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Identifiability of discrete-time linear switched systems

Published:12 April 2010Publication History

ABSTRACT

In this paper we study the identifiability of linear switched systems (LSSs) in discrete-time.The question of identifiability is central to system identification, as it sets the boundaries of applicability of any system identification method; no system identification algorithm can properly estimate the parameters of a system which is not identifiable. We present necessary and sufficient conditions that guarantee structural identifiability for parametrized LSSs. We also introduce the class of semi-algebraic parametrizations, for which these conditions can be checked effectively.

References

  1. L.Bako, G.Merc'ere, and S. Lecoeuche. Online structured subspace identification with application to switched linear systems. International Journal of Control, 82:1496--1515, 2009.Google ScholarGoogle ScholarCross RefCross Ref
  2. L.Bako, G.Merc'ere, R.Vidal, and S.Lecoeuche. Identification of switched linear state space models without minimum dwell time. In IFAC Symposium on System Identification, Saint Malo, France, 2009.Google ScholarGoogle ScholarCross RefCross Ref
  3. S.Basu, R.Pollack, and M.Roy. Algorithms in Real Algebraic Geometry. Springer, 2003. Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. A.Bemporad, A.Garulli, S.Paoletti, and A.Vicino. A bounded-error approach to piecewise affine system identification. IEEE Transactions on Automatic Control}, 50(10):1567--1580, 2005.Google ScholarGoogle ScholarCross RefCross Ref
  5. M.Coste. An Introduction to O-minimal Geometry. Dip. Mat. Univ. Pisa, 2000.Google ScholarGoogle Scholar
  6. M.Coste, J.Bochnak, and M.-F. Roy. Real Algebraic Geometry Ergebnisse der Math. Springer Verlag, 1998.Google ScholarGoogle Scholar
  7. S.Eilenberg.Automata, Languages and Machines Academic Press, New York, London, 1974. Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. H.B. Enderton. A Mathematical Introduction to Logic, Second Edition. Academic Press, 2 edition.Google ScholarGoogle Scholar
  9. G.Ferrari-Trecate, M.Muselli, D.Liberati, and M.Morari. A clustering technique for the identification of piecewise affine systems. Automatica, 39:205--217, 2003. Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. M.Fliess, C.Join, and W.Perruquetti. Real-time estimation for switched linear systems. In IEEE Conf. on Decision and Control, 2008.Google ScholarGoogle ScholarCross RefCross Ref
  11. F.Gecseg and I.Peak. Algebraic theory of automata. Akademiai Kiado, Budapest, 1972.Google ScholarGoogle Scholar
  12. A.Juloski, W.Heemels, G.Ferrari-Trecate, R.Vidal, S.Paoletto, and J.Niessen. Comparison of four procedures for the identification of hybrid systems. In Hybrid Systems: Computation and Control}, LNCS. Springer-Verlag, Berlin, 2005. Google ScholarGoogle ScholarDigital LibraryDigital Library
  13. A.L. Juloski, S.Weiland, and W.Heemels. A bayesian approach to identification of hybrid systems. IEEE Transactions on Automatic Control, 50:1520--1533, 2005.Google ScholarGoogle ScholarCross RefCross Ref
  14. D.Liberzon. Switching in Systems and Control. Birkhauser, Boston, 2003.Google ScholarGoogle ScholarCross RefCross Ref
  15. L.Ljung. System Identification: Theory for the user (2nd Ed.). PTR Prentice Hall., Upper Saddle River, USA, 1999. Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. Y.Ma and R.Vidal. A closed form solution to the identification of hybrid {ARX} models via the identification of algebraic varieties. In Hybrid Systems: Computation and Control, pages 449--465, 2005. Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. H.Nakada, K.Takaba, and T.Katayama. Identification of piecewise affine systems based on statistical clustering technique. Automatica, 41:905--913, 2005. Google ScholarGoogle ScholarDigital LibraryDigital Library
  18. S.Paoletti, A.Juloski, G.Ferrari-Trecate, and R.Vidal. Identification of hybrid systems: A tutorial. European Journal of Control, 13(2-3):242 -- 260, 2007.Google ScholarGoogle ScholarCross RefCross Ref
  19. S.Paoletti, J.Roll, A.Garulli, and A.Vicino. Input/ouput realization of piecewise affine state space models. In IEEE Conf. on Decision. and Control, 2007.Google ScholarGoogle Scholar
  20. M.Petreczky. Realization theory for linear switched systems. In MTNS2004, 2004.Google ScholarGoogle Scholar
  21. M.Petreczky.Hybrid formal power series and their application to realization theory of hybrid systems. In Proc. of International Symposium on Mathematical Theory of Networks and Systems, 2006.Google ScholarGoogle Scholar
  22. M.Petreczky. Realization Theory of Hybrid Systems. PhD thesis, Vrije Universiteit, Amsterdam, 2006. Available at http://www.cwi.nl/mpetrec.Google ScholarGoogle Scholar
  23. M.Petreczky. Realization theory for linear switched systems: Formal power series approach. Systems and Control Letters, 56(9 - 10):588--595, 2007.Google ScholarGoogle Scholar
  24. M. Petreczky and J. H. van Schuppen. Realization theory for linear hybrid systems. Manuscript submitted to IEEE Transactions on Automatic Control, 200 (conditionally accepted).Google ScholarGoogle Scholar
  25. M.Petreczky and J.H. Van Schuppen. Realization theory of discrete-time linear hybrid system. In 15th IFAC Symposium on System Identification, 2009.Google ScholarGoogle ScholarCross RefCross Ref
  26. J.Roll, A.Bemporad, and L.Ljung. Identification of piecewise affine systems via mixed-integer programming. Automatica, 40:37--50, 2004. Google ScholarGoogle ScholarDigital LibraryDigital Library
  27. Z.Sun and S.S. Ge. Switched linear systems : control and design. Springer, London, 2005.Google ScholarGoogle Scholar
  28. P.V. Tan, G.Millerioux, and J.Daafouz. Invertibility, flatness and identifiability of switched linear dynamical systems: an application to secure communications. In 47th IEEE Conf. Decision and Control, 2008.Google ScholarGoogle Scholar
  29. S.Vajda, K.R. Godfrey, and R.Hershcel. Similarity transformation approach to identifiability analysis of nonlinear compartmental models.Mathematical Biosciences, 93:217--248, 1989.Google ScholarGoogle ScholarCross RefCross Ref
  30. J.van den Hof.System theory and system identification of compartmental systems. PhD thesis, University of Groningen, 1996.Google ScholarGoogle Scholar
  31. V.Verdult and M.Verhaegen. Subspace identification of piecewise linear systems. In Proc.Conf. Decision and Control, 2004.Google ScholarGoogle ScholarCross RefCross Ref
  32. R.Vidal.Recursive identification of switched {ARX} systems. Automatica, 44(9):2274 -- 2287, 2008. Google ScholarGoogle ScholarDigital LibraryDigital Library
  33. R.Vidal, A.Chiuso, and S.Sastry. Observability and identifiability of jump linear systems. In Proc. IEEE Conf. Dec. and Control, pages 3614 -- 3619, 2002.Google ScholarGoogle ScholarCross RefCross Ref
  34. E.Walter and Y.Lecourtier.Global approaches to identifiability testing for linear and nonlinear state-space models. Mathematics and Computers in Simulation, XXIV:472--482, 1982.Google ScholarGoogle ScholarCross RefCross Ref
  35. S.Weiland, A.L. Juloski, and B.Vet. On the equivalence of switched affine models and switched {ARX} models.In 45th IEEE Conf. on Decision and Control, 2006.Google ScholarGoogle ScholarCross RefCross Ref

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      • Published in

        cover image ACM Conferences
        HSCC '10: Proceedings of the 13th ACM international conference on Hybrid systems: computation and control
        April 2010
        308 pages
        ISBN:9781605589558
        DOI:10.1145/1755952

        Copyright © 2010 ACM

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        Publication History

        • Published: 12 April 2010

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