This study investigates the shortcomings of existing unscented type Kalman filters (UTKFs) used for state estimation problem of non-linear stochastic dynamic systems. There are three kinds of UTKFs – the traditional unscented Kalman filter, the cubature Kalman filter and the transformed unscented Kalman filter. It was demonstrated in the past that these algorithms could capture the posterior mean and covariance accurately to the second order for any non-linearity when propagated through the true non-linear system. However, they yield different information on the higher order terms. Owing to the dualistic effect of higher order terms on the performance of these algorithms, it is desirable to come up with some solution to preserve the positive effect of this information in a single algorithm. Based on the assumption that the state measurements are Guassian, two methods are proposed in this work as the suitable candidates. The first one is an adaptive method that chooses the UTKF achieving the highest value of the likelihood function. The second method treats these algorithms as sub-filters and uses the partitioning approach to obtain an overall estimate. The numerical simulations show that the proposed methods can have comparable performances as one of the best UTKFs for the problems under consideration.

References

1. 1)
  - 2. Farina, A., Ristic, B., Benvenuti, D.: ‘Tracking a ballistic target: comparison of several nonlinear filters’, IEEE Trans. Aerosp. Electron. Syst., 2002, 38, (3), pp. 854–867 (doi: 10.1109/TAES.2002.1039404).
2. 2)
  - 20. Lainiotis, D.G., Papaparaskeva, P.: ‘A new class of efficient adaptive nonlinear filters (ANLF)’, IEEE Trans. Signal Process., 1998, 46, (6), pp. 1730–1737 (doi: 10.1109/78.678509).
3. 3)
  - 22. Zhai, Y.: ‘Improved nonlinear filtering for target tracking’. Ph.D. dissertation, University of Oklahoma Graduate College, Norman, Oklahoma, 2007.
4. 4)
  - 10. Chang, L.B., Hu, B.Q., Chang, G.B., Li, A.: ‘Huber-based novel robust unscented Kalman filter’, IET Sci. Meas. Technol., 2012, 6, (6), pp. 502–509 (doi: 10.1049/iet-smt.2011.0169).
5. 5)
  - 5. Arasaratnam, I., Haykin, S.: ‘Cubature Kalman filters’, IEEE Trans. Autom. Control, 2009, 54, (6), pp. 1254–1269 (doi: 10.1109/TAC.2009.2019800).
6. 6)
  - 13. Ito, K., Xiong, K.: ‘Gaussian filters for nonlinear filtering problems’, IEEE Trans. Autom. Control, 2000, 45, (5), pp. 910–927 (doi: 10.1109/9.855552).
7. 7)
  - 4. Simon, D.: ‘Optimal state estimation’ (John Wiley & Sons, 2006).
8. 8)
  - 1. Bar-Shalom, Y., Li, X.R., Kirubarajan, T.: ‘Estimation with applications to tracking and navigation’ (New York, Wiley, 2001).
9. 9)
  - 11. Arasaratnam, I., Haykin, S., Hurd, T.R.: ‘Cubature Kalman filtering for continuous-discrete systems: theory and simulations’, IEEE Trans. Signal Process., 2010, 58, (10), pp. 4977–4993 (doi: 10.1109/TSP.2010.2056923).
10. 10)
  - 21. Lainiotis, D.G., Papaparaskeva, P.: ‘Efficient algorithms of clustering adaptive nonlinear filters’, IEEE Trans. Autom. Control, 1999, 44, (7), pp. 1454–1459 (doi: 10.1109/9.774122).
11. 11)
  - 18. Wu, Y.X., Hu, D.W., Wu, M.P., Hu, X.P.: ‘Unscented Kalman filtering for additive noise case: augmented versus nonaugmented’, IEEE Signal Process. Lett., 2005, 12, (5), pp. 357–360 (doi: 10.1109/LSP.2005.845592).
12. 12)
  - 17. Hu, B.Q., Chang, L.B., Li, A., Qin, F.: ‘Comment on ‘highly efficient sigma point filter for spacecraft attitude and rate estimation’, Math. Probl. Eng., 2012, article id 170391, 5 pages. doi: 10.1155/2012/170391.
13. 13)
  - 7. Julier, S., Uhlmann, J., Durrant-Whyte, H.F.: ‘A new method for the nonlinear transformation of means and covariances in filters and estimators’, IEEE Trans. Autom. Control, 2000, 45, (3), pp. 477–482 (doi: 10.1109/9.847726).
14. 14)
  - 12. Chang, L.B., Hu, B.Q., Li, A., Qin, F.: ‘Transformed unscented Kalman filter’, IEEE Trans. Autom. Control, 2013, 58, (1), pp. 252–257 (doi: 10.1109/TAC.2012.2204830).
15. 15)
  - 9. Chang, L.B., Hu, B.Q., Chang, G.B., Li, A.: ‘Multiple outliers suppression derivative-free filter based on unscented transformation’, J. Guid. Control Dyn., 2012, 35, (6), pp. 1902–1907 (doi: 10.2514/1.57576).
16. 16)
  - 19. Dunik, J., Simandl, M., Straka, O.: ‘Unscented Kalman filter: aspects and adaptive setting of scaling parameter’, IEEE Trans. Autom. Control, to be published, doi: 10.1109/TAC.2012.2188424.
17. 17)
  - 23. Chang, L.B., Hu, B.Q., Chang, G.B., Li, A.: ‘Marginalised iterated unscented Kalman filter’, IET Control Theory Appl., 2012, 6, (6), pp. 847–854 (doi: 10.1049/iet-cta.2011.0457).
18. 18)
  - 8. Julier, S., Uhlmann, J.: ‘Unscented filtering and nonlinear estimation’, Proc. IEEE, 2004, 92, (3), pp. 401–422 (doi: 10.1109/JPROC.2003.823141).
19. 19)
  - 3. Merwe, R.v.d.: ‘Sigma-point Kalman filters for probabilistic inference in dynamic state-space models’. Ph.D. OGI School of Sci. & Eng., Oregon Health & Sci. Univ., Portland, USA, 2004.
20. 20)
  - 14. Kotecha, J.H., Djuric, P.M.: ‘Gaussian particle filtering’, IEEE Trans. Signal Process., 2003, 51, (10), pp. 2592–2601 (doi: 10.1109/TSP.2003.816758).
21. 21)
  - 15. Stroud, A.H.: ‘Remarks on the disposition of points in numerical integration formulas’, Math. Comput., 1957, 11, (60), pp. 257–261 (doi: 10.1090/S0025-5718-1957-0093911-3).
22. 22)
  - 16. Xiu, D.B.: ‘Numerical integration formulas of degree two’, Appl. Numer. Math., 2008, 58, pp. 1515–1520 (doi: 10.1016/j.apnum.2007.09.004).
23. 23)
  - 6. Wu, Y.X., Hu, D.W., Wu, M.P., Hu, X.P.: ‘A numerical-integration perspective on Gaussian filters’, IEEE Trans. Signal Process., 2006, 56, (8), pp. 2910–2921 (doi: 10.1109/TSP.2006.875389).

Unscented type Kalman filter: limitation and combination

References

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