图  1  函数$l(x)$曲线图

Fig.  1  The curves of function $l(x)$

图  2  外部环境扰动下船舶的期望轨迹和实际轨迹

Fig.  2  Desired trajectory and actual trajectory under external environment disturbances

图  3  外部环境扰动下期望轨迹${\pmb \eta}_d=[x_d, y_d, \psi_d]^{\rm T}$和文算法实际轨迹${\pmb \eta}=[x, y, \psi]^{\rm T}$历时曲线

Fig.  3  Curves of desired trajectory ${\pmb \eta}_d=[x_d, y_d, \psi_d]^{\rm T}$ and actual trajectory ${\pmb \eta}=[x, y, \psi]^{\rm T}$ with proposed controller versus time under external environment disturbances

图  4  外部环境扰动下前进速度$u$、横移速度$\nu$和艏摇角速度$r$历时曲线

Fig.  4  Curves of surge velocity $u$, sway velocity $\nu$ and yaw rate $r$ versus time under external environment disturbances

图  5  神经网络逼近历时曲线

Fig.  5  Curves of learning behavior of neural networks

图  6  控制器输出

Fig.  6  Curves of controller outputs

图  7  外部环境扰动和逼近误差的界$\delta_1$, $\delta_2$, $\delta_3$及其估计值$\hat{\delta}_1$, $\hat{\delta}_2$, $\hat{\delta}_3$历时曲线

Fig.  7  Curves of the bounds of external environment disturbances and approximation errors $\delta_1$, $\delta_2$, $\delta_3$ and their estimations $\hat{\delta}_1$, $\hat{\delta}_2$, $\hat{\delta}_3$ verse time with proposed controller

图  8  相同海况下本文算法与常规动态面算法跟踪性能比较

Fig.  8  Comparison of tracking errors under the same sea conditions between the proposed method and the dynamic surface control method