Cluster Computing

, Volume 22, Supplement 3, pp 6795–6806 | Cite as

A novel aerodynamic parameter estimation algorithm via sigma point Rauch–Tung–Striebel smoother using expectation maximization

  • Wei ZhangEmail author
  • Hongwei Wang
  • Yilei Liu
  • Junyi Zuo
  • Heping Wang


We consider the problem of aerodynamic parameter estimation for aircraft dynamics modeled by a state space model where the statistic information of both the process and measurement noises are missing. To deal with the missing statistics, we propose in this work a new approach in which an augmented sigma point Rauch–Tung–Striebel (RTS) Kalman smoother is integrated with the expectation maximization (EM) algorithm. We define a new state vector by combining the original states and the unknown aerodynamic parameters. In addition, we impose a Gaussian random walk model for the unknown aerodynamic parameters and then build the extended state space model for the augmented RTS Kalman smoother. The expectation terms in the EM algorithm are approximated by the sigma point rule which is also applied in the augmented RTS Kalman smoother. Moreover, the non-convex optimization problem involved in the EM is solved in analytical forms rather than in numerical approaches. A comparative study of identifying the aerodynamic parameters of the flight test platform HFB-320 shows that the proposed approach achieves a substantial performance improvement over the existing ones, especially in terms of the convergence rate.


Aircraft system identification Parameter estimation Expectation maximization algorithm Rauch–Tung–Striebel smoother Third-degree cubature rule 



This work was supported by the National Natural Science Foundation of China (Nos. 11472222 and 61473227), the National Science Foundation of Shaanxi Province of China (Grant No. 2015JM6304), the Aviation Science Foundation of China (Grant No. 20151353018), and the Aerospace Technology Support Fund of China (Grant No. 2014-HT-XGD).


  1. 1.
    Jategaonkar, R.V.: Flight Vehicle System Identification: A Time Domain Methodology, vol. 216. AIAA, Reston, VA, USA (2006)CrossRefGoogle Scholar
  2. 2.
    Hamel, P.G., Jategaonkar, R.V.: Evolution of flight vehicle system identification. J. Aircr. 33(1), 9–28 (1996)CrossRefGoogle Scholar
  3. 3.
    Owens, D.Bruce, et al.: Overview of dynamic test techniques for flight dynamics research at NASA LaRC. AIAA Paper 3146, 5–8 (2006)Google Scholar
  4. 4.
    Iliff, K.W.: Parameter estimation for flight vehicles. J. Guid. Control Dyn. 12(5), 609–622 (1989)CrossRefGoogle Scholar
  5. 5.
    Kutluay, U., Gokmen, M., Bulent, P.: An application of equation error method to aerodynamic model identification and parameter estimation of a gliding flight vehicle. In: AIAA Atmospheric Flight Mechanics Conference (2009)Google Scholar
  6. 6.
    Rohlfs, M.: Identification of non-linear derivative models from Bo 105 flight test data. Aeronaut. J. 102(1011), 1–8 (1998)Google Scholar
  7. 7.
    Li, C.W., Zou, X.H.: Maximum likelihood method based on interior point algorithm for aircraft parameter identification. J. Aircr. 42(5), 1355–1358 (2005)CrossRefGoogle Scholar
  8. 8.
    Paris, A.C., Alaverdi, O.: Nonlinear aerodynamic model extraction from flight-test data for the S-3B viking. J. Aircr. 42(1), 26–32 (2005)CrossRefGoogle Scholar
  9. 9.
    Jategaonkar, R., Plaetschke, E.: Estimation of Aircraft Parameters Using Filter Error Methods and Extended Kalman Filter. Dt. Forschungs-u. VersuchsanstaltfürLuft-u, Raumfahrt (1988)Google Scholar
  10. 10.
    Garcia-Velo, J., Walker, B.: Aerodynamic parameter estimation for high-performance aircraft using extended Kalman filter. In: AIAA, No. 95–3500 (1995)Google Scholar
  11. 11.
    Chowdhary, Girish, Ravindra, J.: Aerodynamic parameter estimation from flight data applying extended and unscented Kalman filter. Aerosp. Sci. Technol. 14(2), 106–117 (2010)CrossRefGoogle Scholar
  12. 12.
    Smith, P.J: Joint state and parameter estimation using data assimilation with application to morphodynamic modelling. Ph.D. Thesis, University of Reading (2010)Google Scholar
  13. 13.
    Yokoyama, N.: Parameter estimation of aircraft dynamics via unscented smoother with expectation-maximization algorithm. J. Guid. Control Dyn. 34(2), 426–436 (2011)CrossRefGoogle Scholar
  14. 14.
    Santitissadeekorn, N., Chris, J.: Two-state filtering for joint state-parameter estimation (2014). arXiv:1403.5989
  15. 15.
    West, M., Liu, J.: Sequential Monte Carlo in Practices, pp. 197–223. Springer, New York (2001)Google Scholar
  16. 16.
    Julier, S.J., Uhlmann, J.K.: Unscented filtering and nonlinear estimation. Proc. IEEE 92(3), 401–422 (2004)CrossRefGoogle Scholar
  17. 17.
    Ito, K., Xiong, K.: Gaussian filters for nonlinear filtering problems. IEEE Trans. Autom. Control 45(5), 910–927 (2000)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Wang, H., Zhang, W., Zuo, J., Wang, H.: Generalized cubature quadrature Kalman filters: derivations and extensions. J. Syst. Eng. Electron. 28(3), 556–562 (2017)Google Scholar
  19. 19.
    Bell, B.M., James, V.B., Pillonetto, G.: An inequality constrained nonlinear Kalman-Bucy smoother by interior point likelihood maximization. Automatica 45(1), 25–33 (2009)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Aravkin, A.Y., et al.: An \(\ell_{1}\)-Laplace robust Kalman smoother. IEEE Trans. Autom. Control 56(12), 2898–2911 (2011)Google Scholar
  21. 21.
    AravkinA, Y., James, V.B., Pillonetto, G.: Optimization viewpoint on Kalman smoothing with applications to robust and sparse estimation. In: Compressed Sensing & Sparse Filtering, pp. 237–280. Springer, Berlin (2014)Google Scholar
  22. 22.
    Fraser, D., Potter, J.: The optimum linear smoother as a combination of two optimum linear filters. IEEE Trans. Autom. Control 14(4), 387–390 (1969)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Ye, J., Ding, Y.: Controllable keyword search scheme supporting multiple users. Future Gener. Comput. Syst. 81, 433–442 (2018)CrossRefGoogle Scholar
  24. 24.
    Gibbs, R.G.: Square root modified Bryson-Frazier smoother. IEEE Trans. Autom. Control 56(2), 542–546 (2011)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Rauch, H.E., Striebel, C.T., Tung, F.: Maximum likelihood estimates of linear dynamic systems. AIAA J. 3(8), 1445–1450 (1965)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Särkkä, S.: Bayesian Filtering and Smoothing, vol. 3. Cambridge University Press, Cambridge (2013)CrossRefGoogle Scholar
  27. 27.
    Sage, A.P., Husa, G.W.: Adaptive filtering with unknown prior statistics. Jt. Autom. Control Conf. 7, 760–769 (1969)Google Scholar
  28. 28.
    Myers, K., Tapley, B.: Adaptive sequential estimation with unknown noise statistics. IEEE Trans. Autom. Control 21(4), 520–523 (1976)CrossRefGoogle Scholar
  29. 29.
    Schön, T.B., Wills, A., Ninness, B.: System identification of nonlinear state-space models. Automatica 47(1), 39–49 (2011)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Arasaratnam, I., Haykin, S., Elliott, R.J.: Discrete-time nonlinear filtering algorithms using Gauss-Hermite quadrature. Process. IEEE 95(5), 953–977 (2007)CrossRefGoogle Scholar
  31. 31.
    Bhaumik, S.: Cubature quadrature Kalman filter. IET Signal Process. 7(7), 533–541 (2013)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Jia, B., Xin, M., Cheng, Y.: High-degree cubature Kalman filter. Automatica 49(2), 510–518 (2013)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Majeed, M., Narayan, Kar I.: Aerodynamic parameter estimation using adaptive unscented Kalman filter. Aircr. Eng. Aerosp. Technol. 85(4), 267–279 (2013)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Wei Zhang
    • 1
    • 2
    Email author
  • Hongwei Wang
    • 1
    • 2
  • Yilei Liu
    • 1
    • 3
  • Junyi Zuo
    • 1
    • 2
  • Heping Wang
    • 1
  1. 1.School of AeronauticsNorthwestern Polytechnical UniversityXi’anChina
  2. 2.Experimental Aircraft Design and Flight Testing Lab of ShaanxiXi’anChina
  3. 3.AVIC Aircraft Co., LtdXi’anPeople’s Republic of China

Personalised recommendations