Parameter-dependent \(H_{\infty}\) control for time-varying delay polytopic systems. (English) Zbl 1205.93134

Summary: This paper addresses the robust stabilization and \(H_{\infty}\) control problem for a class of linear polytopic systems with continuously distributed delays. The control objective is to design a robust \(H_{\infty}\) controller that satisfies some exponential stability constraints on the closed-loop poles. Using improved parameter-dependent Lyapunov Krasovskii functionals, new delay-dependent conditions for the robust \(H_{\infty}\) control are established in terms of linear matrix inequalities.


93D21 Adaptive or robust stabilization
93B36 \(H^\infty\)-control
93C41 Control/observation systems with incomplete information


LMI toolbox
Full Text: DOI


[1] Zames, G.: Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms, and approximate inverses. IEEE Trans. Automat. Control 26, 301–320 (1981) · Zbl 0474.93025
[2] Francis, B.A.: A Course in H Control Theory. Springer, Berlin (1987) · Zbl 0624.93003
[3] Petersen, I.R., Ugrinovskii, V.A., Savkin, A.V.: Robust Control Design Using H Methods. Springer, London (2000)
[4] Ravi, R., Nagpal, K.M., Khargonekar, P.P.: H control of linear time-varying systems: A state-space approach. SIAM J. Control Optim. 29, 1394–1413 (1991) · Zbl 0741.93017
[5] Kolmanovskii, V.B., Shaikhet, L.E.: Control of Systems with Aftereffect. Translations of Mathematical Monographs, vol. 157. Am. Math. Soc., Providence (1996)
[6] Udwadia, F.E., von Bremen, H., Phohomsiri, P.: Time-delayed control design for active control of structures: Principles and applications. Struct. Control Health Monit. 14, 27–61 (2007)
[7] Udwadia, F.E., Hosseini, M., Chen, Y.: Robust control of uncertain systems with time-varying delays in control input. In: Proc. of the American Control Conference, USA, pp. 3840–3845 (1997)
[8] Fridman, E., Shaked, U.: Delay-dependent stability and H control: constant and time-varying delays. Int. J. Control 76, 48–60 (2003) · Zbl 1023.93032
[9] Kwon, O.M., Park, J.H.: Robust H filtering for uncertain time-delay systems: Matrix inequality approach. J. Optim. Theory Appl. 129, 309–324 (2006) · Zbl 1136.93043
[10] Phat, V.N., Nam, P.T.: Robust stabilization of linear systems with delayed state and control. J. Optim. Theory Appl. 140, 287–299 (2009) · Zbl 1159.93027
[11] Phat, V.N., Ha, Q.P.: H 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
[12] 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) · Zbl 1367.93176
[13] Zhang, J., Xia, Y., Shi, P.: Parameter-dependent robust H filtering for uncertain discrete-time systems. Automatica 45, 560–565 (2009) · Zbl 1158.93406
[14] Colaneri, P., Geromel, J.C.: Parameter dependent Lyapunov function for time-varying polytopic systems. In: Proc. Amer. Contr. Conf., Portland OR, pp. 604–608 (2005)
[15] 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
[16] Nam, P.T., Phat, V.N.: Robust exponential stability and stabilization of linear uncertain polytopic time-delay systems. J. Control Theory Appl. 6, 163–170 (2008)
[17] Niculescu, S.L.: H memoryless control with stability constraint for time-delay systems: an LMI approach. IEEE Trans. Automat. Control 43, 739–743 (1998) · Zbl 0911.93031
[18] Mondié, S., Kharitonov, V.L.: Exponential estimates for retarded time-delay systems: An LMI approach. IEEE Trans. Automat. Control 50, 268–273 (2005) · Zbl 1365.93351
[19] Boyd, S., Ghaoui, L.E., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia (1994) · Zbl 0816.93004
[20] Gu, K.: An integral inequality in the stability problem of time-delay systems. In: Proc. of the 39th IEEE Conf. on Decision and Control, Sydney, Australia, pp. 2805–2810 (2000)
[21] Gahinet, P., Nemirovskii, A., Laub, A.J., Chilali, M.: LMI Control Toolbox: For Use with Matlab. The Math Work, Inc, Natick (1995)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.