The consensus problem in networks with transmission delays

We study discrete- and continuous-time consensus problems on networks in the presence of distributed time delays. We focus on information transmission delays, as opposed to information processing delays, so that each node of the network compares its current state with the past states of its neighbours. We consider directed and weighted networks where the connection structure is described by a normalized Laplacian matrix and show that consensus is achieved if and only if the underlying graph contains a directed spanning tree. This statement holds independently of the transmission delays, which is in contrast to the case of processing delays. Furthermore, we calculate the consensus value explicitly, and show that it is determined by the history of the system over an interval of time, unlike the case of processing delays where the consensus value depends only on the initial state of the system at time zero. This provides the consensus algorithm with improved robustness against noise.

[1]  J. Cronin.  A stability criterion , 1971 .

[2]  M. Degroot.  Reaching a Consensus , 1974 .

[3]  J. Hale.  Theory of Functional Differential Equations , 1977 .

[4]  Yoshiki Kuramoto,et al.  Chemical Oscillations, Waves, and Turbulence , 1984, Springer Series in Synergetics.

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

[6]  Nancy A. Lynch,et al.  Distributed Algorithms , 1992, Lecture Notes in Computer Science.

[7]  R. Merris.  Laplacian matrices of graphs: a survey , 1994 .

[8]  Vicsek,et al.  Novel type of phase transition in a system of self-driven particles. , 1995, Physical review letters.

[9]  Fan Chung,et al.  Spectral Graph Theory , 1996 .

[10]  Richard M. Murray,et al.  INFORMATION FLOW AND COOPERATIVE CONTROL OF VEHICLE FORMATIONS , 2002 .

[11]  M. G. Earl,et al.  Synchronization in oscillator networks with delayed coupling: a stability criterion. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  Richard M. Murray,et al.  Consensus problems in networks of agents with switching topology and time-delays , 2004, IEEE Transactions on Automatic Control.

[13]  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).

[14]  R. Olfati-Saber,et al.  Consensus Filters for Sensor Networks and Distributed Sensor Fusion , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[15]  Silviu-Iulian Niculescu,et al.  Stability of Traffic Flow Behavior with Distributed Delays Modeling the Memory Effects of the Drivers , 2007, SIAM J. Appl. Math..

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

[17]  Fatihcan M Atay,et al.  Clustered chimera states in delay-coupled oscillator systems. , 2008, Physical review letters.

[18]  Karl Henrik Johansson,et al.  Consensus under communication delays , 2008, 2008 47th IEEE Conference on Decision and Control.

[19]  Wim Michiels,et al.  Consensus Problems with Distributed Delays, with Application to Traffic Flow Models , 2009, SIAM J. Control. Optim..

[20]  Fatihcan M. Atay.  Consensus in networks under transmission delays and the normalized Laplacian , 2010 .

[21]  Frank Bauer.  Normalized graph Laplacians for directed graphs , 2011 .

[22]  Wenlian Lu,et al.  Mathematik in den Naturwissenschaften Leipzig Consensus and synchronization in discrete-time networks of multi-agents with stochastically switching topologies and time delays by , 2011 .

[23]  Y. Ershov.  A stability criterion , 2012 .