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Robust stability analysis of uncertain systems with two additive time-varying delay components. (English) Zbl 1173.93024
Summary: This paper is concerned with stability analysis for uncertain systems. The systems are based on a new time-delay model proposed recently, which contains multiple successive delay components in the state. The relationship between the time-varying delay and its upper bound is taken into account when estimating the upper bound of the derivative of Lyapunov functional. As a result, some less conservative stability criteria are established for systems with two successive delay components and parameter uncertainties. Numerical examples show that the proposed criteria are effective and are an improvement over some existing results in the literature.

##### MSC:
 93D09 Robust stability 93C41 Control/observation systems with incomplete information 93C23 Control/observation systems governed by functional-differential equations 93C15 Control/observation systems governed by ordinary differential equations 15A39 Linear inequalities of matrices
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##### References:
 [1] Peng, C.; Tian, Y., Delay-dependent robust stability criteria for uncertain systems with interval time-varying delay, J. comput. appl. math., 214, 2, 480-494, (2008) · Zbl 1136.93437 [2] Peng, C.; Tian, Y., Networked H∞ control of linear systems with state quantization, Inform. sci., 177, 24, 5763-5774, (2007) · Zbl 1126.93338 [3] Fridman, E.; Shaked, U., An improved stabilization method for linear time-delay systems, IEEE trans. automat. control, 47, 11, 1931-1937, (2002) · Zbl 1364.93564 [4] Fridman, E.; Shaked, U., Delay-dependent stability and H∞ control: constant and time-varying delays, Int. J. control, 76, 1, 48-60, (2003) · Zbl 1023.93032 [5] Gu, K.; Kharitonov, V.L.; Chen, J., Stability of time-delay systems, (2003), Birkhäuser Boston · Zbl 1039.34067 [6] He, Y.; Wang, Q.; Lin, C.; Wu, M., Delay-range-dependent stability for systems with time-varying delay, Automatica, 43, 2, 371-376, (2007) · Zbl 1111.93073 [7] He, Y.; Wang, Q.; Xie, L.; Lin, C., Further improvement of free-weighting matrices technique for systems with time-varying delay, IEEE trans. automat. control, 52, 2, 293-299, (2007) · Zbl 1366.34097 [8] He, Y.; Wu, M.; She J, H.; Liu, G.P., Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic-type uncertainties, IEEE trans. automat. control, 49, 5, 828-832, (2004) · Zbl 1365.93368 [9] Jing, X.; Tan, D.; Wang, Y., An LMI approach to stability of systems with severe time-delay, IEEE trans. automat. control, 49, 7, 1192-1195, (2004) · Zbl 1365.93226 [10] Kim, J., Delay and its time-derivative dependent robust stability of time-delayed linear systems with uncertainty, IEEE trans. automat. control, 46, 5, 789-792, (2001) · Zbl 1008.93056 [11] Y. Lee, Y. Moon, W. Kwon, K. Lee, Delay-dependent robust H∞ control for uncertain systems with time-varying state-delay, in: Proceedings of the 40th Conference on Decision Control, vol. 4, Orlando, FL, 2001, pp. 3208-3213. [12] Lin, C.; Wang, Q.; Lee, T., A less conservative robust stability test for linear uncertain time-delay systems, IEEE trans. automat. control, 51, 1, 87-91, (2006) · Zbl 1366.93469 [13] Xu, S.; Lam, J., Improved delay-dependent stability criteria for time-delay systems, IEEE trans. automat. control, 50, 3, 384-387, (2005) · Zbl 1365.93376 [14] Moon, Y.; Park, P.; Kwon, W.; Lee, Y., Delay-dependent robust stabilization of uncertain state-delayed systems, Int. J. control, 74, 14, 1447-1455, (2001) · Zbl 1023.93055 [15] Wu, M.; He, Y.; She, J.; Liu, G., Delay-dependent criteria for robust stability of time-varying delay systems, Automatica, 40, 8, 1435-1439, (2004) · Zbl 1059.93108 [16] H. Yan, X. Huang, H. Zhang, M. Wang, Delay-dependent robust stability criteria of uncertain stochastic systems with time-varying delay, Chaos, Solitons & Fractals, in press, doi:10.1016/j.chaos.2007.09.049. [17] Zhang, Z.; Li, C.; Liao, X., Delay-dependent robust stability analysis for interval linear time-variant systems with delays and application to delayed neural networks, Neurocomputing, 70, 16-18, 2980-2995, (2007) [18] Corless, M., Guaranteed rates of exponential convergence for exponential convergence for uncertain system, J. opt. theor. appl., 67, 3, 481-494, (1990) · Zbl 0682.93040 [19] Karimi, H.R., Robust dynamic parameter-dependent output feedback control of uncertain parameter-dependent state-delayed systems, Nonlinear dynam. syst. theor., 6, 2, 143-158, (2006) · Zbl 1135.93027 [20] Lou, X.Y.; Cui, B.T., Robust stability for nonlinear uncertain neural networks with delay, Nonlinear dynam. syst. theor., 7, 4, 369-378, (2007) · Zbl 1138.93395 [21] Lam, J.; Gao, H.; Wang, C., Stability analysis for continuous systems with two additive time-varying delay components, Syst. control lett., 56, 1, 16-24, (2007) · Zbl 1120.93362 [22] Gao, H.; Chen, T.; Lam, J., A new delay system approach to network-based control, Automatica, 44, 1, 39-52, (2008) · Zbl 1138.93375 [23] He, Y.; Liu, G.P.; Rees, D.; Wu, M., Stability analysis for neural networks with time-varying interval delay, IEEE trans. neural networks, 18, 6, 1850-1854, (2007) [24] Gahinet, P.; Nemirovskii, A.; Laub, A.; Chilali, M., LMI control toolbox user’s guide, (1995), The Math. Works Inc. Natick, MA [25] Ladyzhenskaya, O.A., Boundary value problems of mathematical physics. Moscow: nauka 1973. English transl. the boundary value problems of mathematical physics, (1985), Springer New York · Zbl 0588.35003 [26] Hale, J.K.; Verduyn Lunel, S.M., Introduction of functional differential equations, (1993), Springer New York · Zbl 1052.93028
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