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Sliding-mode \({H_{\infty}}\) synchronization for complex dynamical network systems with Markovian jump parameters and time-varying delays. (English) Zbl 1458.93236

Summary: This paper is devoted to the investigation of the sliding-mode controller design problem for a class of complex dynamical network systems with Markovian jump parameters and time-varying delays. On the basis of an appropriate Lyapunov-Krasovskii functional, a set of new sufficient conditions is developed which not only guarantee the stochastic stability of the sliding-mode dynamics, but also satisfy the \({H_{\infty}}\) performance. Next, an integral sliding surface is designed to guarantee that the closed-loop error system reach the designed sliding surface in a finite time. Finally, an example is given to illustrate the validity of the obtained theoretical results.

MSC:

93E03 Stochastic systems in control theory (general)
93B12 Variable structure systems
93A15 Large-scale systems
60J74 Jump processes on discrete state spaces
93C15 Control/observation systems governed by ordinary differential equations
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[1] Barabasi, A.L., Albert, R., Jeong, H.: Scale-free characteristics of random networks: the topology of the world wide web. Physica A 281(1), 69-77 (2000) · doi:10.1016/S0378-4371(00)00018-2
[2] Watts, D.J., Strogatz, S.H.: Collective dynamics of ‘small-world’ networks. Nature 393, 440-442 (1988) · Zbl 1368.05139 · doi:10.1038/30918
[3] Barbasi, A.L., Albert, R.: Emergence of scaling in random networks. Science 286, 509-512 (1999) · Zbl 1226.05223 · doi:10.1126/science.286.5439.509
[4] Albert, R., Jeong, H., Barabsi, A.L.: Diameter of the world wide web. Nature 401, 130-131 (1999) · doi:10.1038/43601
[5] Strogatz, S.H.: Exploring complex network. Nature 410, 268-276 (2001) · Zbl 1370.90052 · doi:10.1038/35065725
[6] Wang, X.F., Chen, G.R.: Synchronization in scale-free dynamical networks: robustness and fragility. IEEE Trans. Circuits Syst. 149, 54-56 (2002) · Zbl 1368.93576 · doi:10.1109/81.974874
[7] Peng, X., Wu, H.Q., Song, K., Shi, J.X.: Global synchronization in finite time for fractional-order neural networks with discontinuous activations and time delays. Neural Netw. 94, 46-54 (2017) · Zbl 1437.93116 · doi:10.1016/j.neunet.2017.06.011
[8] Liu, M., Wu, H.Q.: Stochastic finite-time synchronization for discontinuous semi-Markovian switching neural networks with time delays and noise disturbance. Neurocomputing 310, 246-264 (2018) · doi:10.1016/j.neucom.2018.03.071
[9] Peng, X., Wu, H.Q., Cao, J.D.: Global nonfragile synchronization in finite time for fractional-order discontinuous neural networks with nonlinear growth activations. IEEE Trans. Neural Netw. Learn. Syst. (2018, in press). https://doi.org/10.1109/TNNLS.2018.2876726 · doi:10.1109/TNNLS.2018.2876726
[10] Zhao, W., Wu, H.Q.: Fixed-time synchronization of semi-Markovian jumping neural networks with time-varying delays. Adv. Differ. Equ. 2018, 213 (2018) · Zbl 1446.93016 · doi:10.1186/s13662-018-1666-z
[11] Ma, Y.C., Ma, N.N., Chen, L.: Synchronization criteria for singular complex networks with Markovian jump and time-varying delays via pinning control. Nonlinear Anal. Hybrid Syst. 28, 85-99 (2018) · Zbl 1388.93083 · doi:10.1016/j.nahs.2017.12.002
[12] Zheng, S.: Projective synchronization in a driven-response dynamical network with coupling time-varying delays. Nonlinear Dyn. 63(3), 1429-1438 (2012) · Zbl 1253.93062 · doi:10.1007/s11071-012-0359-5
[13] Zheng, S.: Projective synchronization analysis of drive-response coupled dynamical network with multiple time-varying delays via impulsive control. Abstr. Appl. Anal. 2014, Article ID581971 (2014) · Zbl 1474.34532
[14] Ma, Y.C., Zheng, Y.Q.: Synchronization of continuous-time Markovian jumping singular complex networks with mixed mode-dependent time delays. Neurocomputing 156, 52-59 (2015) · doi:10.1016/j.neucom.2015.01.001
[15] Huang, X.J., Ma, Y.C.: Finite-time \(H∞{H_{\infty}}\) sampled-data synchronization for Markovian jump complex networks with time-varying delays. Neurocomputing 296, 82-99 (2018) · doi:10.1016/j.neucom.2018.03.024
[16] Wang, Z.B., Wu, H.Q.: Global synchronization in fixed time for semi-Markovian switching complex dynamical networks with hybrid coupling and time-varying delays. Nonlinear Dyn. (2018, in press). https://doi.org/10.1007/s11071-018-4675-2 · doi:10.1007/s11071-018-4675-2
[17] Rankkiyappan, R., Velmurugan, G., Nicholas, G.J., Selvamani, R.: Exponential synchronization of Lur’e complex dynamical networks with uncertain inner coupling and pinning impulsive control. Appl. Math. Comput. 307, 217-231 (2017) · Zbl 1411.34076
[18] Song, Q., Cao, J., Liu, F.: Pinning-controlled synchronization of hybrid-coupled complex dynamical networks with mixed time-delays. Int. J. Robust Nonlinear Control 22(6), 690-706 (2012) · Zbl 1273.93016 · doi:10.1002/rnc.1725
[19] Lee, T.H., Ma, Q., Xu, S., Ju, H.P.: Pinning control for cluster synchronization of complex dynamical networks with semi-Markovian jump topology. Int. J. Control 88(6), 1223-1235 (2015) · Zbl 1316.93013 · doi:10.1080/00207179.2014.1002110
[20] Yang, X.S., Cao, J.D.: Synchronization of complex networks with coupling delay via pinning control. IMA J. Math. Control Inf. 34(2), 579-596 (2017) · Zbl 1397.93182
[21] Feng, J.W., Sun, S.H., Chen, X., Zhao, Y., Wang, J.Y.: The synchronization of general complex dynamical network via pinning control. Nonlinear Dyn. 67(2), 1623-1633 (2012) · Zbl 1242.93053 · doi:10.1007/s11071-011-0092-5
[22] Lee, T.H., Wu, Z.G., Ju, H.P.: Synchronization of a complex dynamical network with coupling time-varying delays via sampled-data control. Appl. Math. Comput. 219(3), 1354-1366 (2012) · Zbl 1291.34120
[23] Liu, X.H., Xi, H.S.: Synchronization of neutral complex dynamical networks with Markovian switching based on sampled-data controller. Neurocomputing 139(2), 163-179 (2014) · doi:10.1016/j.neucom.2014.02.048
[24] Zhao, H., Li, L.X., Peng, H.P., Xiao, J.H., Yang, Y.X., Zhang, M.W.: Impulsive control for synchronization and parameters identification of uncertain multi-links complex network. Nonlinear Dyn. 83(3), 1437-1451 (2016) · Zbl 1351.93041 · doi:10.1007/s11071-015-2416-3
[25] Dai, A.D., Zhou, W.N., Feng, J.W., Xu, S.B.: Exponential synchronization of the coupling delayed switching complex dynamical networks via impulsive control. Adv. Differ. Equ. 2013, 195 (2013) · Zbl 1379.93013 · doi:10.1186/1687-1847-2013-195
[26] Syed, A.M., Yogambigai, J.: Passivity-based synchronization of stochastic switched complex dynamical networks with additive time-varying delays via impulsive control. Neurocomputing 273(17), 209-221 (2018) · doi:10.1016/j.neucom.2017.07.053
[27] Khanzadeh, A., Pourgholi, M.: Fixed-time sliding mode controller design for synchronization of complex dynamical networks. Nonlinear Dyn. 88(4), 2637-2649 (2017) · Zbl 1398.93276 · doi:10.1007/s11071-017-3400-x
[28] Syed, A.M., Yogambigai, J., Cao, J.D.: Synchronization of master – slave Markovian switching complex dynamical networks with time-varying delays in nonlinear function via sliding mode control. Acta Math. Sci. 37(2), 368-384 (2017) · Zbl 1389.93241 · doi:10.1016/S0252-9602(17)30008-5
[29] Jin, X. Z.; Ye, D.; Wang, D., Robust synchronization of a class of complex networks with nonlinear couplings via a sliding mode control method, 1811-1815 (2012)
[30] Wang, Z.B., Wu, H.Q.: Projective synchronization in fixed time for complex dynamical networks with nonidentical nodes via second-order sliding mode control strategy. J. Franklin Inst. 355, 7306-7334 (2018) · Zbl 1398.93037 · doi:10.1016/j.jfranklin.2018.07.018
[31] Han, Y.Q., Kao, Y.G., Gao, C.C.: Robust sliding mode control for uncertain discrete singular systems with time-varying delays and external disturbances. Automatica 75, 210-216 (2017) · Zbl 1351.93035 · doi:10.1016/j.automatica.2016.10.001
[32] Karimi, H.A.: A sliding mode approach to \(H∞{H_{\infty}}\) synchronization of master – slave time-delay systems with Markovian jumping parameters and nonlinear uncertainties. J. Franklin Inst. 349, 1480-1496 (2012) · Zbl 1254.93046 · doi:10.1016/j.jfranklin.2011.09.015
[33] Wu, L.G., Zheng, W.X.: Passivity-based sliding mode control of uncertain singular time-delay systems. Automatica 45, 2120-2127 (2009) · Zbl 1175.93065 · doi:10.1016/j.automatica.2009.05.014
[34] Wu, L.G., Gao, Y.B., Liu, J.X., Yi, H.Y.: Event-triggered sliding mode control of stochastic systems via output feedback. Automatica 82, 79-92 (2017) · Zbl 1376.93030 · doi:10.1016/j.automatica.2017.04.032
[35] Liu, X.H., Vargas, A.N., Yu, X.H., Xu, L.: Stabilizing two-dimensional stochastic systems through sliding mode control. J. Franklin Inst. 354(14), 5813-5824 (2017) · Zbl 1373.93367 · doi:10.1016/j.jfranklin.2017.07.015
[36] Han, Y.Q., Kao, Y.G., Gao, C.C.: Robust sliding mode control for uncertain discrete singular systems with time-varying delays. Int. J. Syst. Sci. 48(4), 818-827 (2017) · Zbl 1358.93047 · doi:10.1080/00207721.2016.1216200
[37] Liu, Y.F., Ma, Y.C., Wang, Y.N.: Reliable finite-time sliding-mode control for singular time-delay system with sensor faults and randomly occurring nonlinearities. Appl. Math. Comput. 320, 341-357 (2018) · Zbl 1426.93042
[38] Liu, L.P., Fu, Z.M., Cai, X.S., Song, X.N.: Non-fragile sliding mode control of discrete singular systems. Commun. Nonlinear Sci. Numer. Simul. 18(3), 735-743 (2013) · Zbl 1277.93022 · doi:10.1016/j.cnsns.2012.08.014
[39] Feng, Z.G., Shi, P.: Sliding mode control of singular stochastic Markov jump systems. IEEE Trans. Autom. Control 62(8), 4266-4273 (2017) · Zbl 1373.93305 · doi:10.1109/TAC.2017.2687048
[40] Zhu, Q., Yu, X.H., Song, A.G., Fei, S.M., Cao, Z.Q., Yang, Y.Q.: On sliding mode control of single input Markovian jump systems. Automatica 50(11), 2897-2904 (2014) · Zbl 1300.93053 · doi:10.1016/j.automatica.2014.10.008
[41] Zhang, Q.L., Li, L., Yan, X.G., Spurgeon, S.K.: Sliding mode control for singular stochastic Markovian jump systems with uncertainties. Automatica 79, 27-34 (2017) · Zbl 1371.93048 · doi:10.1016/j.automatica.2017.01.002
[42] Zhang, D., Zhang, Q.L.: Sliding mode control for T-S fuzzy singular semi-Markovian jump system. Nonlinear Anal. Hybrid Syst. 30, 72-91 (2018) · Zbl 1408.93043 · doi:10.1016/j.nahs.2018.04.006
[43] Cheng, C.C., Chen, S.H.: Adaptive sliding mode controller design based on T-S fuzzy system models. Automatica 42(6), 1005-1010 (2006) · Zbl 1110.93027 · doi:10.1016/j.automatica.2006.02.016
[44] Jing, Y.H., Yang, G.H.: Fuzzy adaptive quantized fault-tolerant control of strict-feedback nonlinear systems with mismatched external disturbance. IEEE Trans. Syst. Man Cybern. Syst. (2018, in press). https://doi.org/10.1109/TSMC.2018.2867100 · doi:10.1109/TSMC.2018.2867100
[45] Ma, Y.C., Ma, N.N.: Finite-time \(H∞{H_{\infty}}\) synchronization for complex dynamical networks with mixed mode-dependent time delays. Neurocomputing 218, 223-233 (2016) · doi:10.1016/j.neucom.2016.08.053
[46] Utkin, V.I.: Variable structure systems with sliding modes. IEEE Trans. Autom. Control 22(2), 212-222 (1997) · Zbl 0382.93036 · doi:10.1109/TAC.1977.1101446
[47] Chen, B., Niu, Y., Zou, Y.: Sliding mode control for stochastic Markovian jumping systems with incomplete transition rate. IET Control Theory Appl. 7(10), 1330-1338 (2013) · doi:10.1049/iet-cta.2013.0083
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