The conventional vector control method is vulnerable to the influence of uncertainties such as load disturbance and parameters perturbation due to conservative selection of sliding mode gain. Thus a robust H∞ sliding mode speed controller and current controller based on Linear Matrix Inequality (LMI) are designed. When designing the speed controller, the solution of the H∞-state feedback controller gain is transformed into the solution of the LMI with constraints, so that the H∞-state feedback controller gain is obtained, and the sliding mode surface is formed. Furthermore, a sliding mode control law based on being norm-bounded is designed. This enables the state of the system to reach the required sliding mode surface within a limited time. When designing the current controller, the sliding mode surface and sliding mode control law are designed in the same way. The composite term in the sliding mode control law is estimated by an adaptive method which eliminates the conservative selection of sliding mode switch gain. Finally, the effectiveness of the proposed control strategy is verified by simulation and experiment. This work is supported by the National Natural Science Foundation of China (No. 51974151) and Guiding Program of Liaoning Natural Science Foundation (No. 20180550438).