Necessary and Sufficient Conditions for Leader-Following Bipartite Consensus With Measurement Noise

Abstract

This paper considers leader-following bipartite consensus of single-integrator multiagent systems in the presence of measurement noise. To attenuate the noise, a time-varying consensus gain ${q}$ ( ${t}$ ) is introduced into the stochastic approximation-type protocol. Necessary and sufficient conditions for ensuring a strong mean square leader-following bipartite consensus are given. In particular, in the absence of measurement noise, the convergence speed of error dynamics is dependent on the eigenvalues of Laplacian and the rate of ${\int ^{{t}}_{0}{q}({s})\text {d}{s}}$ approaching infinity. By appropriately choosing ${q}$ ( ${t}$ ), the speed of leader-following bipartite consensus convergence can be improved in a fixed communication topology. It is proven that conditions for the signed digraph to be structurally balanced and having a spanning tree are necessary and sufficient to ensure leader-following bipartite consensus, regardless of measurement noise.