scholar 

  28. Nahi, N., Optimal Recursive Estimation with Uncertain Observation, IEEE Trans. Inform. Theory, 1969, vol. 15, no. 4, pp. 457–462.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This work was supported by the Russian Foundation for Basic Research, project no. 19-07-00187-a.

Author information

Authors and Affiliations

  1. Institute of Informatics Problems of the Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, Moscow, Russia

    A. V. Bosov & G. B. Miller

  2. Moscow Aviation Institute, Moscow, Russia

    A. V. Bosov

Authors
  1. A. V. Bosov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. B. Miller
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to A. V. Bosov or G. B. Miller.

Additional information

This paper was recommended for publication by M.M. Khrustalev, a member of the Editorial Board

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bosov, A.V., Miller, G.B. Conditionally Minimax Nonlinear Filter and Unscented Kalman Filter: Empirical Analysis and Comparison. Autom Remote Control 80, 1230–1251 (2019). https://doi.org/10.1134/S0005117919070026

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117919070026

Keywords

Search

Navigation