﻿ 基于Super-twisting的欠驱动船舶滑模自抗扰控制
 舰船科学技术  2022, Vol. 44 Issue (8): 73-78    DOI: 10.3404/j.issn.1672-7649.2022.08.015 PDF

Sliding mode active disturbance rejection control for underactuated ship based on super-twisting
HAN Jun-qing, LI Wei, MENG Fan-bin, DU Ya-zhen, ZHU Zhi-jun, XIE Hua-wei
Tianjin Navigation Instruments Research Institute, Tianjin 300131, China
Abstract: To solve the problem of tracking control of underactuated ship under complex sea conditions, a sliding mode active disturbance rejection control method based on Super-twisting was proposed. Firstly, the tracking control was transformed into course control by line of sight (LOS) method, and a tracking differentiator was designed to obtain the first and second derivatives of the expected course. Secondly, based on the nonlinear Nomoto model, an extended state observer was designed to observe and compensate the identification errors, nonlinear terms and disturbance caused by wind and wave under third-order sea conditions to improve the robustness of the controller. A second-order sliding mode control method based on Super-twisting is designed to make the sliding mode variables converge in finite time with the presence of interference observation error, and effectively reduce the chattering phenomenon of control output, and finally achieve high precision tracking control. Finally, the coast guard boat is taken as the controlled object for simulation verification. The simulation results show that the proposed control algorithm can effectively improve the accuracy of the ship's tracking control in the third-level sea condition and enhance the robustness to disturbance.
Key words: trajectory tracking     tracking differentiator     Nomoto model     extended state observer     Super-twisting second-order sliding mode
0 引　言

1 模型描述

 \left\{ \begin{aligned} &\dot x = u\cos \psi - v\sin \psi，\\ &\dot y = u\sin \psi + v\cos \psi，\\ &\dot \psi = r ，\\ &\dot u = \frac{{\left( {m + {m_y}} \right)vr + {X_H} + {X_P} + {X_R} + {X_{{\rm{disturb}}}}}}{{m + {m_x}}} ，\\ &\dot v = \frac{{\left( {m + {m_x}} \right)ur + {Y_H} + {Y_P} + {Y_R} + {Y_{{\rm{disturb}}}}}}{{m + {m_y}}} ，\\ &\dot r = \frac{{{N_H} + {N_P} + {N_R} + {N_{{\rm{disturb}}}}}}{{{I_{zz}} + {J_{zz}}}} 。\\ \end{aligned} \right. (1)

 $\ddot \psi + \frac{K}{T}\left( {\alpha \dot \psi + \beta {{\dot \psi }^3}} \right) = \frac{K}{T}\delta + d 。$ (2)

2 控制器设计

 图 1 控制器结构框图 Fig. 1 Block diagram of controller
2.1 跟踪微分器设计

 \left\{ \begin{aligned} &{{\dot v}_1} = {v_2} ，\\ &{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \vdots \\ &{{\dot v}_{n - 1}} = {v_n}，\\ &{{\dot v}_n} = {r^n}f\left( {{v_1} - v,\frac{{{v_2}}}{r}, \cdots ,\frac{{{v_n}}}{{{r^{n - 1}}}}} \right) 。\end{aligned} \right. (3)

 \left\{ \begin{aligned} &{{\dot v}_1} = {v_2} ，\\ & {{\dot v}_2} = {v_3} ，\\ &{{\dot v}_3} = - {r^3}\left( {{v_1} - {\psi _d}} \right) - 3{r^2}{v_2} - 3r{v_3} 。\end{aligned} \right. (4)

 $G\left( s \right) = \frac{{{r^3}}}{{{s^3} + 3r{s^2} + 3{r^2}s + {r^3}}}。$ (5)

2.2 线性扩张观测器设计

 \left\{ \begin{aligned} &\dot \psi = r ，\\ &\dot r = f\left( {\dot \psi ,d} \right) + {b_0}\delta 。\end{aligned} \right. (6)

 \left\{ \begin{aligned} &{{\dot x}_1} = {x_2} ，\\ &{{\dot x}_2} = {x_3} + {b_0}\delta ，\\ & {{\dot x}_3} = h。\end{aligned} \right. (7)

 \left\{ \begin{aligned} & {{\dot {\hat x}}_1} = {{\hat x}_2} + {l_1}\left( {{x_1} - {{\hat x}_1}} \right)，\\ &{{\dot {\hat x}}_2} = {{\hat x}_3} + {l_2}\left( {{x_1} - {{\hat x}_1}} \right) + {b_0}\delta ，\\ &{{\dot {\hat x}}_3} = {l_3}\left( {{x_1} - {{\hat x}_1}} \right) 。\end{aligned} \right. (8)

2.3 Super-twisting 2阶滑模设计

 ${e}_{\psi }={x}_{1}-{\psi }_{d}\text{，}{\dot{e}}_{\psi }={\widehat{x}}_{2}-{\dot{\psi }}_{d}，$ (9)

 $s = {\dot e_\psi } + c{e_\psi }，$ (10)

 $\begin{split} \dot s =& {{\ddot e}_\psi } + c{{\dot e}_\psi }= {{\dot {\hat x}}_2} - {{\ddot \psi }_d} + c{{\dot e}_\psi } =\\ &{{\hat x}_3} + {b_0}\delta - {{\ddot \psi }_d} + c{{\dot e}_\psi } 。\end{split}$ (11)

 $\dot s = - {k_1}{\left| s \right|^{0.5}}{{\rm{sgn}}} \left( s \right) - {k_2}\int_0^t {{{\rm{sgn}}} \left( s \right){\rm{d}}\tau } 。$ (12)

 $\begin{split} \delta = &\frac{1}{{{b_0}}}({{\ddot \psi }_d} - c{{\dot e}_\psi } - {{\hat x}_3} - {k_1}{\left| s \right|^{0.5}}{{\rm{sgn}}} \left( s \right) \hfill- \\ & {k_2}\int_0^t {{{\rm{sgn}}} \left( s \right){\rm{d}}\tau } ) 。\end{split}$ (13)
3 稳定性证明 3.1 扩张观测器稳定性证明

 \left\{ \begin{aligned} &{{\tilde x}_1}{\text{ = }}{x_1} - {{\hat x}_1} ，\\ &{{\tilde x}_2}{\text{ = }}{x_2} - {{\hat x}_2}，\\ &{{\tilde x}_3}{\text{ = }}{x_3} - {{\hat x}_3} ，\\ &{{\dot {\tilde x}}_1} = {{\tilde x}_2} - {l_1}{{\tilde x}_1} ，\\ &{{\dot {\tilde x}}_2} = {{\tilde x}_3} - {l_2}{{\tilde x}_1}，\\ &{{\dot {\tilde x}}_3} = h - {l_3}{{\tilde x}_1} 。\end{aligned} \right. (14)

 \left\{ \begin{aligned} &{{\dot X}_1} = {X_2} \\ &{{\dot X}_2} = {X_3} \\ &\begin{aligned}{{\dot X}_3} = &{{\dot {\tilde x}}_3} - {l_2}{{\dot {\tilde x}}_2} - {l_1}{{\ddot {\tilde x}}_1}= \\ &- {l_3}{{\tilde x}_1} + h - {l_2}\left( {{{\tilde x}_3} - {l_2}{{\tilde x}_2}} \right) -\\ &{l_1}\left[ {{{\tilde x}_3} - {l_2}{{\tilde x}_2} - {l_1}\left( {{{\tilde x}_2} - {l_1}{{\tilde x}_1}} \right)} \right] = \\ &- {l_1}{X_3} - {l_2}{X_2} - {l_3}{X_1} + h 。\end{aligned} \end{aligned} \right. (15)

4.1 无风浪环境

 图 2 航迹追踪仿真图 Fig. 2 Simulation diagram of track tracking

 图 3 首向控制仿真图 Fig. 3 Simulation diagram of heading control

 图 4 航迹偏差仿真图 Fig. 4 Simulation diagram of tracking error

 图 5 航迹偏差局部放大图 Fig. 5 Partial enlarged detail of tracking error

 图 6 首向误差仿真图 Fig. 6 Simulation diagram of heading error

 图 7 航迹偏差局部放大图 Fig. 7 Partial enlarged detail of heading error

 图 8 输出舵角指令仿真图 Fig. 8 Simulation diagram of rudder angle
4.2 3级海况环境

 图 9 航迹偏差仿真图 Fig. 9 Simulation diagram of tracking error

 图 10 首向误差仿真图 Fig. 10 Simulation diagram of heading error

5 结　语

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