This paper presents an intelligent data-driven control strategy, termed Robust Artificial Neural Network Tracking Control (RANNTC), for enhancing the performance of power converters used in DC microgrid applications interfaced with the AC grid. The RANNTC employs a recurrent radial basis function neural network architecture with an online learning algorithm to provide robust voltage regulation and power sharing capabilities for two distinct converters. The first is a fully controlled rectifier that regulates the DC bus voltage while acting as a slack bus to maintain power balance within the DC network. The second is a bidirectional AC/DC converter that enables controlled power flow between the AC and DC grids while operating at unity power factor. The RANNTC controller design integrates an online learning algorithm based on the gradient-descent method to adapt the neural network weights, centers, and widths, enhancing function approximation accuracy. Comprehensive simulation studies validate the proposed strategy's effectiveness under various operating scenarios, including start-up, step changes in reference voltage, load variations, and parameter uncertainties. Comparative analyses with conventional proportional-integral (PI) control highlight the RANNTC's superior dynamic performance, faster settling times, and lower voltage overshoots/undershoots. A hardware testbed is developed, and real-time implementation of the RANNTC using a dSPACE platform is demonstrated. Experimental results corroborate the simulation findings, showcasing the RANNTC's robustness against parameter variations and its ability to maintain low total harmonic distortion in the AC line currents under diverse operating conditions, offering an effective solution for enhancing DC microgrid operation.