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Least-squares-based iterative and gradient-based iterative estimation algorithms for bilinear systems

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Abstract

For bilinear systems with colored noise, this paper gives the input–output representation of the bilinear systems through eliminating the state variables in the model and derives a three-stage gradient-based iterative algorithm and a three-stage least-squares-based iterative algorithm for identifying the parameters of the input–output representation by means of the hierarchical identification principle. A gradient-based iterative (GI) algorithm is given for comparison. Compared with the GI algorithm, the proposed algorithms have lower computational burden and faster convergence speed. The simulation results indicate that the proposed algorithms are more effective for identifying bilinear systems.

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References

  1. Wang, Y.J., Ding, F.: 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Xu, L.: 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 

  4. Xu, L., Chen, L., Xiong, W.L.: 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 

  5. Xu, L.: 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 

  6. Chaudhary, N.I., Raja, M.A.Z.: Identification of Hammerstein nonlinear ARMAX systems using nonlinear adaptive algorithms. Nonlinear Dyn. 79(2), 1385–1397 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brewick, P.T., Masri, S.F.: An evaluation of data-driven identification strategies for complex nonlinear dynamic systems. Nonlinear Dyn. 85(2), 1297–1318 (2016)

    Article  MathSciNet  Google Scholar 

  8. Mao, Y.W., Ding, F.: A novel parameter separation based identification algorithm for Hammerstein systems. Appl. Math. Lett. 60, 21–27 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Mao, Y.W., Ding, F.: Multi-innovation stochastic gradient identification for Hammerstein controlled autoregressive autoregressive systems based on the filtering technique. Nonlinear Dyn. 79(3), 1745–1755 (2015)

    Article  MATH  Google Scholar 

  10. Chai, J., Tian, B., Zhen, H.L., Sun, W.R.: Conservation laws, bilinear forms and solitons for a fifth-order nonlinear Schrödinger equation for the attosecond pulses in an optical fiber. Ann. Phys. 359, 371–384 (2015)

    Article  MATH  Google Scholar 

  11. Ponce, Y.M., Torres, E.C., Jacas, C.R.G., Barigye, S.J., et al.: Novel 3D bio-macromolecular bilinear descriptors for protein science: predicting protein structural classes. J. Theor. Biol. 374, 125–137 (2015)

    Article  MATH  Google Scholar 

  12. Christoskov, I.D., Petkov, P.T.: A practical procedure of bilinear weighted core kinetics parameters computation for the purpose of experimental reactivity determination. Ann. Nucl. Energy 29(9), 1041–1054 (2002)

    Article  Google Scholar 

  13. Daniel-Berhe, S., Unbehauen, H.: Bilinear continuous-time systems identification via hartley-based modulating functions. Automatica 34(4), 499–503 (1998)

    Article  MATH  Google Scholar 

  14. Kalouptsidis, N., Koukoulas, P., Mathews, V.J.: Blind identification of bilinear systems. IEEE Trans. Signal Process. 51(2), 484–499 (2003)

    Article  MathSciNet  Google Scholar 

  15. Verdult, V., Verhaegen, M.: Kernel methods for subspace identification of multivariable LPV and bilinear systems. Automatica 41(9), 1557–1565 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gibson, S., Wills, A., Ninness, B.: Maximum-likelihood parameter estimation of bilinear systems. IEEE Trans. Autom. Control 50(10), 1581–1596 (2005)

    Article  MathSciNet  Google Scholar 

  17. Li, G.Q., Wen, C.Y., Zhang, A.M.: Fixed point iteration in identifying bilinear models. Syst. Control Lett. 83, 28–37 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Meng, D.D.: Recursive least squares and multi-innovation gradient estimation algorithms for bilinear stochastic systems. Circuits Syst. Signal Process. 36(3), 1052–1065 (2017)

    Article  MathSciNet  Google Scholar 

  19. Pan, J., Jiang, X., Wan, X.K., Ding, W.F.: A filtering based multi-innovation extended stochastic gradient algorithm for multivariable control systems. Int. J. Control Autom. Syst. (2017). doi:10.1007/s12555-016-0081-z

    Google Scholar 

  20. Ghadimi, E., Shames, I., Johansson, M.: Multi-step gradient methods for networked optimization. IEEE Trans. Signal Process. 61(21), 5417–5429 (2013)

    Article  Google Scholar 

  21. Narendra, K.S., Parthasarathy, K.: Gradient methods for the optimization of dynamical systems containing neural networds. IEEE Trans. Neural Netw. 2(2), 252–262 (1991)

    Article  Google Scholar 

  22. Ding, F., Xu, L., Zhu, Q.M.: Performance analysis of the generalised projection identification for time-varying systems. IET Control Theory Appl. 10(18), 2506–2514 (2016)

    Article  Google Scholar 

  23. Xu, L., Ding, F.: The parameter estimation algorithms for dynamical response signals based on the multi-innovation theory and the hierarchical principle. IET Signal Process. (2017). doi:10.1049/iet-spr.2016.0220

    Google Scholar 

  24. Wang, Y.J., Ding, F.: 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 

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

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  27. Hu, Y.B., Liu, B.L., Zhou, Q.: A multi-innovation generalized extended stochastic gradient algorithm for output nonlinear autoregressive moving average systems. Appl. Math. Comput. 247, 218–224 (2014)

    MathSciNet  MATH  Google Scholar 

  28. Mao, Y.W., Ding, F.: A novel data filtering based multi-innovation stochastic gradient algorithm for Hammerstein nonlinear systems. Digit. Signal Process. 46, 215–225 (2015)

    Article  MathSciNet  Google Scholar 

  29. Wan, X.K., Li, Y., Xia, C., Wu, M.H., Liang, J., Wang, N.: A T-wave alternans assessment method based on least squares curve fitting technique. Measurement 86, 93–100 (2016)

    Article  Google Scholar 

  30. Xu, L., Ding, F.: Recursive least squares and multi-innovation stochastic gradient parameter estimation methods for signal modeling. Circuits Syst. Signal Process. 36(4), 1735–1753 (2017)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  32. Wang, D.Q., Zhang, W.: Improved least squares identification algorithm for multivariable Hammerstein systems. J. Frankl. Inst. 352(11), 5292–5307 (2015)

    Article  MathSciNet  Google Scholar 

  33. Tai, C.T.: An iterative method of solving a system of linear equations and its physical interpretation from the point of view of scattering theory. IEEE Trans. Antennas Propag. 18(5), 713–714 (1970)

    Article  Google Scholar 

  34. Feng, Y.T., Varga, A., Anderson, B.D.O., Lovera, M.: A new iterative algorithm to solve periodic Riccati differential equations with sign indefinite quadratic terms. IEEE Trans. Autom. Control 56(4), 929–934 (2011)

    Article  MathSciNet  Google Scholar 

  35. Abebe, A.T., Kang, C.G.: Iterative order recursive least square estimation for exploiting frame-wise sparsity in compressive sensing-based MTC. IEEE Commun. Lett. 20(5), 1018–1021 (2016)

    Article  Google Scholar 

  36. Wang, F.F., Ding, F.: Gradient-based iterative identification methods for multivariate pseudo-linear moving average systems using the data filtering. Nonlinear Dyn. 84(4), 2003–2015 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  38. Ding, F., Liu, X.M., Ma, X.Y.: 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 

  39. Schranz, C., Knöbel, C., Kretschmer, J., Zhao, Z., Möller, K.: Hierarchical parameter identification in models of respiratory mechanics. IEEE Trans. Biomed. Eng. 58(11), 3234–3241 (2011)

  40. Wang, D.Q.: 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 

  41. Dai, H., Sinha, N.K.: Robust recursive least-squares method with modified weights for bilinear system identification. IEE Proc. D Control Theory Appl. 136(3), 122–126 (1989)

    Article  MATH  Google Scholar 

  42. Li, M.H., Liu, X.M., Ding, F.: The gradient based iterative estimation algorithms for bilinear systems with autoregressive noise. Circuits Syst. Signal Process. (2017). doi:10.1007/s00034-017-0527-4

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

    Article  Google Scholar 

  44. Feng, L., Wu, M.H., Li, Q.X., et al.: Array factor forming for image reconstruction of one-dimensional nonuniform aperture synthesis radiometers. IEEE Geosci. Remote Sens. Lett. 13(2), 237–241 (2016)

    Article  Google Scholar 

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

    Article  Google Scholar 

  46. Wang, T.Z., Wu, H., Ni, M.Q., 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 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 61472195) and the Taishan Scholar Project Fund of Shandong Province of China.

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Authors and Affiliations

  1. College of Automation and Electronic Engineering, Qingdao University of Science and Technology, Qingdao, 266042, People’s Republic of China

    Meihang Li, Ximei Liu & Feng Ding

  2. Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Feng Ding

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  1. Meihang Li
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  2. Ximei Liu
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Correspondence to Ximei Liu.

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Li, M., Liu, X. & Ding, F. Least-squares-based iterative and gradient-based iterative estimation algorithms for bilinear systems. Nonlinear Dyn 89, 197–211 (2017). https://doi.org/10.1007/s11071-017-3445-x

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