Skip to main content
Log in

Design of modified fractional adaptive strategies for Hammerstein nonlinear control autoregressive systems

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In this study, modified fractional least mean square (FrLMS) algorithms are formulated for parameter estimation of Hammerstein nonlinear control autoregressive system (HNCAR) by exploiting the fractional calculus concepts in weight adaptation mechanism of the algorithm. In modified FrLMS (MFrLMS) of first kind, forgetting factor is applied to exploit the strength of both standard LMS and FrLMS algorithms. The MFrLMS algorithm of second kind is based on single fractional weight adaptation term in cost function for finding the optimal values instead of considering both fractional and first derivative terms as in FrLMS approach. Performance analysis of the proposed methods are carried out on the basis of optimization ability to the true values of HNCAR system, and comparison is made in terms of accuracy, convergence and complexity. Reliability and effectiveness of the proposed scheme is validated through performance indices based on mean square error, variance account for and Nash Sutcliffe efficiency for sufficient large number of independent runs.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

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

Similar content being viewed by others

References

  1. Wang, C., Tang, T.: Several gradient-based iterative estimation algorithms for a class of nonlinear systems using the filtering technique. Nonlinear Dyn. 77(3), 769–780 (2014)

    Article  Google Scholar 

  2. Han, H., Xie, L., Ding, F., Liu, X.: Hierarchical least-squares based iterative identification for multivariable systems with moving average noises. Math. Comput. Model. 51(9), 1213–1220 (2010)

    Article  MATH  Google Scholar 

  3. Chen, J., Zhang, Y., Ding, R.: Gradient-based parameter estimation for input nonlinear systems with ARMA noises based on the auxiliary model. Nonlinear Dyn. 72(4), 865–871 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ding, F., Liu, P.X., Liu, G.: Gradient based and least-squares based iterative identification methods for OE and OEMA systems. Digt. Signal Process. 20(3), 664–677 (2010)

    Article  Google Scholar 

  5. Zhou, L., Li, X., Pan, F.: Gradient-based iterative identification for Wiener nonlinear systems with non-uniform sampling. Nonlinear Dyn. 76(1), 627–634 (2014)

    Article  MathSciNet  Google Scholar 

  6. Ding, F.: Two-stage least squares based iterative estimation algorithm for CARARMA system modeling. Appl. Math. Model. 37(7), 4798–4808 (2013)

    Article  MathSciNet  Google Scholar 

  7. Hu, P., Ding, F.: 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 

  8. Li, X., Zhou, L., Sheng, J., Ding, R.: Recursive least squares parameter estimation algorithm for dual-rate sampled-data nonlinear systems. Nonlinear Dyn. 76(2), 1327–1334 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ding, F., Chen, T.: Identification of Hammerstein nonlinear ARMAX systems. Automatica 41(9), 1479–1489 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Shen, Q., Ding, F.: 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 

  11. Ding, F.: Hierarchical multi-innovation stochastic gradient algorithm for Hammerstein nonlinear system modeling. Appl. Math. Model. 37(4), 1694–1704 (2013)

    Article  MathSciNet  Google Scholar 

  12. Wang, D., Ding, F., Ximei, L.: Least squares algorithm for an input nonlinear system with a dynamic subspace state space model. Nonlinear Dyn. 75(1–2), 49–61 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ding, F., Shi, Y., Chen, T.: Gradient-based identification methods for Hammerstein nonlinear ARMAX models. Nonlinear Dyn. 45(1–2), 31–43 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ding, F., Liu, X., Chu, J.: Gradient-based and least-squares-based iterative algorithms for Hammerstein systems using the hierarchical identification principle. IET Control Theory Appl. 7(2), 176–184 (2013)

    Article  MathSciNet  Google Scholar 

  15. Ding, F., Deng, K., Liu, X.: Decomposition based Newton iterative identification method for a Hammerstein nonlinear FIR system with ARMA noise. Circuits Syst. Signal Process. 33(9), 2881–2893 (2014)

    Article  MathSciNet  Google Scholar 

  16. Karimi-Ghartemani, M., Iravani, M.R.: A nonlinear adaptive filter for online signal analysis in power systems: applications. IEEE Trans. Power Deliv. 17(2), 617–622 (2002)

    Article  Google Scholar 

  17. Cao, Y.Y., Lin, Z.: Robust stability analysis and fuzzy-scheduling control for nonlinear systems subject to actuator saturation. IEEE Trans. Fuzzy Syst. 11(1), 57–67 (2003)

    Article  Google Scholar 

  18. Hunter, I.W., Korenberg, M.J.: The identification of nonlinear biological systems: Wiener and Hammerstein cascade models. Biol. Cybern. 55(2–3), 135–144 (1986)

    MathSciNet  MATH  Google Scholar 

  19. Fruzzetti, K.P., Palazoğlu, A., McDonald, K.A.: Nonlinear model predictive control using Hammerstein models. J. Process Control 7(1), 31–41 (1997)

    Article  Google Scholar 

  20. Dupont, S., Luettin, J.: Audio-visual speech modeling for continuous speech recognition. IEEE Trans. Multimed. 2(3), 141–151 (2000)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. Chen, H., Ding, F.: Decomposition based recursive least squares parameter estimation for Hammerstein nonlinear controlled autoregressive systems. IEEE American Control Conference (ACC), 2013 (pp. 2436–2441) (2013, June)

  24. Xiong, W., Fan, W., Ding, R.: Least-squares parameter estimation algorithm for a class of input nonlinear systems. J. Appl. Math. (2012)

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

    Article  MathSciNet  Google Scholar 

  26. Ortigueira, M.D.: Introduction to fractional signal processing. Part I: continuous time systems. IEE Proc. Vis. Image Signal Process. 1, 62–70 (2000)

    Article  Google Scholar 

  27. Ortigueira, M.D.: Introduction to fractional signal processing. Part 2: discrete-time systems. IEE Proc. Vis. Image Signal Process. 1, 71–78 (2000)

    Article  Google Scholar 

  28. Ortigueira, M.D., Machado, J.T.: Fractional signal processing and applications. Signal Process. 83(11), 2285–2286 (2003)

    Article  Google Scholar 

  29. Ortigueira, M.D., Machado, J.T.: Fractional calculus applications in signals and systems. Signal Process. 86(10), 2503–2504 (2006)

    Article  MATH  Google Scholar 

  30. Shyu, J.J., Pei, S.C., Chan, C.H.: An iterative method for the design of variable fractional-order FIR differintegrators. Signal Process. 89(3), 320–327 (2009)

    Article  MATH  Google Scholar 

  31. Tseng, C.C., Lee, S.L.: Designs of fractional derivative constrained 1-D and 2-D FIR filters in the complex domain. Signal Process. 95, 111–125 (2014)

    Article  Google Scholar 

  32. Machado, J.T.: Fractional order describing functions. Signal Process. 107, 389–394 (2015). doi:10.1016/j.sigpro.2014.05.012

    Article  Google Scholar 

  33. Petráš, I.: Fractional-order feedback control of a DC motor. J. Electr. Eng. 60(3), 117–128 (2009)

    Google Scholar 

  34. Ortigueira, M.D., Trujillo, J.J., Martynyuk, V.I., Coito, F.J.: A generalized power series and its application in the inversion of transfer functions. Signal Process. 107, 238–245 (2015)

    Article  Google Scholar 

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

    Google Scholar 

  36. Shah, S.M., Samar, R., Raja, M.A.Z., Chambers, J.A.: Fractional normalised filtered-error least mean squares algorithm for application in active noise control systems. Electron. Lett 50(14), 973–975 (2014)

    Article  Google Scholar 

  37. Raja, M.A.Z., Chaudhary, N.I.: Adaptive strategies for parameter estimation of Box–Jenkins systems. IET Signal Process. 8(9), 968–980 (2014). doi:10.1049/iet-spr.2013.0438

    Article  Google Scholar 

  38. Chaudhary, N.I., Raja, M.A.Z.: Design of fractional adaptive strategy for input nonlinear Box–Jenkins systems. Signal Process. (2015). doi:10.1016/j.sigpro.2015.04.015

  39. Chaudhary, N.I., Raja, M.A.Z.: Identification of Hammerstein nonlinear ARMAX systems using nonlinear adaptive algorithms. Nonlinear Dyn. 79(2), 1385–1397 (2015)

    Article  MathSciNet  Google Scholar 

  40. Chaudhary, N.I., Raja, M.A.Z., Khan, J.A., Aslam, M.S.: Identification of input nonlinear control autoregressive systems using fractional signal processing approach. Sci. World J. ID 467276 (2013)

  41. Raja, M.A.Z., Chaudhary, N.I.: Two-stage fractional least mean square identification algorithm for parameter estimation of CARMA systems. Signal Process. 107, 327–339 (2015). doi:10.1016/j.sigpro.2014.06.015

    Article  Google Scholar 

  42. Aslam, M.S., Raja, M.A.Z.: A new adaptive strategy to improve online secondary path modeling in active noise control systems using fractional signal processing approach. Signal Process. 107, 433–443 (2015). doi:10.1016/j.sigpro.2014.04.012

    Article  Google Scholar 

  43. Shah, S.M., Samar, R., Naqvi, S.M.R., Chambers, J.A.: Fractional order constant modulus blind algorithms with application to channel equalisation. Electron. Lett. 50(23), 1702–1704 (2014)

    Article  Google Scholar 

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

    Google Scholar 

  45. Geravanchizadeh, M., Osgouei, S.G.: Dual-channel speech enhancement using normalized fractional least-mean-squares algorithm. IEEE 19th Iranian Conference on Electrical Engineering (ICEE), 2011 (pp. 1–5) (2011, May)

  46. Osgouei, S. G., Geravanchizadeh, M.: Speech enhancement using convex combination of fractional least-mean-squares algorithm. IEEE 5th International Symposium on Telecommunications (IST), 2010 (pp. 869-872) (2010, December)

  47. Dubey, S.K., Rout, N.K.: FLMS algorithm for acoustic echo cancellation and its comparison with LMS. IEEE 1st International Conference on Recent Advances in Information Technology (RAIT), 2012, pp. 852–856 (2012, March)

  48. Akhtar, P., Yasin, M.: : Performance analysis of bessel beamformer and LMS algorithm for smart antenna array in mobile communication system. In Emerging Trends and Applications in Information Communication Technologies, pp. 52–61. Berlin: Springer (2012)

  49. Yasin, M., Akhtar, P.: Performance analysis of bessel beamformer with LMS algorithm for smart antenna array. In:IEEE International Conference on Open Source Systems and Technologies (ICOSST), 2012, pp. 1–5 (2012, December)

  50. Shoaib, B., Qureshi, I. M.: A modified fractional least mean square algorithm for chaotic and nonstationary time series prediction. Chin. Phys. B, 23(3), 030502. 305–312 (2014)

  51. Shoaib, B., Qureshi, I.M.: Adaptive step-size modified fractional least mean square algorithm for chaotic time series prediction. Chin. Phys. B 23(5), 050503 (2014)

    Article  Google Scholar 

  52. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    MATH  Google Scholar 

  53. Anatoly, A.K., Hari, M.S., Juan, J.T.: Theory and Applications of Fractional Differential Equations, North Holland Mathematics Studies, vol. 204. Elsevier, New York (2006)

    Google Scholar 

  54. Ortigueira, M.D.: Fractional calculus for scientists and engineers, vol. 84. Springer, Dordrecht (2011)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Muhammad Asif Zahoor Raja.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chaudhary, N.I., Raja, M.A.Z. & Rehman Khan, A.U. Design of modified fractional adaptive strategies for Hammerstein nonlinear control autoregressive systems. Nonlinear Dyn 82, 1811–1830 (2015). https://doi.org/10.1007/s11071-015-2279-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-015-2279-7

Keywords

Navigation