In the context of weak grids, vector-controlled modular multilevel converters (MMC) suffer from issues such as low inertia, low damping, and poor system stability. To address these challenges, this paper proposes a control strategy for virtual synchronous generators (VSGs) based on a fuzzy logic control algorithm. The conFtrol strategy leverages the capability of fuzzy algorithms to handle the fuzziness and uncertainty of input signals, enabling adaptive control of the virtual inertia and damping coefficient of the VSG, thus empowering the system with autonomous frequency and voltage regulation capabilities. When the system deviates from or approaches the stable operating point, increasing or decreasing the virtual inertia allows for dynamic adjustment of the virtual inertia and damping coefficient in response to load fluctuations during MMC operation. Through simulation verification, it is demonstrated that the proposed control method provides inertia support to the system during sudden changes in active load during MMC grid-connected operation. This control method achieves adaptive adjustment of the virtual inertia and damping coefficient, effectively enhancing system stability. The simulation results validate the effectiveness and correctness of the proposed control strategy.