The Effect of Approximating Distributed Delay Control Laws on Stability

  • Wim Michiels
  • Sabine Mondié
  • Dirk Roose
  • Michel Dambrine
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 38)


An overview of stability results on the implementation of distributed delay control laws, arising in the context of finite spectrum assignment, is given. First the case where distributed delays are approximated with a finite sum of point-wise delays is considered. The instability mechanism is briefly discussed and conditions for a safe implementation are presented. Secondly modifications of the control law to remove the limitations, imposed by these conditions, are outlined. Throughout the chapter eigenvalue plots are used to provide an intuitive explanation for the phenomena and results.


Essential Spectrum Quadrature Rule Instability Mechanism Smith Predictor Unstable Eigenvalue 
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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  1. 1.
    Anstein, Z., “Linear systems with delayed control: a reduction,” IEEE Transactions on Automatic COntrol, vol. 27: 869–879, 1982.CrossRefGoogle Scholar
  2. 2.
    Avellar C.E. and Hale J.K: “On the zeros of exponential polynomials,” Mathematical analysis and applications, vol. 73: 434–452, 1980.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Datko, R., “Two examples of ill-posedness with respect to time delays revisited,” IEEE Transactions on Automatic Control, vol. 42: 434–452, 1997MathSciNetCrossRefGoogle Scholar
  4. 4.
    Oatko, R., “Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks,” SIAM Journal on Control and Optimization, vol. 26: 697–713, 1988.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Datko R., Lagnese J., and Polis M.: “An example on the effect of time delays in boundary feedback stabilization of wave equations,” SIAM Journal on Control and Optimization, vol. 24: 152–156, 1986.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Engelborghs K., Dambrine M. and Roose D.: “Limitations of a class of stabilization methoos for delay equation, IEEE Transactions on Automatic Control, vol. 46(2), 336–339, 2001.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Engelborghs, K., Luzyanina T. and Samaey, G., “DOE-BIFrOOL v. 2.00: a Matlab package for bifurcation analysis of delay differential equations,” Technical Report TW-330, Depanment of Computer Science, K.U.Leuven, Leuven, Belgium, 2001 (Available from Scholar
  8. 8.
    Fattouh, A., Sename, O. and Dion, J.M., “Pulse controller design for linear time-delay systems,” Proceedings of the IFAC WorkshOIJ on System, Structure and Conrol, Prague, the Czech Republic, 2001.Google Scholar
  9. 9.
    Georgiou, T.T. and Smith, M.C., “Graphs, Causality and Stabilizability: Linear, Shift-Invariant Systems on £2(0,∞) Math. Control Signals Systems, vol. 6: 195–223, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10. Hale J.K. and Verduyn Lunel, S.M.: Introduction to functional differential equations vol. 99 of Applied Mathematical Sciences. Springer-Verlag, 1993.Google Scholar
  11. 11.
    Hale, J.K., “Effects of delays on dynamics,” in Topological melhods in differential equations and inclusions, (A. Granas, M. Frigon, G. Sabidussi, Eds.), Kluwer Academic Publishers, 191–238, 1995.Google Scholar
  12. 12.
    Hale, J.K. and Verduyn Lunel, S.M, “Strong stabilization of neutral functional differential equations,” IMA Journal of Mathematical Conrol and Information. vol. 19: 5–23, 2002.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Hannsgen, K.B., Renardy, Y. and Wheeler, R.L., “Effectiveness and robustness with respect to time delays of boundary feedback stabilization in one-dimensional viscoelasticity,” SlAM Journal on Control and Optimization. vol. 26: 1200–1234. 1988MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Hardy G.H. and Wright E.M.: An illlroduction of the theory of numbers. Oxford University Press, 1968Google Scholar
  15. 15.
    Kolmanovskii, V.B. and Nosov, V.R., Stability of functional differential equations, vol. 180, Mathematics in Science and Engineering, Academic Press, 1986.Google Scholar
  16. 16.
    Kwon, W.H. and Pearson, A.E., “Feedback stabilization of linear systems with delayed control,” IEEE Transactions all Auotomatic Control, vol. 25: 266–269, 1980.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Logemann H.: “Destabilizing effects of smalltime-delays on feedback-controlled descriptor systems,” Linear Algebra and its Applications, vol. 272: 131–153, 1998.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Logemann, H. and Rebarber, R., “The effect of smalltime-delays on the closed-loop stability of boundary control systems,” Math. Control Signals Systems, vol. 9: 123–151, 1996.MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Logemann H., Rebarber R. and Weiss G.: “Conditions for robustness and nonrobustness of the stability of feedback control systems with respecllo small delays in the feedback loop,” SIAM Journal and Control and Optimization, vol. 34(2):572–600, 1996.MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Logemann, H. and Townley, S., “The effect of small delays in the feedback loop on the stability of neutral systems.” Systems & Control Letters, vol. 27: 267–274, 1996.MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Manitius A. Z. and Olbrot A.W.: “Finite spectrum assignment problem for systems with Delays,” IEEE Trans. Autom. Contr., vol. AC-24No.4, 541–553, 1979.MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Michiels W., Mondie S. and Roose, D.: “Robust stabilization of time-delay systems with distributed delay control laws: necessary and sufficient conditions for a safe implementation,” Automatica, 2002, submitted.Google Scholar
  23. 23.
    Michiels, W., Mondié. S. and Roose, D.: “Robust stabilization of time-delay systems with distributed delay control laws: necessary and sufficient conditions for a safe implementation.” Technical Report TW-363 Department of Computer Science, K.U.Leuven Belgium, 2003Google Scholar
  24. 24.
    Michiels W., Engelborghs K., Roose D. and Dochain D.: “SenSitivity to infinitesimal delays in neutral equations,” SIAM Jormral on Control and Optimization, 40(4): 1134–1158, 2002.MathSciNetCrossRefGoogle Scholar
  25. 25.
    Michiels W., Engelborghs K., Vansevenant P., and Roose D.: “The continuous pole placement method for delay equations,” Automatica, 38(5):747–761, 2002.MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Michiels W. and Niculescu S.-I.: “On the delay sensitivity of Smith predictors,” International Journal of System Sciences, invited paper in special issue on control theory of time-delay systems, 2003 (to appear).Google Scholar
  27. 27.
    Mirkin, L. and Zhong, Q.-C., “Are distributed delay control laws intrinsically unapproximable,” in tiProceedings of the 4th IFAC Workshop on Time-Delay Systems, INRIA Rocquencourt France, 2003.Google Scholar
  28. 28.
    Mondié S., Dambrine M. and Santos, O.: “Approximations of control laws with distributed delays: a necessary condition for stability,” Kybernetica, vol. 38: 541–551, 2002.zbMATHGoogle Scholar
  29. 29.
    Mondie, S. and Michiels, W., “Finite spectrum assignment of unstable time-delay systems with a safe implementation,” IEEE Transactions on Automatic Control, 2003 (accepted).Google Scholar
  30. 30.
    Niculescu S.-I.: Delay effects on stability: A robust cOntrol approach, Springer Heidelberg, 2001.zbMATHGoogle Scholar
  31. 31.
    Palmor, Z.J., “Time-delay compensation-Smith predictor and its modifications,” in The COntrol Handbook, (CRC and IEEE Press New York), (chapter 10), 224–237, 1996.Google Scholar
  32. 32.
    Rasvan, V. and Popescu, D, “Control of systems with input delay: An elementary approach,” (this volume, part III).Google Scholar
  33. 33.
    Rasvan, V. and Popescu, D., “Feedback stabilization of systems with delays in control,” Control Enginering and Applied Informatics vol. 3: 62–66, 2001.Google Scholar
  34. 34.
    Smith, OJ., “Closer control of loops with dead time,” Chemical Engineering Progress, vol. 53: 217–219, 1957.Google Scholar
  35. 35.
    Van Assche V., Dambrine M., Lafay J.-F. and Richard J.-P.: “Some problems arising in the implementation of distributed-delay control law,” Proceedings of the 39th IEEE Conference on Decision and COntrol, Phoenix, AZ, December 1999.Google Scholar
  36. 36.
    Van Assche v., Dambrine M., Lafay J.-F. and Richard J.-P.: “Implementation of a distributed control law for a class of systems with delay,” Proceedings of the 3rd Workshop Oil Time Delay Systems, 266–271. Santa Fe. NM, December 2001Google Scholar
  37. 37.
    Wang, Q.G. Lee, T.H. and Tan, K.K,Finite Spectrum Assignment for TIme-Delay Systems, Lecture Notes in Control and Information Sciences, vol. 239, Springer-Verlag, dy1999.Google Scholar
  38. 38.
    Watanabe, K. and Ito, M, &#c201C;A process model control for linear systems with delay.” itIEEE Transactions all Automatic Conrol. vol. 26: 1261–1268, 1981zbMATHCrossRefGoogle Scholar
  39. 39.
    Wanatabe, K., “Finite spectrum assignment and observer for multivariable systems with commensurate delays,” IEEE Transactions on Automatic Control, vol. 31: 543–550, 1986CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Wim Michiels
    • 1
  • Sabine Mondié
    • 2
  • Dirk Roose
    • 1
  • Michel Dambrine
    • 3
  1. 1.K.U. Leuven, Department of Computer ScienceHeverleeBelgium
  2. 2.Departemento de Control AutomáticoCINVESTAV-IPNMéxicoMexico
  3. 3.LAIL, UPRESA CNRS 8021, Ecole Centrale de LilleFrance

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