In this paper, a fully distributed hierarchical control strategy is proposed for operating networked grid-supporting inverters (GSIs) in islanded ac microgrids (MGs). The primary control level implements frequency and voltage control of an ac MG through a cascaded structure, consisting of a droop control loop, a virtual impedance control loop, a mixed ${H_2}/{H_\infty }$ -based voltage control loop, and a sliding-mode-control-based current loop. Compared to conventional proportional-plus-integral-based cascaded control, the proposed cascaded control does not require a precise model for the GSI system. The proposed secondary control level implements distributed-consensus-based economic automatic generation control and distributed automatic voltage control, which integrates the conventional secondary control and tertiary control into a single control level by bridging a gap between traditional secondary control and tertiary control. Simulation results demonstrate the effectiveness of the proposed hierarchical control strategy.