Skip to main content
SpringerLink
Log in

A Multi-innovation Recursive Least Squares Algorithm with a Forgetting Factor for Hammerstein CAR Systems with Backlash

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

Abstract

This study addresses the identification of Hammerstein CAR systems with backlash, where the nonlinear backlash is described as one regression identification model using a two switching function mathematical model. In such a case, the Hammerstein CAR systems with backlash can be transformed into a piecewise linearized model. Then, a novel multi-innovation recursive least squares algorithm with a forgetting factor is applied to estimate the parameters of the proposed model. Finally, numerical examples are presented to test the performance of the proposed algorithm.

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

Similar content being viewed by others

References

  1. A. Barreiro, A. Banos, Input-output stability of systems with backlash. Automatica 42(6), 1017–1024 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. S. Bittanti, P. Boizern, M. Campi, Convergence and exponential convergence of identification algorithms with directional forgetting factor. Automatica 26(5), 929–932 (1990)

    Article  MATH  Google Scholar 

  3. V. Cerone, D. Regruto, Bounding the parameters of linear systems with input backlash. IEEE Trans. Autom. Control 52(3), 531–536 (2007)

    Article  MathSciNet  Google Scholar 

  4. J. Chen, X. Wang, R. Ding, Gradient based estimation algorithm for Hammerstein systems with saturation and dead-zone nonlinearities. Appl. Math. Model. 36(1), 238–243 (2012)

    Article  MATH  Google Scholar 

  5. F. Ding, P.X. Liu, G.J. Liu, Auxiliary model based multi-innovation extended stochastic gradient parameter estimation with colored measurement noises. Signal Process. 89(10), 1883–1890 (2009)

    Article  MATH  Google Scholar 

  6. F. Ding, Several multi-innovation identification methods. Digit. Signal Proc. 20(4), 1027–1039 (2010)

    Article  Google Scholar 

  7. 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

    MathSciNet  Google Scholar 

  8. F. Ding, T. Chen, Performance analysis of multi-innovation gradient type identification methods. Automatica 43(1), 1–14 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. F. Ding, J. Ding, Least squares parameter estimation with irregularly missing data. Int. J. Adapt. Control Signal Process. 24(7), 540–553 (2010)

    MathSciNet  MATH  Google Scholar 

  10. F. Ding, X.P. Liu, Parameter estimation with scarce measurements. Automatica 47(8), 1646–1655 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. R. Dong, Q.Y. Tan, Y.H. Tan, Recursive identification algorithm for dynamic systems with output backlash and its convergence. Int. J. Appl. Math. Comput. Sci. 19(4), 631–638 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. T.R. Fortescue, L.S. Kershenbaum, B.E. Ydstie, Implementation of self tuning regulators with variable forgetting factors. Automatica 17(6), 831–835 (1981)

    Article  Google Scholar 

  13. Y. Gao, H. Li, M. Chadli, H.K. Lam, Static output-feedback control for interval type-2 discrete-time fuzzy systems. Complexity (2014). doi:10.1002/cplx.21617

    MathSciNet  Google Scholar 

  14. Y. Gao, P. Shi, H. Li, S.K. Nguang, Output tracking control for fuzzy delta operator systems with time-varying delays. J. Franklin Inst. 352, 2951–2970 (2015)

    Article  MathSciNet  Google Scholar 

  15. L.L. Han, F. Ding, Identification for multirate multi-input systems using the multi-innovation identification theory. Comput. Math. Appl. 57(9), 1438–1449 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. L.L. Han, F. Ding, Multi-innovation stochastic gradient algorithms for multi-input multi-output systems. Digit. Signal Proc. 19(4), 545–554 (2009)

    Article  MathSciNet  Google Scholar 

  17. A. Janczak, Identification of Nonlinear Systems Using Neural Networks and Polynomial Models: A Block-Oriented Approach, vol. 310 (Springer, New York, 2004)

    MATH  Google Scholar 

  18. A.K. Kohli, A. Rai, Numeric variable forgetting factor RLS algorithm for second-order volterra filtering. Circuits Syst. Signal Process. 32(1), 223–232 (2013)

    Article  MathSciNet  Google Scholar 

  19. S.H. Leung, C.F. So, Gradient-based variable forgetting factor RLS algorithm in time-varying environments. IEEE Trans. Signal Process. 53(8), 3141–3150 (2005)

    Article  MathSciNet  Google Scholar 

  20. Y. Li, S. Tong, T. Li, Adaptive fuzzy output feedback control of uncertain nonlinear systems with unknown backlash-like hysteresis. Inf. Sci. 198(1), 130–146 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. X.L. Li, L.C. Zhou, R. Ding, Auxiliary model-based forgetting factor stochastic gradient algorithm for Dual-rate nonlinear systems and its application to a nonlinear analog circuit. Circuits Syst. Signal Process. 33(6), 1957–1969 (2014)

    Article  Google Scholar 

  22. H. Li, C. Wu, P. Shi, Y. Gao, Control of nonlinear networked systems with packet dropouts: interval type-2 fuzzy model-based approach. IEEE Trans. Cybern. (2014). doi:10.1109/TCYB.2014.2371814

    Google Scholar 

  23. Y. Li, S. Tong, T. Li, Observer-based adaptive fuzzy tracking control of MIMO stochastic nonlinear systems with unknown control direction and unknown dead-zones. IEEE Trans. Fuzzy Syst. 23(4), 1228–1240 (2015)

    Article  Google Scholar 

  24. H. Li, Y. Gao, L. Wu, H.K. Lam, Fault detection for T-S fuzzy time-delay systems: delta operator and input-output methods. IEEE Trans. Cybern. 45(2), 229–241 (2015)

    Article  Google Scholar 

  25. H. Li, X. Sun, L. Wu, H.K. Lam, State and output feedback control of a class of fuzzy systems with mismatched membership functions. IEEE Trans. Fuzzy Syst. (2015). doi:10.1109/TFUZZ.2014.2387876

    Google Scholar 

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

    Google Scholar 

  27. H. Li, Y. Pan, Q. Zhou, Filter design for interval type-2 fuzzy systems with D stability constraints under a unified frame. IEEE Trans. Fuzzy Syst. 23(3), 719–725 (2015)

    Article  Google Scholar 

  28. Y. Li, S. Tong, Adaptive fuzzy output-feedback control of pure-feedback uncertain nonlinear systems with unknown dead zone. IEEE Trans. Fuzzy Syst. 22(5), 1341–1347 (2014)

    Article  MathSciNet  Google Scholar 

  29. Y.J. Liu, Y.S. Xiao, X.L. Zhao, Multi-innovation stochastic gradient algorithm for multiple-input single-output systems using the auxiliary model. Appl. Math. Comput. 215(4), 1477–1483 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. Y. Liu, L. Yu, F. Ding, Multi-innovation extended stochastic gradient algorithm and its performance analysis. Circuits Syst. Signal Process. 29(4), 649–667 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. L. Ljung, Recursive identification algorithms. Circuits Syst. Signal Process. 21(1), 57–68 (2002)

    Article  MathSciNet  Google Scholar 

  32. J. Ma, F. Ding, Recursive relations of the cost functions for the least-squares algorithms for multivariable systems. Circuits Syst. Signal Process. 32(1), 83–101 (2013)

    Article  MathSciNet  Google Scholar 

  33. X.Y. Ma, F. Ding, Gradient-based parameter identification algorithms for observer canonical state space systems using state estimates. Circuits Syst. Signal Process. 34(5), 1697–1709 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  34. L. Márton, B. Lantos, Control of mechanical systems with Stribeck friction and backlash. Syst. Control Lett. 58(2), 141–147 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  35. C. Paleologu, J. Benesty, S. Ciochina, A robust variable forgetting factor recursive least-squares algorithm for system identification. IEEE Signal Process. Lett. 15(10), 597–600 (2008)

    Article  Google Scholar 

  36. J. Qiu, G. Feng, H. Gao, Static-output-feedback \(H_\infty \) 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 

  37. J. Qiu, H. Tian, Q. Lu, H. 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 

  38. J. Qiu, Y. Wei, H.R. Karimi, New approach to delay-dependent \(H_\infty \) control for continuous-time Markovian jump systems with time-varying delay and deficient transition descriptions. J. Franklin Inst. 352(1), 189–215 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  40. P. Rostalski, T. Besselmann, M. Barić, F.V. Belzen, M. Morari, A hybrid approach to modelling, control and state estimation of mechanical systems with backlash. Int. J. Control 80(11), 1729–1740 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  41. S. Tong, S.S. Sui, Y. Li, Fuzzy adaptive output feedback control of MIMO nonlinear systems with partial tracking errors constrained. IEEE Trans. Fuzzy Syst. 23(4), 729–742 (2015)

    Article  Google Scholar 

  42. J. Vörös, Modeling and parameter identification of systems with multisegment piecewise-linear Characteristics. IEEE Trans. Autom. Control 47(1), 184–188 (2002)

    Article  MathSciNet  Google Scholar 

  43. J. Vörös, Modeling and identification of systems with backlash. Automatica 46(2), 369–374 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  44. J. Vörös, Identification of nonlinear cascade systems with time-varying backlash. J. Electr. Eng. 62(2), 87–92 (2011)

    Google Scholar 

  45. J. Vörös, Parametric identification of systems with general backlash. Informatica 23(2), 283–298 (2012)

    MathSciNet  MATH  Google Scholar 

  46. T. Wang, H. Gao, J. Qiu, A combined adaptive neural network and nonlinear model predictive control for multirate networked industrial process control. IEEE Trans. Neural Netw. Learn. Syst. (2015). doi:10.1109/TNNLS.2015.2411671

    MathSciNet  Google Scholar 

  47. 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

    Google Scholar 

  48. 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

    Google Scholar 

  49. W.C. Yu, N.Y. Shih, Bi-loop recursive least squares algorithm with forgetting factors. IEEE Signal Process. Lett. 13(8), 505–508 (2006)

    Article  Google Scholar 

  50. J.B. Zhang, F. Ding, Y. Shi, Self-tuning control based on multi-innovation stochastic gradient parameter estimation. Syst. Control Lett. 58(1), 69–75 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  51. R. Zhang, P. Li, A. Xue, A. Jiang, S. Wang, A simplified linear iterative predictive functional control approach for chamber pressure of industrial coke furnace. J. Process Control 20(4), 464–471 (2010)

    Article  Google Scholar 

  52. R. Zhang, A. Xue, S. Wang, Dynamic modeling and nonlinear predictive control based on partitioned model and nonlinear optimization. Ind. Eng. Chem. Res. 50(13), 8110–8121 (2011)

    Article  Google Scholar 

  53. R. Zhang, A. Xue, S. Wnag, J. Zhang, An improved state space model structure and a corresponding predictive functional control design with improved control performance. Int. J. Control 85(8), 1146–1161 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  54. R. Zhang, J. Lu, H. Qu, F. Gao, State space model predictive fault-tolerant control for batch processes with partial actuator failure. J. Process Control 24(5), 613–620 (2014)

    Article  Google Scholar 

  55. R. Zhang, F. Gao, State space model predictive control using partial decoupling and output weighting for improved model/plant mismatch performance. Ind. Eng. Chem. Res. 52(2), 817–829 (2013)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This work was supported in part by the National High Technology Research and Development Program of China (China 863 Program) (Nos. 2014AA041505, 2013AA040405), the National Natural Science Foundation of China (Nos. 61572238, 61573167), the Chinese State Grain Administration Commonwealth Research Project (No. 201313012), and the Fundamental Research Funds for the Central Universities (No. JUSRP51310A).

Author information

Authors and Affiliations

  1. School of Internet of Things Engineering, Jiangnan University, Wuxi, 214122, People’s Republic of China

    Zhenwei Shi, Yan Wang & Zhicheng Ji

  2. Wuxi Electrical and Higher Vocational School, Wuxi, 214028, People’s Republic of China

    Zhenwei Shi

Authors
  1. Zhenwei Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhicheng Ji
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhicheng Ji.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shi, Z., Wang, Y. & Ji, Z. A Multi-innovation Recursive Least Squares Algorithm with a Forgetting Factor for Hammerstein CAR Systems with Backlash. Circuits Syst Signal Process 35, 4271–4289 (2016). https://doi.org/10.1007/s00034-016-0271-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-016-0271-1

Keywords

Search

Navigation