Distributed Finite-Time Average Consensus in Digraphs in the Presence of Time Delays

Most algorithms for distributed averaging only guarantee asymptotic convergence. This paper introduces a distributed protocol that allows nodes to find the exact average of the initial values in a finite and minimum number of steps on interconnection topologies described by strongly connected directed graphs (digraphs). More specifically, under the assumption that each component has knowledge of the number of its outgoing links (i.e., the number of components to which it sends information), we show that the average value can be computed based on local observations over a finite time interval. The average can be obtained in a finite number of steps even when the information exchange is subject to delays. The proposed algorithm is the first in the literature that allows for distributed computation of the exact average in digraphs in finite time, with and without delays.

[1]  Christoforos N. Hadjicostis,et al.  Coordination and Control of Distributed Energy Resources for Provision of Ancillary Services , 2010, 2010 First IEEE International Conference on Smart Grid Communications.

[2]  Paul A. Beardsley,et al.  Image and animation display with multiple mobile robots , 2012, Int. J. Robotics Res..

[3]  Long Wang,et al.  Finite-Time Consensus Problems for Networks of Dynamic Agents , 2007, IEEE Transactions on Automatic Control.

[4]  Kai Cai,et al.  Average consensus on general strongly connected digraphs , 2012, Autom..

[5]  Christoforos N. Hadjicostis,et al.  Average Consensus in the Presence of Delays in Directed Graph Topologies , 2014, IEEE Transactions on Automatic Control.

[6]  Mikael Johansson,et al.  A Randomized Incremental Subgradient Method for Distributed Optimization in Networked Systems , 2009, SIAM J. Optim..

[7]  Jorge Cortés,et al.  Finite-time convergent gradient flows with applications to network consensus , 2006, Autom..

[8]  Raphaël M. Jungers,et al.  Graph diameter, eigenvalues, and minimum-time consensus , 2012, Autom..

[9]  Reza Olfati-Saber,et al.  Flocking for multi-agent dynamic systems: algorithms and theory , 2006, IEEE Transactions on Automatic Control.

[10]  J. Rafael Sendra,et al.  Rank of a Hankel matrix over Z[x1,..., xr] , 2005, Applicable Algebra in Engineering, Communication and Computing.

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

[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]  John N. Tsitsiklis,et al.  Parallel and distributed computation , 1989 .

[14]  Ling Shi,et al.  Decentralised minimum-time consensus , 2013, Autom..

[15]  Jie Lin,et al.  Coordination of groups of mobile autonomous agents using nearest neighbor rules , 2003, IEEE Trans. Autom. Control..

[16]  John N. Tsitsiklis,et al.  On distributed averaging algorithms and quantization effects , 2007, 2008 47th IEEE Conference on Decision and Control.

[17]  Sanjay P. Bhat,et al.  Finite-Time Semistability and Consensus for Nonlinear Dynamical Networks , 2008, IEEE Transactions on Automatic Control.

[18]  Damiano Varagnolo,et al.  Asynchronous Newton-Raphson Consensus for Distributed Convex Optimization , 2012 .

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

[20]  Luca Schenato,et al.  Average TimeSync: a consensus-based protocol for time synchronization in wireless sensor networks1 , 2009 .

[21]  Christoforos N. Hadjicostis,et al.  Asynchronous coordination of distributed energy resources for the provisioning of ancillary services , 2011, 2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[22]  Ling Shi,et al.  Decentralised final value theorem for discrete-time LTI systems with application to minimal-time distributed consensus , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[23]  Leonard J. Schulman,et al.  On matrix factorization and scheduling for finite-time average-consensus , 2010 .

[24]  Leonidas Georgopoulos,et al.  Definitive Consensus for Distributed Data Inference , 2011 .

[25]  Christoforos N. Hadjicostis,et al.  Average consensus in the presence of dynamically changing directed topologies and time delays , 2014, 53rd IEEE Conference on Decision and Control.

[26]  Luca Schenato,et al.  Average TimeSynch: A consensus-based protocol for clock synchronization in wireless sensor networks , 2011, Autom..

[27]  Asuman E. Ozdaglar,et al.  Opinion Fluctuations and Disagreement in Social Networks , 2010, Math. Oper. Res..

[28]  John N. Tsitsiklis,et al.  Distributed Asynchronous Deterministic and Stochastic Gradient Optimization Algorithms , 1984, 1984 American Control Conference.

[29]  Panganamala Ramana Kumar,et al.  Toward a theory of in-network computation in wireless sensor networks , 2006, IEEE Communications Magazine.

[30]  Damiano Varagnolo,et al.  Newton-Raphson Consensus for Distributed Convex Optimization , 2015, IEEE Transactions on Automatic Control.

[31]  Christoforos N. Hadjicostis,et al.  Distributed Weight Balancing Over Digraphs , 2014, IEEE Transactions on Control of Network Systems.

[32]  I. Couzin,et al.  Effective leadership and decision-making in animal groups on the move , 2005, Nature.

[33]  George Vogiatzis,et al.  A probabilistic model for trust and reputation , 2010, AAMAS.

[34]  Alain Y. Kibangou,et al.  Graph Laplacian based matrix design for finite-time distributed average consensus , 2012, 2012 American Control Conference (ACC).

[35]  Dennis S. Bernstein,et al.  Finite-Time Stability of Continuous Autonomous Systems , 2000, SIAM J. Control. Optim..

[36]  C.N. Hadjicostis,et al.  Finite-Time Distributed Consensus in Graphs with Time-Invariant Topologies , 2007, 2007 American Control Conference.

[37]  M Franceschelli,et al.  Distributed Averaging in Sensor Networks Based on Broadcast Gossip Algorithms , 2011, IEEE Sensors Journal.