CN103935480B

CN103935480B - A kind of rudder stabilization method based on analytic modell analytical model Predictive control design

Info

Publication number: CN103935480B
Application number: CN201410105375.4A
Authority: CN
Inventors: 彭秀艳; 贾书丽; 孙涛; 王显峰; 孙宏放
Original assignee: Harbin Engineering University
Current assignee: Harbin Engineering University
Priority date: 2014-05-26
Filing date: 2014-05-26
Publication date: 2016-08-17
Anticipated expiration: 2034-05-26
Also published as: CN103935480A

Abstract

The invention belongs to the fields of ship engineering, control science and control engineering, and relates to a ship rudder anti-rolling method based on analytical model predictive control design. The invention includes: acquiring various coefficients of the control object; acquiring the roll angle signal; acquiring the roll angle velocity signal; and setting the initial value of the controller parameter. The present invention makes full use of the boundedness of uncertain items, and combines analytical model predictive control to effectively solve the parameter uncertainty problem of the rudder stabilization control system. Compared with other methods, the method provided has improved the stabilization efficiency improve. The invention has a simple structure, is easy to implement, and can well meet the needs of practical engineering applications.

Description

A Ship Rudder Stabilization Method Based on Analytical Model Predictive Control Design

技术领域technical field

本发明属于船舶工程、控制科学与控制工程领域，涉及一种基于解析模型预测控制设计的船舶舵减摇方法。The invention belongs to the fields of ship engineering, control science and control engineering, and relates to a ship rudder anti-rolling method based on analytical model predictive control design.

背景技术Background technique

由于船舶横摇运动阻尼很小，使得船舶在风浪中会产生剧烈的横摇，过大的横摇会对船舶航行性能和安全性带来很大的影响，为保证船舶在复杂海况安全航行，舵减横摇技术作为一种新型控制思想，近年来在国内外受到了极大的关注，而且用舵来进行减摇装置的设计，简单方便，价格低廉，因此，舵减摇技术是船舶运动控制领域中一个重要研究课题。Due to the small damping of the rolling motion of the ship, the ship will produce severe rolling in the wind and waves. Excessive rolling will have a great impact on the navigation performance and safety of the ship. In order to ensure the safe navigation of the ship in complex sea conditions, Rudder anti-rolling technology, as a new control idea, has received great attention at home and abroad in recent years, and the design of anti-rolling device with rudder is simple, convenient and cheap. Therefore, rudder anti-rolling technology is an important An important research topic in the field of control.

目前对于舵减横摇系统的控制，国内外都有一定的研究，但尚未发现应用解析模型预测控制进行舵减摇设计的文献和专利成果。At present, there are certain researches on the control of rudder anti-rolling system at home and abroad, but no literature and patent achievements have been found on the application of analytical model predictive control for rudder anti-rolling design.

由于船舶运动本质上是非线性的，从船舶运动环境和自身运动特征可知，其精确的数学模型难以得到，造成模型参数具有不确定性，因而，船舶舵减横摇控制器应该基于非线性控制理论来设计，且必须具有对模型参数摄动的鲁棒性。船舶舵减横摇控制要求在非线性模型具有参数不确定前提下对海浪引起的横摇运动抑制，舵减横摇摇控制器设计必须满足对模型参数摄动的鲁棒性。但传统PID，LQG控制算法，不能有效处理模型非线性和参数不确定性问题。解析模型预测控制是针对非线性模型提出的，具有易于建模、响应迅速、控制性能较好、鲁棒性强和逻辑结构简单等特点，但目前解析模型预测控制在大多情况下，只解决确定性标称系统的控制问题，为此针对非线性系统模型参数不确定性问题，提出一种改进解析模型预测控制方法。首先，将其重定义成带有不确定项标称系统模型，基于解析模型预测控制理论对重定义标称模型进行理论推导，得到含有不确定项的控制律。由于控制规律有界，取其边界值，从理论推导上消除了控制器中的不确定项，有效的解决了模型中的参数不确定性问题。Since the ship's motion is nonlinear in nature, it is difficult to obtain an accurate mathematical model from the ship's motion environment and its own motion characteristics, resulting in uncertainty in the model parameters. Therefore, the ship rudder anti-rolling controller should be based on nonlinear control theory To design, and must be robust to perturbation of model parameters. The anti-rolling control of ship rudder requires the suppression of the rolling motion caused by sea waves under the premise that the nonlinear model has parameter uncertainties. The design of the rudder anti-rolling controller must satisfy the robustness to the perturbation of model parameters. However, traditional PID and LQG control algorithms cannot effectively deal with model nonlinearity and parameter uncertainty. Analytical model predictive control is proposed for nonlinear models. It has the characteristics of easy modeling, rapid response, good control performance, strong robustness, and simple logic structure. However, in most cases, analytical model predictive control only solves certain problems. For the control problem of nonlinear nominal system, an improved analytical model predictive control method is proposed to solve the problem of nonlinear system model parameter uncertainty. First, it is redefined as a nominal system model with uncertain terms, and the redefined nominal model is theoretically deduced based on the analytical model predictive control theory, and the control law with uncertain terms is obtained. Since the control law is bounded, the boundary value is taken, which eliminates the uncertain item in the controller from the theoretical derivation, and effectively solves the parameter uncertainty problem in the model.

船舶舵减横摇控制系统仅能测量船舶横摇角，应用解析模型预测控制方法时须引入状态观测器以获得控制器需要的横摇角速率信息。相比其他观测器，高增益观测器的稳定性和精度有理论保证，计算简便参数少，参数一旦选定，无需调整。由于高增益观测器设计过程中部分函数需满足局部李普希兹条件，这样舵减横摇控制最终需给出其满足设计高增益观测器所需条件，本文证明给出其需要满足条件的详细证明，为其应用提供依据。该方案无需状态可测，通过状态观测器直接获得观测值，解决了横摇角速度不可测问题。The ship rudder anti-rolling control system can only measure the ship's roll angle. When applying the analytical model predictive control method, a state observer must be introduced to obtain the roll angle rate information required by the controller. Compared with other observers, the stability and accuracy of the high-gain observer are theoretically guaranteed, and the calculation is simple and has few parameters. Once the parameters are selected, no adjustment is required. Since some functions in the design process of the high-gain observer need to satisfy the local Lipschitz conditions, the rudder anti-rolling control finally needs to provide the conditions required for the design of the high-gain observer. This paper proves that it needs to satisfy the conditions in detail. provide a basis for its application. This scheme does not need state measurable, and directly obtains the observed value through the state observer, which solves the problem of unmeasurable roll angular velocity.

发明内容Contents of the invention

本发明的目的在于提供能够有效抑制模型的参数摄动的基于解析模型预测控制设计的船舶舵减摇方法。The object of the present invention is to provide a ship rudder stabilization method based on analytical model predictive control design that can effectively suppress the parameter perturbation of the model.

本发明的目的是这样实现的：The purpose of the present invention is achieved like this:

基于解析模型预测控制设计的船舶舵减摇方法包括：The ship rudder stabilization method based on analytical model predictive control design includes:

(1)获取控制对象的各项系数，包括a₁，a₂，a₃，a₄，b及相对应的由航速及横稳性高度变化引起的不确定项的边界值m_i(i＝1,2,3,4)，b₁，控制对象为：(1) Obtain various coefficients of the control object, including a ₁ , a ₂ , a ₃ , a ₄ , b and the corresponding boundary value m _i of uncertain items caused by changes in ship speed and lateral stability altitude (i= 1,2,3,4), b ₁ , the control object is:

$\{\begin{matrix} {\overset{\cdot \cdot}{x x}}_{11} = = {x x}_{22} \\ {\overset{\cdot \cdot}{x x}}_{22} = = (({a a}_{11} + + Δ Δ {a a}_{11})) {x x}_{22} + + (({a a}_{22} + + Δ Δ {a a}_{22})) | | {x x}_{22} | | {x x}_{22} + + (({a a}_{33} + + Δ Δ {a a}_{33})) {x x}_{11} + + (({a a}_{44} + + Δ Δ {a a}_{44})) {x x}_{11}^{33} + + ((b b + + Δb Δb)) u u + + w w \end{matrix},,$

${a a}_{11} = = - - \frac{{K K}_{p p}}{{I I}_{xx xxx} + + {J J}_{xx xxx}},, {a a}_{22} = = - - \frac{{K K}_{pp pp}}{{I I}_{xx xxx} + + {J J}_{xx xxx}},, {a a}_{33} = = - - \frac{W W \cdot \cdot GM GM}{{I I}_{xx xxx} + + {J J}_{xx xxx}},, {a a}_{44} = = \frac{W W \cdot &Center Dot; GM GM}{{I I}_{xx xxx} + + {J J}_{xx xxx}},,$

$b b = = - - {I I}_{δz δz} {U u}^{22} {Y Y}_{δyy δyy} / / (({I I}_{xx xxx} + + {J J}_{xx xxx})),, {I I}_{xx xxx} + + {J J}_{xx xxx} = = \frac{W W}{g g} {[[((0.3085 0.3085 + + 0.0227 0.0227 B B / / d d - - 0.0043 0.0043 L L / / 100100)) B B]]}^{22},,$

K_pp＝3k_b(I_xx+J_xx)/4，k_a、k_b为衰减系数跟船型，及排水量有关，W为排水量，GM为横稳性高，B为船宽，d为吃水深度，L为船长，g为重力加速度； K _pp ＝3k _b (I _xx +J _xx )/4, k _a and k _b are attenuation coefficients related to ship type and displacement, W is displacement, GM is high lateral stability, B is width of ship, d is draft , L is the length of the ship, g is the acceleration due to gravity;

(2)采集横摇角信号 (2) Acquisition of roll angle signal

(3)获取横摇角速度信号取高增益观测器增益p^-1(θ)C₀ ^T＝[2θ,θ²]^T，设定参数θ值；高增益观测器状态为：(3) Obtain the roll angular velocity signal Take the gain of high-gain observer p ^-1 (θ)C ₀ ^T ＝[2θ,θ ² ] ^T , set the value of parameter θ; the state of high-gain observer is:

$\{\begin{matrix} \overset{\overset{\cdot &Center Dot;}{&OverBar; &OverBar;}}{x x} = = [\begin{matrix} 00 & 11 \\ {a a}_{33} + + Δ Δ {a a}_{33} & {a a}_{11} + + Δ Δ {a a}_{11} \end{matrix}] \overset{&OverBar; &OverBar;}{x x} + + [\begin{matrix} 00 \\ 11 \end{matrix}] ((b b + + Δb Δb)) u u + + (({a a}_{22} + + Δ Δ {a a}_{22})) | | {\overset{&OverBar; &OverBar;}{x x}}_{22} | | {\overset{&OverBar; &OverBar;}{x x}}_{22} + + (({a a}_{44} + + Δ Δ {a a}_{44})) {\overset{&OverBar; &OverBar;}{x x}}_{11}^{33} - - {p p}^{- - 11} ((θ θ)) {C C}^{T T} ((\overset{&OverBar; &OverBar;}{y the y} - - y the y)) \\ \overset{&OverBar; &OverBar;}{y the y} = = [\begin{matrix} 11 \\ 00 \end{matrix}] \overset{&OverBar; &OverBar;}{x x} \end{matrix}$

其中，“-”代表观测，为非线性项，p^-1(θ)C₀ ^T＝[2θ,θ²]^T，具有观测器系统为：Among them, "-" stands for observation, is a nonlinear term, p ^-1 (θ)C ₀ ^T ＝[2θ,θ ² ] ^T , and the observer system is:

$\{\begin{matrix} {\overset{\overset{\cdot &Center Dot;}{&OverBar; &OverBar;}}{x x}}_{11} = = {\overset{&OverBar; &OverBar;}{x x}}_{22} - - 22 θϵ θϵ \\ {\overset{\overset{\cdot &Center Dot;}{&OverBar; &OverBar;}}{x x}}_{22} = = (({a a}_{11} + + Δ Δ {a a}_{11})) {\overset{&OverBar; &OverBar;}{x x}}_{22} + + (({a a}_{22} + + Δ Δ {a a}_{22})) | | {\overset{&OverBar; &OverBar;}{x x}}_{22} | | {\overset{&OverBar; &OverBar;}{x x}}_{22} + + (({a a}_{33} + + Δ Δ {a a}_{33})) {\overset{&OverBar; &OverBar;}{x x}}_{11} (({a a}_{44} + + Δ Δ {a a}_{44})) {\overset{&OverBar; &OverBar;}{x x}}_{11}^{33} + + ((b b + + Δb Δb)) {u u}_{r r} ((\overset{&OverBar; &OverBar;}{x x})) - - {θ θ}^{22} ϵ ϵ \\ ϵ ϵ = = \overset{&OverBar; &OverBar;}{y the y} - - y the y \end{matrix};;$

(4)设定控制器参数初值k₁，k₂，以期望横摇角横摇角及横摇角速度为输入信号，计算舵减摇回路的控制输出u₁，(4) Set the initial value of the controller parameters k ₁ , k ₂ to the desired roll angle roll angle and roll rate As the input signal, calculate the control output u ₁ of the rudder stabilization circuit,

$u_{1} (\overset{&OverBar;}{x}) = - \frac{1}{b &PlusMinus; b_{1}} (k_{0} {\overset{&OverBar;}{x}}_{1} + k_{1} {\overset{&OverBar;}{x}}_{2} + (a_{1} &PlusMinus; m_{1}) {\overset{&OverBar;}{x}}_{2} + (a_{2} &PlusMinus; m_{2}) | {\overset{&OverBar;}{x}}_{2} | {\overset{&OverBar;}{x}}_{2} + (a_{3} &PlusMinus; m_{3}) {\overset{&OverBar;}{x}}_{1} + (a_{4} &PlusMinus; m_{4}) {\overset{&OverBar;}{x}}_{1}^{3}),$ 得到基于解析模型预测控制设计的船舶舵减横摇控制系统为： $u_{1} (\overset{&OverBar;}{x}) = - \frac{1}{b &PlusMinus; b_{1}} (k_{0} {\overset{&OverBar;}{x}}_{1} + k_{1} {\overset{&OverBar;}{x}}_{2} + (a_{1} &PlusMinus; m_{1}) {\overset{&OverBar;}{x}}_{2} + (a_{2} &PlusMinus; m_{2}) | {\overset{&OverBar;}{x}}_{2} | {\overset{&OverBar;}{x}}_{2} + (a_{3} &PlusMinus; m_{3}) {\overset{&OverBar;}{x}}_{1} + (a_{4} &PlusMinus; m_{4}) {\overset{&OverBar;}{x}}_{1}^{3}),$ The ship rudder anti-rolling control system designed based on the analytical model predictive control is obtained as follows:

$\{\begin{matrix} {\overset{\overset{\cdot &Center Dot;}{&OverBar; &OverBar;}}{x x}}_{11} = = {\overset{&OverBar; &OverBar;}{x x}}_{22} - - 22 θϵ θϵ \\ {\overset{\overset{\cdot &Center Dot;}{&OverBar; &OverBar;}}{x x}}_{22} = = (({a a}_{11} + + Δ Δ {a a}_{11})) {\overset{&OverBar; &OverBar;}{x x}}_{22} + + (({a a}_{22} + + Δ Δ {a a}_{22})) | | {\overset{&OverBar; &OverBar;}{x x}}_{22} | | {\overset{&OverBar; &OverBar;}{x x}}_{22} + + (({a a}_{33} + + Δ Δ {a a}_{33})) {\overset{&OverBar; &OverBar;}{x x}}_{11} + + (({a a}_{44} + + Δ Δ {a a}_{44})) {\overset{&OverBar; &OverBar;}{x x}}_{11}^{33} + + ((b b + + Δb Δb)) {u u}_{r r} ((\overset{&OverBar; &OverBar;}{x x})) + + w w - - {θ θ}^{22} ϵ ϵ \\ {u u}_{11} ((\overset{&OverBar; &OverBar;}{x x})) = = - - \frac{11}{b b &PlusMinus; &PlusMinus; {b b}_{11}} (({k k}_{00} {\overset{&OverBar; &OverBar;}{x x}}_{11} + + {k k}_{11} {\overset{&OverBar; &OverBar;}{x x}}_{22} + + (({a a}_{11} &PlusMinus; &PlusMinus; {m m}_{11})) {\overset{&OverBar; &OverBar;}{x x}}_{22} + + (({a a}_{22} &PlusMinus; &PlusMinus; {m m}_{22})) | | {\overset{&OverBar; &OverBar;}{x x}}_{22} | | {\overset{&OverBar; &OverBar;}{x x}}_{22} + + (({a a}_{33} &PlusMinus; &PlusMinus; {m m}_{33})) {\overset{&OverBar; &OverBar;}{x x}}_{11} + + (({a a}_{44} &PlusMinus; &PlusMinus; {m m}_{44})) {\overset{&OverBar; &OverBar;}{x x}}_{11}^{33})) \\ ϵ ϵ = = \overset{&OverBar; &OverBar;}{y the y} - - y the y \end{matrix},,$

其中：为x的观测值， in: is the observed value of x,

预测控制器初值设定，预测周期T₁＝60s，控制阶次l＝4，观测器参数θ＝3，控制器参数k₀＝0.05，k₁＝0.375。Prediction controller initial value setting, prediction period T ₁ =60s, control order l=4, observer parameter θ=3, controller parameter k ₀ =0.05, k ₁ =0.375.

本发明的有益效果在于：The beneficial effects of the present invention are:

(1)充分利用了不确定项的有界性，结合解析模型预测控制，有效解决了舵减摇控制系统的参数不确定性问题。(1) By making full use of the boundedness of uncertain items, combined with analytical model predictive control, the problem of parameter uncertainty in the rudder stabilization control system is effectively solved.

(2)本发明所提供的方法相比其他方法，在减摇效率上得到了提高。(2) Compared with other methods, the method provided by the present invention has improved anti-rolling efficiency.

(3)本发明专利结构简单，易于实现，能很好满足实际工程应用的需要。(3) The structure of the patented invention is simple, easy to implement, and can well meet the needs of practical engineering applications.

附图说明Description of drawings

图1一种基于解析模型预测控制设计的船舶舵减摇方法的工作流程图，Fig. 1 is a working flow diagram of a ship rudder anti-rolling method based on analytical model predictive control design,

图2基于解析模型预测控制设计的实际船舶舵减横摇控制系统的组成框图，Figure 2 is a block diagram of the actual ship rudder anti-rolling control system designed based on analytical model predictive control,

图3本文所提出的解析模型预测控制下横摇运动及舵角输出仿真图。Fig. 3 The simulation diagram of the roll motion and rudder angle output under the predictive control of the analytical model proposed in this paper.

具体实施方式detailed description

下面结合附图对本发明做更详细地描述：The present invention is described in more detail below in conjunction with accompanying drawing:

本发明一种基于解析模型预测控制设计的船舶舵减摇方法，包括：The present invention is a ship rudder anti-rolling method based on analytical model predictive control design, comprising:

(1)获取控制对象的各项系数，包括a₁，a₂，a₃，a₄，b及相对应的不确定项的边界值m_i(i＝1,2,3,4)，b₁。(1) Obtain various coefficients of the control object, including a ₁ , a ₂ , a ₃ , a ₄ , b and the boundary values of the corresponding uncertain items m _i (i=1,2,3,4), b ₁ .

控制对象的数学表达式：Mathematical expression of the control object:

其中为横摇的惯性力矩，为横摇阻尼力矩，为横摇恢复力矩，为横摇角，为横摇角速度，为横摇角加速度，K_R＝-I_δzU²Y_δyyδ，K_D为海浪扰动力矩。 $I_{xx} + J_{xx} = \frac{W}{g} {[(0.3085 + 0.0227 B / d - 0.0043 L / 100) B]}^{2}, K_{p} = 2 k_{a} \sqrt{W \cdot GM \cdot (I_{xx} + J_{xx})} / π,$ K_pp＝3k_b(I_xx+J_xx)/4，k_a、k_b为衰减系数跟船型，及排水量有关。W为排水量，GM为横稳性高，B为船宽，d为吃水深度，L为船长，g为重力加速度，为船舶的进水角，U为航速，δ舵角。in is the moment of inertia of rolling, is the rolling damping moment, is the roll restoring moment, is the roll angle, is the rolling angular velocity, is the rolling angular acceleration, K _R =-I _δz U ² Y _δyy δ, K _D is the wave disturbance moment. $I_{xxx} + J_{xxx} = \frac{W}{g} {[(0.3085 + 0.0227 B / d - 0.0043 L / 100) B]}^{2}, K_{p} = 2 k_{a} \sqrt{W &Center Dot; GM &Center Dot; (I_{xx} + J_{xxx})} / π,$ K _pp ＝3k _b (I _xx +J _xx )/4, k _a and k _b are attenuation coefficients related to ship type and displacement. W is displacement, GM is high lateral stability, B is breadth, d is draft, L is length, g is gravity acceleration, is the flooding angle of the ship, U is the speed, and δ is the rudder angle.

将式(1)表示为状态空间形式Express (1) in the state space form

$\{\begin{matrix} \overset{\cdot \cdot}{x x} = = f f ((x x)) + + g g ((x x)) u u + + w w \\ y the y = = h h ((x x)) \end{matrix} - - - - - - ((22))$

其中， $f (x) = [\begin{matrix} x_{2} \\ a_{1} x_{2} + a_{2} | x_{2} | x_{2} + a_{3} x_{1} + a_{4} x_{1}^{3} \end{matrix}],$ g(x)＝b，h(x)＝x₁，w为海浪扰动，a₁，a₂，a₃，a₄为计算出的已知系数。in, $f (x) = [\begin{matrix} x_{2} \\ a_{1} x_{2} + a_{2} | x_{2} | x_{2} + a_{3} x_{1} + a_{4} x_{1}^{3} \end{matrix}],$ g(x)=b, h(x)=x ₁ , w is wave disturbance, a ₁ , a ₂ , a ₃ , a ₄ are known coefficients calculated.

其中， $a_{1} = - \frac{K_{p}}{I_{xx} + J_{xx}}, a_{2} = - \frac{K_{pp}}{I_{xx} + J_{xx}}, a_{3} = - \frac{W \cdot GM}{I_{xx} + J_{xx}}, a_{4} = \frac{W \cdot GM}{I_{xx} + J_{xx}},$ in, $a_{1} = - \frac{K_{p}}{I_{xx} + J_{xx}}, a_{2} = - \frac{K_{pp}}{I_{xx} + J_{xxx}}, a_{3} = - \frac{W \cdot GM}{I_{xxx} + J_{xx}}, a_{4} = \frac{W &Center Dot; GM}{I_{xxx} + J_{xxx}},$

b＝-I_δzU²Y_δyy/(I_xx+J_xx)。b=-I _δz U ² Y _δyy /(I _xx +J _xx ).

而由于速度和初稳性高度变化引起系数的变化量为：Δa₁，Δa₂，Δa₃，Δa₄，Δb，这些量的变化一般不能准确求出，但可以预估计其最大绝对值，记为：m_i(i＝1,2,3,4)及b₁。However, due to the change of speed and initial stability height, the coefficient changes are: Δa ₁ , Δa ₂ , Δa ₃ , Δa ₄ , Δb. Generally, the changes of these quantities cannot be calculated accurately, but the maximum absolute value can be predicted. are: m _i (i=1, 2, 3, 4) and b ₁ .

对照式(2)，则得出船舶舵减横摇的不确定非线性系统为：Comparing with formula (2), it can be concluded that the uncertain nonlinear system of ship rudder roll reduction is:

$\{\begin{matrix} {\overset{\cdot &Center Dot;}{x x}}_{11} = = {x x}_{22} \\ {\overset{\cdot \cdot}{x x}}_{22} = = (({a a}_{11} + + Δ Δ {a a}_{11})) {x x}_{22} + + (({a a}_{22} + + Δ Δ {a a}_{22})) | | {x x}_{22} | | {x x}_{22} + + (({a a}_{33} + + Δ Δ {a a}_{33})) {x x}_{11} + + (({a a}_{44} + + Δ Δ {a a}_{44})) {x x}_{11}^{33} + + ((b b + + Δb Δb)) u u + + w w \\ y the y = = h h ((x x)) \end{matrix} - - - - - - ((33))$

本文为某运输船，该船的各项参数由表1给出。This article is a transport ship, and the parameters of the ship are given in Table 1.

表1 船舶模型各参数Table 1 Parameters of the ship model

初始航速为10m/s，初始横稳性高为0.776m，由表1可计算出模型的各项系数，假设船速在(0.5U,1.5U)之间变化，且初稳性高度也在(0.5GM,1.5GM)之间变化时，我们可以预估计模型系数变化量的最大绝对值见表2。The initial speed is 10m/s, and the initial lateral stability height is 0.776m. The coefficients of the model can be calculated from Table 1. It is assumed that the ship speed changes between (0.5U, 1.5U), and the initial stability height is also When changing between (0.5GM, 1.5GM), we can predict the maximum absolute value of the model coefficient variation as shown in Table 2.

表2 舵减摇运动方程系数和系数变化量Table 2 Coefficients of rudder stabilization motion equation and coefficient variation

(2)采集横摇角信号通过罗经采集横摇角信号。(2) Acquisition of roll angle signal The roll angle signal is collected through the compass.

(3)获取横摇角速度信号取高增益观测器增益p^-1(θ)C₀ ^T＝[2θ,θ²]^T，设定参数θ值。取高增益观测器状态方程如下：(3) Obtain the roll angular velocity signal Take the high-gain observer gain p ^-1 (θ)C ₀ ^T =[2θ,θ ² ] ^T , and set the parameter θ value. The state equation of the high-gain observer is taken as follows:

$\{\begin{matrix} \overset{\overset{\cdot \cdot}{&OverBar; &OverBar;}}{x x} = = A A \overset{&OverBar; &OverBar;}{x x} + + Bb Bb ((\overset{&OverBar; &OverBar;}{x x})) u u + + φ φ ((\overset{&OverBar; &OverBar;}{x x})) - - {p p}^{- - 11} ((θ θ)) {C C}^{T T} ((\overset{&OverBar; &OverBar;}{y the y} - - y the y)) \\ \overset{&OverBar; &OverBar;}{y the y} = = C C \overset{&OverBar; &OverBar;}{x x} \end{matrix} - - - - - - ((44))$

其中，“-”代表观测，为非线性项，p(θ)定义为如下方程的解：Among them, "-" stands for observation, is a nonlinear term, p(θ) is defined as the solution of the following equation:

0＝-θp(θ)-(A^Tp(θ)+p(θ)A)+C^TC (5)0＝-θp(θ)-(A ^T p(θ)+p(θ)A)+C ^T C (5)

将式(3)写成形如式(4)的状态方程为：The state equation written as formula (3) as formula (4) is:

$\{\begin{matrix} \overset{\overset{\cdot \cdot}{&OverBar; &OverBar;}}{x x} = = [\begin{matrix} 00 & 11 \\ {a a}_{33} + + Δ Δ {a a}_{33} & {a a}_{11} + + Δ Δ {a a}_{11} \end{matrix}] \overset{&OverBar; &OverBar;}{x x} + + [\begin{matrix} 00 \\ 11 \end{matrix}] ((b b + + Δb Δb)) u u + + (({a a}_{22} + + Δ Δ {a a}_{22})) | | {\overset{&OverBar; &OverBar;}{x x}}_{22} | | {\overset{&OverBar; &OverBar;}{x x}}_{22} + + (({a a}_{44} + + Δ Δ {a a}_{44})) {\overset{&OverBar; &OverBar;}{x x}}_{11}^{33} - - {p p}^{- - 11} ((θ θ)) {C C}^{T T} ((\overset{&OverBar; &OverBar;}{y the y} - - y the y)) \\ \overset{&OverBar; &OverBar;}{y the y} = = [\begin{matrix} 11 \\ 00 \end{matrix}] \overset{&OverBar; &OverBar;}{x x} \end{matrix} - - - - - - ((66))$

其中，p^-1(θ)C₀ ^T＝[C_n ¹θ,C_n ²θ²,…,C_n ⁿθⁿ]^T。Wherein, p ^-1 (θ)C ₀ ^T =[C _n ¹ θ,C _n ² θ ² ,...,C _n ⁿ θ ⁿ ] ^T .

由于本文中p^-1(θ)C₀ ^T＝[2θ,θ²]^T，式(6)表示为：Since p ^-1 (θ)C ₀ ^T =[2θ,θ ² ] ^T in this paper, formula (6) is expressed as:

$\{\begin{matrix} {\overset{\overset{\cdot &Center Dot;}{&OverBar; &OverBar;}}{x x}}_{11} = = {\overset{&OverBar; &OverBar;}{x x}}_{22} - - 22 θϵ θϵ \\ {\overset{\overset{\cdot &Center Dot;}{&OverBar; &OverBar;}}{x x}}_{22} = = (({a a}_{11} + + Δ Δ {a a}_{11})) {\overset{&OverBar; &OverBar;}{x x}}_{22} + + (({a a}_{22} + + Δ Δ {a a}_{22})) | | {\overset{&OverBar; &OverBar;}{x x}}_{22} | | {\overset{&OverBar; &OverBar;}{x x}}_{22} + + (({a a}_{33} + + Δ Δ {a a}_{33})) {\overset{&OverBar; &OverBar;}{x x}}_{11} (({a a}_{44} + + Δ Δ {a a}_{44})) {\overset{&OverBar; &OverBar;}{x x}}_{11}^{33} + + ((b b + + Δb Δb)) {u u}_{r r} ((\overset{&OverBar; &OverBar;}{x x})) - - {θ θ}^{22} ϵ ϵ \\ ϵ ϵ = = \overset{&OverBar; &OverBar;}{y the y} - - y the y \end{matrix} - - - - - - ((77))$

由式(7)可获取横摇角速度信号的信息。The rolling angular velocity signal can be obtained by formula (7) Information.

(4)设定控制器参数初值k₁，k₂，以及为输入信号，应用解析模型预测控制的控制算法思想，不考虑模型中不确定项存在的情况下，计算舵减摇回路的控制输出u：(4) Set the initial value of the controller parameter k ₁ , k ₂ to and As the input signal, the control algorithm idea of analytical model predictive control is applied, and the control output u of the rudder stabilization loop is calculated without considering the existence of uncertain items in the model:

$u u = = - - \frac{11}{b b} (({k k}_{00} {\overset{&OverBar; &OverBar;}{x x}}_{11} + + {k k}_{11} {\overset{&OverBar; &OverBar;}{x x}}_{22} + + {a a}_{11} {\overset{&OverBar; &OverBar;}{x x}}_{22} + + {a a}_{22} | | {\overset{&OverBar; &OverBar;}{x x}}_{22} | | {\overset{&OverBar; &OverBar;}{x x}}_{22} + + {a a}_{33} {\overset{&OverBar; &OverBar;}{x x}}_{11} + + {a a}_{44} {\overset{&OverBar; &OverBar;}{x x}}_{11}^{33})) - - - - - - ((88))$

由于不确定项的存在且有界，应用本文进一步改进的解析模型预测控制算法，处理不确定项后得到控制输出u₁：Since the uncertain items exist and are bounded, the control output u ₁ is obtained after dealing with the uncertain items by applying the further improved analytical model predictive control algorithm in this paper:

${u u}_{11} ((\overset{&OverBar; &OverBar;}{x x})) = = - - \frac{11}{b b &PlusMinus; &PlusMinus; {b b}_{11}} (({k k}_{00} {\overset{&OverBar; &OverBar;}{x x}}_{11} + + {k k}_{11} {\overset{&OverBar; &OverBar;}{x x}}_{22} + + (({a a}_{11} &PlusMinus; &PlusMinus; {m m}_{11})) {\overset{&OverBar; &OverBar;}{x x}}_{22} + + (({a a}_{22} &PlusMinus; &PlusMinus; {m m}_{22})) | | {\overset{&OverBar; &OverBar;}{x x}}_{22} | | {\overset{&OverBar; &OverBar;}{x x}}_{22} + + (({a a}_{33} &PlusMinus; &PlusMinus; {m m}_{33})) {\overset{&OverBar; &OverBar;}{x x}}_{11} + + (({a a}_{44} &PlusMinus; &PlusMinus; {m m}_{44})) {\overset{&OverBar; &OverBar;}{x x}}_{11}^{33})) - - - - - - ((99))$

由式(3)、(7)、(9)可以得到基于解析模型预测控制设计的船舶舵减横摇控制系统为：From equations (3), (7), and (9), it can be obtained that the ship rudder anti-rolling control system designed based on analytical model predictive control is:

$\{\begin{matrix} {\overset{\overset{\cdot &Center Dot;}{&OverBar; &OverBar;}}{x x}}_{11} = = {\overset{&OverBar; &OverBar;}{x x}}_{22} - - 22 θϵ θϵ \\ {\overset{\overset{\cdot &Center Dot;}{&OverBar; &OverBar;}}{x x}}_{22} = = (({a a}_{11} + + Δ Δ {a a}_{11})) {\overset{&OverBar; &OverBar;}{x x}}_{22} + + (({a a}_{22} + + Δ Δ {a a}_{22})) | | {\overset{&OverBar; &OverBar;}{x x}}_{22} | | {\overset{&OverBar; &OverBar;}{x x}}_{22} + + (({a a}_{33} + + Δ Δ {a a}_{33})) {\overset{&OverBar; &OverBar;}{x x}}_{11} + + (({a a}_{44} + + Δ Δ {a a}_{44})) {\overset{&OverBar; &OverBar;}{x x}}_{11}^{33} + + ((b b + + Δb Δb)) {u u}_{r r} ((\overset{&OverBar; &OverBar;}{x x})) + + w w - - {θ θ}^{22} ϵ ϵ \\ {u u}_{11} ((\overset{&OverBar; &OverBar;}{x x})) = = - - \frac{11}{b b &PlusMinus; &PlusMinus; {b b}_{11}} (({k k}_{00} {\overset{&OverBar; &OverBar;}{x x}}_{11} + + {k k}_{11} {\overset{&OverBar; &OverBar;}{x x}}_{22} + + (({a a}_{11} &PlusMinus; &PlusMinus; {m m}_{11})) {\overset{&OverBar; &OverBar;}{x x}}_{22} + + (({a a}_{22} &PlusMinus; &PlusMinus; {m m}_{22})) | | {\overset{&OverBar; &OverBar;}{x x}}_{22} | | {\overset{&OverBar; &OverBar;}{x x}}_{22} + + (({a a}_{33} &PlusMinus; &PlusMinus; {m m}_{33})) {\overset{&OverBar; &OverBar;}{x x}}_{11} + + (({a a}_{44} &PlusMinus; &PlusMinus; {m m}_{44})) {\overset{&OverBar; &OverBar;}{x x}}_{11}^{33})) \\ ϵ ϵ = = \overset{&OverBar; &OverBar;}{y the y} - - y the y \end{matrix} - - - - - - ((1010))$

其中，±符号选取与其对应前一项符号一致。Among them, the symbol selection of ± is consistent with the symbol of the previous item.

非线性解析模型预测控制控制器设计实现从理论推导上消除了控制器中的不确定项，完成船舶舵减横摇改进非线性解析模型预测控制器设计，具体为The realization of nonlinear analytical model predictive control controller design eliminates the uncertain items in the controller from theoretical derivation, and completes the design of the improved nonlinear analytical model predictive controller for ship rudder anti-rolling, specifically as

a)含有不确定项的解析模型预测控制规律。a) Analytical model predictive control law with uncertain terms.

将带有不确定项的式(2)转换为：Convert equation (2) with uncertain terms into:

$\{\begin{matrix} \overset{\cdot \cdot}{x x} = = {f f}^{* *} ((x x)) + + {g g}^{* *} ((x x)) u u \\ y the y = = h h ((x x)) \end{matrix} - - - - - - ((1111))$

其中，f^*(x)＝f(x)+Δf(x)，g^*(x)＝g(x)+Δg(x)。不失一般性，假设系统(11)的平衡点x^o，有f^*(x^o)＝0，g^*(x^o)≠0，h(x^o)＝0，称式(11)为重定义下的标称模型。Wherein, f ^* (x)=f(x)+Δf(x), g ^* (x)=g(x)+Δg(x). Without loss of generality, assume that the equilibrium point x ^o of system (11) has f ^* (x ^o ) = 0, g ^* (x ^o )≠0, h(x ^o ) = 0, and formula (11) is called Define the nominal model under .

系统(11)的滚动时域的性能函数为:The performance function of the rolling time domain of system (11) is:

$J J = = \frac{11}{22} {&Integral; &Integral;}_{00}^{{T T}_{11}} {((\overset{^^}{y the y} ((t t + + τ τ)) - - {\overset{^^}{y the y}}_{d d} ((t t + + τ τ))))}^{T T} ((\overset{^^}{y the y} ((t t + + τ τ)) - - {\overset{^^}{y the y}}_{d d} ((t t + + τ τ)))) dτ dτ - - - - - - ((1212))$

其中，和分别为输出和参考信号在[t,t+T₁]的预测值，τ∈[0,T₁]，T₁为预测周期。in, and are the predicted values of the output and reference signals at [t,t+T ₁ ], τ∈[0,T ₁ ], and T ₁ is the forecast period.

系统(11)在t时刻的预测控制问题描述为：The predictive control problem of system (11) at time t is described as:

$\{\begin{matrix} \overset{\cdot \cdot}{x x} = = {f f}^{* *} ((\overset{^^}{x x} ((t t + + τ τ)))) + + {g g}^{* *} ((\overset{^^}{x x} ((t t + + τ τ)))) \overset{^^}{u u} ((t t + + τ τ)) \\ y the y = = h h ((\overset{^^}{x x} ((t t + + τ τ)))) \end{matrix} - - - - - - ((1313))$

状态变量的初始值给为：State variables The initial value of is given as:

$\overset{^^}{x x} ((t t)) = = x x ((t t)) - - - - - - ((1414))$

实际控制规律的初始值，即：Actual Control Law The initial value of , namely:

$u u ((t t)) = = \overset{^^}{u u} ((t t + + τ τ)),, τ τ = = 00 - - - - - - ((1515))$

基于解析模型预测控制思想得到：Based on the idea of analytical model predictive control, we get:

${L L}_{{f f}^{* *}}^{ρ ρ} h h ((x x)) + + {L L}_{{g g}^{* *}} {L L}_{{f f}^{* *}}^{ρ ρ - - 11} h h ((x x)) \overset{^^}{u u} {((t t))}^{* *} - - {y the y}_{d d}^{[[ρ ρ]]} + + K K {M m}_{ρ ρ} = = 00 - - - - - - ((1616))$

其中，K＝[k₀,k₁,…,k_ρ-1]代表矩阵Γ_ll ^-1Γ_ρl ^T的第一行元素，由预测周期T₁、控制阶次l及相关度ρ决定，为最优控制律。Among them, K=[k ₀ , k ₁ ,…,k _ρ-1 ] represents the first row element of the matrix Γ _ll ^-1 Γ _ρl ^T , which is determined by the forecast period T ₁ , control order l and correlation degree ρ, is the optimal control law.

${Γ Γ}_{((i i,, j j))} = = \frac{{T T}_{11}^{i i + + j j - - 11}}{((i i - - 11))!! ((j j - - 11))!! ((i i + + j j - - 11))},, i i,, j j = = 11,, . . . . . .,, ρ ρ + + l l + + 11 - - - - - - ((1919))$

由式(16)，可以得到带有不确定项的解析模型预测控制的最优控制律：From formula (16), the optimal control law of analytical model predictive control with uncertain items can be obtained:

$u u ((t t)) = = {\overset{^^}{u u}}^{* *} ((t t)) = = - - {(({L L}_{{g g}^{* *}} {L L}_{{f f}^{* *}}^{ρ ρ - - 11} h h ((x x))))}^{- - 11} \cdot \cdot {{{Σ Σ}_{i i = = 00}^{ρ ρ - - 11} {k k}_{i i} (({L L}_{{f f}^{* *}}^{i i} h h ((x x)) - - {y the y}_{d d}^{[[i i]]} ((t t)))) + + {L L}_{{f f}^{* *}}^{ρ ρ} h h ((x x)) - - {y the y}_{d d}^{[[ρ ρ]]} ((t t))}} - - - - - - ((2020))$

b)消除控制规律中的不确定项。b) Eliminate uncertain items in the control law.

由式(11)可知：It can be known from formula (11):

$\begin{matrix} | | | | {f f}^{* *} ((x x)) | | | | = = | | | | f f ((x x)) + + Δf Δf ((x x)) \\ \leq \leq | | | | f f ((x x)) | | | | + + {Σ Σ}_{i i = = 00}^{p p} {m m}_{i i} {| | | | x x | | | |}^{i i} \end{matrix} - - - - - - ((21 twenty one))$

则知f^*(x)有界，并记 Then we know that f ^* (x) is bounded, and record

$\begin{matrix} | | | | {g g}^{* *} ((x x)) | | | | = = | | | | g g ((x x)) + + Δg Δg ((x x)) | | | | \\ \leq \leq | | | | g g ((x x)) | | | | + + {b b}_{11} \end{matrix} - - - - - - ((22 twenty two))$

则知g^*(x)有界。又有：Then we know that g ^* (x) is bounded. And again:

$\begin{matrix} | | | | {L L}_{{f f}^{* *}}^{k k} h h ((x x)) | | | | = = | | | | {L L}_{f f}^{k k} h h ((x x)) + + {L L}_{Δf Δf}^{k k} h h ((x x)) \\ \leq \leq | | | | {L L}_{f f}^{k k} h h ((x x)) | | | | + + | | | | {L L}_{{f f}_{d d}}^{k k} h h ((x x)) | | | | \end{matrix} - - - - - - ((23 twenty three))$

$\begin{matrix} | | | | {L L}_{{g g}^{* *}} {L L}_{{f f}^{* *}}^{k k} h h ((x x)) | | | | = = | | | | {L L}_{g g} {L L}_{f f}^{k k} h h ((x x)) + + {L L}_{Δg Δ g} {L L}_{f f}^{k k} h h ((x x)) + + {L L}_{g g} {L L}_{Δf Δ f}^{k k} h h ((x x)) + + {L L}_{Δg Δ g} {L L}_{Δf Δ f}^{k k} h h ((x x)) | | | | \\ \leq \leq | | | | {L L}_{g g} {L L}_{f f}^{k k} h h ((x x)) | | | | + + {b b}_{11} | | | | \frac{&PartialD; &PartialD; (({L L}_{f f}^{k k} h h ((x x))))}{&PartialD; &PartialD; x x} | | | | + + | | | | {L L}_{g g} {L L}_{{f f}_{d d}}^{k k} h h ((x x)) | | | | + + {b b}_{11} | | | | \frac{&PartialD; &PartialD; (({L L}_{{f f}_{d d}}^{k k} h h ((x x))))}{&PartialD; &PartialD; x x} | | | | \end{matrix} - - - - - - ((24 twenty four))$

由式(23)、(24)可知与均有界，边界值：From formula (23), (24) we can know and are bounded, Boundary value:

${L L}_{{f f}^{* *}}^{k k} h h ((x x)) = = {L L}_{f f}^{k k} h h ((x x)) + + {L L}_{&PlusMinus; &PlusMinus; {f f}_{d d}}^{k k} h h ((x x)) - - - - - - ((2525))$

其中，式(25)处的±符号选取与f(x)对应项相同。Among them, the selection of the ± sign in formula (25) is the same as the corresponding item of f(x).

$L_{g^{*}} L_{f^{*}}^{k} h (x)$ 边界值： $L_{g^{*}} L_{f^{*}}^{k} h (x)$ Boundary value:

${L L}_{{g g}^{* *}} {L L}_{{f f}^{* *}}^{k k} h h ((x x)) = = {L L}_{g g} {L L}_{f f}^{k k} h h ((x x)) &PlusMinus; &PlusMinus; {b b}_{11} \frac{&PartialD; &PartialD; (({L L}_{f f}^{k k} h h ((x x))))}{&PartialD; &PartialD; x x} + + {L L}_{g g} {L L}_{&PlusMinus; &PlusMinus; {f f}_{d d}}^{k k} h h ((x x)) &PlusMinus; &PlusMinus; {b b}_{11} \frac{&PartialD; &PartialD; (({L L}_{&PlusMinus; &PlusMinus; {f f}_{d d}}^{k k} h h ((x x))))}{&PartialD; &PartialD; x x} - - - - - - ((2626))$

其中，式(26)b₁项前的±符号与g(x)相同，含f_d项前的符号与f(x)对应项相同。Among them, the sign of ± before the b ₁ term in formula (26) is the same as g(x), and the sign before the term including f _d is the same as the corresponding term of f(x).

综上，可知u(t)有界，取u₁为其边界值，得到改进的舵减横摇解析模型预测控制器：In summary, it can be seen that u(t) is bounded, and u ₁ is taken as its boundary value, and the improved rudder roll reduction analytical model predictive controller is obtained:

${u u}_{11} = = - - \frac{11}{b b &PlusMinus; &PlusMinus; {b b}_{11}} (({k k}_{00} {x x}_{11} + + {k k}_{11} {x x}_{22} + + (({a a}_{11} &PlusMinus; &PlusMinus; {m m}_{11})) {x x}_{22} + + (({a a}_{22} &PlusMinus; &PlusMinus; {m m}_{22})) | | {x x}_{22} | | {x x}_{22} + + (({a a}_{33} &PlusMinus; &PlusMinus; {m m}_{33})) {x x}_{11} + + (({a a}_{44} &PlusMinus; &PlusMinus; {m m}_{44})) {x x}_{11}^{33})) - - - - - - ((2727))$

结合高增益观测器和改进的解析模型预测控制律，船舶舵减横摇控制系统为：Combining the high-gain observer and the improved analytical model predictive control law, the ship rudder anti-rolling control system is:

$\{\begin{matrix} {\overset{\overset{\cdot &Center Dot;}{&OverBar; &OverBar;}}{x x}}_{11} = = {\overset{&OverBar; &OverBar;}{x x}}_{22} - - 22 θϵ θϵ \\ {\overset{\overset{\cdot &Center Dot;}{&OverBar; &OverBar;}}{x x}}_{22} = = (({a a}_{11} + + Δ Δ {a a}_{11})) {\overset{&OverBar; &OverBar;}{x x}}_{22} + + (({a a}_{22} + + Δ Δ {a a}_{22})) | | {\overset{&OverBar; &OverBar;}{x x}}_{22} | | {\overset{&OverBar; &OverBar;}{x x}}_{22} + + (({a a}_{33} + + Δ Δ {a a}_{33})) {\overset{&OverBar; &OverBar;}{x x}}_{11} + + (({a a}_{44} + + Δ Δ {a a}_{44})) {\overset{&OverBar; &OverBar;}{x x}}_{11}^{33} + + ((b b + + Δb Δb)) {u u}_{r r} ((\overset{&OverBar; &OverBar;}{x x})) + + w w - - {θ θ}^{22} ϵ ϵ \\ {u u}_{11} ((\overset{&OverBar; &OverBar;}{x x})) = = - - \frac{11}{b b &PlusMinus; &PlusMinus; {b b}_{11}} (({k k}_{00} {\overset{&OverBar; &OverBar;}{x x}}_{11} + + {k k}_{11} {\overset{&OverBar; &OverBar;}{x x}}_{22} + + (({a a}_{11} &PlusMinus; &PlusMinus; {m m}_{11})) {\overset{&OverBar; &OverBar;}{x x}}_{22} + + (({a a}_{22} &PlusMinus; &PlusMinus; {m m}_{22})) | | {\overset{&OverBar; &OverBar;}{x x}}_{22} | | {\overset{&OverBar; &OverBar;}{x x}}_{22} + + (({a a}_{33} &PlusMinus; &PlusMinus; {m m}_{33})) {\overset{&OverBar; &OverBar;}{x x}}_{11} + + (({a a}_{44} &PlusMinus; &PlusMinus; {m m}_{44})) {\overset{&OverBar; &OverBar;}{x x}}_{11}^{33})) \\ ϵ ϵ = = \overset{&OverBar; &OverBar;}{y the y} - - y the y \end{matrix} - - - - - - ((2828))$

其中，u_r为控制器输出经过舵机执行器后的实际舵角。Among them, u _r is the actual rudder angle output by the controller after passing through the actuator of the steering gear.

该控制器能够有效抑制模型的参数摄动。本发明提出的一种基于解析模型预测控制设计的船舶舵减摇方法思路清晰、步骤完整、易于工程实现。同时，根据据劳斯-霍尔维茨判据理论可知，由劳斯-霍尔维茨判据确定的符合取值范围的k_i可以保证闭环系统稳定。The controller can effectively suppress the parameter perturbation of the model. The ship rudder anti-rolling method proposed by the invention based on the analytical model predictive control design has clear thinking, complete steps and easy engineering implementation. At the same time, according to the Routh-Horwitz criterion theory, the _ki determined by the Routh-Horwitz criterion that meets the value range can ensure the stability of the closed-loop system.

结合图1～3，本发明包括以下几个步骤：In conjunction with Fig. 1～3, the present invention comprises the following steps:

步骤一：计算控制对象的各项系数。包括a₁，a₂，a₃，a₄，b及相对应的由航速及横稳性高度变化引起的不确定项的边界值m_i(i＝1,2,3,4)，b₁。控制对象为：Step 1: Calculate the coefficients of the control object. Including a ₁ , a ₂ , a ₃ , a ₄ , b and the corresponding boundary value m _i (i=1,2,3,4), b ₁ of uncertain items caused by changes in ship speed and lateral stability altitude . The control objects are:

$\{\begin{matrix} {\overset{\cdot &Center Dot;}{x x}}_{11} = = {x x}_{22} \\ {\overset{\cdot &Center Dot;}{x x}}_{22} = = (({a a}_{11} + + Δ Δ {a a}_{11})) {x x}_{22} + + (({a a}_{22} + + Δ Δ {a a}_{22})) | | {x x}_{22} | | {x x}_{22} + + (({a a}_{33} + + Δ Δ {a a}_{33})) {x x}_{11} + + (({a a}_{44} + + Δ Δ {a a}_{44})) {x x}_{11}^{33} + + ((b b + + Δb Δb)) u u + + w w \end{matrix}$

步骤二：由罗经采集横摇角信号 Step 2: Collect the roll angle signal from the compass

步骤三：通过高增益观测器获得横摇角速度信号 Step 3: Obtain the roll angular velocity signal through a high-gain observer

步骤四：以期望横摇角、横摇角以及横摇角速度为输入信号，输入到由解析模型预测控制算法所设计的控制器，获得控制舵角，再将该舵角输入到舵机执行器中，输出实际舵角，将实际舵角再输入到控制对象中得到横摇角信息，通过所得横摇角信息值判断是否满足控制效果要求，调节控制器参数k₁，k₂，直到输出的控制效果满意为止。Step 4: Take the desired roll angle, roll angle, and roll angular velocity as input signals to the controller designed by the analytical model predictive control algorithm to obtain the control rudder angle, and then input the rudder angle to the steering gear actuator In , the actual rudder angle is output, and then the actual rudder angle is input into the control object to obtain the information of the roll angle, and the obtained information value of the roll angle is used to judge whether the control effect is met, and the controller parameters k ₁ , k ₂ are adjusted until the output until the control effect is satisfactory.

本发明是由船舶舵减横摇系统的解析模型预测控制器接受期望横摇角信息、高增益观测器信息及罗经获得的系统输出横摇角信息，经过运算输出指令舵角，舵机执行器根据舵角指令输出实际舵角给船体，船舶抑制产生的横摇角，输出的横摇角经由罗经测得并输入给高增益观测器和控制器，如此形成闭环控制系统，如图2所示。In the present invention, the analytical model predictive controller of the ship rudder anti-rolling system receives the expected roll angle information, the high-gain observer information and the system output roll angle information obtained by the compass, and outputs the command rudder angle through calculation, and the steering gear actuator According to the rudder angle command, the actual rudder angle is output to the hull, the ship suppresses the roll angle, and the output roll angle is measured by the compass and input to the high-gain observer and controller, thus forming a closed-loop control system, as shown in Figure 2 .

据此，本文以某船为例，选取预测周期T₁＝60s，控制阶次l＝4，观测器参数θ＝3，控制器参数k₀＝0.05，k₁＝0.375，在有义波高为3m，遭遇角为90°，波浪周期为8s的海况下，给出解析模型预测控制前后横摇运动及舵角输出仿真图如图3所示。Accordingly, this paper takes a certain ship as an example, selects the prediction period T ₁ =60s, the control order l=4, the observer parameter θ=3, the controller parameter k ₀ =0.05, k ₁ =0.375, when the significant wave height is 3m, the encounter angle is 90°, and the wave period is 8s, the simulation diagram of roll motion and rudder angle output before and after the predictive control of the analytical model is given, as shown in Fig. 3.

Claims

1. A ship rudder stabilization method based on analytical model predictive control design is characterized by comprising the following steps:

(1) obtaining each item coefficient of a control object, including a₁，a₂，a₃，a₄B and the corresponding limit value m of the uncertainty term caused by the change in the speed and the stability_i(i＝1,2,3,4)，b₁The control objects are as follows:

\{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = (a_{1} + {Δa}_{1}) x_{2} + (a_{2} + {Δa}_{2}) | x_{2} | x_{2} + (a_{3} + {Δa}_{3}) x_{1} + (a_{4} + {Δa}_{4}) x_{1}^{3} + (b + Δ b) u + w \end{matrix},

a_{1} = - \frac{K_{p}}{I_{x x} + J_{x x}}, a_{2} = - \frac{K_{p p}}{I_{x x} + J_{x x}}, a_{3} = - \frac{W \cdot G M}{I_{x x} + J_{x x}}, a_{4} = \frac{W \cdot G M}{I_{x x} + J_{x x}},

b＝-I_zU²Y_yy/(I_xx+J_xx)，

K_pp＝3k_b(I_xx+J_xx)/4，k_a、k_bin order to make the attenuation coefficient related to the ship shape and the displacement, W is the displacement, GM is high in transverse stability, B is the ship width, d is the draft, L is the ship length, g is the gravity acceleration, W is the wave disturbance, U is the control output of the rudder anti-rolling loop, U is the speed, U is the speed of the ship, and U is the speed of the ship_rOutputting an actual rudder angle passing through a steering engine actuator for the controller;

(2) collecting roll angle signals

(3) Obtaining roll angular velocity signalsGain p of high gain observer^-1(θ)C^T＝[2θ,θ²]^TSetting a parameter theta value;

the high-gain observer state is:

\{\begin{matrix} \overset{\cdot}{\overset{&OverBar;}{x}} = [\begin{matrix} 0 & 1 \\ a_{3} + {Δa}_{3} & a_{1} + {Δa}_{1} \end{matrix}] \overset{&OverBar;}{x} + [\begin{matrix} 0 \\ 1 \end{matrix}] (b + Δ b) u + (a_{2} + Δ a_{2}) | {\overset{&OverBar;}{x}}_{2} | {\overset{&OverBar;}{x}}_{2} + (a_{4} + Δ a_{4}) {\overset{&OverBar;}{x}}_{1}^{3} - p^{- 1} (θ) C^{T} (\overset{&OverBar;}{y} - y) \\ \overset{&OverBar;}{y} = [\begin{matrix} 1 \\ 0 \end{matrix}] \overset{&OverBar;}{x} \end{matrix}

where "-" represents an observation with an observer system of:

\{\begin{matrix} {\overset{\cdot}{\overset{&OverBar;}{x}}}_{1} = {\overset{&OverBar;}{x}}_{2} - 2 θ ϵ \\ {\overset{\cdot}{\overset{&OverBar;}{x}}}_{2} = (a_{1} + {Δa}_{1}) {\overset{&OverBar;}{x}}_{2} + (a_{2} + {Δa}_{2}) | {\overset{&OverBar;}{x}}_{2} | {\overset{&OverBar;}{x}}_{2} + (a_{3} + {Δa}_{3}) {\overset{&OverBar;}{x}}_{1} + (a_{4} + {Δa}_{4}) {\overset{&OverBar;}{x}}_{1}^{3} + (b + Δ b) u_{r} (\overset{&OverBar;}{x}) - θ^{2} ϵ \\ ϵ = \overset{&OverBar;}{y} - y \end{matrix};

(4) setting an initial value k of a controller parameter₀，k₁At a desired roll angleRoll angleAnd roll angular velocityCalculating the control output u of the rudder anti-roll loop for the input signal₁，

The ship rudder rolling reduction control system based on the analytic model predictive control design is obtained by the following steps:

\{\begin{matrix} {\overset{\cdot}{\overset{&OverBar;}{x}}}_{1} = {\overset{&OverBar;}{x}}_{2} - 2 θ ϵ \\ {\overset{\cdot}{\overset{&OverBar;}{x}}}_{2} = (a_{1} + {Δa}_{1}) {\overset{&OverBar;}{x}}_{2} + (a_{2} + {Δa}_{2}) | {\overset{&OverBar;}{x}}_{2} | {\overset{&OverBar;}{x}}_{2} + (a_{3} + {Δa}_{3}) {\overset{&OverBar;}{x}}_{1} + (a_{4} + {Δa}_{4}) {\overset{&OverBar;}{x}}_{1}^{3} + (b + Δ b) u_{r} (\overset{&OverBar;}{x}) + w - θ^{2} ϵ \\ u_{1} (\overset{&OverBar;}{x}) = - \frac{1}{b &PlusMinus; b_{1}} (k_{0} {\overset{&OverBar;}{x}}_{1} + k_{1} {\overset{&OverBar;}{x}}_{2} + (a_{1} &PlusMinus; m_{1}) {\overset{&OverBar;}{x}}_{2} + (a_{2} &PlusMinus; m_{2}) | {\overset{&OverBar;}{x}}_{2} | {\overset{&OverBar;}{x}}_{2} + (a_{3} &PlusMinus; m_{3}) {\overset{&OverBar;}{x}}_{1} + (a_{4} &PlusMinus; m_{4}) {\overset{&OverBar;}{x}}_{1}^{3}) \\ ϵ = \overset{&OverBar;}{y} - y \end{matrix},

wherein:is an observed value of x and is,

2. the ship rudder stabilization method based on analytical model predictive control design according to claim 1, characterized in that: setting the initial value of the prediction controller, predicting the period T₁60s, control order l 4, observer parameter θ 3, and controller parameter k₀＝0.05，k₁＝0.375。