Global Synchronization of Directional Networked Systems With Eventually Dissipative Nodes

In the existing literature related to the global synchronization of networked systems, the V-uniformly decreasing condition on the nodal self-dynamics has often been assumed in advance. However this assumption excludes nonuniformly Lipschitz systems, such as the well-known Lorenz oscillators. In this paper, we adopt the eventual dissipativity condition to relax the V-uniformly decreasing condition such that the obtained results are applicable to some nonuniformly Lipschitz systems. Firstly the concept of synchronization degree is presented to depict the synchronizability of the nodal self-dynamics (NSD) system over its associated attractor. Then a virtual node whose trajectory is proved to ultimately evolve in the same region with that of isolated node is presented to play the reference trajectory. Next a simple global synchronization criterion that highlights the interplay between the NSD system and the network topology is presented, which can be applied for the reducible networks. Finally, a simulation example on a reducible network verifies the analytic results.

[1]  Daniel W. C. Ho,et al.  Globally Exponential Synchronization and Synchronizability for General Dynamical Networks , 2010, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[2]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[3]  Mario di Bernardo,et al.  Novel decentralized adaptive strategies for the synchronization of complex networks , 2009, Autom..

[4]  Ji Xiang,et al.  Global synchronization of networked systems with bidirectional connections , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[5]  David J. Hill,et al.  Passivity-based control and synchronization of general complex dynamical networks , 2009, Autom..

[6]  Fuchen Zhang,et al.  Bounds for a new chaotic system and its application in chaos synchronization , 2011 .

[7]  Xiang Li,et al.  Pinning a complex dynamical network to its equilibrium , 2004, IEEE Trans. Circuits Syst. I Regul. Pap..

[8]  Mauricio Barahona,et al.  Synchronization in small-world systems. , 2002, Physical review letters.

[9]  Guanrong Chen,et al.  A time-varying complex dynamical network model and its controlled synchronization criteria , 2005, IEEE Transactions on Automatic Control.

[10]  Chai Wah Wu,et al.  Synchronization in Coupled Chaotic Circuits and Systems , 2002 .

[11]  Lei Wang,et al.  Synchronization in complex networks with switching topology , 2011 .

[12]  Guanrong Chen,et al.  Estimating the bounds for the Lorenz family of chaotic systems , 2005 .

[13]  T. Carroll,et al.  Master Stability Functions for Synchronized Coupled Systems , 1998 .

[14]  Jurgen Kurths,et al.  Synchronization in complex networks , 2008, 0805.2976.

[15]  Donghua Zhou,et al.  Synchronization in uncertain complex networks. , 2006, Chaos.

[16]  Ji Xiang,et al.  Analysis of Pinning-Controlled Networks: A Renormalization Approach , 2009, IEEE Transactions on Automatic Control.

[17]  Chai Wah Wu,et al.  Synchronization in arrays of coupled nonlinear systems with delay and nonreciprocal time-varying coupling , 2005, IEEE Transactions on Circuits and Systems II: Express Briefs.

[18]  Junan Lu,et al.  Adaptive synchronization of an uncertain complex dynamical network , 2006, IEEE Transactions on Automatic Control.

[19]  Yingmin Jia,et al.  Algebraic criteria for consensus problems of continuous-time networked systems , 2009, Int. J. Control.

[20]  L. Chua,et al.  Synchronization in an array of linearly coupled dynamical systems , 1995 .

[21]  Mario di Bernardo,et al.  Contraction Theory and Master Stability Function: Linking Two Approaches to Study Synchronization of Complex Networks , 2009, IEEE Transactions on Circuits and Systems II: Express Briefs.

[22]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[23]  Xiao Fan Wang,et al.  Synchronization in Small-World Dynamical Networks , 2002, Int. J. Bifurc. Chaos.

[24]  Tianping Chen,et al.  Boundedness and synchronization of y-coupled Lorenz systems with or without controllers ☆ , 2008 .

[25]  Tianping Chen,et al.  New approach to synchronization analysis of linearly coupled ordinary differential systems , 2006 .

[26]  Jean-Jacques E. Slotine,et al.  Methodological remarks on contraction theory , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[27]  Wenwu Yu,et al.  On pinning synchronization of complex dynamical networks , 2009, Autom..

[28]  Guanrong Chen,et al.  A time-varying complex dynamical network model and its controlled synchronization criteria , 2004, IEEE Trans. Autom. Control..

[29]  Hong-yong Yang,et al.  Chaos Synchronization of General Complex Dynamical Networks with Coupling Delays , 2006, 2006 6th World Congress on Intelligent Control and Automation.

[30]  M. Hasler,et al.  Persistent clusters in lattices of coupled nonidentical chaotic systems. , 2003, Chaos.

[31]  Wei Ren,et al.  Information consensus in multivehicle cooperative control , 2007, IEEE Control Systems.

[32]  Erik M. Bollt,et al.  Sufficient Conditions for Fast Switching Synchronization in Time-Varying Network Topologies , 2006, SIAM J. Appl. Dyn. Syst..

[33]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[34]  Jean-Jacques E. Slotine,et al.  On Contraction Analysis for Non-linear Systems , 1998, Autom..

[35]  Tianping Chen,et al.  Exponential synchronization of nonlinear coupled dynamical networks with a delayed coupling , 2007 .

[36]  Maurizio Porfiri,et al.  Criteria for global pinning-controllability of complex networks , 2008, Autom..

[37]  Jinde Cao,et al.  On Pinning Synchronization of Directed and Undirected Complex Dynamical Networks , 2010, IEEE Transactions on Circuits and Systems I: Regular Papers.

[38]  Ji Xiang,et al.  On the V-stability of complex dynamical networks , 2007, Autom..

[39]  Lin Huang,et al.  Consensus of Multiagent Systems and Synchronization of Complex Networks: A Unified Viewpoint , 2016, IEEE Transactions on Circuits and Systems I: Regular Papers.

[40]  Mario di Bernardo,et al.  On QUAD, Lipschitz, and Contracting Vector Fields for Consensus and Synchronization of Networks , 2011, IEEE Transactions on Circuits and Systems I: Regular Papers.