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.

中文翻译：

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带测量噪声的Leader-Follow 二方共识的充要条件

本文考虑了在存在测量噪声的情况下单集成器多智能体系统的领导者跟随二方共识。为了减弱噪音，随时间变化的共识增益 ${q}$ ( ${t}$ ) 被引入到随机近似型协议中。给出了保证强均方领导跟随二方共识的充分必要条件。特别是在没有测量噪声的情况下，误差动力学的收敛速度取决于拉普拉斯算子的特征值和 ${\int ^{{t}}_{0}{q}({s})\text {d}{s}}$ 接近无穷大。By appropriately choosing ${q}$ ( ${t}$ )，在固定的通信拓扑中可以提高leader-follow双方共识收敛的速度。事实证明，有符号有向图在结构上平衡并具有生成树的条件是必要且充分的，以确保领导者遵循两方共识，而不管测量噪声如何。