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On $$H_{\infty}$$ control for linear systems with interval time-varying delay. (English) Zbl 1100.93017
Summary: This paper deals with the problem of delay-dependent robust $$H_{\infty}$$ control for linear time-delay systems with norm-bounded, and possibly time-varying, uncertainty. The time-delay is assumed to be a time-varying continuous function belonging to a given interval, which means that the lower and upper bounds for the time-varying delay are available, and no restriction on the derivative of the time-varying delay is needed, which allows the time-delay to be a fast time-varying function. Based on an integral inequality, which is introduced in this paper, and Lyapunov-Krasovskii functional approach, a delay-dependent bounded real lemma (BRL) is first established without using model transformation and bounding techniques on the related cross product terms. Then employing the obtained BRL, a delay-dependent condition for the existence of a state feedback controller, which ensures asymptotic stability and a prescribed $$H_{\infty}$$ performance level of the closed-loop systems for all admissible uncertainties, is proposed in terms of a linear matrix inequality (LMI). A numerical example is also given to illustrate the effectiveness of the proposed method.

##### MSC:
 93B36 $$H^\infty$$-control 93C23 Control/observation systems governed by functional-differential equations 93D15 Stabilization of systems by feedback
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##### References:
 [1] Cao, Y.-Y.; Sun, Y.-X.; Lam, J., Delay dependent robust $$H_\infty$$ control for uncertain systems with time varying delays, IEE Proceedings: control theory and applications, 143, 338-344, (1998) [2] Chow, M.Y.; Tipsuwan, Y., Network-based control systems: A tutorial, (), 1593-1602 [3] 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 [4] Gao, H.; Wang, C., Comments and further results on “A descriptor system approach to $$H_\infty$$ control of linear time-delay systems”, IEEE transactions on automatic control, 48, 520-525, (2003) · Zbl 1364.93211 [5] Gu, K., Discretization schemes for Lyapunov-krasovskii functions in time-delay systems, Kybernetica, 37, 479-504, (2001) · Zbl 1265.93176 [6] Gu, K.; Han, Q.-L.; Luo, A.C.J.; Niculescu, S.-I., Discretized Lyapunov functional for systems with distributed delay and piecewise constant coefficient, International journal of control, 74, 734-744, (2001) [7] Gu, K.; Kharitonov, V.L.; Chen, J., Stability of time-delay systems, (2003), Birkhauser Boston · Zbl 1039.34067 [8] Han, Q.-L., Stability criteria for a class of linear neutral systems with time-varying discrete and distributed delays, IMA journal of mathematical control and information, 20, 371-386, (2003) · Zbl 1046.93039 [9] 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 [10] 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) [11] Han, Q.-L., & Mehdi, D. (1999). Robust $$H_\infty$$ controller synthesis for uncertain systems with multiple time-varying delays: An LMI approach. Proceedings of the 14th IFAC world congress, Vol. C (pp. 271-276). Beijing, P.R. China. [12] He, Y.; Wu, M.; She, J.H.; Liu, G.P., Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays, Systems and control letters, 51, 57-65, (2004) · Zbl 1157.93467 [13] Li, X.; de Souza, C.E., Criteria for robust stability and stabilization of uncertain linear systems with time delays, Automatica, 33, 1657-1662, (1997) [14] Mehdi, D.; Boukas, E.-K.; Liu, Z.K., Dynamical systems with multiple time-varying delays: stability and stabilizability, Journal of optimization theory and applications, 113, 537-565, (2002) · Zbl 1006.93064 [15] Xu, S.; Lam, J.; Yang, C., $$H_\infty$$ and positive real control for linear neutral delay systems, IEEE transactions on automatic control, 46, 1321-1326, (2001) · Zbl 1008.93033 [16] Yu, L.; Chu, J.; Su, H., Robust memoryless $$H_\infty$$ controller design for linear time-delay systems with norm-bounded time-varying uncertainty, Automatica, 32, 1759-1762, (1996) · Zbl 0875.93102 [17] Yue, D.; Han, Q.-L.; Peng, C., State feedback controller design of networked control systems, IEEE transactions on circuits and systems—II: express briefs, 51, 640-644, (2004) [18] Zhang, W.; Branicky, M.S.; Phillips, S.M., Stability of networked control systems, IEEE control systems magazine, 21, 84-99, (2001) [19] Zhang, X.-M.; Wu, M.; She, J.-H.; He, Y., Delay-dependent stabilization of linear systems with time-varying state and input delays, Automatica, 41, 1405-1412, (2005) · Zbl 1093.93024
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