This paper presents a variable structure control in natural frame for a three-phase voltage source inverter. The proposed control method is based on modifying the converter model in natural reference frame, preserving the low frequency state space variables dynamics. Using this model in a Kalman filter, the system state-space variables are estimated allowing to design three robust current sliding-mode controllers in natural frame. The main closed-loop features of the proposed method are: 1) robustness against grid inductance variations because the proposed model is independent of the grid inductance, 2) the power losses are reduced since physical damping resistors are avoided, and 3) the control bandwidth can be increased due to the combination of a variable hysteresis controller with a Kalman filter. To complete the control scheme, a theoretical stability analysis is developed. Finally, selected experimental results validate the proposed control strategy and permit illustrating all its appealing features.