A linear matrix inequality (LMI)-based robust stabilizing control is proposed in this paper for a three-phase grid-connected inverter (GCI) with L-filtered output. Previous research, such as MPC, required high computational power and precise modeling in order to obtain offset-free performance. Achieving optimal performance in the case of PI control poses a persistent chal-lenge in terms of gain tuning. This proposed control strategy effectively addresses the afore-mentioned issues by the utilization of systematic control design, incorporating integral action to mitigate the presence of offset error. The set of state feedback and integral gain is obtained by solving the LMI-based optimization problem to maximize the convergence rate to a steady state in the presence of uncertainty in the L-filter. The mentioned uncertainties are represent-ed by potential ranges of the inductor values. Output power delivery can be simply regulated by a computed reference state using a given power reference and measured grid current and voltage. The effectiveness of the proposed method is verified through simulations. The pro-posed robust control method demonstrates a significant decrease in ripple, with a reduction of 86.66% when compared to the conventional PI control approach.