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Recursive Least Squares and Multi-innovation Gradient Estimation Algorithms for Bilinear Stochastic Systems

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Abstract

Bilinear systems are a special class of nonlinear systems. Some systems can be described by using bilinear models. This paper considers the parameter identification problems of bilinear stochastic systems. The difficulty of identification is that the model structure of the bilinear systems includes the products of the states and inputs. To this point, this paper gives the input–output representation of the bilinear systems through eliminating the state variables in the model and derives a least squares algorithm and a multi-innovation stochastic gradient algorithm for identifying the parameters of bilinear systems based on the least squares principle and the multi-innovation identification theory. The simulation results indicate that the proposed algorithms are effective for identifying bilinear systems.

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References

  1. A. Alfi, Particle swarm optimization algorithm with dynamic inertia weight for online parameter identification applied to Lorenz chaotic system. Int. Innov. Comput. Inf. Control 8(2), 1191–1203 (2012)

    Google Scholar 

  2. A. Alfi, PSO with adaptive mutation and inertia weight and its application in parameter estimation of dynamic systems. Acta Autom. Sin. 37(5), 541–549 (2011)

    Article  MATH  Google Scholar 

  3. A. Alfi, M.M. Fateh, Identification of nonlinear systems using modified particle swarm optimization: a hydraulic suspension system. J. Veh. Syst. Dyn. 46(6), 871–887 (2011)

    Article  Google Scholar 

  4. A. Alfi, M.M. Fateh, Intelligent identification and control using improved fuzzy particle swarm optimization. Expert Syst. Appl. 38(10), 12312–12317 (2011)

    Article  Google Scholar 

  5. A. Alfi, H. Modares, System identification and control using adaptive particle swarm optimization. Appl. Math. Model. 35, 1210–1221 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Arab, A. Alfi, An adaptive gradient descent-based local search in memetic algorithm for solving engineering optimization problems. Inf. Sci. 299, 117–142 (2015)

    Article  Google Scholar 

  7. Z.X. Cao, Y. Yang, J.Y. Lu, F.R. Gao, Constrained two dimensional recursive least squares model identification for batch processes. J. Process Control 24(6), 871–879 (2014)

    Article  Google Scholar 

  8. X. Cao, D.Q. Zhu, S.X. Yang, Multi-AUV target search based on bioinspired neurodynamics model in 3-D underwater environments. IEEE Trans. Neural Netw. Learn. Syst. (2016). doi:10.1109/TNNLS.2015.2482501

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

    MathSciNet  MATH  Google Scholar 

  10. Z.Z. Chu, D.Q. Zhu, S.X. Yang, Observer-based adaptive neural network trajectory tracking control for remotely operated Vehicle. IEEE Trans. Neural Netw. Learn. Syst. (2016). doi:10.1109/TNNLS.2016.2544786

  11. H. Dai, N.K. Sinha, Robust recursive least-squares method with modified weights for bilinear system identification. IEE Proc. Part D 136(3), 122–126 (1989)

    Article  MATH  Google Scholar 

  12. A. Darabi, A. Alfi, B. Kiumarsi, H. Modares, Employing adaptive PSO algorithm for parameter estimation of an exciter machine. J. Dyn. Syst-T ASME 134(1). doi:10.1115/1.4005371, (2012)

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

    Google Scholar 

  14. 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 

  15. F. Ding, X.M. Liu, Y. Gu, An auxiliary model based least squares algorithm for a dual-rate state space system with time-delay using the data filtering. J. Franklin Inst. 353(2), 398–408 (2016)

    Article  MathSciNet  Google Scholar 

  16. F. Ding, X.M. Liu, M.M. Liu, The recursive least squares identification algorithm for a class of Wiener nonlinear systems. J. Franklin Inst. 353(7), 1518–1526 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. F. Ding, X.M. Liu, X.Y. Ma, Kalman state filtering based least squares iterative parameter estimation for observer canonical state space systems using decomposition. J. Comput. Appl. Math. 301, 135–143 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. 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. 35, (2016). doi:10.1007/s00034-015-0190-6

  19. F. Fnaiech, L. Ljung, Recursive identification of bilinear systems. Int. J. Control 45(2), 453–470 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  20. S. Gibson, A. Wills, B. Ninness, Maximum-likelihood parameter estimation of bilinear systems. IEEE Trans. Autom. Control 50(10), 1581–1596 (2005)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  22. 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)

    MathSciNet  MATH  Google Scholar 

  23. Y. Ji, X.M. Liu, 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 

  24. 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 

  25. S.X. Jing, T.H. Pan, Z.M. Li, Recursive bayesian algorithm with covariance resetting for identification of Box–Jenkins systems with non-uniformly sampled input data. Circuits Syst. Signal Process. 16, 1–14 (2015)

    MATH  Google Scholar 

  26. V.R. Karanam, P.A. Frick, R.R. Mohler, Bilinear system identification by Walsh functions. IEEE Trans. Autom. Control 23, 709–713 (1978)

    Article  MATH  Google Scholar 

  27. G.Q. Li, C.Y. Wen, A.M. Zhang, Fixed point iteration in identifying bilinear models. Syst. Control Lett. 83, 28–37 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. P. Lopes dos Santos, J.A. Ramos, J.L. Martins de Carvalho, Identification of a Benchmark Wiener–Hammerstein: a bilinear and Hammerstein–Bilinear model approach. Control Eng. Pract. 20, 1156–1164 (2012)

    Article  Google Scholar 

  29. H. Modares, A. Alfi, M.N. Sistani, Parameter estimation of bilinear systems based on an adaptive particle swarm optimization. Eng. Appl. Artif. Intell. 23, 1105–1111 (2010)

    Article  Google Scholar 

  30. Y. Mousavi, A. Alfi, A memetic algorithm applied to trajectory control by tuning of fractional order proportional–integral–derivative controllers. Appl. Soft. Comput. 36, 599–617 (2015)

    Article  Google Scholar 

  31. B.Q. Mu, J. Guo, L.Y. Wang, G. Yin, L.J. Xu, W.X. Zheng, Identification of linear continuous-time systems under irregular and random output sampling. Automatica 60, 100–114 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. J. Na, X.M. Ren, Y.Q. Xia, Adaptive parameter identification of linear SISO systems with unknown time-delay. Syst. Control Lett. 66, 43–50 (2014)

    Article  MathSciNet  MATH  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. D.Q. Wang, Least squares-based recursive and iterative estimation for output error moving average systems using data filtering. IET Control Theory Appl. 5(14), 1648–1657 (2011)

    Article  MathSciNet  Google Scholar 

  35. D.W. Wang, F. Ding, Parameter estimation algorithms for multivariable Hammerstein CARMA systems. Inf. Sci. 355, 237–248 (2016)

    Article  MathSciNet  Google Scholar 

  36. Y.J. Wang, F. Ding, Novel data filtering based parameter identification for multiple-input multiple-output systems using the auxiliary model. Automatica (2016). doi:10.1016/j.automatica.2016.05.024

  37. Y.J. Wang, F. Ding, The filtering based iterative identification for multivariable systems. IET Control Theory Appl. 10(8), 894–902 (2016)

    Article  MathSciNet  Google Scholar 

  38. Y.J. Wang, F. Ding, The auxiliary model based hierarchical gradient algorithms and convergence analysis using the filtering technique. Signal Process. 128, 212–221 (2016)

    Article  Google Scholar 

  39. Y.J. Wang, F. Ding, Recursive least squares algorithm and gradient algorithm for Hammerstein–Wiener systems using the data filtering. Nonlinear Dyn. 84(2), 1045–1053 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  40. X.H. Wang, F. Ding, Modelling and multi-innovation parameter identification for Hammerstein nonlinear state space systems using the filtering technique. Math. Comput. Model. Dyn. Syst. 22(2), 113–140 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  41. 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 

  42. C. Wang, T. Tang, 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  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  44. W.G. Zhang, Decomposition based least squares iterative estimation for output error moving average systems. Eng. Comput. 31(4), 709–725 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 61273194). The author is grateful to her Master Supervisor Professor Feng Ding and the main idea of this paper comes from him and his books “System Identification—New Theory and Methods, Science Press, Beijing, 2013,” “System Identification—Performances Analysis for Identification Methods, Science Press, Beijing, 2014” and “System Identification—Multi-Innovation Identification Theory and Methods, Science Press, Beijing, 2016.”

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

    Dandan Meng

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Meng, D. Recursive Least Squares and Multi-innovation Gradient Estimation Algorithms for Bilinear Stochastic Systems. Circuits Syst Signal Process 36, 1052–1065 (2017). https://doi.org/10.1007/s00034-016-0337-0

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