Hybrid AC/DC microgrids are of special interest for energization due to their flexibility, low infrastructure investments, reduced conversion losses, and reliability against failures on the utility grid. In these systems, global economic dispatch needs communication between the AC and DC subgrids to achieve a near-optimal solution since the incremental costs of all generators need to be equalized. Hence, this paper proposes a distributed coordination between generators by means of a finite-time controller for the microgrid's interlinking converter, which ensures an economic operation while taking care of the microgrid power utilization. The latter implies that a multi-objective control is performed by the interlinking converter, which uses shared variables of incremental costs and average powers from distributed generators in AC and DC sides. Also, an adaptive weighting method is proposed to adjust the control effort regarding the average power utilization of a side microgrid. The controller's performance is verified through simulations and experimental setups. Results show that the proposed strategy is able to perform a trade-off between the two control objectives while achieving a finite-time convergence even though communication delays exist.INDEX TERMS Economic dispatch, distributed control, finite-time control, hybrid AC/DC microgrid, multi-objective control.