Synchronisation of heterogeneous Kuramoto oscillators with sampled information and a constant leader

ABSTRACT The Kuramoto oscillator has been widely studied because it can model a wide variety of engineering problems. Conditions for frequency synchronisation of a network of undirected coupled Kuramoto oscillators have been well established. However, the topology that describes the interaction between oscillators may not be connected due to environmental limitations or link failures. In this work we propose a consensus-based control strategy that forces the network to follow a virtual agent with a constant frequency, and where only few agents have access to the leader state. The controller is based on the exchange of information between the oscillators through a communication network. We introduce conditions for synchronisation and phase cohesiveness with the proposed control strategy that depends on the interaction between the physical topology and the communication topology. Finally, we study two applications to illustrate concepts presented in this work: (1) frequency synchronisation in smart grids; and (2) vehicle coordination.

[1]  Zhu Han,et al.  Synchronization of power networks without and with communication infrastructures , 2011, 2011 IEEE International Conference on Smart Grid Communications (SmartGridComm).

[2]  Magnus Egerstedt,et al.  Graph Theoretic Methods in Multiagent Networks , 2010, Princeton Series in Applied Mathematics.

[3]  Nicanor Quijano,et al.  Synchronization of dynamical networks with a communication infrastructure: A smart grid application , 2013, 52nd IEEE Conference on Decision and Control.

[4]  T. Zourntos,et al.  A flocking-based dynamical systems paradigm for smart power system analysis , 2012, 2012 IEEE Power and Energy Society General Meeting.

[5]  L. Moreau,et al.  Stability of continuous-time distributed consensus algorithms , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[6]  Florian Dörfler,et al.  Synchronization in complex networks of phase oscillators: A survey , 2014, Autom..

[7]  M. Sen.  Preserving positive realness through discretization , 2000 .

[8]  S. Strogatz.  From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators , 2000 .

[9]  Yoshiki Kuramoto,et al.  Self-entrainment of a population of coupled non-linear oscillators , 1975 .

[10]  Xiao Fan Wang,et al.  Pinning control of complex networked systems: A decade after and beyond , 2014, Annu. Rev. Control..

[11]  Luis Felipe Giraldo,et al.  Dynamic Task Performance, Cohesion, and Communications in Human Groups , 2016, IEEE Transactions on Cybernetics.

[12]  Eduardo Alberto Canale,et al.  Global Synchronization Properties for Different Classes of Underlying Interconnection Graphs for Kuramoto Coupled Oscillators , 2009, FGIT.

[13]  Florian Dörfler,et al.  Voltage stabilization in microgrids via quadratic droop control , 2013, 52nd IEEE Conference on Decision and Control.

[14]  Hanlei Wang.  Similarity Decomposition Approach to Leader-Follower Oscillatory Synchronization of Networked Mechanical Systems , 2014, ArXiv.

[15]  Xiang Li,et al.  Stability of Synchronous Solutions in a Directed Kuramoto-Oscillator Network With a Pacemaker , 2017, IEEE Transactions on Circuits and Systems II: Express Briefs.

[16]  A. Winfree.  Biological rhythms and the behavior of populations of coupled oscillators. , 1967, Journal of theoretical biology.

[17]  Mihailo R. Jovanovic,et al.  Input-Output Analysis and Decentralized Optimal Control of Inter-Area Oscillations in Power Systems , 2016, IEEE Transactions on Power Systems.

[18]  Naomi Ehrich Leonard,et al.  Stabilization of Planar Collective Motion With Limited Communication , 2008, IEEE Transactions on Automatic Control.

[19]  A. Jadbabaie,et al.  On the stability of the Kuramoto model of coupled nonlinear oscillators , 2005, Proceedings of the 2004 American Control Conference.

[20]  Francesco Bullo,et al.  Breaking the Hierarchy: Distributed Control and Economic Optimality in Microgrids , 2014, IEEE Transactions on Control of Network Systems.

[21]  Enrique Mallada,et al.  Synchronization of phase-coupled oscillators with arbitrary topology , 2010, Proceedings of the 2010 American Control Conference.

[22]  Reza Olfati-Saber,et al.  Consensus and Cooperation in Networked Multi-Agent Systems , 2007, Proceedings of the IEEE.

[23]  Salvatore Monaco,et al.  Average passivity for discrete-time and sampled-data linear systems , 2010, 49th IEEE Conference on Decision and Control (CDC).

[24]  Anuradha M. Annaswamy,et al.  Robust Adaptive Control , 1984, 1984 American Control Conference.

[25]  Tianping Chen,et al.  Pinning Complex Networks by a Single Controller , 2007, IEEE Transactions on Circuits and Systems I: Regular Papers.

[26]  Nicanor Quijano,et al.  Tracking of Kuramoto oscillators with input saturation and applications in smart grids , 2014, 2014 American Control Conference.

[27]  Eduardo D. Sontag,et al.  Synchronization of Interconnected Systems With Applications to Biochemical Networks: An Input-Output Approach , 2009, IEEE Transactions on Automatic Control.

[28]  F. Chung.  Laplacians and the Cheeger Inequality for Directed Graphs , 2005 .

[29]  Celso Freitas,et al.  Partial synchronization in networks of non-linearly coupled oscillators: The Deserter Hubs Model. , 2015, Chaos.

[30]  Sara M Lewis,et al.  Flash signal evolution, mate choice, and predation in fireflies. , 2008, Annual review of entomology.

[31]  Thomas K. D. M. Peron,et al.  The Kuramoto model in complex networks , 2015, 1511.07139.

[32]  Yu Zhao,et al.  Finite‐time consensus tracking for harmonic oscillators using both state feedback control and output feedback control , 2013 .

[33]  Xiang Li,et al.  Synchronizing a Weighted and Weakly-Connected Kuramoto-Oscillator Digraph With a Pacemaker , 2015, IEEE Transactions on Circuits and Systems I: Regular Papers.

[34]  Florian Dörfler,et al.  Synchronization and transient stability in power networks and non-uniform Kuramoto oscillators , 2009, Proceedings of the 2010 American Control Conference.

[35]  Florian Dörfler,et al.  Exploring synchronization in complex oscillator networks , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[36]  R. Sepulchre,et al.  Oscillator Models and Collective Motion , 2007, IEEE Control Systems.

[37]  Manfredi Maggiore,et al.  The Flocking Problem , 2016 .

[38]  Naomi Ehrich Leonard,et al.  Collective motion and oscillator synchronization , 2005 .

[39]  Zhao Wang,et al.  Voltage stability of weak power distribution networks with inverter connected sources , 2013, 2013 American Control Conference.

[40]  Yongqiang Wang,et al.  Exponential Synchronization Rate of Kuramoto Oscillators in the Presence of a Pacemaker , 2012, IEEE Transactions on Automatic Control.

[41]  G. Filatrella,et al.  Analysis of a power grid using a Kuramoto-like model , 2007, 0705.1305.

[42]  Francesco Bullo,et al.  Synchronization and power sharing for droop-controlled inverters in islanded microgrids , 2012, Autom..

[43]  M. Roukes,et al.  Phase synchronization of two anharmonic nanomechanical oscillators. , 2013, Physical review letters.

[44]  Frank L. Lewis,et al.  Cooperative Control of Multi-Agent Systems: Optimal and Adaptive Design Approaches , 2013 .

[45]  W. El-Khattam,et al.  Investigating distributed generation systems performance using Monte Carlo simulation , 2006, IEEE Transactions on Power Systems.

[46]  Karl Henrik Johansson,et al.  A graph-theoretic approach on optimizing informed-node selection in multi-agent tracking control , 2014 .

[47]  Mario di Bernardo,et al.  Multiplex PI control for consensus in networks of heterogeneous linear agents , 2015, Autom..

[48]  X. Xue,et al.  Formation of phase-locked states in a population of locally interacting Kuramoto oscillators , 2013 .