A sliding mode approach to \(H_{\infty }\) synchronization of master-slave time-delay systems with Markovian jumping parameters and nonlinear uncertainties. (English) Zbl 1254.93046

Summary: In this paper, a sliding-mode approach is proposed for exponential \(H_{\infty }\) synchronization problem of a class of master-slave time-delay systems with both discrete and distributed time-delays, norm-bounded nonlinear uncertainties and Markovian switching parameters. Using an appropriate Lyapunov-Krasovskii functional, some delay-dependent sufficient conditions and a synchronization law including the master-slave parameters is established for designing a delay-dependent mode-dependent sliding mode exponential \(H_{\infty }\) synchronization control law in terms of linear matrix inequalities. The controller guarantees the \(H_{\infty }\) synchronization of the two coupled master and slave systems regardless of their initial states. Two numerical examples are given to show the effectiveness of the method.


93B12 Variable structure systems
60J75 Jump processes (MSC2010)
93C10 Nonlinear systems in control theory


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[1] Utkin, V.I., Variable structure systems with sliding modes, IEEE transactions on automatic control, 22, 2, 212-222, (1977) · Zbl 0382.93036
[2] Utkin, V.I.; Guldner, J.; Shi, J.X., Sliding mode control in electromechanical systems, (1999), Taylor and Francis London, U.K
[3] Xia, Y.; Zhu, Z.; Fu, M., Back-stepping sliding mode control for missile systems based on an extended state observer, IET control theory and applications, 5, 1, 93-102, (2011)
[4] Zhang, J.; Shi, P.; Xia, Y., Robust adaptive sliding mode control for fuzzy systems with mismatched uncertainties, IEEE transactions on fuzzy systems, 18, 4, 700-711, (2010)
[5] Basin, M.; Calderon-Alvarez, D., Sliding mode regulator as solution to optimal control problem for non-linear polynomial systems, Journal of the franklin institute, 347, 6, 910-922, (2010) · Zbl 1200.49033
[6] Boukas, E.K.; Benzaouia, A., Stability of discrete-time linear systems with Markovian jumping parameters and constrained control, IEEE transactions on automatic control, 47, 3, 516-521, (2002) · Zbl 1364.93843
[7] Boukas, E.K.; Zhang, Q.; Yin, G., Robust production and maintenance planning in stochastic manufacturing systems, IEEE transactions on automatic control, 40, 6, 1098-1102, (1995) · Zbl 0837.90054
[8] Cao, Y.; Lam, J., Stochastic stabilizability and H∞ control for discrete-time jump linear systems with time delay, Journal of franklin institute, 336, 8, 1263-1281, (1999) · Zbl 0967.93095
[9] Shi, P.; Boukas, E.K., H∞ control for Markovian jumping linear systems with parametric uncertainty, Journal of optimization theory and applications, 95, 1, 75-99, (1997) · Zbl 1026.93504
[10] Mariton, M., Jump linear systems in automatic control, (1990), Marcel Dekker Inc New York
[11] Mahmoud, M.S.; Shi, P., Robust control for Markovian jump linear discrete-time systems with unknown nonlinearities, IEEE transactions on circuits and systems—I, 49, 4, 538-542, (2002) · Zbl 1368.93154
[12] Malmoud, M.S.; Shi, P.; Shi, Y., H∞ and robust control of interconnected systems with Markovian jump parameters, Journal of discrete and continuous dynamical systems, series B, 5, 2, 365-384, (2005) · Zbl 1075.93042
[13] Mahmoud, M.S.; Shi, P., Robust stability, stabilization and H∞ control of time-delay systems with Markovian jump parameters, Journal of robust and nonlinear control, 13, 8, 755-784, (2003) · Zbl 1029.93063
[14] Mahmoud, M.S.; Shi, P., Output feedback stabilization and disturbance attenuation of time-delay jumping systems, IMA journal of mathematical control and information, 20, 2, 179-199, (2003) · Zbl 1087.93049
[15] Zhang, L.; Boukas, E., Stability and stabilization of Markovian jump linear systems with partly unknown transition probability, Automatica, 45, 2, 463-468, (2009) · Zbl 1158.93414
[16] Zhang, L.; Lam, J., Necessary and sufficient conditions for analysis and synthesis of Markov jump linear systems with incomplete transition descriptions, IEEE transactions on automatic control, 55, 7, 1695-1701, (2010) · Zbl 1368.93782
[17] Hale, J.; Verduyn Lunel, S.M., Introduction to functional differential equations, (1993), Springer Verlag New York, NY · Zbl 0787.34002
[18] Karimi, H.R.; Zapateiro, M.; Luo, N., A linear matrix inequality approach to robust fault detection filter design of linear systems with mixed time-varying delays and nonlinear perturbations, Journal of franklin institute, 347, 957-973, (2010) · Zbl 1201.93033
[19] Wang, Z.; Lam, J.; Liu, X.H., Nonlinear filtering for state delayed systems with Markovian switching, IEEE transactions on signal processing, 51, 2321-2328, (2003) · Zbl 1369.94314
[20] Xu, S.; Chen, T.; Lam, J., Robust H∞ filtering for uncertain Markovian jump systems with mode-dependent time delays, IEEE transactions on automatic control, 48, 5, 900-907, (2003) · Zbl 1364.93816
[21] Wang, Z.; Lam, J.; Liu, X., Exponential filtering for uncertain Markovian jump time-delay systems with nonlinear disturbances, IEEE transactions on circuits and systems, II: express briefs, 51, 5, 262-268, (2004)
[22] Mahmoud, M.S.; Shi, P., Robust Kalman filtering for continuous time-lag systems with Markovian jump parameters, IEEE transactions on circuits and systems—I, 50, 98-105, (2003) · Zbl 1368.93725
[23] Yang, H.; Xia, Y.; Shi, P., Observer-based sliding mode control for a class of discrete systems via delta operator approach, Journal of franklin institute, 347, 7, 1199-1213, (2010) · Zbl 1202.93066
[24] Karimi, H.R., Robust delay-dependent H∞ control of uncertain time-delay systems with mixed neutral, discrete and distributed time-delays and Markovian switching parameters, IEEE transactions on circuits and systems I, 58, 8, 1910-1923, (2011)
[25] D. Chen, W. Zhang, Sliding mode control of uncertain neutral stochastic systems with multiple delays, Mathematical Problems in Engineering 2008 (2008), Article ID 761342, 1-9, doi:10.1155/2008/761342.
[26] Xia, Y.; Jia, Y., Robust sliding-mode control for uncertain time-delay systems: an LMI approach, IEEE transactions on automatic control, 48, 1086-1092, (2003) · Zbl 1364.93608
[27] Shi, P.; Xia, Y.; Liu, G.P.; Rees, D., On designing of sliding-mode control for stochastic jump systems, IEEE transactions on automatic control, 51, 97-103, (2006) · Zbl 1366.93682
[28] Zhang, L.; Boukas, E., Mode-dependent H∞ filtering for discrete-time Markovian jump linear systems with partly unknown transition probability, Automatica, 45, 6, 1462-1467, (2009) · Zbl 1166.93378
[29] Zhang, L., H∞ estimation for piecewise homogeneous Markov jump linear systems, Automatica, 45, 11, 2570-2576, (2009) · Zbl 1180.93100
[30] Lin, Z.; Xia, Y.; Shi, P.; Wu, H., Robust sliding mode control for uncertain linear discrete systems independent of time-delay, International journal of innovative computing information and control, 7, 2, 869-881, (2011)
[31] Y. Niu, D.W.C. Ho, Stabilization of discrete-time stochastic systems via sliding mode technique, Journal of the Franklin Institute, in Press, doi:10.1016/j.jfranklin.2011.06.005. · Zbl 1254.93156
[32] Mahmoud, M.S.; Shi, P., Methodologies for control of jumping time-delay systems, (2003), Kluwer Academic Publishers Amsterdam
[33] Wu, L.; Shi, P.; Gao, H., State estimation and sliding mode control of Markovian jump singular systems, IEEE transactions on automatic control, 55, 5, 1213-1219, (2010) · Zbl 1368.93696
[34] Wu, L.; Ho, D.W.C., Sliding mode control of singular stochastic hybrid systems, Automatica, 46, 779-783, (2010) · Zbl 1193.93184
[35] Niu, Y.; Ho, D.W.C.; Wang, X., Sliding mode control for Itô stochastic systems with Markovian switching, Automatica, 43, 1784-1790, (2007) · Zbl 1119.93063
[36] Pecora, L.M.; Carroll, T.L., Synchronization in chaotic systems, Physical review letters, 64, 821-824, (1990) · Zbl 0938.37019
[37] Femat, R.; Alvarez-Ramírez, J.; Fernández-Anaya, G., Adaptive synchronization of high-order chaotic systems: a feedback with low-order parametrization, Physica D: nonlinear phenomena, 139, 3-4, 231-246, (2000) · Zbl 0954.34037
[38] Liao, T.L.; Tsai, S.H., Adaptive synchronization of chaotic systems and its application to secure communication, Chaos, solitons and fractals, 11, 9, 1387-1396, (2000) · Zbl 0967.93059
[39] Feki, M., An adaptive chaos synchronization scheme applied to secure communication, Chaos, solitons and fractals, 18, 141-148, (2003) · Zbl 1048.93508
[40] Wang, Y.W.; Wen, C.; Soh, Y.C.; Xiao, J.W., Adaptive control and synchronization for a class of nonlinear chaotic systems using partial system states, Physics letters A, 351, 1-2, 79-84, (2006) · Zbl 1234.93087
[41] Gao, H.; Lam, J.; Chen, G., New criteria for synchronization stability of general complex dynamical networks with coupling delays, Physics letters A, 360, 2, 263-273, (2006) · Zbl 1236.34069
[42] Karimi, H.R.; Maass, P., Delay-range-dependent exponential H∞ synchronization of a class of delayed neural networks, Chaos, solitons & fractals, 41, 3, 1125-1135, (2009) · Zbl 1198.93179
[43] Wen, G.; Wang, Q.G.; Lin, C.; Han, X.; Li, G., Synthesis for robust synchronization of chaotic systems under output feedback control with multiple random delays, Chaos, solitons & fractals, 29, 5, 1142-1146, (2006) · Zbl 1142.93430
[44] Cao, J.; Lu, J., Adaptive synchronization of neural networks with or without time-varying delay, Chaos, 16, 013133, (2006) · Zbl 1144.37331
[45] Karimi, H.R.; Gao, H., New delay-dependent exponential H∞ synchronization for uncertain neural networks with mixed time delays, IEEE transactions on systems, man, cybernetics—part B, 40, 1, 173-185, (2010)
[46] Wang, L.; Cao, J., Global robust point dissipativity of interval neural networks with mixed time-varying delays, Nonlinear dynamics, 55, 1-2, 169-178, (2009) · Zbl 1169.92005
[47] Park, J.H., Synchronization of Genesio chaotic system via backstepping approach, Chaos solitons and fractals, 27, 1369-1375, (2006) · Zbl 1091.93028
[48] Kolmanovskii, V.; Koroleva, N.; Maizenberg, T.; Mao, X.; Matasov, A., ‘neutral stochastic differential delay equations with Markovian switching.’, Stochastic analysis and applications, 21, 819-847, (2003) · Zbl 1025.60028
[49] Gahinet, P.; Nemirovsky, A.; Laub, A.J.; Chilali, M., ‘LMI control toolbox: for use with Matlab, (1995), The MATH Works, Inc Natik, MA
[50] Karimi, H.R.; Yazdanpanah, M.J.; Patel, R.V.; Khorasani, K., Modeling and control of linear two-time scale systems: applied to single-link flexible manipulators, Journal of intelligent & robotic systems, 45, 3, 235-265, (2006)
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