Optimal damping in decentralized formation control

We address the problem of formation control in multi–agent systems tasked with tracking a leader–imposed trajectory. Inter–agent interactions are encoded by a graph, with each node representing an agent that uses only relative position and velocity information from its neighbors for control. A subset of agents, designated as dampers, also utilizes their absolute velocity to improve damping in the formation. We formulate an optimization problem to minimize the maximum real part of the closed-loop eigenvalues, a criterion that directly governs both stability margins and the rate at which the formation recovers from disturbances. We derive an explicit solution for a line graph configuration where the leader and a damper occupy the endpoints. We also provide the globally optimal solution for a generic graph, assuming that each agent applies a damping action that must be the same across all nodes. We prove the overall robustness of the protocol against network topology variations. Theoretical findings are validated through both simulations and real-world experiments with a quadrotor swarm, which confirm that damping agents significantly improve formation integrity under dynamic conditions.