The finite control set model predictive current control(FCS-MPCC)can significantly improve the dynamic performance of the five-phase permanent magnet synchronous motor (5P-PMSM), but its control performance highly depends on the accuracy of the model parameters. The parameter error compensation based on the first-order sliding mode observer can improve the robustness of FCS-MPCC under Model parameter mismatch. The introduction of the sign function leads to chattering in the observation results. In order to eliminate this chattering, a first-order low-pass filter needs to be added for compensation. However, the introduction of filters will increase system design difficulty and bring system delays, which will affect the compensation effect. Therefore, a second-order sliding mode observer, based on the variable-gain Super-Twisting algorithm, is proposed to realize the parameter error estimation and one-beat delay compensation in this paper. By establishing the Lyapunov function, the stability proof of the proposed observer, and the calculation method of the observer parameters are given. Besides, the third harmonic current can be suppressed, and the computational burden of the control unit can be reduced by controlling the proportion of medium vector and large vector reasonably. According to the deadbeat control, the optimal virtual voltage vector action time in each cycle is obtained, which improves the tracking ability of the reference current. At last, Extensive experimental results validate that i) the proposed observer can significantly improve the robustness of model predictive current control even under parameters mismatch; ii) the proposed control method can effectively enhance the dynamic performance of the 5-phase PMSM.