Multiple-input single-output nonlinear system identification using Bezier- Bernstein polynomials with noise cancellation
Subject Areas : Electrical Engineering
1 - Department of Electrical Engineering, Langarud Branch, Islamic Azad University, Langarud.
Keywords: Nonlinear System Identification, Multi Input- Single Output Hammerstein Model, Bezier-Bernstein Polynomial, Modified Genetic Algorithm,
Abstract :
This article deals with an identification method for the fractional multiple-input single-output model. It is considered the Hammerstein model to separate dynamic linear and static nonlinear behaviors. Which Bezier-Bernstein polynomials are used to approximate the nonlinear functions and the fractional order transfer function is applied to estimate the linear part. A hybrid identification method based on a modified evolutionary algorithm and a recursive classic method is presented. As an advantage, this method can also correctly identify the system in the presence of output noise. A photovoltaic experimental system and a numerical example are used to illustrate the efficiency and performance of the proposed scheme.
References:
[1] Birs, I. R., Muresan, C. I., Folea, S., & Prodan, O. (2016). A comparison between integer and fractional order pd controllers for vibration suppression. Applied Mathematics and Nonlinear Sciences, 1(1), 273-282.
[2] Yang Q, Chen D, Zhao T, Chen Y. Fractional calculus in image processing: a review. Fractional Calculus and Applied Analysis. 2016 Oct 1;19(5):1222-49.
[3] Sheng H, Chen Y, Qiu T. An Overview of Fractional Processes and Fractional-Order Signal Processing Techniques. In Fractional Processes and Fractional-Order Signal Processing 2012 (pp. 31-46). Springer London. ISBN: 978-1-4471-2232-6
[4] Baleanu D, Golmankhaneh AK, Nigmatullin R, Golmankhaneh AK. Fractional newtonian mechanics. Central European Journal of Physics. 2010 Feb 1;8(1):120-125.
[5] Yousfi N, Melchior P, Rekik C, Derbel N, Oustaloup A. Path tracking design by fractional prefilter using a combined QFT/H∞ design for TDOF uncertain feedback systems. J. Appl. Nonlinear Dyn. 2012;1(3):239-61.
[6] Monje CA, Vinagre BM, Feliu V, Chen Y. Tuning and auto-tuning of fractional order controllers for industry applications. Control engineering practice. 2008 Jul 31;16(7):798-812.
[7] Gabano JD, Poinot T, Kanoun H. Identification of a thermal system using continuous linear parameter-varying fractional modelling. IET Control Theory & Applications. 2011 May 5;5(7):889-99.
[8] Cois O, Oustaloup A, Battaglia E, Battaglia JL. Non integer model from modal decomposition for time domain system identification. IFAC Proceedings Volumes. 2000 Jun 30;33(15):989-94.
[9] Poinot T, Trigeassou JC. Identification of fractional systems using an output-error technique. Nonlinear Dynamics. 2004 Dec 1;38(1):133-54.
[10] Lin J, Poinot T, Li ST, Trigeassou JC. Identification of non-integer-order systems in frequency domain. Journal Control Theory Appl.2008;25:517-20.
[11] Valerio D, Sa da Costa J. Identifying digital and fractional transfer functions from a frequency response. International Journal of Control. 2011 Mar 1;84(3):445-57.
[12] Petras I. Fractional-order nonlinear systems: modeling, analysis and simulation. Springer Science & Business Media; 2011 May 30. ISBN: 978-3-642-18100-9.
[13] Ivanov DV. Identification discrete fractional order Hammerstein systems. In Control and Communications (SIBCON), 2015 International Siberian Conference on 2015 May 21 (pp. 1-4). IEEE.
[14] Rattanawaorahirunkul R, Sanposh P, Panjapornpon C. Nonlinear system identification of pH process using Hammerstein-Wiener model. InElectronics, Information, and Communications (ICEIC), 2016 International Conference on 2016 Jan 27 (pp. 1-4). IEEE.
[15] Alonge F, D'Ippolito F, Raimondi FM, Tumminaro S. Nonlinear modeling of DC/DC converters using the Hammerstein's approach. IEEE transactions on power electronics. 2007 Jul;22(4):1210-21.
[16] Huo HB, Zhu XJ, Hu WQ, Tu HY, Li J, Yang J. Nonlinear model predictive control of SOFC based on a Hammerstein model. Journal of Power Sources. 2008 Oct 15;185(1):338-44.
[17] Turunen J, Tanttu JT, Loula P. Hammerstein model for speech coding. EURASIP Journal on Applied Signal Processing. 2003 Jan 1;2003:1238-49.
[18] Balestrino A, Landi A, Ould-Zmirli M, Sani L. Automatic nonlinear auto-tuning method for Hammerstein modeling of electrical drives. IEEE Transactions on Industrial Electronics. 2001 Jun;48(3):645-55.
[19] Le F, Markovsky I, Freeman CT, Rogers E. Recursive identification of Hammerstein systems with application to electrically stimulated muscle. Control Engineering Practice. 2012 Apr 30;20(4):386-96.
[20] Taringou F, Hammi O, Srinivasan B, Malhame R, Ghannouchi F M, Behaviour modelling of wideband RF transmitters using Hammerstein–Wiener models, IET circuits, devices & systems, 2010, 4(4), 282--290.
[21] Bai EW, Fu M. A blind approach to Hammerstein model identification. IEEE Transactions on Signal Processing. 2002 Jul;50(7):1610-9.
[22]Hasiewicz Z, Mzyk G. Hammerstein system identification by non-parametric instrumental variables. International Journal of Control. 2009 Mar 1;82(3):440-55.
[23] Greblicki W. Stochastic approximation in nonparametric identification of Hammerstein systems. IEEE Transactions on Automatic Control. 2002 Nov;47(11):1800-10.
[24] Chen H F, Pathwise convergence of recursive identification algorithms for Hammerstein systems, IEEE Transactions on Automatic Control, 2004, 49(10), 1641--1649.
[25] Goethals I, Pelckmans K, Suykens JA, De Moor B. Subspace identification of Hammerstein systems using least squares support vector machines. IEEE Transactions on Automatic Control. 2005 Oct;50(10):1509-19.
[26] Mete S, Ozer S, Zorlu H. System identification using Hammerstein model optimized with differential evolution algorithm. AEU-International Journal of Electronics and Communications. 2016 Dec 31;70(12):1667-75.
[27] Mohamed AO, Malti R, Olivier CO, Oustaloup A. System Identification Using Fractional Hammerstein Models. IFAC Proceedings Volumes. 2002 Dec 31;35(1):265-9.
[28] Zhao Y, Li Y, Chen Y. Complete parametric identification of fractional order Hammerstein systems. InFractional Differentiation and Its Applications (ICFDA), 2014 International Conference on 2014 Jun 23 (pp. 1-6). IEEE.
[29] Vanbeylen L. A fractional approach to identify Wiener–Hammerstein systems. Automatica. 2014 Mar 31;50(3):903-9.
[30] Hammar K, Djamah T, Bettayeb M. Fractional Hammerstein system identification using polynomial non-linear state space model. InControl, Engineering & Information Technology (CEIT), 2015 3rd International Conference on 2015 May 25 (pp. 1-6). IEEE.
[31] Hammar K, Djamah T, Bettayeb M. Fractional hammerstein system identification using particle swarm optimization. InModelling, Identification and Control (ICMIC), 2015 7th International Conference on 2015 Dec 18 (pp. 1-6). IEEE.
[32] Hong X, Mitchell RJ. Hammerstein model identification algorithm using Bezier–Bernstein approximation. IET Control Theory & Applications. 2007 Jul 1;1(4):1149-59.
[33] Farin G. Curves and surfaces for computer-aided geometric design: a practical guide. Elsevier; 2014 Jun 28.
[34] Ahmadi M, Mojallali H. Identification of multiple-input single-output Hammerstein models using Bezier curves and Bernstein polynomials. Applied Mathematical Modelling. 2011 Apr 30;35(4):1969-82.
[35] Farouki R, Goodman T. On the optimal stability of the Bernstein basis. Mathematics of Computation of the American Mathematical Society. 1996;65(216):1553-66.
[36] Jahani Moghaddam, M., Mojallali, H., &Teshnehlab, M. (2018). Recursive identification of multiple-input single-output fractional-order Hammerstein model with time delay. Applied Soft Computing, 70, 486-500.
[37] Jahani Moghaddam, M., Mojallali, H., &Teshnehlab, M. (2018). A multiple‐input–single‐output fractional‐order Hammerstein model identification based on modified neural network. Mathematical Methods in the Applied Sciences, 41(16), 6252-6271.
[38] Sundari S, Nachiappan A. Online Identification Using RLS Algorithm and Kaczmarz’s Projection Algorithm for a Bioreactor Process. International Journal of Engineering and Computer Science. 2014;3:7974-8.
[39] Vahidi A, Stefanopoulou A, Peng H. Recursive least squares with forgetting for online estimation of vehicle mass and road grade: theory and experiments. Vehicle System Dynamics. 2005 Jan 1;43(1):31-55.
[40] Hussain MN, Omar AM, Samat AA. Identification of multiple input-single output (miso) model for mppt of photovoltaic system. InControl System, Computing and Engineering (ICCSCE), 2011 IEEE International Conference on 2011 Nov 25 (pp. 49-53). IEEE.
[41] Eskinat E, Johnson SH, Luyben WL. Use of Hammerstein models in identification of nonlinear systems. AIChE Journal. 1991 Feb 1;37(2):255-68.
[1] Birs, I. R., Muresan, C. I., Folea, S., & Prodan, O. (2016). A comparison between integer and fractional order pd controllers for vibration suppression. Applied Mathematics and Nonlinear Sciences, 1(1), 273-282.
[2] Yang Q, Chen D, Zhao T, Chen Y. Fractional calculus in image processing: a review. Fractional Calculus and Applied Analysis. 2016 Oct 1;19(5):1222-49.
[3] Sheng H, Chen Y, Qiu T. An Overview of Fractional Processes and Fractional-Order Signal Processing Techniques. In Fractional Processes and Fractional-Order Signal Processing 2012 (pp. 31-46). Springer London. ISBN: 978-1-4471-2232-6
[4] Baleanu D, Golmankhaneh AK, Nigmatullin R, Golmankhaneh AK. Fractional newtonian mechanics. Central European Journal of Physics. 2010 Feb 1;8(1):120-125.
[5] Yousfi N, Melchior P, Rekik C, Derbel N, Oustaloup A. Path tracking design by fractional prefilter using a combined QFT/H∞ design for TDOF uncertain feedback systems. J. Appl. Nonlinear Dyn. 2012;1(3):239-61.
[6] Monje CA, Vinagre BM, Feliu V, Chen Y. Tuning and auto-tuning of fractional order controllers for industry applications. Control engineering practice. 2008 Jul 31;16(7):798-812.
[7] Gabano JD, Poinot T, Kanoun H. Identification of a thermal system using continuous linear parameter-varying fractional modelling. IET Control Theory & Applications. 2011 May 5;5(7):889-99.
[8] Cois O, Oustaloup A, Battaglia E, Battaglia JL. Non integer model from modal decomposition for time domain system identification. IFAC Proceedings Volumes. 2000 Jun 30;33(15):989-94.
[9] Poinot T, Trigeassou JC. Identification of fractional systems using an output-error technique. Nonlinear Dynamics. 2004 Dec 1;38(1):133-54.
[10] Lin J, Poinot T, Li ST, Trigeassou JC. Identification of non-integer-order systems in frequency domain. Journal Control Theory Appl.2008;25:517-20.
[11] Valerio D, Sa da Costa J. Identifying digital and fractional transfer functions from a frequency response. International Journal of Control. 2011 Mar 1;84(3):445-57.
[12] Petras I. Fractional-order nonlinear systems: modeling, analysis and simulation. Springer Science & Business Media; 2011 May 30. ISBN: 978-3-642-18100-9.
[13] Ivanov DV. Identification discrete fractional order Hammerstein systems. In Control and Communications (SIBCON), 2015 International Siberian Conference on 2015 May 21 (pp. 1-4). IEEE.
[14] Rattanawaorahirunkul R, Sanposh P, Panjapornpon C. Nonlinear system identification of pH process using Hammerstein-Wiener model. InElectronics, Information, and Communications (ICEIC), 2016 International Conference on 2016 Jan 27 (pp. 1-4). IEEE.
[15] Alonge F, D'Ippolito F, Raimondi FM, Tumminaro S. Nonlinear modeling of DC/DC converters using the Hammerstein's approach. IEEE transactions on power electronics. 2007 Jul;22(4):1210-21.
[16] Huo HB, Zhu XJ, Hu WQ, Tu HY, Li J, Yang J. Nonlinear model predictive control of SOFC based on a Hammerstein model. Journal of Power Sources. 2008 Oct 15;185(1):338-44.
[17] Turunen J, Tanttu JT, Loula P. Hammerstein model for speech coding. EURASIP Journal on Applied Signal Processing. 2003 Jan 1;2003:1238-49.
[18] Balestrino A, Landi A, Ould-Zmirli M, Sani L. Automatic nonlinear auto-tuning method for Hammerstein modeling of electrical drives. IEEE Transactions on Industrial Electronics. 2001 Jun;48(3):645-55.
[19] Le F, Markovsky I, Freeman CT, Rogers E. Recursive identification of Hammerstein systems with application to electrically stimulated muscle. Control Engineering Practice. 2012 Apr 30;20(4):386-96.
[20] Taringou F, Hammi O, Srinivasan B, Malhame R, Ghannouchi F M, Behaviour modelling of wideband RF transmitters using Hammerstein–Wiener models, IET circuits, devices & systems, 2010, 4(4), 282--290.
[21] Bai EW, Fu M. A blind approach to Hammerstein model identification. IEEE Transactions on Signal Processing. 2002 Jul;50(7):1610-9.
[22]Hasiewicz Z, Mzyk G. Hammerstein system identification by non-parametric instrumental variables. International Journal of Control. 2009 Mar 1;82(3):440-55.
[23] Greblicki W. Stochastic approximation in nonparametric identification of Hammerstein systems. IEEE Transactions on Automatic Control. 2002 Nov;47(11):1800-10.
[24] Chen H F, Pathwise convergence of recursive identification algorithms for Hammerstein systems, IEEE Transactions on Automatic Control, 2004, 49(10), 1641--1649.
[25] Goethals I, Pelckmans K, Suykens JA, De Moor B. Subspace identification of Hammerstein systems using least squares support vector machines. IEEE Transactions on Automatic Control. 2005 Oct;50(10):1509-19.
[26] Mete S, Ozer S, Zorlu H. System identification using Hammerstein model optimized with differential evolution algorithm. AEU-International Journal of Electronics and Communications. 2016 Dec 31;70(12):1667-75.
[27] Mohamed AO, Malti R, Olivier CO, Oustaloup A. System Identification Using Fractional Hammerstein Models. IFAC Proceedings Volumes. 2002 Dec 31;35(1):265-9.
[28] Zhao Y, Li Y, Chen Y. Complete parametric identification of fractional order Hammerstein systems. InFractional Differentiation and Its Applications (ICFDA), 2014 International Conference on 2014 Jun 23 (pp. 1-6). IEEE.
[29] Vanbeylen L. A fractional approach to identify Wiener–Hammerstein systems. Automatica. 2014 Mar 31;50(3):903-9.
[30] Hammar K, Djamah T, Bettayeb M. Fractional Hammerstein system identification using polynomial non-linear state space model. InControl, Engineering & Information Technology (CEIT), 2015 3rd International Conference on 2015 May 25 (pp. 1-6). IEEE.
[31] Hammar K, Djamah T, Bettayeb M. Fractional hammerstein system identification using particle swarm optimization. InModelling, Identification and Control (ICMIC), 2015 7th International Conference on 2015 Dec 18 (pp. 1-6). IEEE.
[32] Hong X, Mitchell RJ. Hammerstein model identification algorithm using Bezier–Bernstein approximation. IET Control Theory & Applications. 2007 Jul 1;1(4):1149-59.
[33] Farin G. Curves and surfaces for computer-aided geometric design: a practical guide. Elsevier; 2014 Jun 28.
[34] Ahmadi M, Mojallali H. Identification of multiple-input single-output Hammerstein models using Bezier curves and Bernstein polynomials. Applied Mathematical Modelling. 2011 Apr 30;35(4):1969-82.
[35] Farouki R, Goodman T. On the optimal stability of the Bernstein basis. Mathematics of Computation of the American Mathematical Society. 1996;65(216):1553-66.
[36] Jahani Moghaddam, M., Mojallali, H., &Teshnehlab, M. (2018). Recursive identification of multiple-input single-output fractional-order Hammerstein model with time delay. Applied Soft Computing, 70, 486-500.
[37] Jahani Moghaddam, M., Mojallali, H., &Teshnehlab, M. (2018). A multiple‐input–single‐output fractional‐order Hammerstein model identification based on modified neural network. Mathematical Methods in the Applied Sciences, 41(16), 6252-6271.
[38] Sundari S, Nachiappan A. Online Identification Using RLS Algorithm and Kaczmarz’s Projection Algorithm for a Bioreactor Process. International Journal of Engineering and Computer Science. 2014;3:7974-8.
[39] Vahidi A, Stefanopoulou A, Peng H. Recursive least squares with forgetting for online estimation of vehicle mass and road grade: theory and experiments. Vehicle System Dynamics. 2005 Jan 1;43(1):31-55.
[40] Hussain MN, Omar AM, Samat AA. Identification of multiple input-single output (miso) model for mppt of photovoltaic system. InControl System, Computing and Engineering (ICCSCE), 2011 IEEE International Conference on 2011 Nov 25 (pp. 49-53). IEEE.
[41] Eskinat E, Johnson SH, Luyben WL. Use of Hammerstein models in identification of nonlinear systems. AIChE Journal. 1991 Feb 1;37(2):255-68.