﻿ 应用关联规则的半潜式航行体稳态控制技术
 舰船科学技术  2022, Vol. 44 Issue (16): 74-78    DOI: 10.3404/j.issn.1672-7649.2022.16.014 PDF

Steady-state control technology of semi-submersible vehicle using association rules
WAN Jun
The 710 Research Institute of CSSC, Yichang 443003, China
Abstract: The semi submersible vehicle is easy to be disturbed to produce steady-state error. The steady-state control technology of semi submersible vehicle based on association rules is proposed. The mathematical model of space motion of semi submersible vehicle is established, its dynamic characteristics and the transformation relationship between ground and fluid coordinate system are analyzed, the Apriori algorithm in association rules is applied and the matrix idea is introduced to improve it, the mathematical model of space motion of semi submersible vehicle is excavated, the control parameters are selected, and a PID controller with adjustable weight is designed by using this parameter, and the semi submersible vehicle control is completed by using PID controller, The PID controller parameters with high accuracy are obtained by convolution neural network prediction, and the PID controller is continuously optimized by using this parameter to complete the steady-state optimal control of semi submersible vehicle. The experimental results show that the turning radius of semi submersible vehicle is negatively correlated with the vertical rudder angle. The maximum and minimum turning radius are about 100 m and 25 m respectively. The sailing depth is directly proportional to the vertical rudder angle. This method has small steady-state control overshoot and fast control response speed.
Key words: association rules     semi-submersible vehicle     steady state control     PID controller     online feedback learning     convolutional neural network
0 引　言

1 半潜式航行体稳态控制 1.1 半潜式航行体空间运动数学模型 1.1.1 坐标系

 $\begin{split} {\boldsymbol{C}}_b^0 =\;& \left( \begin{array}{*{20}{c}} {\cos \theta \cos \psi }&{\sin \theta } \\ { - \sin \theta \cos \psi \cos \varphi + \sin \psi \sin \varphi }&{\cos \theta \cos \varphi }\\ {\sin \theta \cos \psi \cos \varphi + \sin \psi \cos \varphi }&{ - \cos \theta \sin \varphi }\end{array}\right.\\ &\left. \begin{array}{*{20}{c}} { - \cos \theta \sin \psi }\\ { \sin \theta \sin \psi \cos \varphi + \cos \psi \sin \varphi } \\ { - \sin \theta \sin \psi \sin \varphi + \cos \psi \sin \varphi } \end{array} \right) 。\end{split}$ (1)

 $\begin{split} {\boldsymbol{C}}_0^b =\;& \left( \begin{array}{*{20}{c}} {\cos \theta \cos \psi }&{ - \sin \theta \cos \psi \cos \varphi + \sin \psi \cos \varphi }\\ {\sin \theta }&{\cos \theta \cos \varphi }\\ { - \sin \psi \cos \theta }&{\sin \theta \sin \psi \cos \varphi + \cos \psi \sin \varphi }\end{array}\right.\\ &\left. \begin{array}{*{20}{c}} {\sin \theta \cos \psi \sin \varphi + \sin \psi \cos \varphi } \\ { - \cos \theta \sin \varphi } \\ { - \sin \theta \sin \psi \sin \varphi + \cos \psi \cos \varphi } \end{array} \right)。\end{split}$ (2)
1.1.2 空间运动方程组

 $\begin{split} & \left( {m + {\lambda _{11}}} \right){{\dot v}_x} - m{y_c}{{\dot \omega }_z} + m{z_c}{{\dot \omega }_y} + \\ & m\left[ {{v_z}{\omega _y} - {v_y}{\omega _z} - {x_c}\left( {\omega _y^2 + \omega _z^2} \right) + {y_c}{\omega _x}{\omega _y} + {z_c}{\omega _z}{\omega _x}} \right] \\ & = - \left( {G - B} \right)\sin \theta + T + X\left( {m + {\lambda _{22}}} \right){{\dot v}_y}+ \\ & \left( {m{x_c} + {\lambda _{26}}} \right){{\dot \omega }_z} - m{z_c}{{\dot \omega }_x} + \\ & m\left[ {{v_x}{\omega _z} - {v_z}{\omega _x} + {x_c}{\omega _x}{\omega _y} + {z_c}{{\dot \omega }_y}{\omega _z} - {y_c}\left( {\omega _x^2 + \omega _z^2} \right)} \right] \\ & = - \left( {G - B} \right)\cos \theta \cos \varphi + Y ，\end{split}$ (3)
 $\begin{split} & \left( {m + {\lambda _{33}}} \right){{\dot v}_z} - \left( {m{x_c} - {\lambda _{35}}} \right){{\dot \omega }_y} + m{y_c}{{\dot \omega }_x} + \\ & m\left[ {{v_y}{\omega _x} - {v_x}{\omega _y} - {x_c}{\omega _z}{\omega _x} + {y_c}{\omega _y}{\omega _z} + {z_c}\left( {\omega _x^2 + \omega _z^2} \right)} \right] \\ & = \left( {G - B} \right)cos\theta \sin \varphi + Z\left( {{J_{xx}} + {\lambda _{44}}} \right){{\dot \omega }_x} + m{y_c}{{\dot v}_z} - \\ & m{z_c}{{\dot v}_y} + m{y_c}\left( {{v_y}{\omega _x} - {v_x}{\omega _y}} \right) + m{z_c}\left( {{v_z}{\omega _x} - {v_x}{\omega _z}} \right) + \\ & \left( {{J_{zz}} - {J_{yy}}} \right){\omega _y}{\omega _z} = G\cos \theta \left( {{y_c}\sin \varphi + {z_c}\cos \varphi } \right)- \\ & B\cos \theta \left( {{y_b}\sin \varphi + {z_b}\cos \varphi } \right) + {M_x} ，\end{split}$ (4)
 $\begin{split} & \left( {{J_{yy}} + {\lambda _{55}}} \right){{\dot \omega }_y} + m{z_c}{{\dot v}_x} - \left( {m{x_c} - {\lambda _{35}}} \right){{\dot v}_z}+ \\ & m{z_c}\left( {{v_z}{\omega _y} - {v_y}{\omega _z}} \right) + m{x_c}\left( {{v_x}{\omega _y} - {v_y}{\omega _x}} \right)+ \\ & \left( {{J_{xx}} - {J_{zz}}} \right){\omega _z}{\omega _x} = - G\left( {{x_c}cos\theta \sin \varphi + {z_c}\sin \varphi } \right) + \\ & B\left( {{x_b}\cos \theta \sin \varphi + {z_b}\sin \varphi } \right) + {M_y} ，\end{split}$ (5)
 $\begin{split} & \left( {{J_{zz}} + {\lambda _{66}}} \right){{\dot \omega }_z} + m{y_c}{{\dot v}_x} - \left( {m{x_c} - {\lambda _{26}}} \right){{\dot v}_y}+ \\ & m{x_c}\left( {{v_x}{\omega _z} - {v_z}{\omega _x}} \right) + m{y_c}\left( {{v_y}{\omega _z} - {v_z}{\omega _y}} \right)+ \\ & \left( {{J_{zz}} - {J_{yy}}} \right){\omega _x}{\omega _y} = G\left( {{y_c}\sin \theta - + {x_c}cos\theta cos\varphi } \right) - \\ & B\left( {{y_b}\sin \theta - + {x_b}cos\theta \cos \varphi } \right) + {M_z}。\end{split}$ (6)

${\omega _x}$ ${\omega _y}$ ${\omega _z}$ 与姿态角变化率 $\dot \psi$ $\dot \theta$ $\dot \varphi$ 的关系为：

 $\left[ {\begin{array}{*{20}{c}} {\dot \psi } \\ {\dot \theta } \\ {\dot \varphi } \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0&{\sec \theta \cos \varphi }&{\sec \theta \sin \varphi } \\ 0&{\sin \varphi }&{\cos \varphi } \\ 1&{ - {\rm{tg}}\theta \cos \varphi }&{{\rm{tg}}\theta \sin \varphi } \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{\omega _x}} \\ {{\omega _y}} \\ {{\omega _z}} \end{array}} \right] ，$ (7)

 $\left( {\begin{array}{*{20}{c}} {{\rm{d}}{x_0}/{\rm{d}}t} \\ {{\rm{d}}{y_0}/{\rm{d}}t} \\ {{\rm{d}}{z_0}/{\rm{d}}t} \end{array}} \right) = C_0^b\left( {\begin{array}{*{20}{c}} {{v_x}} \\ {{v_y}} \\ {{v_z}} \end{array}} \right)，$ (8)

 $\alpha = - {\rm{arctg}}\left( {{v_y}/{v_x}} \right),$ (9)
 $\beta = {\rm{arctg}}\left( {{v_z}/\sqrt {v_x^2 + v_y^2} } \right),$ (10)
 $v = \sqrt {v_x^2 + v_y^2 + v_z^2}。$ (11)
1.2 应用关联规则的控制参数挖掘 1.2.1 关联规则算法的基本原理

1）项目和项集

2）事务和事务数据库

$I$ 的一个子集为所有事务 $T$ ，通过事务ID区别差异事务，所有事务集表示为事务数据库 $D$ ，所含事务数为 $\left| D \right|$

3）项集的支持度

$D$ 中含有个事务数，则有项集 $X$ $X \subset I$ ，且 ${\rm{count}}\left( {X \subseteq T} \right)$ ，则 $X$ 的支持度为：

 ${\rm support} \left( X \right) = \frac{{{\rm{count}}\left( {X \subseteq T} \right)}}{{\left| D \right|}}。$ (12)

4）项集最小支持度与频繁集

5）关联规则

 $R:X \Rightarrow Y。$ (13)

6）关联规则的支持度

 ${\rm support} \left( {X \Rightarrow Y} \right) = \frac{{{\rm{count}}\left( {X \cup Y} \right)}}{{\left| D \right|}}。$ (14)

7）关联规则的可信度

 ${\rm{confidence}} \left( {X \Rightarrow Y} \right) = \frac{{{\rm support} \left( {X \cup Y} \right)}}{{{\rm support} \left( X \right)}}。$ (15)

8）连接和剪枝

1.2.2 基于矩阵的Apriori算法

 ${M_{ij}} = \left\{ {\begin{array}{*{20}{c}} {1，\mathop {}\nolimits_{} {I_i} \in {T_j}}，\\ {0，\mathop {}\nolimits_{} {I_i} \notin {T_j}}。\end{array}} \right.$ (16)

$D$ 中所有项 ${I_i}$ 的向量为：

 ${L_i} = \left( {{m_{i1}},{m_{i2}}, \cdots ,{m_{in}}} \right)，$ (17)

$I$ 的支持度计数为：

 ${\rm{{support}}} \_{\rm{count}}\left( {{I_i}} \right) = \sum\limits_{i = 1}^n {{m_{ij}}}。$ (18)

2项集 $\left\{ {{I_i},{I_j}} \right\}$ 的向量为：

 ${L_{ij}} = {L_i} \wedge {L_j} = \left( {{m_{i1}} \wedge {m_{j1}},{m_{i2}} \wedge {m_{j2}}, \cdots ,{m_{in}} \wedge {m_{jm}}} \right)。$ (19)

 $\begin{split} & {{\rm{support}}} \_{{\rm{{count}}}}\left( {{L_{ij}}} \right) = \sum\limits_{i = 1}^n {\left( {{m_{ik}} \wedge {m_{jk}}} \right)} =\\ & {m_{i1}} \wedge {m_{j1}} + {m_{i2}} \wedge {m_{j2}} + \cdots + {m_{in}} \wedge {m_{jm}}。\end{split}$ (20)

$K$ 项集 $\left\{ {{I_1},{I_2}, \cdots ,{I_k}} \right\}$ 的向量为 ${L_{12 \cdots k}} = {L_1} \wedge {L_2} \wedge \cdots \wedge {L_k}$ ，则 $K$ 项集 $\left\{ {{I_1},{I_2}, \cdots ,{I_k}} \right\}$ 的支持度计数描述为：

 ${{\rm{support}}} \_{\rm{count}}\left( {{L_{1,2, \cdots k}}} \right) = \sum\limits_{i = 1}^n {\left( {{m_{1i}} \wedge {m_{2i}} \wedge \cdots {m_{ki}}} \right)}。$ (21)

1.3 基于卷积神经网络的PID控制器设计

 $\Delta {w_{ij}}\left( k \right) = \eta \left( {{d_j}\left( k \right) - {o_j}\left( k \right)} \right){o_i}\left( k \right){o_j}\left( k \right)。$ (22)

 $\Delta u\left( k \right) = k\left( {{{\omega '}_1}{x_1} + {{\omega '}_2}{x_2} + {{\omega '}_3}{x_3}} \right)，$ (23)
 ${w'_i}\left( k \right) = \frac{{{w_i}\left( k \right)}}{{\displaystyle\sum\limits_{i = 1}^3 {\left| {{w_i}\left( k \right)} \right|} }}，$ (24)
 $\left\{ {\begin{array}{*{20}{c}} {{w_1}\left( k \right) = {w_1}\left( {k - 1} \right) + {\eta _I}z\left( k \right)u\left( k \right){x_1}\left( k \right)}，\\ {{w_2}\left( k \right) = {w_2}\left( {k - 1} \right) + {\eta _P}z\left( k \right)u\left( k \right){x_2}\left( k \right)}，\\ {{w_3}\left( k \right) = {w_3}\left( {k - 1} \right) + {\eta _D}z\left( k \right)u\left( k \right){x_3}\left( k \right)}。\end{array}} \right.$ (25)

 $u\left( k \right) = u\left( {k - 1} \right) + K\sum\limits_{i = 1}^3 {{{w'}_i}\left( k \right){x_i}\left( k \right)}。$ (26)

2 实验分析

 图 1 半潜式航行体水平面航迹 Fig. 1 Horizontal track of semi-submersible vehicle

 图 2 半潜式航行体深度 Fig. 2 Depth of semi-submersible vehicle

 图 3 半潜式航行体姿态角 Fig. 3 Attitude angle of semi-submersible vehicle

 图 4 姿态控制通道响应 Fig. 4 Attitude control channel response

 图 5 船舶行驶控制效果 Fig. 5 Control effect of ship driving

3 结　语

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