Stabilization of nonlinear systems with varying parameter by a control Lyapunov function

Wajdi Kallel; Thouraya Kharrat

Kybernetika (2017)

  • Volume: 53, Issue: 5, page 853-867
  • ISSN: 0023-5954

Abstract

top
In this paper, we provide an explicit homogeneous feedback control with the requirement that a control Lyapunov function exists for affine in control systems with bounded parameter that satisfies an homogeneous condition. We use a modified version of the Sontag's formula to achieve our main goal. Moreover, we prove that the existence of an homogeneous control Lyapunov function for an homogeneous system leads to an homogeneous closed-loop system which is asymptotically stable by an homogeneous feedback control. In addition, we study the finite time stability for affine in control systems with varying parameter.

How to cite

top

Kallel, Wajdi, and Kharrat, Thouraya. "Stabilization of nonlinear systems with varying parameter by a control Lyapunov function." Kybernetika 53.5 (2017): 853-867. <http://eudml.org/doc/294394>.

@article{Kallel2017,
abstract = {In this paper, we provide an explicit homogeneous feedback control with the requirement that a control Lyapunov function exists for affine in control systems with bounded parameter that satisfies an homogeneous condition. We use a modified version of the Sontag's formula to achieve our main goal. Moreover, we prove that the existence of an homogeneous control Lyapunov function for an homogeneous system leads to an homogeneous closed-loop system which is asymptotically stable by an homogeneous feedback control. In addition, we study the finite time stability for affine in control systems with varying parameter.},
author = {Kallel, Wajdi, Kharrat, Thouraya},
journal = {Kybernetika},
keywords = {feedback stabilization; homogeneous system; nonlinear control systems; Lyapunov function; finite time stability},
language = {eng},
number = {5},
pages = {853-867},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Stabilization of nonlinear systems with varying parameter by a control Lyapunov function},
url = {http://eudml.org/doc/294394},
volume = {53},
year = {2017},
}

TY - JOUR
AU - Kallel, Wajdi
AU - Kharrat, Thouraya
TI - Stabilization of nonlinear systems with varying parameter by a control Lyapunov function
JO - Kybernetika
PY - 2017
PB - Institute of Information Theory and Automation AS CR
VL - 53
IS - 5
SP - 853
EP - 867
AB - In this paper, we provide an explicit homogeneous feedback control with the requirement that a control Lyapunov function exists for affine in control systems with bounded parameter that satisfies an homogeneous condition. We use a modified version of the Sontag's formula to achieve our main goal. Moreover, we prove that the existence of an homogeneous control Lyapunov function for an homogeneous system leads to an homogeneous closed-loop system which is asymptotically stable by an homogeneous feedback control. In addition, we study the finite time stability for affine in control systems with varying parameter.
LA - eng
KW - feedback stabilization; homogeneous system; nonlinear control systems; Lyapunov function; finite time stability
UR - http://eudml.org/doc/294394
ER -

References

top
  1. Artstein, Z., 10.1016/0362-546x(83)90049-4, Nonlinear Anal. TMA 7 (1983), 1163-1173. Zbl0525.93053MR0721403DOI10.1016/0362-546x(83)90049-4
  2. Bhat, S. P., Bernstein, D. S., 10.1109/9.668834, IEEE Trans. Automat. Control 43 (1998), 678-682. Zbl0925.93821MR1618028DOI10.1109/9.668834
  3. Cai, X. S., Han, Z. Z., Zhang, W., 10.3724/sp.j.1004.2009.00206, Acta Automat. Sinica 35 (2009), 206-209. MR2531861DOI10.3724/sp.j.1004.2009.00206
  4. Čelikovský, S., Aranda-Bricaire, E., 10.1016/s0167-6911(98)00062-0, Systems Control Lett. 36 (1999), 21-37. MR1750623DOI10.1016/s0167-6911(98)00062-0
  5. Huang, J., Yu, L., Xia, S., 10.1007/s00034-014-9741-5, Circuits Systems Signal Process. 33 (2015), 2319-2331. MR3217482DOI10.1007/s00034-014-9741-5
  6. Hong, Y., Wang, J., Cheng, D., 10.1109/tac.2006.875006, IEEE Trans. Automat. Control 51 (2006), 858-862. MR2232614DOI10.1109/tac.2006.875006
  7. Jerbi, H., 10.1016/s0167-6911(01)00172-4, Systems Control Lett. 41 (2002), 173-178. MR2072233DOI10.1016/s0167-6911(01)00172-4
  8. Jerbi, H., Kallel, W., Kharrat, T., 10.1007/s10883-008-9053-9, J. Dynamical Control Syst. 14 (2008), 595-606. MR2448693DOI10.1007/s10883-008-9053-9
  9. Jerbi, H., Kharrat, T., Only a level set of a control Lyapunov function for homogeneous systems., Kybernetika 41 (2005), 593-600. MR2192425
  10. Krstic, M., Kokotovic, P. V., 10.1016/0167-6911(94)00107-7, Systems Control Lett. 26 (1995), 17-23. MR1347637DOI10.1016/0167-6911(94)00107-7
  11. Massera, J. L., 10.2307/1969955, Ann. Math. 64 (1956), 182-206. MR0079179DOI10.2307/1969955
  12. Moulay, E., 10.1016/j.automatica.2008.05.003, Automatica 44 (2008), 2981-2984. MR2527229DOI10.1016/j.automatica.2008.05.003
  13. Moulay, E., Perruquetti, W., 10.1016/j.jmaa.2005.11.046, J. Math. Anal. Appl. 323 (2006), 1430-1443. Zbl1131.93043MR2260193DOI10.1016/j.jmaa.2005.11.046
  14. Rosier, L., 10.1016/0167-6911(92)90078-7, Systems Control Lett. 19 (1992), 467-473. Zbl0762.34032MR1195304DOI10.1016/0167-6911(92)90078-7
  15. Sepulchre, R., Aeyels, D., 10.1007/bf01211517, Math. Control Signals Syst. 9 (1996), 34-58. MR1410047DOI10.1007/bf01211517
  16. Shafiei, M. H., Yazdanpanah, M. J., 10.1016/j.isatra.2009.11.004, ISA Trans. 49 (2010), 215-221. DOI10.1016/j.isatra.2009.11.004
  17. Sontag, E. D., 10.1016/0167-6911(89)90028-5, Systems Control Lett. 13 (1989), 117-123. MR1014237DOI10.1016/0167-6911(89)90028-5
  18. Sontag, E. D., 10.1137/0321028, SIAM J. Control Optim. 21 (1983), 462-471. MR0696908DOI10.1137/0321028
  19. Tsinias, J., 10.1016/0362-546x(88)90060-0, Nonlinear Anal. TMA 12 (1988), 1283-1296. Zbl0662.93055MR0969506DOI10.1016/0362-546x(88)90060-0
  20. Tsinias, J., 10.1007/bf02551276, Math. Control Signals Syst. 2 (1989), 343-357. Zbl0688.93048MR1015672DOI10.1007/bf02551276
  21. Zhang, W., Su, H., Cai, X., Guo, H., 10.1007/s00034-014-9848-8, Circuits Systems Signal Process. 34 (2015), 341-352. MR3299171DOI10.1007/s00034-014-9848-8
  22. Wang, H., Han, Z., Zhang, W., Xie, Q., 10.1016/j.nonrwa.2008.08.010, Nonlinear Analysis: Real World Appl. 10 (2009), 2842-2849. MR2523247DOI10.1016/j.nonrwa.2008.08.010
  23. Wang, H., Han, Z., Zhang, W., Xie, Q., 10.1016/j.jsv.2008.07.023, J. Sound Vibration 320 (2009), 365-372. MR2335867DOI10.1016/j.jsv.2008.07.023

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.