Abstract
This paper investigates the fixed-time synchronization of complex dynamical networks with nonidentical nodes in the presence of bounded uncertainties and disturbances using sliding mode control technique. Firstly, a novel sliding surface is introduced and fixed-time stability of the sliding mode dynamics is proven by benefiting from Gudermannian function. Then, a novel sliding mode controller is proposed whereby fixed-time stability of the reaching mode is guaranteed. The outstanding feature of the proposed controller is that fixed convergence times of reaching and sliding modes are independent design parameters explicitly existing in the control law. This allows us not only to set the reaching time of reaching mode and settling time of sliding mode at any desired values in advance but also to adjust them independent of each other and in the most straightforward possible way. Finally, simulation results are reported in order to show the effectiveness of the proposed controller.
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References
Zhu, J., Yang, G.: Robust \(H_{\infty }\) dynamic output feedback synchronization for complex dynamical networks with disturbances. Neurocomputing 175, 287–292 (2016)
Zhang, L., Wang, Y., Huang, Y.: Synchronization for non-dissipatively coupled time-varying complex dynamical networks with delayed coupling nodes. Nonlinear Dyn. 82(2), 1581–1593 (2015)
Li, K., Yu, W., Ding, Y.: Successive lag synchronization on nonlinear dynamical networks via linear feedback control. Nonlinear Dyn. 80(1), 421–430 (2015)
Du, H.: Function projective synchronization in complex dynamical networks with and without external disturbances via error feedback control. Neurocomputing 173(2), 1443–1449 (2016)
Qiu, J., Cheng, L., Chen, X., Lu, J., He, H.: Semi-periodically intermittent control for synchronization of switched complex networks: a mode-dependent average dwell time approach. Nonlinear Dyn. 83(2), 1757–1771 (2016)
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(5), 690–706 (2012)
Wen, G., Yu, W., Hu, G., Cao, J., Yu, X.: Pinning synchronization of directed networks with switching topologies: a multiple Lyapunov functions approach. IEEE Trans. Neural Net. Learn. Syst. 26(10), 3239–3250 (2015)
Zhou, L., Wang, C., Zhou, L.: Cluster synchronization on multiple sub-networks of complex networks with nonidentical nodes via pinning control. Nonlinear Dyn. 83(1), 1079–1100 (2016)
Yu, W., Lellis, P., Chen, G., Bernardo, M., Kurths, J.: Distributed adaptive control of synchronization in complex networks. IEEE Trans. Autom. Control 57(7), 2153–2158 (2012)
Tan, M., Pan, Q., Zhou, X.: Adaptive stabilization and synchronization of non-diffusively coupled complex networks with nonidentical nodes of different dimensions. Nonlinear Dyn. 85(1), 303–316 (2016)
Shi, H., Sun, Y., Miao, L., Duan, Z.: Outer synchronization of uncertain complex delayed networks with noise coupling. Nonlinear Dyn. (2016). doi:10.1007/s11071-016-2836-8
Bagheri, A., Ozgoli, S.: Exponentially impulsive projective and lag synchronization between uncertain complex networks. Nonlinear Dyn. 84(3), 2043–2055 (2016)
Feng, J., Yu, F., Zhao, Y.: Exponentially synchronization of nonlinearly coupled complex networks with hybrid time-varying delays via impulsive control. Nonlinear Dyn. 85(1), 621–632 (2016)
Zhao, H., Li, L., Peng, H., Xiao, J., Yang, Y., Zheng, M.: Impulsive control for synchronization and parameters identification of uncertain multi-links complex network. Nonlinear Dyn. 83(2), 1437–1451 (2016)
Lu, L., Yu, M., Li, C., Liu, S., Yan, B., Chang, H., Zhou, J., Liu, Y.: Projective synchronization of a class of complex network based on high-order sliding mode control. Nonlinear Dyn. 73(1), 411–416 (2013)
Yu, Y., Gao, Y., Han, X., Bi, Q.: Modified function projective bursting synchronization for fast-slow systems with uncertainties and external disturbances. Nonlinear Dyn. 79(3), 2359–2369 (2015)
Hou, H., Zhang, Q., Zheng, M.: Cluster synchronization in nonlinear complex networks under sliding mode control. Nonlinear Dyn. 83(1), 739–749 (2016)
Bhat, S.P., Bernstein, D.S.: Finite-time stability of continuous autonomous systems. SIAM J. Control Optim. 38(2), 751–766 (2000)
Sun, J., Shen, Y., Wang, X., Chen, J.: Finite-time combination-combination synchronization of four different chaotic systems with unknown parameters via sliding mode control. Nonlinear Dyn. 76(1), 383–397 (2014)
Sun, J., Wang, Y., Wang, Y., Shen, Y.: Finite-time synchronization between two complex-variable chaotic systems with unknown parameters via nonsingular terminal sliding mode control. Nonlinear Dyn. 85(2), 1105–1117 (2016)
Tan, M., Tian, W.: Finite-time stabilization and synchronization of complex dynamical networks with nonidentical nodes of different dimensions. Nonlinear Dyn. 79(1), 731–741 (2015)
Mei, J., Jiang, M., Xu, W., Wang, B.: Finite-time synchronization control of complex dynamical networks with time delays. Commun. Nonlinear Sci. Numer. Simul. 18(9), 2462–2478 (2013)
Li, L., Tu, Z., Mei, J., Jian, J.: Finite-time synchronization of complex delayed networks via intermittent control with multiple switched periods. Nonlinear Dyn. 85(1), 375–388 (2016)
Jing, T., Chen, F., Li, Q.: Finite-time outer synchronization of complex networks with time-varying delay and unknown parameters. Appl. Math. Model. 39(23–24), 7734–7743 (2015)
Wang, X., Fang, J., Mao, H., Dai, A.: Finite-time global synchronization for a class of Markovian jump complex networks with partially unknown transition rates under feedback control. Nonlinear Dyn. 79(1), 47–61 (2014)
Cui, W., Sun, S., Fang, J., Xu, Y., Zhao, L.: Finite-time synchronization of Markovian jump complex networks with partially unknown transition rates. J. Frankl. Inst. 351(4), 2543–2561 (2014)
Xie, Q., Si, G., Zhang, Y., Yuan, Y., Yao, R.: Finite-time synchronization and identification of complex delayed networks with Markovian jumping parameters and stochastic perturbations. Chaos Solitons Fractals 86, 35–49 (2016)
Yang, X., Wu, Z., Cao, J.: Finite-time synchronization of complex networks with nonidentical discontinuous nodes. Nonlinear Dyn. 73(3), 2313–2327 (2013)
Liu, X., Yu, W., Cao, J., Alsaadi, F.: Finite-time synchronization control of complex networks via non-smooth analysis. IET Control Theory Appl. 9(7), 1245–1253 (2015)
Yang, X., Ho, D.W.C., Lu, J., Song, Q.: Finite-time cluster synchronization of T-S fuzzy complex networks with discontinuous subsystems and random coupling delays. IEEE Trans. Fuzzy Syst. 23(5), 2302–2316 (2015)
Mei, J., Jiang, M., Wu, Z., Wang, X.: Periodically intermittent controlling for finite-time synchronization of complex dynamical networks. Nonlinear Dyn. 79(1), 295–305 (2015)
Jing, Y., Chen, F., Zhang, X.: Finite-time lag synchronization of time-varying delayed complex networks via periodically intermittent control and sliding mode control. Neurocomputing 199, 178–184 (2016)
Liu, S., Chen, L.: Second-order terminal sliding mode control for networks synchronization. Nonlinear Dyn. 79(1), 205–213 (2015)
Polyakov, A.: Fixed-time stabilization of linear systems via sliding mode control. 12th International Workshop on Variable Structure Systems. Mumbai, Maharashtra: IEEE, pp. 1–6 (2012)
Polyakov, A.: Nonlinear feedback design for fixed-time stabilization of linear control systems. IEEE Trans. Autom. Control 57(7), 2106–2110 (2012)
Liu, X., Chen, T.: Fixed-time cluster synchronization for complex networks via pinning control. ArXiv Preprint arXiv:1509.03350v1
Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010)
Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities. Cambridge University Press, Cambridge (1952)
Utkin, V.I.: Sliding Mode in Control and Optimization. Springer, Berlin (1992)
Tigan, G., Opris, D.: Analysis of a 3D chaotic system. Chaos Solitons Fractals 36(4), 1315–1319 (2008)
Genesio, R., Tesi, A.: Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems. Automatica 28(2), 531–548 (1992)
Vaidyanathan, S., Sampath, S., Azar, A.T.: Global chaos synchronization of identical chaotic systems via novel sliding mode control method and its application to Zhu system. Int. J. Model. Identif. Control 23(1), 92–100 (2015)
Park, J.H.: Chaos synchronization between two different chaotic dynamical systems. Chaos Solitons Fractals 27(2), 549–554 (2006)
Agiza, H.N., Yassen, M.T.: Synchronization of Rossler and Chen Chaotic dynamical systems using active control. Phys. Lett. A 278(3), 191–197 (2001)
Yau, H., Yan, J.: Design of sliding mode controller for Lorenz chaotic system with nonlinear input. Chaos Solitons Fractals 19(3), 891–898 (2004)
Liu, C., Liu, T., Liu, L., Liu, K.: A new chaotic attractor. Chaos Solitons Fractals 22(4), 1031–1038 (2004)
Xiao-Qun, W., Jun-An, L.: Parameter identification and backstepping control of uncertain Lu system. Chaos Solitons Fractals 18(3), 721–729 (2003)
Chen, J., Chen, H., Lin, Y.: Synchronization and anti-synchronization coexist in Chen-Lee chaotic systems. Chaos Solitons Fractals 39(2), 707–716 (2009)
Hua, C., Guan, X., Shi, P.: Adaptive feedback control for a class of chaotic systems. Chaos Solitons Fractals 23(2), 757–756 (2005)
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Khanzadeh, A., Pourgholi, M. Fixed-time sliding mode controller design for synchronization of complex dynamical networks. Nonlinear Dyn 88, 2637–2649 (2017). https://doi.org/10.1007/s11071-017-3400-x
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DOI: https://doi.org/10.1007/s11071-017-3400-x