Discretization schemes for Lyapunov-Krasovskii functionals in time-delay systems

Keqin Gu

Kybernetika (2001)

  • Volume: 37, Issue: 4, page [479]-504
  • ISSN: 0023-5954

Abstract

top
This article gives an overview of discretized Lyapunov functional methods for time-delay systems. Quadratic Lyapunov–Krasovskii functionals are discretized by choosing the kernel to be piecewise linear. As a result, the stability conditions may be written in the form of linear matrix inequalities. Conservatism may be reduced by choosing a finer mesh. Simplification techniques, including elimination of variables and using integral inequalities are also discussed. Systems with multiple delays and distributed delays are also treated. Finally, the treatment of uncertainties and input-output performance requirements are discussed.

How to cite

top

Gu, Keqin. "Discretization schemes for Lyapunov-Krasovskii functionals in time-delay systems." Kybernetika 37.4 (2001): [479]-504. <http://eudml.org/doc/33547>.

@article{Gu2001,
abstract = {This article gives an overview of discretized Lyapunov functional methods for time-delay systems. Quadratic Lyapunov–Krasovskii functionals are discretized by choosing the kernel to be piecewise linear. As a result, the stability conditions may be written in the form of linear matrix inequalities. Conservatism may be reduced by choosing a finer mesh. Simplification techniques, including elimination of variables and using integral inequalities are also discussed. Systems with multiple delays and distributed delays are also treated. Finally, the treatment of uncertainties and input-output performance requirements are discussed.},
author = {Gu, Keqin},
journal = {Kybernetika},
keywords = {time-delay system; Lyapunov-Krasovskii functional; multiple delays; time-delay system; Lyapunov-Krasovskii functional; multiple delays},
language = {eng},
number = {4},
pages = {[479]-504},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Discretization schemes for Lyapunov-Krasovskii functionals in time-delay systems},
url = {http://eudml.org/doc/33547},
volume = {37},
year = {2001},
}

TY - JOUR
AU - Gu, Keqin
TI - Discretization schemes for Lyapunov-Krasovskii functionals in time-delay systems
JO - Kybernetika
PY - 2001
PB - Institute of Information Theory and Automation AS CR
VL - 37
IS - 4
SP - [479]
EP - 504
AB - This article gives an overview of discretized Lyapunov functional methods for time-delay systems. Quadratic Lyapunov–Krasovskii functionals are discretized by choosing the kernel to be piecewise linear. As a result, the stability conditions may be written in the form of linear matrix inequalities. Conservatism may be reduced by choosing a finer mesh. Simplification techniques, including elimination of variables and using integral inequalities are also discussed. Systems with multiple delays and distributed delays are also treated. Finally, the treatment of uncertainties and input-output performance requirements are discussed.
LA - eng
KW - time-delay system; Lyapunov-Krasovskii functional; multiple delays; time-delay system; Lyapunov-Krasovskii functional; multiple delays
UR - http://eudml.org/doc/33547
ER -

References

top
  1. Boyd S., Ghaoui L. El, Feron, E., Balakrishnan V., Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia 1994 Zbl0816.93004MR1284712
  2. Boyd S., Yang Q., 10.1080/00207178908559769, Internat. J. Control 49 (1989), 2215–2240 (1989) Zbl0729.93067MR1007704DOI10.1080/00207178908559769
  3. Souza C. E. De, Li X., 10.1016/S0005-1098(99)00025-4, Automatica 35 (1999), 1313–1321 (1999) MR1829975DOI10.1016/S0005-1098(99)00025-4
  4. Doyle J. C., Wall, J., Stein G., Performance and robustness analysis for structured uncertainty, In: IEEE Conference on Decision and Control 1982, pp. 629–636 (1982) 
  5. Gahinet P., Nemirovski A., Laub A. J., Chilali M., LMI Control Toolbox for use with MATLAB, Natick, MA MathWorks, 1995 
  6. Gu K., 10.1080/002071797223406, Internat. J. Control 68 (1997), 923–934 (1997) MR1689707DOI10.1080/002071797223406
  7. Gu K., Stability of linear time-delay systems with block-diagonal uncertainty, In: 1998 American Control Conference, Philadelphia 1998, pp. 1943–1947 (1998) 
  8. Gu K., 10.1080/002071799220092, Internat. J. Control 72 (1999), 16, 1436–1445 (1999) Zbl0959.93053MR1723330DOI10.1080/002071799220092
  9. Gu K., Partial solution of LMI in stability problem of time-delay systems, In: Proc. 38th IEEE Conference on Decision and Control 1999, pp. 227–232 (1999) 
  10. Gu K., 10.1002/(SICI)1099-1239(199901)9:1<1::AID-RNC382>3.0.CO;2-S, Internat. J. Robust and Nonlinear Control 9 (1999), 1–14 (1999) Zbl0923.93046MR1669772DOI10.1002/(SICI)1099-1239(199901)9:1<1::AID-RNC382>3.0.CO;2-S
  11. Gu K., An integral inequality in the stability problem of time-delay systems, In: Proc. 39th IEEE Conference on Decision and Control 2000 
  12. Gu K., 10.1080/00207170110047190, Internat. J. Control 74 (2001), 10, 967–976 Zbl1015.93053MR1847365DOI10.1080/00207170110047190
  13. Gu K., Han Q.-L., Discretized Lyapunov functional for linear uncertain systems with time-varying delay, In: 2000 American Control Conference, Chicago 2000 
  14. Gu K., Han Q.-L., Luo A. C. J., Niculescu S.-I., 10.1080/00207170010031486, Internat. J. Control 74 (2001), 7, 737–744 Zbl1015.34061MR1826755DOI10.1080/00207170010031486
  15. Gu K., Luo A. C. J., Niculescu S.-I., Discretized Lyapunov functional for systems with distributed delay, In: 1999 European Control Conference, Karlsruhe 1999 (1999) 
  16. Gu K., Niculescu S.-I., 10.1109/9.847747, IEEE Trans. Automat. Control 45 (2000), 572–575 Zbl0986.34066MR1762880DOI10.1109/9.847747
  17. Gu K., Niculescu S.-I., Further remarks on additional dynamics in various model transformations of linear delay systems, In: 2000 American Control Conference, Chicago 2000 Zbl1056.93511MR1819827
  18. Hale J. K., Lunel S. M. Verduyn, Introduction to Functional Differential Equations, Springer–Verlag, New York 1993 MR1243878
  19. Han Q.-L., Gu K., On robust stability of time-delay systems with norm-bounded uncertainty, IEEE Trans. Automat. Control, accepted Zbl1006.93054MR1853685
  20. Han Q.-L., Gu K., Stability of linear systems with time-varying delay: a generalized discretized Lyapunov functional approach, Asian J. of Control, accepted 
  21. Huang W., 10.1016/0022-247X(89)90166-2, J. Math. Anal. Appl. 142 (1989), 83–94 (1989) Zbl0705.34084MR1011411DOI10.1016/0022-247X(89)90166-2
  22. Infante E. F., Castelan W. V., 10.1016/0022-0396(78)90051-7, J. Differential Equations 29 (1978), 439–451 (1978) MR0507489DOI10.1016/0022-0396(78)90051-7
  23. Kharitonov V., Robust stability analysis of time delay systems: A survey, In: Proc. IFAC System Structure Control, Nantes 1998 
  24. Kharitonov V. L., Melchor–Aguilar D. A., Some remarks on model transformations used for stability and robust stability analysis of time-delay systems, In: Proc. 38th IEEE Conference on Decision and Control, Phoenix 1999, pp. 1142–1147 (1999) 
  25. Kolmanovskii V. B., Niculescu S.-I., Gu K., Delay effects on stability: a survey, In: Proc. 38th IEEE Conference on Decision and Control, Phoenix 1999, pp. 1993–1998 (1999) 
  26. Kolmanovskii V. B., Richard J.-P., Stability of some systems with distributed delays, In: JESA, special issue on “Analysis and Control of Time-delay Systems”, 31 (1997), 971–982 (1997) 
  27. Krasovskii N. N., Stability of Motion, Stanford University Press, 1963 Zbl0109.06001MR0147744
  28. Li X., Souza C. E. de, 10.1016/S0005-1098(97)00082-4, Automatica 33 (1997), 1657–1662 (1997) MR1481824DOI10.1016/S0005-1098(97)00082-4
  29. Nesterov Y., Nemirovskii A., Interior–Point Polynomial Algorithms in Convex Programming SIAM, Philadelphia 199 MR1258086
  30. Niculescu S. I., Dugard, L., Dion J. M., Stabilité et stabilisation robustes des systèmes à retard, In: Proc. Journées Robustesse, Toulouse 1995 
  31. Niculescu S. I., Souza C. E. de, Dion J. M., Dugard L., Robust stability and stabilization of uncertain linear systems with state delay: single delay case (I), and Multiple delays case (II), In: Proc. IFAC Workshop Robust Control Design, Rio de Janeiro 1994, pp. 469–474 and 475–480 (1994) 
  32. Niculescu S. I., Verriest E. I., Dugard, L., Dion J. M., Stability and robust stability of time-delay systems: A guided tour, In: Stability and Control of Time-Delay Systems (L. Dugard and E. I. Verriest, eds., Lecture Notes in Control and Information Sciences), Springer–Verlag, London 1997, pp. 1–71 (1997) MR1482571
  33. Packard A., Doyle J., 10.1016/0005-1098(93)90175-S, Automatica 29 (1993), 71–109 (1993) Zbl0772.93023MR1200542DOI10.1016/0005-1098(93)90175-S
  34. Zhou K., Doyle J. C., Glover K., Robust and Optimal Control, Prentice Hall, Englewood Cliffs, N.J. 1996 Zbl0999.49500

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.