Abstract
In this paper, the problem of delay-dependent stability for uncertain dynamic systems with time-varying delays is considered. The parameter uncertainties are assumed to be norm-bounded. Using a new augmented Lyapunov functional, novel delay-dependent stability criteria for such systems are established in terms of LMIs (linear matrix inequalities), which can be solved easily by the application of convex optimization algorithms. Three numerical examples are given to show the superiority of the proposed method.
Similar content being viewed by others
References
Udwadia, F.E., Weber, H.I., Leitmann, G.: Dynamical Systems and Control. Chapman & Hall/CRC Press, London, Boca Raton (2004)
Leitmann, G., Udwadia, F.E., Kryazhimskii, A.V.: Dynamics and Control. CRC Press, Boca Raton (1999)
Hale, J., Lunel, S.M.V.: Introduction to Functional Differential Equations. Springer, New York (1993)
Kolmanovskii, V.B., Myshkis, A.: Applied Theory of Functional Differential Equations. Kluwer Academic, Dordrecht (1992)
Phohomsiri, P., Udwadia, F.E., Von Bremmen, H.: Time-delayed positive velocity feedback control design for active control of structures,. J. Eng. Mech. 132, 690–703 (2006)
Udwadia, F.E., Hosseini, M.A.M., Chen, Y.H.: Robust control of uncertain systems with time varying delays in control input. In: Proceedings of the American Control Conference, Albuquerque, New Mexico, pp. 3641–3644 (1997)
Gu, K.: An integral inequality in the stability problem of time-delay systems. In: Proceedings of the IEEE Conf. Decision Control, Sydney, Australia, pp. 2805–2810 (2000)
Gu, K.: Discretized Lyapunov functional for uncertain systems with multiple time-delay. Int. J. Control 72, 1436–1445 (1999)
Zhu, X.L., Wang, G.H.: Networked-based robust H ∞ control of continuous-time systems with uncertainty. Asian J. Control 11, 21–30 (2009)
Lien, C.H.: Guaranteed cost observer-based controls for a class of uncertain neutral time-delay systems. J. Optim. Theory Appl. 126, 137–156 (2005)
Kwon, O.M., Park, J.H.: Matrix inequality approach to novel stability criterion for time delay systems with nonlinear uncertainties. J. Optim. Theory Appl. 126, 643–656 (2005)
Kwon, O.M., Park, J.H.: Exponential stability for time-delay systems with interval time-varying delays and nonlinear perturbations. J. Optim. Theory Appl. 139, 277–293 (2008)
Park, J.H.: Convex optimization approach to dynamic output feedback control for delay differential systems of neutral type. J. Optim. Theory Appl. 127, 411–423 (2005)
Parlakçi, N.N.A.: Extensively augmented Lyapunov functional approach for the stability of neutral time-delay systems. IET Control Theory Appl. 2, 431–436 (2008)
Park, P.G.: A delay-dependent stability criterion for systems with uncertain linear state-delayed systems. IEEE Trans. Automat. Control 35, 876–877 (1999)
Xu, S., Lam, J.: Improved delay-dependent stability criteria for time-delay systems. IEEE Trans. Automat. Control 50, 384–387 (2005)
Yue, D., Won, S., Kwon, O.: Delay dependent stability of neutral systems with time delay: an LMI approach. IEE Proc., Control Theory Appl. 150, 23–27 (2003)
Kwon, O.M., Park, J.H.: On improved delay-dependent robust control for uncertain time-delay systems. IEEE Trans. Automat. Control 49, 1991–1995 (2004)
Li, T., Guo, L., Wu, L.: Simplified approach to the asymptotical stability of linear systems with interval time-varying delay. IET Control Theory Appl. 3, 252–260 (2009)
Parlakçi, N.N.A.: Robust stability of uncertain time-varying state-delayed systems. IEE Proc., Control Theory Appl. 153, 469–477 (2006)
Qian, W., Cong, S., Sun, Y., Fei, S.: Novel robust stability criteria for uncertain systems with time-varying delay. Appl. Math. Comput. 215, 866–872 (2009)
Boyd, S., Ghaoui, L. El, Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia (1994)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by F.E. Udwadia.
This research was supported by the MKE (The Ministry of Knowledge Economy), Korea, under the ITRC (Information Technology Research Center) support program supervised by the IITA (Institute for Information Technology Advancement) (IITA-2009-C1090-0904-0007). Also, the research of J.H. Park was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0009373).
Rights and permissions
About this article
Cite this article
Kwon, O.M., Lee, S.M. & Park, J.H. Linear Matrix Inequality Approach to New Delay-Dependent Stability Criteria for Uncertain Dynamic Systems with Time-Varying Delays. J Optim Theory Appl 149, 630–646 (2011). https://doi.org/10.1007/s10957-011-9795-5
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-011-9795-5