Convergence rate for consensus with delays

We study the problem of reaching a consensus in the values of a distributed system of agents with time-varying connectivity in the presence of delays. We consider a widely studied consensus algorithm, in which at each time step, every agent forms a weighted average of its own value with values received from the neighboring agents. We study an asynchronous operation of this algorithm using delayed agent values. Our focus is on establishing convergence rate results for this algorithm. In particular, we first show convergence to consensus under a bounded delay condition and some connectivity and intercommunication conditions imposed on the multi-agent system. We then provide a bound on the time required to reach the consensus. Our bound is given as an explicit function of the system parameters including the delay bound and the bound on agents’ intercommunication intervals.

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

[2]  D. Angeli,et al.  Convergence Speed of Unsteady Distributed Consensus: Decay Estimate Along the Settling Spanning-Trees , 2006, SIAM J. Control. Optim..

[3]  Pierre-Alexandre Bliman,et al.  Average consensus problems in networks of agents with delayed communications , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[4]  John N. Tsitsiklis,et al.  Problems in decentralized decision making and computation , 1984 .

[5]  M. Cao,et al.  A Lower Bound on Convergence of a Distributed Network Consensus Algorithm , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[6]  Stephen P. Boyd,et al.  Fast linear iterations for distributed averaging , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[7]  Angelia Nedic,et al.  On the rate of convergence of distributed subgradient methods for multi-agent optimization , 2007, 2007 46th IEEE Conference on Decision and Control.

[8]  J.N. Tsitsiklis,et al.  Convergence in Multiagent Coordination, Consensus, and Flocking , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

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

[10]  R. Srikant,et al.  Consensus with Quantized Information Updates , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

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

[12]  F. Fagnani,et al.  Communication constraints in coordinated consensus problems , 2006, 2006 American Control Conference.

[13]  John N. Tsitsiklis,et al.  Convergence Speed in Distributed Consensus and Averaging , 2009, SIAM J. Control. Optim..

[14]  Asuman E. Ozdaglar,et al.  Distributed Subgradient Methods for Multi-Agent Optimization , 2009, IEEE Transactions on Automatic Control.

[15]  Michael Athans,et al.  Convergence and asymptotic agreement in distributed decision problems , 1982, 1982 21st IEEE Conference on Decision and Control.

[16]  R. Carli,et al.  Average consensus on networks with transmission noise or quantization , 2007, 2007 European Control Conference (ECC).

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

[18]  John N. Tsitsiklis,et al.  Convergence Speed in Distributed Consensus and Averaging , 2009, SIAM J. Control. Optim..

[19]  Benjamin Van Roy,et al.  Consensus Propagation , 2005, IEEE Transactions on Information Theory.

[20]  Stephen P. Boyd,et al.  Gossip algorithms: design, analysis and applications , 2005, Proceedings IEEE 24th Annual Joint Conference of the IEEE Computer and Communications Societies..

[21]  J.N. Tsitsiklis,et al.  Convergence Rates in Distributed Consensus and Averaging , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[22]  R. Srikant,et al.  Quantized Consensus , 2006, 2006 IEEE International Symposium on Information Theory.

[23]  Tamer Basar,et al.  Asymptotic agreement and convergence of asynchronous stochastic algorithms , 1986, 1986 25th IEEE Conference on Decision and Control.