CN103935480A - Rudder stabilizing method based on analytic model prediction control design - Google Patents

Abstract

The invention belongs to the field of ship engineering, control science and control engineering, and relates to a rudder stabilizing method based on the analytic model prediction control design. The rudder stabilizing method comprises the step of acquiring various coefficients of a control target, the step of collecting rolling angle signals, the step of acquiring rolling angular speed signals, and the step of setting parameter initial values of a controller. According to the rudder stabilizing method, the boundness of uncertain items is fully utilized, analytic model prediction control is used, and the problem that the parameters of a rudder stabilizing control system are uncertain is effectively solved. Compared with other methods, the rudder stabilizing method has the advantages that stabilizing efficiency is improved. According to the rudder stabilizing method, the structure is simple. The rudder stabilizing method is easy to achieve and can meet the requirements of the actual engineering application very well.

Description

A kind of boats and ships rudder stabilization method based on analytic model Predictive control design

Technical field

The invention belongs to Marine engineering, control science and dominant project field, relate to a kind of boats and ships rudder stabilization method based on analytic model Predictive control design.

Background technology

Because ship rolling motion damping is very little, make boats and ships can produce violent rolling in stormy waves, excessive rolling meeting brings very large impact to ship's navigation performance and safety, for ensureing that boats and ships are at complicated sea situation safe navigation, rudder roll damping technology is as the novel control thought of one, at home and abroad have been a great concern in recent years, and carry out the design of antirolling apparatus with rudder, simple and convenient, cheap, therefore, rudder stabilization technology is an important subject in motion of ship control field.

At present, for the control of rudder roll damping system, there is certain research both at home and abroad, but not yet find that analytic application Model Predictive Control carries out document and the patent achievement of rudder stabilization design.

Because motion of ship is nonlinear in essence, from motion of ship environment and displacement feature, its accurate math modeling is difficult to obtain, cause model parameter to there is uncertainty, thereby, rudder rollstabilization controller should design based on nonlinear control theory, and must have the robustness to model parameter perturbation.The control of rudder rollstabilization requires to have at nonlinear model the rolling motion under the uncertain prerequisite of parameter, wave being caused and suppresses, and rudder roll damping shakes controller design must meet the robustness to model parameter perturbation.But traditional PI D, LQG control algorithm, the effectively non-linear and parameter uncertainty problem of transaction module.Analytic model predictive control proposes for nonlinear model, have the modeling of being easy to, response rapidly, controller performance better, strong robustness and the feature such as logical organization is simple, but analytic model predictive control is in most situation at present, only solve the control problem of certainty nominal system, for nonlinear system model parameter uncertainty problem, a kind of analytic model forecast Control Algorithm of improving is proposed for this reason.First, redefined into indeterminate nominal system model, carried out theory derivation based on analytic model predictive control theory to redefining nominal model, obtained the control law that contains indeterminate.Due to control law bounded, get its boundary value, to derive and eliminated the indeterminate in controller from theory, actv. has solved the parameter uncertainty problem in model.

Rudder rollstabilization control system only can be measured boats and ships roll angle, when analytic application model predictive control method, must introduce the rollrate information that state observer needs to obtain controller.Compare other observers, the stability of High-gain observer and precision have theoretical guarantee, calculate easy parameter few, once parameter is selected, without adjustment.Because partial function in High-gain observer design process need meet Local Lipschitz condition, rudder roll damping control finally need provide the required condition of its satisfied design High-gain observer like this, prove herein to provide the detailed proof that it need to satisfy condition, for its application provides foundation.This scheme can be surveyed without state, directly obtains observed value by state observer, has solved angular velocity in roll and can not survey problem.

Summary of the invention

The object of the present invention is to provide the effectively boats and ships rudder stabilization method based on analytic model Predictive control design of the Parameter Perturbation of inhibition.

The object of the present invention is achieved like this:

Boats and ships rudder stabilization method based on analytic model Predictive control design comprises:

(1) obtain every coefficient of control object, comprise a ₁, a ₂, a ₃, a ₄, b and the corresponding boundary value m that is changed the indeterminate causing by the speed of a ship or plane and transverse metacentric height _i(i=1,2,3,4), b ₁, control object is:

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

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

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

k _pp=3k _b(I _xx+ J _xx)/4, k _a, k _bfor extinguishing coefficient is with ship type, and displacement is relevant, and W is displacement, and GM is that transverse stability is high, and B is the beam, and d is draft, and L is captain, and g is acceleration due to gravity;

(2) gather roll angle signal

(3) obtain angular velocity in roll signal get High-gain observer gain p ^-1(θ) C ₀ ^t=[2 θ, θ ²] ^t, setup parameter θ value; High-gain observer state is:

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

Wherein, "-" representative observation, for nonlinear terms, p ^-1(θ) C ₀ ^t=[2 θ, θ ²] ^t, there is observer system to be:

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

(4) setting controller parameter initial value k ₁, k ₂, to expect roll angle roll angle and angular velocity in roll for incoming signal, calculate the control output u in rudder stabilization loop ₁,

$u_1\left(\stackrel{\‾}{x}\right)=-\frac{1}{b\±b_1}(k_0{\stackrel{\‾}{x}}_1+k_1{\stackrel{\‾}{x}}_2+(a_1\±m_1){\stackrel{\‾}{x}}_2+(a_2\±m_2)|{\stackrel{\‾}{x}}_2|{\stackrel{\‾}{x}}_2+(a_3\±m_3){\stackrel{\‾}{x}}_1+(a_4\±m_4){\stackrel{\‾}{x}}_1^3),$ The rudder rollstabilization control system obtaining based on analytic model Predictive control design is:

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

Wherein: for the observed value of x,

Predictive controller initial value design, predetermined period T ₁=60s, controls order l=4, observer parameter θ=3, controller parameter k ₀=0.05, k ₁=0.375.

Beneficial effect of the present invention is:

(1) take full advantage of the boundedness of indeterminate, in conjunction with analytic model predictive control, efficiently solve the parameter uncertainty problem of rudder stabilization control system.

(2) method provided by the present invention is compared additive method, on anti-rolling efficiency, is improved.

(3) patent structure of the present invention is simple, is easy to realize, can the fine needs that meet practical engineering application.

Brief description of the drawings

The workflow diagram of a kind of boats and ships rudder stabilization method based on analytic model Predictive control design of Fig. 1,

The composition frame chart of the actual boats and ships rudder roll damping control system of Fig. 2 based on analytic model Predictive control design,

Rolling motion and rudder angle Output simulation figure under the analytic model predictive control that Fig. 3 proposes herein.

Detailed description of the invention

Below in conjunction with accompanying drawing, the present invention is described in more detail:

A kind of boats and ships rudder stabilization method based on analytic model Predictive control design of the present invention, comprising:

(1) obtain every coefficient of control object, comprise a ₁, a ₂, a ₃, a ₄, the boundary value m of b and corresponding indeterminate _i(i=1,2,3,4), b ₁.

The mathematic(al) representation of control object:

Wherein for the moment of inertia of rolling, for roll-damping moment, for rolling countermoment, for roll angle, for angular velocity in roll, for roll angle acceleration/accel, K _r=-I _{δ z}u ²y _{δ yy}δ, K _dfor wave disturbance moment. $I_{\mathrm{xx}}+J_{\mathrm{xx}}=\frac{W}{g}{\left[(0.3085+0.0227B/d-0.0043L/100)B\right]}^2,K_p=2k_a\sqrt{W\·\mathrm{GM}\·(I_{\mathrm{xx}}+J_{\mathrm{xx}})}/\mathrm{\π},$ K _pp=3k _b(I _xx+ J _xx)/4, k _a, k _bfor extinguishing coefficient is with ship type, and displacement is relevant.W is displacement, and GM is that transverse stability is high, and B is the beam, and d is draft, and L is captain, and g is acceleration due to gravity, for the angle of attack of boats and ships, U is the speed of a ship or plane, δ rudder angle.

Formula (1) is expressed as to 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)=x ₁, w is wave disturbance, a ₁, a ₂, a ₃, a ₄for the known coefficient calculating.

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 _δzU ²Y _δyy/(I _xx+J _xx)。

And cause that because speed and initial metacentric height change the variable quantity of coefficient is: Δ a ₁, Δ a ₂, Δ a ₃, Δ a ₄, Δ b, the variation of this tittle generally can not accurately be obtained, but can its maximum value of pre-estimation, is designated as: m _i(i=1,2,3,4) and b ₁.

Contrast formula (2), show that the uncertain nonlinear system of rudder rollstabilization is:

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

Be certain carrier herein, the parameters of this ship is provided by table 1.

The each parameter of table 1 dummy ship

The initial speed of a ship or plane is 10m/s, initial horizontal stability height is 0.776m, can be calculated every coefficient of model by table 1, suppose that ship's speed is at (0.5U, 1.5U), change, and when initial metacentric height also changes between (0.5GM, 1.5GM), the maximum value that we can pre-estimation model coefficient variable quantity is in table 2.

Table 2 rudder stabilization equation of motion coefficient and index variation amount

(2) gather roll angle signal gather roll angle signal by compass.

(3) obtain angular velocity in roll signal get High-gain observer gain p ^-1(θ) C ₀ ^t=[2 θ, θ ²] ^t, setup parameter θ value.Get High-gain observer equation of state as follows:

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

Wherein, "-" representative observation, for nonlinear terms, p (θ) is defined as the solution of following equation:

0＝-θp(θ)-(A ^Tp(θ)+p(θ)A)+C ^TC (5)

Formula (3) is write to be shaped and suc as formula the equation of state of (4) is:

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

Wherein, p ^-1(θ) C ₀ ^t=[C _n ¹θ, C _n ²θ ²..., C _n ⁿθ ⁿ] ^t.

Due to p herein ^-1(θ) C ₀ ^t=[2 θ, θ ²] ^t, formula (6) is expressed as:

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

Can obtain angular velocity in roll signal by formula (7) information.

(4) setting controller parameter initial value k ₁, k ₂, with and for incoming signal, the control algorithm thought of analytic application Model Predictive Control, does not consider in the situation of indeterminate existence in model, calculates the control output u in rudder stabilization loop:

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

Due to existence and the bounded of indeterminate, application is further improved analytic model predictive control algorithm herein, controlled output u after processing indeterminate ₁:

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

The rudder rollstabilization control system that can be obtained based on analytic model Predictive control design by formula (3), (7), (9) is:

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

Wherein, ± symbol is chosen consistent with its corresponding last symbol.

The design of Nonlinear Analytical Model Predictive Control controller realizes from theory derives and has eliminated the indeterminate in controller, completes rudder rollstabilization and improves the design of Nonlinear Analytical model predictive controller, is specially

A) the analytic model predictive control rule that contains indeterminate.

Formula with indeterminate (2) is converted 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 ^*(x)=f (x)+Δ f (x), g ^*(x)=g (x)+Δ g (x).Without loss of generality, the equilibrium point x of supposing the system (11) ^o, have f ^*(x ^o)=0, g ^*(x ^o) ≠ 0, h (x ^o)=0, title formula (11) is the nominal model under redefining.

The performance function of the rolling time domain of 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, with be respectively output and reference signal at [t, t+T ₁] predictor, τ ∈ [0, T ₁], T ₁for predetermined period.

System (11) is described as in the predictive control problem in t moment:

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

State variable initial value give be:

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

Working control rule initial value, that is:

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

Obtain based on analytic model predictive control thought:

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

Wherein, K=[k ₀, k ₁..., k _ρ-1] represent matrix Γ _ll ^-1Γ _{ρ l} ^tthe first row element, by predetermined period T ₁, control order l and degree of correlation ρ and determine, for optimal control law.

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

By formula (16), can obtain the optimal control law with the analytic model predictive control of indeterminate:

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

B) eliminate the indeterminate in control law.

From formula (11):

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

Know f ^*(x) bounded, and note

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

Know g ^*(x) bounded.Have again:

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

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

From formula (23), (24) with all bounded, 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, locate ± symbol of formula (25) is chosen identical with f (x) respective items.

$L_{g^{*}}{L}_{f^{*}}^{k}h\left(x\right)$ 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) b ₁before ± symbol is identical with g (x), containing f _dsymbol before is identical with f (x) respective items.

To sum up, known u (t) bounded, gets u ₁for its boundary value, the rudder roll damping analytic model predictive controller being improved:

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

Wherein, ± symbol is chosen consistent with its corresponding last symbol.

In conjunction with High-gain observer and improved analytic model Predictive control law, rudder rollstabilization control system is:

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

Wherein, u _rfor controller is exported the actual rudder angle after steering wheel actr.

This controller is the Parameter Perturbation of inhibition effectively.The present invention propose a kind of boats and ships rudder stabilization method based on analytic model Predictive control design clear, step is complete, be easy to Project Realization.Meanwhile, according to known according to Louth-Hall dimension thatch criterion theory, the k that meets span being determined by Louth-Hall dimension thatch criterion _ican ensure that closed loop system is stable.

In conjunction with Fig. 1～3, the present invention includes following step:

Step 1: the every coefficient that calculates control object.Comprise a ₁, a ₂, a ₃, a ₄, b and the corresponding boundary value m that is changed the indeterminate causing by the speed of a ship or plane and transverse metacentric height _i(i=1,2,3,4), b ₁.Control object is:

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

Step 2: gather roll angle signal by compass

Step 3: obtain angular velocity in roll signal by High-gain observer

Step 4: to expect that roll angle, roll angle and angular velocity in roll are as incoming signal, be input to the controller designed by analytic model predictive control algorithm, obtain and control rudder angle, again this rudder angle is input in steering wheel actr, output actual rudder angle, actual rudder angle is input in control object again and obtains roll angle information, judge whether to meet control effect requirements, adjustment control parameter k by the gained roll angle value of information ₁, k ₂, until the control good results of output.

The present invention accepts by the analytic model predictive controller of rudder rollstabilization system the system output roll angle information of expecting that roll angle information, High-gain observer information and compass obtain, through computing output command rudder angle, steering wheel actr is exported actual rudder angle to hull according to rudder angle instruction, boats and ships suppress the roll angle producing, the roll angle of output records and inputs to High-gain observer and controller via compass, so form closed loop control system, as shown in Figure 2.

Accordingly, herein taking certain ship as example, choose predetermined period T ₁=60s, controls order l=4, observer parameter θ=3, controller parameter k ₀=0.05, k ₁=0.375, be 3m there being adopted wave height, wave encounter angle is 90 °, under the sea situation that period of a wave is 8s, provides analytic model predictive control front and back rolling motion and rudder angle Output simulation figure as shown in Figure 3.

Claims (2)

1. the boats and ships rudder stabilization method based on analytic model Predictive control design, is characterized in that:

(1) obtain every coefficient of control object, comprise a ₁, a ₂, a ₃, a ₄, b and the corresponding boundary value m that is changed the indeterminate causing by the speed of a ship or plane and transverse metacentric height _i(i=1,2,3,4), b ₁, control object is:

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

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

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

k _pp=3k _b(I _xx+ J _xx)/4, k _a, k _bfor extinguishing coefficient is with ship type, and displacement is relevant, and W is displacement, and GM is that transverse stability is high, and B is the beam, and d is draft, and L is captain, and g is acceleration due to gravity;

(2) gather roll angle signal

(3) obtain angular velocity in roll signal get High-gain observer gain p ^-1(θ) C ₀ ^t=[2 θ, θ ²] ^t, setup parameter θ value; High-gain observer state is:

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

Wherein, "-" representative observation, for nonlinear terms, p ^-1(θ) C ₀ ^t=[2 θ, θ ²] ^t, there is observer system to be:

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

(4) setting controller parameter initial value k ₁, k ₂, to expect roll angle roll angle and angular velocity in roll for incoming signal, calculate the control output u in rudder stabilization loop ₁,

$u_1\left(\stackrel{\‾}{x}\right)=-\frac{1}{b\±b_1}(k_0{\stackrel{\‾}{x}}_1+k_1{\stackrel{\‾}{x}}_2+(a_1\±m_1){\stackrel{\‾}{x}}_2+(a_2\±m_2)|{\stackrel{\‾}{x}}_2|{\stackrel{\‾}{x}}_2+(a_3\±m_3){\stackrel{\‾}{x}}_1+(a_4\±m_4){\stackrel{\‾}{x}}_1^3),$ The rudder rollstabilization control system obtaining based on analytic model Predictive control design is:

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

Wherein: for the observed value of x,

2. a kind of boats and ships rudder stabilization method based on analytic model Predictive control design according to claim 1, is characterized in that: described predictive controller initial value design, predetermined period T ₁=60s, controls order l=4, observer parameter θ=3, controller parameter k ₀=0.05, k ₁=0.375.