Circuits, Systems, and Signal Processing

, Volume 38, Issue 2, pp 590–610 | Cite as

Auxiliary Model-Based Recursive Generalized Least Squares Algorithm for Multivariate Output-Error Autoregressive Systems Using the Data Filtering

  • Qinyao Liu
  • Feng DingEmail author


This paper focuses on the parameter estimation problem of multivariate output-error autoregressive systems. Based on the data filtering technique and the auxiliary model identification idea, we derive a filtering-based auxiliary model recursive generalized least squares algorithm. The key is to filter the input–output data and to derive two identification models, one of which includes the system parameters and the other contains the noise parameters. Compared with the auxiliary model-based recursive generalized least squares algorithm, the proposed algorithm requires less computational burden and can generate more accurate parameter estimates. Finally, an illustrative example is provided to verify the effectiveness of the proposed algorithm.


Filtering technique Parameter estimation Recursive least squares Multivariate system Auxiliary model 



This work was supported by the National Natural Science Foundation of China (No. 61273194) and the 111 Project (B12018).


  1. 1.
    A. Al-Smadi, A least-squares-based algorithm for identification of non-Gaussian ARMA models. Circuits Syst. Signal Process. 26(5), 715–731 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Y. Cao, L.C. Ma, S. Xiao et al., Standard analysis for transfer delay in CTCS-3. Chin. J. Electron. 26(5), 1057–1063 (2017)CrossRefGoogle Scholar
  3. 3.
    H.B. Chen, Y.S. Xiao et al., Hierarchical gradient parameter estimation algorithm for Hammerstein nonlinear systems using the key term separation principle. Appl. Math. Comput. 247, 1202–1210 (2014)MathSciNetzbMATHGoogle Scholar
  4. 4.
    J.L. Ding, Recursive and iterative least squares parameter estimation algorithms for multiple-input-output-error systems with autoregressive noise. Circuits, Syst. Signal Process. 37(5), 1884–1906 (2018)MathSciNetCrossRefGoogle Scholar
  5. 5.
    F. Ding, H.B. Chen, L. Xu, J.Y. Dai, Q.S. Li, T. Hayat, A hierarchical least squares identification algorithm for Hammerstein nonlinear systems using the key term separation. J. Franklin Inst. 355(8), 3737–3752 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    F. Ding, D.D. Meng, J.Y. Dai, Q.S. Li, A. Alsaedi, T. Hayat, Least squares based iterative parameter identification for stochastic dynamical systems with ARMA noise using the model equivalence. Int. J. Control Autom. Syst. 16(2), 630–639 (2018)CrossRefGoogle Scholar
  7. 7.
    F. Ding, F.F. Wang, L. Xu et al., Decomposition based least squares iterative identification algorithm for multivariate pseudo-linear ARMA systems using the data filtering. J. Franklin Inst. 354(3), 1321–1339 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    F. Ding, L. Xu, F.E. Alsaadi, T. Hayat, Iterative parameter identification for pseudo-linear systems with ARMA noise using the filtering technique. IET Control Theory Appl. 12(7), 892–899 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    F. Ding, L. Xu, Q.M. Zhu, Performance analysis of the generalised projection identification for time-varying systems. IET Control Theory Appl. 10(18), 2506–2514 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    M. Gan, C.L.P. Chen, G.Y. Chen, L. Chen, On some separated algorithms for separable nonlinear squares problems. IEEE Trans. Cybern. (2018).
  11. 11.
    M. Gan, H.X. Li, H. Peng, A variable projection approach for efficient estimation of RBF-ARX model. IEEE Trans. Cybern. 45(3), 462–471 (2015)CrossRefGoogle Scholar
  12. 12.
    H.J. Gao, X.W. Li, J.B. Qiu, Finite frequency \(H_{\infty }\) deconvolution with application to approximated bandlimited signal recovery. IEEE Trans. Autom. Control 63(1), 203–210 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    H.L. Gao, C.C. Yin, The perturbed sparre Andersen model with a threshold dividend strategy. J. Comput. Appl. Math. 220(1–2), 394–408 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    W. Greblicki, M. Pawlak, Hammerstein system identification with the nearest neighbor algorithm. IEEE Trans. Inf. Theory 63(8), 4746–4757 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    P.C. Gong, W.Q. Wang, F.C. Li, H. Cheung, Sparsity-aware transmit beamspace design for FDA-MIMO radar. Signal Process. 144, 99–103 (2018)CrossRefGoogle Scholar
  16. 16.
    P. Li, R. Dargaville, Y. Cao et al., Storage aided system property enhancing and hybrid robust smoothing for large-scale PV systems. IEEE Trans. Smart Grid 8(6), 2871–2879 (2017)CrossRefGoogle Scholar
  17. 17.
    X.W. Li, J. Lam, H.J. Gao et al., \(H_{\infty }\) and \(H_2\) filtering for linear systems with uncertain Markov transitions. Automatica 67, 252–266 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    P. Li, R.X. Li, Y. Cao, G. Xie, Multi-objective sizing optimization for island microgrids using triangular aggregation model and Levy-Harmony algorithm. IEEE Trans. Ind. Inform. (2018).
  19. 19.
    M.H. Li, X.M. Liu, Auxiliary model based least squares iterative algorithms for parameter estimation of bilinear systems using interval-varying measurements. IEEE Access 6, 21518–21529 (2018)CrossRefGoogle Scholar
  20. 20.
    M.H. Li, X.M. Liu, The least squares based iterative algorithms for parameter estimation of a bilinear system with autoregressive noise using the data filtering technique. Signal Process. 147, 23–34 (2018)CrossRefGoogle Scholar
  21. 21.
    M.H. Li, X.M. Liu et al., The maximum likelihood least squares based iterative estimation algorithm for bilinear systems with autoregressive moving average noise. J. Franklin Inst. 354(12), 4861–4881 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    J.H. Li, W. Zheng, J.P. Gu, L. Hua, A recursive identification algorithm for Wiener nonlinear systems with linear state-space subsystem. Circuits Syst. Signal Process. 37(6), 2374–2393 (2018)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Y. Li, W.H. Zhang, X.K. Liu, H-index for discrete-time stochastic systems with Markovian jump and multiplicative noise. Automatica 90, 286–293 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Y. Lin, W. Zhang, Necessary/sufficient conditions for pareto optimum in cooperative difference game. Optim. Control, Appl. Methods 39(2), 1043–1060 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    F. Liu, Continuity and approximate differentiability of multisublinear fractional maximal functions. Math. Inequal. Appl. 21(1), 25–40 (2018)MathSciNetzbMATHGoogle Scholar
  26. 26.
    F. Liu, On the Triebel–Lizorkin space boundedness of Marcinkiewicz integrals along compound surfaces. Math. Inequal. Appl. 20(2), 515–535 (2017)MathSciNetzbMATHGoogle Scholar
  27. 27.
    F. Liu, H.X. Wu, Singular integrals related to homogeneous mappings in triebel–lizorkin spaces. J. Math. Inequal. 11(4), 1075–1097 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    F. Liu, H.X. Wu, Regularity of discrete multisublinear fractional maximal functions. Sci. China–Math. 60(8), 1461–1476 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    F. Liu, H.X. Wu, On the regularity of maximal operators supported by submanifolds. J. Math. Anal. Appl. 453(1), 144–158 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    F. Liu, Q.Y. Xue, K. Yabuta, Rough maximal singular integral and maximal operators supported by subvarieties on Triebel–Lizorkin spaces. Nonlinear Anal. 171, 41–72 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Q.Y. Liu, F. Ding, The data filtering based generalized stochastic gradient parameter estimation algorithms for multivariate output-error autoregressive systems using the auxiliary model. Multidimens. Syst. Signal Process.
  32. 32.
    L. Ljung, System Identification: Theory for the User, 2nd edn. (Prentice Hall, Englewood Cliffs, NJ, 1999)zbMATHGoogle Scholar
  33. 33.
    P. Ma, F. Ding, Q.M. Zhu, Decomposition-based recursive least squares identification methods for multivariate pseudolinear systems using the multi-innovation. Int. J. Syst. Sci. 49(5), 920–928 (2018)CrossRefGoogle Scholar
  34. 34.
    Y.W. Mao, F. Ding, A novel parameter separation based identification algorithm for Hammerstein systems. Appl. Math. Lett. 60, 21–27 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    B.Q. Mu, E.W. Bai, W.X. Zheng et al., A globally consistent nonlinear least squares estimator for identification of nonlinear rational systems. Automatica 77, 322–335 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Z.H. Rao, C.Y. Zeng, M.H. Wu et al., Research on a handwritten character recognition algorithm based on an extended nonlinear kernel residual network. KSII Trans. Int. Inf. Syst. 12(1), 413–435 (2018)Google Scholar
  37. 37.
    Y.J. Wang, F. Ding, A filtering based multi-innovation gradient estimation algorithm and performance analysis for nonlinear dynamical systems. IMA J. Appl. Math. 82(6), 1171–1191 (2017)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Y.J. Wang, F. Ding, L. Xu, Some new results of designing an IIR filter with colored noise for signal processing. Dig. Signal Process. 72, 44–58 (2018)MathSciNetCrossRefGoogle Scholar
  39. 39.
    D.Q. Wang, L. Mao et al., Recasted models based hierarchical extended stochastic gradient method for MIMO nonlinear systems. IET Control Theory Appl. 11(4), 476–485 (2017)MathSciNetCrossRefGoogle Scholar
  40. 40.
    D.Q. Wang, Z. Zhang, J.Y. Yuan, Maximum likelihood estimation method for dual-rate Hammerstein systems. Int. J. Control Autom. Syst. 15(2), 698–705 (2017)CrossRefGoogle Scholar
  41. 41.
    L. Xu, The parameter estimation algorithms based on the dynamical response measurement data. Adv. Mech. Eng. 9(11), 1–12 (2017). Google Scholar
  42. 42.
    L. Xu, F. Ding, Iterative parameter estimation for signal models based on measured data. Circuits Syst. Signal Process. 37(7), 3046–3069 (2018)MathSciNetCrossRefGoogle Scholar
  43. 43.
    L. Xu, F. Ding, Parameter estimation algorithms for dynamical response signals based on the multi-innovation theory and the hierarchical principle. IET Signal Process. 11(2), 228–237 (2017)CrossRefGoogle Scholar
  44. 44.
    L. Xu, F. Ding, Parameter estimation for control systems based on impulse responses. Int. J. Control Autom. Syst. 15(6), 2471–2479 (2017)CrossRefGoogle Scholar
  45. 45.
    L. Xu, F. Ding, Recursive least squares and multi-innovation stochastic gradient parameter estimation methods for signal modeling. Circuits Syst. Signal Process. 36(4), 1735–1753 (2017)CrossRefzbMATHGoogle Scholar
  46. 46.
    L. Xu, F. Ding, Y. Gu, A. Alsaedi, T. Hayat, A multi-innovation state and parameter estimation algorithm for a state space system with d-step state-delay. Signal Process. 140, 97–103 (2017)CrossRefGoogle Scholar
  47. 47.
    G.H. Xu, Y. Shekofteh, A. Akgul, C.B. Li, S. Panahi, A new chaotic system with a self-excited attractor: entropy measurement, signal encryption, and parameter estimation. Entropy 20(2), 86 (2018). CrossRefGoogle Scholar
  48. 48.
    C.C. Yin, C.W. Wang, The perturbed compound Poisson risk process with investment and debit interest. Methodol. Comput. Appl. Probab. 12(3), 391–413 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    C.C. Yin, Y.Z. Wen, Exit problems for jump processes with applications to dividend problems. J. Comput. Appl. Math. 245, 30–52 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    C.C. Yin, Y.Z. Wen, Optimal dividend problem with a terminal value for spectrally positive Levy processes. Insur. Math. Econ. 53(3), 769–773 (2013)CrossRefzbMATHGoogle Scholar
  51. 51.
    C.C. Yin, K.C. Yuen, Optimality of the threshold dividend strategy for the compound Poisson model. Stat. Probab. Lett. 81(12), 1841–1846 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    C.C. Yin, J.S. Zhao, Nonexponential asymptotics for the solutions of renewal equations, with applications. J. Appl. Probab. 43(3), 815–824 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Y.Z. Zhang, Y. Cao, Y.H. Wen, L. Liang, F. Zou, Optimization of information interaction protocols in cooperative vehicle-infrastructure systems. Chin. J. Electron. 27(2), 439–444 (2018)CrossRefGoogle Scholar
  54. 54.
    X. Zhang, F. Ding, A. Alsaadi, T. Hayat, Recursive parameter identification of the dynamical models for bilinear state space systems. Nonlinear Dyn. 89(4), 2415–2429 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    W. Zhang, X. Lin, B.S. Chen, LaSalle-type theorem and its applications to infinite horizon optimal control of discrete-time nonlinear stochastic systems. IEEE Trans. Automatic Control 62(1), 250–261 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    E. Zhang, R. Pintelon, Identification of multivariable dynamic errors-in-variables system with arbitrary inputs. Automatica 82, 69–78 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    X. Zhang, L. Xu et al., Combined state and parameter estimation for a bilinear state space system with moving average noise. J. Franklin Inst. 355(6), 3079–3103 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    N. Zhao, R. Liu, Y. Chen, M. Wu, Y. Jiang, W. Xiong, C. Liu, Contract design for relay incentive mechanism under dual asymmetric information in cooperative networks. Wireless Netw. (2018).
  59. 59.
    D.Q. Zhu, X. Cao, B. Sun, C.M. Luo, Biologically inspired self-organizing map applied to task assignment and path planning of an AUV system. IEEE Trans. Cognit. Dev. Syst. 10(2), 304–313 (2018)CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education)Jiangnan UniversityWuxiPeople’s Republic of China
  2. 2.College of Automation and Electronic EngineeringQingdao University of Science and TechnologyQingdaoPeople’s Republic of China

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