Designing Communication Topologies for Optimal Synchronization Trajectories of HomogeneousLinear Multi-Agent Systems

In this paper the synchronization of homogeneous linear multi-agent systems is considered. In this process, all agents are required to converge to a common trajectory called synchronization trajectory. Without synchronization every sin- gle agent would have followed its own autonomous trajectory determined by its initial state. Hence, we interpret the initial states of the agents as their respective preferences. We define a cost function that penalizes the compromise each agent has to make when the synchronization trajectory differs from the agent's preferred trajectory. Using classical synchronization controllers, the synchronization trajectory essentially depends on the communication topology and the initial states of the agents. Therefore, we pose optimization problems concerned with finding the communication topology yielding minimal cost, i.e. the synchronization trajectory that constitutes an optimal compromise forall agents. In this respect, we minimize the cost for a given initial state, for the worst-case as well as for the average over all admissible initial states of the agents. The introduced minimization problems are reformulated as semidefinite programs so that they can be efficiently solved. A numerical example illustrates the results.