The transition to sustainable energy systems demands robust and reliable control strategies to ensure stability and performance under dynamic and uncertain conditions. This paper presents an integral quadratic constraint (IQC) based framework for robust control synthesis and analysis in a microgrid, with guarantees of robust stability and performance. The IQC theory offers a powerful mathematical framework for analyzing the impact of uncertainties and nonlinearities in energy systems, providing more generalized and flexible tools compared to the widely used structured singular value i.e., μ-synthesis. The application in the microgrid control system synthesis demonstrates the superior performance and robustness of the proposed IQC-based framework in the presence of parameter variations and exogenous disturbances. This paper bridges the gap between advanced control theory and its practical applications in microgrids, offering a new perspective for researchers and stakeholders seeking innovative solutions for the control system design and analysis.