Abstract.
Fractional calculus extends the scope of adaptive algorithms supporting the design of novel fractional methods that outperform standard strategies in various applications arising in applied physics and engineering. In this study, a momentum fractional least-mean-square (M-FLMS) algorithm for nonlinear system identification using a first and fractional-order gradient information is proposed. The M-FLMS avoids being trapped in local minima and provides faster convergence than the standard FLMS. The convergence and complexity analysis of the M-FLMS are given along with simulation results of a benchmark nonlinear system identification problem. The M-FLMS accuracy is verified through a parameter estimation problem for a nonlinear Hammerstein structure, modeling an electrically stimulated muscle (ESM) for rehabilitation of paralyzed muscles. The proposed method is studied in detail for different levels of noise variance, fractional orders and proportion of gradients used in the current update.
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References
S.A. Billings, Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains (John Wiley and Sons, Ltd, UK, 2013)
F. Chen, F. Ding, J. Comput. Nonlinear Dyn. 11, 021005 (2016)
V.Z. Filipovic, Nonlinear Dyn. 90, 1427 (2017)
M. Lawryńczuk, Nonlinear Dyn. 86, 1193 (2016)
F. Alonge et al., IEEE Trans. Ind. Appl. 51, 3975 (2015)
C.M. Holcomb, R.A. de Callafon, R.R Bitmead, IFAC Proc. 47, 493 (2014)
O.A. Maatallah et al., Appl. Energy 145, 191 (2015)
F. Le, Identification of electrically stimulated muscle after stroke, Doctoral dissertation (University of Southampton, 2011)
F. Le et al., Control Eng. Practice 20, 386 (2012)
F. Le et al., Control Eng. Practice 18, 396 (2010)
K. Narendra, P. Gallman, IEEE Trans. Autom. Control 11, 546 (1966)
W. Greblicki, M. Pawlak, Int. J. Control 45, 343 (1987)
A. Mehmood et al., Signal, Image Video Proc. 12, 1603 (2018)
M.A.Z. Raja et al., Neural Comput. Appl. 29, 1455 (2018)
D. Comminiello et al., Signal Process. 135, 168 (2017)
C. Wang, T. Tang, Nonlinear Dyn. 77, 769 (2014)
F. Ding, X.P. Liu, G. Liu, Digit. Signal Process. 21, 215 (2011)
P. Cao, X. Luo, Digit. Signal Process. 56, 15 (2016)
Q. Shen, F. Ding, Nonlinear Dyn. 85, 499 (2016)
Y. Wang, F. Ding, Automatica 71, 308 (2016)
F. Ding, X. Liu, J. Chu, IET Control Theory Appl. 7, 176 (2013)
G. Toth et al., J. Comput. Phys. 231, 870 (2012)
L. Zhuo et al., Chin. Phys. 14, 1095 (2005)
F. Tobar et al., Pattern Recog. Lett. 105, 200 (2018)
Y. Zhao et al., Infrared Phys. Technol. 65, 17 (2014)
Y. Tan, Z. He, B. Tian, IEEE Signal Process. Lett. 22, 1244 (2015)
N.I. Chaudhary et al., Neural Comput. Appl. 29, 41 (2018)
S. Cheng et al., Signal Process. 133, 260 (2017)
M. Geravanchizadeh, S.G. Osgouei, Iran. J. Elect. Electron. Eng. 10, 256 (2014)
S. Zubair et al., Signal Process. 142, 441 (2018)
J.T. Machado, V. Kiryakova, F. Mainardi, Commun. Nonlinear Sci. Numer. Simul. 16, 1140 (2011)
S. He, S. Banerjee, B. Yan, Complexity 2018, 4140762 (2018)
S. He, S. Banerjee, Physica A 501, 408 (2018)
H. Sun et al., Commun. Nonlinear Sci. Numer. Simul. 64, 213 (2018)
Y. Wang, Eur. Phys. J. Plus 133, 481 (2018)
V.F. Morales-Delgado et al., Eur. Phys. J. Plus 133, 200 (2018)
J.F. Gómez-Aguilar et al., Eur. Phys. J. Plus 133, 103 (2018)
R. Roohi et al., Eur. Phys. J. Plus 133, 412 (2018)
N.I. Chaudhary, M.A.Z. Raja, Nonlinear Dyn. 79, 1385 (2015)
N.I. Chaudhary, M.A.Z. Raja, A.U.R. Khan, Nonlinear Dyn. 82, 1811 (2015)
M.S. Aslam, N.I. Chaudhary, M.A.Z. Raja, Nonlinear Dyn. 87, 519 (2017)
S.S. Haykin, Adaptive Filter Theory (Pearson Education, India, 2008)
I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications (Elsevier, 1998)
S. Roy, J.J. Shynk, IEEE Trans. Acoust., Speech, Signal Process. 38, 2088 (1990)
N.I. Chaudhary, S. Zubair, M.A.Z. Raja, Neural Comput. Appl. 30, 1133 (2018)
T. Kailath, Linear Systems (Prentice-Hall, Englewood Cliffs, NJ, 1980)
S. Qin et al., Med. Phys. 43, 3388 (2016)
K. Hammar, T. Djamah, M. Bettayeb, Nonlinear Dyn. 96, 2613 (2019)
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Chaudhary, N.I., Zubair, S., Aslam, M.S. et al. Design of momentum fractional LMS for Hammerstein nonlinear system identification with application to electrically stimulated muscle model. Eur. Phys. J. Plus 134, 407 (2019). https://doi.org/10.1140/epjp/i2019-12785-8
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DOI: https://doi.org/10.1140/epjp/i2019-12785-8