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This paper analyzes the stability problem of the grid-connected voltage-source inverter (VSI) with LC filters, which demonstrates that the possible grid-impedance variations have a significant influence on the system stability when conventional proportional-integrator (PI) controller is used for grid current control. As the grid inductive impedance increases, the low-frequency gain and bandwidth of the PI controller have to be decreased to keep the system stable, thus degrading the tracking performance and disturbance rejection capability. To deal with this stability problem, an H∞ controller with explicit robustness in terms of grid-impedance variations is proposed to incorporate the desired tracking performance and the stability margin. By properly selecting the weighting functions, the synthesized H∞ controller exhibits high gains at the vicinity of the line frequency, similar to the traditional proportional-resonant controller; meanwhile, it has enough high-frequency attenuation to keep the control loop stable. An inner inverter-output-current loop with high bandwidth is also designed to get better disturbance rejection capability. The selection of weighting functions, inner inverter-output-current loop design, and system disturbance rejection capability are discussed in detail in this paper. Both simulation and experimental results of the proposed H∞ controller as well as the conventional PI controller are given and compared, which validates the performance of the proposed control scheme.