Distributed shortest distance consensus problem in multi-agent systems

In this paper, a distributed shortest-distance consensus problem is studied under dynamically changing network topologies. Here each agent is assigned to a certain convex region and has information about only its own region. All these regions might not have an intersection. Through local interaction with their neighbors, multiple agents need to collectively converge to a common location that has the shortest total squared distances to these regions. First, a distributed algorithm with a signum function is introduced. A corresponding condition is given to guarantee that all agents finally reach consensus while minimizing the distance performance function. Second, a continuous approximation algorithm is then proposed to remove possible chattering. It is proved that all agents finally converge to a consensus zone where the distance between every two connected agents is no larger than the algorithm parameter. A bound of the difference between the given distance performance function and its minimum is derived, which is proportional to the algorithm parameter. Third, a distributed algorithm is proposed to take into account the case where the agents´ regions are time varying. It is shown that the agents reach consensus while minimizing the distance performance function when the regions do not vary abruptly.