A State Space Approach to System Identification

  • Dionisio Bernal
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 502)


System identification is the process of obtaining descriptions of dynamical systems based on the analysis of experimental observations. This chapter presents, in sufficient detail to allow a numerical implementation by the interested reader, an algorithm for identifying a state space description of a time invariant system that behaves linearly. The chapter begins by clarifying the connection between the discrete time models that are identified from sampled data and the underlying continuous time physical systems. The (formal) connection is shown to be strictly related to the assumed parameterization of the excitation in the inter-sample and it is noted that the commonly used zero order hold premise (constant load within the sampling time), ubiquitous in control applications, is usually suboptimal in off-line identification. Specifically, it is contended that accuracy in the transfer from discrete to continuous time can be promoted by operating on the premise that the input is a band limited function. The notions of controllability and observability, which relate the sensor deployment to the modes that can be identified from observations, are discussed following the discrete to continuous transfer section.

The central part of the chapter examines the extraction of pulse response functions for multiple input multiple output testing setups and the use of these functions in the formulation of a discrete time state space model by means of the Eigensystem Realization Algorithm. The chapter continues by illustrating the connection between the realization matrices and the modal properties of the system, i.e., modal frequencies, damping ratios and mode shapes. The need to separate system modes from modes that appear in the computations due to inevitable approximations and sensor noise is also discussed and some specific guidelines to discriminate between the two are given. The identification part of the chapter closes with an introduction to the identification of systems where the excitation signals are not deterministically known. An appendix presenting a technique to localize damage in structural or mechanical systems from changes in realization results concludes the chapter.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. P.J. Antsaklis and A.N. Michel Linear systems, Birkhuser, Boston 2006.zbMATHGoogle Scholar
  2. D. Bernal. Discrete to continuous time transfer for band limited inputs. Journal of Engineering Mechanics, ASCE, 133 (12): 1370–1377, 2007a.CrossRefGoogle Scholar
  3. D. Bernal. Damage localization from the null space of changes in transfer matrices. AIAA Journal, 45 (2):374–381, 2007b.CrossRefGoogle Scholar
  4. D. Bernal Load vectors for damage localization. Journal of Engineering Mechanics, ASCE, 128(1):7–14, 2002.CrossRefMathSciNetGoogle Scholar
  5. D. Bernal. Flexibility-based damage localization from stochastic realization results. Journal of Engineering Mechanics, ASCE, 132(6):651–658, 2006.CrossRefMathSciNetGoogle Scholar
  6. D.J. Ewins. Modal testing: theory and practice, 1984. Research Studies Press Ltd, England.Google Scholar
  7. G. Golub and C. Van Loan. Matrix computation, 1989. The Jonhs Hopkins University Press, Baltimore, Maryland.Google Scholar
  8. H. Hanselmann. Implementation of digital controllers a survey. Automatica, 23(1):7–32, 1987.zbMATHCrossRefGoogle Scholar
  9. W. Heylen, S. Lammens, and P. Sas. Modal analysis theory and testing, 1998. Katholieke Universiteit Leuven.Google Scholar
  10. B.L. Ho and R.E. Kalman. Effective construction of linear state-variable models from input output functions. Regelungstechnik 12:545548, 1965.Google Scholar
  11. J. Juang and R. Pappa. An eigensystem realization algorithm for modal parameter identification and model reduction. Journal of Guidance Control and Dynamics, 8(5):610–627, 1985.CrossRefGoogle Scholar
  12. J.N. Juang. Applied system identification, 1994. Prentice Hall, Inc., Englewood Cliffs, New Jersey.zbMATHGoogle Scholar
  13. T. Kailath, Linear systems, 1990. Englewood Cliffs, Prentice-Hall, N.J.Google Scholar
  14. T. Katayama. Subspace methods for system identification, 2005. Springer.Google Scholar
  15. P. Lancaster. Lambda-matrices and Vibrating Systems, 1966. Pergamon Press.Google Scholar
  16. L. Junjg. System identification: theory for the user, 1987. Prentice Hall, Englewood-Cliffs.Google Scholar
  17. R. S. Pappa and K.B. Elliot. Consistent mode indicator for the eigensystem realization algorithm. Journal of guidance and Control, 16(5):832–838, 1993.CrossRefGoogle Scholar
  18. J. Schoukens, R. Pintelon and H. Van Hamme. Identification of linear dynamic systems using piecewise constant excitations: use, misuse and alternatives. Automatica, 30(7):1153–1169, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  19. T. Södertröm, and P Stoica. System identification, 2001. Prentice Hall International, Series in Systems and Control Engineering, UK.Google Scholar
  20. H. Ubenhauen and G.P. Rao Identification of continuous systems, 1987. North-Holland, Amsterdam.Google Scholar
  21. P. Van Overschee and B.L. De Moor. Subspace identification for linear systems: theory, implementation, applications, 1996. Kluwer Academic Publishers, Boston.zbMATHGoogle Scholar

Copyright information

© CISM, Udine 2008

Authors and Affiliations

  • Dionisio Bernal
    • 1
  1. 1.Civil and Environmental Engineering, Center for Digital Signal ProcessingNortheastern UniversityBoston

Personalised recommendations