﻿ 基于FDLQR的喷流舵船舶航向横摇控制研究
 舰船科学技术  2020, Vol. 42 Issue (8): 74-81    DOI: 10.3404/j.issn.1672-7649.2020.08.014 PDF

1. 哈尔滨工程大学 自动化学院，黑龙江 哈尔滨 150001;
2. 中国船舶科学研究中心，江苏 无锡 214082

Research on course and roll control of ship using jet rudder based on FDLQR
JIN Zhong-jia1,2, SI Chao-shan2, QIU Geng-yao2, XIA Xian2
1. School of Automation, Harbin Engineering University, Harbin 150001, China;
2. China Shipping Research Center, Wuxi 214082, China
Abstract: In this paper, roll stabilization control of term vessel using jet rudder is investigated in the random sea waves. Firstly, the hydrodynamic characteristics of jet rudder are introduced, and the ideal control input is obtained by linear interpolation. Secondly, the four-degree-of-freedom non-linear coupling model of surface ship is simplified to three-degree-of-freedom linear model with direct navigation, and a frequency division linear quadratic regulation (FDLQR) control method based on the linear model is proposed to solve the rudder roll damping (RRD) problem of single rudder. The mathematical model of integrated control simulation is constructed. Finally, the comparative simulation of jet rudder control is carried out under under various conditions. The results show that the jet rudder can achieve better heading/rolling control performance. The designed FDLQR controller has strong performance of tracking ability, with consideration of control cost. The control using jet rudder based on FDLQR has important reference value for course and rolling control of the full scale ship, especially at low speed.
Key words: jet rudder     frequency division     LQR     roll stabilization
0 引　言

1 喷流舵流体动力特性分析

 图 1 喷流舵流体动力原理 Fig. 1 Hydrodynamic principle of jet rudder

 ${C_{\mu} }=\frac{{{\rho _J}Q{V_j}}}{{\frac{1}{2}{\rho _{\infty} }V_0^2A}}=\frac{{2{Q^2}}}{{V_0^2bS\Delta hS'}}{\text{。}}$ (1)

 图 2 喷流舵喷流形式 Fig. 2 Patterns of jet rudder

 图 3 喷流舵升力系数特性图 Fig. 3 Characteristic chart of lift coefficient of jet rudder
 ${C_{L - {\rm{JR}}}}=f({C_{\mu} }(\delta ),\delta )={L'_{{\rm{J}}\delta }}\delta \;{\text{。}}$ (2)

2 船舶运动数学模型 2.1 船舶非线性耦合运动模型

1）运动坐标系原点 ${o_b}$ 与船舶重心G重合；

2） ${o_{_b}}{{{x}}_{{_b}}}$ ${o_{_b}}{{{y}}_{{_b}}}$ ${o_{_b}}{{{z}}_{{_b}}}$ 轴是船体的3个惯性主轴。

 $\left\{ \begin{array}{l} m(\dot u - vr)={X_{\sum} } \;{\text{，}} \\ m(\dot v + ur)={Y_{\sum} } \;{\text{，}}\\ {I_x}\ddot \phi ={K_{\sum} } \;{\text{，}}\\ {I_z}\dot r={N_{\sum} } \;{\text{。}}\\ \end{array} \right.$ (3)

 $\left\{ \begin{array}{l} (m + {m_x})\dot u - (m + {m_y})vr=X {\text{，}}\\ (m + {m_y})\dot v + (m + {m_x})ur + {m_y}{\alpha _y}\dot r - {m_y}{l_y}\ddot \phi =Y {\text{，}}\\ ({I_x} + {J_x})\ddot \phi - {m_y}{l_y}\dot v - {m_x}{l_x}ur + W \cdot \overline {GM} \cdot \phi =K {\text{，}}\\ ({I_z} + {J_z})\dot r + {m_y}{\alpha _y}\dot v=N {\text{。}}\\ \end{array} \right.$ (4)

 ${\dot{ x}}={{f}}({{x}},{{u}},{{d}}){\text{。}}$ (5)

 $\left[ {\begin{array}{*{20}{c}} {\dot u} \\ {\dot v} \\ {\dot r} \\ {\dot \psi } \\ {\dot p} \\ {\dot \phi } \end{array}} \right]{\rm{=}}\left[ {\begin{array}{*{20}{c}} {\frac{{{\tau _X}}}{{m + {m_x}}}} \\ {\frac{{({I_x} + {J_x})({I_z} + {J_z}){\tau _Y} + ({I_z} + {J_z}){l_y}{m_y}{\tau _K} - ({I_x} + {J_x}){\alpha _y}{m_y}{\tau _N}}}{{({I_x} + {J_x})({I_z} + {J_z})(m + {m_y}) - ({I_x} + {J_x})\alpha _y^2m_y^2 - ({I_z} + {J_z})l_y^2m_y^2}}} \\ {\frac{{ - ({I_x} + {J_x}){\alpha _y}{m_y}{\tau _Y} - {\alpha _y}{l_y}m_y^2{\tau _K} + (({I_x} + {J_x})(m + {m_y}) - l_y^2m_y^2){\tau _N}}}{{({I_x} + {J_x})({I_z} + {J_z})(m + {m_y}) - ({I_x} + {J_x})\alpha _y^2m_y^2 - ({I_z} + {J_z})l_y^2m_y^2}}} \\ {r\cos \phi } \\ {\frac{{({I_z} + {J_z}){l_y}{m_y}{\tau _Y} + (({I_z} + {J_z})(m + {m_y}) - \alpha _y^2m_y^2){\tau _K} - {\alpha _y}{l_y}m_y^2{\tau _N}}}{{({I_x} + {J_x})({I_z} + {J_z})(m + {m_y}) - ({I_x} + {J_x})\alpha _y^2m_y^2 - ({I_z} + {J_z})l_y^2m_y^2}}} \\ p \end{array}} \right]\;{\text{。}}$ (6)

 $\begin{split} {\tau _X}=\;&(m + {m_y})vr + {X_{uu}}{u^2} + {X_{vr}}vr + {X_{vv}}{v^2} + {X_{rr}}{r^2} +\\ &{X_{\phi \phi }}{\phi ^2} + (1 - {t_{\rm{p}}})\rho {n^2}D_{\rm{P}}^4{k_T} - \frac{1}{2}(1 - {t_R})\rho {A_{\rm{R}}}{f_a}U_{\rm{R}}^2\\ &\sin {\alpha _{\rm{R}}}\cos \delta + {X_{{\rm{env}}}} \;{\text{，}}\\ {\tau _Y}= \;&- (m + {m_x})ur + {Y_v}v + {Y_r}r + {Y_p}p + {Y_{\phi} }\phi + {Y_{vvv}}{v^3} + \\ &{Y_{rrr}}{r^3} + {Y_{vvr}}{v^2}r + {Y_{vrr}}v{r^2} + {Y_{vv\phi }}{v^2}\phi + {Y_{v\phi \phi }}v{\phi ^2} +\\ & {Y_{rr\phi }}{r^2}\phi +{Y_{r\phi \phi }}r{\phi ^2} - \frac{1}{2}(1 + {a_{\rm{H}}})\rho {A_{\rm{R}}}{f_a}U_{\rm{R}}^2\\ &\sin {\alpha _{\rm{R}}}\cos \delta {\rm{ + }}{Y_{{\rm{env}}}} \;{\text{，}}\\ {\tau _K}=\;&{m_x}{l_x}ur - WGM\phi + {K_v}v + {K_r}r + {K_p}p + {K_{\phi} }\phi +\\ & {K_{vvv}}{v^3} + {K_{rrr}}{r^3} + {K_{vvr}}{v^2}r + {K_{vrr}}v{r^2} + {K_{vv\phi }}{v^2}\phi + \\ &{K_{v\phi \phi }}v{\phi ^2} + {K_{rr\phi }}{r^2}\phi + {K_{r\phi \phi }}r{\phi ^2} + \frac{1}{2}(1 + {a_{\rm{H}}})\\ &{z_{\rm{R}}}\rho {A_{\rm{R}}}{f_a}U_{\rm{R}}^2\sin {\alpha _{\rm{R}}}\cos {\delta _{\rm{R}}} + {K_{{\rm{env}}}} \;{\text{，}}\\ {\tau _N}=\;&{N_v}v + {N_r}r + {N_p}p + {N_\varphi }\varphi + {N_{vvv}}{v^3} + {N_{rrr}}{r^3} + \\ &{N_{vvr}}{v^2}r + {N_{vrr}}v{r^2} + {N_{vv\phi }}{v^2}\phi + {N_{v\phi \phi }}v{\phi ^2} +\\ &{N_{rr\phi }}{r^2}\phi + {N_{r\phi \phi }}r{\phi ^2} - \frac{1}{2}({x_{\rm{R}}} + {a_{\rm{H}}}{x_{\rm{H}}})\\ &\rho {A_{\rm{R}}}{f_a}U_{\rm{R}}^2\sin {\alpha _{\rm{R}}}\cos \delta + {N_{{\rm{env}}}} \;{\text{。}} \end{split}$

2.2 船舶直航运动模型

 ${{M}}{\dot{ \nu}} + {{N}}({u_0}){{\nu}} + {{G}}{{\eta}} ={{b}}\delta\;{\text{。}}$ (7)

 $\begin{split}{{TM}}'{{{T}}^{ - 1}}{\dot{ \nu}} + \frac{U}{L}{{TN}}'({u_0}){{{T}}^{ - 1}}{{\nu}} +& {\left(\frac{U}{L}\right)^2}{{TG}}'{{{T}}^{ - 1}}{{\eta}} =\\ & \frac{{{U^2}}}{L}{{Tb}}'\delta {\text{。}} \end{split}$ (8)

 $\begin{array}{l} {{x}}={{Ax}} + {{Bu}} \;{\text{,}}\\ {{y}}={{Cx}} \;{\text{。}}\\ \end{array}$ (9)

 ${{A}}=\left[ {\begin{array}{*{20}{c}} {{{({{{A}}_{11}})}_{3 \times 3}}}&{{{({{{A}}_{12}})}_{3 \times 2}}} \\ {{{({{{A}}_{21}})}_{2 \times 3}}}&{{{({{{A}}_{22}})}_{2 \times 2}}} \end{array}} \right]{\text{，}}$
 ${({{{A}}_{11}})_{3 \times 3}}=- {{{M}}^{ - 1}} \frac{{{U_0}}}{L}{{TN}}{{{T}}^{ - 1}}{\text{，}}$
 ${({{{A}}_{12}})_{3 \times 2}}=- {{{M}}^{ - 1}}\frac{{{U_0}}}{L}{{TG}}{{{T}}^{ - 1}} (:,[2,3]){\text{，}}$
 ${({{{A}}_{21}})_{2 \times 3}}=\left[ \begin{array}{*{20}{c}} 0&1&0 \\ 0&0&1 \end{array} \right] {\text{，}}{({{{A}}_{22}})_{2 \times 2}} =\left[ {\begin{array}{*{20}{c}} 0&0 \\ 0&0 \end{array}} \right] {\text{，}}$
 ${{B}}={\left[ {\begin{array}{*{20}{c}} {{{({B_1})}_{3 \times 1}}}&0&0 \end{array}} \right]^{\rm{T}}}{\text{，}}{B_1}={{{M}}^{ - 1}}\frac{{U_0^2}}{L}{{Tb}}'{\text{，}}$
 ${{C}}=\left[ \begin{array}{*{20}{c}} 0&1&0&0&0 \\ 0&0&1&0&0 \\ 0&0&0&1&0 \\ 0&0&0&0&1 \end{array} \right]{\text{。}}$

2.3 海浪干扰力矩建模

 图 4 波浪干扰力模拟与运动响应关系 Fig. 4 The relationship between wave disturbance force simulation and motion response

 $\begin{array}{l} {{{\dot{ x}}}_{\rm{w}}}={{{A}}_{\rm{w}}}{{{x}}_{\rm{w}}} + {{{B}}_{\rm{w}}}u \;{\text{，}}\\ {{{y}}_{\rm{w}}}{\rm{=}}{{{C}}_{\rm{w}}}{{{x}}_{\rm{w}}} \;{\text{。}}\\ \end{array}$ (10)

3 基于FDLQR的RRD控制器设计 3.1 线性二次最优RRD控制系统设计

RRD系统的控制目标是同时控制航向和横摇，即使得目标航向角 ${\psi _{{d}}}$ 为常数，目标横摇角 ${\phi _{{d}}}$ 、目标横摇角速度 ${p_{{d}}}$ 满足 ${\phi _{{d}}}$ = ${p_{{d}}}$ =0。从控制角度看，引入输入项控制，实际上增大了系统的自振频率和阻尼。设定 ${{{y}}_{{d}}}={\left[ {\begin{array}{*{20}{c}} 0\quad0\quad0\quad{{\psi _{{d}}}} \end{array}} \right]{\rm^{T}}}$ ，当目标输出量作用于系统时，要求系统产生一控制向量，使系统实际输出向量 ${{y}}$ 始终跟踪目标输出，并使得性能指标最小化，这是一个典型输出跟踪系统优化问题。因此，取输出跟踪系性能指标[6]如下：

 ${{J}}=\min \left\{ {\frac{1}{2}\int_0^{\rm{T}} {({{{\tilde{ y}}}^{\rm{T}}}{{Q}}{\tilde{ y}} + {{{u}}^{\rm{T}}}{{Ru}}){{{\rm{d}}}}\tau } } \right\}\;{\text{。}}$ (11)

 图 5 全状态反馈LQ跟踪控制原理图 Fig. 5 Principle diagram of full state feedback LQ tracking control

 ${{u}}={{{G}}_1}{{x}} + {{{G}}_{\bf{2}}}{{{y}}_{\rm{d}}} \;{\text{。}}$ (12)

 图 6 常规舵开环系统特征根 Fig. 6 Open-loop system eigenvalues of conventional rudder

 图 9 喷流舵闭环系统特征根 Fig. 9 Close-loop system eigenvalues of jet rudder

 图 7 常规舵闭环系统特征根 Fig. 7 Close-loop system eigenvalues of conventional rudder

 图 8 喷流舵开环系统特征根 Fig. 8 Open-loop system eigenvalues of jet rudder
3.2 舵减摇分频控制

 ${h_l}(s)=\frac{{{\psi _{{\rm{filter}}}}}}{\psi }=\frac{1}{{{T_l}s + 1}}\;,$ (13)

 ${h_h}(s)=\frac{{{\phi _{{\rm{filter}}}}}}{\phi }=\frac{{{T_h}s}}{{{T_h}s + 1}}\;{\text{。}}$ (14)

 $\begin{split} & {{{x}}_{\psi f}}(k + 1)={e^{ - h/{T_l}}}{{{x}}_{\psi f}}(k) + (1 - {e^{ - h/{T_l}}}){{{u}}_{\psi} }(k) \;{\text{，}}\\ & {{{y}}_{\psi f}}(k)={{{x}}_{\psi f}}(k) \;{\text{，}}\\ & {{{u}}_{\psi} }(k){\rm{=}}{{{x}}_{\psi} }(k) \;{\text{。}} \end{split}$ (15)

 $\begin{split} & {{{x}}_{\phi f}}(k + 1)={e^{ - h/{T_h}}}{{{x}}_{\phi f}}(k) + ({e^{ - h/{T_h}}} - 1){{{u}}_{\phi} }(k) \;{\text{，}}\\ & {{{y}}_{\phi f}}(k)={{{x}}_{\phi f}}(k) + {{{x}}_{\phi} }(k) \;{\text{，}}\\ & {{{u}}_{\phi} }(k){\rm{=}}{{{x}}_{\phi} }(k) \;{\text{。}} \end{split}$ (16)

 ${{u}}={G_1}{{{x}}_f} + {G_{\bf{2}}}{{{y}}_{\rm{d}}\;{\text{。}}}$ (17)

 图 10 基于FDLQR的航向横摇控制仿真模型示意图 Fig. 10 Schematic diagram of simulation model for course rolling control based on FDLQR
4 系统仿真与分析

 图 11 常规舵和喷流舵船舶航向角时间历程（U=7.4 m/s） Fig. 11 The ship course angle time history of ships using conventional rudder and jet rudder (U=7.4 m/s)

 图 14 常规舵和喷流舵角速度时间历程（U=7.4 m/s） Fig. 14 The angle rate time history of conventional rudder and jet rudder (U=7.4 m/s）

 图 12 常规舵和喷流舵船舶横摇角时间历程（U=7.4 m/s） Fig. 12 The ship rolling angle time history of ships using conventional rudder and jet rudder (U=7.4 m/s)

 图 13 常规舵和喷流舵角时间历程（U=7.4 m/s） Fig. 13 The angle time history of conventional rudder and jet rudder (U=7.4 m/s)

 图 15 常规舵和喷流舵船舶航向角时间历程（U=4 m/s） Fig. 15 The ship course angle time history of ships using conventional rudder and jet rudder (U=4 m/s)

 图 18 常规舵和喷流舵角速度时间历程（U=4 m/s） Fig. 18 The angle rate time history of conventional rudder and jet rudder (U=4 m/s)

 图 16 常规舵和喷流舵船舶横摇角时间历程（U=4 m/s） Fig. 16 The ship rolling angle time history of ships using conventional rudder and jet rudder (U=4 m/s)

 图 17 常规舵和喷流舵角时间历程（U=4 m/s） Fig. 17 The angle time history of conventional rudder and jet rudder (U=4 m/s)

5 结　语

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