Hybrid AC-DC Microgrid systems have recently emerged as a promising method for connecting AC loads with AC Microgrids (ACM) and DC loads with DC Microgrids (DCM). The Bidirectional Power Converter (BPC) allows the AC-DC Microgrid to distribute power between the ACM and DCM. Effective supervisory power sharing and robust control methods are needed to share power among the AC-DC Microgrid sources and ensure seamless operation during different uncertain conditions. Also, suitable islanding and grid synchronisation techniques are needed to connect and disconnect the Microgrid with the utility grid seamlessly. In this study, an Optimal Power Management Strategy (OPMS) is considered for the sharing of power among the sources. To implement the OPMS, an equivalent robust optimization problem for a restricted data set is used. The goal of optimization is to distribute power while maximizing the use of renewable energy, minimizing the use of conventional energy, and using minimal energy transfer through the BPC. To achieve robust tracking commitment and robust control during the parameter uncertainty, a nested loop Robust Control Scheme (RCS) is proposed. A Mixed H ∞ /H 2 Optimal (MO) controller is used in the outer loop of the RCS, while a Robust LQR (RLQR) optimal controller is used in the inner loop, the nested loop control scheme is described as MORLQR controller. An Optimal Linear Kalman filter-based SRFPLL (OLKF-SRFPLL) is proposed to achieve grid synchronization. To implement islanding detection, a Synchro-Extraction Transform along with an Optimal Support Vector Machine (SET-OSVM) classification is considered. Considering different operational conditions and uncertainty scenarios, the performance and robustness of the proposed AC-DC Microgrid with proposed power-sharing and control in both GCM and IMs and during the transition between both modes is analyzed using MATLAB/SIMULINK.