﻿ 基于扩张观测器的船舶动力定位系统反演滑模变结构控制
 舰船科学技术  2017, Vol. 39 Issue (2): 103-107 PDF

Back-stepping sliding mode control of ship dynamic positioning system based on extended state observer
JIN Yue, YU Meng-hong, YUAN Wei, FAN Ji-sheng
School of Electronic and Information, Jiangsu University of Science and Technology, Zhenjiang 212003, China
Abstract: Aiming at the ship dynamic positioning control system in the offshore platform, In view of the advantages of back-stepping sliding mode control and extended observer, a sliding mode control method for dynamic positioning of ship based on extended observer is proposed. Considering the system of unknown external disturbances and parameters of ship model uncertainty, the system is divided as the observer of the inner ring and the outer ring controller. First, using the extended state observer to estimate the unknown state and uncertainty of the system which are compensated in the outer ring of the back-stepping sliding mode control then, finally Lyapunov method is used to demonstrate the system's stability. The ship point control simulation experiment shows that back-stepping sliding mode controller based on extended state observer with strong robustness and control performance can make the ship's surge position, sway position and swing angle gradually keep in the expected value. It can effectively suppress the chattering problem in the conventional sliding mode control and is beneficial to the application of ship engineering.
Key words: dynamic positioning control     extended state observer     back-stepping sliding mode control
0 引言

1 船舶数学模型

 图 1 大地坐标系和船体坐标系 Fig. 1 Geodetic coordinate system and ship coordinate system

 $\left\{ \begin{array}{l} \dot \eta = R\left( \psi \right)v \text{，}\\ M\dot \nu+D\nu = {R^{\rm T}}(\psi )b+\tau \text{。} \end{array} \right.$ (1)

 $\begin{array}{l} { R}(\psi ) = \left[ {\begin{array}{*{20}{l}} {\cos \psi } & { - \sin \psi } & 0\\ {\sin \psi } & {\cos \psi } & 0\\ 0 & 0 & 1 \end{array}} \right]\text{，}\\ { M} = \left[ {\begin{array}{*{20}{c}} {m - {X_{\dot u}}} & 0 & 0\\ 0 & {m - {Y_{\dot v}}} & {m{x_g} - {Y_{\dot r}}}\\ 0 & {m{x_g} - {N_{\dot v}}} & {{I_z} - {N_{\dot r}}} \end{array}} \right]\text{，}\\ { D} = \left[ {\begin{array}{*{20}{c}} { - {X_u}} & 0 & 0\\ 0 & { - {Y_v}} & { - {Y_r}}\\ 0 & { - {N_v}} & { - {N_r}} \end{array}} \right]\text{。} \end{array}$

2 基于扩张观测器的反演滑模变结构控制系统的设计

2.1 扩张观测器的设计

 $\left\{ \begin{array}{l} {x^{\left( n \right)}} = f\left( {x,{x^{\left( 1 \right)}}, \cdots ,{x^{\left( {n - 1} \right)}},t} \right)+w\left( t \right)+bu \text{，}\\ y = x\left( t \right)\text{。} \end{array} \right.$ (2)

$a\left( t \right) = f\left( {x,{x^{\left( 1 \right)}}, \cdots ,{x^{\left( {n - 1} \right)}},t} \right)+w\left( t \right)$，则n+1 阶 ESO 的一般形式为：

 $\left\{ \begin{array}{l} {{\dot z}_1} = {z_2} - {g_1}\left( {{z_1} - x\left( t \right)} \right)\text{，}\\ \vdots \\ {{\dot z}_n} = {z_{n+1}} - {g_n}\left( {{z_1} - x\left( t \right)} \right)+bu \text{，}\\ {{\dot z}_{n+1}} = - {g_{n+1}}\left( {{z_1} - x\left( t \right)} \right)\text{。} \end{array} \right.$ (3)

 $\begin{array}{l} fal\left( {\varepsilon ,\alpha ,\delta } \right) = \left\{ \begin{array}{l} {\left| \varepsilon \right|^\alpha }sat\left( \varepsilon \right),\left| \varepsilon \right| > \delta \text{，}\\ sign\left( \varepsilon \right),\varepsilon \leqslant \delta \text{，} \end{array} \right.\\ sat\left( \varepsilon \right) = \left\{ \begin{array}{l} \varepsilon /\varsigma ,\left| \varepsilon \right| \leqslant \varsigma \text{，}\\ sign\left( \varepsilon \right),\left| \varepsilon \right| > \varsigma \text{。} \end{array} \right. \end{array}$ (4)

 $\left\{ \begin{array}{l} {{\dot z}_1} = {z_2} - {\beta _1}{\varepsilon _1}\text{，}\\ {{\dot z}_2} = {z_3} - {\beta _2}fal\left( {{\varepsilon _1},{\alpha _1},\delta } \right)+bu\text{，}\\ {{\dot z}_2} = - {\beta _3}fal\left( {{\varepsilon _1},{\alpha _2},\delta } \right)\text{。} \end{array} \right.$ (5)

2.2 反演滑模控制器的设计

 ${ M}{{ R}^{ - 1}}\left( \varphi \right)\ddot \eta+{ M}{\dot { R}^{ - 1}}\left( \varphi \right)\dot \eta+{ D}{{ R}^{ - 1}}\left( \varphi \right)\dot \eta = \tau+d\left( t \right)\text{，}$ (6)

 $\ddot \eta = { A}\dot \eta+{ B}\left( {\tau+d\left( t \right)} \right)\text{，}$ (7)

${v^*} \!=\! { R}\left( \varphi \right)v \Rightarrow \dot \eta \!=\! {v^*}$，对式（1）进行简单调整：

 $\left\{ \begin{array}{l} \dot \eta = {v^*}\text{，}\\ {{\dot v}^*} = { A}{v^*}+{ B}\left( {\tau+d\left( t \right)} \right)\text{。} \end{array} \right.$ (8)

1） 船舶轨迹跟踪误差为z1 =η-ηdηd 为船舶目标位置，则 ${\dot z_1} = {v^*} - {\dot \eta _d}$

 ${V_1} = \frac{1}{2}z_1^2\text{，}$ (9)

 ${\dot V_1} = {z_1}{\dot z_1} = {z_1}{z_2} - {c_1}z_1^2\text{，}$ (10)

 $\sigma = {k_1}{z_1}+{z_2}\text{。}$ (11)

 $\sigma \!=\! {k_1}{z_1} \!+\! {z_2} \!=\! {k_1}{z_1} \!+\! {\dot z_1}+{c_1}{z_1} = \left( {{k_1}+{c_1}} \right){z_1}+{\dot z_1}\text{。}$ (12)

2） 定义 Lyapunov 函数：

 ${V_2} = {V_1}+\frac{1}{2}{\sigma ^2}\text{，}$ (13)

 $\begin{split}\\[-12pt] {{\dot V}_2} = & {{\dot V}_1}+\sigma \dot \sigma = {z_1}{z_2} - {c_1}z_1^2+\sigma \dot \sigma = \\ & {z_1}{z_2} - {c_1}z_1^2+\sigma \left( {{k_1}{{\dot z}_1}+{{\dot z}_2}} \right) = \\ & {z_1}{z_2} \!-\! {c_1}z_1^2 \!+\! \sigma \left( {{k_1}\left( {{z_2} \!-\! {c_1}{z_1}} \right) \!+\! {v^*} \!-\! {{\ddot \eta }_d} \!+\! {c_1}{{\dot z}_1}} \right) = \\ & {z_1}{z_2}\! - \!{c_1}z_1^2 \!+\! \sigma \left( {{k_1}\left( {{z_2} \!-\! {c_1}{z_1}} \right) \!+\! A\left( {{z_2} \!+\! {{\dot \eta }_d} \!-\! {c_1}{z_1}} \right)} +\right.\\ & \left. { Bu+F - {{\ddot \eta }_d}+{c_1}{{\dot z}_1}} \right)\text{。} \end{split}$ (14)

 $\begin{split}\\[-12pt] u = & {{ B}^{ - 1}}\left( { - {k_1}\left( {{z_2} - {c_1}{z_1}} \right) - A\left( {{z_2}+{{\dot \eta }_d} - {c_1}{z_1}} \right) - } \right.\\ & \left. {\bar F{\mathop{\rm sgn}} \left( \sigma \right)+{{\ddot \eta }_d} - {c_1}{{\dot z}_1} - h\left( {\sigma+\beta {\mathop{\rm sgn}} \left( \sigma \right)} \right)} \right)\text{。} \end{split}$ (15)

3 系统稳定性分析

 $\begin{split}\\[-12pt] {{\dot V}_2} = & {z_1}{z_2} - {c_1}z_1^2 - h{\sigma ^2} - h\beta \left| \sigma \right|+F\sigma - \\ & \bar F\left| \sigma \right| \leqslant - {c_1}z_1^2+{z_1}{z_2} - h{\sigma ^2} - h\beta \left| \sigma \right|\text{，} \end{split}$ (16)

 ${ Q} = \left[ {\begin{array}{*{20}{c}} {{c_1}+hk_1^2} & {h{k_1} - \frac{1}{2}}\\ {h{k_1} - \frac{1}{2}} & h \end{array}} \right]\text{，}$ (17)

 $\begin{split}\\[-12pt] {z^{\rm T}}{ Q}z = & \left[ \!\!\!{\begin{array}{*{20}{c}} {{z_1}} & {{z_2}} \end{array}}\!\! \right]\left[ \!\!{\begin{array}{*{20}{c}} {{c_1}+hk_1^2} & {h{k_1} - \frac{1}{2}}\\ {h{k_1} - \frac{1}{2}} & h \end{array}} \!\! \right]{\left[ \!\!{\begin{array}{*{20}{c}} {{z_1}} & {{z_2}} \end{array}}\!\! \right]^{\bf T}}= \\ & {\rm{ }} {c_1}z_1^2 - {z_1}{z_2}+hk_1^2z_1^2+2h{k_1}{z_1}{z_2}+hz_2^2 =\\ & {\rm{ }} {c_1}z_1^2 - {z_1}{z_2}+h{\sigma ^2}\text{，} \end{split}$ (18)

 ${\dot V_2} \leqslant - {z^{\bf T}}{ Q}z - h\beta \left| \sigma \right| \leqslant 0 \text{，}$ (19)

 $\left| { Q} \right| = h\left( {{c_1}+hk_1^2} \right) - {\left( {h{k_1} - \frac{1}{2}} \right)^2} = h\left( {{c_1}+{k_1}} \right) - \frac{1}{4}\text{。}$ (20)

4 仿真研究及结果

 $\begin{array}{l} { M} = \left[ {\begin{array}{*{20}{c}} {0.9270} & 0 & 0\\ 0 & {1.7502} & { - 0.1754}\\ 0 & { - 0.1754} & {0.1578} \end{array}} \right] \text{，}\\ { D} = \left[ {\begin{array}{*{20}{c}} {0.0558} & 0 & 0\\ 0 & {0.1183} & { - 0.0402}\\ 0 & { - 0.0151} & {0.0506} \end{array}} \right]\text{。} \end{array}$

 图 2 船舶纵荡位置输出 Fig. 2 Output of the ship longitudinal swing position

 图 3 船舶横荡位置输出 Fig. 3 Output of ship swing position

 图 4 船舶艏摇角度输出 Fig. 4 Output of ship yaw angle

 图 5 船舶运动轨迹 Fig. 5 The track of the ship motion

 图 6 船舶控制力、力矩 Fig. 6 Ship control force and control torque

 图 7 船舶外界扰动力观测值 Fig. 7 Observation values of external disturbance
5 结语

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