Abstract
This paper studies the problem of the robustly exponential stabilization for uncertain Markovian jump systems with mode-dependent time-varying state delays. The contribution of this paper is two-fold. Firstly, by constructing a modified Lyapunov functional and using free-weighting matrices technique, some delay-dependent robustly exponential stability criteria of such systems are obtained in terms of linear matrix inequalities (LMIs), which are less conservative than some existing ones. Secondly, a state feedback controller is designed, which can guarantee the robustly exponential stability of the uncertain closed-loop systems. Some illustrative numerical examples are given to demonstrate the reduced conservatism and applicability of the obtained results.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
Boyd, B., Ghaoui, L.E., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in Systems and Control Theory. SIAM, Philadelphia (1994)
Boukas, E.K., Liu, Z.K., Liu, G.X.: Delay-dependent robust stability and H ∞ control of jump linear systems with time-delay. Int. J. Control 74(4), 329–340 (2001)
Boukas, E.K., Liu, Z.K.: Robust stability and stabilizability of Markov jump linear uncertain systems with mode-dependent time delays. J. Optim. Theory Appl. 109(3), 587–600 (2001)
Chen, Y., Xue, A., Zhao, X.: Novel stability analysis for delay Markovian jump systems. In: Proc. ACC the 7th, Hong Kong, China, 27–29 August (2009)
Chen, B., Li, H., Shi, P., Lin, C., Zhou, Q.: Delay-dependent stability analysis and controller synthesis for Markovian jump systems with state and input delays. Inf. Sci. 179(16), 2851–2860 (2009)
Cao, Y., Lam, J.: Robust H ∞ control of uncertain Markovian jump systems with time-delay. IEEE Trans. Autom. Control 45(1), 77–83 (2000)
Cao, Y., Lam, J., Hu, L.: Delay-dependent stochastic stability and H ∞ analysis for time-delay systems with Markovian jumping parameters. J. Franklin Inst. 340(6–7), 423–434 (2003)
Cao, Y., Yan, W., Xue, A.: Improve delay-dependent stability conditions and H ∞ analysis for time-delay systems. In: Proc. 43rd IEEE Conf. Decis. Control, Atlanties, Bahamas, pp. 4527–4532 (2004)
Chen, W., Guan, Z., Lu, X.: Delay-dependent output feedback stabilization of Markovian jump system with time-delay. IEE Proc., Control Theory Appl. 151(5), 561–566 (2004)
Fujisaki, H.: On correlation values of m-phase spreading sequences of Markov chains. IEEE Trans. Circuits Syst. I, Regul. Pap. 49(12), 1745–1750 (2002)
Fei, Z., Gao, H., Shi, P.: New results on stabilization of Markovian jump systems with time delay. Automatica 45(10), 2300–2306 (2009)
Gu, K., Kharitonov, V.L., Chen, J.: Stability of Time-Delay Systems. Birkhauser, Boston (2003)
Gouaisbaut, F., Peaucelle, D.: Delay-dependent stability analysis of linear time delay systems. In: IFAC Workshop on Time Delay System (2006)
He, Y., Wu, M., She, J.: Delay-dependent exponential stability for delayed neural networks with time-varying delay. IEEE Trans. Circuits Syst. II, Express Briefs 53(7), 230–234 (2006)
He, Y., Wu, M., She, J., Liu, G.: Delay-dependent criteria for robust stability criteria for uncertain neutral systems with mixed delays. Syst. Control Lett. 51(1), 57–65 (2004)
Lian, J., Zhao, J., Dimirovski, G.M.: Robust H ∞ sliding mode control for a class of uncertain switched delay systems. Int. J. Syst. Sci. 40(8), 855–866 (2009)
Lian, J., Zhao, J.: Adaptive variable structure control for uncertain switched delay systems. Circuits Syst. Signal Process. 29(6), 1089–1102 (2010)
Ma, L., Da, F., Zhang, K.: Mean-square exponential stability of stochastic Markovian jump systems with mode-dependent time-varying delays. In: Proc. 48th ICDC and CCC, Shanghai, P.R. China, pp. 16–18 (2009)
Ma, L., Da, F., Wu, L.: Delayed-state-feedback exponential stabilization of stochastic Markovian jump systems with mode-dependent time-varying state delays. Acta Autom. Sin. 36(11), 1601–1608 (2010)
Moon, Y.S., Park, P., Kwon, W.H.: Robust stabilization of uncertain input-delayed systems using reduction method. Automatica 37(2), 307–312 (2001)
Mao, X., Yuan, C.: Stochastic Differential Equations with Markovian Switching. Imperial College Press, London (2006)
Sun, M., Lam, J., Xu, S., Zou, Y.: Robust exponential stabilization for Markovian jump systems with mode-dependent input delay. Automatica 43(10), 1799–1807 (2007)
Shu, Z., Lam, J., Xu, S.: Robust stabilization of Markovian delay systems with delay-dependent exponential estimates. Automatica 42(11), 2001–2008 (2006)
Shen, H., Xu, S., Song, X., Chu, Y.: Delay-dependent H ∞ filtering for stochastic systems with Markovian switching and mixed mode-dependent delays. Nonlinear Anal. Hybrid Syst. 4(1), 122–133 (2010)
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)
Wang, Z., Lam, J., Liu, X.: Exponential filtering for uncertain Markovian jump time-delay systems with nonlinear disturbances. IEEE Trans. Cir. Syst. Part II: Express Briefs 51(5), 262–268 (2004)
Wu, Z., Su, H., Chu, J.: Delay-dependent H ∞ filtering for singular Markovian jump time-delay systems. Signal Process. 90(6), 1815–1824 (2010)
Wang, M., Zhao, J., Dimirovski, G.M.: Stabilisation and L 2-gain analysis for a class of uncertain switched non-linear systems. Int. J. Syst. Sci. 40(10), 1077–1085 (2009)
Xu, S., Lam, J., Mao, X.: Delay-dependent H ∞ control and filtering for uncertain Markovian jump system with time-varying delays. IEEE Trans. Circuits Syst. I, Regul. Pap. 54(9), 2070–2077 (2007)
Xu, S., Chen, T., Lam, J.: Robust H ∞ filtering for uncertain Markovian jump systems with mode-dependent time delays. IEEE Trans. Autom. Control 48(5), 900–907 (2003)
Xiong, J., Lam, J.: Stabilization of discrete-time Markovian jump linear systems via time-delayed controller. Automatica 42(5), 747–753 (2006)
Yue, D., Han, Q.: Delay-dependent exponential stability of stochastic systems with time-varying delay, nonlinearity, and Markovian switching. IEEE Trans. Autom. Control 50(2), 217–222 (2005)
Zhang, Y., He, Y., Wu, M., Zhang, J.: Stabilization for Markovian jump systems with partial information on transition probability based on free-connection weighting matrices. Automatica 47(1), 79–84 (2010)
Zhao, X., Ling, M., Zeng, Q.: Delay-dependent stability criterion and H ∞ analysis for Markovian jump systems with time-varying delays. Asian J. Control 13(2), 1–8 (2011)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chen, H., Zhu, C., Hu, P. et al. Delayed-state-feedback exponential stabilization for uncertain Markovian jump systems with mode-dependent time-varying state delays. Nonlinear Dyn 69, 1023–1039 (2012). https://doi.org/10.1007/s11071-012-0324-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-012-0324-3