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Identification and nonlinearity compensation of hysteresis using NARX models

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Abstract

This paper deals with two problems: the identification and compensation of hysteresis nonlinearity in dynamical systems using nonlinear polynomial autoregressive models with exogenous inputs (NARX). First, based on gray-box identification techniques, some constraints on the structure and parameters of NARX models are proposed to ensure that the identified models display a key feature of hysteresis. In addition, a more general framework is developed to explain how hysteresis occurs in such models. Second, two strategies to design hysteresis compensators are presented. In one strategy, the compensation law is obtained through simple algebraic manipulations performed on the identified models. In the second strategy, the compensation law is directly identified from the data. Both numerical and experimental results are presented to illustrate the efficiency of the proposed procedures. Also, it has been found that the compensators based on gray-box models outperform the cases with models identified using black-box techniques.

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References

  1. Aguirre, L.A.: Some remarks on structure selection for nonlinear models. Int. J. Bifurc. Chaos 4(6), 1707–1714 (1994)

    MATH  Google Scholar 

  2. Aguirre, L.A.: Identification of smooth nonlinear dynamical systems with non-smooth steady-state features. Automatica 50(4), 1160–1166 (2014)

    MathSciNet  MATH  Google Scholar 

  3. Aguirre, L.A.: A Bird‘s Eye View of Nonlinear System Identification. arXiv:1907.06803 [eess.SY] (2019)

  4. Aguirre, L.A., Alves, G.B., Corrêa, M.V.: Steady-state performance constraints for dynamical models based on RBF networks. Eng. Appl. Artif. Intel. 20, 924–935 (2007)

    Google Scholar 

  5. Aguirre, L.A., Barroso, M.F.S., Saldanha, R.R., Mendes, E.M.A.M.: Imposing steady-state performance on identified nonlinear polynomial models by means of constrained parameter estimation. IEE Proc. Control Theory Appl. 151(2), 174–179 (2004)

    Google Scholar 

  6. Aguirre, L.A., Lopes, R.A.M., Amaral, G., Letellier, C.: Constraining the topology of neural networks to ensure dynamics with symmetry properties. Phys. Rev. 69, 026701 (2004)

    Google Scholar 

  7. Aguirre, L.A., Mendes, E.M.A.M.: Global nonlinear polynomial models: structure, term clusters and fixed points. Int. J. Bifurc. Chaos 6(2), 279–294 (1996)

    MathSciNet  MATH  Google Scholar 

  8. Akaike, H.: A new look at the statistical model identification. IEEE Trans. Autom. Control 19(6), 716–723 (1974)

    MathSciNet  MATH  Google Scholar 

  9. Araújo, I.B.Q., Guimarães, J.P.F., Fontes, A.I.R., Linhares, L.L.S., Martins, A.M., Araújo, F.M.U.: NARX model identification using correntropy criterion in the presence of non-Gaussian noise. J. Control Autom. Electr. Syst. 30(4), 453–464 (2019)

    Google Scholar 

  10. Ayala, H.V.H., Habineza, D., Rakotondrabe, M., Klein, C.E., Coelho, L.S.: Nonlinear black-box system identification through neural networks of a hysteretic piezoelectric robotic micromanipulator. IFAC-PapersOnLine 48(28), 409–414 (2015)

    Google Scholar 

  11. Baeza, J.R., Garcia, C.: Friction compensation in pneumatic control valves through feedback linearization. J. Control Autom. Electr. Syst. 29(3), 303–317 (2018)

    Google Scholar 

  12. Bernstein, D.S.: Ivory ghost (ask the experts). IEEE Control Syst. Mag. 27(5), 16–17 (2007)

    MathSciNet  Google Scholar 

  13. Billings, S.A., Chen, S.: Extended model set, global data and threshold model identification of severely non-linear systems. Int. J. Control 50(5), 1897–1923 (1989)

    MATH  Google Scholar 

  14. Brewick, P.T., Masri, S.F., Carboni, B., Lacarbonara, W.: Data-based nonlinear identification and constitutive modeling of hysteresis in NiTiNOL and steel strands. J. Eng. Mech. 142(12), 04016107 (2016)

    Google Scholar 

  15. Brokate, M., Sprekels, J.: Hysteresis and Phase Transitions. Springer, New York (1996)

    MATH  Google Scholar 

  16. Carboni, B., Lacarbonara, W., Brewick, P.T., Masri, S.F.: Dynamical response identificaiton of a class of nonlinear hysteretic systems. J. Intel. Mater. Syst. Struct. 29(13), 1–16 (2018)

    Google Scholar 

  17. Chan, R.W.K., Yuen, J.K.K., Lee, E.W.M., Arashpour, M.: Application of nonlinear-autoregressive-exogenous model to predict the hysteretic behaviour of passive control systems. Eng. Struct. 85, 1–10 (2015)

    Google Scholar 

  18. Chaoui, H., Gualous, H.: Adaptive control of piezoelectric actuators with hysteresis and disturbance compensation. J. Control Autom. Electr. Syst. 27(6), 579–586 (2016)

    Google Scholar 

  19. Chen, S., Billings, S.A., Luo, W.: Orthogonal least squares methods and their application to non-linear system identification. Int. J. Control 50(5), 1873–1896 (1989)

    MATH  Google Scholar 

  20. Choudhury, M.A.A.S., Shah, S.L., Thornhill, N.F.: Diagnosis of Process Nonlinearities and Valve Stiction: Data Driven Approaches. Springer, Heidelberg (2008)

    MATH  Google Scholar 

  21. Deng, L., Tan, Y.: Modeling hysteresis in piezoelectric actuators using NARMAX models. Sens. Actuators A Phys. 149(1), 106–112 (2009)

    Google Scholar 

  22. Dong, R., Tan, Y.: Inverse hysteresis modeling and nonlinear compensation of ionic polymer metal composite sensors. In: Proceeding of the 11th World Congress on Intelligent Control and Automation, pp. 2121–2125. Shenyang, China (2014)

  23. Draper, N.R., Smith, H.: Applied Regression Analysis, 3rd edn. Wiley, New York (1998)

    MATH  Google Scholar 

  24. Falsone, A., Piroddi, L., Prandini, M.: A randomized algorithm for nonlinear model structure selection. Automatica 60, 227–238 (2015)

    MathSciNet  MATH  Google Scholar 

  25. Fu, J., Liao, G., Yu, M., Li, P., Lai, J.: NARX neural network modeling and robustness analysis of magnetorheological elastomer isolator. Smart Mater. Struct. 25(12), 125019 (2016)

    Google Scholar 

  26. Ge, P., Jouaneh, M.: Tracking control of a piezoceramic actuator. IEEE Trans. Control Syst. Technol. 4(3), 209–216 (1996)

    Google Scholar 

  27. Gu, G.Y., Yang, M.J., Zhu, L.M.: Real-time inverse hysteresis compensation of piezoelectric actuators with a modified Prandtl–Ishlinskii model. Rev. Sci. Instrum. 83(6), 065106 (2012)

    Google Scholar 

  28. Hassani, V., Tjahjowidodo, T., Do, T.N.: A survey on hysteresis modeling, identification and control. Mech. Syst. Signal Process. 49(1–2), 209–233 (2014)

    Google Scholar 

  29. Ikhouane, F., Rodellar, J.: Systems with Hysteresis: Analysis, Identification and Control Using the Bouc-Wen Model. Wiley, New York (2007)

    MATH  Google Scholar 

  30. Jayakumar, P.: Modeling and identification in structural dynamics. Technical Report EERL-87-01, California Institute of Technology, Pasadena, CA (1987)

  31. Kyprianou, A., Worden, K., Panet, M.: Identification of hysteretic systems using the differential evolution algorithm. J. Sound Vib. 248(2), 289–314 (2001)

    Google Scholar 

  32. Lacerda Júnior, W.R., Martins, S.A.M., Nepomuceno, E.G.: Influence of sampling rate and discretization methods in the parameter identification of systems with hysteresis. J. Appl. Nonlinear Dyn. 6(4), 509–520 (2017)

    MathSciNet  Google Scholar 

  33. Lacerda Júnior, W.R., Martins, S.A.M., Nepomuceno, E.G., Lacerda, M.J.: Control of Hysteretic Systems Through an Analytical Inverse Compensation based on a NARX model. IEEE Access pp. 1–1 (2019)

  34. Leontaritis, I.J., Billings, S.A.: Input–output parametric models for non-linear systems part I: deterministic non-linear systems. Int. J. Control 41(2), 303–328 (1985)

    MATH  Google Scholar 

  35. Leontaritis, I.J., Billings, S.A.: Input–output parametric models for non-linear systems part II: stochastic non-linear systems. Int. J. Control 41(2), 329–344 (1985)

    MATH  Google Scholar 

  36. Leva, A., Piroddi, L.: NARX-based technique for the modelling of Magneto–Rheological damping devices. Smart Mater. Struct. 11(1), 79–88 (2002)

    Google Scholar 

  37. Martins, S.A.M., Aguirre, L.A.: Sufficient conditions for rate-independent hysteresis in autoregressive identified models. Mech. Syst. Signal Process. 75, 607–617 (2016)

    Google Scholar 

  38. Martins, S.A.M., Nepomuceno, E.G., Barroso, M.F.S.: Improved structure detection for polynomial NARX models using a multiobjective error reduction ratio. J. Control Autom. Electr. Syst. 24(6), 764–772 (2013)

    Google Scholar 

  39. Masri, S.F., Caffrey, J.P., Caughey, T.K., Smyth, A.W., Chassiakos, A.G.: Identification of the state equation in complex non-linear systems. Int. J. Non-Linear Mech. 39(7), 1111–1127 (2004)

    MATH  Google Scholar 

  40. Mayergoyz, I.D.: Mathematical Models of Hysteresis. Springer, New York (1991)

    MATH  Google Scholar 

  41. Morris, K.A.: What is hysteresis? Appl. Mech. Rev. 64(5), 050801 (2011)

    Google Scholar 

  42. Oh, J., Bernstein, D.S.: Semilinear Duhem model for rate-independent and rate-dependent hysteresis. IEEE Trans. Autom. Control 50(5), 631–645 (2005)

    MathSciNet  MATH  Google Scholar 

  43. Parlitz, U., Hornstein, A., Engster, D., Al-Bender, F., Lampaert, V., Tjahjowidodo, T., Fassois, S.D., Rizos, D., Wong, C.X., Worden, K., Manson, G.: Identification of pre-sliding friction dynamics. Chaos 14(2), 420–430 (2004)

    MathSciNet  MATH  Google Scholar 

  44. Pearson, R.K.: Discrete-Time Dynamic Models. Oxford University Press, Oxford (1999)

    MATH  Google Scholar 

  45. Pei, J.S., Wright, J.P., Smyth, A.W.: Mapping polynomial fitting into feedforward neural networks for modeling nonlinear dynamic systems and beyond. Comput. Methods Appl. Mech. Eng. 194(42), 4481–4505 (2005)

    MathSciNet  MATH  Google Scholar 

  46. Peng, J., Chen, X.: A survey of modeling and control of piezoelectric actuators. Mod. Mech. Eng. 3(1), 1–20 (2013)

    Google Scholar 

  47. Piroddi, L.: Simulation error minimisation methods for NARX model identification. Int. J. Model. Identif. Control 3(4), 392–403 (2008)

    Google Scholar 

  48. Quaranta, G., Lacarbonara, W., Masri, S.F.: A review on computational intelligence for identification of nonlinear dynamical systems. Nonlinear Dyn. 99, 1709–1761 (2020)

    Google Scholar 

  49. Rakotondrabe, M.: Bouc–Wen modeling and inverse multiplicative structure to compensate hysteresis nonlinearity in piezoelectric actuators. IEEE Trans. Autom. Sci. Eng. 8(2), 428–431 (2011)

    Google Scholar 

  50. Rakotondrabe, M.: Smart Materials-Based Actuators at the Micro/Nano-Scale: Characterization Control and Applications. Springer, New York (2013)

    Google Scholar 

  51. Retes, P.F.L., Aguirre, L.A.: NARMAX model identification using a randomised approach. Int. J. Model. Identif. Control 31(3), 205–216 (2019)

    Google Scholar 

  52. Romano, R.A., Garcia, C.: Valve friction and nonlinear process model closed-loop identification. J. Process Control 21(4), 667–677 (2011)

    Google Scholar 

  53. Smyth, A.W., Masri, S.F., Kosmatopoulos, E.B., Chassiakos, A.G., Caughey, T.K.: Development of adaptive modeling techniques for non-linear hysteretic systems. Int. J. Non-Linear Mech. 37(8), 1435–1451 (2002)

    MATH  Google Scholar 

  54. Srinivasan, R., Rengaswamy, R.: Stiction compensation in process control loops: a framework for integrating stiction measure and compensation. Ind. Eng. Chem. Res. 44(24), 9164–9174 (2005)

    Google Scholar 

  55. Tao, G., Kokotovic, P.V.: Adaptive control of plants with unknown hystereses. IEEE Trans. Autom. Control 40(2), 200–212 (1995)

    MathSciNet  MATH  Google Scholar 

  56. Visintin, A.: Differential Models of Hysteresis. Springer, Berlin (1994)

    MATH  Google Scholar 

  57. Visone, C.: Hysteresis modelling and compensation for smart sensors and actuators. J. Phys. Conf. Ser. 138(1), 012028 (2008)

    Google Scholar 

  58. Wen, Y.K.: Method for random vibration of hysteretic systems. J. Eng. Mech. Div. 102(2), 249–263 (1976)

    Google Scholar 

  59. Worden, K., Barthorpe, R.J.: Identification of hysteretic systems using NARX models, Part I: evolutionary identification. In: Simmermacher, T., Cogan, S., Horta, L.G., Barthorpe, R. (eds.) Topics in Model Validation and Uncertainty Quantification, vol. 4, pp. 49–56. Springer (2012)

  60. Worden, K., Hensman, J.J.: Parameter estimation and model selection for a class of hysteretic systems using Bayeisan inference. Mech. Syst. Signal Process. 32, 153–169 (2012)

    Google Scholar 

  61. Worden, K., Wong, C.X., Parlitz, U., Hornstein, A., Engster, D., Tjahjowidodo, T., Al-Bender, F., Rizos, D.D., Fassois, S.D.: Identification of pre-sliding and sliding friction dynamics: grey box and black-box models. Mech. Syst. Signal Process. 21(1), 514–534 (2007)

    MATH  Google Scholar 

  62. Xia, P.Q.: An inverse model of MR damper using optimal neural network and system identification. J. Sound Vib. 266(5), 1009–1023 (2003)

    Google Scholar 

  63. Yi, S., Yang, B., Meng, G.: Ill-conditioned dynamic hysteresis compensation for a low-frequency magnetostrictive vibration shaker. Nonlinear Dyn. 96(1), 535–551 (2019)

    MATH  Google Scholar 

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Acknowledgements

The authors would like to thank Arthur N. Montanari for the insightful discussions. PEOGBA, BOST, and LAA gratefully acknowledge financial support from CNPq (Grant Nos. 142194/2017-4, 310848/2017-2 and 303412/2019-4) and FAPEMIG (TEC-1217/98).

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Authors and Affiliations

  1. Graduate Program in Electrical Engineering, Universidade Federal de Minas Gerais, Av. Antônio Carlos 6627, Belo Horizonte, MG, 31270-901, Brazil

    Petrus E. O. G. B. Abreu & Lucas A. Tavares

  2. Department of Electronic Engineering, Universidade Federal de Minas Gerais, Av. Antônio Carlos 6627, Belo Horizonte, MG, 31270-901, Brazil

    Bruno O. S. Teixeira & Luis A. Aguirre

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  1. Petrus E. O. G. B. Abreu
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  2. Lucas A. Tavares
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  3. Bruno O. S. Teixeira
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Correspondence to Petrus E. O. G. B. Abreu.

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Abreu, P.E.O.G.B., Tavares, L.A., Teixeira, B.O.S. et al. Identification and nonlinearity compensation of hysteresis using NARX models. Nonlinear Dyn 102, 285–301 (2020). https://doi.org/10.1007/s11071-020-05936-5

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