Advertisement

Automation and Remote Control

, Volume 80, Issue 7, pp 1230–1251 | Cite as

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

  • A. V. BosovEmail author
  • G. B. MillerEmail author
Stochastic Systems

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.

Keywords

nonlinear stochastic observation system unscented transform unscented Kalman filter conditionally optimal filtering conditionally minimax nonlinear filter simulation modeling 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

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

References

  1. 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. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Julier, S.J., The Scaled Unscented Transformation, Proc. IEEE Am. Control Conf. (ACC’02), 2002, pp. 4555–4559.Google Scholar
  4. 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.CrossRefGoogle Scholar
  5. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 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.CrossRefGoogle Scholar
  7. 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.CrossRefGoogle Scholar
  8. 8.
    Scardua, L.A. and da Cruz, J.J., Complete Offline Tuning of the Unscented Kalman Filter, Automatica, 2017, vol. 80, pp. 54–61.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 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. 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.CrossRefGoogle Scholar
  15. 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.CrossRefGoogle Scholar
  16. 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.CrossRefGoogle Scholar
  17. 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. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 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. 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.MathSciNetzbMATHGoogle Scholar
  21. 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.MathSciNetzbMATHGoogle Scholar
  22. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 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.CrossRefGoogle Scholar
  25. 25.
    Shiryaev, A.N., Veroyatnost’ (Probability), Moscow: Nauka, 1989.zbMATHGoogle Scholar
  26. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    May, R.M., Simple Mathematical Models with Very Complicated Dynamics, Nature, 1976, vol. 261, pp. 459–467.CrossRefzbMATHGoogle Scholar
  28. 28.
    Nahi, N., Optimal Recursive Estimation with Uncertain Observation, IEEE Trans. Inform. Theory, 1969, vol. 15, no. 4, pp. 457–462.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Institute of Informatics Problems of the Federal Research Center “Computer Science and Control” of the Russian Academy of SciencesMoscowRussia
  2. 2.Moscow Aviation InstituteMoscowRussia

Personalised recommendations