Fractional frequency transmission system (FFTS) is an offshore wind power energy transmission scheme, the AC/AC converter is the most important equipment in FFTS system, and modular multilevel matrix converter (MMMC) is an AC/AC converter with high-voltage and high-power. For the current control of MMMC input side and output side, the traditional proportional integral (PI) control is generally used, which has many control parameters and unsatisfactory control effect, while Lyapunov function control strategy is superior to PI control in the number of controllers, control complexity and control effect. Therefore, the Lyapunov control strategy of MMMC currents on input side and output side is proposed. According to the topology of MMMC, the current decoupling models of MMMC input side and output side are derived，and combined with Lyapunov control theory, the current mathematical model of MMMC input side and output side Lyapunov function control is established ultimately, and the global asymptotic stability of the proposed Lyapunov control is proved, the imprecision of Lyapunov function control and the parameter selection of Lyapunov control is discussed. Finally, the proposed control strategy and the traditional control strategy are applied to current control of MMMC input side and output side on MATLAB respectively to conduct experimental comparison under different working conditions, and the simulation results verify the correctness and superiority of the proposed Lyapunov control method.