Finite-time consensus on the median value with robustness properties

In this paper we propose a novel continuous-time protocol that solves the consensus problem on the median value, i.e., it provides distributed agreement in networked multi-agent systems where the quantity of interest is the median value of the agents’ initial values. In contrast to the average value, the median value is a statistical measure inherently robust to the presence of outliers, which is a significant robustness issue in large-scale sensor and multi-agent networks. The proposed protocol requires only binary information regarding the relative state differences among the neighboring agents and achieves consensus on the median value in finite time by exploiting a suitable ad-hoc discontinuous local interaction rule. In addition, we characterize certain resiliency properties of the proposed protocol against the presence of uncooperative agents which do not implement the underlying local interaction rule whereas they interact with their neighbors thus influencing the network. In particular, we prove that despite the persistent influence of (at most) a certain number of uncooperative agents, the cooperative agents achieve finite time consensus on a value lying inside the convex hull of the cooperative agents’ initial conditions, provided that the special class of so-called “ k -safe” network topology is considered. Capabilities of the proposed consensus protocol and its effectiveness are supported by numerical studies.