Encyclopedia of Systems and Control

Living Edition
| Editors: John Baillieul, Tariq Samad

Subspace Techniques in System Identification

  • Michel VerhaegenEmail author
Living reference work entry

Latest version View entry history

DOI: https://doi.org/10.1007/978-1-4471-5102-9_107-2


An overview is given of the class of subspace techniques (STs) for identifying linear, time-invariant state-space models from input-output data. STs do not require a parametrization of the system matrices and as a consequence do not suffer from problems related to local minima that often hamper successful application of parametric optimization- based identification methods.

The overview follows the historic line of development. It starts from Kronecker’s result on the representation of an infinite power series by a rational function and then addresses, respectively, the deterministic realization problem, its stochastic variant, and finally the identification of a state-space model given in innovation form.

The overview summarizes the fundamental principles of the algorithms to solve the problems and summarizes the results about the statistical properties of the estimates as well as the practical issues like choice of weighting matrices and the selection of dimension parameters in using these STs in practice. The overview concludes with probing some future challenges and makes suggestions for further reading.


Extended observability matrix Hankel matrix Innovation model State-space model Singular value decomposition (SVD) 
This is a preview of subscription content, log in to check access.


  1. Bauer D (2005) Asymptotic properties of subspace estimators. Automatica 41(3):359–376MathSciNetCrossRefGoogle Scholar
  2. Bauer D, Ljung L (2002) Some facts about the choice of the weighting matrices in larimore type of subspace algorithms. Automatica 38(5):763–773MathSciNetCrossRefGoogle Scholar
  3. Cauberghe B, Guillaume P, Pintelon R, Verboven P (2006) Frequency-domain subspace identification using {FRF} data from arbitrary signals. J Sound Vib 290(3–5):555–571CrossRefGoogle Scholar
  4. Chiuso A (2010) Asymptotic properties of closed-loop cca-type subspace identification. IEEE-TAC 55(3):634–649MathSciNetzbMATHGoogle Scholar
  5. Jansson M, Wahlberg B (1996) A linear regression approach to state-space subspace systems. Signal Process 52:103–129CrossRefGoogle Scholar
  6. Juang J-N, Pappa RS (1985) Approximate linear realizations of given dimension via Ho’s algorithm. J Guid Control Dyn 8(5):620–627CrossRefGoogle Scholar
  7. Katayama T (2005) Subspace methods for system identification. Springer, LondonCrossRefGoogle Scholar
  8. Kronecker L (1890) Algebraische reduktion der schaaren bilinearer formen. S.B. Akad. Berlin, pp 663–776Google Scholar
  9. Larimore W (1990) Canonical variate analysis in identification, filtering, and adaptive control. In: Proceedings of the 29th IEEE conference on decision and control, 1990, Honolulu, vol 2, pp 596–604Google Scholar
  10. Liu Z, Vandenberghe L (2010) Interior-point method for nuclear norm approximation with application to system identification. SIAM J Matrix Anal Appl 31(3):1235–1256MathSciNetCrossRefGoogle Scholar
  11. Ljung L (2007) The system identification toolbox: the manual. The MathWorks Inc., Natick. 1st edition 1986, 7th edition 2007Google Scholar
  12. Peternell K, Scherrer W, Deistler M (1996) Statistical analysis of novel subspace identification methods. Signal Process 52(2):161–177CrossRefGoogle Scholar
  13. Schutter BD (2000) Minimal state space realization in linear system theory: an overview. J Comput Appl Math 121(1–2):331–354MathSciNetCrossRefGoogle Scholar
  14. van der Veen GJ, van Wingerden JW, Bergamasco M, Lovera M, Verhaegen M (2013) Closed-loop subspace identification methods: an overview. IET Control Theory Appl 7(10):1339–1358MathSciNetCrossRefGoogle Scholar
  15. Van Overschee P, De Moor B (1993) Subspace algorithms for the stochastic identification problem. Automatica 29(3):649–660MathSciNetCrossRefGoogle Scholar
  16. Van Overschee P, De Moor B (1994) N4sid: subspace algorithms for the identification of combined deterministic-stochastic systems. Automatica 30(1):75–93MathSciNetCrossRefGoogle Scholar
  17. Van Overschee P, De Moor B (1995) A unifying theorem for three subspace system identification algorithms. Automatica 31(12):1853–1864MathSciNetCrossRefGoogle Scholar
  18. Van Overschee P, De Moor B (1996) Identification for linear systems: theory – implementation – applications. Kluwer Academic Publisher Group, DordrechtCrossRefGoogle Scholar
  19. van Wingerden J, Verhaegen M (2009) Subspace identification of bilinear and LPV systems for open and closed loop data. Automatica 45(2):372–381MathSciNetCrossRefGoogle Scholar
  20. Verhaegen M (1994) Identification of the deterministic part of mimo state space models given in innovations form from input–output data. Automatica 30(1):61–74MathSciNetCrossRefGoogle Scholar
  21. Verhaegen M, Verdult V (2007) Filtering and identification: a least squares approach. Cambridge University Press, Cambridge/New YorkCrossRefGoogle Scholar
  22. Verhaegen M, Yu X (1995) A class of subspace model identification algorithms to identify periodically and arbitrarily time-varying systems. Automatica 31(2):201–216MathSciNetCrossRefGoogle Scholar
  23. Viberg M (1995) Subspace-based methods for the identification of linear time-invariant systems. Automatica 31(12):1835–1851MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  1. 1.Delft Center for Systems and ControlDelft UniversityDelftThe Netherlands

Section editors and affiliations

  • Lennart Ljung
    • 1
  1. 1.Division of Automatic Control, Department of Electrical EngineeringLinköping UniversityLinköpingSweden