This study addresses development and implementation of robust control methods for a three-phase grid-connected voltage source inverter (VSI) accompanied by an inductive-capacitive-inductive (LCL) filter. A challenge of current control for the VSI is LCL filter resonance near to the control stability boundary, which interacts with the inverter control switching actions and creates the possibility of instability. In general, active damping is needed to stabilize the system and ensure robust performance in steady-state and dynamic responses. While many active damping methods have been proposed to resolve this issue, capacitor-current-feedback active damping has been most widely used for its simple implementation. There has been no clear consensus regarding design of a control system including capacitor-current-feedback active damping. This is due to the fact that simulation/experiment results are not congruent with the design analyses on which the control is designed. This study explains the incoherence between theory and practice when it comes to a capacitor-currents-feedback active damping system. Proposed capacitor-current-estimate active damping utilizing a developed posteriori Kalman estimator gives coherent simulation results as expected from the design analyses. This reveals that the highly oscillatory capacitor currents containing the inverter switching effects bring about uncertainty in the system performance. The switching effects are not incorporated in the analyses and control system design. Therefore, it is required to remove the switching noise from the capacitor currents in order to yield consistent results. It has been confirmed that the proportional-negative feedback of the capacitor current is equivalent to virtual impedance connected in parallel with the filter capacitor. In a digitally controlled system, the computation delay causes the equivalent resistance of the virtual impedance to become negative in the frequency range of fs/6 to fs/2, which produces a pair of open-loop unstable poles in RHP. This happens when the displaced resonance peak by active damping is in that region. Thus, an a priori Kalman estimator has been developed to generate one-sample-ahead state variable estimates to reconstruct the capacitor currents for active damping, which can compensate for the delay. The one-sample-ahead capacitor-current estimates are computed from the inverter-side and grid-side current estimates. The proposed method provides extended limits of the active damping gain that improve robustness against system parameter variation. It also allows strong active damping which can sufficiently attenuate the resonance. Grid condition is another significant factor affecting the stability of the system. In particular, a weak grid tends to provide high impedance. The system employing the proposed active damping method stably operates in a weak grid, ensuring robustness under grid impedance variation. The developed Kalman estimators offer an effective and easy way of determining the stability status of a system in addition to the functions of filtering and estimation. Stability analysis can be easily made since state variable estimates go to infinity when a system is unstable. As a promising approach, model predictive control (MPC) has been designed for the system. This study suggests that MPC including active damping can be employed for a grid-connected VSI with an LCL filter with good dynamic performance.