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Identifiability and Identification of Linear Systems with Delays

  • Lotfi Belkoura
  • Michel Dambrine
  • Yuri Orlov
  • Jean-Pierre Richard
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 38)

Abstract

Parameter identifiability and identification are studied for linear differential delay equations of neutral type and with distributed delays. It is shown how the identifiability property can be fonnulated in terms of controllability conditions, namely approximate controllability for the general case, and weak controllability for the retarded case with finitely many lumped delays in the state vector and control input. The notion of sufficiently rich input, which enforces identifiability, is also addressed, and the results are obtained assuming knowledge of the solution on a bounded time interval. Once the parameter identifiability is guaranteed, synthesis of an adaptive parameter identifier is developed for systems with finitely many lumped delays in the state vector and control input. Theoretical results arc supported by numerical simulations.

Keywords

Delay System Functional Differential Equation Bounded Time Interval Neutral Type Parameter Identifiability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Agarwal, M. and Canudas, C.: On line estimation of time delay and continuous-time process parameters, Int. Journal of Control, [1987], 46, pp. 295–311.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bai, E.-W. and Chyung, D.H. Improving delay estimates derived from least-squares algorithms and Pade approximations, International Journal of Systems Science, [1993], 24, 745–756.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    L. Belkoura and Y. Orlov, Identifiability analysis of linear delay-differential systems, IMA J. of Mathematical Control and Information, 19:73–81, [2002].zbMATHCrossRefGoogle Scholar
  4. 4.
    F. Bianchini and E.P. Ryan. A Razumikhin-type lemma for functional differential equations with application to adaptive control. Automatica, 35(5):809–818, [1999].MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Biswas, K.K. and Singh, G. ([1978]). Identification of stochastic time delay systems, IEEE Transactions on Automatic Control, AC-23, 504–505.CrossRefGoogle Scholar
  6. 6.
    H.H. Choi and M.J. Chung. Observer-based H∞ controller design for state delayed linear systems. Automatica, 32(7): 1073–1075, [1996].MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    H.H. Choi and M.J. Chung. Robust obscrver-based H∞ controller dcsign for tinear uncertain time-delay systems. Automatica, 33(9): 1749–1752, [1997].MathSciNetCrossRefGoogle Scholar
  8. 8.
    M. Darouach. Linear functional observers for systems with delays in state variables. IEEE Trans. Aut. Control, 46(3):491–496, [March 2001].MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    M. Darouach, P. Pierrot, and E. Richard. Design of reduced-order observers without internal delays. IEEE Trans. Aut. Cont., 44(9): 1711–1713, [Sept. 1999].MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    C.E. DeSouza, R.E. Palhares, and P.L.D. Peres. Robust H∞ filtering for uncertain linear systems with multiple time-varying state delays: An LMI approach. In 38th IEEE CDC99 (Conf. on Dec. and Control), pages 2023–2028, Phoenix, AZ, [Dec. 1999].Google Scholar
  11. 11.
    S. Diop, I. Kolmanovsky, P. Moraal, and M. vanNieuwstadt. Preserving stability/performance when facing an unknown time delay. Control Eng. Practice, 9: 1319–1325, [Dec. 2001].CrossRefGoogle Scholar
  12. 12.
    L. Ehrenpreis. Solutions of some problems of division, part IV, Invertible and elliptic operators, Amer. Journal of Mathematics. 82:522–588, [1960]MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    F.W. Fairmar and A. Kumar. Delay-less observers for systems with delay. IEEE Trans. Aut. Control, 31(3):258–259, [March 1986].CrossRefGoogle Scholar
  14. 14.
    Fernandes, J.M. and Ferriera, A.R. An all-pass approximation to time delay. UKACC International Conference on CONTROL 96, [1996] 1208–1213.CrossRefGoogle Scholar
  15. 15.
    S.G Foda and M.S. Mahmoud. Adaptive stabilization of delay differential systems with unknown uncertainty bounds. Int. J. Control, 71(2):259–275, [1998]MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Gawthrop, P.J., Nihtila, M. and Rad A.B.: Recursive parameter estimation of continuous-time systems with unknown time delay, Control-Theory and Advanced Technology, [1989], 5, pp. 227–248.MathSciNetGoogle Scholar
  17. 17.
    A. Gennani, C. Manes, and P. Pepe. A state observer for nonlinear delay systems. In 37th IEEE CDC98 Conf. on Dec. and Control), pages 355–360, Tampa, FL, [Dec. 1998].Google Scholar
  18. 18.
    J.K. Hale and S.M. Verduyn-Lunel. Introduction to Functional Differential Equations, volume 99 of Applied Math. Sciences, Springer NY, [1993].Google Scholar
  19. 19.
    Y.P. Huang and K. Zhou. Robust stability of uncertain time delay systems. IEEE Trans. Aut. Control, 45(11):2169–2173, [Nov. 2000].MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    V.B. Kolmanovskii and A. Myshkis. Introduction to the theory and applications of functional differential equations. Kluwer Acad. Dordreeht, [1999].zbMATHCrossRefGoogle Scholar
  21. 21.
    V.B. Kolmanovskii and V. R. Nosov. Stability of functional differential equations. Academic Press London, [1986].zbMATHGoogle Scholar
  22. 22.
    Kurl, H. and Goedecke, W. ([1981]). Digital parameter-adaptive control of processes with unknown dead time, Automatica, 17, 245–252.CrossRefGoogle Scholar
  23. 23.
    J. Leyva-Ramos and A.E. Pearson. An asymptotic modal observer for linear autonomous time lag systems. IEEE Trans. Aut. Cont., 40: 1291–1294, [July 1995].MathSciNetzbMATHGoogle Scholar
  24. 24.
    Liu, G. Adaptive predictor for slowly time-varying systems with variable time-delay, Advances in modelling and simulation, Vol. 20, pp. 9–21, [1990]MathSciNetGoogle Scholar
  25. 25.
    S.M.V. Lunel. Parameter identifiability of differential delay equations, Int J. of Adapt. Control Signal Process, 15: [2001].Google Scholar
  26. 26.
    S. Majhi and Atherton D.P. A novel identifcation method for time delay processes. In ECC99, European Control Conf., Karslruhe, Gennany, [1999].Google Scholar
  27. 27.
    L. Mirkin and G. Tadmor. H∞ control of systems with I/O delay: A review of some problem-oriented methods. IMA J. Math. Control Information, 19(1), [2002].Google Scholar
  28. 28.
    S.I. Niculescu. Delay Effects on Stability, volume 269 of LNCIS. Springer, [2001].Google Scholar
  29. 29.
    Y. Orlov, L. Belkoura, M. Dambrine and J.P. Richard, On identifiability of linear timedelay systems, IEEE Trans. Aut. Conlrol, 47(8):1319–1324, [Aug. 2002].MathSciNetCrossRefGoogle Scholar
  30. 30.
    Y. Orlov, L. Belkoura, J.P. Richard, and M. Dambrine, On-Line Parameter Identification of Linear Time-Delay Systems, to appear in Proc. IEEE CDC 02, Las Vegas.Google Scholar
  31. 31.
    Pourboghrat F., Chyung, D.H., Parameter identification of linear delay systems, Int. J. Control, vol. 49no. 2, pp. 595–627, [1989]MathSciNetzbMATHGoogle Scholar
  32. 32.
    Rad, A.B.: Self-tuning control of systems wilh unknown time delay: a continuous-time approach, Control Theory and Advanced Technology, [1994], 10, pp. 479–497.MathSciNetGoogle Scholar
  33. 33.
    O. Sename. New trends in design of observers for time-delay systems. Kybermetica, 37(4):427–458, [2001].Google Scholar
  34. 34.
    J.P. Richard. Time delay systems: An overview of some recent advances and open problems. Automatica, 39(9), [2003] (to appear).Google Scholar
  35. 35.
    K.K. Tan, Q.K. Wang, and T.H. Lee. Finite spectrum assignment control of unstable time delay processes with relay tuning. Ind. Eng. Chem. Res., 37(4): 1351–1357, 1998.CrossRefGoogle Scholar
  36. 36.
    Tuch, J., Feuer, A. and Palmor Z.J. Time delay estimation in continuous linear time-invariant systems, IEEE Trans. Aut. Control, Vol. 39no. 4, pp. 823–827, [1994]MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    S.M. Verduyn-Lunel. Identification problems in functional differential equations. In 36th IEEE CDC97 (Conf. on Dec. and Control), pages 4409–4413, San Diego, CA. [Dec. 1997].Google Scholar
  38. 38.
    E.I. Verriest. Robust stability and adaptive control of time-varying neutral systems. In 38th IEEE CDC99 (Conf. on Dec. and Control), pages 4690–4695. Phocnix, AZ, [Dec. 1999].Google Scholar
  39. 39.
    Z. Wang, B. Huang, and H. Unbchausen. Robust H∞ observer design for uncertain time-delay systems: (I) the continuous case. In IFAC 14th World Congress, pages 231–236, Beijing, China, [1999].Google Scholar
  40. 40.
    Z. Wang, B. Huang, and H. Unbchausen. Robust H∞ observer design of linear state delayed systems with parametric uncertainty: The discrete-time case. Automatica. 35(6):1161–1167, [1999].MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    K. Watanabe, E. Nobuyama, and K. Kojima. Recent advances in control of time-delay systems a tutorial review. In 35th IEEE CDC96 (Conf. on Dec. and Control), pages 2083–2089. Kobe, Japan, [Dec. 1996].Google Scholar
  42. 42.
    Y.X. Yao, Y.M. Zhang, and R. Kovacevic. Functional observer and state feed back for input time-delay systems. Int. J. Control, 66(4):603–617, [1997].MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Y. Yamamoto. Reachability of a class of infinite-dimensional linear systems: An external approach with application to general neutral systems, SIAM J. of Control and Optimization, 27: 217–234, [1989]zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Lotfi Belkoura
    • 1
  • Michel Dambrine
    • 2
  • Yuri Orlov
    • 3
  • Jean-Pierre Richard
    • 2
  1. 1.LAIL, Universite des Sciences el Technologies de LilleFrance
  2. 2.LAIL, Ecole Centrale de LilleFrance
  3. 3.CICESE Research Center, Electronics and Telecom Dpt.San DiegoUSA

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