﻿ 滑模控制船舶动力定位控制系统研究
 舰船科学技术  2018, Vol. 40 Issue (3): 61-65 PDF

Design of ship dynamic positioning control system based on sliding model control
GUAN Ke-ping, ZHANG Xin-fang
Merchant Marine College, Shanghai Maritime University, Shanghai 201306, China
Abstract: To improve the accuracy of dynamic positioning ship or operating platform in the complicated sea conditions, the controller of Dynamic Positioning System (DPS) is designed. Ship motion mathematical model with three degrees of freedom is set up. The method of Sliding Model Control (SMC) is adopted to design the controller. And the function of Lyapunov is used to analyze the stability of the controller. At last, one dynamic positioning ship is used as the simulation object. Under the condition of the interference of the external environment, the results of simulation show that the sliding model controller has good ability of control performance, as well as the stability and robustness. The controller is considered as a useful reference for the Dynamic Positioning System.
Key words: sliding model control     mathematical model     dynamic positioning system     controller
0 引　言

 图 1 船舶动力定位系统框图 Fig. 1 Block diagram of DPS

1 船舶动力定位系统的数学模型

1.1 建立坐标系

 图 2 大地坐标系和随船坐标系 Fig. 2 Earth-fixed frame and body frame

2种坐标系的转换关系为[5]

 $\dot \eta = { J}\left( \varphi \right)\upsilon \text{，}$ (1)

 ${ J}\left( \varphi \right) = \left[ {\begin{array}{*{20}{c}}{{{cos}}\varphi } & { - {{sin}}\varphi } & 0\\{{{sin}}\varphi } & {{{cos}}\varphi } & 0\\0 & 0 & 1\end{array}} \right]\text{。}$ (2)
1.2 船舶运动的数学模型

 \begin{aligned}{ M}\dot \upsilon + & { D}\left( \upsilon \right)\upsilon = {\tau _c} + {\tau _s}\text{，}\\ & \dot \eta = { J}\left( \varphi \right)\upsilon \text{。}\end{aligned}

 $\begin{array}{l}{ M} = \left[ {\begin{array}{*{20}{c}}{m - {X_{\dot u}}} & 0 & 0\\0 & {m - {Y_{\dot \nu }}} & 0\\0 & 0 & {{I_z} - {N_{\dot r}}}\end{array}} \right]\text{，}\\[10pt]{ D}\left( \upsilon \right) = \left[ {\begin{array}{*{20}{c}}{ - {X_u}} & 0 & 0\\0 & { - {Y_v}} & 0\\0 & 0 & { - {N_r}}\end{array}} \right]\text{。}\end{array}$

2 船舶动力定位系统滑模控制器的设计

2.1 滑模变结构控制理论

2.2 滑模变结构控制器的设计

 图 3 双环滑模动力定位系统框图 Fig. 3 Block diagram of bicyclic SMC system

 $e = \eta - {\eta _d}\text{，}$ (4)

 $\dot e = \dot \eta - {\dot \eta _d} = \upsilon - {\upsilon _d}\text{，}$ (5)

 $\ddot e = \dot \upsilon - {\dot \upsilon _d}\text{，}$ (6)

 ${s_o} = \eta + { \varLambda _1}\mathop \smallint \nolimits_0^t \eta {{d}}t,\;\;\;\;{s_0} \in {R^3}\text{，}$ (7)

 ${\dot s_o} = \dot \eta + {\varLambda _1}\eta$ (8)

 ${\dot s_o} = { J}\left( \varphi \right)\dot e + { J}\left( \varphi \right){\upsilon _d} + {\varLambda _1}\eta \text{，}$ (9)

 ${\upsilon _d} = {{ J}^{ - 1}}\left( \varphi \right)\left( { - {\varLambda _1}\eta - {\rho _1}{{sgn}}\left( {{s_o}} \right)} \right)\text{，}$ (10)

 ${\dot s_o} = { J}\left( \varphi \right)\dot e - {\rho _1}{{sgn}}\left( {{s_o}} \right)\text{。}$ (11)

 ${s_i} = \dot e + {\varLambda _2}\mathop \smallint \nolimits_0^t \dot e{{d}}t\text{，}$ (12)

 ${\dot s_i} = \ddot e + {\varLambda _2}\dot e \text{，}$ (13)

 ${\dot s_i} = - {{ M}^{ - 1}}{ D}\upsilon + {{ M}^{ - 1}}\left( {{\tau _c} + {\tau _s}} \right) - {\dot \upsilon _d} + {\varLambda _2}\dot e\text{，}$ (14)

 ${\tau _c} = M\left( {{{\dot s}_i} + {{\dot \upsilon }_d} - {\varLambda _2}\dot e} \right) + D\upsilon - {\tau _s}\text{，}$ (15)

 ${\dot s_i} = {\tau _s} - {\rho _2}{{sgn}}\left( {{\tau _{\max}}{s_i}} \right)\text{，}$ (16)

 ${\tau _c} = { M}\left( {{{\dot s}_i} + {{\dot \upsilon }_d} - {\varLambda _2}\dot e} \right) + { D}\upsilon - {\dot s_i} - {\rho _2}{{sgn}}\left( {{\tau _{\max}}{s_i}} \right)\text{。}$ (17)
2.3 稳定性分析

Lyapunov函数常作为判断系统稳定性的重要工具，本文用来判断所设计控制器的稳定性[11]

 ${V_o} = \frac{1}{2}{s_o}^{{T}}{s_o}\text{，}$ (18)

 ${\dot V_o} = {s_o}^{{T}}{\dot s_o}\text{，}$ (19)

 ${\dot V_o} = {s_o}^{{T}}{ J}\left( \psi \right)\dot e - {\rho _1}{s_o}^{{T}}{{sgn}}\left( {{s_o}} \right)\text{。}$ (20)

 ${\dot V_o} = - {\rho _1}{s_o}^{{T}}{{sgn}}\left( {{s_o}} \right) \leqslant 0\text{。}$ (21)

 ${V_i} = \frac{1}{2}{s_i}^{{T}}{s_i}\text{，}$ (22)

 ${\dot V_i} = {s_i}^{{T}}{\dot s_i}\text{，}$ (23)

 ${\dot V_i} = {s_i}^{{T}}{\tau _s} - {\rho _2}{s_i}^{{T}}{{sgn}}\left( {{\tau _{\max}}{s_i}} \right)\leqslant 0 \text{。}$ (24)

3 仿真试验

 $\dot \nu = - {{ M}^{ - 1}}{ D}\left( \upsilon \right)\upsilon + {{ M}^{ - 1}}\left( {{\tau _c} + {\tau _s}} \right)\text{，}$ (25)

 $\dot \upsilon = { A}\upsilon + { B}\left( {{\tau _c} + {\tau _s}} \right)\text{，}$ (26)

 \begin{aligned}{ M} = \left[ {\begin{array}{*{20}{c}}{1.127 \ 4} & 0 & 0\\0 & {1.890 \ 2} & 0\\0 & 0 & {0.127 \ 8}\end{array}} \right]\text{，}\\{ D} = \left[ {\begin{array}{*{20}{c}}{0.035 \ 8} & 0 & 0\\0 & {0.118 \ 3} & 0\\0 & 0 & { - 0.030 \ 8}\end{array}} \right]\text{。}\end{aligned}

 \begin{aligned}{ A} = \left[ {\begin{array}{*{20}{c}}{ - 0.031 \ 8} & 0 & 0\\0 & { - 0.062 \ 8} & 0\\0 & 0 & {0.250 \ 6}\end{array}} \right],\\{ B} = \left[ {\begin{array}{*{20}{c}}{ - 0.887 \ 0} & 0 & 0\\0 & { - 0.541 \ 5} & 0\\0 & 0 & { - 8.008 \ 2}\end{array}} \right]\end{aligned}

 图 4 船舶位置和艏向的变化曲线 Fig. 4 Curve of ship positioning and heading

 图 5 船舶速度的变化曲线 Fig. 5 Curve of ship speed

 图 6 控制律的变化曲线 Fig. 6 Curve of control law

4 结　语

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