The Distributed Biased Min-Consensus Protocol Revisited: Pre-Specified Finite Time Control Strategies and Small-Gain Based Analysis

Unlike the classical distributed consensus protocols that enable a group of agents to reach an agreement regarding a certain quantity of interest in a distributed fashion, the distributed biased min-consensus protocol (DBMC) has been proven to handle the advanced complexity of solving the shortest path problem. Such a protocol is commonly incorporated as the first step of a hierarchical architecture in real applications, such as robot path planning and the management of dispersed computing services. However, a major limitation of DBMC is the lack of results regarding its convergence within a user-assigned time frame. In this paper, we first propose two control strategies to ensure that the state error of DBMC decreases exactly to zero or a desired level within a finite time specified by the user. This paper further investigates the nominal DBMC itself. By leveraging small-gain based stability tools and embedding DBMC into a framework applicable to such tools, this paper also proves the global exponential input-to-state stability of DBMC, outperforming its current stability results. Simulations are provided to validate the efficacy of our theoretical results.