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Convergence of the auxiliary model-based multi-innovation generalized extended stochastic gradient algorithm for Box–Jenkins systems

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Abstract

This paper focuses on the parameter estimation problem of Box–Jenkins systems. Using the multi-innovation identification theory, an auxiliary model-based multi-innovation generalized extended stochastic gradient algorithm is derived. The convergence of the proposed algorithm is analyzed based on the stochastic martingale theory. It is proved that the parameter estimation errors converge to zero under persistent excitation conditions. Two simulation examples are provided to confirm the convergence results.

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References

  1. Ding, F.: System Identification—New Theory and Methods. Science Press, Beijing (2013)

    Google Scholar 

  2. Ding, F.: System Identification—Performances Analysis for Identification Methods. Science Press, Beijing (2014)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Pintelon, R., Schoukens, J., Rolain, Y.: Box–Jenkins continuous-time modeling. Automatica 36(7), 983–991 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, J., Lu, J.A., Zhou, J.: Topology identification of complex networks from noisy time series using ROC curve analysis. Nonlinear Dyn. 75(4), 761–768 (2014)

    Article  MathSciNet  Google Scholar 

  6. Arrieta, A.F., Neild, S.A., Wagg, D.J.: Nonlinear dynamic response and modeling of a bi-stable composite plate for applications to adaptive structures. Nonlinear Dyn.58(1–2), 259–272 (2009)

  7. Zhu, D.Q., Liu, Q., Hu, Z.: Fault-tolerant control algorithm of the manned submarine with multi-thruster based on quantum behaved particle swarm optimization. Int. J. Control 84(11), 1817–1829 (2012)

    Article  MathSciNet  Google Scholar 

  8. Zhu, D.Q., Huang, H., Yang, S.X.: Dynamic task assignment and path planning of multi-AUV system based on an improved self-organizing map and velocity synthesis method in three-dimensional underwater workspace. IEEE Trans. Cybern. 43(2), 504–514 (2013)

    Article  Google Scholar 

  9. Sun, B., Zhu, D.Q., Yang, S.X.: A bio-inspired filtered backstepping cascaded tracking control of 7000m manned submarine vehicle. IEEE Trans. Ind. Electron. 61(7), 3682–3692 (2014)

    Article  Google Scholar 

  10. Wang, X.H., Ding, F.: Performance analysis of the recursive parameter estimation algorithms for multivariable Box–Jenkins systems. J. Franklin Inst. Eng. Appl. Math. 351(10), 4749–4764 (2014)

    Article  MathSciNet  Google Scholar 

  11. Zhang, Y., Yang, H.Z.: Bias compensation recursive least squares identification for output error systems with colored noises. Acta Automat. Sin. 33(10), 1053–1060 (2007)

    MATH  Google Scholar 

  12. Wu, A.G., Qian, Y.Y., Wu, W.J.: Bias compensation-based recursive least-squares estimation with forgetting factors for output error moving average systems. IET Signal Process. 8(5), 483–494 (2014)

    Article  Google Scholar 

  13. Ding, F., Wang, Y.J., Ding, J.: Recursive least squares parameter estimation algorithms for systems with colored noise using the filtering technique and the auxiliary model. Digit. Signal Process. 37, 100–108 (2015)

    Article  Google Scholar 

  14. Xu, X.P., Wang, F., Liu, G.J., Qian, F.C.: Identification of Hammerstein systems using key-term separation principle, auxiliary model and improved particle swarm optimisation algorithm. IET Signal Process. 7(8), 766–773 (2013)

    Article  Google Scholar 

  15. Xie, L., Yang, H.Z.: Interactive parameter estimation for output error moving average systems. Trans. Inst. Meas. Control 35(1), 34–43 (2013)

    Article  Google Scholar 

  16. Ding, J., Fan, C.X., Lin, J.X.: Auxiliary model based parameter estimation for dual-rate output error systems with colored noise. Appl. Math. Model. 37(6), 4051–4058 (2013)

    Article  MathSciNet  Google Scholar 

  17. Wang, C., Tang, T.: Recursive least squares estimation algorithm applied to a class of linear-in-parameters output error moving average systems. Appl. Math. Lett. 29, 36–41 (2014)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  19. Wu, D.H., Li, Y.Y.: Fault diagnosis of variable pitch for wind turbines based on the multi-innovation forgetting gradient identification algorithm. Nonlinear Dyn. 79(3), 2069–2077 (2015)

    Article  Google Scholar 

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

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Bai, E.W., Li, K.: Convergence of the iterative algorithm for a general Hammerstein system identification. Automatica 46(11), 1891–1896 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hu, Y.B.: Iterative and recursive least squares estimation algorithms for moving average systems. Simul. Model. Pract. Theory 34, 12–19 (2013)

  24. Liu, Y.J., Yu, L., et al.: Multi-innovation extended stochastic gradient algorithm and its performance analysis. Circuits Syst. Signal Process. 29(4), 649–667 (2010)

    Article  MATH  Google Scholar 

  25. Forssell, U., Ljung, L.: Identification of unstable systems using output error and Box–Jenkins model structures. IEEE Trans. Autom. Control 45(1), 137–141 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. Liu, X.G., Lu, J.: Least squares based iterative identification for a class of multirate systems. Automatica 46(3), 549–554 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Chen, J., Wang, X.: Multi-innovation generalized extended stochastic gradient algorithm for multi-input multi-output nonlinear Box–Jenkins systems based on the auxiliary model. Life System Modelling and Intelligent Computing Lecture Notes in Computer Science, vol. 6328, pp. 136–146 (2010)

  28. Wang, D.Q., et al.: Performance analysis of the auxiliary models based multi-innovation stochastic gradient estimation algorithm for output error systems. Digit. Signal Process. 20(3), 750–762 (2010)

    Article  Google Scholar 

  29. Ding, F.: State filtering and parameter identification for state space systems with scarce measurements. Signal Process. 104, 369–380 (2014)

    Article  Google Scholar 

  30. Liu, Y.J., Wang, D.Q., et al.: Least squares based iterative algorithms for identifying Box–Jenkins models with finite measurement data. Digit. Signal Process. 20(5), 1458–1467 (2010)

    Article  Google Scholar 

  31. Wang, L.Y., Xie, L., Wang, X.F.: The residual based interactive stochastic gradient algorithms for controlled moving average models. Appl. Math. Comput. 211(2), 442–449 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Ding, F., Gu, Y.: Performance analysis of the auxiliary model-based stochastic gradient parameter estimation algorithm for state space systems with one-step state delay. Circuits Syst. Signal Process. 32(2), 585–599 (2013)

    Article  MathSciNet  Google Scholar 

  33. Ljung, L.: System Identification: Theory for the User, 2nd edn. Prentice Hall, Englewood Cliffs, NJ (1999)

    Google Scholar 

  34. 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 

  35. Vörös, J.: Recursive identification of Hammerstein systems with discontinuous nonlinearities containing dead-zones. IEEE Trans. Autom. Control 48(12), 2203–2206 (2003)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  37. Hu, Y.B., Liu, B.L., Zhou, Q., Yang, C.: 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 

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

    Article  MathSciNet  Google Scholar 

  39. Ding, J., Lin, J.X.: Modified subspace identification for periodically non-uniformly sampled systems by using the lifting technique. Circuits Syst. Signal Process. 33(5), 1439–1449 (2014)

    Article  Google Scholar 

  40. Liu, Y.J., Ding, F., Shi, Y.: An efficient hierarchical identification method for general dual-rate sampled-data systems. Automatica 50(3), 962–970 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  41. Gu, Y., Ding, F., Li, J.H.: State filtering and parameter estimation for linear systems with d-step state-delay. IET Signal Process. 8(6), 639–646 (2014)

    Article  Google Scholar 

  42. Gu, Y., Ding, F., Li, J.H.: 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 

  43. Shi, Y., Yu, B.: Robust mixed H-2/H-infinity control of networked control systems with random time delays in both forward and backward communication links. Automatica 47(4), 754–760 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  44. Li, H., Shi, Y.: Robust H-infty filtering for nonlinear stochastic systems with uncertainties and random delays modeled by Markov chains. Automatica 48(1), 159–166 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  45. Ding, F.: Hierarchical parameter estimation algorithms for multivariable systems using measurement information. Inform. Sci. 277, 396–405 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 61273194) and the PAPD of Jiangsu Higher Education Institutions.

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

    Xuehai Wang

Authors
  1. Xuehai Wang
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  2. Feng Ding
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Correspondence to Feng Ding.

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Wang, X., Ding, F. Convergence of the auxiliary model-based multi-innovation generalized extended stochastic gradient algorithm for Box–Jenkins systems. Nonlinear Dyn 82, 269–280 (2015). https://doi.org/10.1007/s11071-015-2155-5

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  • DOI: https://doi.org/10.1007/s11071-015-2155-5

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