Islanded microgrids have low real and reactive power generation capacity and low synchronous inertia. This makes them susceptible to large frequency and voltage deviations caused by switching of large loads or equipment failures. These events deteriorate power quality and can cause frequency or voltage collapse. Grid-supporting lithium-ion battery energy storage units are a possible solution to this problem as they are able to respond quickly to changes of their real and reactive power set-points. In this paper, a decentralized grid-supporting inverter control structure for lithium-ion battery energy-storage units is proposed. A fixed-structure controller manipulates the energy storage unit's real and reactive power set-points, enabling fast response to voltage and frequency fluctuations of the microgrid. To tune the controller, no parametric model of the microgrid is required. Only the frequency-domain data, computed from a signal injection approach, are used to minimize H1 performance criteria. The performance of the proposed controller is verified through realtime software-in-the-loop simulation on a detailed microgrid model, where it is compared with a conventional inverse-droop controller. It is found that the proposed controller can effectively minimize the impact of a wide range of disturbances on the microgrid's voltage and frequency.