Analysis and design of distributed algorithms for X-consensus

This paper presents analysis and design results for distributed consensus algorithms in multi-agent networks. We consider arbitrary 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. 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 conclude with two distributed algorithms that achieve, respectively, max and min consensus in finite time