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Hierarchical Least Squares Identification for Hammerstein Nonlinear Controlled Autoregressive Systems

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Abstract

This paper considers the parametric identification problems of a Hammerstein nonlinear system which consists of a static nonlinear block followed by a linear dynamic subsystem. A hierarchical least squares algorithm is developed by using the hierarchical identification principle, which decomposes a nonlinear system into several subsystems with smaller dimensions and fewer variables and estimates the parameters of each subsystem, respectively. The performance analysis indicates that the parameter estimates given by the proposed algorithm converge to their true values and the proposed algorithm requires higher computational efficiencies compared with the recursive least squares algorithm.

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References

  1. E.W. Bai, A blind approach to the Hammerstein–Wiener model identification. Automatica 38(6), 967–979 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  2. E.W. Bai, Decoupling the linear and nonlinear parts in Hammerstein model identification. Automatica 40(4), 671–676 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. V. Cerone, D. Regruto, Parameter bounds for discrete-time Hammerstein models with bounded output errors. IEEE Trans. Autom. Control 48(10), 1855–1860 (2003)

    Article  MathSciNet  Google Scholar 

  4. H.B. Chen, F. Ding, Decomposition based recursive least squares parameter estimation for Hammerstein nonlinear controlled autoregressive systems, 2013 American Control Conference (ACC), June 17–19, 2013, Washington, USA, pp. 2442–2447

  5. M. Dehghan, M. Hajarian, An iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices. Appl. Math. Model. 34(3), 639–654 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  6. M. Dehghan, M. Hajarian, Matrix equations over (R, S)-symmetric and (R, S)-skew symmetric matrices. Comput. Math. Appl. 59(11), 3583–3594 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  7. M. Dehghan, M. Hajarian, Two algorithms for finding the Hermitian reflexive and skew-Hermitian solutions of Sylvester matrix equations. Appl. Math. Lett. 24(4), 444–449 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  8. M. Dehghan, M. Hajarian, Analysis of an iterative algorithm to solve the generalized coupled Sylvester matrix equations. Appl. Math. Model. 37(5), 3285–3300 (2011)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  10. F. Ding, Combined state and least squares parameter estimation algorithms for dynamic systems. Appl. Math. Model. 38(1), 403–412 (2014)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  12. F. Ding, Hierarchical parameter estimation algorithms for multivariable systems using measurement. Info. Sci. 277, 396–405 (2014)

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

    Article  MATH  MathSciNet  Google Scholar 

  14. F. Ding, T. Chen, Hierarchical gradient-based identification of multivariable discrete-time systems. Automatica 41(2), 315–325 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. F. Ding, Y. Gu, 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 

  16. F. Ding, X.M. Liu, H.B. Chen, G.Y. Yao, Hierarchical gradient based and hierarchical least squares based iterative parameter identification for CARARMA systems. Signal Process. 97, 31–39 (2014)

    Article  Google Scholar 

  17. J. Ding, J.X. Lin, 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 

  18. J. Ding, F. Ding, X.P. Liu, G. Liu, Hierarchical least squares identification for linear SISO systems with dual-rate sampled-data. IEEE Trans. Autom. Control 56(11), 2677–2683 (2011)

    Article  MathSciNet  Google Scholar 

  19. J. Ding, C.X. Fan, J.X. Lin, 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 

  20. D. Fan, K. Lo, Identification for disturbed MIMO Wiener systems. Nonlinear Dyn. 55(1–2), 31–42 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  21. Y. Gu, X.L. Lu, R.F. Ding, Parameter and state estimation algorithm for a state space model with a one-unit state delay. Circuits Syst. Signal Process. 32(5), 2267–2280 (2013)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  23. W.H. Ho, J.H. Chou, C.Y. Guo, Parameter identification of chaotic systems using improved differential evolution algorithm. Nonlinear Dyn. 61(1–2), 29–41 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  24. J.H. Li, Parameter estimation for Hammerstein CARARMA systems based on the Newton iteration. Appl. Math. Lett. 26(1), 91–96 (2013)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  27. Y.J. Liu, F. Ding, Y. Shi, An efficient hierarchical identification method for general dual-rate sampled-data systems. Automatica 50(3), 962–973 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  28. Y.J. Liu, F. Ding, Y. Shi, Least squares estimation for a class of non-uniformly sampled systems based on the hierarchical identification principle. Circuits Syst. Signal Process. 31(6), 1985–2000 (2012)

    Article  MATH  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  MATH  MathSciNet  Google Scholar 

  30. X.L. Luan, P. Shi, F. Liu, Stabilization of networked control systems with random delays. IEEE Trans. Ind. Electron. 58(9), 4323–4330 (2011)

    Article  Google Scholar 

  31. X.L. Luan, S.Y. Zhao, F. Liu, \(H_\infty \) control for discrete-time markov jump systems with uncertain transition probabilities. IEEE Trans. Autom. Control 58(6), 1566–1572 (2013)

    Article  MathSciNet  Google Scholar 

  32. M.D. Narayanan, S. Narayanan, C. Padmanabhan, Parametric identification of nonlinear systems using multiple trials. Nonlinear Dyn. 48(4), 341–360 (2007)

    Article  MATH  Google Scholar 

  33. Y. Shi, H. Fang, Kalman filter based identification for systems with randomly missing measurements in a network environment. Int. J. Control 83(3), 538–551 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  34. Y. Shi, B. Yu, Output feedback stabilization of networked control systems with random delays modeled by Markov chains. IEEE Trans. Autom. Control 54(7), 1668–1674 (2009)

    Article  MathSciNet  Google Scholar 

  35. P. Shi, X.L. Luan, F. Liu, \(H_\infty \) filtering for discrete-time systems with stochastic incomplete measurement and mixed delays. IEEE Trans. Ind. Electron. 59(6), 2732–2739 (2012)

    Article  Google Scholar 

  36. W. Silva, Identification of nonlinear aeroelastic systems based on the Volterra theory: progress and opportunities. Nonlinear Dyn. 39(1–2), 25–62 (2005)

    Article  MATH  Google Scholar 

  37. S. Yin, S.X. Ding, A. Haghani, H.Y. Hao, P. Zhang, A comparison study of basic data-driven fault diagnosis and process monitoring methods on the benchmark Tennessee Eastman process. J. Process Control 22(9), 1567–1581 (2012)

    Article  Google Scholar 

  38. S. Yin, S.X. Ding, A. Haghani, H.Y. Hao, Data-driven monitoring for stochastic systems and its application on batch process. Int. J. Syst. Sci. 44(7), 1366–1376 (2013)

    Article  MATH  Google Scholar 

  39. S. Yin, H. Luo, S.X. Ding, Real-time implementation of fault-tolerant control systems with performance optimization. IEEE Trans. Ind. Electron. 61(5), 2402–2411 (2013)

    Article  Google Scholar 

  40. Y. Zhang, Unbiased identification of a class of multi-input single-output systems with correlated disturbances using bias compensation methods. Math. Comput. Model. 53(9–10), 1810–1819 (2011)

    Article  MATH  Google Scholar 

  41. Y. Zhang, G.M. Cui, Bias compensation methods for stochastic systems with colored noise. Appl. Math. Model. 35(4), 1709–1716 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  42. D.Q. Zhu, H. Huang, S.X. Yang, Dynamic task assignment and path planning of multi-auv system based on an improved self-organizing map and velocity synthesis method in 3D underwater workspace. IEEE Trans. Cybern. 43(2), 504–514 (2013)

    Article  Google Scholar 

  43. D.Q. Zhu, Q. Liu, Z. Hu, 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 

  44. D.Q. Zhu, J. Liu, S.X. Yang, Particle swarm optimization approach to thruster fault-tolerant control of unmanned underwater vehicles. Int. J. Robot. Autom. 26(3), 426–432 (2011)

    Google Scholar 

  45. D.Q. Zhu, Y. Zhao, M.Z. Yan, A Bio-inspired neurodynamics based backstepping path-following control of an AUV with ocean current. Int. J. Robot. Autom. 27(3), 280–287 (2012)

    Google Scholar 

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Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 61273194), the Natural Science Foundation of Jiangsu Province (China, BK2012549) 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

    Huibo Chen & Feng Ding

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  1. Huibo Chen
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  2. Feng Ding
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Correspondence to Feng Ding.

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Chen, H., Ding, F. Hierarchical Least Squares Identification for Hammerstein Nonlinear Controlled Autoregressive Systems. Circuits Syst Signal Process 34, 61–75 (2015). https://doi.org/10.1007/s00034-014-9839-9

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  • DOI: https://doi.org/10.1007/s00034-014-9839-9

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