Design of sign fractional optimization paradigms for parameter estimation of nonlinear Hammerstein systems

Abstract

Fractional calculus plays a fundamental role in understanding the physics of nonlinear systems due to its heritage of uncertainty, nonlocality and complexity. In this study, novel sign fractional least mean square (F-LMS) algorithms are designed for ease in hardware implementation by applying sign function to input data and estimation error corresponding to first and fractional-order derivative terms in weight update mechanism of the standard F-LMS method. Theoretical expressions are derived for proposed sign F-LMS and its variants; strength of methods for different fractional orders is evaluated numerically through computer simulations for parameter estimation problem based on nonlinear Hammerstein system for low and high signal–noise variations. Comparison of the results from true parameters of the model illustrates the worth of the scheme in terms of accuracy, convergence and robustness. The stability and viability of design methodologies are examined through statistical observations on sufficiently large number of independent runs through mean square deviation and Nash–Sutcliffe efficiency performance indices.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Abbreviations

J :

Cost function

w :

Weights of a filter

x :

Input signal

µ 1 :

Step size for first-order gradient

v :

Noise signal

w :

Weight vector

\({\varvec{\uptheta}}\) :

Parameter vector

\(\delta\) :

Fitness function

E :

Error signal

M :

Number of filter taps

fr :

Fractional order

µ 2 :

Step size for fractional-order gradient

y :

Output signal

φ :

Information vector

\({\hat{\boldsymbol{\uptheta}}}\) :

Estimated parameter vector

σ 2 :

Variance

References

  1. 1.

    Pu YF, Zhou JL, Zhang Y, Zhang N, Huang G, Siarry P (2015) Fractional extreme value adaptive training method: fractional steepest descent approach. IEEE Trans Neural Netw Learn Syst 26(4):653–662

    MathSciNet  Google Scholar 

  2. 2.

    Cheng S, Wei Y, Chen Y, Li Y, Wang Y (2017) An innovative fractional order LMS based on variable initial value and gradient order. Sig Process 133:260–269

    Google Scholar 

  3. 3.

    Cheng S, Wei Y, Chen Y, Liang S, Wang Y (2017) A universal modified LMS algorithm with iteration order hybrid switching. ISA Trans 67:67–75

    Google Scholar 

  4. 4.

    Shah SM, Samar R, Raja MAZ (2018) Fractional-order algorithms for tracking Rayleigh fading channels. Nonlinear Dyn 92(3):1243–1259

    Google Scholar 

  5. 5.

    Chaudhary NI, Raja MAZ, Aslam MS, Ahmed N (2018) Novel generalization of Volterra LMS algorithm to fractional order with application to system identification. Neural Comput Appl 29(6):41–58

    Google Scholar 

  6. 6.

    Raja MAZ, Chaudhary NI (2015) Two-stage fractional least mean square identification algorithm for parameter estimation of CARMA systems. Sig Process 107(2015):327–339

    Google Scholar 

  7. 7.

    Chaudhary NI, Raja MAZ (2015) Identification of Hammerstein nonlinear ARMAX systems using nonlinear adaptive algorithms. Nonlinear Dyn 79(2):1385–1397

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Chaudhary NI, Raja MAZ (2015) Design of fractional adaptive strategy for input nonlinear Box–Jenkins systems. Sig Process 116:141–151

    Google Scholar 

  9. 9.

    Cheng S, Wei Y, Sheng D, Chen Y, Wang Y (2018) Identification for Hammerstein nonlinear ARMAX systems based on multi-innovation fractional order stochastic gradient. Sig Process 142:1–10

    Google Scholar 

  10. 10.

    Aslam MS, Chaudhary NI, Raja MAZ (2017) A sliding-window approximation-based fractional adaptive strategy for Hammerstein nonlinear ARMAX systems. Nonlinear Dyn 87(1):519–533

    MATH  Google Scholar 

  11. 11.

    Sharafian A, Ghasemi R (2019) Fractional neural observer design for a class of nonlinear fractional chaotic systems. Neural Comput Appl 31(4):1201–1213

    Google Scholar 

  12. 12.

    Arya Y (2019) AGC of restructured multi-area multi-source hydrothermal power systems incorporating energy storage units via optimal fractional-order fuzzy PID controller. Neural Comput Appl 31(3):851–872

    Google Scholar 

  13. 13.

    Atangana A, Koca I (2016) Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order. Chaos Solitons Fractals 89:447–454

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Dokuyucu MA, Celik E, Bulut H, Baskonus HM (2018) Cancer treatment model with the Caputo-Fabrizio fractional derivative. Eur Phys J Plus 133(3):92

    Google Scholar 

  15. 15.

    Atangana A (2018) Non validity of index law in fractional calculus: a fractional differential operator with Markovian and non-Markovian properties. Physica A 505:688–706

    MathSciNet  Google Scholar 

  16. 16.

    Ortigueira MD, Machado JT (2006) Fractional calculus applications in signals and systems. Sig Process 86(10):2503–2504

    MATH  Google Scholar 

  17. 17.

    Ortigueira MD, Ionescu CM, Machado JT, Trujillo JJ (2015) Fractional signal processing and applications. Sig Process 107:197

    Google Scholar 

  18. 18.

    Singh J, Kumar D, Hammouch Z, Atangana A (2018) A fractional epidemiological model for computer viruses pertaining to a new fractional derivative. Appl Math Comput 316:504–515

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Lodhi S, Manzar MA, Raja MAZ (2019) Fractional neural network models for nonlinear Riccati systems. Neural Comput Appl 31(1):359–378

    Google Scholar 

  20. 20.

    Stamov G, Stamova I (2017) Impulsive fractional-order neural networks with time-varying delays: almost periodic solutions. Neural Comput Appl 28(11):3307–3316

    MATH  Google Scholar 

  21. 21.

    Wang F, Yang Y, Xu X, Li L (2017) Global asymptotic stability of impulsive fractional-order BAM neural networks with time delay. Neural Comput Appl 28(2):345–352

    Google Scholar 

  22. 22.

    Monje CA, Chen Y, Vinagre BM, Xue D, Feliu-Batlle V (2010) Fractional-order systems and controls: fundamentals and applications. Springer, Berlin

    Google Scholar 

  23. 23.

    Chen Y, Xue D, Visioli A (2016) Guest editorial for special issue on fractional order systems and controls. IEEE/CAA J Autom Sin 3(3):255–256

    MathSciNet  Google Scholar 

  24. 24.

    Yin C, Cheng Y, Chen Y, Stark B, Zhong S (2015) Adaptive fractional-order switching-type control method design for 3D fractional-order nonlinear systems. Nonlinear Dyn 82(1–2):39–52

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Jafarian A, Mokhtarpour M, Baleanu D (2017) Artificial neural network approach for a class of fractional ordinary differential equation. Neural Comput Appl 28(4):765–773

    Google Scholar 

  26. 26.

    Saad KM, Baleanu D, Atangana A (2018) New fractional derivatives applied to the Korteweg–de Vries and Korteweg–de Vries–Burger’s equations. Comput Appl Math 37(4):5203–5216

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Yousefi F, Rivaz A, Chen W (2019) The construction of operational matrix of fractional integration for solving fractional differential and integro-differential equations. Neural Comput Appl 31(6):1867–1878

    Google Scholar 

  28. 28.

    Allwright A, Atangana A (2018) Fractal advection-dispersion equation for groundwater transport in fractured aquifers with self-similarities. Eur Phys J Plus 133(2):48

    Google Scholar 

  29. 29.

    Talaei Y, Asgari M (2018) An operational matrix based on Chelyshkov polynomials for solving multi-order fractional differential equations. Neural Comput Appl 30(5):1369–1376

    Google Scholar 

  30. 30.

    Psychalinos C, Elwakil AS, Radwan AG, Biswas K (2016) Guest editorial: fractional-order circuits and systems: theory, design, and applications. Circuits Syst Signal Process 35(6):1807–1813

    MathSciNet  Google Scholar 

  31. 31.

    Elwakil AS (2010) Fractional-order circuits and systems: an emerging interdisciplinary research area. IEEE Circuits Syst Mag 10(4):40–50

    Google Scholar 

  32. 32.

    Chen D, Sun S, Zhang C, Chen Y, Xue D (2013) Fractional-order TV-L2 model for image denoising. Cent Eur J Phys 11(10):1414–1422

    Google Scholar 

  33. 33.

    Chen D, Chen Y, Xue D (2015) Fractional-order total variation image denoising based on proximity algorithm. Appl Math Comput 257:537–545

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Raja MAZ, Qureshi IM (2009) A modified least mean square algorithm using fractional derivative and its application to system identification. Eur J Sci Res 35(1):14–21

    Google Scholar 

  35. 35.

    Raja MAZ, Chaudhary NI (2014) Adaptive strategies for parameter estimation of Box–Jenkins systems. IET Signal Proc 8(9):968–980

    Google Scholar 

  36. 36.

    Zubair S, Chaudhary NI, Khan ZA, Wang W (2018) Momentum fractional LMS for power signal parameter estimation. Sig Process 142:441–449

    Google Scholar 

  37. 37.

    Chaudhary NI, Ahmed M, Khan ZA, Zubair S, Raja MAZ, Dedovic N (2018) Design of normalized fractional adaptive algorithms for parameter estimation of control autoregressive autoregressive systems. Appl Math Model 55:698–715

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Aslam MS, Raja MAZ (2015) A new adaptive strategy to improve online secondary path modeling in active noise control systems using fractional signal processing approach. Sig Process 107:433–443

    Google Scholar 

  39. 39.

    Shah SM, Samar R, Naqvi SMR, Chambers JA (2014) Fractional order constant modulus blind algorithms with application to channel equalisation. Electron Lett 50(23):1702–1704

    Google Scholar 

  40. 40.

    Geravanchizadeh M, Ghalami Osgouei S (2014) Speech enhancement by modified convex combination of fractional adaptive filtering. Iran J Electr Electron Eng 10(4):256–266

    Google Scholar 

  41. 41.

    Chaudhary NI, Zubair S, Raja MAZ (2017) A new computing approach for power signal modeling using fractional adaptive algorithms. ISA Trans 68:189–202

    Google Scholar 

  42. 42.

    Shoaib B, Qureshi IM (2014) A modified fractional least mean square algorithm for chaotic and nonstationary time series prediction. Chin Phys B 23(3):030502

    Google Scholar 

  43. 43.

    Chaudhary NI, Aslam MS, Raja MAZ (2017) Modified Volterra LMS algorithm to fractional order for identification of Hammerstein non-linear system. IET Signal Proc 11(8):975–985

    Google Scholar 

  44. 44.

    Chaudhary NI, Raja MAZ, Khan AUR (2015) Design of modified fractional adaptive strategies for Hammerstein nonlinear control autoregressive systems. Nonlinear Dyn 82(4):1811–1830

    Google Scholar 

  45. 45.

    Billings SA (2013) Nonlinear system identification: NARMAX methods in the time, frequency, and spatio-temporal domains. Wiley, Chichester

    Google Scholar 

  46. 46.

    Vörös J (2015) Iterative identification of nonlinear dynamic systems with output backlash using three-block cascade models. Nonlinear Dyn 79(3):2187–2195

    MathSciNet  Google Scholar 

  47. 47.

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

    MathSciNet  Google Scholar 

  48. 48.

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

    MathSciNet  MATH  Google Scholar 

  49. 49.

    Wang DQ, Liu HB, Ding F (2015) Highly efficient identification methods for dual-rate Hammerstein systems. IEEE Trans Control Syst Technol 23(5):1952–1960

    Google Scholar 

  50. 50.

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

    MathSciNet  MATH  Google Scholar 

  51. 51.

    Khani F, Haeri M (2015) Robust model predictive control of nonlinear processes represented by Wiener or Hammerstein models. Chem Eng Sci 129:223–231

    Google Scholar 

  52. 52.

    Ławryńczuk M (2016) Nonlinear predictive control of dynamic systems represented by Wiener–Hammerstein models. Nonlinear Dyn 86(2):1193–1214

    MathSciNet  MATH  Google Scholar 

  53. 53.

    Zhang J, Chin KS, Ławryńczuk M (2018) Nonlinear model predictive control based on piecewise linear Hammerstein models. Nonlinear Dyn 92(3):1001–1021

    MATH  Google Scholar 

  54. 54.

    Togun N, Baysec S, Kara T (2012) Nonlinear modeling and identification of a spark ignition engine torque. Mech Syst Signal Process 26:294–304

    Google Scholar 

  55. 55.

    Holcomb CM, de Callafon RA, Bitmead RR (2014) Closed-loop identification of Hammerstein systems with application to gas turbines. IFAC Proc 47(3):493–498

    Google Scholar 

  56. 56.

    Alonge F, Rabbeni R, Pucci M, Vitale G (2015) Identification and robust control of a quadratic DC/DC boost converter by Hammerstein model. IEEE Trans Ind Appl 51(5):3975–3985

    Google Scholar 

  57. 57.

    Zhao Y, Jiang Y, Feng J, Wu L (2016) Modeling of memristor-based chaotic systems using nonlinear Wiener adaptive filters based on backslash operator. Chaos Solitons Fractals 87:12–16

    MathSciNet  MATH  Google Scholar 

  58. 58.

    Le F, Markovsky I, Freeman CT, Rogers E (2012) Recursive identification of Hammerstein systems with application to electrically stimulated muscle. Control Eng Pract 20(4):386–396

    Google Scholar 

  59. 59.

    Le F, Markovsky I, Freeman CT, Rogers E (2010) Identification of electrically stimulated muscle models of stroke patients. Control Eng Pract 18(4):396–407

    Google Scholar 

  60. 60.

    Rébillat M, Hennequin R, Corteel E, Katz BF (2011) Identification of cascade of Hammerstein models for the description of nonlinearities in vibrating devices. J Sound Vib 330(5):1018–1038

    Google Scholar 

  61. 61.

    Maatallah OA, Achuthan A, Janoyan K, Marzocca P (2015) Recursive wind speed forecasting based on Hammerstein auto-regressive model. Appl Energy 145:191–197

    Google Scholar 

  62. 62.

    Hu H, Ding R (2014) Least squares based iterative identification algorithms for input nonlinear controlled autoregressive systems based on the auxiliary model. Nonlinear Dyn 76(1):777–784

    MathSciNet  MATH  Google Scholar 

  63. 63.

    Li G, Wen C, Zheng WX, Chen Y (2011) Identification of a class of nonlinear autoregressive models with exogenous inputs based on kernel machines. IEEE Trans Signal Process 59(5):2146–2159

    MathSciNet  MATH  Google Scholar 

  64. 64.

    Xiao Y, Song G, Liao Y, Ding R (2012) Multi-innovation stochastic gradient parameter estimation for input nonlinear controlled autoregressive models. Int J Control Autom Syst 10(3):639–643

    Google Scholar 

  65. 65.

    Chen H, Ding F (2015) Hierarchical least squares identification for Hammerstein nonlinear controlled autoregressive systems. Circuits Syst Signal Process 34(1):61–75

    MathSciNet  MATH  Google Scholar 

  66. 66.

    Chen H, Ding F, Xiao Y (2015) Decomposition-based least squares parameter estimation algorithm for input nonlinear systems using the key term separation technique. Nonlinear Dyn 79(3):2027–2035

    MATH  Google Scholar 

  67. 67.

    Raja MAZ, Shah AA, Mehmood A, Chaudhary NI, Aslam MS (2018) Bio-inspired computational heuristics for parameter estimation of nonlinear Hammerstein controlled autoregressive system. Neural Comput Appl 29(12):1455–1474

    Google Scholar 

  68. 68.

    Chaudhary NI, Zubair S, Raja MAZ (2016) Design of momentum LMS adaptive strategy for parameter estimation of Hammerstein controlled autoregressive systems. Neural Comput Appl. https://doi.org/10.1007/s00521-016-2762-1

    Article  Google Scholar 

  69. 69.

    Chaudhary NI, Raja MAZ, Khan JA, Aslam MS (2013) Identification of input nonlinear control autoregressive systems using fractional signal processing approach. Sci World J 2013:467276. https://doi.org/10.1155/2013/467276

    Article  Google Scholar 

  70. 70.

    Chaudhary NI, Manzar MA, Raja MAZ (2018) Fractional Volterra LMS algorithm with application to Hammerstein control autoregressive model identification. Neural Comput Appl. https://doi.org/10.1007/s00521-018-3362-z

    Article  Google Scholar 

  71. 71.

    Haykin S (2008) Adaptive filter theory. Pearson Education India, New Delhi

    Google Scholar 

  72. 72.

    Podlubny I (1999) Fractional differential equations. Academic Press, New York

    Google Scholar 

  73. 73.

    Kilbas A, Aleksandrovich A, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations, vol 204. Elsevier, Hoboken

    Google Scholar 

  74. 74.

    Merrikh-Bayat F, Bagheri-Shouraki S (2011) Mixed analog-digital crossbar-based hardware implementation of sign–sign LMS adaptive filter. Analog Integr Circ Sig Process 66(1):41–48

    Google Scholar 

  75. 75.

    Rahman MZU, Shaik RA, Reddy DRK (2011) Efficient sign based normalized adaptive filtering techniques for cancelation of artifacts in ECG signals: application to wireless biotelemetry. Sig Process 91(2):225–239

    MATH  Google Scholar 

  76. 76.

    Eweda E (1999) Transient performance degradation of the LMS, RLS, sign, signed regressor, and sign-sign algorithms with data correlation. IEEE Trans Circuits Syst II Analog Digit Signal Process 46(8):1055–1062

    MATH  Google Scholar 

  77. 77.

    Lotfizad M, Yazdi HS (2005) Modified clipped LMS algorithm. EURASIP J Appl Sig Process 2005:1229–1234

    MATH  Google Scholar 

  78. 78.

    Lu L, Zhao H, Li K, Chen B (2016) A novel normalized sign algorithm for system identification under impulsive noise interference. Circuits Syst Signal Process 35(9):3244–3265

    MathSciNet  MATH  Google Scholar 

  79. 79.

    Abdel-Nasser M, Mahmoud K, Kashef H (2018) A novel smart grid state estimation method based on neural networks. Int J Interact Multimedia Artif Intell 5(1):92–100

    Google Scholar 

  80. 80.

    Bouchra N, Aouatif A, Mohammed N, Nabil H, Benkaddour FZ, Taghezout N et al (2018) Deep belief network and auto-encoder for face classification. Int J Interact Multimedia Artif Intell. https://doi.org/10.9781/ijimai.2018.06.004

    Article  Google Scholar 

  81. 81.

    Romero Á, Dorronsoro JR, Díaz J (2018) Day-ahead price forecasting for the Spanish Electricity Market. Int J Interact Multimedia Artif Intell. https://doi.org/10.9781/ijimai.2018.04.008(in press)

    Article  Google Scholar 

  82. 82.

    Croda RMC, Romero DEG, Morales SOC (2018) Sales prediction through neural networks for a small dataset. Int J Interact Multimedia Artif Intell. https://doi.org/10.9781/ijimai.2018.04.003(in press)

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Muhammad Saeed Aslam.

Ethics declarations

Conflict of interest

All the authors of the manuscript declared that there are no potential conflicts of interest.

Human and animal rights statements

All the authors of the manuscript declared that there is no research involving human participants and/or animal.

Informed consent

All the authors of the manuscript declared that there is no material that required informed consent.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chaudhary, N.I., Aslam, M.S., Baleanu, D. et al. Design of sign fractional optimization paradigms for parameter estimation of nonlinear Hammerstein systems. Neural Comput & Applic 32, 8381–8399 (2020). https://doi.org/10.1007/s00521-019-04328-0

Download citation

Keywords

  • Nonlinear system identification
  • Sign regressors
  • Fractional adaptive signal processing
  • Hammerstein models
  • Control structures