CN103336427A - Method of dynamic modeling and controller design for three-dimensional output probability density function - Google Patents

Abstract

The invention discloses a method of dynamic modeling and controller design for a three-dimensional output probability density function, which belongs to the field of random distribution control theory. The method comprises the following steps: step 1: building a three-dimensional output PDF dynamic model based on a square root B-spline model; step 2, building an input-output model of the three-dimensional output PDF by utilizing input-output data collected in a real system and recursive least squares algorithm; step 3: selecting and using instant square root performance figures to design the controller, and designing controlling quantity of the controller by optimizing the instant square root performance figures, so as to realize a situation that a system output PDF distribution shape tracks a given output PDF distribution shape. The method designs a conventional optimization controller, realizes the tracking of the given output PDF by the output PDF by optimizing the square root performance figures, enriches the three-dimensional output PDF control theory, and provides the new method for industry processes with three-dimensional output distribution characters.

Description

A kind of dynamic modeling of three-dimensional output probability density function and controller design method

Technical field

The invention belongs to stochastic distribution control theory field, relate in particular to a kind of dynamic modeling and controller design method of three-dimensional output probability density function.

Background technology

The stochastic system control theory is one of important branch of control theory and application, mainly is because most industrial processs all are subjected to the interference of random signal, at this practical problems, has formed the stochastic control theory of system.Its early stage achievement in research concentrates on the statistical property of control system variable self, the control target is average and the variance of system, in existent method, stochastic variable Gaussian distributed in most of hypothesis stochastic system, yet the not realistic application of this hypothesis, for example fiber length distribution in the paper-making process, the distribution of the grain in the grain processing and boiler flame temperature distribution etc.The general random system distributes with exporting PDF(probability density function) probability density function represents, when stochastic variable is gaussian variable, average and variance by control system can realize the output PDF of system is controlled, do not satisfy the system of Gaussian distribution for stochastic variable, its average and variance can not comprise the full detail of system, can not realize this system is exported the control of PDF to the control of system's average and variance.At this type systematic, Wang Hong teaches the method that proposed direct control system output PDF shape in 1998, namely exports PDF control.The direct CONTROLLER DESIGN of these class methods is so that the output PDF of system distribution shape tracing preset PDF distribution shape.

Output PDF can approach by the B SPLINE NEURAL NETWORK, realized the decoupling zero of original complicated coupling system so to a certain extent, output PDF control for this class decoupling zero is referred to as SDC(stochastic distribution control) stochastic system distribution control.These class methods have broken through the limitation of STOCHASTIC CONTROL research, traditional STOCHASTIC CONTROL problem are converted into the method for setting up advantages of simple model, the efficient suitable control algolithm of design.Realization is described from the state-space model that the partial differential equation of the implicit descriptive system dynamic perfromance of complexity is converted into decoupling zero, finally reaches to be similar to determine the same dynamic behaviour that can describe the stochastic distribution system with model more accurately of system.SDC compares more with stochastic control theory in the past and tallies with the actual situation, and therefore, this theory is dissolved into new industrial circle, will give its great vitality.

At the stochastic distribution system with two-dimensional characteristics, set up comparatively perfect theoretical system.As aspect the system modelling, linear B batten model, reasonable B batten model, square root B batten model and reasonable square root B batten model have been set up, input and output ARMAX model, neural network PDF model etc.At the controller design aspect, instantaneous optimal tracking control algolithm, optimal tracking control algolithm, model reference adaptive control algolithm, predictive control algorithm, structuring controller algorithm, Iterative Learning Control Algorithm etc. have been realized.In recent years, scholars have done a large amount of work at the aspects such as robust control, minimum entropy control, fault diagnosis and wave filter design of stochastic distribution control.

In sum, obtained very big progress at two dimension output distribution control problem, also there is the three-dimensional output of class distribution problem in the actual industrial process, as the distributed in three dimensions of material concentration in the fire coal circulating fluid bed boiler in three-dimensional temperature field, power station that characterizes boiler flame temperature etc.These distributed in three dimensions and industrial process operation conditions are closely bound up, and this enhances productivity, reduces aspect such as environmental pollution and all has important use and be worth to whole industrial process.Along with the high speed development of Computer Image Processing and sensor technology, the detection of three-dimensional being exported distribution situation has obtained development rapidly.But the method that obtains system's output distribution by the advanced technology on-line measurement implements very complicated, required apparatus expensive.

The three-dimensional output PDF control problem of the present invention's research is an important component part of SDC theory, yet imperfection is gone back in the modeling of three-dimensional PDF and the research of control problem, article " Modeling and control of the flame temperature distribution using probability density function shaping " has carried out static modeling and controller design to three-dimensional output PDF, select two-dimentional B spline base function for use and set up the static model of three-dimensional output PDF by least-squares algorithm, optimize quadratic performance index, method with gradient has obtained system's local optimum control input, and Computer Simulation research has obtained legitimate result.So far, rarely have for the research of the three-dimensional output of dynamic process distribution control aspect and to deliver, but not have directly three-dimensional output to be distributed reports as the research of controlling.

In order further to improve three-dimensional output PDF control theory, make three-dimensional random distribution control problem realize becoming possibility, the present invention has at first set up the instantaneous square root B batten model of three-dimensional output PDF, the dynamic change that adds weights on instantaneous square root B batten model basis has partly constituted based on the three-dimensional of square root B batten model exports the PDF dynamic model, realize the dynamic decoupling between the weights, analyzed the condition that three-dimensional output PDF dynamic model satisfies the nature constraint; Set up three-dimensional output PDF input according to system's inputoutput data by recursive least squares then; Select instantaneous square root performance index at last, designed conventional optimal controller.The present invention has carried out perfect to three-dimensional output PDF control theory, the control of exporting distribution problem for three-dimensional provides new method and thinking.

Summary of the invention

The present invention is directed to three-dimensional and export the needs of being left to be desired of PDF theory and actual industrial process, proposed a kind of dynamic modeling and controller design method of three-dimensional output probability density function.

A kind of dynamic modeling of three-dimensional output probability density function and controller design method, this method may further comprise the steps:

Step 1: make up the three-dimensional output PDF dynamic model based on square root B batten model;

Described structure may further comprise the steps based on the three-dimensional output PDF dynamic model of square root B batten model:

Step S1: the instantaneous square root B batten model that makes up three-dimensional output PDF according to two-dimentional B-spline function;

As follows with two one dimension B-spline function tensor product representation two dimension B-spline functions:

$B_{j,i}(x,r)=B_{j_x{i}_x}\left(x\right)B_{j_r{i}_r}\left(r\right)$

Wherein,

Obtained by following recursion formula:

$B_{1,i_x}\left(x\right)=\left\{\begin{array}{cc}1,& x\∈[x_{i_x},x_{i_x+1})\\ 0,& x\∉[x_{i_x},x_{i_x+1})\end{array}\right.$

$B_{j_x,i_x}\left(x\right)=\frac{x-x_{i_x}}{x_{i_x+j_x-1}-x_{i_x}}{B}_{j_x-1,i_x}\left(x\right)+\frac{x_{i_x+j_x}-x}{x_{i_x+j_x}-x_{i_x+1}}{B}_{j_x-1,i_x+1}\left(x\right)$

Obtained by following recursion formula:

${\mathrm{<figure-callout\; id="1"\; label="B"\; filenames="HDA00003372491300011.png,HDA00003372491300012.png"\; state="\{\{state\}\}">B</figure-callout>}}_{1,i_r}\left(r\right)=\left\{\begin{array}{cc}1,& r\&Element;{[r}_{i_r},r_{i_r+1})\\ 0,& r\&NotElement;[r_{i_r},r_{i_r+1})\end{array}\right.$

$B_{j_r,i_r}\left(r\right)=\frac{r-r_{i_r}}{r_{i_r+j_r-1}-r_{i_r}}{B}_{j_r-1,i_r}\left(r\right)+\frac{r_{i_r+j_r}-r}{r_{i_r+j_r}-r_{i_r+1}}{B}_{j_r-1,i_r+1}\left(r\right)$

Wherein, B _{J, i}(x r) is two-dimentional B spline base function;

Be one dimension B spline base function;

Be one dimension B spline base function; X, r are respectively the variable of definition spatially, x ∈ [a ₁, b ₁], r ∈ [a ₂, b ₂]; a ₁Be the interior lower limit of setting between the setting district; b ₁Be the interior higher limit of setting between the setting district; [a ₁, b ₁] for comprising a ₁And b ₁An interval; a ₂Be the interior lower limit of setting between the setting district; b ₂Be the interior higher limit of setting between the setting district; [a ₂, b ₂] for comprising a ₂And b ₂An interval; J represents the order of two-dimentional B batten; I represents two-dimentional B spline base function number; j _xOrder for the basis function chosen on the X-axis; i _xNumber for the basis function chosen on the X-axis; j _rOrder for the basis function chosen on the R axle; i _rNumber for the basis function chosen on the R axle;

Be 1 rank i _xIndividual B-spline function;

Be j _x-1 rank i _xIndividual B-spline function;

Be j _x-1 rank i _x+ 1 B spline base function;

Be nodal value, and have $x_{j_x+1}<\&CenterDot;\&CenterDot;\&CenterDot;<a_1=x_0<\&CenterDot;\&CenterDot;\&CenterDot;<x_{m_x+1}=b_1<\&CenterDot;\&CenterDot;\&CenterDot;<x_{m_x+j_x};$ m _xBe interval [a ₁, b ₁] interior effective node number, j _x-1 counts for the acromere of the interval left and right sides;

For comprising

With

An interval;

Be 1 rank i _rIndividual B-spline function;

Be j _r-1 rank i _rIndividual B-spline function; Be j _r-1 rank i _r+ 1 B spline base function;

Be nodal value, and have $r_{j_r+1}<\&CenterDot;\&CenterDot;\&CenterDot;<a_2={\mathrm{<figure-callout\; id="0"\; label="r"\; filenames="HDA00003372491300011.png,HDA00003372491300012.png"\; state="\{\{state\}\}">r</figure-callout>}}_0<\&CenterDot;\&CenterDot;\&CenterDot;<r_{m_r+1}={\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HDA00003372491300042.png"\; state="\{\{state\}\}">b</figure-callout>}}_2<\&CenterDot;\&CenterDot;\&CenterDot;<r_{m_r+j_r};$ m _rBe interval [a ₂, b ₂] interior effective node number, j _r-1 counts for the acromere of the interval left and right sides;

For comprising With An interval;

With two-dimentional B-spline function B _{J, i}(x r), omits the order j of B batten, i.e. two-dimentional B-spline function B _{J, i}(x r) is designated as B _i(x, r);

The instantaneous square root B batten model that obtains three-dimensional output PDF based on two-dimentional square root B-spline function is:

$\sqrt{\mathrm{\&gamma;}(x,r,u_k)}=C_0(x,r)V_k+B_n(x,r){\mathrm{\&omega;}}_n\left(V_k\right)$

Wherein,

γ (x, r, u _k) be three-dimensional output probability density function;

The two-dimentional B-spline function number of n for selecting, k is sampling instant;

C ₀(x, r)=[B ₁(x, r), B ₂(x, r) ..., B _N-1(x, r)], wherein, C ₀(x r) is 1 * (n-1) dimension basis function conversion vector;

B _i(x r) is two-dimentional B-spline function;

V _k=[ω ₁(u _k), ω ₂(u _k) ..., ω _N-1(u _k)] ^T, wherein, V _kBe k corresponding (n-1) * 1 right-safeguarding value vector of the moment;

ω _i(u _k) for depending on u _kWeights, u _kBe k corresponding control action of the moment.

B _n(x r) is two-dimentional B-spline function;

ω _n(V _k) be the weights of n basis function correspondence.

Step S2: on the basis of step S1, add the dynamic change part of weights, obtain the dynamic model based on the three-dimensional output of square root B batten model PDF;

The weights dynamic part of supposing adding is:

V _k=AV _k-1+Bu _k-1

Wherein, A is (n-1) * (n-1) dimension parameter matrix of expression system dynamic relationship, and B is (the n-1) * 1 dimension parameter matrix of expression system dynamic relationship; V _K-1Be k-1 corresponding n-1 right-safeguarding value vector of the moment, u _K-1Be k-1 corresponding controlled quentity controlled variable of the moment;

Three-dimensional output PDF dynamic model based on square root B batten model is:

$V_k={\mathrm{AV}}_{k-1}+{\mathrm{Bu}}_{k-1}\sqrt{\mathrm{\&gamma;}(x,r,u_k)}=C_0(x,r)V_k+B_n(x,r){\mathrm{\&omega;}}_n\left(V_k\right)$

Step S3: the weight vector V of system _kThere is nonlinear relationship in the weights corresponding with n basis function, analyze the dynamic decoupling that obtains between the three-dimensional output PDF dynamic model weights;

Decoupling zero formula between the described three-dimensional output PDF dynamic model weights is:

$\left[\begin{array}{c}{V}_k\\ {\mathrm{\&omega;}}_n\left(V_k\right)\end{array}\right]=Q^{-1}\left[\begin{array}{c}{C}_1\\ {\mathrm{<figure-callout\; id="2"\; label="C"\; filenames="HDA00003372491300042.png"\; state="\{\{state\}\}">C</figure-callout>}}_2\end{array}\right]$

Wherein,

Wherein, C ₁Integration for k moment output probability density function root mean square and basis function transformation matrix product;

Wherein, C ₂Be k moment output probability density function root mean square and n two dimensional basis functions B _n(x, integration r);

$Q=\left[\begin{array}{cc}{\mathrm{\&Sigma;}}_0& {{\mathrm{\&Sigma;}}_1}^T\\ {\mathrm{\&Sigma;}}_1& {\mathrm{\&Sigma;}}_2\end{array}\right],$ Wherein, Q is Σ ₀, Σ ₁, Σ ₂Variation, when basis function was selected the inputoutput data of back and known real system, Q was known quantity.

Wherein, Σ ₀Be basis function conversion vector C ₀(x, the integration of square value r) in its field of definition;

Wherein, Σ ₁Be basis function conversion vector C ₀(x is r) with the integration of n basis function product in its field of definition scope;

Wherein, Σ ₂Be n basis function B _n(x, r) square integration in its field of definition scope;

a ₁Be the interior lower limit of setting between the X-axis setting district; b ₁Be the interior higher limit of setting between the X-axis setting district;

a ₂Be the interior lower limit of setting between R axle setting district; b ₂Be the interior higher limit of setting between R axle setting district;

Σ ₁ ^TBe Σ ₁Transposed matrix.

Step S4: analyzing three-dimensional output PDF dynamic model satisfies the condition that the nature constraint possesses and is:

||V _k|| _Σ≤1

Wherein, || V _k|| _Σ=V _k ^TΣ V _k,

More than in two formulas, V _kBe k corresponding n-1 right-safeguarding value vector of the moment; V _k ^TBe V _kTransposed matrix; Σ is Σ ₀, Σ ₁And Σ ₂The conversion vector;

Step 2: utilize the inputoutput data that collects in the real system to set up the input of three-dimensional output PDF by recursive least squares;

The input of the three-dimensional output PDF that sets up is:

$f(x,r,u_k)=\underset{i=1}{\overset{n-1}{\mathrm{\&Sigma;}}}{a}_{i}f(x,r,u_{k-i})+\underset{j=0}{\overset{n-2}{\mathrm{\&Sigma;}}}{C}_0(x,r)D_j{u}_{k-j-1}$

Wherein, $f(x,r,u_k)=\sqrt{\mathrm{\&gamma;}(x,r,u_k)}-B_n(x,r){\mathrm{\&omega;}}_n\left(V_k\right);$

More than in two formulas, f (x, r, u _k) be the k variation of corresponding output probability density function constantly; a _iBe corresponding f (x, r, u of the k-i moment _K-i) coefficient; F (x, r, u _K-i) be the k-i variation of corresponding output probability density function constantly; u _K-iBe k-i corresponding control action of the moment; u _K-j-1Be k-j-1 corresponding control action of the moment; D _j=[d _J1..., d _Ji..., d _{J (n-1)}] ^TFor needing the parameter of identification; d _JiFor with C ₀(x, r) the corresponding coefficient of item in.

Step 3: select instantaneous square root performance index CONTROLLER DESIGN for use, by the controlled quentity controlled variable of the instantaneous square root performance index of optimization CONTROLLER DESIGN, the output PDF of realization system distribution shape tracing preset distributes and exports the shape that PDF distributes.

The instantaneous square root performance index of described selection are:

$J={\&Integral;}_{a_2}^{{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HDA00003372491300042.png"\; state="\{\{state\}\}">b</figure-callout>}}_2}{\&Integral;}_{a_1}^{b_1}{(\sqrt{\mathrm{\&gamma;}(x,r,u_{k+1})}-\sqrt{g(x,r)})}^2\mathrm{dxdr}+{{\mathrm{<figure-callout\; id="2"\; label="Ru\; k"\; filenames="HDA00003372491300042.png"\; state="\{\{state\}\}">Ru</figure-callout>}}_k}^2$

Wherein, J is instantaneous square root performance index value; γ (x, r, u _K+1) be three-dimensional output probability density function; (x r) is given three-dimensional output PDF distribution function to g; R is the constraint constant of control action; a ₁Be the interior lower limit of setting between the X-axis setting district; b ₁Be the interior higher limit of setting between the X-axis setting district; a ₂Be the interior lower limit of setting between R axle setting district; b ₂Be the interior higher limit of setting between R axle setting district.

As follows by the controlled amount of the instantaneous square root performance index of optimization:

$u_k=-\frac{{\&Integral;}_{a_2}^{{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HDA00003372491300042.png"\; state="\{\{state\}\}">b</figure-callout>}}_2}{\&Integral;}_{a_1}^{b_1}\left(C_0(x,r)D_0\hat{g}(x,r)\right)\mathrm{dxdr}}{{\&Integral;}_{a_2}^{{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HDA00003372491300042.png"\; state="\{\{state\}\}">b</figure-callout>}}_2}{\&Integral;}_{a_1}^{b_1}{\left(C_0(x,r)D_0\right)}^2\mathrm{dxdr}+R}$

Wherein,

$\hat{g}(x,r)=\underset{i=2}{\overset{n-1}{\mathrm{\&Sigma;}}}(a_{i}f(x,r,k-i+1)+C_0(x,r)D_{i-1}{u}_{k-i+1})+a_{1}f(x,r,k)+B_n(x,r){\mathrm{\&omega;}}_n\left(V_k\right)-\sqrt{g(x,r)};$

Wherein,

Variation for known quantity and parameter; (x, r k-i+1) are the k-i+1 variation of corresponding output probability density function constantly to f; D ₀Be the parameter value that picks out.

Described controlled quentity controlled variable u _kBe by right

In f (x, r, k) f (x, r, k-1) ... f (x, r, k-n+2) ω _n(V _k) and u _K-1u _K-2U _K-n+2The adjustment of value, the shape that the output PDF of realization system distribution shape tracing preset output PDF distributes.

Beneficial effect of the present invention: 1, the present invention has ensured that the output PDF of system greater than 1 constraint condition, has analyzed the output PDF of system and has been divided at 1 o'clock in its field of definition inner product, the constraint condition that weights should satisfy; 2, set up the dynamic model of three-dimensional output PDF among the present invention according to square root B batten model, then the dynamic model of setting up based on the square root B-spline function has been carried out conversion, according to the inputoutput data of gathering, set up the input of system; 3, the present invention has designed the optimization routine controller, by optimizing the square root performance index, obtains the control action of system, realizes that output PDF distribution shape is to the tracking of given output PDF distribution shape.The present invention has enriched three-dimensional output PDF control theory, for the industrial process with three-dimensional output distribution character provides new method.

Description of drawings

Fig. 1 is two-dimentional B-spline function image;

Fig. 2 is that the initial p DF of Three-Dimensional Dynamic system distributes;

Fig. 3 is that the given output PDF of Three-Dimensional Dynamic system distributes;

Fig. 4 is the output PDF response curved surface of Three-Dimensional Dynamic system;

Fig. 5 exports PDF to given PDF tracking error for last moment control;

Fig. 6 is the response curve of controlled quentity controlled variable in the control procedure;

Fig. 7 is performance index change curve in the control procedure;

Fig. 8 is overall flow figure of the present invention.

Embodiment

In order to deepen the understanding of the present invention, below in conjunction with accompanying drawing specific embodiments of the invention are described in further detail.Should be emphasized that following explanation only is exemplary, the scope that is not meant to limit the present invention and application thereof.

For actual industrial process needs, output PDF control theory is applied to have the system of distributed in three dimensions characteristic, adopt the mechanism method to set up the complicacy that system model and controller design cause in order to simplify.The present invention proposes a kind of dynamic modeling and controller design method of three-dimensional output probability density function.Be used for realization to the tracking of whole output PDF distribution shape.

The present invention is divided into following a few step:

One, makes up the instantaneous square root B batten model of three-dimensional output PDF, the dynamic change that adds weights on instantaneous square root B batten model basis has partly constituted based on the three-dimensional of square root B batten model exports the PDF dynamic model, realize the dynamic decoupling between the three-dimensional output PDF dynamic model weights, analyzing three-dimensional output PDF dynamic model satisfies the condition that the nature constraint possesses;

Two, on the basis of step 1 for the ease of CONTROLLER DESIGN, above-mentioned dynamic model is carried out conversion, utilize the inputoutput data that collects in the real system to set up the input of three-dimensional output PDF by recursive least squares;

Three, select instantaneous square root performance index CONTROLLER DESIGN for use on the basis of step 2, by adjusting the control action that controller obtains, reach the shape that the output PDF of system distribution shape tracing preset distribution output PDF distributes.

Specifically be divided into:

1, two-dimentional B-spline function method for expressing

The two dimension B-spline function is with the tensor product representation of two one dimension B-spline functions:

$B_{j,i}(x,r)=B_{j_x{i}_x}\left(x\right)B_{j_r{i}_r}\left(r\right)---\left(1\right)$

Wherein,

Computing formula is obtained by following recursion formula:

${\mathrm{<figure-callout\; id="1"\; label="B"\; filenames="HDA00003372491300011.png,HDA00003372491300012.png"\; state="\{\{state\}\}">B</figure-callout>}}_{1,i_x}\left(x\right)=\left\{\begin{array}{cc}1,& x\&Element;[x_{i_x},x_{i_x+1})\\ 0,& x\&NotElement;[x_{i_x},x_{i_x+1})\end{array}\right.---\left(2\right)$

$B_{j_x,i_x}\left(x\right)=\frac{x-x_{i_x}}{x_{i_x+j_x-1}-x_{i_x}}{B}_{j_x-1,i_x}\left(x\right)+\frac{x_{i_x+j_x}-x}{x_{i_x+j_x}-x_{i_x+1}}{B}_{j_x-1,i_x+1}\left(x\right)---\left(3\right)$

Obtained by following recursion formula:

${\mathrm{<figure-callout\; id="1"\; label="B"\; filenames="HDA00003372491300011.png,HDA00003372491300012.png"\; state="\{\{state\}\}">B</figure-callout>}}_{1,i_r}\left(r\right)=\left\{\begin{array}{cc}1,& r\&Element;{[r}_{i_r},r_{i_r+1})\\ 0,& r\&NotElement;[r_{i_r},r_{i_r+1})\end{array}\right.---\left(4\right)$

$B_{j_r,i_r}\left(r\right)=\frac{r-r_{i_r}}{r_{i_r+j_r-1}-r_{i_r}}{B}_{j_r-1,i_r}\left(r\right)+\frac{r_{i_r+j_r}-r}{r_{i_r+j_r}-r_{i_r+1}}{B}_{j_r-1,i_r+1}\left(r\right)---\left(5\right)$

Wherein, B _{J, i}(x r) is two-dimentional B spline base function;

Be one dimension B spline base function;

Be one dimension B spline base function; X, r are respectively the variable of definition spatially, x ∈ [a ₁, b ₁], r ∈ [a ₂, b ₂]; a ₁Be the interior lower limit of setting between the setting district; b ₁Be the interior higher limit of setting between the setting district; [a ₁, b ₁] for comprising a ₁And b ₁An interval; a ₂Be the interior lower limit of setting between the setting district; b ₂Be the interior higher limit of setting between the setting district; [a ₂, b ₂] for comprising a ₂And b ₂An interval; J represents the order of two-dimentional B batten; I represents two-dimentional B spline base function number; j _xOrder for the basis function chosen on the X-axis; i _xNumber for the basis function chosen on the X-axis; j _rOrder for the basis function chosen on the R axle; i _rNumber for the basis function chosen on the R axle;

Be 1 rank i _xIndividual B-spline function; Be j _x-1 rank i _xIndividual B-spline function;

Be j _x-1 rank i _x+ 1 B spline base function;

Be nodal value, and have $x_{j_x+1}<\&CenterDot;\&CenterDot;\&CenterDot;<a_1=x_0<\&CenterDot;\&CenterDot;\&CenterDot;<x_{m_x+1}={\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="HDA00003372491300011.png,HDA00003372491300012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1<\&CenterDot;\&CenterDot;\&CenterDot;<x_{m_x+j_x};$ m _xBe interval [a ₁, b ₁] interior effective node number, j _x-1 counts for the acromere of the interval left and right sides;

For comprising

With

An interval;

Be 1 rank i _rIndividual B-spline function;

Be j _r-1 rank i _rIndividual B-spline function;

Be j _r-1 rank i _r+ 1 B spline base function;

Be nodal value, and have $r_{j_r+1}<\&CenterDot;\&CenterDot;\&CenterDot;<a_2={\mathrm{<figure-callout\; id="0"\; label="r"\; filenames="HDA00003372491300011.png,HDA00003372491300012.png"\; state="\{\{state\}\}">r</figure-callout>}}_0<\&CenterDot;\&CenterDot;\&CenterDot;<r_{m_r+1}={\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HDA00003372491300042.png"\; state="\{\{state\}\}">b</figure-callout>}}_2<\&CenterDot;\&CenterDot;\&CenterDot;<r_{m_r+j_r};$ m _rBe interval [a ₂, b ₂] interior effective node number, j _r-1 counts for the acromere of the interval left and right sides;

For comprising

With

An interval;

2, make up the instantaneous square root model of three-dimensional output PDF

The square root model namely approaches the square root of the output PDF of system with two-dimentional B-spline function, to guarantee the nonnegativity of output PDF in control procedure of stochastic system.

The discrete form of the instantaneous square root model of three-dimensional output PDF is expressed as:

$\sqrt{\mathrm{\&gamma;}(x,r{,u}_k)}=\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&omega;}}_i\left(u_k\right)B_i(x,r)+e_0---\left(6\right)$

Wherein,

γ (x, r, u _k) be the output probability density function;

The two-dimentional B-spline function number of n for selecting, k is sampling instant;

B _i(x r) is two-dimentional B-spline function, wherein, has omitted the order j of B batten;

ω _i(u _k) for depending on u _kWeights; u _kBe k corresponding control action of the moment.

e ₀Approximate error for system;

Generally ignore e ₀Then the instantaneous square root model representation of three-dimensional output PDF is:

$\sqrt{\mathrm{\&gamma;}(x,r,u_k)}=\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&omega;}}_i\left(u_k\right)B_i(x,r)---\left(7\right)$

For given three-dimensional output PDF function, formula (7) is unique, gets x ∈ [a ₁, b ₁] r ∈ [a ₂, b ₂] be the stochastic variable span, the instantaneous square root model of three-dimensional being exported PDF according to (7) formula further is expressed as:

$\sqrt{\mathrm{\&gamma;}(x,r,u_k)}=C_0(x,r)V_k+B_n(x,r){\mathrm{\&omega;}}_n\left(V_k\right)---\left(8\right)$

Wherein,

C ₀(x, r)=[B ₁(x, r), B ₂(x, r) ..., B _N-1(x, r)], wherein, C ₀(x r) is the conversion vector of 1 * (n-1) dimension basis function, B _N-1(x r) is n-1 basis function; V _k=[ω ₁(u _k), ω ₂(u _k) ..., ω _N-1(u _k)] ^T, wherein, V _kBe (n-1) * 1 right-safeguarding value vector, ω _N-1(u _k) be the weights of n-1 basis function correspondence; [ω ₁(u _k), ω ₂(u _k) ..., ω _N-1(u _k)] ^TBe [ω ₁(u _k), ω ₂(u _k) ..., ω _N-1(u _k)] transposed matrix; ω _n(V _k) be the weights of n weights correspondence; B _n(x r) is n basis function.

3, the dynamic change of adding weights partly constitutes the three-dimensional output PDF dynamic model based on square root B batten model

The three-dimensional output PDF square root model of above-mentioned design does not relate to weights to be changed, and under a lot of situations, is dynamic relationship between output PDF and the input.General hypothesis V _kWith control input u _kBetween be that linear dynamic is relevant, the dynamic change of hypothesis weights here partly is expressed as:

V _k=AV _k-1+Bu _k-1 (9)

In the formula (9), A is (n-1) * (n-1) dimension parameter matrix of expression system dynamic relationship, and B is (the n-1) * 1 dimension parameter matrix of expression system dynamic relationship; V _K-1Be k-1 corresponding n-1 right-safeguarding value vector of the moment, u _K-1Be k-1 corresponding controlled quentity controlled variable of the moment.

So, based on the three-dimensional output PDF dynamic model expression of square root B batten model be:

V _k=AV _k-1+Bu _k-1 (10)

$\sqrt{\mathrm{\&gamma;}(x,r,u_k)}=C_0(x,r)V_k+B_n(x,r){\mathrm{\&omega;}}_n\left(V_k\right)---\left(11\right)$

4, the dynamic decoupling between the three-dimensional output PDF dynamic model weights;

Being drawn between the weights of basis function correspondence by (10), (11) formula is nonlinear relationship, in order to address this problem, need do as down conversion, to realize the dynamic decoupling of weights.

[C is multiply by on formula (11) both sides together ₀ ^T(x, r) B _n(x, r)] ^TCan get:

$\left[\begin{array}{c}{C_0}^T(x,r)\\ B_n(x,r)\end{array}\right]\sqrt{\mathrm{\&gamma;}(x,r,u_k)}=\left[\begin{array}{cc}{C_0}^T(x,r)C_0(x,r)& {C_0}^T(x,r)B_n(x,r)\\ B_n(x,r)C_0(x,r)& {B_n}^2(x,r)\end{array}\right]\left[\begin{array}{c}{V}_k\\ {\mathrm{\&omega;}}_n\left(V_k\right)\end{array}\right]---\left(12\right)$

Get x ∈ [a ₁, b ₁] r ∈ [a ₂, b ₂] be the stochastic variable span, following formula both sides integration can get:

$\left[\begin{array}{c}{C}_1\\ {\mathrm{<figure-callout\; id="2"\; label="C"\; filenames="HDA00003372491300042.png"\; state="\{\{state\}\}">C</figure-callout>}}_2\end{array}\right]=Q\left[\begin{array}{c}{V}_k\\ {\mathrm{\&omega;}}_n\left(V_k\right)\end{array}\right]---\left(13\right)$

Wherein,

Wherein, C ₁Integration for k moment output probability density function root mean square and basis function transformation matrix product;

Wherein, C ₂Be k moment output probability density function root mean square and n two dimensional basis functions B _n(x, integration r);

$Q=\left[\begin{array}{cc}{\mathrm{\&Sigma;}}_0& {{\mathrm{\&Sigma;}}_1}^T\\ {\mathrm{\&Sigma;}}_1& {\mathrm{\&Sigma;}}_2\end{array}\right],$ Wherein, Q is Σ ₀, Σ ₁, Σ ₂Variation, when basis function was selected the inputoutput data of back and known real system, Q was known quantity.

Wherein, Σ ₀Be basis function conversion vector C ₀(x, the integration of square value r) in its field of definition;

Wherein, Σ ₁Be basis function conversion vector C ₀(x is r) with the integration of n basis function product in its field of definition scope;

Wherein, Σ ₂Be n basis function B _n(x, r) square integration in its field of definition scope;

a ₁Be the interior lower limit of setting between the X-axis setting district; b ₁Be the interior higher limit of setting between the X-axis setting district;

a ₂Be the interior lower limit of setting between R axle setting district; b ₂Be the interior higher limit of setting between R axle setting district;

Σ ₁ ^TBe Σ ₁Transposed matrix.

When the B batten quadrature obtained, the Q inverse of a matrix always exists in the formula (13), and formula (13) is expressed as:

$\left[\begin{array}{c}{V}_k\\ {\mathrm{\&omega;}}_n\left(V_k\right)\end{array}\right]=Q^{-1}\left[\begin{array}{c}{C}_1\\ C_2\end{array}\right]---\left(14\right)$

Formula (14) has realized ω _n(V _k) and V _kDynamic decoupling.

4, three-dimensional output PDF dynamic model satisfies nature constraint condition and derives as follows:

Because γ is (x, r, u _k) be the output probability density function, then satisfying integration its field of definition scope planted agent is one constraint:

${\&Integral;}_{a_2}^{{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HDA00003372491300042.png"\; state="\{\{state\}\}">b</figure-callout>}}_2}{\&Integral;}_{a_1}^{{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="HDA00003372491300011.png,HDA00003372491300012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1}\mathrm{\&gamma;}(x,r,u_k)\mathrm{dxdr}={\&Integral;}_{a_2}^{{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HDA00003372491300042.png"\; state="\{\{state\}\}">b</figure-callout>}}_2}{\&Integral;}_{a_1}^{b_1}{\left(\sqrt{\mathrm{\&gamma;}(x,r,u_k)}\right)}^2\mathrm{dxdr}=1---\left(15\right)$

Release according to formula (15):

V _k ^TΣ ₀V _k+2Σ ₁V _kω _n(u _k)+Σ ₂ω _n ²(u _k)=1 (16)

Solution formula (16) can get

${\mathrm{\&omega;}}_n\left(u_k\right)=\frac{1}{{\mathrm{\&Sigma;}}_2}(-{\mathrm{\&Sigma;}}_1\&PlusMinus;\sqrt{{V_k}^T{{\mathrm{\&Sigma;}}_1}^T{\mathrm{\&Sigma;}}_1{V}_k-{V_k}^T{\mathrm{\&Sigma;}}_0{\mathrm{\&Sigma;}}_2{V}_k-{\mathrm{\&Sigma;}}_2}---\left(17\right)$

Draw ω by following formula _n(u _k) and V _kBetween be nonlinear relationship, and ω _n(u _k) can remember ω with other n-1 right of freedom value representation _n(u _k)=h (V _k).In order to guarantee that following formula has solution, should satisfy following formula:

V _k ^TΣ ₁ ^TΣ ₁ ^TV _k-V _k ^TΣ ₀Σ ₂V _k-Σ ₂≥0 (18)

Abbreviation formula (18) can get following non-linear constrain

||V _k|| _Σ≤1 (19)

Wherein, V _kBe (n-1) * 1 right-safeguarding value vector; || V _k|| _Σ=V _k ^TΣ V _k, expression V _kThe Σ norm;

Σ is Σ ₀, Σ ₁And Σ ₂The conversion vector.

As long as system's weights that calculating is tried to achieve satisfy the condition that formula (19) possesses, just can satisfy output PDF and be divided into 1 natural constraint condition in its field of definition inner product.

Obtaining three-dimensional output PDF dynamic model by above-mentioned derivation satisfies the condition that the nature constraint possesses and is:

||V _k|| _Σ≤1

5, the input of three-dimensional output PDF

The weights dynamic relationship that above-mentioned (9) formula is represented is not easy to obtain in real system, so, need carry out a down conversion to the three-dimensional output PDF dynamic model of setting up based on square root B batten model, the dynamic model expression that (10) (11) formula is represented becomes input.Order

$f(x,r,u_k)=\sqrt{\mathrm{\&gamma;}(x,r,u_k)}-B_n(x,r)h\left(V_k\right)=C_0(x,r)V_k---\left(20\right)$

Wherein, f (x, r, u _k) be the output function of equal value with output characteristics.Weights dynamic equation (10) is introduced displacement operator z ^-1, formula (20) is rewritten as:

f(x,r,u _k)=C ₀(x,r)(I-Az ^-1) ^-1Bu _k-1 (21)

Then according to (I-Az ^-1) ^-1The expansion of B namely obtains the input of three-dimensional output PDF with the following formula abbreviation:

$f(x,r,u_k)=\underset{i=1}{\overset{n-1}{\mathrm{\&Sigma;}}}{a}_{i}f(x,r,u_{k-i})+\underset{j=0}{\overset{n-2}{\mathrm{\&Sigma;}}}{C}_0(x,r)D_j{u}_{k-j-1}---\left(22\right)$

Wherein, I is that (n-1) * (n-1) is unit matrix; N is the basis function number of choosing; F (x, r, u _K-i) be the k-i output function of equal value of correspondence constantly; u _K-iBe k-i corresponding control action of the moment; u _K-j-1Be k-j-1 corresponding control action of the moment; a _iBe corresponding f (x, r, u of the k-i moment _K-i) coefficient, D _j=[d _J1..., d _Ji..., d _{J (n-1)}] ^TFor needing the parameter of identification, d _JiFor with C ₀(x, r) the corresponding coefficient of item in.

D _jAnd a _iAll be the unknown quantity in (22) formula, adopt the unknown parameter in the input of the three-dimensional output of recursive least squares identification PDF according to the form of (22) formula.Identification process is as follows:

Definition

θ=[a ₁,…,a _n-1,d ₀₁,…,d _0(n-1),d ₁₁,…,d _1(n-1),…,d _(n-2)1,…,d _(n-2)(n-1)] ^T (23)

φ(x,r,k)=[f(x,r,k-1),…,f(x,r,k-n+1),u _k-1C ₀₁(x,r),…,u _k-1C _0(n-1)(x,r),…,

u _k-n+1C ₀₁(x,r),…,u _0(k-n+1)C _0(n-1)(x,r)] ^T (24)

Wherein, θ is the weights that need identification; (x, r k) are known quantity to φ, when the inputoutput data that collects in the real system is determined.

Get x ∈ [a ₁, b ₁] r ∈ [a ₂, b ₂] be the stochastic variable span, in its field of definition scope, select N respectively _xAnd N _rIndividual sampled point is formed f (x _i, r _j, k):

f(x _i,r _j,k)=θ ^Tφ(x _i,r _j,k) (25)

x _i, r _jBe the sampled point of X-axis and R axle, i=1,2 ..., N _x, j=1,2 ..., N _r

Recursive least squares is defined as follows:

$\mathrm{\&theta;}(i+1,j+1)=\mathrm{\&theta;}(i,j)+\frac{P(i,j)\mathrm{\&phi;}(x_i,r_j,k)\mathrm{\&epsiv;}(i,j)}{1+{\mathrm{\&phi;}}^T(x_i,r_j,k)P(i,j)\mathrm{\&phi;}(x_i,r_j,k)}---\left(26\right)$

ε(i,j)=f(x _i,r _j,k)-θ ^T(i,j)φ(x _i,r _j,k) (27)

$P(i+1,j+1)=(I-\frac{P(i,j)\mathrm{\&phi;}(x_i,r_j,k){\mathrm{\&phi;}}^T(x_i,r_j,k)}{1+{\mathrm{\&phi;}}^T(x_i,r_j,k)P(i,j)\mathrm{\&phi;}(x_i,r_j,k)})P(i,j)---\left(28\right)$

Wherein, P (i, required transformation matrix when j) being identified parameters; (i j) is the identification error to ε; P (1,1)=10 ^3-6I _{N (n-1)}Be initial matrix, wherein, I _{N (n-1)}For n * (n-1) is unit matrix; θ (1,1)=θ ₀Be the initial weight vector.

The input parameter identification step of three-dimensional output PDF is as follows:

(1) selects suitable two-dimentional B spline base function, calculate basis function B _i(x, r) (i=1,2 ..., value n).

(2) (k 〉=n), { u is imported in collection system control at sampling instant k _K-1,, u _K-n+1And the interior sample point γ (x of field of definition scope _i, r _j, u _K-1) ..., γ (x _i, r _j, u _K-n+1) value;

(3) calculate h (V by formula (14) _K-1) ..., h (V _K-n-1) value, calculate f (x by definition _i, r _j, u _K-1) ..., f (x _i, r _j, u _K-n+1) and φ (x _i, r _j, value k);

(4) according to formula (26)-(28), estimated parameter θ, note θ (N _x, N _r) for the estimated value of sampling instant k;

(5) if k less than N, then k increases by 1, turns to for second step.

After the identification, θ is known quantity, then obtains the input of three-dimensional output PDF:

$f(x,r,u_k)=\underset{i=1}{\overset{n-1}{\mathrm{\&Sigma;}}}{a}_{i}f(x,r,u_{k-i})+\underset{j=0}{\overset{n-2}{\mathrm{\&Sigma;}}}{C}_0(x,r)D_j{u}_{k-j-1}$

6, instantaneous optimization Tracking Control Design

Input according to the three-dimensional output PDF that sets up above can carry out corresponding controller design.The purpose of CONTROLLER DESIGN is to select suitable control input to make the output PDF distribution shape of system's reality approach expectation PDF distribution shape as much as possible, consider the model of foundation, the weights of hypothesized model satisfy constraint condition, so select the square root quadratic performance index:

$J={\&Integral;}_{a_2}^{{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HDA00003372491300042.png"\; state="\{\{state\}\}">b</figure-callout>}}_2}{\&Integral;}_{a_1}^{b_1}(\sqrt{\mathrm{\&gamma;}(x,r,u_{k+1})}-{\sqrt{g(x,r)})}^2\mathrm{dxdr}+{{\mathrm{Ru}}_k}^2---\left(29\right)$

Be rewritten as according to formula (20) (22) following formula:

$J\left(u_k\right)={\&Integral;}_{a_2}^{{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HDA00003372491300042.png"\; state="\{\{state\}\}">b</figure-callout>}}_2}{\&Integral;}_{a_1}^{b_1}{(C_0(x,r)D_0{u}_k+\hat{g}(x,r))}^2\mathrm{dxdr}+{{\mathrm{Ru}}_k}^2---\left(30\right)$

Wherein:

$\hat{g}(x,r)=\underset{i=2}{\overset{n-1}{\mathrm{\&Sigma;}}}(a_{i}f(x,r,k-i+1)+C_0(x,r)D_{i-1}{u}_{k-i+1})+a_{1}f(x,r,k)+B_n(x,r)h\left(V_k\right)-\sqrt{g(x,r)}$

Expansion (30):

$J\left(u_k\right)={{\mathrm{<figure-callout\; id="2"\; label="u\; k"\; filenames="HDA00003372491300042.png"\; state="\{\{state\}\}">u</figure-callout>}}_k}^2{\&Integral;}_{a_2}^{{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HDA00003372491300042.png"\; state="\{\{state\}\}">b</figure-callout>}}_2}{\&Integral;}_{a_1}^{b_1}{\left(C_0(x,r)D_0\right)}^2\mathrm{dxdr}+2u_k{\&Integral;}_{a_2}^{{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HDA00003372491300042.png"\; state="\{\{state\}\}">b</figure-callout>}}_2}{\&Integral;}_{a_1}^{b_1}{C}_0(x,r)D_0\hat{g}(x,r)\mathrm{dxdr}+{\&Integral;}_{a_2}^{{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HDA00003372491300042.png"\; state="\{\{state\}\}">b</figure-callout>}}_2}{\&Integral;}_{a_1}^{b_1}\hat{g}{(x,r)}^2\mathrm{dxdr}+{{\mathrm{<figure-callout\; id="2"\; label="Ru\; k"\; filenames="HDA00003372491300042.png"\; state="\{\{state\}\}">Ru</figure-callout>}}_k}^2$

$\left(31\right)$

Adopt the algorithm of optimal control, to u _kAsk local derviation, order

Then obtain:

$u_k=-\frac{{\&Integral;}_{a_2}^{{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HDA00003372491300042.png"\; state="\{\{state\}\}">b</figure-callout>}}_2}{\&Integral;}_{a_1}^{b_1}\left(C_0(x,r)D_0\hat{g}(x,r)\right)\mathrm{dxdr}}{{\&Integral;}_{a_2}^{{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HDA00003372491300042.png"\; state="\{\{state\}\}">b</figure-callout>}}_2}{\&Integral;}_{a_1}^{b_1}{\left(C_0(x,r)D_0\right)}^2\mathrm{dxdr}+R}---\left(32\right)$

Since then, in the present invention, a kind of dynamic modeling and controller design method of three-dimensional output probability density function finish.

Embodiment is as follows:

Because experiment condition is limited, be difficult to obtain the input and output data of real system, dynamic vector A, the B of supposing the system are known, and following surface construction based on the three-dimensional output PDF dynamic model of square root B batten model is:

V _k=AV _k-1+Bu _k-1

$\sqrt{\mathrm{\&gamma;}(x,r,u_k)}=C_0(x,r)V_k+B_n(x,r)h\left(V_k\right)$

Wherein

$A=\left(\begin{array}{ccccc}0.991& 0& 0& 0& 0\\ 0& 0.994& 0& 0& 0\\ 0.704& 0.056& 0.887& 0& 0\\ 0& 0& 0& 0.947& 0\\ 0& 0& 0& 0.057& 0.976\end{array}\right);$

B=[0.0209 0.0448 0.0246 0.0292 0.0305] ^T；

According to the definition of two-dimentional B spline base function, select one dimension B spline base function to construct two-dimentional B-spline function here.Suppose x, the span of r is x ∈ [0,1] r ∈ [0,1], and the basis function on x and the r axle is defined as follows:

(1) basis function on the X-axis:

B ₁(x)=xI _x1+(2-x)I _x2

B ₂(x)=(x-1)I _x2+(3-x)I _x3

B ₃(x)=(x-2)I _x3+(4-x)I _x4

Wherein I=1,2,3,4

(2) basis function on the R axle

$B_1\left(r\right)=\frac{1}{2}{r}^2{I}_{r1}+(-{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA00003372491300042.png"\; state="\{\{state\}\}">r</figure-callout>}}^2+3r-\frac{3}{2})I_{r2}+\frac{1}{2}{(r-3)}^2{I}_{r3}$

$B_2\left(r\right)=\frac{1}{2}{(r-1)}^2{I}_{r2}+(-{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA00003372491300042.png"\; state="\{\{state\}\}">r</figure-callout>}}^2+5r-\frac{11}{2})I_{r3}+\frac{1}{2}{(r-4)}^2{I}_{r4}$

Wherein

I=1,2,3,4

The known quantity that the controller design needs is:

Initial weight V ₀=[0.688 2.129 1.551 0.166 0.792] ^T

The input constraint factor is R=0.0005;

The control action span is u ∈ [0,1];

Initial control action is u ₀=0.3;

The control of desired output PDF correspondence is input as u=0.65;

Here Xuan two-dimentional B spline base function as shown in Figure 1.Fig. 2, the initial output PDF that Fig. 3 has provided three-dimensional linear system respectively distributes and desired output PDF distributed image, the output PDF of the system response curved surface that Fig. 4 distributes for tracing preset output, the tracking error of last moment desired output PDF and control output PDF as shown in Figure 5, Fig. 6 provides is the response curve of control input action in the control procedure, can be restrained and close to the expectation input by the controlled input of figure.Fig. 7 is the change curve of performance index in the control procedure.

Claims (1)

1. the dynamic modeling of a three-dimensional output probability density function and controller design method is characterized in that, this method may further comprise the steps:

Step 1: make up the three-dimensional output PDF dynamic model based on square root B batten model;

Described structure may further comprise the steps based on the three-dimensional output PDF dynamic model of square root B batten model:

Step S1: the instantaneous square root B batten model that makes up three-dimensional output PDF according to two-dimentional B-spline function;

As follows with two one dimension B-spline function tensor product representation two dimension B-spline functions:

$B_{j,i}(x,r)=B_{j_x{i}_x}\left(x\right)B_{j_r{i}_r}\left(r\right)$

Wherein,

Obtained by following recursion formula:

$B_{1,i_x}\left(x\right)=\left\{\begin{array}{cc}1,& x\&Element;[x_{i_x},x_{i_x+1})\\ 0,& x\&NotElement;[x_{i_x},x_{i_x+1})\end{array}\right.$

$B_{j_x,i_x}\left(x\right)=\frac{x-x_{i_x}}{x_{i_x+j_x-1}-x_{i_x}}{B}_{j_x-1,i_x}\left(x\right)+\frac{x_{i_x+j_x}-x}{x_{i_x+j_x}-x_{i_x+1}}{B}_{j_x-1,i_x+1}\left(x\right)$

Obtained by following recursion formula:

$B_{1,i_r}\left(r\right)=\left\{\begin{array}{cc}1,& r\&Element;{[r}_{i_r},r_{i_r+1})\\ 0,& r\&NotElement;[r_{i_r},r_{i_r+1})\end{array}\right.$

$B_{j_r,i_r}\left(r\right)=\frac{r-r_{i_r}}{r_{i_r+j_r-1}-r_{i_r}}{B}_{j_r-1,i_r}\left(r\right)+\frac{r_{i_r+j_r}-r}{r_{i_r+j_r}-r_{i_r+1}}{B}_{j_r-1,i_r+1}\left(r\right)$

Wherein, B _{J, i}(x r) is two-dimentional B spline base function;

Be one dimension B spline base function;

Be one dimension B spline base function; X, r are respectively the variable of definition spatially, x ∈ [a ₁, b ₁], r ∈ [a ₂, b ₂]; a ₁Be the interior lower limit of setting between the setting district; b ₁Be the interior higher limit of setting between the setting district; [a ₁, b ₁] for comprising a ₁And b ₁An interval; a ₂Be the interior lower limit of setting between the setting district; b ₂Be the interior higher limit of setting between the setting district; [a ₂, b ₂] for comprising a ₂And b ₂An interval; J represents the order of two-dimentional B batten; I represents two-dimentional B spline base function number; j _xOrder for the basis function chosen on the X-axis; i _xNumber for the basis function chosen on the X-axis; j _rOrder for the basis function chosen on the R axle; i _rNumber for the basis function chosen on the R axle;

Be 1 rank i _xIndividual B-spline function;

Be j _x-1 rank i _xIndividual B-spline function;

Be j _x-1 rank i _x+ 1 B spline base function;

Be nodal value, and have $x_{j_x+1}<\&CenterDot;\&CenterDot;\&CenterDot;<a_1=x_0<\&CenterDot;\&CenterDot;\&CenterDot;<x_{m_x+1}=b_1<\&CenterDot;\&CenterDot;\&CenterDot;<x_{m_x+j_x};$ m _xBe interval [a ₁, b ₁] interior effective node number, j _x-1 counts for the acromere of the interval left and right sides;

For comprising

With

An interval;

Be 1 rank i _rIndividual B-spline function;

Be j _r-1 rank i _rIndividual B-spline function;

Be j _r-1 rank i _r+ 1 B spline base function;

Be nodal value, and have $r_{j_r+1}<\&CenterDot;\&CenterDot;\&CenterDot;<a_2=r_0<\&CenterDot;\&CenterDot;\&CenterDot;<r_{m_r+1}=b_2<\&CenterDot;\&CenterDot;\&CenterDot;<r_{m_r+j_r};$ m _rBe interval [a ₂, b ₂] interior effective node number, j _r-1 counts for the acromere of the interval left and right sides;

For comprising

With

An interval;

With two-dimentional B-spline function B _{J, i}(x r), omits the order j of B batten, i.e. two-dimentional B-spline function B _{J, i}(x r) is designated as B _i(x, r);

The instantaneous square root B batten model that obtains three-dimensional output PDF based on two-dimentional square root B-spline function is:

$\sqrt{\mathrm{\&gamma;}(x,r,u_k)}=C_0(x,r)V_k+B_n(x,r){\mathrm{\&omega;}}_n\left(V_k\right)$

Wherein,

γ (x, r, u _k) be three-dimensional output probability density function;

The two-dimentional B-spline function number of n for selecting, k is sampling instant;

C ₀(x, r)=[B ₁(x, r), B ₂(x, r) ..., B _N-1(x, r)], wherein, C ₀(x r) is 1 * (n-1) dimension basis function conversion vector;

B _i(x r) is two-dimentional B-spline function;

V _k=[ω ₁(u _k), ω ₂(u _k) ..., ω _N-1(u _k)] ^T, wherein, V _kBe k corresponding (n-1) * 1 right-safeguarding value vector of the moment;

ω _i(u _k) for depending on u _kWeights, u _kBe k corresponding control action of the moment;

B _n(x r) is two-dimentional B-spline function;

ω _n(V _k) be the weights of n basis function correspondence;

Step S2: on the basis of step S1, add the dynamic change part of weights, obtain the dynamic model based on the three-dimensional output of square root B batten model PDF;

The weights dynamic part of supposing adding is:

V _k=AV _k-1+Bu _k-1

Wherein, A is (n-1) * (n-1) dimension parameter matrix of expression system dynamic relationship, and B is (the n-1) * 1 dimension parameter matrix of expression system dynamic relationship; V _K-1Be k-1 corresponding n-1 right-safeguarding value vector of the moment, u _K-1Be k-1 corresponding controlled quentity controlled variable of the moment;

Three-dimensional output PDF dynamic model based on square root B batten model is:

$V_k={\mathrm{AV}}_{k-1}+{\mathrm{Bu}}_{k-1}\sqrt{\mathrm{\&gamma;}(x,r,u_k)}=C_0(x,r)V_k+B_n(x,r){\mathrm{\&omega;}}_n\left(V_k\right)$

Step S3: the weight vector V of system _kThere is nonlinear relationship in the weights corresponding with n basis function, analyze the dynamic decoupling that obtains between the three-dimensional output PDF dynamic model weights;

Decoupling zero formula between the described three-dimensional output PDF dynamic model weights is:

$\left[\begin{array}{c}{V}_k\\ {\mathrm{\&omega;}}_n\left(V_k\right)\end{array}\right]=Q^{-1}\left[\begin{array}{c}{C}_1\\ C_2\end{array}\right]$

Wherein,

Wherein, C ₁Integration for k moment output probability density function root mean square and basis function transformation matrix product;

Wherein, C ₂Be k moment output probability density function root mean square and n two dimensional basis functions B _n(x, integration r);

$Q=\left[\begin{array}{cc}{\mathrm{\&Sigma;}}_0& {{\mathrm{\&Sigma;}}_1}^T\\ {\mathrm{\&Sigma;}}_1& {\mathrm{\&Sigma;}}_2\end{array}\right],$ Wherein, Q is Σ ₀, Σ ₁, Σ ₂Variation, when basis function was selected the inputoutput data of back and known real system, Q was known quantity;

Wherein, Σ ₀Be basis function conversion vector C ₀(x, the integration of square value r) in its field of definition;

Wherein, Σ ₁Be basis function conversion vector C ₀(x is r) with the integration of n basis function product in its field of definition scope;

Wherein, Σ ₂Be n basis function B _n(x, r) square integration in its field of definition scope;

a ₁Be the interior lower limit of setting between the X-axis setting district; b ₁Be the interior higher limit of setting between the X-axis setting district;

a ₂Be the interior lower limit of setting between R axle setting district; b ₂Be the interior higher limit of setting between R axle setting district;

Σ ₁ ^TBe Σ ₁Transposed matrix;

Step S4: analyzing three-dimensional output PDF dynamic model satisfies the condition that the nature constraint possesses and is:

||V _k|| _Σ≤1

Wherein, || V _k|| _Σ=V _k ^TΣ V _k,

More than in two formulas, V _kBe k corresponding n-1 right-safeguarding value vector of the moment; V _k ^TBe V _kTransposed matrix; Σ is Σ ₀, Σ ₁And Σ ₂The conversion vector;

Step 2: utilize the inputoutput data that collects in the real system to set up the input of three-dimensional output PDF by recursive least squares;

The input of the three-dimensional output PDF that sets up is:

$f(x,r,u_k)=\underset{i=1}{\overset{n-1}{\mathrm{\&Sigma;}}}{a}_{i}f(x,r,u_{k-i})+\underset{j=0}{\overset{n-2}{\mathrm{\&Sigma;}}}{C}_0(x,r)D_j{u}_{k-j-1}$

Wherein, $f(x,r,u_k)=\sqrt{\mathrm{\&gamma;}(x,r,u_k)}-B_n(x,r){\mathrm{\&omega;}}_n\left(V_k\right);$

More than in two formulas, f (x, r, u _k) be the k variation of corresponding output probability density function constantly; a _iBe corresponding f (x, r, u of the k-i moment _K-i) coefficient; F (x, r, u _K-i) be the k-i variation of corresponding output probability density function constantly; u _K-iBe k-i corresponding control action of the moment; u _K-j-1Be k-j-1 corresponding control action of the moment; D _j=[d _J1..., d _Ji..., d _{J (n-1)}] ^TFor needing the parameter of identification; d _JiFor with C ₀(x, r) the corresponding coefficient of item in;

Step 3: select instantaneous square root performance index CONTROLLER DESIGN for use, by the controlled quentity controlled variable of the instantaneous square root performance index of optimization CONTROLLER DESIGN, the output PDF of realization system distribution shape tracing preset distributes and exports the shape that PDF distributes;

The instantaneous square root performance index of described selection are:

$J={\&Integral;}_{a_2}^{b_2}{\&Integral;}_{a_1}^{b_1}{(\sqrt{\mathrm{\&gamma;}(x,r,u_{k+1})}-\sqrt{g(x,r)})}^2\mathrm{dxdr}+{{\mathrm{Ru}}_k}^2$

Wherein, J is instantaneous square root performance index value; γ (x, r, u _K+1) be three-dimensional output probability density function; (x r) is given three-dimensional output PDF distribution function to g; R is the constraint constant of control action; a ₁Be the interior lower limit of setting between the X-axis setting district; b ₁Be the interior higher limit of setting between the X-axis setting district; a ₂Be the interior lower limit of setting between R axle setting district; b ₂Be the interior higher limit of setting between R axle setting district;

As follows by the controlled amount of the instantaneous square root performance index of optimization:

$u_k=-\frac{{\&Integral;}_{a_2}^{b_2}{\&Integral;}_{a_1}^{b_1}\left(C_0(x,r)D_0\hat{g}(x,r)\right)\mathrm{dxdr}}{{\&Integral;}_{a_2}^{b_2}{\&Integral;}_{a_1}^{b_1}{\left(C_0(x,r)D_0\right)}^2\mathrm{dxdr}+R}$

Wherein,

$\hat{g}(x,r)=\underset{i=2}{\overset{n-1}{\mathrm{\&Sigma;}}}(a_{i}f(x,r,k-i+1)+C_0(x,r)D_{i-1}{u}_{k-i+1})+a_{1}f(x,r,k)+B_n(x,r){\mathrm{\&omega;}}_n\left(V_k\right)-\sqrt{g(x,r)};$

Wherein,

Variation for known quantity and parameter; (x, r k-i+1) are the k-i+1 variation of corresponding output probability density function constantly to f; D ₀Be the parameter value that picks out;

Described controlled quentity controlled variable u _kBe by to g

In f (x, r, k) f (x, r, k-1) ... f (x, r, k-n+2) ω _n(V _k) and u _K-1u _K-2U _K-n+2The adjustment of value, the shape that the output PDF of realization system distribution shape tracing preset output PDF distributes.

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