Skip to main content
SpringerLink
Log in

Recursive Relations of the Cost Functions for the Least-Squares Algorithms for Multivariable Systems

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

Abstract

In this paper, we focus on the recursive computation of the cost functions for the least-squares-type algorithms for multivariable linear regressive models. It is shown that the proposed recursive computation formulas for the cost functions can also be extended to the estimation algorithm of the multivariable equation error models, e.g., the controlled autoregressive models. The simulation results indicate that the proposed algorithm is effective.

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

Similar content being viewed by others

References

  1. M.S. Ahmad, O. Kukrer, A. Hocanin, Robust recursive inverse adaptive algorithm in impulsive noise. Circuits Syst. Signal Process. 31(2), 703–710 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Al–Smadi, A least-squares-based algorithm for identification of non-Gaussian ARMA models. Circuits Syst. Signal Process. 26(5), 715–731 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. H.I. Ashiba, K.H. Awadalla, S.M. EI-Halfawy, F.E. Abd EI-Samie, Adaptive least squares interpolation of infrared images. Circuits Syst. Signal Process. 30(3), 543–551 (2011)

    Article  MATH  Google Scholar 

  4. M. Ayatai, Adaptive fuzzy control of nonlinear in parameters uncertain chaotic systems using improved speed gradient method. Circuits Syst. Signal Process. 31(2), 911–926 (2012)

    Article  Google Scholar 

  5. B. Bao, Y.Q. Xu, J. Sheng, R.F. Ding, Least squares based iterative parameter estimation algorithm for multivariable controlled ARMA system modelling with finite measurement data. Math. Comput. Model. 53(9–10), 1664–1669 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Y.N. Cao, Z.Q. Liu, Signal frequency and parameter estimation for power systems using the hierarchical identification principle. Math. Comput. Model. 51(5–6), 854–861 (2010)

    Article  MathSciNet  Google Scholar 

  7. J. Chen, Y. Zhang, R.F. Ding, Auxiliary model based multi-innovation algorithms for multivariable nonlinear systems. Math. Comput. Model. 52(9–10), 1428–1434 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. F. Ding, Hierarchical multi-innovation stochastic gradient algorithm for Hammerstein nonlinear system modeling. Appl. Math. Model. (2012). doi:10.1016/j.apm.2012.04.039

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

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. F. Ding, T. Chen, Hierarchical least squares identification methods for multivariable systems. IEEE Trans. Autom. Control 50(3), 397–402 (2005)

    Article  MathSciNet  Google Scholar 

  12. F. Ding, T. Chen, Hierarchical identification of lifted state-space models for general dual-rate systems. IEEE Trans. Circuits Syst. I, Regul. Pap. 52(6), 1179–1187 (2005)

    Article  MathSciNet  Google Scholar 

  13. J. Ding, F. Ding, Bias compensation based parameter estimation for output error moving average systems. Int. J. Adapt. Control Signal Process. 25(12), 1100–1111 (2011)

    Article  Google Scholar 

  14. 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 

  15. J. Ding, L.L. Han, X.M. Chen, Time series AR modeling with missing observations based on the polynomial transformation. Math. Comput. Model. 51(5–6), 527–536 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. F. Ding, P.X. Liu, G. Liu, Multi-innovation least squares identification for linear and pseudo-linear regression models. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 40(3), 767–778 (2010)

    Article  Google Scholar 

  17. F. Ding, G. Liu, X.P. Liu, Partially coupled stochastic gradient identification methods for non-uniformly sampled systems. IEEE Trans. Autom. Control 55(8), 1976–1981 (2010)

    Article  MathSciNet  Google Scholar 

  18. F. Ding, Y.J. Liu, B. Bao, Gradient based and least squares based iterative estimation algorithms for multi-input multi-output systems. Proc. Inst. Mech. Eng., Part I, J. Syst. Control Eng. 226(1), 43–55 (2012)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. F. Ding, X.P. Liu, G. Liu, Identification methods for Hammerstein nonlinear systems. Digit. Signal Process. 21(2), 215–238 (2011)

    Article  Google Scholar 

  21. J. Ding, Y. Shi et al., A modified stochastic gradient based parameter estimation algorithm for dual-rate sampled-data systems. Digit. Signal Process. 20(4), 1238–1249 (2010)

    Article  Google Scholar 

  22. F. Ding, Y.S. Xiao, A finite-data-windows least squares algorithm with a forgetting factor for dynamical modeling. Appl. Math. Comput. 186(1), 184–192 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. W. Dong, J.D. Li, Z. Lu, Parameter estimation for correlated MIMO channels with frequency-selective fading. Wirel. Pers. Commun. 52(4), 813–828 (2010)

    Article  Google Scholar 

  24. W. Dong, J.D. Li, Z. Lu, W. Zhang, Joint frequency offset and channel estimation for MIMO systems in the presence of timing error. Circuits Syst. Signal Process. 29(2), 275–293 (2010)

    Article  MATH  Google Scholar 

  25. F. Felici, J. van Wingerden, M. Verhaegen, Subspace identification of MIMO LPV systems using a periodic scheduling sequence. Automatica 43(10), 1684–1697 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. S. Gibson, B. Ninness, Robust maximum-likelihood estimation of multivariable dynamic systems. Automatica 41(10), 1667–1682 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  29. L.L. Han, J. Sheng et al., Auxiliary models based recursive least squares identification for multirate multi-input systems. Math. Comput. Model. 50(7–8), 1100–1106 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. H.Q. Han, L. Xie et al., Hierarchical least squares based iterative identification for multivariable systems with moving average noises. Math. Comput. Model. 51(9–10), 1213–1220 (2010)

    Article  MATH  Google Scholar 

  31. J.H. Li, F. Ding, Maximum likelihood stochastic gradient estimation for Hammerstein systems with colored noise based on the key term separation technique. Comput. Math. Appl. 62(11), 4170–4177 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  32. J.H. Li, F. Ding, G.W. Yang, Maximum likelihood least squares identification method for input nonlinear finite impulse response moving average systems. Math. Comput. Model. 55(3–4), 442–450 (2012)

    Article  MathSciNet  Google Scholar 

  33. 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 Proces. (2012). doi:10.1007/s00034-012-9421-2

  34. Y.J. Liu, J. Sheng, R.F. Ding, Convergence of stochastic gradient estimation algorithm for multivariable ARX-like systems. Comput. Math. Appl. 59(8), 2615–2627 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. 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 

  36. Y.J. Liu, L. Yu 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 

  37. J.X. Ma, W.L. Xiong, E. Ding, Recursive relations of the criterion functions for the recursive least squares algorithms, in The 2012 Chinese Control and Decision Conference (2012 CCDC), May 23–25, 2012, Taiyuan, China (2012), pp. 2081–2086

    Google Scholar 

  38. T.J. Moir, A unified family of recursive algorithms using feedback. Circuits Syst. Signal Process. 30(5), 1047–1054 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  39. R. Pintelon, J. Schoukens, P. Guillaume, Box–Jenkins identification revisited—part III: multivariable systems. Automatica 43(5), 868–875 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  40. M. Rachid, B. Maamar, D. Said, Multivariable fractional system approximation with initial conditions using integral state space representation. Comput. Math. Appl. 59(5), 1842–1851 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  41. M.S. Radenkovic, T. Bose, A recursive blind adaptive equalizer for IIR channels with common zeros. Circuits Syst. Signal Process. 28(3), 467–486 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  42. T. Schon, A. Wills, B. Ninness, System identification of nonlinear state-space models. Automatica 47(1), 39–49 (2011)

    Article  MathSciNet  Google Scholar 

  43. R.M. Udrea, D.N. Vizireanu, Quantized multiple sinusoids signal estimation algorithm. J. Instrum. 3(2), 1–7 (2008)

    Article  Google Scholar 

  44. M. Viberg, Subspace-based state-space system identification. Circuits Syst. Signal Process. 21(1), 23–27 (2002)

    Article  MathSciNet  Google Scholar 

  45. D.N. Vizireanu, Quantized sine signals estimation algorithm for portable DSP based instrumentation. Int. J. Electron. 96(11), 1175–1181 (2009)

    Article  Google Scholar 

  46. D.N. Vizireanu, A simple and precise real-time four point single sinusoid signals instantaneous frequency estimation method for portable DSP based instrumentation. Measurement 44(2), 500–502 (2011)

    Article  Google Scholar 

  47. D.N. Vizireanu, A fast, simple and accurate time-varying frequency estimation method for single-phase electric power systems. Measurement 45(5), 1331–1333 (2012)

    Article  Google Scholar 

  48. D.N. Vizireanu, S.V. Halunga, Simple, fast and accurate eight points amplitude estimation method of sinusoidal signals for DSP based instrumentation. J. Instrum. 7(2), 1–10 (2012)

    Google Scholar 

  49. J. Vörös, Iterative algorithm for parameter identification of Hammerstein systems with two-segment nonlinearities. IEEE Trans. Autom. Control 44(11), 2145–2149 (1999)

    Article  MATH  Google Scholar 

  50. 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 

  51. D.Q. Wang, F. Ding, Least squares based and gradient based iterative identification for Wiener nonlinear systems. Signal Process. 91(5), 1182–1189 (2011)

    Article  MATH  Google Scholar 

  52. W. Wang, F. Ding, J.Y. Dai, Maximum likelihood least squares identification for systems with autoregressive moving average noise. Appl. Math. Model. 36(5), 1842–1853 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  53. W. Wang, J.H. Li, R.F. Ding, Maximum likelihood parameter estimation algorithm for controlled autoregressive models. Int. J. Comput. Math. 88(16), 3458–3467 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  54. Y.S. Xiao, F. Ding, Y. Zhou, M. Li, J.Y. Dai, On consistency of recursive least squares identification algorithms for controlled auto-regression models. Appl. Math. Model. 32(11), 2207–2215 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  55. 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 

  56. Z.N. Zhang, F. Ding, X.G. Liu, Hierarchical gradient based iterative parameter estimation algorithm for multivariable output error moving average systems. Comput. Math. Appl. 61(3), 672–682 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  57. 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 

  58. Z.N. Zhang, J. Jia, R.F. Ding, Hierarchical least squares based iterative estimation algorithm for multivariable Box–Jenkins-like systems using the auxiliary model. Appl. Math. Comput. 218(9), 5580–5587 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  59. Z. Zheng, G. Li, Y. Teng, Simplified estimation of 2D DOA for coherently distributed sources. Wirel. Pers. Commun. 62(4), 907–922 (2012)

    Article  Google Scholar 

  60. L. Zhong, Robust maximum-likelihood parameter estimation of stochastic state-space systems based on EM algorithm. Prog. Nat. Sci. 17(9), 1095–1103 (2007)

    Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 60973043) and the 111 Project (B12018).

Author information

Authors and Affiliations

  1. Key Laboratory of Advanced Process Control for Light Industry, Ministry of Education, Jiangnan University, Wuxi, 214122, P.R. China

    Junxia Ma

  2. Control Science and Engineering Research Center, Jiangnan University, Wuxi, 214122, P.R. China

    Feng Ding

Authors
  1. Junxia Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Feng Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Feng Ding.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ma, J., Ding, F. Recursive Relations of the Cost Functions for the Least-Squares Algorithms for Multivariable Systems. Circuits Syst Signal Process 32, 83–101 (2013). https://doi.org/10.1007/s00034-012-9448-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-012-9448-4

Keywords

Search

Navigation