Delay-Dependent Robust Stability of Uncertain Time-Delayed Linear Systems

불확정성 시간지연 선형시스템의 지연 종속 강인 안정성

  • 이연규 (충북대학교 대학원 제어계측공학과) ;
  • 김진훈 (충북대학교 전기전자공학부)
  • Published : 2006.04.01

Abstract

In this paper, we propose a new delay-dependent criterion on the robust stability of time-delayed linear systems having norm bounded uncertainties. Based on new form of Lyapunov-Krasovskii functional and the Newton-Leibniz formula, we drive a result in the form of LMI which guarantees the robust stability without any model transformation. The Newton-Leibniz equation was used to relate the cross terms with free matrices. Finally, we show the usefulness of our result by two numerical examples.

Keywords

References

  1. S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory. Philadelphia, PA : SIAM, 1994
  2. W. J. Rogh, Linear System Theory. Upper Saddle River, New Jersey:Prentice Hall, 1996
  3. Niculescu, S.-I., Neto, A. T., Dion, J.-M., and Dugard, L., 1995, 'Delay-Dependent Stability of Linear Systems with Delayed state : an LMI Approach,' Proc. 34th IEEE Conf. on Decision and Control, PP. 1495-1497 https://doi.org/10.1109/CDC.1995.480347
  4. E. Fridman and U. Shaked, 'New bounded real lemma for time-delay systems and their applications,' IEEE Trans. Autom. Control, vol. 46, no. 12, PP. 1973-1978, Dec. 2001 https://doi.org/10.1109/9.975503
  5. E. Fridman and U. Shaked, 'An improved stabilization method for linear time-delay systems,' IEEE Trans. Autom. Control, Vol. 47, no. 11, PP. 1931-1937, Nov. 2002 https://doi.org/10.1109/TAC.2002.804462
  6. X. Li and C. E. de Souza, 'Delay-dependent robust stability and stabilization of uncertain linear delay systems : A linear matrix inequality approach,' IEEE Trans. Autom. Control, Vol. 42, no. 8, PP. 1144-1148, Aug. 1997 https://doi.org/10.1109/9.618244
  7. Y. S. Moon, P. Park, W. H. Kwon and Y. S. Lee, 'Delay-dependent robust stabilization of uncertain state-delayed systems,' Int. J. Control, Vol. 74 PP. 1447-1455, 2001 https://doi.org/10.1080/00207170110067116
  8. P. Park, 'A delay-dependent stability criterion for system with uncertain time-invariant delays,' IEEE. Trans. Autom. Control, Vol. 44,no. 4, PP. 876-877, Apr, 1999 https://doi.org/10.1109/9.754838
  9. S.-I. Niculescu, 'Memoryless control with an ${\alpha}$-stability constrint for time-delay systems : An LMI approach,' IEEE. Trans. Autom. Control, Vol. 43, no. 5, PP. 739-743, May, 1998 https://doi.org/10.1109/9.668850
  10. J. H. Kim, 'Delay and its time-derivative dependent robust stability of time-delayed linear systems with uncertainty,' IEEE. Trans. Autom. Control, Vol. 46, no. 5, PP. 789-792, May, 2001 https://doi.org/10.1109/9.920802
  11. H. Gao and C. Wang, 'Delay-dependent robust $H_{\infty}$ and $L_1$ - $L_2$ filtering for a class of uncertain nonlinear time-delay systems,' IEEE. Trans. Autom. Control, Vol. 48, no. 9, PP. 1661-1666, Sep, 2003 https://doi.org/10.1109/TAC.2003.817012
  12. Y. xia and Y. Jia, 'Robust control of state delayed systems with polytopic type uncertainties via parameter-dependent Lyapunov functionals,' Syst. Control Lett., Vol. 50, PP. 183-193, 2003 https://doi.org/10.1016/S0167-6911(03)00153-1
  13. Y. He, M. Wu, J. H. She and G. P. Liu 'Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic-type uncertainties,' IEEE. Trans. Autom. Control, Vol. 49, no. 5, PP. 789-792, May, 2004 https://doi.org/10.1109/TAC.2004.828317
  14. Y.S. Lee, Y.S. Moon, W.H. Kwon and P.G. Park 'Delay-dependent robust H1 control for uncertain systems with a state-delay,' Automatica, Vol. 40, PP. 65-72, Jan, 2004 https://doi.org/10.1016/j.automatica.2003.07.004
  15. M. Wu, Y. He, J. H. She and G. P. Liu 'Delay-dependent criteria for robust stability of time-varying delay systems,' Automatica, Vol. 40, PP. 1435-1439, Mar, 2004 https://doi.org/10.1016/j.automatica.2004.03.004
  16. S. Xu and J. Lam 'Improved delay-dependent stability criteria for time-delay systems,' IEEE. Trans. Autom. Control, Vol. 50, no. 3, PP. 384-387, Mar, 2005 https://doi.org/10.1109/TAC.2005.843873
  17. M. S. Mahmoud and A. Ismail 'New results on delay-dependent control of time-delay systems,' IEEE. Trans. Autom. Control, Vol. 50, no. 1, PP. 95-100, Jan, 2005 https://doi.org/10.1109/TAC.2004.841130(410)50
  18. X. M. Zhang, M. Wu, J. H. She and Y. He 'Delay-dependent stabilization of linear systems with time-varying state and input delays,' Automatica, Vol. 41, PP. 1405-1412, May, 2005 https://doi.org/10.1016/j.automatica.2005.03.009