Advertisement

Nonlinear system identification using fractional Hammerstein–Wiener models

  • Karima HammarEmail author
  • Tounsia Djamah
  • Maamar Bettayeb
Original paper

Abstract

The paper deals with identification of fractional order nonlinear systems based on Hammerstein–Wiener models. An output error approach is developed using the robust Levenberg–Marquardt algorithm. It presents the difficulty of the parametric sensitivity functions calculation which requires a heavy computational load at each iteration. To overcome this drawback, the fractional nonlinear system is reformulated under a regression form, and the gradient and the Hessian can be obtained in a closed form without using the parametric sensitivity functions. The method’s efficiency is confirmed on numerical simulations, and its feasibility is illustrated with its application to the modeling of an experimental arm robot.

Keywords

Identification Nonlinear system Fractional order systems Hammerstein–Wiener model Levenberg–Marquardt algorithm Regression 

Notes

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest

References

  1. 1.
    Chen, F., Ding, F.: Recursive least squares identification algorithms for multiple-input nonlinear Box-Jenkins systems using the maximum likelihood principle. J. Comput. Nonlinear Dyn. 11(2), 021005 (2016)CrossRefGoogle Scholar
  2. 2.
    Zhao, W., Chen, H.F.: Identification of Wiener, Hammerstein, and NARX systems as Markov chains with improved estimates for their nonlinearities. Syst. Control Lett. 61(12), 1175–1186 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Jia, L., Li, X., Chiu, M.S.: The identification of neuro-fuzzy based MIMO Hammerstein model with separable input signals. Neurocomputing 174, 530–541 (2016)CrossRefGoogle Scholar
  4. 4.
    Mathews, V.J., Sicuranza, G.: Polynomial Signal Processing, p. 452. Wiley, New York (2000)Google Scholar
  5. 5.
    Vörös, J.: Recursive identification of time-varying non-linear cascade systems with static input and dynamic output non-linearities. Trans. Inst. Meas. Control 40(3), 896–902 (2018)CrossRefGoogle Scholar
  6. 6.
    Taringou, F., Srinivasan, B., Malhame, R., Ghannouchi, F.: Hammerstein-Wiener model for wideband RF transmitters using base-band data. In: Microwave Conference, Asia-Pacific (2007)Google Scholar
  7. 7.
    Crama, P., Schoukens, J.: Hammerstein-Wiener system estimator initialization. Automatica 40(9), 1543–1550 (2004)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Luo, X.S., Song, Y.D.: Data-driven predictive control of Hammerstein-Wiener systems based on subspace identification. Inf. Sci. 422, 446–447 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kumar, P., Devanand, R., Schoen, M. P.: sEMG and skeletal muscle force modeling: a nonlinear hammerstein-wiener model, multiple regression model and entropy based threshold approach. In: 2nd International Electronic Conference on Entropy and Its Applications. Multidisciplinary Digital Publishing Institute (2015)Google Scholar
  10. 10.
    Zambrano, D., Tayamon, S., Carlsson, B., Wigren, T.: Identification of a discrete-time nonlinear Hammerstein-Wiener model for a selective catalytic reduction system. Am. Control Conf. 2011, 78–83 (2011)Google Scholar
  11. 11.
    Roodgar Amoli, E., Salehinia, S., Ghoreishi, M.: Comparative study of expert predictive models based on adaptive neuro fuzzy inference system, nonlinear autoregressive exogenous and Hammerstein-Wiener approaches for electrical discharge machining performance: Material removal rate and surface roughness. J. Eng. Manuf. 230(9), 1690–1701 (2016)CrossRefGoogle Scholar
  12. 12.
    Nadimi, E.S., Green, O., Blanes-Vidal, V., Larsen, J.J., Christensen, L.P.: Hammerstein-Wiener model for the prediction of temperature variations inside silage stack-bales using wireless sensor networks. Biosyt. Eng. 112(3), 236–247 (2012)CrossRefGoogle Scholar
  13. 13.
    Abu Arqub, O.: Solutions of time-fractional Tricomi and Keldysh equations of Dirichlet functions types in Hilbert space. Numer. Methods Partial Differ. Equ. 34(5), 1759–1780 (2018)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Arqub, O.A., Maayah, B.: Numerical solutions of integrodifferential equations of Fredholm operator type in the sense of the Atangana-Baleanu fractional operator. Chaos, Solitons Fractals 117, 117–124 (2018)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Abu Arqub, O.: Numerical algorithm for the solutions of fractional order systems of Dirichlet function types with comparative analysis. Fundam. Inf. 166(2), 111–137 (2019)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Abu Arqub, O.: Application of residual power series method for the solution of time-fractional Schrödinger equations in one-dimensional space. Fundam. Inf. 166(2), 87–110 (2019)CrossRefGoogle Scholar
  17. 17.
    Burov, S., Barkai, E.: Fractional Langevin equation: overdamped, underdamped, and critical behaviors. Phys. Rev. E 78(3), 031112 (2008)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Tang, Y., Zhang, X., Hua, C., Li, L., Yang, Y.: Parameter identification of commensurate fractional-order chaotic system via differential evolution. Phys. Lett. A. 376(4), 457–464 (2012)CrossRefGoogle Scholar
  19. 19.
    Djamah, T., Mansouri, R., Djennoune, S., Bettayeb, M.: Heat transfer modeling and identification using fractional order state space models. Journal europaan des systemès automatisès. 42(6–8), 939–951 (2008)zbMATHGoogle Scholar
  20. 20.
    Isfer, L.A.D., Lenzi, M.K., Lenzi, E.K.: Identification of biochemical reactors using fractional differential equations. Lat. Am. Appl. Res. 40(2), 193–198 (2010)Google Scholar
  21. 21.
    Wang, Y., Ding, F.: Recursive least squares algorithm and gradient algorithm for Hammerstein-Wiener systems using the data filtering. Nonlinear Dyn. 84(2), 1045–1053 (2016)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Wang, D.Q., Ding, F.: Hierarchical least squares estimation algorithm for Hammerstein-Wiener systems. IEEE Signal Process. Lett. 19(12), 825–828 (2012)CrossRefGoogle Scholar
  23. 23.
    Brouri, A., Kadi, L., Slassi, S.: Frequency identification of Hammerstein-Wiener systems with backlash input nonlinearity. Int. J. Control Autom. Syst. 15(5), 2222–2232 (2017)CrossRefGoogle Scholar
  24. 24.
    Ni, B., Gilson, M., Garnier, H.: Refined instrumental variable method for Hammerstein-Wiener continuous-time model identification. IET Control Theory. A. 7(9), 1276–1286 (2013)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Hammar, K., Djamah, T., Bettayeb, M. (2015, December). Fractional Hammerstein system identification using particle swarm optimization. In: The 7th International Conference on Modelling, Identification and Control ( ICMIC), Sousse, Tunisia (2015)Google Scholar
  26. 26.
    Moghaddam, M.J., Mojallali, H., Teshnehlab, M.: Recursive identification of multiple-input single-output fractional-order Hammerstein model with time delay. Appl. Soft Comput. 70, 486–500 (2018)CrossRefGoogle Scholar
  27. 27.
    Rahmani, M.R., Farrokhi, M.: Identification of neuro-fractional Hammerstein systems: a hybrid frequency-/time-domain approach. Soft Comput. 22(24), 8097–8106 (2017)CrossRefGoogle Scholar
  28. 28.
    Sersour, L., Djamah, T., Bettayeb, M.: Nonlinear system identification of fractional Wiener models. Nonlinear Dyn. 92(4), 1493–1505 (2018)CrossRefGoogle Scholar
  29. 29.
    Sersour, L., Djamah, T., Bettayeb, M.: Identification of Wiener fractional model using self-adaptive velocity particle swarm optimization. In: The 7th International Conference on Modelling, Identification and Control ( ICMIC 2015), Sousse, Tunisia (2015)Google Scholar
  30. 30.
    Allafi, W., Zajic, I., Uddin, K., Burnham, K.J.: Parameter estimation of the fractional-order Hammerstein-Wiener model using simplified refined instrumental variable fractional-order continuous time. IET. Control Theory A. 11(15), 2591–2598 (2017)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Ray, S.S., Atangana, A., Noutchie, S.C., Kurulay, M., Bildik, N., Kilicman, A.: Fractional calculus and its applications in applied mathematics and other sciences. Math. Probl. Eng. 2014, 2 (2014)CrossRefGoogle Scholar
  32. 32.
    Ionescu, C., Kelly, J.F.: Fractional calculus for respiratory mechanics: power law impedance, viscoelasticity, and tissue heterogeneity. Chaos Solitons Fractals 102, 433–440 (2017)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Podlubny, I.: Fractional-order systems and \(PI^{\lambda }D^{\mu }\) controllers. IEEE Trans. Autom. Control 44(1), 208–214 (1999)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, San Fransisco (1993)zbMATHGoogle Scholar
  35. 35.
    Bai, E.W.: An optimal two-stage identification algorithm for Hammerstein-Wiener nonlinear systems. Automatica 34(3), 333–338 (1998)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Wang, D., Ding, F.: Extended stochastic gradient identification algorithms for Hammerstein-Wiener ARMAX systems. Comput. Math. Appl. 56(12), 3157–3164 (2008)MathSciNetCrossRefGoogle Scholar
  37. 37.
    De Moor, B., De Gersem, P., De Schutter, B., Favoreel, W.: DAISY: A database for identification of systems. J A 38(3), 4–5 (1997)Google Scholar
  38. 38.
    Schoukens, J., Suykens, J., Ljung, L.: Wiener-hammerstein benchmark, 15th IFAC Symposium on System Identification. St. Malo, France (2009)Google Scholar
  39. 39.
    Ugalde, H.M.R., Carmona, J.C., Reyes-Reyes, J., Alvarado, V.M., Mantilla, J.: Computational cost improvement of neural network models in black box nonlinear system identification. Neurocomputing 166, 96–108 (2015)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • Karima Hammar
    • 1
    Email author
  • Tounsia Djamah
    • 1
  • Maamar Bettayeb
    • 2
  1. 1.Laboratoire de Conception et Conduite des Systèmes de, Production (L2CSP)UMMTOTizi-OuzouAlgérie
  2. 2.Department of Electrical and Computer EngineeringUniversity of Sharjah UAE and (CEIES), King Abdulaziz UniversityJeddahSaudi Arabia

Personalised recommendations