﻿ 基于分数阶微积分的无人船运动非线性数学控制模型
 舰船科学技术  2022, Vol. 44 Issue (7): 78-81    DOI: 10.3404/j.issn.1672-7649.2022.07.015 PDF

Nonlinear control of unmanned ship based on Fractional Calculus
ZHAO Xin-yu
School of Mathematics and Information Science of Henan Normal University, Xinxiang 453007, China
Abstract: In case of severe weather such as strong wind and waves, the unmanned ship is easy to be affected and nonlinear motion occurs. If the stability control cannot be implemented in time, it is easy to capsize. To solve the above problems, a nonlinear mathematical control model of unmanned ship motion based on fractional calculus is studied. In this study, six types of nonlinear motions of unmanned ship are described, namely surge, sway, heave, roll, pitch and yaw. Collect ship motion parameters through various equipment to clarify the state of unmanned ship. Combined with fractional calculus function and operator, the nonlinear mathematical control equation of unmanned ship motion is established and solved, the control quantities in three directions of x-axis, y-axis and z-axis are obtained, and the control model is constructed. The results show that compared with the control model based on neural network, the control method based on feedback linearization and fuzzy control system, the model has stronger anti rolling and anti surge effect, and can better ensure the stability of the ship in strong wind and big waves.
Key words: fractional calculus     unmanned ship     motion parameters     nonlinear mathematical control model
0 引　言

1 无人船运动非线性数学控制模型设计

1.1 无人船非线性运动描述

 图 1 无人船非线性运动参考坐标系 Fig. 1 Nonlinear motion reference coordinate system of unmanned ship

 $\begin{split} & F\left( {A,A1,A2,B,B1,B2} \right) = \hfill \\ & \frac{{g\left[ \begin{gathered} m\left( {X{\text{ + }}Y + Z} \right) + L\left( {u + v + w} \right) \hfill \\ + m\left( {K + M{\text{ + }}N} \right) + L\left( {p + q + r} \right) \hfill \\ \end{gathered} \right]}}{{m \cdot Q \cdot U}} 。\end{split}$ (1)

1.2 无人船运动参数采集

1）输入原始含噪无人船运动参数数据为Yi

2）利用主成分分析方法对Yi进行处理，并转换为矩阵的形式，记为R，公式为：

 ${\boldsymbol{R}} = \left[ \begin{array}{*{20}{c}} {r_{11}}&{{r_{12}}}&{...}&{{r_{1n}}} \\ {r_{21}}&{{r_{22}}}&{...}&{{r_{2n}}} \\ ... \\ {r_{m1}}&{{r_{m2}}}&{...}&{{r_{mn}}} \\ \end{array} \right]。$ (2)

3）选择矩阵中前 $k$ 个主成分，并让其保持不变。

4）将除了 $k$ 个主成分外的所有主成分分量转换为二维图像形式，记为 $H$

5）利用K-SVD 算法对 $H$ 进行去噪处理，去噪公式如下：

 $H' = \frac{{KT\sum {{r_{ij}}} }}{{\sqrt m }} 。$ (3)

6）对去噪后的二维图像进行逆变换。

7）得到去噪后的无人船运动参数数据，完成去噪处理。此外，为保证不同数据具有统一性，需要去除其量纲。公式如下：

 $x' = \frac{{x - \phi }}{\varphi } 。$ (4)

1.3 无人船运动非线性数学控制实现

 $\left\{ \begin{gathered} {X_{t + 1}} = \frac{{P_\beta ^\alpha \left( {{V_t} + {C_t}\sqrt L + {G_t}} \right)}}{2}，\hfill \\ {Y_{t + 1}} = \sqrt {\frac{{f\left( {{D_t} \cdot {C_t}} \right)}}{2}} ，\hfill \\ {Z_{t + 1}} = \frac{{P\left( {{K_t} + {M_t} + {N_t}} \right)}}{{pqr}} 。\end{gathered} \right.$ (5)

1）分数阶微积分算子描述为：

 P_\beta ^{\alpha} = \left\{ \begin{aligned} & \dfrac{{{d^\alpha }}}{{d{t^\beta }}}, \alpha > 0，\\ &{1,} {\alpha = 0}，\\ &{\int_a^{\beta} {{{(d\eta )}^{\alpha} }} ,} {{\alpha} < 0} 。\end{aligned} \right. (6)

2）分数阶微积分基本函数描述为：

 $f = \sum\limits_{k = 0}^\infty {\frac{{{B^k}}}{{\lambda (\alpha k + \beta )}}} 。$ (7)

 图 2 无人船运动非线性数学控制实现过程 Fig. 2 Realization process of nonlinear mathematical control of unmanned ship motion
2 模型测试与分析

2.1 测试环境搭建

2.2 工况相关参数设置

2.3 无人船运动控制量

 图 3 无人船运动控制量 Fig. 3 Motion control quantity of unmanned ship
2.4 控制效果分析

3 结　语

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