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Least Squares Identification for Hammerstein Multi-input Multi-output Systems Based on the Key-Term Separation Technique

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Abstract

System modeling and parameter estimation are basic for system analysis and controller design. This paper considers the parameter identification problem of a Hammerstein multi-input multi-output (H-MIMO) system. In order to avoid the product terms in the identification model, we derive a pseudo-linear identification model of the H-MIMO system through separating a key term from the output equation of the system and present a hierarchical generalized least squares (LS) algorithm for estimating the parameters of the system. Moreover, we present a new LS algorithm to reduce the computational burden. The proposed algorithms are simple in principle and can achieve a higher computational efficiency than the over-parameterization-based LS estimation algorithm. Finally, we test the proposed algorithms by the simulation example and show their effectiveness.

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References

  1. Z.J. Cai, E.W. Bai, Making parametric Hammerstein system identification a linear problem. Automatica 47(8), 1806–1812 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. J. Chen, Y.X. Ni, Parameter identification methods for an additive nonlinear system. Circuits Syst. Signal Process. 33(10), 3053–3064 (2014)

    Article  Google Scholar 

  3. M.M. da Silva, T. Wigren, T. Mendonça, Nonlinear identification of a minimal neuromuscular blockade model in anesthesia. IEEE Trans. Control Syst. Technol. 20(1), 181–188 (2012)

    Google Scholar 

  4. S. da Silva, S. Cogan, E. Foltte, Nonlinear identification in structural dynamics based on Wiener series and Kautz filters. Mech. Syst. Signal Process. 24(1), 52–58 (2010)

    Article  Google Scholar 

  5. M. Dehghan, M. Hajarian, Fourth-order variants of Newton’s method without second derivatives for solving non-linear equations. Eng. Comput. 29(4), 356–365 (2012)

    Article  Google Scholar 

  6. M. Dehghan, M. Hajarian, Iterative algorithms for the generalized centro-symmetric and central anti-symmetric solutions of general coupled matrix equations. Eng. Comput. 29(5), 528–560 (2012)

    Article  Google Scholar 

  7. M. Dehghan, M. Hajarian, SSHI methods for solving general linear matrix equations. Eng. Comput. 28(8), 1028–1043 (2011)

    Article  Google Scholar 

  8. F. Ding, System Identification—New Theory and Methods (Science Press, Beijing, 2013)

    Google Scholar 

  9. F. Ding, System Identification—Performances Analysis for Identification Methods (Science Press, Beijing, 2014)

    Google Scholar 

  10. F. Ding, T. Chen, Hierarchical gradient-based identification of multivariable discrete-time systems. Automatica 41(2), 315–325 (2005)

    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. (2015). doi:10.1007/s00034-015-0190-6

  12. F. Ding, Y.J. Wang, J. Ding, Recursive least squares parameter identification for systems with colored noise using the filtering technique and the auxiliary model. Digital Signal Process. 37, 100–108 (2015)

    Article  Google Scholar 

  13. S. Gibson, B. Ninness, Robust maximum-likelihood estimation of multivariable dynamic systems. Automatica 41(10), 1667–1682 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Y. Gu, F. Ding, J.H. Li, States based iterative parameter estimation for a state space model with multi-state delays using decomposition. Signal Process. 106, 294–300 (2015)

    Article  Google Scholar 

  15. V.F. Gubarev, S.V. Melnichuk, Identification of multivariable systems using steady-state mode parameters. J. Autom. Inf. Sci. 44(9), 24–42 (2012)

    Article  Google Scholar 

  16. P.P. Hu, F. Ding, Multistage least squares based iterative estimation for feedback nonlinear systems with moving average noises using the hierarchical identification principle. Nonlinear Dyn. 73(1–2), 583–592 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Y.B. Hu, B.L. Liu, Q. Zhou, C. Yang, Recursive extended least squares parameter estimation for Wiener nonlinear systems with moving average noises. Circuits Syst. Signal Process. 33(2), 655–664 (2013)

    Article  MathSciNet  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  Google Scholar 

  19. Y. Ji, X.M. Liu et al., 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 

  20. J.H. Li, Parameter estimation for Hammerstein CARARMA systems based on the Newton iteration. Appl. Math. Lett. 26(1), 91–96 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Y. Liu, E.W. Bai, Iterative identification of Hammerstein systems. Automatica 43(2), 346–354 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. X.H. Lü, X.M. Ren, Identification of Hammerstein systems with asymmetric dead-zone nonlinearities using canonical parameterized model. J. Control Theory Appl. 10(4), 511–516 (2012)

    Article  MathSciNet  Google Scholar 

  23. G. Mercére, L. Bako, Parameterization and identification of multivariable state-space systems: A canonical approach. Automatica 47(8), 1547–1555 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. T. Oomen, O. Bosgra, System identification for achieving robust performance. Automatica 48(9), 1975–1987 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. J.Q. Qiu, M.F. Ren, Y.R. Niu, Y.C. Zhao, Y.M. Guo, Fault estimation for nonlinear dynamic systems. Circuits Syst. Signal Process. 31(2), 655–664 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. J. Reyland, E.W. Bai, Generalized Wiener system identification: general backlash nonlinearity and finite impulse response linear part. Int. J. Adapt. Control Signal Process. 28(11), 1174–1188 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. G. Shafiee, M. Arefi, M. Jahed-Motlagh, A. Jalali, Nonlinear predictive control of a polymerization reactor based on piecewise linear Wiener model. Chem. Eng. J. 143(1), 282–292 (2008)

    Article  Google Scholar 

  28. Q.Y. Shen, F. Ding, Iterative estimation methods for Hammerstein controlled autoregressive moving average systems based on the key-term separation principle. Nonlinear Dyn. 75(4), 709–716 (2014)

    Article  MATH  Google Scholar 

  29. Q.Y. Shen, F. Ding, Iterative identification methods for input nonlinear multivariable systems using the key-term separation principle. J. Franklin Inst. 352(7), 2847–2865 (2015)

    Article  MathSciNet  Google Scholar 

  30. Q.Y. Shen, F. Ding, Multi-innovation parameter estimation for Hammerstein MIMO output-error systems based on the key-term separation. IFAC-PapersOnLine 48(8), 457–462 (2015)

    Article  Google Scholar 

  31. Y. Shi, H. Fang, Kalman filter based identification for systems with randomly missing measurements in a network environment. Int. J. Control 83(3), 538–551 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. S.W. Sung, T. Wigren, J. Lee, Modeling and control of Wiener-type processes. Chem. Eng. Sci. 59(7), 1515–1521 (2004)

    Article  Google Scholar 

  33. J. Vörös, Parameter identification of discontinuous Hammerstein systems. Automatica 33(6), 1141–1146 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  34. J. Vörös, Recursive identification of Hammerstein systems with discontinuous nonlinearities containing dead-zones. IEEE Trans. Autom. Control 48(12), 2203–2206 (2003)

    Article  MathSciNet  Google Scholar 

  35. Y.J. Wang, F. Ding, Iterative estimation for a nonlinear IIR filter with moving average noise by means of the data filtering technique. IMA J. Math. Control Inf. (2016). doi:10.1093/imamci/dnv067

    Google Scholar 

  36. X.H. Wang, F. Ding, Convergence of the recursive identification algorithms for multivariate pseudo-linear regressive systems. Int. J. Adapt. Control Signal Process. 2016, 30(x). doi:10.1002/acs.2642

  37. D.Q. Wang, H.B. Liu et al., Highly efficient identification methods for dual-rate Hammerstein systems. IEEE Trans. Control Syst. Technol. 23(5), 1952–1960 (2015)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  39. Z. Xu, J. Zhao, J. Qian, Y. Zhu, Nonlinear MPC using an identified LPV model. Ind. Eng. Chem. Res. 48(6), 3043–3051 (2009)

    Article  Google Scholar 

  40. B. Yu, H. Fang, Y. Lin, Y. Shi, Identification of Hammerstein output-error systems with two-segment nonlinearities: algorithm and applications. Control Intell. Syst. 38(4), 194–201 (2010)

    MathSciNet  MATH  Google Scholar 

  41. B. Yu, Y. Shi, H. Huang, \(l_2\)-\(l_\infty \) Filtering for multirate systems based on lifted models. Circuits Syst. Signal Process. 27(5), 699–711 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 61273194), the Research Innovation Program for College Graduates of Jiangsu Province (No. KYLX\(\_1121\)) and the PAPD of Jiangsu Higher Education Institutions.

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Authors and Affiliations

  1. Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Jiangnan University, Wuxi, 214122, People’s Republic of China

    Qianyan Shen & Feng Ding

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  1. Qianyan Shen
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  2. Feng Ding
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Correspondence to Feng Ding.

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Shen, Q., Ding, F. Least Squares Identification for Hammerstein Multi-input Multi-output Systems Based on the Key-Term Separation Technique. Circuits Syst Signal Process 35, 3745–3758 (2016). https://doi.org/10.1007/s00034-015-0211-5

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  • DOI: https://doi.org/10.1007/s00034-015-0211-5

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