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.