An Improved Consistent Subspace Identification Method Using Parity Space for State-space Models

  • Jie HouEmail author
  • Fengwei Chen
  • Penghua Li
  • Zhiqin Zhu
  • Fei Liu


In this paper, an alternative consistent subspace identification method using parity space is proposed. The future/past input data and the past output data are used to construct the instrument variable to eliminate the noise effect on consistent estimation. The extended observability matrix and the triangular block-Toeplitz matrix are then retrieved from a parity space of the noise-free matrix using a singular value decomposition based method. The system matrices are finally estimated from the above estimated matrices. The consistency of the proposed method for estimation of the extended observability matrix and the triangular block-Toeplitz matrix is established. Compared with the classical SIMs using parity space like SIMPCA and SIMPCA-Wc, the proposed method generally enhances the estimated model efficiency/accuracy thanks to the use of future input data. Two examples are presented to illustrate the effectiveness and merit of the proposed method.


Consistency instrumental variables parity space rank condition subspace identification 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    P. V. Overschee and B. L. De Moor, Subspace Identification for Linear Systems: Theory-implementation-applications, Springer, Berlin, 2012.zbMATHGoogle Scholar
  2. [2]
    T. Katayama, Subspace Methods for System Identification, Springer, Berlin, 2006.zbMATHGoogle Scholar
  3. [3]
    S. J. Qin, “An overview of subspace identification,” Computers & Chemical Engineering, vol. 30, no. 10–12, pp. 1502–1513, September 2006.CrossRefGoogle Scholar
  4. [4]
    M. Verhaegen, “Identification of the deterministic part of MIMO state-space models given in innovations form from input-output data,” Automatica, vol. 30, no. 1, pp. 61–74, January 1994.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    W. E. Larimore, “System identification, reduced-order filtering and modeling via canonical variate analysis,” American Control Conference, pp. 445–451, June 1983.Google Scholar
  6. [6]
    P. V. Overschee and B. L. De Moor, “N4SID: subspace algorithms for the identification of combined deterministicstochastic systems,” Automatica, vol. 30, no. 1, pp. 75–93, January 1994.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    M. Jansson and B. Wahlberg, “On consistency of subspace methods for system identification,” Automatica, vol. 34, no. 12, pp. 1507–1519, December 1998.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    A. Chiuso and G. Picci, “The asymptotic variance of subspace estimates,” Journal of Econometrics, vol. 118, no. 1–2, pp. 257–291, JanuaryFebruary 2004.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    D. Bauer, “Estimating ARMAX systems for multivariate time series using the state approach to subspace algorithms,” Journal of Multivariate Analysis, vol. 100, no. 3, pp. 397–421, March 2009.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    J. Hou, T. Liu, and F. Chen, “Orthogonal projection based subspace identification against colored noise,” Control Theory and Technology, vol. 15, no. 1, pp. 69–77, February 2017.MathSciNetCrossRefGoogle Scholar
  11. [11]
    W. Li and S. J. Qin, “Consistent dynamic PCA based on errors-in-variables subspace identification,” Journal of Process Control, vol. 11, no. 6, pp. 661–678, December 2001.CrossRefGoogle Scholar
  12. [12]
    J. Wang and S. J. Qin, “A new subspace identification approach based on principal component analysis,” Journal of Process Control, vol. 12, no. 8, pp. 841–855, December 2002.CrossRefGoogle Scholar
  13. [13]
    J. Wang and S. J. Qin, “Closed-loop subspace identification using the parity space,” Automatica, vol. 42, no. 2, pp. 315–320, February 2006.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    W. Li, H. Raghavan, and S. Shah, “Subspace identification of continuous time models for process fault detection and isolation,” Journal of Process Control, vol. 13, no. 5, pp. 407–421, August 2003.CrossRefGoogle Scholar
  15. [15]
    A. Micchi and G. Pannocchia, “Comparison of input signals in subspace identification of multivariable illconditioned systems,” Journal of Process Control, vol. 18, no. 6, pp. 582–593, July 2008.CrossRefGoogle Scholar
  16. [16]
    H. Yang, S. Li, and K. Li, “Order estimation of multivariable ill-conditioned processes based on PCA method,” Journal of Process Control, vol. 22, no. 7, pp. 1397–1403, August 2012.CrossRefGoogle Scholar
  17. [17]
    Z. Liao, Z. Zhu, S. Liang, C. Peng, and Y. Wang, “Subspace identification for fractional order Hammerstein systems based on instrumental variables,” International Journal of Control, Automation and Systems, vol. 10, no. 5, pp. 947–953, September 2012.CrossRefGoogle Scholar
  18. [18]
    P. Wu, H. P. Pan, J. Ren, and C. Yang, “A new subspace identification approach based on principal component analysis and noise estimation,” Industrial & Engineering Chemistry Research, vol. 54, no. 18, pp. 5106–5114, April 2015.CrossRefGoogle Scholar
  19. [19]
    D. Maurya, A. K. Tangirala, and S. Narasimhan, “Identification of Errors-in-Variables models using dynamic iterative principal component analysis,” Industrial & Engineering Chemistry Research, vol. 57, no. 35, pp. 11939–11954, August 2018.CrossRefGoogle Scholar
  20. [20]
    J. Hou, T. Liu, B. Wahlberg, and M. Jansson, “Subspace Hammerstein model identification under periodic disturbance,” IFAC-PapersOnLine, vol. 51, no. 15, pp. 335–340, July 2018.CrossRefGoogle Scholar
  21. [21]
    S. X. Ding, P. Zhang, A. Naik, E. L. Ding, and B. Huang, “Subspace method aided data-driven design of fault detection and isolation systems,” Journal of Process Control, vol. 19, no. 9, pp. 1496–1510, October 2009.CrossRefGoogle Scholar
  22. [22]
    S. X. Ding, “Data-driven design of monitoring and diagnosis systems for dynamic processes: a review of subspace technique based schemes and some recent results,” Journal of Process Control, vol. 24, no. 2, pp. 431–449, February 2014.CrossRefGoogle Scholar
  23. [23]
    S. Yin, G. Wang, and H. R. Karimi, “Data-driven design of robust fault detection system for wind turbines,” Mechatronics, vol. 24, no. 4, pp. 298–306, June 2014.CrossRefGoogle Scholar
  24. [24]
    S. Yin, S. X. Ding, X. Xie, and H. Luo, “A review on basic data-driven approaches for industrial process monitoring,” IEEE Transactions on Industrial Electronics, vol. 61, no. 11, pp. 6418–6428, November 2014.CrossRefGoogle Scholar
  25. [25]
    S. Yin, S. X. Ding, A. H. Abandan Sari, and H. Hao, “Datadriven monitoring for stochastic systems and its application on batch process,” International Journal of Systems Science, vol. 44, no. 7, pp. 1366–1376, July 2013.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    K. Peng, M. Wang, and J. Dong, “Event-triggered fault detection framework based on subspace identification method for the networked control systems,” Neurocomputing, vol. 239, no. 24, pp. 257–267, May 2017.CrossRefGoogle Scholar
  27. [27]
    G. Wang and Z. Huang, “Data-driven fault-tolerant control design for wind turbines with robust residual generator,” IET Control Theory & Applications, vol. 9, no. 7, pp. 1173–1179, April 2015.CrossRefGoogle Scholar
  28. [28]
    J. S. Wang and G. H. Yang, “Data-driven output-feedback fault-tolerant compensation control for digital PID control systems with unknown dynamics,” IEEE Transactions on Industrial Electronics, vol. 63, no. 11, pp. 7029–7039, November 2016.CrossRefGoogle Scholar
  29. [29]
    Y.Wang, H. Zhang, S.Wei, D. Zhou, and B. Huang, “Control performance assessment for ILC-controlled batch processes in a 2-D system framework,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 48, no. 9, pp. 1493–1504, September 2017.Google Scholar
  30. [30]
    J. S. Wang and G. H. Yang, “Data-driven compensation method for sensor drift faults in digital PID systems with unknown dynamics,” Journal of Process Control, vol. 65, pp. 15–33, May 2018.CrossRefGoogle Scholar
  31. [31]
    H. Oku and H. Kimura, “A recursive 4SID from the inputoutput point of view,” Asian Journal of Control, vol. 1, no. 4, pp. 258–269, December 1999.CrossRefGoogle Scholar
  32. [32]
    D. S. Bernstein, Matrix Mathematics, 2nd ed., Princeton University Press, Princeton, 2009.CrossRefzbMATHGoogle Scholar
  33. [33]
    V. Filipovic, N. Nedic, and V. Stojanovic, “Robust identification of pneumatic servo actuators in the real situations,” Forschung im Ingenieurwesen, vol. 75, no. 4, pp. 183–196, December 2011.CrossRefGoogle Scholar
  34. [34]
    V. Stojanovic and N. Nedic, “Robust identification of OE model with constrained output using optimal input design,” Journal of the Franklin Institute, vol. 353, no. 2, pp. 576–593, January 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    V. Stojanovic and N. Nedic, “Joint state and parameter robust estimation of stochastic nonlinear systems,” International Journal of Robust and Nonlinear Control, vol. 26, no. 14, pp. 3058–3074, September 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    P. V. Overschee and B. L. De Moor, “A unifying theorem for three subspace system identification algorithms,” Automatica, vol. 31, no. 12, pp. 1853–1864, December 1995.MathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    T. Liu, B. Huang, and S. J. Qin, “Bias-eliminated subspace model identification under time-varying deterministic type load disturbance,” Journal of Process Control, vol. 25, pp. 41–49, January 2015.CrossRefGoogle Scholar
  38. [38]
    J. Hou, T. Liu, and Q. G. Wang, “Recursive subspace identification subject to relatively slow time-varying load disturbance,” International Journal of Control, vol. 91, no. 3, pp. 648–664, May 2018.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© ICROS, KIEE and Springer 2019

Authors and Affiliations

  • Jie Hou
    • 1
    Email author
  • Fengwei Chen
    • 2
  • Penghua Li
    • 1
  • Zhiqin Zhu
    • 1
  • Fei Liu
    • 3
  1. 1.College of AutomationChongqing University of Posts and TelecommunicationsChongqingChina
  2. 2.Department of AutomationWuhan UniversityWuhan, HubeiChina
  3. 3.National Robot Test and Assessment Center (Chongqing), Chongqing Dexin Robot Testing Center Co., LtdChongqingChina

Personalised recommendations