﻿ 船舶矢量舵减横摇控制系统
 舰船科学技术  2019, Vol. 41 Issue (4): 76-82 PDF

1. 山东省船舶控制工程与智能系统工程技术研究中心，山东 荣成 264300;
2. 哈尔滨工程大学，黑龙江 哈尔滨 150001

Vessel vector rudder reduction roll control system
LIU Sheng1,2, TAN Yin-chao1
1. Shandong Shipbuilding Control Engineering and Intelligent System Engineering Technology Research Center, Rongcheng 264300, China;
2. Harbin Engineering University, Harbin 150001, China
Abstract: For the roll motion problem of ships without anti-rolling device system, this paper proposes that the rudder and fin rudder constitute two relatively independent vector control plane roll stability control systems. The vector rudder control torque and the m-order regression model of torque, rudder angle and wing rudder angle are established accuracy. A system μ-robust controller is designed. The intelligent coordinator for rudder angle/wing rudder angle is designed based on the improved genetic algorithm. The simulation results show that the vector rudder is decelerated and the rudder is reduced. The rod control system can effectively reduce roll, reduce system energy consumption, and enhance the robustness against system parameter perturbation.
Key words: vessel vector rudder     roll-off reduction     μ-robust controller     intelligent coordination decision maker
0 引　言

1 船舶矢量舵减横摇控制系统建模

 图 1 船舶矢量舵减横摇控制系统结构原理图 Fig. 1 Schematic diagram of the structure of ship vector rudder anti-roll control system

 \left\{ {\begin{aligned} & \begin{array}{l} {\rm{m}}(\dot v + ur) = - {m_y}\dot y - {m_x}ur + {Y_v}v + {Y_r}r + {Y_{v\left| v \right|}}v\left| v \right| +\\ {Y_{r\left| r \right|}}r\left| r \right|+ {Y_{vvr}}{v^2}r + {Y_{vrr}}v{r^2} + {Y_r}({\alpha _r},{\beta _r}) + {Y_d}\text{，} \end{array}\\ & \begin{array}{l} {I_{\rm{z}}}\dot r = - \Delta {I_z}\dot r + {N_v}v + {N_r}r + {N_{v\left| v \right|}}v\left| v \right| + \\ {N_{r\left| r \right|}}r\left| r \right| + {N_{vvr}}{v^2}r + {N_{vrr}}v{r^2}+ {N_\varphi }\varphi + {N_{v\left| \varphi \right|}}v\left| \varphi \right| +\\ {N_{r\left| \varphi \right|}}r\left| \varphi \right| + {N_r}({\alpha _r},{\beta _r}) + {N_d}\text{，} \end{array}\\ & \begin{array}{l} {I_x}\dot p = - \Delta {I_x}\dot p - 2{N_p}p - Wh\varphi - {z_H}( - {m_y}\dot v -\\ {m_x}ur + {Y_v}v + {Y_r}r + {Y_{v\left| v \right|}}v\left| v \right|+ {Y_{r\left| r \right|}}r\left| r \right| + {Y_{vvr}}{v^2}r +\\ {Y_{vrr}}v{r^2}){\rm{ + }} {K_r}({\alpha _r},{\beta _r}) + {K_d}\text{。} \end{array} \end{aligned}} \right. (1)

 $\begin{split} & {Y_r}({\alpha _r},{\beta _r}) = (1 + {\alpha _H}){P_N}\cos {\alpha _r} \text{，}\\ & {N_r}({\alpha _r},{\beta _r}) = - (1 + {\alpha _H}){x_R}{P_N}\cos {\alpha _r} \text{，} \\ & {K_r}({\alpha _r},{\beta _r}) = (1 + {\alpha _H}){z_R}{P_N}\cos {\alpha _r} \text{。} \end{split}$ (2)

 ${\alpha _H}{\rm{ = 1}}{\rm{.052}}{{\rm{C}}_b} + 0.125\text{。}$ (3)

$P\cos {\alpha _r}{\rm{ = }}{P_y}$ , ${P_y}$ 为舵/翼舵的升力。近似计算公式为：

 $\begin{split} & {Y_r}({\alpha _r},{\beta _r}) = (1 + {\alpha _H}){P_y} \text{，}\\ & {N_r}({\alpha _r},{\beta _r}) = - (1 + {\alpha _H}){x_R}{P_y} \text{，}\\ & {K_r}({\alpha _r},{\beta _r}) = (1 + {\alpha _H}){z_R}{P_y} \text{，}\\ & {P_y}{\rm{ = }}\frac{1}{2}\rho {\upsilon ^2}{S_p}{C_{yr}}\left( {{\alpha _r},{\beta _r}} \right) \text{。} \end{split}$ (4)

 $\begin{split} {u_r}({\alpha _r},{\beta _r}) = & {A_0} + {a_1}{\alpha _r} + {a_2}{\alpha _r}^2 + \cdots + {a_m}{\alpha _r}^m +\\ & {b_1}{\beta _r} + {b_2}{\beta _r}^2 +\cdots + {b_m}{\beta _r}^m +\\ & {c_1}{a_r}{\beta _r} + {c_2}{\alpha _r}^2{\beta _r} + \cdots + {c_{m - 1}}{\alpha _r}^{m - 1}{\beta _r} + \\ & {d_2}{a_r}{\beta _r}^2 + {d_3}{\alpha _r}{\beta _r}^3 + \cdots + {d_{m - 1}}{\alpha _r}{\beta _r}^{m - 1} \text{。} \end{split}$ (5)

 图 2 舵/翼舵升力系数图谱 Fig. 2 Rudder / wing rudder lift coefficient map
 $\begin{split} {c_r}({\alpha _r},{\beta _r}) = & 1.47{\alpha _r} + 3.04{\alpha _r}^2 - 3.82{\alpha _r}^3 - 1.46{\alpha _r}^4 + \\ & 0.29{\beta _r} + 3.86{\beta _r}^2 - 10.45{\beta _r}^3{\rm{ + }}8.37{\beta _r}^4 + \\ & {\rm{ 0}}{\rm{.50}}{a_r}{\beta _r} - 4.09{\alpha _r}^2{\beta _r} + 7.61{\alpha _r}^3{\beta _r} + \\ & 4.43{\alpha _r}{\beta _r}^2 - 4.59{\alpha _r}{\beta _r}^3 \text{。} \end{split}$ (6)

 ${u_r}({\alpha _r},{\beta _r}) = \frac{1}{2}\rho {V^2}{S_r}{L_r}{c_r}\left( {{\alpha _r},{\beta _r}} \right)\text{。}$ (7)

 图 3 舵扭矩系数图谱 Fig. 3 Torque factor map
 $\begin{split} {c_{mr}}({\alpha _r},{\beta _r}) =& 0.356\;6{\alpha _r} + 0.143\;4{\alpha _r}^2 + 1.210\;3{\alpha _r}^3 - \\ & 1.482\;8{\alpha _r}^4 + 0.381\;2{\beta _r} + 0.374\;1{\beta _r}^2 -\\ & 0.971\;2{\beta _r}^3{\rm{ + 0}}{\rm{.693\;9}}{\beta _r}^4 {\rm{ + 0}}{\rm{.111\;3}}{a_r}{\beta _r} + \\ &\! 0.720\;1{\alpha _r}^2{\beta _r}\! -\! 1.194\;8{\alpha _r}^3{\beta _r} \!+\! 2.257\;6{\alpha _r}{\beta _r}^2 -\!\!\!\!\!\!\!\!\!\! \\ & 3.103\;8{\alpha _r}{\beta _r}^3 - 0.623\;1{\alpha _r}^2{\beta _r}^2 \text{。} \end{split}$ (8)

 图 4 舵扭矩系数图谱 Fig. 4 Torque factor map
 $\begin{split} {c_{m3r}}({\alpha _r},{\beta _r}) =& 0.432\;4{\alpha _r} + 0.110\;0{\alpha _r}^2 - 0.511\;6{\alpha _r}^3+ \\ & 0.350\;6{\alpha _r}^4 + 0.347\;4{\beta _r} + 0.232\;1{\beta _r}^2 +\\ & 0.942\;7{\beta _r}^3 - 1.333\;5{\beta _r}^4 {\rm{ + 0}}{\rm{.241\;9}}{a_r}{\beta _r} +\\ & 1.892\;3{\alpha _r}^2{\beta _r} \!-\! 2.443\;2{\alpha _r}^3{\beta _r} \!+\! 0.424\;6{\alpha _r}{\beta _r}^2-\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \\ & 0.752\;8{\alpha _r}{\beta _r}^3 - 0.731\;0{\alpha _r}^2{\beta _r}^2 \text{。} \end{split}$ (9)

 图 5 舵/翼舵升力系数相对误差曲面 Fig. 5 Relative error surface of rudder/finger rudder lift coefficient

 图 6 舵扭矩系数相对误差曲面 Fig. 6 Relative error surface of rudder torque coefficient

 图 7 翼舵扭矩系数相对误差曲面 Fig. 7 Relative error surface of torque coefficient of wing rudder
2 $\mu$ -鲁棒控制策略设计

 $\left\{ \!\!\!\!\begin{array}{l} \left( {{I_x}{\rm{ + }}{a_{\varphi \varphi }}} \right)\ddot \varphi + {b_{\varphi \varphi }}\dot \varphi + {c_{\varphi \varphi }}\varphi + {a_{\varphi \psi }}\ddot \psi + {b_{\varphi \psi }}\dot \psi = {K_r}({\alpha _r},{\beta _r}) + {K_d}\text{，} \\ \left( {{I_z}{\rm{ + }}{a_{\psi \psi }}} \right)\ddot \psi + {b_{\psi \psi }}\dot \psi + {a_{\psi \varphi }}\ddot \varphi + {b_{\psi \varphi }}\dot \varphi = {N_r}({\alpha _r},{\beta _r}) + {N_d}\text{。} \\ \end{array} \right.$ (10)

${x_1} = \varphi ,{x_2} = \dot \varphi ,{x_3} = \psi ,X = {\left[ {{x_1}{x_2}{x_3}} \right]^{\rm T}}$ ，则系统的状态方程为：

 $\begin{split} & \dot X = \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} 0&1&0 \end{array}} \\ { - E_1^{ - 1}{E_2}} \end{array}} \right]X + \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} 0&0 \end{array}} \\ {E_1^{ - 1}} \end{array}} \right]\left[ \begin{array}{l} {K_r}({\alpha _r},{\beta _r}) \\ {N_r}({\alpha _r},{\beta _r}) \\ \end{array} \right] +\\ & \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} 0&0 \end{array}} \\ {E_1^{ - 1}} \end{array}} \right]\left[ \begin{array}{l} {K_d} \\ {N_d} \\ \end{array} \right] y = CX,C = \left[ {\begin{array}{*{20}{c}} 1&0&0 \end{array}} \right] \text{。} \end{split}$ (11)

 ${E_1} = \left[\!\!\!\! {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{I_x}{\rm{ + }}{a_{\varphi \varphi }}}&{{a_{\varphi \psi }}} \end{array}} \\ {\begin{array}{*{20}{c}} {{a_{\psi \varphi }}}&{{I_z}{\rm{ + }}{a_{\psi \psi }}} \end{array}} \end{array}} \!\!\!\!\right], {E_2} = \left[\!\!\!\! {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{c_{\varphi \varphi }}}&{{b_{\varphi \varphi }}}&{{b_{\varphi \psi }}} \end{array}} \\ {\begin{array}{*{20}{c}} 0&{{b_{\psi \varphi }}}&{{b_{\psi \psi }}} \end{array}} \end{array}}\!\!\!\!\right]\text{。}$ (12)

 \begin{aligned} & {{\dot X}_1} = \left[\!\!\!\! {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} 0&1&0 \end{array}} \\ { - E_1^{ - 1}{E_2}} \end{array}} \!\!\!\!\right]{X_1} + \left[\!\!\!\! {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} 0&0 \end{array}} \\ {E_1^{ - 1}} \end{array}} \!\!\!\!\right]\left[ \!\!\!\!\begin{array}{l} {K_r}({\alpha _r},{\beta _r}) \\ {N_r}({\alpha _r},{\beta _r}) \\ \end{array} \!\!\!\! \right] \text{，} \\ & {{\dot X}_2} = \left[\!\!\!\! {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} 0&1&0 \end{array}} \\ { - E_1^{ - 1}{E_2}} \end{array}} \!\!\!\!\right]{X_2} + \left[\!\!\!\! {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} 0&0 \end{array}} \\ {E_1^{ - 1}} \end{array}} \!\!\!\!\right]\left[ \!\!\!\!\begin{array}{l} {K_d} \\ {N_d} \\ \end{array} \!\!\!\!\right]\text{，} \\ & y = CX = C{X_1} + C{X_2} \text{。} \end{aligned} (13)

 $\left[ \!\!\!\!\begin{array}{l} {K_r}({\alpha _r},{\beta _r}) \\ {N_r}({\alpha _r},{\beta _r}) \\ \end{array} \!\!\!\! \right]{\rm{ = }}\left[\!\!\!\! \begin{array}{l} \left( {1{\rm{ + }}{a_H}} \right){z_R} \\ - \left( {1{\rm{ + }}{a_H}} \right){x_R} \\ \end{array} \!\!\!\! \right]{P_y}({\alpha _r},{\beta _r}) = {B_R}{P_y}({\alpha _r},{\beta _r})\text{，}$ (14)

 $\left\{ \begin{split} & {{\dot X}_1} = \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} 0&1&0 \end{array}} \\ { - E_1^{ - 1}{E_2}} \end{array}} \right]{X_1} + \left[ {\begin{array}{*{20}{c}} 0 \\ {E_1^{ - 1}{B_R}} \end{array}} \right]{P_y}({\alpha _r},{\beta _r})\text{，} \\ & y = C{X_1} + \omega\text{。} \\ \end{split} \right.$ (15)

${q_1} = \left( {{c_{\varphi \varphi }},{b_{\varphi \varphi }},{b_{\varphi \psi }},{b_{\psi \varphi }},{b_{\psi \psi }}} \right)$ ${q_2} = \big( {{a_{\varphi \varphi }},{a_{\varphi \psi }},{a_{\psi \varphi }},}$ ${{a_{\psi \psi }}} \big),q = \left( {{q_1},{q_2}} \right)$ ，得

 $M\left( q \right) = \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {A\left( q \right)}&{B\left( q \right)} \end{array}} \\ {\begin{array}{*{20}{c}} {C\left( q \right)}&{D\left( q \right)} \end{array}} \end{array}} \right]\text{。}$ (16)

 $\begin{split} & M(g) = {F_u}({P_M},\Delta ) \text{，} \\ & \Delta = diag({c_{\varphi \varphi }},{b_{\varphi \varphi }},{b_{\varphi \psi }},{b_{\psi \varphi }},{b_{\psi \psi }},{a_{\varphi \varphi }},{a_{\varphi \psi }},{a_{\psi \varphi }},{a_{\psi \psi }}) \text{，} \\ \end{split}$ (17)
 ${P_M} = \left[ {\begin{array}{*{20}{c}} {{P_{M11}}}&{{P_{M12}}} \\ {{P_{M21}}}&{{P_{M22}}} \end{array}} \right]\text{，}$ (18)

 $\Delta = {F_u}({P_\Delta },{\Delta _\delta })\text{，}$ (19)
 ${\Delta _\delta } = diag({\delta _1},{\delta _2},{\delta _3},{\delta _4},{\delta _5},{\delta _6},{\delta _7},{\delta _8},{\delta _9})\text{。}$ (20)

$\left| {{\delta _i}} \right| \leqslant 1$ ${P_\Delta } = \left[ {\begin{array}{*{20}{c}} {{P_{\Delta 11}}}&{{P_{\Delta 12}}} \\ {{P_{\Delta 21}}}&{{P_{\Delta 22}}} \end{array}} \right]$ ，利用“星积运算”，得到 $M(\delta )$ 的线性分式变换模型为：

 $M(\delta ) = {F_u}(P,{\Delta _\delta })\text{。}$ (21)

 $P = \left[ {\begin{split} {{F_l}({P_\Delta },{P_{M11}})}\;\;\;\;{{P_{\Delta 12}}{{(I - {P_{M11}}{P_{\Delta 22}})}^{ - 1}}{P_{M12}}} \\ {{P_{M21}}{{(I - {P_{\Delta 22}}{P_{M11}})}^{ - 1}}{P_{\Delta 12}}}\;\;\;\;\;{{F_u}({P_M},{P_{\Delta 21}})} \end{split}} \right]\text{。}$ (22)

 图 8 船舶舵/翼舵减摇 $\mu$ -鲁棒控制系统原理图 Fig. 8 Ship rudder/wing rudder roll-robust control system schematic

 ${W_p} = \frac{{0.39s}}{{{s^4} + 2.46{s^3} + 1.29{s^2} + 0.84s + 0.19}}\text{，}$ (23)
 ${W_r} = \frac{{0.03{s^2} + 0.03s + 0.01}}{{{s^2} + 100s + 0.001}}\text{。}$ (24)

 ${k_{rk}}\left( s \right) = \frac{{ - 8.89(s + 100)(s + 0.94)({s^2} + 0.08s + 0.003)}}{{(s + 5.10)(s + 0.63)(s + 0.05)({s^2} + 5.01s + 21.4)}}\text{。}$ (25)
3 舵角/翼舵角智能协调决策器设计

 $\begin{split} {J_r}\left( {k + 1} \right) =& \int {_{{\alpha _r}\left( k \right)}^{{\alpha _r}\left( {k + 1} \right)}} {M_{\alpha r}}\left( {\theta ,{\beta _r}\left( k \right)} \right){\rm d}\theta + \\ & \int {_{{\alpha _r}\left( {k + 1} \right) + {\beta _r}\left( k \right)}^{{\alpha _r}\left( {k + 1} \right) + {\beta _r}\left( {k + 1} \right)}} {M_{\beta r}}\left( {{\alpha _r}\left( {k + 1} \right),\theta } \right){\rm d}\theta \text{，} \end{split}$ (26)
 ${M_{\alpha r}}{\rm{ + }}{M_{\alpha N}} + {M_{\alpha J}} + {M_{\alpha h}} + {M_{\alpha f}} = 0\text{，}$ (27)
 ${M_{\beta r}}{\rm{ + }}{M_{\beta N}} + {M_{\beta J}} + {M_{\beta h}} + {M_{\beta f}} = 0\text{。}$ (28)

 $\begin{split} {J_r}\left( {k + 1} \right) =& \displaystyle\frac{1}{2}\rho {V^2}{S_{\alpha r}}{L_{\alpha r}}\int {_{{\alpha _r}\left( k \right)}^{{\alpha _r}\left( {k + 1} \right)}} {c_{\alpha r}}\left( {\theta ,{\beta _r}\left( {k{\rm{ + }}1} \right)} \right){\rm d}\theta + \\ & \left( {{M_{\alpha J}} + {M_{\alpha h}} + {M_{\alpha f}}} \right)\left( {{\alpha _r}\left( {k{\rm{ + }}1} \right) - {\alpha _r}\left( k \right)} \right) + \\ & \displaystyle\frac{1}{2}\rho {V^2}{S_{\beta r}}{L_{\beta r}}\int {_{{\alpha _r}\left( {k + 1} \right) + {\beta _r}\left( k \right)}^{{\alpha _r}\left( {k + 1} \right) + {\beta _r}\left( {k + 1} \right)}} {c_{\beta N}}\times \\ &\left( {{\alpha _r}\left( {k{\rm{ + }}1} \right),\theta } \right)d\theta + \left( {{M_{\beta J}} + {M_{\beta h}} + {M_{\beta f}}} \right) \times\\ &\left( {{\beta _r}\left( {k{\rm{ + }}1} \right) - {\beta _r}\left( k \right)} \right) \text{。} \end{split}$ (29)

 ${u_r}({\alpha _r}(k + 1),{\beta _r}(k + 1)) = {K_r}(k + 1)\text{，}$ (30)

 $\begin{split} & \left| {{\alpha _r}(k + 1)} \right| \leqslant {\alpha _{r\max }} \text{，} \\ & \left| {{{\dot \alpha }_r}(k + 1)} \right| \leqslant {{\dot \alpha }_{r\max }} \text{，} \\ & \left| {{\beta _r}(k + 1)} \right| \leqslant {\beta _{r\max }} \text{，} \\ & \left| {{{\dot \beta }_r}(k + 1)} \right| \leqslant {{\dot \beta }_{r\max }} \text{。} \end{split}$ (31)

 $\begin{split} & {J_r}(k + 1)\min \text{，} \\ & {u_r}({\alpha _r}(k + 1),{\beta _r}(k + 1)) = {N_r}(k + 1) \text{，} \\ & \left| {{\alpha _r}(k + 1)} \right| \leqslant {\alpha _{r\max }} \text{，} \\ & \left| {{{\dot \alpha }_r}(k + 1)} \right| \leqslant {{\dot \alpha }_{r\max }} \text{，} \\ & \left| {{\beta _r}(k + 1)} \right| \leqslant {\beta _{r\max }} \text{，} \\ & \left| {{{\dot \beta }_r}(k + 1)} \right| \leqslant {{\dot \beta }_{r\max }} \text{。} \end{split}$ (32)

 图 9 舵角/翼舵角改进GA智能协调决策分配器实现流程图 Fig. 9 Flow diagram of the improved intelligent coordination decision distributor for rudder angle / wing rudder angle
4 系统仿真试验及结果

5 结　语

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