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Global asymptotical stability and generalized synchronization of phase synchronous dynamical networks. (English) Zbl 1183.70039
Summary: This paper examines the global asymptotical stability of the phase synchronous dynamical networks composed by a class of nonlinear pendulum-like systems with multiple equilibria. Sufficient conditions for the determination of global asymptotical stability are given in terms of linear matrix inequalities (LMIs). Furthermore, a concept of generalized synchronization is introduced, and the criterion of which is proposed in a simple form. Those results are of particular convenience for networks that possess large numbers of nodes, and they can be used to discuss controller design problems as well. Numerical simulations and analytical results are in excellent agreement with each other.
MSC:
70K20Stability of nonlinear oscillations (general mechanics)
93D20Asymptotic stability of control systems
References:
[1]Watts, D.J., Strogatz, S.H.: Collective dynamics of ’small-world’ networks. Nature 393, 440–442 (1998) · doi:10.1038/30918
[2]Albert, R., Barabasi, A.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74(1), 47–97 (2002) · Zbl 1205.82086 · doi:10.1103/RevModPhys.74.47
[3]Wang, X.F.: Complex networks: topology, dynamics, and synchronization. Int. J. Bifurc. Chaos 5(12), 885–916 (2002)
[4]Barahona, M., Pecora, L.M.: Synchronization in small-world systems. Phys. Rev. Lett. 89(5), 054101 (2002) · doi:10.1103/PhysRevLett.89.054101
[5]Wu, C.W.: Synchronization in Complex Networks of Nonlinear Dynamical Systems. World Scientific, Singapore (2006)
[6]Li, C.G., Chen, G.R.: Synchronization in general complex dynamical networks with coupling delays. Physica A 343, 263–277 (2004) · doi:10.1016/j.physa.2004.05.058
[7]Wang, X.F., Chen, G.R.: Pinning control of scale-free networks. Physica A 310, 521–531 (2002) · Zbl 0995.90008 · doi:10.1016/S0378-4371(02)00772-0
[8]Wang, X.F., Chen, G.R.: Synchronization in scale-free networks: robustness and fragility. IEEE Trans. CAS-1 49, 54–62 (2002) · doi:10.1109/81.974874
[9]Liu, X., Wang, J.Z., Huang, L.: Global synchronization for a class of dynamical complex networks. Physica A 386, 543–556 (2007) · doi:10.1016/j.physa.2007.08.029
[10]Xu, S.Y., Yang, Y.: Synchronization for a class of complex dynamical networks with time-delay. Commun. Nonlinear Sci. Numer. Simul. 14, 3230–3238 (2009) · Zbl 1221.34205 · doi:10.1016/j.cnsns.2008.12.022
[11]Leonov, G.A., Burkin, I.M., Shepeljavyi, A.I.: Frequency Methods in Oscillation Theory. Kluwer Academic, Dordrecht (1992)
[12]Leonov, G.A., Reitmann, V., Smirnova, V.B.: Non-local methods for pendulum-like feedback systems. Teubner-Texte zur Mathematik Bd. 132, B.G., Teubner Stuttgart-Leipzig (1992)
[13]Leonov, G.A., Ponomarenko, D.V., Smirnova, V.B.: Frequency-Domain Methods for Nonlinear Analysis. World Scientific, Singapore (1996)
[14]Rantzer, A.: On the Kalman–Yakubovich–Popov lemma. Syst. Control Lett. 28, 7–10 (1996) · Zbl 0866.93052 · doi:10.1016/0167-6911(95)00063-1
[15]Yang, Y., Huang, L.: H controller synthesis for pendulum-like systems. Syst. Control Lett. 50, 263–276 (2003) · Zbl 1157.93375 · doi:10.1016/S0167-6911(03)00159-2
[16]Duan, Z.S., Wang, J.Z., Huang, L.: Criteria for dichotomy and gradient-like behavior of a class of nonlinear systems with multiple equilibria. Automatica 43, 1583–1589 (2007) · Zbl 1128.93313 · doi:10.1016/j.automatica.2007.02.003
[17]Yang, Y., Fu, R., Huang, L.: Robust analysis and synthesis for a class of uncertain nonlinear systems with multiple equilibria. Syst. Control Lett. 53, 89–105 (2004) · Zbl 1157.93382 · doi:10.1016/j.sysconle.2004.02.024
[18]Duan, Z.S., Wang, J.Z., Huang, L.: Input and output coupled nonlinear systems. IEEE Trans. CAS-1 52(3), 567–575 (2007) · doi:10.1109/TCSI.2004.842873
[19]Yang, Y., Duan, Z.S., Huang, L.: Global convergence of a class of discrete-time interconnected pendulum-like systems. J. Optim. Theory Appl. 133, 257–273 (2007) · Zbl 1145.93033 · doi:10.1007/s10957-007-9172-6
[20]Wu, C.W.: Synchronization in Coupled Chaotic Circuits and Systems. World Scientific, Singapore (2002)
[21]Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge Univ. Press, Cambridge (1985)
[22]Wu, C.W.: Application of Kronecker products to the analysis of systems with uniform linear coupling. IEEE Trans. CAS-1 42(10), 775–778 (1995) · doi:10.1109/81.473586
[23]Balakrishnan, V., Vandenberghe, L.: Semidefinite programming duality and linear time-invariant systems. IEEE Trans. Autom. Control. 48, 30–41 (2003) · doi:10.1109/TAC.2002.806652
[24]Lu, P.L., Yang, Y., Li, Z.K., Huang, L.: Decentralized dynamic output feedback for globally asymptotic stabilization of a class of dynamic networks. Int. J. Control 81(7), 1054–1061 (2008) · Zbl 1152.93302 · doi:10.1080/00207170701635213
[25]Boyd, S., ELGhaoui, L., Feron, E., Balakrishnam, V.: Linear Matrix Inequalities in Systems and Control. SIAM, Philadelphia (1994)
[26]Gardner, F.M.: Phaselock Techniques. Wiley, New York (1979)
[27]Ware, K.M., Lee, H.S., Sodini, C.G.: A 200-mhz cmos phase-locked loop with dual phase detectors. IEEE J. Solid-State Circ. 24, 1560–1568 (1989) · doi:10.1109/4.44991
[28]Abramovitch, D.Y.: Lyapunov redesign of analog phase-lock loops. IEEE Trans. Commun. 38, 2197–2202 (1990) · doi:10.1109/26.64662
[29]Buckwalter, J., York, R.A.: Time delay considerations in high-frequency phase-locked loop. In: IEEE Radio Frequency Integrated Circuits Symposium, pp. 181–184 (2000)
[30]Harb, B.A., Harb, A.M.: Chaos and bifurcation in a third-order phase locked loop. Chaos Soliton Fractals 19, 667–672 (2004) · Zbl 1085.93514 · doi:10.1016/S0960-0779(03)00197-8