Distributed algorithms for reaching consensus on general functions

This paper presents analysis and design results for distributed consensus algorithms in multi-agent networks. We consider continuous consensus functions of the initial state of the network agents. Under mild smoothness assumptions, we obtain necessary and sufficient conditions characterizing any algorithm that asymptotically achieves consensus. This characterization is the building block to obtain various design results for networks with weighted, directed interconnection topologies. We first identify a class of smooth functions for which one can synthesize in a systematic way distributed algorithms that achieve consensus. We apply this result to the family of weighted power mean functions, and characterize the exponential convergence properties of the resulting algorithms. We establish the validity of these results for scenarios with switching interconnection topologies. Finally, we conclude with two discontinuous distributed algorithms that achieve, respectively, max and min consensus in finite time.

[1]  P. Bullen.  Handbook of means and their inequalities , 1987 .

[2]  A. Bacciotti,et al.  Stability and Stabilization of Discontinuous Systems and Nonsmooth Lyapunov Functions , 1999 .

[3]  Seif Haridi,et al.  Distributed Algorithms , 1992, Lecture Notes in Computer Science.

[4]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[5]  Peng Gao.  Certain Bounds for the Differences of Means , 2002 .

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

[7]  Éva Tardos,et al.  Network games , 2004, STOC '04.

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

[9]  J. Shamma,et al.  Belief consensus and distributed hypothesis testing in sensor networks , 2006 .

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

[11]  Loh Hooi-Tong,et al.  On a Class of Directed Graphs - With an Application to Traffic-Flow Problems , 1970, Oper. Res..

[12]  Luc Moreau,et al.  Stability of multiagent systems with time-dependent communication links , 2005, IEEE Transactions on Automatic Control.

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

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

[15]  Brian D. O. Anderson,et al.  The Multi-Agent Rendezvous Problem. Part 1: The Synchronous Case , 2007, SIAM J. Control. Optim..

[16]  Masafumi Yamashita,et al.  Distributed memoryless point convergence algorithm for mobile robots with limited visibility , 1999, IEEE Trans. Robotics Autom..

[17]  Randal W. Beard,et al.  Consensus seeking in multiagent systems under dynamically changing interaction topologies , 2005, IEEE Transactions on Automatic Control.

[18]  Kristi A. Morgansen,et al.  Discrete time Kuramoto models with delay , 2006 .

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

[20]  Mehdi Behzad,et al.  Graphs and Digraphs , 1981, The Mathematical Gazette.

[21]  Mireille E. Broucke,et al.  Local control strategies for groups of mobile autonomous agents , 2004, IEEE Transactions on Automatic Control.

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

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

[24]  Brian D. O. Anderson,et al.  The Multi-Agent Rendezvous Problem. Part 2: The Asynchronous Case , 2007, SIAM J. Control. Optim..

[25]  Stephen P. Boyd,et al.  A scheme for robust distributed sensor fusion based on average consensus , 2005, IPSN 2005. Fourth International Symposium on Information Processing in Sensor Networks, 2005..

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

[28]  Audra E. Kosh,et al.  Linear Algebra and its Applications , 1992 .

[29]  Richard M. Murray,et al.  Approximate distributed Kalman filtering in sensor networks with quantifiable performance , 2005, IPSN 2005. Fourth International Symposium on Information Processing in Sensor Networks, 2005..

[30]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[31]  Sonia Martínez,et al.  Robust rendezvous for mobile autonomous agents via proximity graphs in arbitrary dimensions , 2006, IEEE Transactions on Automatic Control.

[32]  R. Pesenti,et al.  Distributed Consensus in Networks of Dynamic Agents , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.