﻿ 基于速度观测器的欠驱动船舶自适应滑模控制
 舰船科学技术  2023, Vol. 45 Issue (13): 48-52    DOI: 10.3404/j.issn.1672-7649.2023.13.010 PDF

Adaptive sliding mode control of underactuated ship based on velocity observer
SUN Hao-nan, CHEN Shi-cai, ZHANG Jian, LIU Yan-zhi
Dalian Maritime University, Dalian 116026, China
Abstract: Underactuated ship has some problems such as unknown disturbance and difficult speed measurement when sailing at sea. These have brought problems to ship motion controller design. In order to solve the above problems, an adaptive sliding mode path following controller with integral sliding mode surface is proposed in this paper. Adaptive law is introduced to compensate the disturbance and perturbation of model parameters, and an improved nonlinear observer is used to observe the velocity which is difficult to measure. System stability is proved by Lyapunov theory. Finally, a comparative simulation of the path following control is carried out in a simulated ocean environment, which proves the effectiveness and superiority of the control algorithm.
Key words: underactuated ship     path following     integral sliding mode surface     adaptive sliding mode control     nonlinear observer
0 引　言

1 船舶数学模型

 图 1 船舶运动平面图 Fig. 1 Ship motion plan

 \left\{ \begin{aligned} & \dot x = u\cos \varphi - v\sin \varphi {\text{ = }}\sqrt {{u^2} + {v^2}} \cos (\varphi + \beta ) ，\\& \dot y = u\sin \varphi + vcos\varphi {\text{ = }}\sqrt {{u^2} + {v^2}} \sin (\varphi + \beta ) ，\\& u = \frac{{{m_{22}}}}{{{m_{11}}}}vr - \frac{{{X_u}}}{{{m_{11}}}}u - \frac{{{X_{\left| u \right|u}}}}{{{m_{11}}}}\left| u \right|u + \frac{1}{{{m_{11}}}}{F_u} + {d_{wu}} ，\\& v = \frac{{{m_{11}}}}{{{m_{22}}}}ur - \frac{{{Y_v}}}{{{m_{22}}}}v - \frac{{{Y_{\left| v \right|v}}}}{{{m_{22}}}}\left| v \right|v + {d_{wv}} ，\\& r = \frac{{{m_{11}} - {m_{22}}}}{{{m_{33}}}}uv - \frac{{{N_r}}}{{{m_{33}}}}r - \frac{{{N_{\left| r \right|r}}}}{{{m_{33}}}}\left| r \right|r + \frac{1}{{{m_{33}}}}{T_r} + {d_{wr}} 。\end{aligned} \right. (1)

 \left\{ \begin{aligned} & \dot \varphi {\text{ = }}r ，\\& \ddot \varphi + \frac{1}{T}\dot \varphi + \frac{\alpha }{T}{{\dot \varphi }^3}{\text{ = }}\frac{K}{T}\delta + f ，\\& f = (\vartriangle K{\text{ + }}\vartriangle T{\text{ + }}d) ，\\& \dot \delta = - \frac{{{K_e}}}{{{T_e}}}\delta + \frac{{{K_e}}}{{{T_e}}}{\delta _r} 。\end{aligned} \right. (2)

 \left\{ \begin{aligned} &{{\dot x}_1} = {x_2} ，\\& {{\dot x}_2} = - \frac{\alpha }{T}{x_2}^3 - \frac{1}{T}{x_2} + \frac{K}{T}u + f 。\end{aligned} \right. (3)
2 控制器设计

1）船舶质量均匀，忽略海浪对船体的影响作用。

2）船舶的实时位置（xy）是可以被获得的，且路径光滑。

3）船舶速度和外界干扰难以测量。对于前进速度u、横漂速度vm、外界海洋环境扰动f存在未知常量 $\bar u$ $\bar f$ ，满足 ${v_m} \ll u \lt \bar u \lt \infty$ $f \lt \bar f \lt \infty$

4）船舶加速度远小于船舶位移。

2.1 自适应滑模控制器设计

 ${y_e} = y - {y_d} 。$ (4)

 ${\dot V_0} = {y_e}\dot y = {y_e}\sqrt {{u^2} + {v^2}} sin(\varphi {\text{ + }}\beta )。$ (5)

 ${\varphi _{\text{d}}}{\text{ = - }}{{{k}}_{\text{0}}}\arcsin ({k_1}{y_e}) - \beta 。$ (6)

 $\left\{ \begin{gathered} e = {x_1} - {\varphi _d} ，\\ \dot e = {x_2} - {{\dot \varphi }_d} ，\\ \ddot e = {{\dot x}_2} - {{\ddot \varphi }_d} 。\\ \end{gathered} \right.$ (7)

 $s = {x_2} - \int [ {\ddot \varphi _d} - {b_0}\dot e - {b_1}e]{\rm{d}}t ，$ (8)

 $\dot s = {\dot x_2} - {\ddot \varphi _d}{\text{ + }}{b_0}\dot e{\text{ + }}{{\text{b}}_1}e = \ddot e + {b_0}\dot e + {b_1}e。$ (9)

 ${{\rm{sgn}}} (t) = \left\{ \begin{gathered} - 1,t \lt 0 ，\\ 0,t = 0 ，\\ 1,t \gt 0 。\\ \end{gathered} \right.$ (10)

 $\begin{split} & - \frac{\alpha }{T}{x_2}^3 - \frac{1}{T}{x_2} + \frac{K}{T}u + f - {{\ddot \varphi }_d} + {b_0}\dot e + {b_1}e =\\& - \eta {{\rm{sgn}}} (s) - ks。\end{split}$ (11)

 $\begin{split} & u = \frac{T}{K}\Bigg(\frac{1}{T}{x_2} + \frac{\alpha }{T}{x_2}^3 - \hat f + {{\ddot \varphi }_d} - {b_0}\dot e - {b_1}e -\\& \eta {{\rm{sgn}}} (s) - ks\Bigg)。\end{split}$ (12)

 $\begin{gathered} {{\dot V}_1} = s \dot s + \frac{1}{\gamma }\tilde f \dot {\hat f } = s(\ddot e + {b_0} \dot e + {b_1}e) + \frac{1}{\gamma } \tilde f \dot{ \hat f} 。\\ \end{gathered}$ (13)

 \begin{aligned}[b] & {{\dot V}_1} = s( - \eta {{\rm{sgn}}} (s) - ks + f - \hat f) - \frac{1}{\gamma }\tilde f\dot {\hat f } =\\ & - \eta \left| s \right| - k{s^2} - \tilde f\Bigg(\frac{1}{\gamma }\dot{ \hat f} - s\Bigg) 。\end{aligned} (14)

 $\dot{ \hat f }= \gamma s 。$ (15)

 \left\{ \begin{aligned} &\hat u = \hat {\dot x}\cos \varphi + \hat {\dot y}\sin \varphi，\\ & {{\hat v}_m} = - \hat {\dot x}\sin \varphi + \hat {\dot y}\cos \varphi 。\\ \end{aligned} \right. (16)

x为例参考文献[14]，设计一种改进的非线性观测器。

 \left\{ \begin{aligned} &\dot {\hat x} = \hat {\dot x} - {c_0}(\hat x - x)，\\ & \dot {\hat {\dot x}} = - {c_1}{\left| {\hat x - x} \right|^q}{{\rm{sgn}}} (\hat x - x) 。\\ \end{aligned} \right. (17)

2.2 控制器稳定性分析

 $\begin{gathered} {V_2} = \frac{1}{2}{y_e} + \frac{1}{2}{s^2} + \frac{1}{{2\gamma }}{{\tilde f}^2} + \frac{1}{2}{(\hat {\dot x} - \dot x)^2} + \frac{1}{2}\lambda {(\hat x - x)^2} ，\\ \end{gathered}$ (18)

 $\begin{split} {{\dot V}_2} =& {y_e}{{\dot y}_e} + s \dot s + \frac{1}{\gamma }{\tilde f} \dot {\tilde f} + ({\hat {\dot x}} - \dot x)(\dot {\hat {\dot x}} - \ddot x) + \\ &\lambda (\hat x - x)(\dot {\hat x} - \dot x) 。\end{split}$ (19)

 $\begin{split} & {{\dot V}_2} = - {y_e}\sqrt {{u^2} + {v_m}^2} \sin ({a_0}\arcsin ({a_1}{y_e})) - \\&ks - \eta s{{\rm{sgn}}} (s) - \lambda {c_0}{(\hat x - x)^2}- \\& (\hat {\dot x }- {\dot x})({c_1}{\left| {\hat x - x} \right|^q}{{\rm{sgn}}} (\hat x - x) -\\& \lambda (\hat x - x)) - \ddot x(\hat {\dot x} - \dot x) 。\end{split}$ (20)

3 仿真实验 3.1 仿真实验对象

 图 2 风向、流向示意图 Fig. 2 Wind and flow direction diagram
3.2 控制器设计参数

 图 3 控制器设计流程图 Fig. 3 Controller design flow diagram
3.3 仿真实验结果

 图 4 路径跟踪误差图 Fig. 4 Path following error diagram

 图 8 横漂速度观测值 Fig. 8 Drifting velocity observed value

 图 5 路径偏差图 Fig. 5 Path deviation diagram

 图 6 舵角变化图 Fig. 6 Rudder Angle change diagram

 图 7 前进速度观测值 Fig. 7 Forward velocity observed value
4 结　语

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