Invariant factors assignment for a class of time-delay systems

Jean-Jacques Loiseau

Kybernetika (2001)

  • Volume: 37, Issue: 3, page [265]-275
  • ISSN: 0023-5954

Abstract

top
It is well–known that every system with commensurable delays can be assigned a finite spectrum by feedback, provided that it is spectrally controllable. In general, the feedback involves distributed delays, and it is defined in terms of a Volterra equation. In the case of multivariable time–delay systems, one would be interested in assigning not only the location of the poles of the closed–loop system, but also their multiplicities, or, equivalently, the invariant factors of the closed–loop system. We answer this question. Our basic tool is the ring of operators that includes derivatives, localized and distributed delays. This ring is a Bezout ring. It is also an elementary divisor ring, and finally one can show that every matrix over this ring can be brought in column reduced form using right unimodular transformations. The formulation of the result we finally obtain in the case of time-delay systems differs from the well–known fundamental theorem of state feedback for finite dimensional systems, mainly because the reduced column degrees of a matrix of operators are not uniquely defined in general.

How to cite

top

Loiseau, Jean-Jacques. "Invariant factors assignment for a class of time-delay systems." Kybernetika 37.3 (2001): [265]-275. <http://eudml.org/doc/33534>.

@article{Loiseau2001,
abstract = {It is well–known that every system with commensurable delays can be assigned a finite spectrum by feedback, provided that it is spectrally controllable. In general, the feedback involves distributed delays, and it is defined in terms of a Volterra equation. In the case of multivariable time–delay systems, one would be interested in assigning not only the location of the poles of the closed–loop system, but also their multiplicities, or, equivalently, the invariant factors of the closed–loop system. We answer this question. Our basic tool is the ring of operators that includes derivatives, localized and distributed delays. This ring is a Bezout ring. It is also an elementary divisor ring, and finally one can show that every matrix over this ring can be brought in column reduced form using right unimodular transformations. The formulation of the result we finally obtain in the case of time-delay systems differs from the well–known fundamental theorem of state feedback for finite dimensional systems, mainly because the reduced column degrees of a matrix of operators are not uniquely defined in general.},
author = {Loiseau, Jean-Jacques},
journal = {Kybernetika},
keywords = {time-delay system; feedback; spectrum; right unimodular transformations; Invariant factors assignment; spectral controllability; Bezout ring; Volterra equation; time-delay system; feedback; spectrum; right unimodular transformations; Invariant factors assignment; spectral controllability; Bezout ring; Volterra equation},
language = {eng},
number = {3},
pages = {[265]-275},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Invariant factors assignment for a class of time-delay systems},
url = {http://eudml.org/doc/33534},
volume = {37},
year = {2001},
}

TY - JOUR
AU - Loiseau, Jean-Jacques
TI - Invariant factors assignment for a class of time-delay systems
JO - Kybernetika
PY - 2001
PB - Institute of Information Theory and Automation AS CR
VL - 37
IS - 3
SP - [265]
EP - 275
AB - It is well–known that every system with commensurable delays can be assigned a finite spectrum by feedback, provided that it is spectrally controllable. In general, the feedback involves distributed delays, and it is defined in terms of a Volterra equation. In the case of multivariable time–delay systems, one would be interested in assigning not only the location of the poles of the closed–loop system, but also their multiplicities, or, equivalently, the invariant factors of the closed–loop system. We answer this question. Our basic tool is the ring of operators that includes derivatives, localized and distributed delays. This ring is a Bezout ring. It is also an elementary divisor ring, and finally one can show that every matrix over this ring can be brought in column reduced form using right unimodular transformations. The formulation of the result we finally obtain in the case of time-delay systems differs from the well–known fundamental theorem of state feedback for finite dimensional systems, mainly because the reduced column degrees of a matrix of operators are not uniquely defined in general.
LA - eng
KW - time-delay system; feedback; spectrum; right unimodular transformations; Invariant factors assignment; spectral controllability; Bezout ring; Volterra equation; time-delay system; feedback; spectrum; right unimodular transformations; Invariant factors assignment; spectral controllability; Bezout ring; Volterra equation
UR - http://eudml.org/doc/33534
ER -

References

top
  1. Brethé D., Contribution à l’étude de la stabilisation des systèmes linéaires à retards, Thèse de Doctorat, École Centrale de Nantes et Université de Nantes, 1997 
  2. Brethé D., Loiseau J. J., 10.1016/S0378-4754(97)00113-4, Math. Comput. Simulation 45 (1998), 339–348 (1998) Zbl1017.93506MR1622412DOI10.1016/S0378-4754(97)00113-4
  3. Eising R., 10.1109/TAC.1978.1101861, IEEE Trans. Automat. Control 23 (1978), 793–799 (1978) Zbl0397.93022MR0507790DOI10.1109/TAC.1978.1101861
  4. Glüsing–Lüerßen H., 10.1137/S0363012995281869, SIAM J. Control Optim. 35 (1997), 480–499 (1997) Zbl0876.93022MR1436634DOI10.1137/S0363012995281869
  5. Habets L., Algebraic and Computational Aspects of Time–delay Systems, Ph. D. Thesis, Eindhoven 1994 Zbl0804.93031MR1276720
  6. Kailath T., Linear Systems, Prentice Hall, Englewood Cliffs, N. J. 1980 Zbl0870.93013MR0569473
  7. Kamen E. W., 10.1007/BF01698126, Math. Systems Theory 9 (1974), 57–74 (1974) MR0395953DOI10.1007/BF01698126
  8. Kamen E. W., Khargonekar P. P., Tannenbaum A., 10.1080/00207178608933506, Internat. J. Control 43 (1986), 837–857 (1986) Zbl0599.93047MR0828360DOI10.1080/00207178608933506
  9. Kaplansky I., 10.1090/S0002-9947-1949-0031470-3, Trans. American Mathematical Society 66 (1949), 464–491 (1949) Zbl0036.01903MR0031470DOI10.1090/S0002-9947-1949-0031470-3
  10. Kučera V., Analysis and Design of Discrete Linear Control Systems, Prentice–Hall, London, and Academia, Prague 1991 Zbl0762.93060MR1182311
  11. Leborgne D., Calcul différentiel complexe, Presses Universitaires de France, Paris, 1991 Zbl0731.30001MR1091546
  12. Loiseau J. J., Algebraic tools for the control and stabilization of time–delay systems, In: Proc. 1st IFAC Workshop on Linear Time Delay Systems, Grenoble 1998, pp. 235–249 (1998) 
  13. Manitius A. Z., Olbrot A. W., 10.1109/TAC.1979.1102124, IEEE Trans. Automat. Control 35 (1979), 541–553 (1979) Zbl0425.93029MR0538808DOI10.1109/TAC.1979.1102124
  14. Morf M., Lévy B. C., Kung S.-Y., New results in 2-D systems theory, Part I: 2-D polynomial matrices, factorizations, and coprimeness. Proc. IEEE 65 (1977), 861–872 (1977) 
  15. Olbrot A. W., 10.1109/TAC.1978.1101879, IEEE Trans. Automat. Control 23 (1978), 887–890 (1978) MR0528786DOI10.1109/TAC.1978.1101879
  16. Rosenbrock H. H., State–space and Multivariable Theory, Wiley, New York 1970 Zbl0246.93010MR0325201
  17. Assche V. Van, Dambrine M., Lafay J.-F., Richard J.-P., Some problems arising in the implementation of distributed–delay control laws, In: Proc. 38th IEEE Conference on Decision and Control, Phoenix 1999 
  18. Watanabe K., 10.1109/TAC.1986.1104336, IEEE Trans. Automat. Control 31 (1986), 543–550 (1986) Zbl0596.93009MR0839083DOI10.1109/TAC.1986.1104336

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.