This study focuses on addressing the the coupling problem of the vertical and horizontal plane of an autonomous underwater vehicle (AUV) with an X-rudder. To guarantee the steering performance of the AUV, a depth and course control algorithm based on an $L_1$ adaptive control algorithm (L1AC) is proposed. Firstly, the linear quadratic regulator (LQR) is designed on the basis of an AUV vertical and horizontal linear model, and the system has optimal control quality for both of the nominal vertical and horizontal models. Secondly, the nonlinear state predictor, disturbance estimator, and adaptive controller are designed separately, which is used for the compensation of the uncertainty and disturbance of the AUV dynamic. Finally, the rudder angle allocation algorithm is designed to realize the rudder angle command of the cross rudder to the X-rudder. The simulation and experiment results show that the adaptive control algorithm proposed in this paper can effectively control AUV to complete high-performance course and depth control. Based on the experiments, the prospects and shortcomings of the L1AC, LQR and MRAC algorithm in AUV control are compared and analyzed.