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Identification of fractional Hammerstein system with application to a heating process

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Abstract

In this paper, fractional Hammerstein system identification is considered, where the linear block is of fractional order. The original discrete Hammerstein system is first converted to a fractional polynomial nonlinear state-space model (PNLSS), which allows a better parameterization of the model. An output-error identification approach is developed based on the robust Levenberg–Marquardt algorithm, whose nevralgic point is the calculation of parametric sensitivity functions. These last are developed as a multivariable fractional PNLSS model which effectively reduces the computational effort. Various simulations are used to test the method’s efficiency and its statistical performance is analyzed using Monte Carlo simulation. Finally, the method is evaluated through a heating experimental benchmark. The obtained results show good agreement with the real system outputs.

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References

  1. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier Science, Amsterdam (2006)

    MATH  Google Scholar 

  2. Podlubny, I.: Fractional-order systems and \(PI^{\lambda }D^{\mu }\) controllers. IEEE Trans. Autom. Contr. 44(1), 208–214 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Machado, J.T., Guest, J.A.T.: Special issue on fractional calculus and applications. Nonlinear Dyn. 29(1–4), 3–22 (2002)

    Google Scholar 

  4. Mansouri, R., Bettayeb, M., Djamah, T., Djennoune, S.: Vector fitting fractional system identification using particle swarm optimization. Appl. Math. Comput. 206, 510–520 (2008)

    MathSciNet  MATH  Google Scholar 

  5. Dai, Y., Wei, Y., Hu, Y., Wang, Y.: Modulating function-based identification for fractional order systems. Neurocomputing 173, 1959–1966 (2016)

    Article  Google Scholar 

  6. Cui, R., Wei, Y., Chen, Y., Cheng, S., Wang, Y.: An innovative parameter estimation for fractional-order systems in the presence of outliers. Nonlinear Dyn. 89(1), 453–463 (2017)

    Article  MATH  Google Scholar 

  7. Cois, O., Oustaloup, A., Poinot, T., Battaglia, J. L.: Fractional state variable filter for system identification by fractional model. In: Proceedings of European Control Conference (ECC), Porto, Portugal (2001)

  8. Djamah, T., Mansouri, R., Djennoune, S., Bettayeb, M.: Heat transfer modeling and identification using fractional order state space models. J. Eur. Des. Syst. Autom. 42(6–8), 939–951 (2008)

    MATH  Google Scholar 

  9. Djamah, T., Bettayeb, M., Djennoune, S.: Identification of multivariable fractional order systems. Asian J. Control 15(3), 741–750 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. 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)

    Article  Google Scholar 

  11. 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)

    Article  MathSciNet  MATH  Google Scholar 

  12. 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)

    Article  Google Scholar 

  13. Schoukens, M., Pintelon, R., Rolain, Y.: Identification of Wiener-Hammerstein systems by a nonparametric separation of the best linear approximation. Automatica 50(2), 628–634 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Smith, J.G., Kamat, S., Madhavan, K.P.: Modeling of pH process using wavelet based Hammerstein model. J. Process Contr. 17(6), 551–561 (2007)

    Article  Google Scholar 

  15. Hunter, I.W., Korenberg, M.J.: The identification of nonlinear biological systems: Wiener and Hammerstein cascade models. Biol. Cybern. 55(2–3), 135–144 (1986)

    MathSciNet  MATH  Google Scholar 

  16. Wang, D.: Hierarchical parameter estimation for a class of MIMO Hammerstein systems based on the reframed models. Appl. Math. Lett. 57, 13–19 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Wang, D., Ding, F., Ximei, L.: Least squares algorithm for an input nonlinear system with a dynamic subspace state space model. Nonlinear Dyn. 75(1–2), 49–61 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zhang, H.T., Li, H.X., Chen, G.: Dual-mode predictive control algorithm for constrained Hammerstein systems. Int. J. Control 81(10), 1609–1625 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Deng, K., Ding, F.: Newton iterative identification method for an input nonlinear finite impulse response system with moving average noise using the key variables separation technique. Nonlinear Dyn. 76(2), 1195–1202 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Shen, Q., Ding, F.: Least squares identification for Hammerstein multi-input multi-output systems based on the key-term separation technique. Circ. Syst. Signal Process. 35(10), 3745–3758 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Chen, J., Zhang, Y., Ding, R.: Gradient-based parameter estimation for input nonlinear systems with ARMA noises based on the auxiliary model. Nonlinear Dyn. 72(4), 865–871 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Cheng, S., Wei, Y., Sheng, D., Chen, Y., Wang, Y.: Identification for Hammerstein nonlinear ARMAX systems based on multi-innovation fractional order stochastic gradient. Signal Process. 142, 1–10 (2018)

    Article  Google Scholar 

  23. Rahmani, M.R., Farrokhi, M.: Identification of neuro-fractional Hammerstein systems: a hybrid frequency-/time-domain approach. Soft Comput. 22(24), 8097–8106 (2017)

    Article  MATH  Google Scholar 

  24. Liao, Z., Zhu, Z., Liang, S., Peng, C., Wang, Y.: Subspace identification for fractional order Hammerstein systems based on instrumental variables. Int. J. Control Autom. 10(5), 947–953 (2012)

    Article  Google Scholar 

  25. Ivanov, D.V.: Identification discrete fractional order Hammerstein systems. In: International Siberian Conference on Control and Communications (SIBCON), Omsk, Russia (2015)

  26. Aoun, M., Malti, R., Olivier, C., Oustaloup, A.: System identification using fractional Hammerstein models. IFAC Proc. 35(1), 265–269 (2002)

    Article  Google Scholar 

  27. Sersour, L., Djamah, T., Bettayeb, M.: Nonlinear system identification of fractional Wiener models. Nonlinear Dyn. 92(4), 1493–1505 (2018)

    Article  Google Scholar 

  28. Zhao, Y., Li, Y., Zhou, F., Zhou, Z., Chen, Y.: An iterative learning approach to identify fractional order KiBaM model. IEEE/CAA J. Autom. Sin. 4(2), 322–331 (2017)

    Article  MathSciNet  Google Scholar 

  29. Zhao, Y., Li, Y., Chen, Y.: Complete parametric identification of fractional order Hammerstein systems. In: Proceedings of International Conference on Fractional Differentiation and Its Applications (ICFDA), Catania, Italy (2014)

  30. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, San Fransisco (1993)

    MATH  Google Scholar 

  31. Rossikhin, Y.A., Shitikova, M.V.: Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results. Appl. Mech. Rev. 63(1), 010801 (2010)

    Article  Google Scholar 

  32. Magin, R.L.: Fractional Calculus in Bioengineering. Begell House Inc., Chicago (2006)

    Google Scholar 

  33. Oldham, K.B., Spanier, J.: Application of Fractional Calculus for Dynamic Problems of Solid Mechanics: Novel Trends and Recent Results the Fractional Calculus. Academic Press, New York (1974)

    Google Scholar 

  34. Zhao, Y., Li, Y., Zhou, F.: Adaptive memory identification of fractional order systems. Discontin. Nonlinearity Complex. 4(4), 413–428 (2015)

    Article  MATH  Google Scholar 

  35. Dzieliński, A., Sierociuk, D.: Stability of discrete fractional order state-space systems. J. Vib. Control 14(9–10), 1543–1556 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  36. Paduart, J., Lauwers, L., Pintelon, R., Schoukens, J.: Identification of a Wiener-Hammerstein system using the polynomial nonlinear state space approach. Control Eng. Pract. 20(11), 1133–1139 (2012)

    Article  Google Scholar 

  37. Marquardt, D.W.: An algorithm for least squares estimation of nonlinear parameters. J. Soc. Ind. Appl. Math. 11(2), 413–441 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  38. De Moor, B.L.R.: Daisy: database for the identification of systems, department of electrical engineering (ed) ESAT/SISTA, KU Leuven, Belgium (2015). http://homes.Esat.Kuleuven.Be/smc/daisy

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Authors and Affiliations

  1. Laboratoire de Conception et Conduite des Systèemes de Production (L2CSP), UMMTO, BP 17 RP, 15000, Tizi-Ouzou, Algeria

    Karima Hammar & Tounsia Djamah

  2. Department of Electrical and Computer Engineering, University of Sharjah UAE and (CEIES), King Abdulaziz University, Jeddah, Saudi Arabia

    Maamar Bettayeb

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  1. Karima Hammar
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  2. Tounsia Djamah
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  3. Maamar Bettayeb
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Correspondence to Karima Hammar.

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Hammar, K., Djamah, T. & Bettayeb, M. Identification of fractional Hammerstein system with application to a heating process. Nonlinear Dyn 96, 2613–2626 (2019). https://doi.org/10.1007/s11071-019-04946-2

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