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Modeling a nonlinear process using the exponential autoregressive time series model

  • Huan Xu
  • Feng Ding
  • Erfu Yang
Original Paper
  • 46 Downloads

Abstract

The parameter estimation methods for the nonlinear exponential autoregressive (ExpAR) model are investigated in this work. Combining the hierarchical identification principle with the negative gradient search, we derive a hierarchical stochastic gradient algorithm. Inspired by the multi-innovation identification theory, we develop a hierarchical-based multi-innovation identification algorithm for the ExpAR model. Introducing two forgetting factors, a variant of the hierarchical-based multi-innovation identification algorithm is proposed. Moreover, to compare and demonstrate the serviceability of these algorithms, a nonlinear ExpAR process is taken as an example in the simulation.

Keywords

Nonlinear ExpAR model Parameter estimation Hierarchical identification Multi-innovation identification Negative gradient search 

Notes

Acknowledgements

This work was supported by the 111 Project (B12018), the National Natural Science Foundation of China (No. 61273194) and the National First-Class Discipline Program of Light Industry Technology and Engineering (LITE2018-26).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Gan, M., Li, H.X., Peng, H.: A variable projection approach for efficient estimation of RBF-ARX model. IEEE Trans. Cybern. 45(3), 462–471 (2015)CrossRefGoogle Scholar
  2. 2.
    Ozaki, T.: Non-linear time series models for non-linear random vibrations. J. Appl. Probab. 17(1), 84–93 (1980)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ozaki, T.: The statistical analysis of perturbed limit cycle processes using nonlinear time series models. J. Time Ser. Anal. 3(1), 29–41 (1982)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Teräsvirta, T.: Specification, estimation, and evaluation of smooth transition autoregressive models. J. Am. Stat. Assoc. 89(425), 208–218 (1994)zbMATHGoogle Scholar
  5. 5.
    Merzougui, M., Dridi, H., Chadli, A.: Test for periodicity in restrictive EXPAR models. Commun. Stat. Theory Methods 45(9), 2770–2783 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen, G.Y., Gan, M., Chen, G.L.: Generalized exponential autoregressive models for nonlinear time series: stationarity, estimation and applications. Inf. Sci. 438, 46–57 (2018)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Zhou, Z.P., Liu, X.F.: State and fault estimation of sandwich systems with hysteresis. Int. J. Robust Nonlinear Control 28(13), 3974–3986 (2018)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Yu, C.P., Verhaegen, M., Hansson, A.: Subspace identification of local systems in one-dimensional homogeneous networks. IEEE Trans. Autom. Control 63(4), 1126–1131 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Pan, J., Ma, H., Jiang, X., et al.: Adaptive gradient-based iterative algorithm for multivariate controlled autoregressive moving average systems using the data filtering technique. Complexity (2018).  https://doi.org/10.1155/2018/9598307
  10. 10.
    Schoukens, M., Tiels, K.: Identification of block-oriented nonlinear systems starting from linear approximations: a survey. Automatica 85, 272–292 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Arqub, O.A., Abo-Hammour, Z.: Numerical solution of systems of second-order boundary value problems using continuous genetic algorithm. Inf. Sci. 279, 396–415 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Yu, C.P., Verhaegen, M.: Blind multivariable ARMA subspace identification. Automatica 66, 3–14 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Yu, C.P., Verhaegen, M.: Data-driven fault estimation of non-minimum phase LTI systems. Automatica 92, 181–187 (2018)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Chen, J., Jiang, B.: Modified stochastic gradient parameter estimation algorithms for a nonlinear two-variable difference system. Int. J. Control Autom. Syst. 14(6), 1493–1500 (2016)CrossRefGoogle Scholar
  15. 15.
    Chen, F.W., Garnier, H., Gilson, M.: Robust identification of continuous-time models with arbitrary time-delay from irregularly sampled data. J. Process Control 25, 19–27 (2015)CrossRefGoogle Scholar
  16. 16.
    Ding, F., Xu, L., Liu, X.M.: Signal modeling—part F: hierarchical iterative parameter estimation for multi-frequency signal models. J. Qingdao Univ. Sci. Technol. (Nat. Sci Ed.) 38(6), 1–13 (2017)Google Scholar
  17. 17.
    Ding, F.: Several multi-innovation identification methods. Digit. Signal Process. 20(4), 1027–1039 (2010)CrossRefGoogle Scholar
  18. 18.
    Li, L.W., Ren, X.M., Guo, F.M.: Modified multi-innovation stochastic gradient algorithm for Wiener–Hammerstein systems with backlash. J. Franklin Inst. 355(9), 4050–4075 (2018)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Cheng, S.S., Wei, Y.H., Sheng, D., Chen, Y.Q., Wang, Y.: Identification for Hammerstein nonlinear ARMAX systems based on multi-innovation fractional order stochastic gradient. Signal Process. 142, 1–10 (2018)CrossRefGoogle Scholar
  20. 20.
    Ding, F., Xu, L., Liu, X.M.: Signal modeling—part E: hierarchical parameter estimation for multi-frequency signal models. J. Qingdao Univ. Sci. Technol. (Nat. Sci Ed.) 38(5), 1–15 (2017)Google Scholar
  21. 21.
    Ding, F.: System Identification—Multi-Innovation Identification Theory and Methods. Science Press, Beijing (2016)Google Scholar
  22. 22.
    Zhang, B., Billings, S.A.: Identification of continuous-time nonlinear systems: the nonlinear difference equation with moving average noise (NDEMA) framework. Mech. Syst. Signal Process. 60–61, 810–835 (2015)CrossRefGoogle Scholar
  23. 23.
    El-Ajou, A., Arqub, O.A., Al-Smadi, M.: A general form of the generalized Taylor’s formula with some applications. Appl. Math. Comput. 256, 851–859 (2015)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Li, X., Zhu, D.Q.: An improved SOM neural network method to adaptive leader-follower formation control of AUVs. IEEE Trans. Ind. Electron. 65(10), 8260–8270 (2018)Google Scholar
  25. 25.
    Chen, M.Z., Zhu, D.Q.: A workload balanced algorithm for task assignment and path planning of inhomogeneous autonomous underwater vehicle system. IEEE Trans. Cognit. Dev. Syst. (2018).  https://doi.org/10.1109/TCDS.2018.2866984
  26. 26.
    Geng, F.Z., Qian, S.P.: An optimal reproducing kernel method for linear nonlocal boundary value problems. Appl. Math. Lett. 77, 49–56 (2018)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Li, X.Y., Wu, B.Y.: A new reproducing kernel collocation method for nonlocal fractional boundary value problems with non-smooth solutions. Appl. Math. Lett. 86, 194–199 (2018)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Pan, J., Li, W., Zhang, H.P.: Control algorithms of magnetic suspension systems based on the improved double exponential reaching law of sliding mode control. Int. J. Control Autom. Syst. 16(6), 2878–2887 (2018)CrossRefGoogle Scholar
  29. 29.
    Xu, L.: The parameter estimation algorithms based on the dynamical response measurement data. Adv. Mech. Eng. 9(11), 1–12 (2017).  https://doi.org/10.1177/1687814017730003 CrossRefGoogle Scholar
  30. 30.
    Xu, L., Ding, F.: Parameter estimation for control systems based on impulse responses. Int. J. Control Autom. Syst. 15(6), 2471–2479 (2017)CrossRefGoogle Scholar
  31. 31.
    Xu, L., Ding, F.: Iterative parameter estimation for signal models based on measured data. Circuits Syst. Signal Process. 37(7), 3046–3069 (2018)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.School of Internet of Things EngineeringJiangnan UniversityWuxiPeople’s Republic of China
  2. 2.College of Automation and Electronic EngineeringQingdao University of Science and TechnologyQingdaoPeople’s Republic of China
  3. 3.Space Mechatronic Systems Technology LaboratoryUniversity of StrathclydeGlasgowUnited Kingdom

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