A modified observer/Kalman filter identification (OKID) algorithm employing output residuals

  • Ahmed AlenanyEmail author
  • David Westwick
  • Helen Shang


The observer/Kalman filter identification (OKID) is an algorithm widely used for the identification of state space models. The standard OKID algorithm involves the estimation of the Kalman filter and system Markov parameters, followed by the realization of a state space model of the system using the eigensystem realization algorithm (ERA). In this paper, a modified and conceptually simple version of the OKID algorithm, termed the residual-based observer/Kalman filter identification (ROKID), is proposed. The ROKID algorithm uses ordinary least square method twice to solve two linear regression problems yielding the Kalman filter residuals and the system Markov parameters, respectively. Finally, the ERA algorithm is used to obtain a state space model of the system. The efficacy of the proposed algorithm is examined and compared with the standard OKID algorithm and the recently proposed OKID with deterministic projection (OKID/DP) algorithm via a simulation example. The results show that the proposed algorithm outperforms the standard OKID algorithm. Although its performance is less than that of the OKID/DP algorithm, due to its simplicity, the proposed algorithm represents a useful tool for linear state space model identification.


State-space models Kalman filtering Markov parameters Linear systems Closed-loop identification 



The authors gratefully acknowledge the financial support provided by the Egyptian Government and the Natural Sciences and Engineering Research Council of Canada (NSERC). The authors would also like to express their sincere thanks to the anonymous reviewers for their valuable comments and suggestions that greatly improved the quality and presentation of the manuscript.


  1. 1.
    Van Overschee P, De Moor B (1996) Subspace identification for linear systems: theory, implementation, and applications. Kluwer Academic Publishers, DordrechtCrossRefzbMATHGoogle Scholar
  2. 2.
    Verhaegen M, Verdult V (2007) Filtering and system identification: a least squares approach. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  3. 3.
    Juang JN, Phan MQ, Horta LG, Longman RW (1993) Identification of observer/Kalman filter Markov parameters: theory and experiments. J Guid Control Dyn 16(2):320–329CrossRefzbMATHGoogle Scholar
  4. 4.
    Chen F, Tsaia J, Liao Y, Guo S, Ho M, Shaw F, Shie L (2014) An improvement on the transient response of tracking for the sampled-data system based on an improved PD-type iterative learning control. J Frankl Inst 351(2):1130–1150CrossRefzbMATHGoogle Scholar
  5. 5.
    Valasek J, Chen W (2003) Observer/Kalman filter identification for online system identification of aircraft. J Guid Control Dyn 26(2):347–353CrossRefGoogle Scholar
  6. 6.
    Tiano A, Sutton R, Lozowicki A, Naeem W (2007) Observer Kalman filter identification of an autonomous underwater vehicle. Control Eng Pract 15(6):727–739CrossRefGoogle Scholar
  7. 7.
    Sun G, Richard WL, Betti R, Chen Z, Xue S (2017) Observer Kalman filter identification of suspen-dome. Math Prob Eng 2017:1–9Google Scholar
  8. 8.
    Lavu BC, Schoen MP, Mahajan A (2005) Adaptive intelligent control of ionic polymermetal composites. Smart Mater Struct 14(4):466–474CrossRefGoogle Scholar
  9. 9.
    Nadimi ES, Sogaard HT (2009) Observer Kalman filter identification and multiple-model adaptive estimation technique for classifying animal behaviour using wireless sensor networks. Comput Electron Agric 68(1):9–17CrossRefGoogle Scholar
  10. 10.
    Qin SJ (2006) An overview of subspace identification. Comput Chem Eng 30(10):1502–1513CrossRefGoogle Scholar
  11. 11.
    Vicario F, Phan MQ, Betti R, Longman RW (2017) OKID via output residuals: a converter from stochastic to deterministic system identification. J Guid Control Dyn 40(12):3226–3238CrossRefGoogle Scholar
  12. 12.
    Doraiswami R, Cheded L (2018) Robust Kalman filter-based least squares identification of a multivariable system. IET Control Theory Appl 12(8):1064–1074MathSciNetCrossRefGoogle Scholar
  13. 13.
    Juang JN, Phan MQ (2001) Identification and control of mechanical systems. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  14. 14.
    Zhao Y, Qin SJ (2014) Subspace identification with non-steady Kalman filter parametrization. J Process Control 24(9):1337–1345CrossRefGoogle Scholar
  15. 15.
    Kung S (1978) A new identification and model reduction algorithm via singular value decompositions. In: 12th Asilomar conference on circuits systems and computers, pp 705–714Google Scholar
  16. 16.
    Van den Hof PMJ, Schrama RJP (1993) An indirect method for transfer function estimation from closed loop data. Automatica 29(6):1523–1527MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Katayamaa T, Kawauchia H, Picci G (2005) Subspace identification of closed loop systems by the orthogonal decomposition method. Automatica 41(5):863–872MathSciNetCrossRefGoogle Scholar
  18. 18.
    Phan MQ, Vicario F, Longman RW, Betti R (2018) State-space model and Kalman filter gain identification by a Kalman filter of a Kalman filter. J Dyn Syst Meas Control 140(3):030902CrossRefGoogle Scholar
  19. 19.
    Vicario F, Phan MQ, Betti R, Longman RW (2015) Output-only observer/Kalman filter identification (\(\text{ O }^3\)KID). Struct Control Health Monit 22(5):847–872CrossRefGoogle Scholar
  20. 20.
    Phan MQ, Vicario F, Longman RW, Betti R (2015) Optimal bilinear observers for bilinear state-space models by interaction matrices. Int J Control 88(8):1504–1522MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Sankar Rao C, Chidambaram M (2015) Subspace identification of transfer function models for an unstable bioreactor. Chem Eng Commun 202(10):1296–1303CrossRefGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Computer and Systems EngineeringZagazig UniversityZagazigEgypt
  2. 2.Department of Computer ScienceNahda University in Beni SuefNew Benisuef CityEgypt
  3. 3.Department of Electrical and Computer EngineeringUniversity of CalgaryCalgaryCanada
  4. 4.School of EngineeringLaurentian UniversitySudburyCanada

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