舰船科学技术  2017, Vol. 39 Issue (10): 70-74 PDF

Dynamic surface active-disturbance rejection control over dynamic positioning of semi-submersible offshore platforms
HE Hong-lei, WANG Yu-long
School of Electronics and Information, Jiangsu University of Science and Technology, Zhenjiang 212003, China
Abstract: By introducing the dynamic surface control to reform the extended state observer and the non-linear state error feedback control law, a dynamic surface active-disturbance rejection controller is designed for dynamic positioning systems of offshore platforms. The dynamic surface extended-state observer is designed to strengthen the estimation ability to the disturbance of systems, while the dynamic surface non-linear state error feedback control law is designed to improve the stability and control efficiency of systems. Simulation results illustrate that the improved dynamic surface active-disturbance rejection-based dynamic positioning control systems can significantly improve the estimation ability to the disturbance, the robustness of the systems is improved greatly, and the systems can provide better control quality and faster response to achieve the enhancement of positioning accuracy of offshore platforms.
Key words: offshore platforms     dynamic positioning     dynamic surface control     active disturbance rejection control
0 引　言

1 海洋平台动力定位常规自抗扰控制方案 1.1 海洋平台低频运动模型的 2 种形式

 $\left\{ {\begin{array}{*{20}{l}}{{\dot \eta } = {J}(\eta ){\nu }}\text{，}\\{{M\dot \nu} + {D\nu } = {\tau } + {\omega }}\text{。} \end{array}} \right.$ (1)

 \begin{aligned}& { J}(\eta ) = \left[ {\begin{array}{*{20}{c}}{\cos \psi }& { - \sin \psi } & 0 \\{\sin \psi }& {\cos \psi } & 0\\0 & 0 & 1\end{array}} \right]\text{，}\\[6pt]& { 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} - {Y_{\dot r}}} & {{I_z} - {N_{\dot r}}}\end{array}} \right]\text{，}\\[6pt]& { 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{aligned}

 ${\ddot \eta } = {{M}_\eta }^{ - 1}(\eta )[{{J}^{ - {\rm{T}}}}(\eta )({\tau } + {\omega }) - {{D}_\eta }{\dot \eta }]\text{，}$ (2)

 ${\ddot \eta } = {{M}_\eta }^{ - 1}(\eta )[{{J}^{ - {\rm{T}}}}(\eta ){\omega } - {{D}_\eta }{\dot \eta }] + {U}\text{。}$ (3)
1.2 自抗扰控制原理

 \left\{ {\begin{aligned}& {y = {x_1}}\text{，}\\& {{{\dot x}_1} = {x_2}}\text{，}\\& {{{\dot x}_2} = f\left( {{x_1},{x_2},t} \right) + \omega \left( t \right) + bu}\text{。}\end{aligned}} \right. (4)

1）跟踪微分器

 $\left\{ {\begin{array}{*{20}{l}}\!\!\!{{{\dot v}_1} = {v_1} + T{v_2}}\text{，}\\{\begin{array}{*{20}{l}}\!\!\!\!\!\!\!{{{\dot v}_2} = {v_2} + Tfh}\text{，}\\\!\!\!\!\!\!{fh = fhan\left( {{v_1} - v,{v_2},r,h} \right)}\text{。}\end{array}}\end{array}} \right.$ (5)

2）扩张状态观测器

 $\left\{ {\begin{array}{*{20}{l}}{e = {z_1} - y}\text{，}\\{{{\dot z}_1} = {z_2} - {\beta _{01}}e}\text{，}\\{{{\dot z}_2} = {z_3} - {\beta _{02}}fal\left( {e,{a_1},\delta } \right) + {b_0}u}\text{，}\\{{{\dot z}_3} = - {\beta _{03}}fal\left( {e,{a_2},\delta } \right)}\text{。}\end{array}} \right.$ (6)

3）非线性状态误差反馈控制律

 $\left\{ {\begin{array}{*{20}{l}}{{e_1} = {v_1} - {z_1}}\text{，}\\{{e_2} = {v_2} - {z_2}}\text{，}\\{{u_0} = {k_1}fal\left( {{e_1},{\alpha _1},\delta } \right) + {k_2}fal\left( {{e_2},{\alpha _2},\delta } \right)}\text{，}\\{u = {u_0} - {z_3}/{b_0}}\text{。}\end{array}} \right.$ (7)

2 动态面自抗扰控制器设计

2.1 动态面扩张状态观测器的设计

 $\left\{ {\begin{array}{*{20}{l}}{{{\dot z}_1} = {z_2}}\text{，}\\{{{\dot z}_2} = {z_3} + {b_0}u}\text{，}\\{{{\dot z}_3} = - {f_3} - {{\dot f}_2} - {{\ddot f}_1}}\text{。}\end{array}} \right.$ (8)

1）定义系统（8）的第 1 个子系统 ${\dot z_1} = {z_2}$ 的动态面方程为

 ${S_1} = {z_1} - {x_1}\text{，}$ (9)

 ${\dot S_1} = {z_2} - {x_2}\text{，}$ (10)

 ${\bar u_1} = - {k_1}{S_1} + {x_2}\text{。}$ (11)

 ${\tau _1}{\dot x_{1d}} + {x_{1d}} = {\bar u_1}\text{。}$ (12)

2）第 2 个子系统 ${\dot z_2} = {z_3} + {b_0}u$ 的动态面方程为：

 ${S_2} = {z_2} - {x_{1d}}\text{，}$ (13)

S2 求导得：

 ${\dot S_2} = {z_3} + {b_0}u - {x_{1d}}\text{，}$ (14)

 ${\bar u_2} = - {k_2}{S_2} - {b_0}u + {\dot x_{1d}}\text{。}$ (15)

 ${\tau _2}{\dot x_{2d}} + {x_{2d}} = {\bar u_2}\text{，}$ (16)

3）第 3 个子系统 ${\dot z_3} = \bar u$ 的动态面方程为：

 ${S_3} = {z_3} - {x_{2d}}\text{，}$ (17)

S3 求导得

 ${\dot S_3} = \bar u - {\dot x_{2d}}\text{，}$ (18)

 $\bar u = - {k_3}{S_3} + {\dot x_{2d}}\text{，}$ (19)

 ${\dot S_1} = {S_2} + {x_{1d}} - {x_2}\text{，}$ (20)

 ${\dot S_1} = {S_2} + {\zeta _1} - {k_1}{S_1}\text{，}$ (21)

 $\begin{array}{l}{{\dot S}_2} = {S_3} + {\zeta _2} - {k_2}{S_2}\text{，}\\[5pt]{{\dot S}_3} = - {k_3}{S_3}\text{。}\end{array}$ (22)

 $\begin{array}{c}{{\dot \zeta }_1} = {{\dot x}_{1d}} + {k_1}{{\dot S}_1} - {{\ddot x}_1} = - \displaystyle\frac{{{\zeta _1}}}{{{\tau _1}}} + {g_1}\text{，}\end{array}$ (23)
 $\begin{array}{c}{{\dot \zeta }_2} = {{\dot x}_{2d}} + {k_2}{{\dot S}_2} + {b_0}\displaystyle\frac{{\partial u}}{{\partial {x_2}}}{{\dot x}_2} + \displaystyle\frac{{{\zeta _1}}}{{{\tau _1}}} = - \displaystyle\frac{{{\zeta _2}}}{{{\tau _2}}} + {g_2}\text{，}\end{array}$ (24)

 \begin{aligned}& {V_{is}} = \frac{{S_i^2}}{2},i = 1,2,3\text{，}\\& {V_{i\zeta }} = \frac{{\zeta _i^2}}{2},i = 1,2\text{，}\end{aligned} (25)

 \left\{ {\begin{aligned}& {{{\dot V}_{is}} = {S_i}({S_{i + 1}} + {\zeta _i} - {k_i}{S_i})=}\\& \;\;\;\;\;\;{ - {k_i}S_i^2 + {S_i}{S_{i + 1}} + {S_i}{\zeta _i},i = 1,2}\text{，}\\& {{{\dot V}_{3s}} = - {k_3}S_3^2}\text{，}\\& {{{\dot V}_{i\zeta }} = - \frac{{\zeta _i^2}}{{{\tau _i}}} + {\zeta _i}{g_i},i = 1,2}\text{，}\end{aligned}} \right. (26)

 $V = \sum\limits_{i = 1}^3 {{V_{is}}} + \sum\limits_{j = 1}^2 {{V_{j\zeta }}} \text{。}$ (27)

 \begin{aligned}& \dot V \leqslant - (2 + {\alpha _0})\sum\limits_{i = 1}^3 {S_i^2} + \sum\limits_{i = 1}^2 {[\frac{{2S_i^2 + S_{i + 1}^2 + \zeta _i^2}}{2}} - \\& (1 + \frac{{g_{i\max }^2}}{{2\varepsilon }} + {\alpha _0})\zeta _i^2 + \frac{{g_{i\max }^2\zeta _i^2}}{{2\varepsilon }}\frac{{g_i^2}}{{g_{i\max }^2}}] + \varepsilon \leqslant - 2{\alpha _0}V + 2\varepsilon\text{。} \end{aligned} (28)

$V = p$ ${\alpha _0} > \varepsilon /p$ 时，有 $\dot V < 0$ 。即只要参数 k1k2k3τ1τ2 选择适当，系统收敛。

2.2 动态面非线性状态误差反馈控制律的设计

 $\left\{ {\begin{array}{*{20}{l}}{{S_4} = y - {v_1}}\text{，}\\{{{\bar u}_3} = - {k_4}{S_4} + {v_2}}\text{，}\\{{\tau _3}{{\dot x}_{3d}} + {x_{3d}} = {{\bar u}_3}}\text{，}\\{{S_5} = {x_2} - {x_{3d}}}\text{，}\\{{u_0} = - {k_5}{S_5} - f({x_1},{x_2},t) - \omega (t) + {{\dot x}_{3d}}}\text{，}\\{u = {u_0} - {z_3}/{b_0}}\text{。}\end{array}} \right.$ (29)

 图 1 二阶动态面自抗扰控制器结构图 Fig. 1 The structure of second-order DS-ADRC
3 仿真实验

 $\begin{array}{l}{M} = \left[ {\begin{array}{*{20}{c}}{6.74 \times {{10}^7}} & 0 & 0 \\0 & {9.15 \times {{10}^7}} & { - 0.608 \times {{10}^7}}\\0 & { - 0.608 \times {{10}^7}} & {1.08 \times {{10}^{11}}}\end{array}} \right]\text{，}\\[6pt]{D} = \left[ {\begin{array}{*{20}{c}}{6.76 \times {{10}^5}} & 0 & 0 \\0 & {5.319 \times {{10}^5}} & { - 1.56 \times {{10}^5}}\\0 & { - 1.56 \times {{10}^5}} & {1.7313 \times {{10}^9}}\end{array}} \right]\text{。}\end{array}$

${T} \!=\! \left[ \!\!{\begin{array}{*{20}{c}} {0.1} \!\!\! & \!\!\!\!\! {0.1} \!\!\!\!\! & \!\!\! {0.1} \end{array}} \!\!\!\!\right]\text{，}$ ${r} \!\!= \!\!\left[ \!\!\! {\begin{array}{*{20}{c}} {100} \!\!\! & \!\!\! {100}\!\!\! & \!\!\!{100} \end{array}} \!\!\!\right]\text{，}$ ${h} = \left[ {\begin{array}{*{20}{c}} {10} & {10} & {10} \end{array}} \right]\text{，}$ ${{\tau }_1} = \left[ {\begin{array}{*{20}{c}} {0.25} & {0.25} & {0.05} \end{array}} \right]\text{，}$ ${{\tau }_2} = \left[ {\begin{array}{*{20}{c}} {0.25} & {0.25} & {0.05} \end{array}} \right]\text{，}$ ${{\tau }_3} = \left[ {\begin{array}{*{20}{c}} {0.1} & {0.1} & {0.1} \end{array}} \right]\text{，}$ ${{k}_1} = \left[ {\begin{array}{*{20}{c}} {0.5} & {0.5} & {0.2} \end{array}} \right]\text{，}$ ${{k}_2} = \left[ {\begin{array}{*{20}{c}} {0.5} & {0.5} & {0.2} \end{array}} \right]\text{，}$ ${{k}_3} = \left[ {\begin{array}{*{20}{c}} 1 & 1 & 5 \end{array}} \right]\text{，}$ ${{k}_4} = \left[ {\begin{array}{*{20}{c}} {10} & {10} & {0.2} \end{array}} \right]\text{，}$ ${{k}_5} = \left[ {\begin{array}{*{20}{c}} 1 & 2 & 2 \end{array}} \right]\text{，}$ ${{b}_0} = \left[ {\begin{array}{*{20}{c}} 1 & 1 & 1 \end{array}} \right]\text{。}$

 图 2 ADRC 的系统总扰动及估计曲线 Fig. 2 The total disturbance and estimation curves of ADRC

 图 3 DS-ADRC 的系统总扰动及估计曲线 Fig. 3 The total disturbance and estimation curves of DS-ADRC