This paper presents a frequency-adaptive current control design for a grid-connected inverter (GCI) with an inductive-capacitive-inductive (LCL) filter in the presence of grid disturbance such as the grid frequency variation and grid voltage harmonic distortion as well as polytopic uncertainties in the LCL filter parameters. The grid current control is achieved by augmenting integral and resonant terms into the LCL-filtered inverter system model to constitute integral-resonant full-state feedback control for zero steady-state error and current harmonic attenuation. To realize the full-state feedback control, the information on all the state variables is essential. However, additional sensors for state measurements increase the implementation cost as well as the complexity. To overcome this issue, a full-state discrete-time observer is employed in the stationary reference frame. Furthermore, to maintain the quality of grid currents injected into the grid, a frequency-adaptive current control is introduced. For this aim, the grid frequency is estimated through an adaptive observer rapidly and precisely. Then, the estimated grid frequency is used to adaptively change the frequency information in the augmented resonant controller for the purpose of producing high-quality grid currents even under both distorted grid voltages and grid frequency variation. In addition, to ensure the robustness against LCL filter parameter perturbation, a linear matrix inequality-linear quadratic regulator (LMI-LQR) approach is proposed for polytopic uncertainties in the LCL filter parameters to design full-state feedback control as well as a full-state observer. To verify the effectiveness of the proposed control scheme, the simulation and experimental results are given.