This paper considers an optimal formation tracking problem for a network of agents with multiple noncooperative and moving targets. The agents intend to not only form a desired geometric pattern but also track the unknown and time-varying centroid of the targets at an optimal distance. A hierarchical framework, which is composed of a centroid estimation level and an optimal formation tracking level, is proposed to deal with the formulated problem. In the centroid estimation level, a novel centroid estimation algorithm is established by using consensus-like protocols. To achieve accurate centroid estimation, the agents are divided into two subsets, i.e., an observation subset and a leader-following subset. The agents in the observation subset estimate the centroid of the moving targets by using their observed trajectories of the targets and the agents in the leader-following subset get an estimate on the centroid of the targets by following the references provided by the observation agents. Based on the estimated centroid, a time-varying aggregative game is designed to model the optimal formation tracking problem. By using average consensus protocols, gradient optimization algorithms, and regularization techniques, a time-varying Nash equilibrium seeking strategy is proposed. It is shown through Lyapunov stability analysis that the centroid estimation algorithm and the time-varying Nash equilibrium seeking strategy are effective. A distinguished feature of the proposed method is that formation and global time-varying optimization can be simultaneously achieved. A numerical example is provided to validate the effectiveness of the proposed method.
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