This article presents a voltage sensorless control design for an LCL-filtered grid-connected inverter in discrete-time domain. The proposed scheme comprises a frequency adaptive current controller and a discrete current-type observer based on the linear quadratic regulator approach, which is robust against the harmonic distortion and frequency variation of grid voltage. A frequency adaptive observer is studied in detail by means of Lyapunov stability theory to ensure high accuracy tracking performance of estimated system variables and grid voltages under different grid conditions. Since the proposed method also generates precise grid frequency and phase angle to facilitate the synchronization task, the conventional phase-locked loop structure can be effectively eliminated. Simulation and experimental results are presented to validate the feasibility of the proposed control scheme.