Automation and Remote Control

, Volume 77, Issue 2, pp 261–276 | Cite as

Efficient adaptation of design parameters of derivative-free filters

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Abstract

The paper deals with state estimation of nonlinear discrete time stochastic dynamic systems with a focus on derivative-free filters. Design parameters of the filters are treated and an efficient way for their adaptation is proposed. The efficiency is based on observing a degree of nonlinearity of the nonlinear state and measurement functions at the working point by means of a non-Gaussianity measure. The adaptation is executed only if the nonlinearity is severe and the design parameter adaptation may bring a significant improvement of the estimate quality. Otherwise the adaptation is switched off to keep computational complexity of the filter low. The developed algorithm is illustrated using a numerical example of bearings-only target tracking.

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References

  1. 1.
    Blasch, E. and Kahler, D., Multiresolution EO/IR Target Track and ID, Proc. Int. Conf. Information Fusion, 2005.Google Scholar
  2. 2.
    Haykin, S. and Arasaratnam, I., Adaptive Signal Processing, Hoboken: Wiley, 2010.Google Scholar
  3. 3.
    Isermann Rolf, Fault-Diagnosis Systems. An Introduction from Fault Detection to Fault Tolerance, Berlin: Springer, 2006.Google Scholar
  4. 4.
    Stepanov, O.A., Primenenie teorii nelineinoi filtratsii v zadachakh obrabotki navigatsionnoi informatsii (Application of Nonlinear Filtering Theory for Processing Navigation Information), St. Petersburg: Elektropribor, 2003.Google Scholar
  5. 5.
    Grachev, A.N. and Shuryigin, S.V., Metodika sinteza iteratsionnykh algoritmov sovmestnogo otsenivaniya parametrov i sostoyaniya lineinykh diskretnykh sistem (The Technique of Synthesis of Iterative Algorithms for Joint Estimation of the Parameters and State of Linear Discrete Systems), in Tr. VII Mezhd. konf. “Identifikatsiya sistem i zadachi upravleniya” (Proc. VII Int. Conf. “System Identification and Control Problems”), Moscow, 2008, 204–219.Google Scholar
  6. 6.
    Volynskii, M.A., Gurov, I.P., and Zakharov, A.S., Dynamic Analysis of the Signals in Optical Coherent Tomography by the Method of Nonlinear Kalman Filtering, J. Opt. Technol., 2008, vol. 75, no. 10, pp. 682–686.CrossRefGoogle Scholar
  7. 7.
    Koshaev, D.A., Kalman Filter-Based Multialternative Method for Fault Detection and Estimation, Autom. Remote Control, 2010, vol. 71, no. 5, pp. 790–802.CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Stepanov, O.A. and Toropov, A.B., A Comparison of Linear and Nonlinear Optimal Estimators in Nonlinear Navigation Problems, Gyroscopy Navigation, 2010, vol. 1, no. 3, pp. 183–190.CrossRefGoogle Scholar
  9. 9.
    Stepanov, O.A. and Toropov, A.B., Linear Optimal Algorithms in Problems of Estimation with Nonlinear Measurements. Relation with the Kalman Type Algorithms, Izv. TulGU, Tekh. Vestn., 2012, no. 7, pp. 172–189.Google Scholar
  10. 10.
    Hmarskiy, P.A. and Solonar, A.S., Features of the UKF in the Polar Coordinates, Dokl. BGUIR, 2013, no. 2, pp. 79–86.Google Scholar
  11. 11.
    Stepanov, O.A. and Toropov, A.B., Application of the Monte Carlo Methods with Partial Analytical Integration Techniques to the Problem of Navigation System Aiding, in Proc. 20th St. Petersburg Int. Conf. on Integrated Navigation Systems, ICINS 2013, St. Petersburg, 2013, 298–301.Google Scholar
  12. 12.
    Berkovskii, N.A. and Stepanov, O.A., Error of Calculating the Optimal Bayesian Estimate Using the Monte Carlo Method in Nonlinear Problems, J. Comput. Syst. Sci. Int., 2013, vol. 52, no. 3, pp. 342–353.CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Mikhailov, N.V. and Koshaev, D.A., Navigation in a Geostationary Orbit Using GNSS Receiver Aided by the Satellite Dynamic Model, Gyroscopy Navigation, 2015, vol. 6, no. 2, pp. 87–100.CrossRefGoogle Scholar
  14. 14.
    Candy, J.V., Bayesian Signal Processing, Hoboken: Wiley, 2008.Google Scholar
  15. 15.
    Sorenson, H.W. and Alspach, D.L., Recursive Bayesian Estimation Using Gaussian Sums, Automatica, 1971, no. 7, pp. 465–479.CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    Anderson, B.D.O. and Moore, J.B., Optimal Filtering, Englewood Cliffs: Prentice Hall, 1979.MATHGoogle Scholar
  17. 17.
    Sarmavuori, J. and Sarkka, S., Fourier–Hermite Kalman Filter, IEEE Trans. Automat. Control, 2012, vol. 57, no. 6, pp. 1511–1515.CrossRefMathSciNetGoogle Scholar
  18. 18.
    Norgaard, M., Poulsen, N.K., and Ravn, O., New Developments in State Estimation for Nonlinear Systems, Automatica, 2000, vol. 36, no. 11, pp. 1627–1638.CrossRefMathSciNetGoogle Scholar
  19. 19.
    Julier, S.J. and Uhlmann, J.K., Unscented Filtering and Nonlinear Estimation, IEEE Rev., 2004, vol. 92, no. 3, pp. 401–421.CrossRefGoogle Scholar
  20. 20.
    Ito, K. and Xiong, K., Gaussian Filters for Nonlinear Filtering Problems, IEEE Trans. Automat. Control, 2000, vol. 45, no. 5, pp. 910–927.CrossRefMathSciNetMATHGoogle Scholar
  21. 21.
    Arasaratnam, I. and Haykin, S., Cubature Kalman Filters, IEEE Trans. Automat. Control, 2009, vol. 54, no. 6, pp. 1254–1269.CrossRefMathSciNetGoogle Scholar
  22. 22.
    Duník, J., Straka, O., and Šimandl, M., Stochastic Integration Filter, IEEE Trans. Automat. Control, 2013, vol. 58, no. 6, pp. 1561–1566.CrossRefMathSciNetGoogle Scholar
  23. 23.
    Sakai, A. and Kuroda, Y., Discriminatively Trained Unscented Kalman Filter for Mobile Robot Localization, J. Adv. Res. Mechan. Eng., 2010, vol. 1, no. 3, pp. 153–161.Google Scholar
  24. 24.
    Duník, J., Šimandl, M., and Straka, O., Adaptive Choice of Scaling Parameter in Derivative-Free Local Filters, Proc. Int. Conf. Information Fusion, Edinburgh, Great Britain, 2010, 22416.Google Scholar
  25. 25.
    Turner, R. and Rasmussen, C.E., Model Based Learning of Sigma Points in Unscented Kalman Filter, Neurocomput., 2012, vol. 80, pp. 47–53.CrossRefGoogle Scholar
  26. 26.
    Duník, J., Šimandl, M., and Straka, O., Unscented Kalman Filter Aspects and Adaptive Setting of Scaling Parameter, IEEE Trans. Automat. Control, 2012, vol. 57, no. 9, pp. 2411–2416.CrossRefMathSciNetGoogle Scholar
  27. 27.
    Chang, L., Hu, B., and Qin, F., Unscented Type Kalman Filter Limitation and Combination, IET Signal Process., 2013, vol. 7, no. 3, pp. 167–176.CrossRefMathSciNetGoogle Scholar
  28. 28.
    Duník, J., Straka, O., and Šimandl, M., Sigma-Point Set Rotation in Unscented Kalman Filter: Analysis and Adaptation, Proc. 19 IFAC World Congress, Cape Town, South Africa, 2014, 5951–5956.Google Scholar
  29. 29.
    Duník, J., Straka, O., and Šimandl, M., On Sigma-Point Set Rotation in Derivative-Free Filters, Proc. Int. Conf. Information Fusion, Salamanca, 2014, 1–8.Google Scholar
  30. 30.
    Rhudy, M., Gu, Y., Gross, J., et al., Evaluation of Matrix Square Root Operations for UKF within a UAV GPS/INS Sensor Fusion Application, Int. J. Navigation Observation, 2012, no. 1, pp. 1–11.Google Scholar
  31. 31.
    Straka, O., Duník, J., Šimandl, M., et al., Aspects and Comparison of Matrix Decompositions in Unscented Kalman Filter, Proc. 2013 American Control Conf., Washington, USA, 2013.Google Scholar
  32. 32.
    Straka, O., Duník, J., and Šimandl, M., Unscented Kalman Filter with Controlled Adaptation, Proc. 16 IFAC Sympos. on System Identification (SYSID), Brussels, 2012, vol. 16, pp. 906–911.Google Scholar
  33. 33.
    Šimandl, M. and Duník, J., Derivative-Free Estimation Methods New Results and Performance Analysis, Automatica, 2009, vol. 45, no. 7, pp. 1749–1757.CrossRefMathSciNetMATHGoogle Scholar
  34. 34.
    Zarei, J., Shokri, E., and Karimi, H.R., Convergence Analysis of Cubature Kalman Filter, Eur. Control Conf. (ECC), Strasbourg, 2014, 1367–1372.Google Scholar
  35. 35.
    Julier, S.J., Uhlmann, J.K., and Durrant-Whyte, H.F., A New Method for the Nonlinear Transformation of Means and Covariances in Filters and Estimators, IEEE Trans. Automat. Control, 2000, vol. 45, no. 3, pp. 477–482.CrossRefMathSciNetMATHGoogle Scholar
  36. 36.
    Jia, B., Xin, M., and Cheng, Y., Sparse Gauss–Hermite Quadrature Filter with Application to Spacecraft Attitude Estimation, J. Guidance, Control, Dynamics, 2011, vol. 27, no. 2, pp. 367–379.CrossRefGoogle Scholar
  37. 37.
    Stroud, A.H., Approximate Calculation of Multiple Integrals, Englewood Cliffs: Prentice Hall, 1971.MATHGoogle Scholar
  38. 38.
    Meyer, C.D., Matrix Analysis and Applied Linear Algebra, Philadelphia: SIAM, 2000.CrossRefMATHGoogle Scholar
  39. 39.
    Groves, P.D., Principles of GNSS, Inertial, and Multisensor Integrated Navigation Systems, Boston: Artech House, 2013, 2nd ed.Google Scholar
  40. 40.
    Straka, O., Duník, J., and Šimandl, M., Unscented Kalman Filter with Advanced Adaptation of Scaling Parameter, Automatica, 2014, vol. 5, no. 10, pp. 2657–2664.CrossRefGoogle Scholar
  41. 41.
    Straka, O., Duník, J., and Šimandl, M., Scaling Parameter in Unscented Transformation Analysis and Specification, Proc. 2012 American Control Conf., Montreal, Canada, 2012.Google Scholar
  42. 42.
    Walter, É. and Pronzato, L., Identification of Parametric Models from Experimental Data, New York: Springer, 1997.MATHGoogle Scholar
  43. 43.
    Li, X.R., Measure of Nonlinearity for Stochastic Systems, Proc. 15 Int. Conf. Information Fusion, Singapore, 2012.Google Scholar
  44. 44.
    Jones, E., Scalzo, M., Bubalo, A., et al., Measures of Nonlinearity for Single Target Tracking Problem, in Proc. SPIE, Signal Processing, Sensor Fusion, and Target Recognition, Orlando, Florida, USA, 2011.Google Scholar
  45. 45.
    Duník, J., Straka, O., and Šimandl, M., Nonlinearity and non-Gaussianity Measures for Stochastic Dynamic Systems, Proc. 16 Int. Conf. Information Fusion, Istanbul, 2013.Google Scholar
  46. 46.
    Brewer, J.W., Kronecker Products and Matrix Calculus in System Theory, IEEE Trans. Circ. Syst., 1978, vol. 25, no. 9, pp. 772–781.CrossRefMathSciNetMATHGoogle Scholar
  47. 47.
    Genz, A. and Monahan, J., A Stochastic Algorithm for High-Dimensional Integrals over Unbounded Regions with Gaussian Weight J. Comput. Appl. Math., 1999, vol. 112, no. 1, pp. 71–81.CrossRefMathSciNetMATHGoogle Scholar
  48. 48.
    Ristic, B., Arulampalam, S., and Gordon, N., Beyond the Kalman Filter Particle Filters for Tracking Applications, Boston: Artech House, 2004.MATHGoogle Scholar
  49. 49.
    Bar-Shalom, Y., Li, X.R., and Kirubarajan, T., Estimation with Applications to Tracking and Navigation Theory Algorithms and Software, New York: Wiley, 2001.CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.University of West BohemiaPilsenCzech Republic

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