Skip to main content
SpringerLink
Log in

Convergence Analysis of the Hierarchical Least Squares Algorithm for Bilinear-in-Parameter Systems

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

This paper studies the convergence of the hierarchical identification algorithm for bilinear-in-parameter systems. By replacing the unknown variables in the information vector with their estimates, a hierarchical least squares algorithm is derived based on the model decomposition. The proposed algorithm has higher computational efficiency than the over-parameterization model-based recursive least squares algorithm. The performance analysis shows that the parameter estimation errors converge to zero under persistent excitation conditions. The effectiveness of the proposed algorithm is verified by simulation examples.

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

Access this article

Log in via an institution

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
Fig. 8

Similar content being viewed by others

References

  1. R. Abrahamsson, S.M. Kay, P. Stoica, Estimation of the parameters of a bilinear model with applications to submarine detection and system identification. Digit. Signal Process. 17(4), 756–773 (2007)

    Article  Google Scholar 

  2. E.W. Bai, An optimal two-stage identification algorithm for Hammerstein–Wiener nonlinear systems. Automatica 34(3), 333–338 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. E.W. Bai, Y. Liu, Least squares solutions of bilinear equations. Syst. Control Lett. 55(6), 466–472 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. H.B. Chen, Y.S. Xiao, F. Ding, Hierarchical gradient parameter estimation algorithm for Hammerstein nonlinear systems using the key term separation principle. Appl. Math. Comput. 247, 1202–1210 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. F. Ding, K.P. Deng, X.M. Liu, 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. F. Ding, X.P. Liu, G. Liu, Identification methods for Hammerstein nonlinear systems. Digit. Signal Process. 21(2), 215–238 (2011)

    Article  Google Scholar 

  8. F. Ding, X.H. Wang, Q.J. Chen, Y.S. Xiao, Recursive least squares parameter estimation for a class of output nonlinear systems based on the model decomposition. Circuits Syst. Signal Process. (2016). doi:10.1007/s00034-015-0190-6

  9. G.C. Goodwin, K.S. Sin, Adaptive Filtering Prediction and Control (Prentice Hall, Englewood Cliffs, 1984)

    MATH  Google Scholar 

  10. G.H. Golub, C.F. Van Loan, Matrix Computations, 3rd edn. (Johns Hopkins University Press, Baltimore, 1996)

    MATH  Google Scholar 

  11. Y. Gu, F. Ding, J.H. Li, States based iterative parameter estimation for a state space model with multi-state delays using decomposition. Signal Process. 106, 294–300 (2015)

    Article  Google Scholar 

  12. A. Hagenblad, L. Ljung, A. Wills, Maximum likelihood identification of Wiener models. Automatica 44(11), 2697–2705 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Y.B. Hu, B.L. Liu, Q. Zhou, A multi-innovation generalized extended stochastic gradient algorithm for output nonlinear autoregressive moving average systems. Appl. Math. Comput. 247, 218–224 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Y.B. Hu, B.L. Liu, Q. Zhou, C. Yang, Recursive extended least squares parameter estimation for Wiener nonlinear systems with moving average noises. Circuits Syst. Signal Process. 33(2), 655–664 (2014)

    Article  MathSciNet  Google Scholar 

  15. J. Huang, Y. Shi, H.N. Huang, Z. Li, l-2-l-infinity filtering for multirate nonlinear sampled-data systems using T–S fuzzy models. Digit. Signal Process. 23(1), 418–426 (2013)

    Article  MathSciNet  Google Scholar 

  16. M. Jafari, M. Salimifard, M. Dehghani, Identification of multivariable nonlinear systems in the presence of colored noises using iterative hierarchical least squares algorithm. ISA Trans. 53(4), 1243–1252 (2014)

    Article  Google Scholar 

  17. Y. Ji, X.M. Liu, Unified synchronization criteria for hybrid switching-impulsive dynamical networks. Circuits Syst. Signal Process. 34(5), 1499–1517 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Y. Ji, X.M. Liu et al., New criteria for the robust impulsive synchronization of uncertain chaotic delayed nonlinear systems. Nonlinear Dyn. 79(1), 1–9 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Y. Liu, E.W. Bai, Iterative identification of Hammerstein systems. Automatica 43(2), 346–354 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. H. Li, Y. Gao, P. Shi, H.K. Lam, Observer-based fault detection for nonlinear systems with sensor fault and limited communication capacity. IEEE Trans. Autom. Control (2015). doi:10.1109/TAC.2015.2503566

  21. H. Li, P. Shi, D. Yao, L. Wu, Observer-based adaptive sliding mode control of nonlinear Markovian jump systems. Automatica 64, 133–142 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. H. Li, C.W. Wu, P. Shi, Y.B. Gao, Control of nonlinear networked systems with packet dropouts: interval type-2 fuzzy model-based approach. IEEE Trans. Cybern. 45(11), 2378–2389 (2015)

    Article  Google Scholar 

  23. H. Li, C.W. Wu, L.G. Wu, H.K. Lam, Y.B. Gao, Filtering of interval type-2 fuzzy systems with intermittent measurements. IEEE Trans. Cybern. (2015). doi:10.1109/TCYB.2015.2413134

  24. H. Li, S. Yin, Y.N. Pan, H.K. Lam, Model reduction for interval type-2 Takagi–Sugeno fuzzy systems. Automatica 61, 308–314 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. J.B. Qiu, S.X. Ding, H.J. Gao, S. Yin, Fuzzy-model-based reliable static output feedback H-infinity control of nonlinear hyperbolic PDE systems. IEEE Trans. Fuzzy Syst. (2015). doi:10.1109/TFUZZ.2015.2457934

  26. J.B. Qiu, G. Feng, H.J. Gao, Static-output-feedback H-infinity control of continuous-time T–S fuzzy affine systems via piecewise Lyapunov functions. IEEE Trans. Fuzzy Syst. 21(2), 245–261 (2013)

    Article  Google Scholar 

  27. J.B. Qiu, H. Tian, Q.G. Lu, H.J. Gao, Nonsynchronized robust filtering design for continuous-time T–S fuzzy affine dynamic systems based on piecewise Lyapunov functions. IEEE Trans. Cybern. 43(6), 1755–1766 (2013)

    Article  Google Scholar 

  28. J.B. Qiu, Y.L. Wei, H.R. Karimi, New approach to delay-dependent H-infinity control for continuous-time Markovian jump systems with time-varying delay and deficient transition descriptions. J. Frankl. Inst. Eng. Appl. Math. 352(1), 189–215 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. J.G. Smith, S. Kamat, K.P. Madhavan, Modeling of pH process using wavenet based Hammerstein model. J. Process Control 17(6), 551–561 (2007)

    Article  Google Scholar 

  30. M. Tadeusiewicz, A. Kuczynski, S. Halgas, Catastrophic fault diagnosis of a certain class of nonlinear analog circuits. Circuits Syst. Signal Process. 34(2), 353–375 (2015)

    Article  Google Scholar 

  31. J. Vörös, Parameter identification of Wiener systems with multisegment piecewise-linear nonlinearities. Syst. Control Lett. 56(2), 99–105 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  32. J. Vörös, Identification of nonlinear dynamic systems with input saturation and output backlash using three-block cascade models. J. Frankl. Inst. Eng. Appl. Math. 351(12), 5455–5466 (2014)

    Article  MathSciNet  Google Scholar 

  33. D.Q. Wang, Hierarchical parameter estimation for a class of MIMO Hammerstein systems based on the reframed models. Appl. Math. Lett. 57, 13–19 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  34. Y.J. Wang, F. Ding, Iterative estimation for a nonlinear IIR filter with moving average noise by means of the data filtering technique. IMA J. Math. Control Inf. (2016). doi:10.1093/imamci/dnv067

  35. Y.J. Wang, F. Ding, Recursive least squares algorithm and gradient algorithm for Hammerstein–Wiener systems using the data filtering. Nonlinear Dyn. (2016). doi:10.1007/s11071-015-2548-5

  36. Y.J. Wang, F. Ding, Recursive parameter estimation algorithms and convergence for a class of nonlinear systems with colored noise. Circuits Syst. Signal Process. (2016). doi:10.1007/s00034-015-0210-6

  37. X.H. Wang, F. Ding, Decomposition based recursive identification algorithms for bilinear-parameter models. The 11th World Congress on Intelligent Control and Automation, June 29–July 4, Shenyang, China, pp. 6107–6111 (2014)

  38. X.H. Wang, F. Ding, Recursive parameter and state estimation for an input nonlinear state space system using the hierarchical identification principle. Signal Process. 117, 208–218 (2015)

    Article  Google Scholar 

  39. X.H. Wang, F. Ding, Convergence of the recursive identification algorithms for multivariate pseudo-linear regressive systems. Int. J. Adapt. Control Signal Process. (2016). doi:10.1002/acs.2642

  40. X.H. Wang, F. Ding, Modelling and multi-innovation parameter identification for Hammerstein nonlinear state space systems using the filtering technique. Math. Comput. Modell. Dyn. Syst. (2016). doi:10.1080/13873954.2016.1142455

  41. T. Wang, H.J. Gao, J.B. Qiu, A combined adaptive neural network and nonlinear model predictive control for multirate networked industrial process control. IEEE Trans. Neural Netw. Learn. Syst. 27(2), 416–425 (2015)

    Article  MathSciNet  Google Scholar 

  42. D.Q. Wang, H.B. Liu et al., Highly efficient identification methods for dual-rate Hammerstein systems. IEEE Trans. Control Syst. Technol. 23(5), 1952–1960 (2015)

    Article  Google Scholar 

  43. D.Q. Wang, W. Zhang, Improved least squares identification algorithm for multivariable Hammerstein systems. J. Frankl. Inst. Eng. Appl. Math. 352(11), 5292–5370 (2015)

    Article  MathSciNet  Google Scholar 

  44. S. Yin, Z.H. Huang, Performance monitoring for vehicle suspension system via fuzzy positivistic C-means clustering based on accelerometer measurements. IEEE/ASME Trans. Mechatronics 20(5), 2613–2620 (2015)

    Article  Google Scholar 

  45. S. Yin, X.W. Li, H.J. Gao, O. Kaynak, Data-based techniques focused on modern industry: an overview. IEEE Trans. Ind. Electron. 62(1), 657–667 (2015)

    Article  Google Scholar 

  46. F. Yu, Z.Z. Mao, M.X. Jia, P. Yuan, Recursive parameter identification of Hammerstein-Wiener systems with measurement noise. Signal Process. 105, 137–147 (2014)

    Article  Google Scholar 

  47. S. Yin, X.P. Zhu, O. Kaynak, Improved PLS focused on key-performance-indicator-related fault diagnosis. IEEE Trans. Ind. Electron. 62(3), 1651–1658 (2015)

    Article  Google Scholar 

  48. Q. Zhou, H. Li, P. Shi, Decentralized adaptive fuzzy tracking control for robot finger dynamics. IEEE Trans. Fuzzy Syst. 23(3), 501–510 (2015)

    Article  Google Scholar 

  49. Q. Zhou, H. Li, P. Shi, Approximation-based adaptive tracking control for MIMO nonlinear systems with input saturation. IEEE Trans. Cybern. 45(10), 2119–2128 (2015)

    Article  Google Scholar 

Download references

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 61273194) and the 111 Project (B12018), the Graduate Research Innovation Program of Jiangsu Province (No. KYLX15_1166) and the Key Research Project of Henan Higher Education Institutions (No. 16A120010).

Author information

Authors and Affiliations

  1. Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Jiangnan University, Wuxi, 214122, People’s Republic of China

    Xuehai Wang & Feng Ding

  2. Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Feng Ding, Fuad E. Alsaadi & Tasawar Hayat

  3. Department of Mathematics, Quaid-I-Azam University, Islamabad, 44000, Pakistan

    Tasawar Hayat

  4. School of Mathematics and Physics, Henan University of Urban Construction, Pingdingshan, 467036, People’s Republic of China

    Xuehai Wang

Authors
  1. Xuehai Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Feng Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fuad E. Alsaadi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Tasawar Hayat
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Feng Ding.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, X., Ding, F., Alsaadi, F.E. et al. Convergence Analysis of the Hierarchical Least Squares Algorithm for Bilinear-in-Parameter Systems. Circuits Syst Signal Process 35, 4307–4330 (2016). https://doi.org/10.1007/s00034-016-0278-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-016-0278-7

Keywords

Search

Navigation