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Hierarchical Stochastic Gradient Algorithm and its Performance Analysis for a Class of Bilinear-in-Parameter Systems

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Abstract

This paper considers the parameter identification for a special class of nonlinear systems, i.e., bilinear-in-parameter systems. Based on the hierarchical identification principle, a hierarchical stochastic gradient (HSG) estimation algorithm is presented. The basic idea is to decompose a bilinear-in-parameter system into two subsystems and to derive the HSG identification algorithm for estimating the system parameters by replacing the unknown variables in the information vectors with their estimates obtained at the previous time. The convergence analysis of the proposed algorithm indicates that the parameter estimation errors converge to zero under persistent excitation conditions. The simulation results show that the proposed algorithm is effective.

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References

  1. R. Abrahamssona, S.M. Kay, P. Stoica, Estimation of the parameters of a bilinear model with applications to submarine detection and system identification. Digit. Signal Process. 17(4), 756–773 (2007)

    Article  Google Scholar 

  2. A. Atitallah, S. Bedoui, K. Abderrahim, Identification of wiener time delay systems based on hierarchical gradient approach. in The 8th Vienna International Conference on Mathematical Modelling—MATHMOD, IFAC-Papers OnLine 48(1), 403–408 (2015)

  3. E.W. Bai, An optimal two-stage identification algorithm for Hammerstein–Wiener nonlinear systems. Automatica 34(3), 333–338 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. E.W. Bai, A blind approach to the Hammerstein–Wiener model identification. Automatica 38(6), 967–979 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. E.W. Bai, Y. Liu, Least squares solutions of bilinear equations. Syst. Control Lett. 55(6), 466–472 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. X. Cao, D.Q. Zhu, S.X. Yang, Multi-AUV target search based on bioinspired neurodynamics model in 3-D underwater environments. IEEE Trans. Neural Netw. Learn. Syst. (2016). doi:10.1109/TNNLS.2015.2482501

    MathSciNet  Google Scholar 

  7. Z.Z. Chu, D.Q. Zhu, S.X. Yang, Observer-based adaptive neural network trajectory tracking control for remotely operated Vehicle. IEEE Trans. Neural Netw. Learn. Syst. (2016). doi:10.1109/TNNLS

    Google Scholar 

  8. F. Ding, G.J. Liu, X.P. Liu, Parameter estimation with scarce measurements. Automatica 47(8), 1646–1655 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. F. Ding, X.M. Liu, M.M. Liu, The recursive least squares identification algorithm for a class of Wiener nonlinear systems. J. Franklin Inst. 353(7), 1518–1526 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. F. Ding, X.M. Liu, X.Y. Ma, Kalman state filtering based least squares iterative parameter estimation for observer canonical state space systems using decomposition. J. Comput. Appl. Math. 301, 135–143 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. F. Ding, X.H. Wang, Q.J. Chen, Y.S. Xiao, Recursive least squares parameter estimation for a class of output nonlinear systems based on the model decomposition. Circuits Syst. Signal Process. (2016). doi:10.1007/s00034-015-0190-6

    MathSciNet  MATH  Google Scholar 

  12. F. Ding, X.M. Liu, Y. Gu, An auxiliary model based least squares algorithm for a dual-rate state space system with time-delay using the data filtering. J. Franklin Inst. 353(2), 398–408 (2016)

    Article  MathSciNet  Google Scholar 

  13. F. Ding, Y. Gu, Performance analysis of the auxiliary model-based stochastic gradient parameter estimation algorithm for state-space systems with one-step state delay. Circuits Syst. Signal Process. 32(2), 585–599 (2013)

    Article  MathSciNet  Google Scholar 

  14. M. Gilson, P. Van den Hof, Instrumental variable methods for closed-loop system identification. Automatica 41(2), 241–249 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. G.C. Goodwin, K.S. Sin, Adaptive Filtering Prediction and Control (Prentice-Hall, Englewood Cliffs, 1984)

    MATH  Google Scholar 

  16. A. Haryanto, K.S. Hong, Maximum likelihood identification of Wiener–Hammerstein models. Mech. Syst. Signal Process. 41(1–2), 54–70 (2013)

    Article  Google Scholar 

  17. Y. Ji, X.M. Liu, F. Ding, New criteria for the robust impulsive synchronization of uncertain chaotic delayed nonlinear systems. Nonlinear Dyn. 79(1), 1–9 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Y. Ji, X.M. Liu, Unified synchronization criteria for hybrid switching-impulsive dynamical networks. Circuits Syst. Signal Process. 34(5), 1499–1517 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. H. Li, Y. Shi, W. Yan, On neighbor information utilization in distributed receding horizon control for consensus-seeking. IEEE Trans. Cybern. (2016). doi:10.1109/TCYB.2015.2459719

    Google Scholar 

  20. H. Li, Y. Shi, W. Yan, Distributed receding horizon control of constrained nonlinear vehicle formations with guaranteed \(\gamma \)-gain stability. Automatica 68, 148–154 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. H. Li, Y. Shi, Robust H-infinity filtering for nonlinear stochastic systems with uncertainties and random delays modeled by Markov chains. Automatica 48(1), 159–166 (2012)

    Article  MathSciNet  Google Scholar 

  22. L. Ljung, System Identification: Theory for the User, 2nd edn. (Prentice Hall, Englewood Cliffs, 1999)

  23. J. Pan, X.H. Yang, H.F. Cai, B.X. Mu, Image noise smoothing using a modified Kalman filter. Neurocomputing 173, 1625–1629 (2016)

    Article  Google Scholar 

  24. J.D. Wang, Q.H. Zhang, L. Ljung, Revisiting Hammerstein system identification through the two-stage algorithm for bilinear parameter estimation. Automatica 45(11), 2627–2633 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. D.Q. Wang, 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 

  26. X.H. Wang, F. Ding, F.E. Alsaadi, T. Hayat, Convergence analysis of the hierarchical least squares algorithm for bilinear-in-parameter systems. Circuits Syst. Signal Process. (2016). doi:10.1007/s00034-016-0278-7

    MathSciNet  Google Scholar 

  27. T.Z. Wang, J. Qi, H. Xu et al., Fault diagnosis method based on FFT–RPCA–SVM for cascaded-multilevel inverter. ISA Trans. 60, 156–163 (2016)

    Article  Google Scholar 

  28. T.Z. Wang, H. Wu, M.Q. Ni et al., An adaptive confidence limit for periodic non-steady conditions fault detection. Mech. Syst. Signal Process. 72–73, 328–345 (2016)

    Article  Google Scholar 

  29. X.H. Wang, F. Ding, Recursive parameter and state estimation for an input nonlinear state space system using the hierarchical identification principle. Signal Process. 117, 208–218 (2015)

    Article  Google Scholar 

  30. D.Q. Wang, F. Ding, Parameter estimation algorithms for multivariable Hammerstein CARMA systems. Inf. Sci. 355, 237–248 (2016)

    Article  MathSciNet  Google Scholar 

  31. Y.J. Wang, F. Ding, Recursive least squares algorithm and gradient algorithm for Hammerstein–Wiener systems using the data filtering. Nonlinear Dyn. 84(2), 1045–1053 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. Y.J. Wang, F. Ding, Novel data filtering based parameter identification for multiple-input multiple-output systems using the auxiliary model. Automatica 71, 308–313 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  33. Y.J. Wang, F. Ding, The filtering based iterative identification for multivariable systems. IET Control Theory Appl. 10(8), 894–902 (2016)

    Article  MathSciNet  Google Scholar 

  34. Y.J. Wang, F. Ding, The auxiliary model based hierarchical gradient algorithms and convergence analysis using the filtering technique. Signal Process. 128, 212–221 (2016)

    Article  Google Scholar 

  35. X.H. Wang, F. Ding, Modelling and multi-innovation parameter identification for Hammerstein nonlinear state space systems using the filtering technique. Math. Comput. Modell. Dyn. Syst. 22(2), 113–140 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  36. C. Wang, T. Tang, Recursive least squares estimation algorithm applied to a class of linear-in-parameters output error moving average systems. Appl. Math. Lett. 29, 36–41 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. C. Wang, T. Tang, Several gradient-based iterative estimation algorithms for a class of nonlinear systems using the filtering technique. Nonlinear Dyn. 77(3), 769–780 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  38. D.Q. Wang, W. Zhang, Improved least squares identification algorithm for multivariable Hammerstein systems. J. Franklin Inst. 352(11), 5292–5370 (2015)

    Article  MathSciNet  Google Scholar 

  39. C. Wang, L. Zhu, Parameter identification of a class of nonlinear systems based on the multi-innovation identification theory. J. Franklin Inst. 352(10), 4624–4637 (2015)

    Article  MathSciNet  Google Scholar 

  40. A. Wills, T.B. Schön, L. Ljung et al., Identification of Hammerstein–Wiener models. Automatica 49(1), 70–81 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  41. W.L. Xiong, J.X. Ma, R.F. Ding, An iterative numerical algorithm for modeling a class of Wiener nonlinear systems. Appl. Math. Lett. 26(4), 487–493 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  42. X.P. Xu, F. Wang, G.J. Liu, Identification of Hammerstein systems using key-term separation principle, auxiliary model and improved particle swarm optimisation algorithm. IET Signal Process. 7(8), 766–773 (2013)

    Article  Google Scholar 

  43. L. Xu, A proportional differential control method for a time-delay system using the Taylor expansion approximation. Appl. Math. Comput. 236, 391–399 (2014)

    MathSciNet  MATH  Google Scholar 

  44. L. Xu, Application of the Newton iteration algorithm to the parameter estimation for dynamical systems. J. Comput. Appl. Math. 288, 33–43 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  45. L. Xu, L. Chen, W.L. Xiong, Parameter estimation and controller design for dynamic systems from the step responses based on the Newton iteration. Nonlinear Dyn. 79(3), 2155–2163 (2015)

    Article  MathSciNet  Google Scholar 

  46. L. Xu, The damping iterative parameter identification method for dynamical systems based on the sine signal measurement. Signal Process. 120, 660–667 (2016)

    Article  Google Scholar 

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Acknowledgments

This work was supported by the National Natural Science Foundation of China (Nos. 61164015, 60474039) and the Key Research Project of Henan Higher Education Institutions (No. 16A120010).

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Authors and Affiliations

  1. School of Information Engineering, Nanchang Hangkong University, Nanchang, 330063, People’s Republic of China

    Feng Ding

  2. College of Mathematics and Information Science, Xinyang Normal University, Xinyang, 464000, People’s Republic of China

    Xuehai Wang

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  1. Feng Ding
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Correspondence to Feng Ding.

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Ding, F., Wang, X. Hierarchical Stochastic Gradient Algorithm and its Performance Analysis for a Class of Bilinear-in-Parameter Systems. Circuits Syst Signal Process 36, 1393–1405 (2017). https://doi.org/10.1007/s00034-016-0367-7

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  • DOI: https://doi.org/10.1007/s00034-016-0367-7

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