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$H_\infty $ control for nonlinear time-varying delay systems with convex polytopic uncertainties. (English) Zbl 1189.93041
Summary: This paper investigates $H_\infty $ control for a class of nonlinear systems with time-varying delays and convex polytopic uncertainties. A new type of parameter-dependent Lyapunov-Krasovskii functional is introduced to derive delay-dependent sufficient conditions for the $H_\infty $ optimal control with exponential stability. All the conditions developed in this paper are formulated in terms of linear matrix inequalities. Finally, a numerical example shows the effectiveness of the proposed methodology.

93C20Control systems governed by PDE
93C41Control problems with incomplete information
93C15Control systems governed by ODE
34H05ODE in connection with control problems
Full Text: DOI
[1] Agarwal, R. P.; Bohner, M.; Li, W. T.: Nonoscillation and oscillation: theory of functional differential equations, (2004) · Zbl 1068.34002
[2] Chukwu, E. N.: Stability and time-optimal control of hereditary systems, (1992) · Zbl 0751.93067
[3] Lakshmikantham, V.; Leela, S.: A technique in stability theory for delay differential equations, Nonlinear anal. TMA 3, 317-323 (1979) · Zbl 0413.34074 · doi:10.1016/0362-546X(79)90021-X
[4] Lakshmikantham, V.; Leela, S.; Martynyuk, M.: Stability analysis of nonlinear systems, (1989) · Zbl 0676.34003
[5] Liu, X.: Delay-dependent H$\infty $control for uncertain fuzzy systems with time-varying delays, Nonlinear anal. TMA 68, 1352-1361 (2008) · Zbl 1137.93021 · doi:10.1016/j.na.2006.12.029
[6] Phat, V. N.; Niamsup, P.: Stability analysis for a class of functional differential equations and applications, Nonlinear anal. TMA 71, 6265-6275 (2009) · Zbl 1201.34121 · doi:10.1016/j.na.2009.06.028
[7] Francis, B. A.: A course in H$\infty $Control theory, (1987)
[8] Petersen, I. R.; Ugrinovskii, V. A.; Savkin, A. V.: Robust control design using H$\infty $Methods, (2000)
[9] P. Colaneri, J.C. Geromel, Parameter dependent Lyapunov function for time-varying polytopic systems, in: Proc. Amer. Contr. Conf., Portland, OR, vol. 6, 2005, pp. 604--608.
[10] A.G. Spark, Analysis of affine parameter-varying systems using parameter dependent Lyapunov functions, in: Proc. of 36th IEEE CDC, California, USA, December, 1997, pp. 990--991.
[11] Ravi, R.; Nagpal, K. M.; Khargonekar, P. P.: H$\infty $control of linear time-varying systems: A state-space approach, SIAM J. Control optim. 29, 1394-1413 (1991) · Zbl 0741.93017 · doi:10.1137/0329071
[12] Cao, Y. Y.; Sun, Y. X.; Lam, J.: Delay dependent robust H$\infty $control for uncertain systems with time-varying delays, IET contr. Theory appl. 145, 338-344 (1988)
[13] Fridman, E.; Shaked, U.: Finite horizon H$\infty $state-feedback control of continuous-time systems with state delays, IEEE trans. Automat. control 45, 2406-2411 (2000) · Zbl 0990.93026 · doi:10.1109/9.895584
[14] Kanoh, H.; Itoh, T.; Abe, N.: Nonlinear H$\infty $control for heat exchangers controlled by the manipulation of flow rate, Nonlinear anal. TMA 30, 2237-2248 (1997) · Zbl 0898.93021 · doi:10.1016/S0362-546X(97)00135-1
[15] Phat, V. N.; Ha, Q. P.: H$\infty $control and exponential stability for a class of nonlinear non-autonomous systems with time-varying delay, J. optim. Theory appl. 142, 603-618 (2009) · Zbl 1178.93047 · doi:10.1007/s10957-009-9512-9
[16] Xu, S.; Lam, J.; Xie, L.: New results on delay-dependent robust H$\infty $control for systems with time varying delays, Automatica 42, 43-348 (2006) · Zbl 1099.93010 · doi:10.1016/j.automatica.2005.09.013
[17] Phat, V. N.; Nam, P. T.: Exponential stability and stabilization of uncertain linear time-varying systems using parameter dependent Lyapunov function, Int. J. Control 8, 1333-1341 (2007) · Zbl 1133.93358 · doi:10.1080/00207170701338867
[18] Park, Ju H.; Kwon, O.: Novel stability criterion of time delay systems with nonlinear uncertainties, Appl. math. Lett. 18, 683-688 (2005) · Zbl 1089.34549 · doi:10.1016/j.aml.2004.04.013
[19] Park, Ju H.; Kwon, O.; Won, S.: LMI approach to robust H$\infty $filtering for neutral delay differential systems, Appl. math. Comput. 150, 235-244 (2004) · Zbl 1040.93067 · doi:10.1016/S0096-3003(03)00223-6
[20] Park, Ju H.: Design of robust H$\infty $filter for a class of neutral systems: LMI optimization approach, Math. comput. Simulation 70, 99-109 (2005) · Zbl 1093.93010 · doi:10.1016/j.matcom.2005.05.002
[21] Park, Ju H.: Novel robust stability criterion for a class of neutral systems with mixed delays and nonlinear perturbations, Appl. math. Comput. 161, 413-421 (2005) · Zbl 1065.34076 · doi:10.1016/j.amc.2003.12.036
[22] Mori, T.; Kokame, H.: A parameter-dependent Lyapunov function for a polytope of matrices, IEEE trans. Automat. control 45, 1516-1519 (2000) · Zbl 0988.93065 · doi:10.1109/9.871762
[23] He, Y.; Wu, M.; She, J. H.; Liu, G. P.: Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic-type uncertainties, IEEE trans. Automat. control 49, 828-832 (2003)
[24] Gao, H.; Meng, X.; Chen, T.: A parameter-dependent approach to robust filtering for time-delay systems, IEEE trans. Automat. control 53, 2420-2425 (2008)
[25] Kwon, O. M.; Park, J. H.: Robust H$\infty $filtering for uncertain time-delay systems: matrix inequality approach, J. optim. Theory appl. 129, 309-324 (2006) · Zbl 1136.93043 · doi:10.1007/s10957-006-9064-1
[26] Zhang, J.; Xia, Y.; Shi, P.: Parameter-dependent robust H$\infty $filtering for uncertain discrete-time systems, Automatica 45, 560-565 (2009) · Zbl 1158.93406 · doi:10.1016/j.automatica.2008.09.005
[27] Boyd, S.; Ghaoui, L. E.; Feron, E.; Balakrishnan, V.: Linear matrix inequalities in system and control theory, (1994) · Zbl 0816.93004
[28] Hale, J. K.; Lunel, Sm. Verduyn: Introduction to functional differential equations, (1993)
[29] Mondie, S.; Kharitonov, V. L.: Exponential estimates for retarded time-delay systems: an LMI approach, IEEE trans. Automat. control 50, 268-273 (2005)