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

Abstract

The invention belongs to Marine engineering, control science and control engineering field, relating to a kind of rudder stabilization method based on analytic modell analytical model Predictive control design.The present invention includes: obtain each term coefficient of control object；Gather roll angle signal；Obtain angular velocity in roll signal；Setting controller initial parameter values.The present invention takes full advantage of the boundedness of indeterminate, in conjunction with analytic modell analytical model PREDICTIVE CONTROL, efficiently solves the Parameter uncertainties sex chromosome mosaicism of rudder stabilization control system, and the method provided is compared additive method, is improved on anti-rolling efficiency.Present configuration is simple, it is easy to accomplish, the needs of practical engineering application can be met very well.

Description

Ship rudder stabilization method based on analytic model predictive control design

Technical Field

The invention belongs to the fields of ship engineering, control science and control engineering, and relates to a ship rudder stabilizing method based on analytical model predictive control design.

Background

The damping of the rolling motion of the ship is very small, so that the ship can generate severe rolling in stormy waves, the overlarge rolling can bring great influence on the navigation performance and safety of the ship, the rudder rolling reduction technology is taken as a novel control idea for ensuring the safe navigation of the ship under complex sea conditions, great attention is paid to the rudder rolling reduction technology at home and abroad in recent years, and the rudder is used for designing the rolling reduction device, so that the rudder rolling reduction technology is simple and convenient and has low price, and therefore, the rudder rolling reduction technology is an important research subject in the field of ship motion control.

At present, a certain amount of research is carried out at home and abroad on the control of a rudder roll reducing system, but documents and patent achievements for rudder roll reducing design by applying analytical model prediction control are not found.

Since the ship motion is nonlinear in nature, an accurate mathematical model is difficult to obtain from the ship motion environment and the motion characteristics of the ship, so that model parameters have uncertainty, and therefore, the ship rudder roll reducing controller should be designed based on a nonlinear control theory and must have robustness on perturbation of the model parameters. The ship rudder roll reduction control requires that roll motion caused by sea waves is inhibited on the premise that a nonlinear model has uncertain parameters, and the design of a rudder roll reduction controller must meet the robustness of perturbation on model parameters. However, the traditional PID and LQG control algorithm cannot effectively solve the problems of model nonlinearity and parameter uncertainty. The analytical model predictive control is provided for a nonlinear model, and has the characteristics of easiness in modeling, quick response, better control performance, strong robustness, simple logic structure and the like, but the control problem of a deterministic nominal system is only solved under most conditions of the current analytical model predictive control, so that an improved analytical model predictive control method is provided for the problem of uncertainty of model parameters of the nonlinear system. Firstly, redefining the model to a nominal system model with an uncertain item, and theoretically deducing the redefined nominal model based on an analytic model predictive control theory to obtain a control law with the uncertain item. Because the control rule is bounded, the boundary value is taken, the uncertainty in the controller is eliminated from theoretical derivation, and the problem of parameter uncertainty in the model is effectively solved.

The ship rudder roll reducing control system can only measure the ship roll angle, and a state observer needs to be introduced to obtain roll angle rate information required by the controller when an analytic model predictive control method is applied. Compared with other observers, the stability and the precision of the high-gain observer are theoretically guaranteed, the calculation is simple and convenient, the parameters are few, and once the parameters are selected, the parameters do not need to be adjusted. Because part of functions in the design process of the high-gain observer needs to meet the local Leptoschiz condition, the rudder roll reduction control needs to give the condition which meets the requirement of designing the high-gain observer finally, and the detailed proof that the condition needs to be met is provided for the application of the high-gain observer. According to the scheme, the state can be measured without a state observer, the observed value is directly obtained through the state observer, and the problem that the roll angular velocity cannot be measured is solved.

Disclosure of Invention

The invention aims to provide a ship rudder stabilization method based on analytical model predictive control design, which can effectively inhibit parameter perturbation of a model.

The purpose of the invention is realized as follows:

the ship rudder stabilization method based on analytical model predictive control design comprises 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:

$\left\{\begin{array}{c}{\stackrel{\·}{x}}_1=x_2\\ {\stackrel{\·}{x}}_2=(a_1+\mathrm{\Δ}{a}_1)x_2+(a_2+\mathrm{\Δ}{a}_2)\left|x_2\right|x_2+(a_3+\mathrm{\Δ}{a}_3)x_1+(a_4+\mathrm{\Δ}{a}_4)x_1^3+(b+\mathrm{\Δb})u+w\end{array}\right.,$

$a_1=-\frac{K_p}{I_{\mathrm{xx}}+J_{\mathrm{xx}}},a_2=-\frac{K_{\mathrm{pp}}}{I_{\mathrm{xx}}+J_{\mathrm{xx}}},a_3=-\frac{W\·\mathrm{GM}}{I_{\mathrm{xx}}+J_{\mathrm{xx}}},a_4=\frac{W\·\mathrm{GM}}{I_{\mathrm{xx}}+J_{\mathrm{xx}}},$

$b=-I_{\mathrm{\δz}}{U}^2{Y}_{\mathrm{\δyy}}/(I_{\mathrm{xx}}+J_{\mathrm{xx}}),I_{\mathrm{xx}}+J_{\mathrm{xx}}=\frac{W}{g}{\left[(0.3085+0.0227B/d-0.0043L/100)B\right]}^2,$

K_pp＝3k_b(I_xx+J_xx)/4，k_a、k_bthe attenuation coefficient is related to the ship shape and the water displacement, W is the water displacement, GM is the transverse stability and the height, B is the ship width, d is the draft, L is the ship length, and g is the gravity acceleration;

(2) collecting roll angle signals

(3) Obtaining roll angular velocity signalsGain p of high gain observer^-1(θ)C₀ ^T＝[2θ,θ²]^TSetting a parameter theta value; the high-gain observer state is:

$\left\{\begin{array}{c}\stackrel{\stackrel{\·}{\‾}}{x}=\left[\begin{array}{cc}0& 1\\ a_3+\mathrm{\Δ}{a}_3& a_1+\mathrm{\Δ}{a}_1\end{array}\right]\stackrel{\‾}{x}+\left[\begin{array}{c}0\\ 1\end{array}\right](b+\mathrm{\Δb})u+(a_2+\mathrm{\Δ}{a}_2)\left|{\stackrel{\‾}{x}}_2\right|{\stackrel{\‾}{x}}_2+(a_4+\mathrm{\Δ}{a}_4){\stackrel{\‾}{x}}_1^3-p^{-1}\left(\mathrm{\θ}\right)C^T(\stackrel{\‾}{y}-y)\\ \stackrel{\‾}{y}=\left[\begin{array}{c}1\\ 0\end{array}\right]\stackrel{\‾}{x}\end{array}\right.$

where "-" represents an observation,is a non-linear term, p^-1(θ)C₀ ^T＝[2θ,θ²]^TThe observer system is provided with:

$\left\{\begin{array}{c}{\stackrel{\stackrel{\·}{\‾}}{x}}_1={\stackrel{\‾}{x}}_2-2\mathrm{\θ\ϵ}\\ {\stackrel{\stackrel{\·}{\‾}}{x}}_2=(a_1+\mathrm{\Δ}{a}_1){\stackrel{\‾}{x}}_2+(a_2+\mathrm{\Δ}{a}_2)\left|{\stackrel{\‾}{x}}_2\right|{\stackrel{\‾}{x}}_2+(a_3+\mathrm{\Δ}{a}_3){\stackrel{\‾}{x}}_1(a_4+\mathrm{\Δ}{a}_4){\stackrel{\‾}{x}}_1^3+(b+\mathrm{\Δb})u_r\left(\stackrel{\‾}{x}\right)-{\mathrm{\θ}}^2\mathrm{\ϵ}\\ \mathrm{\ϵ}=\stackrel{\‾}{y}-y\end{array}\right.;$

(4) setting an initial value k of a controller parameter₁，k₂At a desired roll angleRoll angleAnd roll angular velocityCalculating the control output u of the rudder anti-roll loop for the input signal₁，

$u_1\left(\stackrel{\‾}{x}\right)=-\frac{1}{b\±b_1}(k_0{\stackrel{\‾}{x}}_1+k_1{\stackrel{\‾}{x}}_2+(a_1\±m_1){\stackrel{\‾}{x}}_2+(a_2\±m_2)|{\stackrel{\‾}{x}}_2|{\stackrel{\‾}{x}}_2+(a_3\±m_3){\stackrel{\‾}{x}}_1+(a_4\±m_4){\stackrel{\‾}{x}}_1^3),$ The ship rudder rolling reduction control system based on the analytic model predictive control design is obtained by the following steps:

$\left\{\begin{array}{c}{\stackrel{\stackrel{\·}{\‾}}{x}}_1={\stackrel{\‾}{x}}_2-2\mathrm{\θ\ϵ}\\ {\stackrel{\stackrel{\·}{\‾}}{x}}_2=(a_1+\mathrm{\Δ}{a}_1){\stackrel{\‾}{x}}_2+(a_2+\mathrm{\Δ}{a}_2)\left|{\stackrel{\‾}{x}}_2\right|{\stackrel{\‾}{x}}_2+(a_3+\mathrm{\Δ}{a}_3){\stackrel{\‾}{x}}_1+(a_4+\mathrm{\Δ}{a}_4){\stackrel{\‾}{x}}_1^3+(b+\mathrm{\Δb})u_r\left(\stackrel{\‾}{x}\right)+w-{\mathrm{\θ}}^2\mathrm{\ϵ}\\ u_1\left(\stackrel{\‾}{x}\right)=-\frac{1}{b\±b_1}(k_0{\stackrel{\‾}{x}}_1+k_1{\stackrel{\‾}{x}}_2+(a_1\±m_1){\stackrel{\‾}{x}}_2+(a_2\±m_2)|{\stackrel{\‾}{x}}_2|{\stackrel{\‾}{x}}_2+(a_3\±m_3){\stackrel{\‾}{x}}_1+(a_4\±m_4){\stackrel{\‾}{x}}_1^3)\\ \mathrm{\ϵ}=\stackrel{\‾}{y}-y\end{array}\right.,$

wherein:is an observed value of x and is,

initial value setting of predictive controller, prediction period T₁60s, control order l 4, observer parameter θ 3, and controller parameter k₀＝0.05，k₁＝0.375。

The invention has the beneficial effects that:

(1) the method has the advantages that the bounded property of the uncertain items is fully utilized, and the problem of parameter uncertainty of the rudder stabilization control system is effectively solved by combining analytic model predictive control.

(2) Compared with other methods, the method provided by the invention has the advantage that the anti-rolling efficiency is improved.

(3) The invention has simple structure, is easy to realize and can well meet the requirement of practical engineering application.

Drawings

FIG. 1 is a flow chart of a method for stabilizing a rudder of a ship based on analytical model predictive control design,

FIG. 2 is a block diagram of the actual ship rudder roll reducing control system designed based on analytical model predictive control,

FIG. 3 is a simulation of roll motion and rudder angle output under the predictive control of an analytical model as set forth herein.

Detailed Description

The invention is described in more detail below with reference to the accompanying drawings:

the invention relates to a ship rudder stabilizing method based on analytical model predictive control design, which comprises the following steps:

(1) obtaining each item coefficient of a control object, including a₁，a₂，a₃，a₄B and the boundary value m of the corresponding uncertainty term_i(i＝1,2,3,4)，b₁。

Mathematical expression of the control object:

whereinIs the moment of inertia for the roll and,in order to provide a roll damping moment,in order to restore the moment in the event of roll,in order to change the transverse rocking angle,in order to obtain the roll angular velocity,for roll angular acceleration, K_R＝-I_zU²Y_yy，K_DIs the disturbance torque of sea waves. $I_{\mathrm{xx}}+J_{\mathrm{xx}}=\frac{W}{g}{\left[(0.3085+0.0227B/d-0.0043L/100)B\right]}^2,K_p=2k_a\sqrt{W\·\mathrm{GM}\·(I_{\mathrm{xx}}+J_{\mathrm{xx}})}/\mathrm{\π},$ K_pp＝3k_b(I_xx+J_xx)/4，k_a、k_bThe attenuation coefficient is dependent on the ship shape and the displacement. W is the displacement, GM is the transverse stability, B is the width of the ship, d is the draft, L is the length of the ship, g is the acceleration of gravity,the U is the water inlet angle of the ship, and the speed and rudder angle.

Expression of formula (1) as a state space form

$\left\{\begin{array}{c}\stackrel{\·}{x}=f\left(x\right)+g\left(x\right)u+w\\ y=h\left(x\right)\end{array}\right.---\left(2\right)$

Wherein, $f\left(x\right)=\left[\begin{array}{c}{x}_2\\ a_1{x}_2+a_2\left|x_2\right|x_2+a_3{x}_1+a_4{x}_1^3\end{array}\right],$ g(x)＝b，h(x)＝x₁w is the wave disturbance, a₁，a₂，a₃，a₄Is the calculated known coefficient.

Wherein, $a_1=-\frac{K_p}{I_{\mathrm{xx}}+J_{\mathrm{xx}}},a_2=-\frac{K_{\mathrm{pp}}}{I_{\mathrm{xx}}+J_{\mathrm{xx}}},a_3=-\frac{W\·\mathrm{GM}}{I_{\mathrm{xx}}+J_{\mathrm{xx}}},a_4=\frac{W\·\mathrm{GM}}{I_{\mathrm{xx}}+J_{\mathrm{xx}}},$

b＝-I_zU²Y_yy/(I_xx+J_xx)。

and the variation of the coefficient caused by the variation of the speed and the initial stability is as follows: Δ a₁，Δa₂，Δa₃，Δa₄Δ b, the variation of these quantities is generally not exactly solved, but its maximum absolute value can be estimated in advance, written as: m is_i(i-1, 2,3,4) and b₁。

By comparing the formula (2), the uncertain nonlinear system for reducing the rolling of the ship rudder is obtained as follows:

$\left\{\begin{array}{c}{\stackrel{\·}{x}}_1=x_2\\ {\stackrel{\·}{x}}_2=(a_1+\mathrm{\Δ}{a}_1)x_2+(a_2+\mathrm{\Δ}{a}_2)\left|x_2\right|x_2+(a_3+\mathrm{\Δ}{a}_3)x_1+(a_4+\mathrm{\Δ}{a}_4)x_1^3+(b+\mathrm{\Δb})u+w\\ y=h\left(x\right)\end{array}\right.---\left(3\right)$

in this context a transport vessel, the parameters of which are given in table 1.

TABLE 1 Ship model parameters

The initial speed is 10m/s, the initial transverse stability is 0.776m, the coefficients of the model can be calculated from table 1, and if the ship speed changes between (0.5U,1.5U) and the initial stability is also changed between (0.5GM,1.5GM), the maximum absolute value of the coefficient change of the model can be estimated in advance as shown in table 2.

TABLE 2 Rudder roll reduction equation coefficients and coefficient variation

(2) Collecting roll angle signalsAnd collecting a roll angle signal through a compass.

(3) Obtaining roll angular velocity signalsGain p of high gain observer^-1(θ)C₀ ^T＝[2θ,θ²]^TThe value of the parameter θ is set. Taking the state equation of the high-gain observer as follows:

$\left\{\begin{array}{c}\stackrel{\stackrel{\·}{\‾}}{x}=A\stackrel{\‾}{x}+\mathrm{Bb}\left(\stackrel{\‾}{x}\right)u+\mathrm{\φ}\left(\stackrel{\‾}{x}\right)-p^{-1}\left(\mathrm{\θ}\right)C^T(\stackrel{\‾}{y}-y)\\ \stackrel{\‾}{y}=C\stackrel{\‾}{x}\end{array}\right.---\left(4\right)$

where "-" represents an observation,for non-linear terms, p (θ) is defined as the solution of the following equation:

0＝-θp(θ)-(A^Tp(θ)+p(θ)A)+C^TC (5)

writing equation (3) to form the equation of state of equation (4) as:

$\left\{\begin{array}{c}\stackrel{\stackrel{\·}{\‾}}{x}=\left[\begin{array}{cc}0& 1\\ a_3+\mathrm{\Δ}{a}_3& a_1+\mathrm{\Δ}{a}_1\end{array}\right]\stackrel{\‾}{x}+\left[\begin{array}{c}0\\ 1\end{array}\right](b+\mathrm{\Δb})u+(a_2+\mathrm{\Δ}{a}_2)\left|{\stackrel{\‾}{x}}_2\right|{\stackrel{\‾}{x}}_2+(a_4+\mathrm{\Δ}{a}_4){\stackrel{\‾}{x}}_1^3-p^{-1}\left(\mathrm{\θ}\right)C^T(\stackrel{\‾}{y}-y)\\ \stackrel{\‾}{y}=\left[\begin{array}{c}1\\ 0\end{array}\right]\stackrel{\‾}{x}\end{array}\right.---\left(6\right)$

wherein p is^-1(θ)C₀ ^T＝[C_n ¹θ,C_n ²θ²,…,C_n ⁿθⁿ]^T。

Since in this text p^-1(θ)C₀ ^T＝[2θ,θ²]^TThe formula (6) is represented as:

$\left\{\begin{array}{c}{\stackrel{\stackrel{\·}{\‾}}{x}}_1={\stackrel{\‾}{x}}_2-2\mathrm{\θ\ϵ}\\ {\stackrel{\stackrel{\·}{\‾}}{x}}_2=(a_1+\mathrm{\Δ}{a}_1){\stackrel{\‾}{x}}_2+(a_2+\mathrm{\Δ}{a}_2)\left|{\stackrel{\‾}{x}}_2\right|{\stackrel{\‾}{x}}_2+(a_3+\mathrm{\Δ}{a}_3){\stackrel{\‾}{x}}_1(a_4+\mathrm{\Δ}{a}_4){\stackrel{\‾}{x}}_1^3+(b+\mathrm{\Δb})u_r\left(\stackrel{\‾}{x}\right)-{\mathrm{\θ}}^2\mathrm{\ϵ}\\ \mathrm{\ϵ}=\stackrel{\‾}{y}-y\end{array}\right.---\left(7\right)$

the roll angular velocity can be obtained from equation (7)SignalThe information of (1).

(4) Setting an initial value k of a controller parameter₁，k₂To do so byAndand (3) for an input signal, applying a control algorithm idea of analytical model predictive control, and calculating the control output u of the rudder stabilizing loop under the condition that an uncertain item in the model is not considered:

$u=-\frac{1}{b}(k_0{\stackrel{\‾}{x}}_1+k_1{\stackrel{\‾}{x}}_2+a_1{\stackrel{\‾}{x}}_2+a_2|{\stackrel{\‾}{x}}_2|{\stackrel{\‾}{x}}_2+a_3{\stackrel{\‾}{x}}_1+a_4{\stackrel{\‾}{x}}_1^3)---\left(8\right)$

because of the presence and the bounds of the uncertainty, processing the uncertainty using analytical model predictive control algorithms further developed hereinAfter term setting, a control output u is obtained₁：

$u_1\left(\stackrel{\‾}{x}\right)=-\frac{1}{b\±b_1}(k_0{\stackrel{\‾}{x}}_1+k_1{\stackrel{\‾}{x}}_2+(a_1\±m_1){\stackrel{\‾}{x}}_2+(a_2\±m_2)|{\stackrel{\‾}{x}}_2|{\stackrel{\‾}{x}}_2+(a_3\±m_3){\stackrel{\‾}{x}}_1+(a_4\±m_4){\stackrel{\‾}{x}}_1^3)---\left(9\right)$

The ship rudder rolling reduction control system based on analytical model predictive control design can be obtained by the following equations (3), (7) and (9):

$\left\{\begin{array}{c}{\stackrel{\stackrel{\·}{\‾}}{x}}_1={\stackrel{\‾}{x}}_2-2\mathrm{\θ\ϵ}\\ {\stackrel{\stackrel{\·}{\‾}}{x}}_2=(a_1+\mathrm{\Δ}{a}_1){\stackrel{\‾}{x}}_2+(a_2+\mathrm{\Δ}{a}_2)\left|{\stackrel{\‾}{x}}_2\right|{\stackrel{\‾}{x}}_2+(a_3+\mathrm{\Δ}{a}_3){\stackrel{\‾}{x}}_1+(a_4+\mathrm{\Δ}{a}_4){\stackrel{\‾}{x}}_1^3+(b+\mathrm{\Δb})u_r\left(\stackrel{\‾}{x}\right)+w-{\mathrm{\θ}}^2\mathrm{\ϵ}\\ u_1\left(\stackrel{\‾}{x}\right)=-\frac{1}{b\±b_1}(k_0{\stackrel{\‾}{x}}_1+k_1{\stackrel{\‾}{x}}_2+(a_1\±m_1){\stackrel{\‾}{x}}_2+(a_2\±m_2)|{\stackrel{\‾}{x}}_2|{\stackrel{\‾}{x}}_2+(a_3\±m_3){\stackrel{\‾}{x}}_1+(a_4\±m_4){\stackrel{\‾}{x}}_1^3)\\ \mathrm{\ϵ}=\stackrel{\‾}{y}-y\end{array}\right.---\left(10\right)$

wherein, the plus or minus sign is selected to be consistent with the corresponding previous sign.

The design of the nonlinear analytical model predictive control controller realizes that uncertainty items in the controller are eliminated through theoretical derivation, and the design of the nonlinear analytical model predictive controller for improving the ship rudder rolling reduction is completed, specifically

a) And (3) predicting a control rule by using an analytic model containing an uncertain item.

Converting equation (2) with the uncertainty term to:

$\left\{\begin{array}{c}\stackrel{\·}{x}=f^{*}\left(x\right)+g^{*}\left(x\right)u\\ y=h\left(x\right)\end{array}\right.---\left(11\right)$

wherein f is^*(x)＝f(x)+Δf(x)，g^*(x) (x) Δ g (x). Without loss of generality, assume the equilibrium point x of the system (11)^oHas f^*(x^o)＝0，g^*(x^o)≠0，h(x^o) When the value is 0, the nominal model in redefinition is referred to as equation (11).

The performance function of the rolling horizon of the system (11) is:

$J=\frac{1}{2}{\∫}_0^{T_1}{(\hat{y}(t+\mathrm{\τ})-{\hat{y}}_d(t+\mathrm{\τ}))}^T(\hat{y}(t+\mathrm{\τ})-{\hat{y}}_d(t+\mathrm{\τ}))\mathrm{d\τ}---\left(12\right)$

wherein,andat [ T, T + T ] for the output and reference signals, respectively₁]Predicted value of τ ∈ [0, T₁]，T₁Is a prediction period.

The predictive control problem of the system (11) at time t is described as:

$\left\{\begin{array}{c}\stackrel{\·}{x}=f^{*}\left(\hat{x}(t+\mathrm{\τ})\right)+g^{*}\left(\hat{x}(t+\mathrm{\τ})\right)\hat{u}(t+\mathrm{\τ})\\ y=h\left(\hat{x}(t+\mathrm{\τ})\right)\end{array}\right.---\left(13\right)$

variable of stateThe initial values of (a) are given as:

$\hat{x}\left(t\right)=x\left(t\right)---\left(14\right)$

law of actual controlThe initial values of (a) are:

$u\left(t\right)=\hat{u}(t+\mathrm{\τ}),\mathrm{\τ}=0---\left(15\right)$

the method is obtained based on analytical model prediction control idea:

${L_{f^{*}}}^{\mathrm{\ρ}}h\left(x\right)+L_{g^{*}}{L_{f^{*}}}^{\mathrm{\ρ}-1}h\left(x\right)\hat{u}{\left(t\right)}^{*}-{y_d}^{\left[\mathrm{\ρ}\right]}+K{M}_{\mathrm{\ρ}}=0---\left(16\right)$

wherein K ═ K₀,k₁,…,k_ρ-1]Representative matrix_ll ^-1 _ρl ^TOf the first row of elements of (2), from the prediction period T₁Control order l and correlation rho,is the optimal control law.

${\mathrm{\Γ}}_{(i,j)}=\frac{T_1^{i+j-1}}{(i-1)!(j-1)!(i+j-1)},i,j=1,...,\mathrm{\ρ}+l+1---\left(19\right)$

From equation (16), the optimal control law for analytical model predictive control with uncertainty can be obtained:

$u\left(t\right)={\hat{u}}^{*}\left(t\right)=-{\left(L_{g^{*}}{L}_{f^{*}}^{\mathrm{\ρ}-1}h\left(x\right)\right)}^{-1}\·\{\underset{i=0}{\overset{\mathrm{\ρ}-1}{\mathrm{\Σ}}}{k}_i(L_{f^{*}}^{i}h\left(x\right)-y_d^{\left[i\right]}\left(t\right))+L_{f^{*}}^{\mathrm{\ρ}}h\left(x\right)-y_d^{\left[\mathrm{\ρ}\right]}\left(t\right)\}---\left(20\right)$

b) and eliminating the uncertain item in the control law.

From the formula (11):

$\begin{array}{c}\left|\right|f^{*}\left(x\right)\left|\right|=\left|\right|f\left(x\right)+\mathrm{\Δf}\left(x\right)\\ \≤\left|\right|f\left(x\right)\left|\right|+\underset{i=0}{\overset{p}{\mathrm{\Σ}}}{m}_i{\left|\right|x\left|\right|}^i\end{array}---\left(21\right)$

then f is known^*(x) Has a boundary and records

$\begin{array}{c}\left|\right|g^{*}\left(x\right)\left|\right|=\left|\right|g\left(x\right)+\mathrm{\Δg}\left(x\right)\left|\right|\\ \≤\left|\right|g\left(x\right)\left|\right|+b_1\end{array}---\left(22\right)$

Then g is known^*(x) Is bounded. The following steps are provided:

$\begin{array}{c}\left|\right|L_{f^{*}}^{k}h\left(x\right)\left|\right|=\left|\right|L_f^{k}h\left(x\right)+L_{\mathrm{\Δf}}^{k}h\left(x\right)\\ \≤\left|\right|L_f^{k}h\left(x\right)\left|\right|+\left|\right|L_{f_d}^{k}h\left(x\right)\left|\right|\end{array}---\left(23\right)$

$\begin{array}{c}\left|\right|L_{g^{*}}{L}_{f^{*}}^{k}h\left(x\right)\left|\right|=\left|\right|L_g{L}_f^{k}h\left(x\right)+L_{\mathrm{\Δg}}{L}_f^{k}h\left(x\right)+L_g{L}_{\mathrm{\Δf}}^{k}h\left(x\right)+L_{\mathrm{\Δg}}{L}_{\mathrm{\Δf}}^{k}h\left(x\right)\left|\right|\\ \≤\left|\right|L_g{L}_f^{k}h\left(x\right)\left|\right|+b_1\left|\right|\frac{\∂\left(L_f^{k}h\left(x\right)\right)}{\∂x}\left|\right|+\left|\right|L_g{L}_{f_d}^{k}h\left(x\right)\left|\right|+b_1\left|\right|\frac{\∂\left(L_{f_d}^{k}h\left(x\right)\right)}{\∂x}\left|\right|\end{array}---\left(24\right)$

from the formulas (23) and (24)Andthe two-dimensional display screen is provided with a bounded area,boundary value:

$L_{f^{*}}^{k}h\left(x\right)=L_f^{k}h\left(x\right)+L_{\±f_d}^{k}h\left(x\right)---\left(25\right)$

wherein, the + -symbol of the formula (25) is selected to be the same as the corresponding term of f (x).

$L_{g^{*}}{L}_{f^{*}}^{k}h\left(x\right)$ Boundary value:

$L_{g^{*}}{L}_{f^{*}}^{k}h\left(x\right)=L_g{L}_f^{k}h\left(x\right)\±b_1\frac{\∂\left(L_f^{k}h\left(x\right)\right)}{\∂x}+L_g{L}_{\±f_d}^{k}h\left(x\right)\±b_1\frac{\∂\left(L_{\±f_d}^{k}h\left(x\right)\right)}{\∂x}---\left(26\right)$

wherein, formula (26) b₁Symbols before the item are the same as g (x), containing f_dThe notation before the term is the same as the f (x) corresponding term.

In summary, it can be seen that u (t) is bounded, and u is taken₁For its boundary values, an improved rudder roll reduction analytic model predictive controller is obtained:

$u_1=-\frac{1}{b\±b_1}(k_0{x}_1+k_1{x}_2+(a_1\±m_1)x_2+(a_2\±m_2)|x_2|x_2+(a_3\±m_3)x_1+(a_4\±m_4)x_1^3)---\left(27\right)$

wherein, the plus or minus sign is selected to be consistent with the corresponding previous sign.

By combining a high-gain observer and an improved analytical model prediction control law, the ship rudder rolling reduction control system comprises:

$\left\{\begin{array}{c}{\stackrel{\stackrel{\·}{\‾}}{x}}_1={\stackrel{\‾}{x}}_2-2\mathrm{\θ\ϵ}\\ {\stackrel{\stackrel{\·}{\‾}}{x}}_2=(a_1+\mathrm{\Δ}{a}_1){\stackrel{\‾}{x}}_2+(a_2+\mathrm{\Δ}{a}_2)\left|{\stackrel{\‾}{x}}_2\right|{\stackrel{\‾}{x}}_2+(a_3+\mathrm{\Δ}{a}_3){\stackrel{\‾}{x}}_1+(a_4+\mathrm{\Δ}{a}_4){\stackrel{\‾}{x}}_1^3+(b+\mathrm{\Δb})u_r\left(\stackrel{\‾}{x}\right)+w-{\mathrm{\θ}}^2\mathrm{\ϵ}\\ u_1\left(\stackrel{\‾}{x}\right)=-\frac{1}{b\±b_1}(k_0{\stackrel{\‾}{x}}_1+k_1{\stackrel{\‾}{x}}_2+(a_1\±m_1){\stackrel{\‾}{x}}_2+(a_2\±m_2)|{\stackrel{\‾}{x}}_2|{\stackrel{\‾}{x}}_2+(a_3\±m_3){\stackrel{\‾}{x}}_1+(a_4\±m_4){\stackrel{\‾}{x}}_1^3)\\ \mathrm{\ϵ}=\stackrel{\‾}{y}-y\end{array}\right.---\left(28\right)$

wherein，u_rAnd outputting the actual rudder angle passing through the actuator of the steering engine for the controller.

The controller can effectively restrain the parameter perturbation of the model. The ship rudder stabilization method based on the analytic model predictive control design provided by the invention is clear in thought, complete in steps and easy to realize in engineering. Meanwhile, according to the Laos-Hall wiz criterion theory, the k which is determined by the Laos-Hall wiz criterion and accords with the value range is known_iThe stability of the closed loop system can be ensured.

With reference to fig. 1 to 3, the present invention comprises the following steps:

the method comprises the following steps: and calculating each coefficient of the control object. Comprises 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)，b₁. The control objects are as follows:

$\left\{\begin{array}{c}{\stackrel{\·}{x}}_1=x_2\\ {\stackrel{\·}{x}}_2=(a_1+\mathrm{\Δ}{a}_1)x_2+(a_2+\mathrm{\Δ}{a}_2)\left|x_2\right|x_2+(a_3+\mathrm{\Δ}{a}_3)x_1+(a_4+\mathrm{\Δ}{a}_4)x_1^3+(b+\mathrm{\Δb})u+w\end{array}\right.$

step two: collecting roll angle signal by compass

Step three: obtaining roll angular velocity signals by a high gain observer

Step four: the method comprises the steps of taking a desired roll angle, a roll angle and a roll angle speed as input signals, inputting the input signals into a controller designed by an analytic model predictive control algorithm to obtain a control rudder angle, inputting the rudder angle into a steering engine actuator, outputting an actual rudder angle, inputting the actual rudder angle into a control object to obtain roll angle information, judging whether the control effect requirement is met or not according to the obtained roll angle information value, and adjusting a controller parameter k₁，k₂Until the control effect of the output is satisfactory.

The invention is that the analytic 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, outputs the instruction rudder angle through calculation, the steering engine actuator outputs the actual rudder angle to the ship body according to the rudder angle instruction, the ship restrains the generated roll angle, the output roll angle is measured by the compass and input to the high gain observer and the controller, thus forming a closed loop control system, as shown in figure 2.

Accordingly, taking a ship as an example, the prediction period T is selected₁60s, control order l 4, observer parameter θ 3, and controller parameter k₀＝0.05，k₁0.375, 3m at the sense wave height, 90 ° at the encounter angle, and 8s at the wave periodFig. 3 shows a simulation diagram of the output of the pitch motion and rudder angle before and after the predictive control by the analytical model.

Claims (2)

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:

$\left\{\begin{array}{c}{\stackrel{\·}{x}}_1=x_2\\ {\stackrel{\·}{x}}_2=(a_1+{\mathrm{\Δa}}_1)x_2+(a_2+{\mathrm{\Δa}}_2)\left|x_2\right|x_2+(a_3+{\mathrm{\Δa}}_3)x_1+(a_4+{\mathrm{\Δa}}_4)x_1^3+(b+\mathrm{\Δ}b)u+w\end{array}\right.,$

$a_1=-\frac{K_p}{I_{xx}+J_{xx}},a_2=-\frac{K_{pp}}{I_{xx}+J_{xx}},a_3=-\frac{W\·GM}{I_{xx}+J_{xx}},a_4=\frac{W\·GM}{I_{xx}+J_{xx}},$

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:

$\left\{\begin{array}{c}\stackrel{\·}{\stackrel{\‾}{x}}=\left[\begin{array}{cc}0& 1\\ a_3+{\mathrm{\Δa}}_3& a_1+{\mathrm{\Δa}}_1\end{array}\right]\stackrel{\‾}{x}+\left[\begin{array}{c}0\\ 1\end{array}\right](b+\mathrm{\Δ}b)u+(a_2+\mathrm{\Δ}{a}_2)\left|{\stackrel{\‾}{x}}_2\right|{\stackrel{\‾}{x}}_2+(a_4+\mathrm{\Δ}{a}_4){\stackrel{\‾}{x}}_1^3-p^{-1}\left(\mathrm{\θ}\right)C^T(\stackrel{\‾}{y}-y)\\ \stackrel{\‾}{y}=\left[\begin{array}{c}1\\ 0\end{array}\right]\stackrel{\‾}{x}\end{array}\right.$

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

$\left\{\begin{array}{c}{\stackrel{\·}{\stackrel{\‾}{x}}}_1={\stackrel{\‾}{x}}_2-2\mathrm{\θ}\mathrm{\ϵ}\\ {\stackrel{\·}{\stackrel{\‾}{x}}}_2=(a_1+{\mathrm{\Δa}}_1){\stackrel{\‾}{x}}_2+(a_2+{\mathrm{\Δa}}_2)\left|{\stackrel{\‾}{x}}_2\right|{\stackrel{\‾}{x}}_2+(a_3+{\mathrm{\Δa}}_3){\stackrel{\‾}{x}}_1+(a_4+{\mathrm{\Δa}}_4){\stackrel{\‾}{x}}_1^3+(b+\mathrm{\Δ}b)u_r\left(\stackrel{\‾}{x}\right)-{\mathrm{\θ}}^2\mathrm{\ϵ}\\ \mathrm{\ϵ}=\stackrel{\‾}{y}-y\end{array}\right.;$

(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:

$\left\{\begin{array}{c}{\stackrel{\·}{\stackrel{\‾}{x}}}_1={\stackrel{\‾}{x}}_2-2\mathrm{\θ}\mathrm{\ϵ}\\ {\stackrel{\·}{\stackrel{\‾}{x}}}_2=(a_1+{\mathrm{\Δa}}_1){\stackrel{\‾}{x}}_2+(a_2+{\mathrm{\Δa}}_2)\left|{\stackrel{\‾}{x}}_2\right|{\stackrel{\‾}{x}}_2+(a_3+{\mathrm{\Δa}}_3){\stackrel{\‾}{x}}_1+(a_4+{\mathrm{\Δa}}_4){\stackrel{\‾}{x}}_1^3+(b+\mathrm{\Δ}b)u_r\left(\stackrel{\‾}{x}\right)+w-{\mathrm{\θ}}^2\mathrm{\ϵ}\\ u_1\left(\stackrel{\‾}{x}\right)=-\frac{1}{b\±b_1}(k_0{\stackrel{\‾}{x}}_1+k_1{\stackrel{\‾}{x}}_2+(a_1\±m_1){\stackrel{\‾}{x}}_2+(a_2\±m_2\left)\right|{\stackrel{\‾}{x}}_2|{\stackrel{\‾}{x}}_2+(a_3\±m_3){\stackrel{\‾}{x}}_1+(a_4\±m_4\left){\stackrel{\‾}{x}}_1^3\right)\\ \mathrm{\ϵ}=\stackrel{\‾}{y}-y\end{array}\right.,$

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。