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Delay-range-dependent stability for systems with time-varying delay. (English) Zbl 1111.93073
Summary: This paper is concerned with the stability analysis for systems with time-varying delay in a range. An appropriate type of Lyapunov functionals is proposed to investigate the delay-range-dependent stability problem. The present results may improve the existing ones due to a method to estimate the upper bound of the derivative of Lyapunov functional without ignoring some useful terms and the introduction of additional terms into the proposed Lyapunov functional, which take into account the range of delay. Numerical examples are given to demonstrate the effectiveness and the benefits of the proposed method.

##### MSC:
 93D30 Scalar and vector Lyapunov functions 93C05 Linear control systems 93C15 Control systems governed by ODE
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##### References:
 [1] Fridman, E.; Shaked, U.: An improved stabilization method for linear time-delay systems. IEEE transactions on automatic control 47, 1931-1937 (2002) [2] Fridman, E.; Shaked, U.: Delay-dependent stability and H$\infty$control: constant and time-varying delays. International journal of control 76, 48-60 (2003) · Zbl 1023.93032 [3] Gao, H.; Lam, J.; Wang, C.; Wang, Y.: Delay-dependent output-feedback stabilisation of discrete-time systems with time-varying state delay. IEE Proceedings--control theory and applications 151, 691-698 (2004) [4] Gu, K.; Kharitonov, V. L.; Chen, J.: Stability of time-delay systems. (2003) · Zbl 1039.34067 [5] Gu, K.; Niculescu, S. -I.: Additional dynamics in transformed time delay systems. IEEE transactions on automatic control 45, 572-575 (2000) · Zbl 0986.34066 [6] Hale, J. K.; Lunel, S. M. Verduyn: Introduction of functional differential equations. (1993) · Zbl 0787.34002 [7] Han, Q. L.: On robust stability of neutral systems with time-varying discrete delay and norm-bounded uncertainty. Automatica 40, 1087-1092 (2004) · Zbl 1073.93043 [8] Han, Q. L.: A descriptor system approach to robust stability of uncertain neutral systems with discrete and distributed delays. Automatica 40, 1791-1796 (2004) · Zbl 1075.93032 [9] Han, Q. L.; Gu, K.: Stability of linear systems with time-varying delay: A generalized discretized Lyapunov functional approach. Asian journal of control 3, 170-180 (2001) [10] He, Y.; Wu, M.; She, J. H.; Liu, G. P.: Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays. Systems & control letters 51, 57-65 (2004) · Zbl 1157.93467 [11] 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 transactions on automatic control 49, 828-832 (2004) [12] Jiang, X.; Han, Q. L.: On H$\infty$control for linear systems with interval time-varying delay. Automatica 41, 2099-2106 (2005) · Zbl 1100.93017 [13] Kharitonov, V. L.; Niculescu, S. -I.: On the stability of linear systems with uncertain delay. IEEE transactions on automatic control 48, 127-132 (2003) [14] Michiels, W.; Assche, V. V.; Niculescu, S. -I.: Stabilization of time-delay systems with a controlled time-varying delay and applications. IEEE transactions on automatic control 50, 493-504 (2005) [15] Moon, Y. S.; Park, P.; Kwon, W. H.; Lee, Y. S.: Delay-dependent robust stabilization of uncertain state-delayed systems. International journal of control 74, 1447-1455 (2001) · Zbl 1023.93055 [16] Park, P.: A delay-dependent stability criterion for systems with uncertain time-invariant delays. IEEE transactions on automatic control 44, 876-877 (1999) · Zbl 0957.34069 [17] Wu, M.; He, Y.; She, J. H.: New delay-dependent stability criteria and stabilizing method for neutral systems. IEEE transactions on automatic control 49, 2266-2271 (2004) [18] Wu, M.; He, Y.; She, J. H.; Liu, G. P.: Delay-dependent criteria for robust stability of time-varying delay systems. Automatica 40, 1435-1439 (2004) · Zbl 1059.93108 [19] Xu, S.; Lam, J.: Improved delay-dependent stability criteria for time-delay systems. IEEE transactions on automatic control 50, 384-387 (2005) [20] Xu, S.; Lam, J.; Zou, Y.: Simplified descriptor system approach to delay-dependent stability and performance analyses for time-delay systems. IEE Proceedings--control theory and applications 152, 147-151 (2005)