CN103336427B

CN103336427B - A kind of dynamic modeling of three-dimensional output probability density function and controller design method

Info

Publication number: CN103336427B
Application number: CN201310244539.7A
Authority: CN
Inventors: 张金芳; 许曼
Original assignee: North China Electric Power University
Current assignee: North China Electric Power University
Priority date: 2013-06-19
Filing date: 2013-06-19
Publication date: 2015-08-12
Anticipated expiration: 2033-06-19
Also published as: CN103336427A

Abstract

The invention discloses a method for dynamic modeling and controller design of a three-dimensional output probability density function in the field of stochastic distribution control theory. The method comprises the following steps: Step 1: Constructing a three-dimensional output PDF dynamic model based on a square root B-spline model; Step 2: Using the input and output data collected in an actual system to establish an input and output model of a three-dimensional output PDF through a recursive least squares algorithm Step 3: select the instantaneous square root performance index design controller, by optimizing the control amount of the instantaneous square root performance index design controller, realize the shape of the system output PDF distribution shape tracking given output PDF distribution; the present invention has designed a conventional optimization controller , by optimizing the square root performance metric, the output PDF distribution is tracked for a given output PDF. The invention enriches the three-dimensional output PDF control theory and provides a new method for the industrial process with three-dimensional output distribution characteristics.

Description

A Dynamic Modeling and Controller Design Method for 3D Output Probability Density Function

技术领域technical field

本发明属于随机分布控制理论领域，尤其涉及一种三维输出概率密度函数的动态建模与控制器设计方法。The invention belongs to the field of stochastic distribution control theory, and in particular relates to a dynamic modeling and controller design method of a three-dimensional output probability density function.

背景技术Background technique

随机系统控制理论是控制理论与应用的重要分支之一，主要是因为绝大多数工业过程都受到随机信号的干扰，针对这一实际问题，已经形成系统的随机控制理论。其早期的研究成果集中于控制系统变量自身的统计特性，控制目标是系统的均值和方差，在已有的方法中，大部分假设随机系统中随机变量服从高斯分布，然而这一假设并不符合实际应用，例如造纸过程中的纤维长度分布、粮食加工中的粮食颗粒分布及锅炉火焰温度分布等。一般随机系统分布用输出PDF(probability density function)概率密度函数表示，当随机变量是高斯变量时，通过控制系统的均值和方差可以实现对系统的输出PDF控制，对于随机变量不满足高斯分布的系统，其均值和方差不能包含系统的全部信息，对系统均值和方差的控制不能实现对这个系统输出PDF的控制。针对这类系统，王宏教授于1998年提出直接控制系统输出PDF形状的方法，即输出PDF控制。该类方法直接设计控制器以使系统输出PDF分布形状跟踪给定PDF分布形状。Stochastic system control theory is one of the important branches of control theory and application, mainly because most industrial processes are disturbed by random signals. Aiming at this practical problem, a systematic stochastic control theory has been formed. Its early research results focused on the statistical characteristics of the control system variables themselves. The control target is the mean and variance of the system. In the existing methods, most of the random variables in the random system are assumed to obey the Gaussian distribution. However, this assumption does not meet the Practical applications, such as fiber length distribution in papermaking, grain particle distribution in grain processing, and boiler flame temperature distribution, etc. Generally, the random system distribution is represented by the output PDF (probability density function) probability density function. When the random variable is a Gaussian variable, the output PDF control of the system can be realized by controlling the mean and variance of the system. For the system whose random variable does not satisfy the Gaussian distribution , its mean value and variance cannot contain all the information of the system, and the control of the system mean value and variance cannot realize the control of the output PDF of this system. For this type of system, Professor Wang Hong proposed a method of directly controlling the output PDF shape of the system in 1998, that is, output PDF control. This type of method directly designs the controller so that the output PDF distribution shape of the system follows the given PDF distribution shape.

输出PDF可以通过B样条神经网络进行逼近，这样在一定程度上实现了原来复杂耦合系统的解耦，对于这类解耦的输出PDF控制，称之为SDC(stochasticdistribution control)随机系统分布控制。该类方法突破了随机控制研究的局限性，将传统的随机控制问题转化为建立简单合理模型、设计高效合适控制算法的方法。实现从复杂隐含描述系统动态特性的偏微分方程转化为解耦的状态空间模型描述，最终达到类似于确定系统一样能用比较准确的模型来描述随机分布系统的动态行为。SDC与以往随机控制理论相比更加符合实际情况，因此，将该理论融入到新的工业领域，将赋予其强大的生命力。The output PDF can be approximated by the B-spline neural network, which realizes the decoupling of the original complex coupling system to a certain extent. For this kind of decoupled output PDF control, it is called SDC (stochastic distribution control) stochastic system distribution control. This type of method breaks through the limitations of stochastic control research, and transforms the traditional stochastic control problem into a method of establishing a simple and reasonable model and designing an efficient and suitable control algorithm. Realize the transformation from complex implicit partial differential equations that describe the dynamic characteristics of the system to decoupled state space model descriptions, and finally achieve a more accurate model to describe the dynamic behavior of random distribution systems similar to deterministic systems. Compared with the previous stochastic control theory, SDC is more in line with the actual situation. Therefore, integrating this theory into new industrial fields will give it strong vitality.

针对具有二维特性的随机分布系统，已经建立了较为完善的理论体系。如在系统建模方面，建立了线性B样条模型、有理B样条模型、平方根B样条模型和有理平方根B样条模型，输入输出ARMAX模型，神经网络PDF模型等。在控制器设计方面，实现了瞬时最优跟踪控制算法、最优跟踪控制算法、模型参考自适应控制算法、预测控制算法、结构化控制器算法、迭代学习控制算法等。近年来，学者们在随机分布控制的鲁棒控制、最小熵控制、故障诊断及滤波器设计等方面做了大量的工作。A relatively complete theoretical system has been established for stochastic distribution systems with two-dimensional properties. For example, in terms of system modeling, linear B-spline models, rational B-spline models, square root B-spline models and rational square root B-spline models, input and output ARMAX models, neural network PDF models, etc. have been established. In terms of controller design, the instantaneous optimal tracking control algorithm, optimal tracking control algorithm, model reference adaptive control algorithm, predictive control algorithm, structured controller algorithm, iterative learning control algorithm, etc. have been realized. In recent years, scholars have done a lot of work on robust control of stochastic distribution control, minimum entropy control, fault diagnosis and filter design.

综上所述，针对二维输出分布控制问题已经取得了很大的进展，可是实际工业过程中还存在一类三维输出分布问题，如表征锅炉火焰温度的三维温度场、电站燃煤循环流化床锅炉中物料浓度的三维分布等。这些三维分布与工业过程运行状况息息相关，这对整个工业过程提高生产效率、减少环境污染等方面都具有重要的应用价值。随着计算机图像处理和传感器技术的高速发展，对三维输出分布状况的检测得到了迅速的发展。但是，通过先进技术在线测量得到系统输出分布的方法实现起来很复杂，所需的设备昂贵。To sum up, great progress has been made in the two-dimensional output distribution control problem, but there is still a kind of three-dimensional output distribution problem in the actual industrial process, such as the three-dimensional temperature field representing the boiler flame temperature, the coal-fired circulation fluidization of power station Three-dimensional distribution of material concentration in bed boiler, etc. These three-dimensional distributions are closely related to the operation status of the industrial process, which has important application value for improving production efficiency and reducing environmental pollution in the entire industrial process. With the rapid development of computer image processing and sensor technology, the detection of three-dimensional output distribution has been developed rapidly. However, the method of obtaining the system output distribution through on-line measurement with advanced technology is very complicated to implement, and the required equipment is expensive.

本发明研究的三维输出PDF控制问题是SDC理论的一个重要组成部分，然而对三维PDF的建模与控制问题的研究还不完善，文章《Modeling and control ofthe flame temperature distribution using probability density function shaping》对三维输出PDF进行了静态建模与控制器设计，选用二维B样条基函数并通过最小二乘算法建立了三维输出PDF的静态模型，优化二次性能指标，用梯度的方法得到了系统局部最优控制输入，计算机仿真研究得到了合理结果。目前为止，对于动态过程三维输出分布控制方面的研究鲜有发表，但是没有直接将三维输出分布作为控制的研究报道。The three-dimensional output PDF control problem studied by the present invention is an important part of the SDC theory, but the research on the modeling and control problem of three-dimensional PDF is not perfect. The article "Modeling and control of the flame temperature distribution using probability density function shaping" The static modeling and controller design of the 3D output PDF were carried out, the static model of the 3D output PDF was established by using the 2D B-spline basis function and the least square algorithm, the secondary performance index was optimized, and the system local Optimal control input, computer simulation research has obtained reasonable results. So far, few studies on the control of three-dimensional output distribution in dynamic processes have been published, but there is no research report on the direct control of three-dimensional output distribution.

为了进一步完善三维输出PDF控制理论，使得三维随机分布控制问题实现成为可能，本发明首先建立了三维输出PDF的瞬时平方根B样条模型，在瞬时平方根B样条模型基础上加入权值的动态变化部分构成了基于平方根B样条模型的三维输出PDF动态模型，实现了权值之间的动态解耦，分析了三维输出PDF动态模型满足自然约束的条件；然后根据系统输入输出数据通过递归最小二乘算法建立了三维输出PDF输入输出模型；最后选择瞬时平方根性能指标，设计了常规最优控制器。本发明对三维输出PDF控制理论进行了完善，为三维输出分布问题的控制提供了新的方法和思路。In order to further improve the three-dimensional output PDF control theory and make it possible to realize the three-dimensional random distribution control problem, the present invention first establishes the instantaneous square root B-spline model of the three-dimensional output PDF, and adds the dynamic change of the weight on the basis of the instantaneous square root B-spline model A three-dimensional output PDF dynamic model based on the square root B-spline model is partially constructed, which realizes the dynamic decoupling between weights, and analyzes the condition that the three-dimensional output PDF dynamic model satisfies the natural constraints; The three-dimensional output PDF input-output model is established by the multiplication algorithm; finally, the instantaneous square root performance index is selected, and a conventional optimal controller is designed. The invention perfects the three-dimensional output PDF control theory, and provides a new method and idea for the control of the three-dimensional output distribution problem.

发明内容Contents of the invention

本发明针对三维输出PDF理论的有待完善以及实际工业过程的需要，提出了一种三维输出概率密度函数的动态建模与控制器设计方法。The invention proposes a method for dynamic modeling and controller design of a three-dimensional output probability density function, aiming at the need to perfect the three-dimensional output PDF theory and the needs of the actual industrial process.

一种三维输出概率密度函数的动态建模与控制器设计方法，该方法包括以下步骤：A method for dynamic modeling and controller design of a three-dimensional output probability density function, the method comprising the following steps:

步骤1：构建基于平方根B样条模型的三维输出PDF动态模型；Step 1: Construct a 3D output PDF dynamic model based on the square root B-spline model;

所述构建基于平方根B样条模型的三维输出PDF动态模型包括以下步骤：The three-dimensional output PDF dynamic model based on the square root B-spline model comprises the following steps:

步骤S1：根据二维B样条函数构建三维输出PDF的瞬时平方根B样条模型；Step S1: constructing an instantaneous square root B-spline model of the three-dimensional output PDF according to the two-dimensional B-spline function;

用两个一维B样条函数张量积表示二维B样条函数如下：The two-dimensional B-spline function is represented by the tensor product of two one-dimensional B-spline functions as follows:

${B B}_{j j,, i i} ((x x,, r r)) = = {B B}_{{j j}_{x x} {i i}_{x x}} ((x x)) {B B}_{{j j}_{r r} {i i}_{r r}} ((r r))$

其中，由如下递推公式得到：in, It is obtained by the following recursive formula:

${B B}_{11,, {i i}_{x x}} ((x x)) = = \{\begin{matrix} 11,, & x x &Element; &Element; [[{x x}_{{i i}_{x x} + + 11},, {x x}_{{i i}_{x x} + + 11})) \\ 00,, & x x &NotElement; &NotElement; [[{x x}_{{i i}_{x x}},, {x x}_{{i i}_{x x} + + 11})) \end{matrix}$

${B B}_{{j j}_{x x},, {i i}_{x x}} ((x x)) = = \frac{x x - - {x x}_{{i i}_{x x}}}{{x x}_{{i i}_{x x} + + {j j}_{x x} - - 11} - - {x x}_{{i i}_{x x}}} {B B}_{{j j}_{x x} - - 11,, {i i}_{x x}} ((x x)) + + \frac{{x x}_{{i i}_{x x} + + {j j}_{x x}} - - x x}{{x x}_{{i i}_{x x} + + {j j}_{x x}} - - {x x}_{{i i}_{x x} + + 11}} {B B}_{{j j}_{x x} - - 11,, {i i}_{x x} + + 11} ((x x))$

由如下递推公式得到： It is obtained by the following recursive formula:

${B B}_{11,, {i i}_{r r}} ((r r)) = = \{\begin{matrix} 11,, & r r &Element; &Element; [[{r r}_{{i i}_{r r} + + 11},, {r r}_{{i i}_{r r} + + 11})) \\ 00,, & r r &NotElement; &NotElement; [[{r r}_{{i i}_{r r}},, {r r}_{{i i}_{r r} + + 11})) \end{matrix}$

${B B}_{{j j}_{r r},, {i i}_{r r}} ((r r)) = = \frac{r r - - {r r}_{{i i}_{r r}}}{{r r}_{{i i}_{r r} + + {j j}_{r r} - - 11} - - {r r}_{{i i}_{r r}}} {B B}_{{j j}_{r r} - - 11,, {i i}_{r r}} ((r r)) + + \frac{{r r}_{{i i}_{r r} + + {j j}_{r r}} - - r r}{{r r}_{{i i}_{r r} + + {j j}_{r r}} - - {r r}_{{i i}_{r r} + + 11}} {B B}_{{j j}_{r r} - - 11,, {i i}_{r r} + + 11} ((r r))$

其中，B_j,i(x,r)为二维B样条基函数；为一维B样条基函数；为一维B样条基函数；x,r分别为在空间上定义的变量，x∈[a₁,b₁]、r∈[a₂,b₂]；a₁为X轴设定区间内设定的下限值；b₁为X轴设定区间内设定的上限值；[a₁,b₁]为包括a₁和b₁的一个区间；a₂为R轴设定区间内设定的下限值；b₂为R轴设定区间内设定的上限值；[a₂,b₂]为包括a₂和b₂的一个区间；j表示二维B样条的阶次；i表示二维B样条基函数个数；j_x为X轴上选取的基函数的阶次；i_x为X轴上选取的基函数的个数；j_r为R轴上选取的基函数的阶次；i_r为R轴上选取的基函数的个数；Among them, B _j,i (x, r) is a two-dimensional B-spline basis function; is a one-dimensional B-spline basis function; is a one-dimensional B-spline basis function; x, r are variables defined in space, x∈[a ₁ ,b ₁ ], r∈[a ₂ ,b ₂ ]; a ₁ is the X-axis setting interval The lower limit value of the setting; b ₁ is the upper limit value set in the X-axis setting interval; [a ₁ ,b ₁ ] is an interval including a ₁ and b ₁ ; a ₂ is in the R-axis setting interval The lower limit value set; b ₂ is the upper limit value set in the R-axis setting interval; [a ₂ ,b ₂ ] is an interval including a ₂ and b ₂ ; j represents the order of the two-dimensional B-spline i represents the number of two-dimensional B-spline basis functions; j _x is the order of the basis functions selected on the X-axis; i _x is the number of basis functions selected on the X-axis; j _r is the order of the basis functions selected on the R-axis The order of the basis function; i _r is the number of basis functions selected on the R axis;

为1阶第i_x个B样条函数；为j_x-1阶第i_x个B样条函数；为j_x-1阶第i_x+1个B样条基函数；为节点值，且有m_x为区间[a₁,b₁]内的有效节点数，j_x-1为区间左右两侧的外节点数；为包括但不包括的一个区间； is the i _xth B-spline function of order 1; is the i _xth B-spline function of order j _x -1; is the i _x +1th B-spline basis function of order j _x -1; is the node value, and has m _x is the number of effective nodes in the interval [a ₁ , b ₁ ], j _x -1 is the number of outer nodes on the left and right sides of the interval; to include but does not include an interval of

为1阶第i_r个B样条函数；为j_r-1阶第i_r个B样条函数；为j_r-1阶第i_r+1个B样条基函数；为节点值，且有m_r为区间[a₂,b₂]内的有效节点数，j_r-1为区间左右两侧的外节点数；为包括但不包括的一个区间； is the i _rth B-spline function of order 1; is the i _rth B-spline function of order j _r -1; is the i _r +1th B-spline basis function of order j _r -1; is the node value, and has m _r is the number of effective nodes in the interval [a ₂ , b ₂ ], j _r -1 is the number of outer nodes on the left and right sides of the interval; to include but does not include an interval of

将二维B样条函数B_j,i(x,r)，省略B样条的阶次j，即二维B样条函数B_j,i(x,r)记为B_i(x,r)；The two-dimensional B-spline function B _j,i (x,r), omitting the order j of the B-spline, that is, the two-dimensional B-spline function B _j,i (x,r) is recorded as B _i (x,r );

基于二维平方根B样条函数得到三维输出PDF的瞬时平方根B样条模型为：The instantaneous square root B-spline model of the three-dimensional output PDF based on the two-dimensional square root B-spline function is:

$\sqrt{γ γ ((x x,, r r,, {u u}_{k k}))} = = {C C}_{00} ((x x,, r r)) {V V}_{k k} + + {B B}_{n no} ((x x,, r r)) {ω ω}_{n no} (({V V}_{k k}))$

其中，in,

γ(x,r,u_k)为三维输出概率密度函数；γ(x,r,u _k ) is the three-dimensional output probability density function;

n为选择的二维B样条函数个数，k为采样时刻；n is the number of selected two-dimensional B-spline functions, and k is the sampling time;

C₀(x,r)＝[B₁(x,r),B₂(x,r),…,B_n-1(x,r)]，其中，C₀(x,r)为1×(n-1)维基函数变换向量；C ₀ (x,r)=[B ₁ (x,r), B ₂ (x,r),…,B _n-1 (x,r)], where C ₀ (x,r) is 1× (n-1) wiki function transformation vector;

B_i(x,r)为二维B样条函数；B _i (x, r) is a two-dimensional B-spline function;

V_k＝[ω₁(u_k),ω₂(u_k),…,ω_n-1(u_k)]^T，其中，V_k为k时刻对应的(n-1)×1维权值向量；V _k ＝[ω ₁ (u _k ),ω ₂ (u _k ),…,ω _n-1 (u _k )] ^T , where V _k is the (n-1)×1-dimensional weight vector corresponding to time k ;

ω_i(u_k)为依赖于u_k的权值，u_k为k时刻对应的控制作用。ω _i (u _k ) is the weight dependent on u _k , and u _k is the corresponding control action at time k.

B_n(x,r)为二维B样条函数；B _n (x, r) is a two-dimensional B-spline function;

ω_n(V_k)为第n个基函数对应的权值。ω _n (V _k ) is the weight corresponding to the nth basis function.

步骤S2：在步骤S1的基础上，加入权值的动态变化部分，得到基于平方根B样条模型三维输出PDF动态模型；Step S2: On the basis of step S1, adding the dynamic change part of the weight value to obtain a three-dimensional output PDF dynamic model based on the square root B-spline model;

假定加入的权值动态部分为：Assume that the dynamic part of the added weight is:

V_k＝AV_k-1+Bu_k-1 V _k ＝AV _k-1 +Bu _k-1

其中，A为表示系统动态关系的(n-1)×(n-1)维参数矩阵,B为表示系统动态关系的(n-1)×1维参数矩阵；V_k-1为k-1时刻对应的n-1维权值向量，u_k-1为k-1时刻对应的控制量；Among them, A is a (n-1)×(n-1)-dimensional parameter matrix representing the dynamic relationship of the system, B is a (n-1)×1-dimensional parameter matrix representing the dynamic relationship of the system; V _k-1 is k-1 The n-1 dimension weight vector corresponding to the moment, u _k-1 is the control amount corresponding to the k-1 moment;

基于平方根B样条模型的三维输出PDF动态模型为：The 3D output PDF dynamic model based on the square root B-spline model is:

${V V}_{k k} = = {AV AV}_{k k - - 11} + + {Bu Bu}_{k k - - 11} \sqrt{γ γ ((x x,, r r,, {u u}_{k k}))} = = {C C}_{00} ((x x,, r r)) {V V}_{k k} + + {B B}_{n no} ((x x,, r r)) {ω ω}_{n no} (({V V}_{k k}))$

步骤S3：系统的权值向量V_k与第n个基函数对应的权值存在非线性关系，分析得到三维输出PDF动态模型权值之间的动态解耦；Step S3: There is a nonlinear relationship between the weight vector V _k of the system and the weight corresponding to the nth basis function, and the dynamic decoupling between the weights of the three-dimensional output PDF dynamic model is obtained through analysis;

所述三维输出PDF动态模型权值之间的解耦公式为：The decoupling formula between the weights of the three-dimensional output PDF dynamic model is:

$[\begin{matrix} {V V}_{k k} \\ {ω ω}_{n no} (({V V}_{k k})) \end{matrix}] = = {Q Q}^{- - 11} [\begin{matrix} {C C}_{11} \\ {C C}_{22} \end{matrix}]$

其中，in,

其中，C₁为k时刻输出概率密度函数均方根与基函数变换矩阵乘积的积分； Among them, C ₁ is the integral of the root mean square of the output probability density function and the product of the basis function transformation matrix at time k;

其中，C₂为k时刻输出概率密度函数均方根与第n个二维基函数B_n(x,r)的积分； Among them, C ₂ is the integral of the root mean square of the output probability density function at time k and the nth two-dimensional basis function B _n (x, r);

$Q = [\begin{matrix} Σ_{0} & {Σ_{1}}^{T} \\ Σ_{1} & Σ_{2} \end{matrix}],$ 其中，Q为∑₀、∑₁、∑₂变换形式，当基函数选定后并已知实际系统的输入输出数据时，Q为已知量。 $Q = [\begin{matrix} Σ_{0} & {Σ_{1}}^{T} \\ Σ_{1} & Σ_{2} \end{matrix}],$ Among them, Q is the transformation form of ∑ ₀ , ∑ ₁ , and ∑ _2. When the basis function is selected and the input and output data of the actual system are known, Q is a known quantity.

其中，∑₀为基函数变换向量C₀(x,r)的平方值在其定义域内的积分； Wherein, ∑ ₀ is the integral of the square value of the basis function transformation vector C ₀ (x, r) in its domain of definition;

其中，∑₁为基函数变换向量C₀(x,r)与第n个基函数乘积在其定义域范围的积分； Among them, ∑ ₁ is the integral of the product of the basis function transformation vector C ₀ (x, r) and the nth basis function within its domain;

其中，∑₂为第n个基函数B_n(x,r)平方在其定义域范围的积分； Among them, ∑ ₂ is the integral of the square of the nth basis function B _n (x, r) in its domain;

a₁为X轴设定区间内设定的下限值；b₁为X轴设定区间内设定的上限值；a ₁ is the lower limit value set in the X-axis setting interval; b ₁ is the upper limit value set in the X-axis setting interval;

a₂为R轴设定区间内设定的下限值；b₂为R轴设定区间内设定的上限值；a ₂ is the lower limit value set in the R-axis setting interval; b ₂ is the upper limit value set in the R-axis setting interval;

∑₁ ^T为∑₁的转置矩阵。∑ ₁ ^T is the transpose matrix of ∑ ₁ .

步骤S4：分析三维输出PDF动态模型满足自然约束具备的条件为：Step S4: Analyze the three-dimensional output PDF dynamic model to meet the conditions of natural constraints:

||V_k||_∑≤1||V _k || _∑ ≤1

其中，||V_k||_∑＝V_k ^T∑V_k， Where, ||V _k || _∑ ＝V _k ^T ∑V _k ,

以上两式中，V_k为k时刻对应的n-1维权值向量；V_k ^T为V_k的转置矩阵；∑为∑₀、∑₁和∑₂的变换向量；In the above two formulas, V _k is the n-1 dimension weight vector corresponding to the k moment; V _k ^T is the transposition matrix of V _k ; ∑ is the transformation vector of ∑ ₀ , ∑ ₁ and ∑ ₂ ;

步骤2：利用实际系统中采集到的输入输出数据通过递归最小二乘算法建立三维输出PDF的输入输出模型；Step 2: Use the input and output data collected in the actual system to establish the input and output model of the three-dimensional output PDF through the recursive least squares algorithm;

建立的三维输出PDF的输入输出模型为：The input and output model of the established 3D output PDF is:

$f f ((x x,, r r,, {u u}_{k k})) = = {Σ Σ}_{i i = = 11}^{n no - - 11} {a a}_{i i} f f ((x x,, r r,, {u u}_{k k - - i i})) + + {Σ Σ}_{j j = = 00}^{n no - - 22} {C C}_{00} ((x x,, r r)) {D D.}_{j j} {u u}_{k k - - j j - - 11}$

其中， $f (x, r, u_{k}) = \sqrt{γ (x, r, u_{k})} - B_{n} (x, r) ω_{n} (V_{k});$ in, $f (x, r, u_{k}) = \sqrt{γ (x, r, u_{k})} - B_{no} (x, r) ω_{no} (V_{k});$

以上两式中，f(x,r,u_k)为k时刻对应的输出概率密度函数的变换形式；a_i为k-i时刻对应的f(x,r,u_k-i)的系数；f(x,r,u_k-i)为k-i时刻对应的输出概率密度函数的变换形式；u_k-i为k-i时刻对应的控制作用；u_k-j-1为k-j-1时刻对应的控制作用；D_j＝[d_j1,…,d_ji,…,d_j(n-1)]^T为需要辨识的参数；d_ji为与C₀(x,r)中的项相对应的系数。In the above two formulas, f(x,r,u _k ) is the transformed form of the output probability density function corresponding to time k; a _i is the coefficient of f(x,r,u _ki ) corresponding to time ki; f(x, r,u _ki ) is the transformed form of the output probability density function corresponding to time ki; u _ki is the control action corresponding to time ki; u _kj-1 is the control action corresponding to time kj-1; D _j ＝[d _j1 ,… ,d _ji ,…,d _j(n-1) ] ^T is the parameter to be identified; d _ji is the coefficient corresponding to the item in C ₀ (x,r).

步骤3：选用瞬时平方根性能指标设计控制器，通过最优化瞬时平方根性能指标设计控制器的控制量，实现系统输出PDF分布形状跟踪给定分布输出PDF分布的形状。Step 3: Design the controller by selecting the instantaneous square root performance index, and design the control quantity of the controller by optimizing the instantaneous square root performance index, so as to realize the output PDF distribution shape of the system tracking the shape of the given distribution output PDF distribution.

所述选择的瞬时平方根性能指标为：The instantaneous square root performance index of the selection is:

$J J = = {&Integral; &Integral;}_{{a a}_{11}}^{{b b}_{22}} {&Integral; &Integral;}_{{a a}_{11}}^{{b b}_{11}} {((\sqrt{γ γ ((x x,, r r,, {u u}_{k k + + 11}))} - - \sqrt{g g ((x x,, r r))}))}^{22} dxdr wxya + + {Ru Ru}_{k k}^{22}$

其中，J为瞬时平方根性能指标值；γ(x,r,u_k+1)为三维输出概率密度函数；g(x,r)为给定三维输出PDF分布函数；R为控制作用的约束常量；a₁为X轴设定区间内设定的下限值；b₁为X轴设定区间内设定的上限值；a₂为R轴设定区间内设定的下限值；b₂为R轴设定区间内设定的上限值。Among them, J is the instantaneous square root performance index value; γ(x,r,u _k+1 ) is the three-dimensional output probability density function; g(x,r) is the given three-dimensional output PDF distribution function; R is the constraint constant of the control effect ; a ₁ is the lower limit value set in the X-axis setting interval; b ₁ is the upper limit value set in the X-axis setting interval; a ₂ is the lower limit value set in the R-axis setting interval; b ₂ is the upper limit value set in the R-axis setting interval.

通过最优化瞬时平方根性能指标得到控制量如下：The control quantity obtained by optimizing the instantaneous square root performance index is as follows:

${u u}_{k k} = = - - \frac{{&Integral; &Integral;}_{{a a}_{22}}^{{b b}_{22}} {&Integral; &Integral;}_{{a a}_{11}}^{{b b}_{11}} (({C C}_{00} ((x x,, r r)) {D D.}_{00} \overset{^^}{g g} ((x x,, r r)))) dxdr wxya}{{&Integral; &Integral;}_{{a a}_{22}}^{{b b}_{22}} {&Integral; &Integral;}_{{a a}_{11}}^{{b b}_{11}} {(({C C}_{00} ((x x,, r r)) {D D.}_{00}))}^{22} dxdr wxya + + R R}$

其中，in,

$\overset{^^}{g g} ((x x,, r r)) = = {Σ Σ}_{i i = = 22}^{n no - - 11} (({a a}_{i i} f f ((x x,, r r,, k k - - i i + + 11)) + + {C C}_{00} ((x x,, r r)) {D D.}_{i i - - 11} {u u}_{k k - - i i + + 11})) + + {a a}_{11} f f ((x x,, r r,, k k)) + + {B B}_{n no} ((x x,, r r)) {ω ω}_{n no} (({V V}_{k k})) - - \sqrt{g g ((x x,, r r))};;$

其中，为已知量与参数的变换形式；f(x,r,k-i+1)为k-i+1时刻对应的输出概率密度函数的变换形式；D₀为辨识出的参数值。in, is the transformation form of known quantities and parameters; f(x,r,k-i+1) is the transformation form of the output probability density function corresponding to time k-i+1; D ₀ is the identified parameter value.

所述控制量u_k是通过对中f(x,r,k),f(x,r,k-1),…f(x,r,k-n+2),ω_n(V_k)和u_k-1,u_k-2,…,u_k-n+2值的调整，实现系统输出PDF分布形状跟踪给定输出PDF分布的形状。The control quantity u _k is passed to In f(x,r,k),f(x,r,k-1),…f(x,r,k-n+2),ω _n (V _k ) and u _k-1 ,u _{k- 2} ,...,u _k-n+2 are adjusted to realize the system output PDF distribution shape tracking the given output PDF distribution shape.

本发明的有益效果：1、本发明保障了系统输出PDF大于1的约束条件，分析了系统输出PDF在其定义域内积分为1时，权值应满足的约束条件；2、本发明中根据平方根B样条模型建立了三维输出PDF的动态模型，然后对建立的基于平方根B样条函数的动态模型进行了变换，根据采集的输入输出数据，建立了系统的输入输出模型；3、本发明设计了常规优化控制器，通过优化平方根性能指标，得到系统的控制作用，实现输出PDF分布形状对给定输出PDF分布形状的跟踪。本发明丰富了三维输出PDF控制理论，为具有三维输出分布特性的工业过程提供了新的方法。Beneficial effects of the present invention: 1, the present invention guarantees the constraint condition that the system output PDF is greater than 1, and analyzes the constraint condition that the system output PDF should be satisfied when the integral is 1 in its domain of definition; 2, in the present invention, according to the square root The B-spline model has set up the dynamic model of three-dimensional output PDF, then transformed the dynamic model based on the square root B-spline function, and established the input-output model of the system according to the input-output data collected; 3, the design of the present invention A conventional optimization controller is adopted, and the control effect of the system is obtained by optimizing the square root performance index, and the output PDF distribution shape is tracked to a given output PDF distribution shape. The invention enriches the three-dimensional output PDF control theory and provides a new method for the industrial process with three-dimensional output distribution characteristics.

附图说明Description of drawings

图1为二维B样条函数图像；Fig. 1 is a two-dimensional B-spline function image;

图2为三维动态系统的初始PDF分布；Figure 2 is the initial PDF distribution of the 3D dynamic system;

图3为三维动态系统的给定输出PDF分布；Fig. 3 is the given output PDF distribution of the three-dimensional dynamic system;

图4为三维动态系统的输出PDF响应曲面；Fig. 4 is the output PDF response surface of the three-dimensional dynamic system;

图5为最后时刻控制输出PDF对给定PDF跟踪误差；Fig. 5 is the tracking error of the given PDF for the control output PDF at the last moment;

图6为控制过程中控制量的响应曲线；Fig. 6 is the response curve of the control quantity in the control process;

图7为控制过程中性能指标变化曲线；Fig. 7 is the change curve of performance index in the control process;

图8为本发明的整体流程图。Fig. 8 is an overall flowchart of the present invention.

具体实施方式Detailed ways

为了加深对本发明的理解，下面结合附图对本发明的具体实施例作进一步的详细说明。应该强调的是，下述说明仅仅是实例性的，并不是为了限制本发明的范围及其应用。In order to deepen the understanding of the present invention, the specific embodiments of the present invention will be further described in detail below in conjunction with the accompanying drawings. It should be emphasized that the following descriptions are only examples and are not intended to limit the scope of the invention and its application.

为了实际工业过程需要，将输出PDF控制理论应用于具有三维分布特性的系统，以便简化采用机理法建立系统模型与控制器设计引起的复杂性。本发明提出了一种三维输出概率密度函数的动态建模与控制器设计方法。用于实现对整个输出PDF分布形状的跟踪。For the needs of the actual industrial process, the output PDF control theory is applied to the system with three-dimensional distribution characteristics, so as to simplify the complexity caused by the establishment of the system model and the controller design by the mechanism method. The invention proposes a method for dynamic modeling and controller design of a three-dimensional output probability density function. Used to enable tracking of the shape of the entire output PDF distribution.

本发明分为以下几步：The present invention is divided into the following steps:

一、构建三维输出PDF的瞬时平方根B样条模型，在瞬时平方根B样条模型基础上加入权值的动态变化部分构成了基于平方根B样条模型的三维输出PDF动态模型，实现三维输出PDF动态模型权值之间的动态解耦，分析三维输出PDF动态模型满足自然约束具备的条件；1. Construct the instantaneous square root B-spline model of the three-dimensional output PDF, and add the dynamic change part of the weight on the basis of the instantaneous square root B-spline model to form a three-dimensional output PDF dynamic model based on the square root B-spline model to realize the dynamic three-dimensional output PDF Dynamic decoupling between model weights, analyzing the 3D output PDF dynamic model to meet the conditions of natural constraints;

二、在步骤一的基础上为了便于设计控制器，对上述动态模型进行变换，利用实际系统中采集到的输入输出数据通过递归最小二乘算法建立三维输出PDF的输入输出模型；2. On the basis of step 1, in order to facilitate the design of the controller, the above dynamic model is transformed, and the input and output data collected in the actual system are used to establish the input and output model of the three-dimensional output PDF through the recursive least squares algorithm;

三、在步骤二的基础上选用瞬时平方根性能指标设计控制器，通过调整控制器得到的控制作用，达到系统输出PDF分布形状跟踪给定分布输出PDF分布的形状。3. On the basis of step 2, select the instantaneous square root performance index to design the controller, and adjust the control effect obtained by the controller to achieve the output PDF distribution shape of the system to track the given distribution and output the shape of the PDF distribution.

具体分为：Specifically divided into:

1、二维B样条函数表示方法1. Two-dimensional B-spline function representation method

二维B样条函数用两个一维B样条函数的张量积表示：A two-dimensional B-spline function is represented by the tensor product of two one-dimensional B-spline functions:

其中，计算公式由如下递推公式得到：in, The calculation formula is obtained by the following recursive formula:

${B B}_{11,, {i i}_{x x}} ((x x)) = = \{\begin{matrix} 11,, & x x &Element; &Element; [[{x x}_{{i i}_{x x} + + 11},, {x x}_{{i i}_{x x} + + 11})) \\ 00,, & x x &NotElement; &NotElement; [[{x x}_{{i i}_{x x}},, {x x}_{{i i}_{x x} + + 11})) \end{matrix} - - - - - - ((22))$

${B B}_{{j j}_{x x},, {i i}_{x x}} ((x x)) = = \frac{x x - - {x x}_{{i i}_{x x}}}{{x x}_{{i i}_{x x} + + {j j}_{x x} - - 11} - - {x x}_{{i i}_{x x}}} {B B}_{{j j}_{x x} - - 11,, {i i}_{x x}} ((x x)) + + \frac{{x x}_{{i i}_{x x} + + {j j}_{x x}} - - x x}{{x x}_{{i i}_{x x} + + {j j}_{x x}} - - {x x}_{{i i}_{x x} + + 11}} {B B}_{{j j}_{x x} - - 11,, {i i}_{x x} + + 11} ((x x)) - - - - - - ((33))$

${B B}_{11,, {i i}_{r r}} ((r r)) = = \{\begin{matrix} 11,, & r r &Element; &Element; [[{r r}_{{i i}_{r r} + + 11},, {r r}_{{i i}_{r r} + + 11})) \\ 00,, & r r &NotElement; &NotElement; [[{r r}_{{i i}_{r r}},, {r r}_{{i i}_{r r} + + 11})) \end{matrix} - - - - - - ((44))$

${B B}_{{j j}_{r r},, {i i}_{r r}} ((r r)) = = \frac{r r - - {r r}_{{i i}_{r r}}}{{r r}_{{i i}_{r r} + + {j j}_{r r} - - 11} - - {r r}_{{i i}_{r r}}} {B B}_{{j j}_{r r} - - 11,, {i i}_{r r}} ((r r)) + + \frac{{r r}_{{i i}_{r r} + + {j j}_{r r}} - - r r}{{r r}_{{i i}_{r r} + + {j j}_{r r}} - - {r r}_{{i i}_{r r} + + 11}} {B B}_{{j j}_{r r} - - 11,, {i i}_{r r} + + 11} ((r r)) - - - - - - ((55))$

其中，B_j,i(x,r)为二维B样条基函数；为一维B样条基函数；为一维B样条基函数；x,r分别为在空间上定义的变量，x∈[a₁,b₁]、r∈[a₂,b₂]；a₁为设定区间内设定的下限值；b₁为设定区间内设定的上限值；[a₁,b₁]为包括a₁和b₁的一个区间；a₂为设定区间内设定的下限值；b₂为设定区间内设定的上限值；[a₂,b₂]为包括a₂和b₂的一个区间；j表示二维B样条的阶次；i表示二维B样条基函数个数；j_x为X轴上选取的基函数的阶次；i_x为X轴上选取的基函数的个数；j_r为R轴上选取的基函数的阶次；i_r为R轴上选取的基函数的个数；Among them, B _j,i (x, r) is a two-dimensional B-spline basis function; is a one-dimensional B-spline basis function; is a one-dimensional B-spline basis function; x, r are variables defined in space, x∈[a ₁ ,b ₁ ], r∈[a ₂ ,b ₂ ]; a ₁ is set in the set interval The lower limit value of b ₁ is the upper limit value set in the set interval; [a ₁ , b ₁ ] is an interval including a ₁ and b ₁ ; a ₂ is the lower limit value set in the set interval ; b ₂ is the upper limit value set in the set interval; [a ₂ , b ₂ ] is an interval including a ₂ and b ₂ ; j represents the order of the two-dimensional B-spline; i represents the two-dimensional B-spline The number of basis functions; j _x is the order of the basis functions selected on the X-axis; i _x is the number of basis functions selected on the X-axis; j _r is the order of the basis functions selected on the R-axis; i _r is the number of basis functions selected on the R axis;

为1阶第i_x个B样条函数；为j_x-1阶第i_x个B样条函数；为j_x-1阶第i_x+1个B样条基函数；为节点值，且有m_x为区间[a₁,b₁]内的有效节点数，j_x-1为区间左右两侧的外节点数；为包括和的一个区间； is the i _xth B-spline function of order 1; is the i _xth B-spline function of order j _x -1; is the i _x +1th B-spline basis function of order j _x -1; is the node value, and has m _x is the number of effective nodes in the interval [a ₁ , b ₁ ], j _x -1 is the number of outer nodes on the left and right sides of the interval; to include and an interval of

为1阶第i_r个B样条函数；为j_r-1阶第i_r个B样条函数；为j_r-1阶第i_r+1个B样条基函数；为节点值，且有m_r为区间[a₂,b₂]内的有效节点数，j_r-1为区间左右两侧的外节点数；为包括和的一个区间； is the i _rth B-spline function of order 1; is the i _rth B-spline function of order j _r -1; is the i _r +1th B-spline basis function of order j _r -1; is the node value, and has m _r is the number of effective nodes in the interval [a ₂ , b ₂ ], j _r -1 is the number of outer nodes on the left and right sides of the interval; to include and an interval of

2、构建三维输出PDF的瞬时平方根模型2. Construct the instantaneous square root model of the 3D output PDF

平方根模型即用二维B样条函数逼近系统输出PDF的平方根，以保证随机系统的输出PDF在控制过程中的非负性。The square root model uses a two-dimensional B-spline function to approximate the square root of the output PDF of the system to ensure the non-negativity of the output PDF of the stochastic system in the control process.

三维输出PDF的瞬时平方根模型的离散形式表示为：The discrete form of the instantaneous square root model of the 3D output PDF is expressed as:

$\sqrt{γ γ ((x x,, r r,, {u u}_{k k}))} = = {Σ Σ}_{i i = = 11}^{n no} {ω ω}_{i i} (({u u}_{k k})) {B B}_{i i} ((x x,, r r)) + + {e e}_{00} - - - - - - ((66))$

其中，in,

γ(x,r,u_k)为输出概率密度函数；γ(x,r,u _k ) is the output probability density function;

B_i(x,r)为二维B样条函数，其中，省略了B样条的阶次j；B _i (x, r) is a two-dimensional B-spline function, where the order j of the B-spline is omitted;

ω_i(u_k)为依赖于u_k的权值；u_k为k时刻对应的控制作用。ω _i (u _k ) is the weight dependent on u _k ; u _k is the corresponding control action at time k.

e₀为系统的近似误差；e ₀ is the approximate error of the system;

通常情况下忽略e₀。则三维输出PDF的瞬时平方根模型表示为：Normally e ₀ is ignored. Then the instantaneous square root model of the three-dimensional output PDF is expressed as:

$\sqrt{γ γ ((x x,, r r,, {u u}_{k k}))} = = {Σ Σ}_{i i = = 11}^{n no} {ω ω}_{i i} (({u u}_{k k})) {B B}_{i i} ((x x,, r r)) - - - - - - ((77))$

对于给定三维输出PDF函数，式(7)是唯一的，取x∈[a₁,b₁]r∈[a₂,b₂]为随机变量取值范围，根据(7)式将三维输出PDF的瞬时平方根模型进一步表示为：For a given three-dimensional output PDF function, formula (7) is unique, taking x∈[a ₁ ,b ₁ ]r∈[a ₂ ,b ₂ ] as the value range of random variables, according to formula (7) the three-dimensional output The instantaneous square root model of PDF is further expressed as:

$\sqrt{γ γ ((x x,, r r,, {u u}_{k k}))} = = {C C}_{00} ((x x,, r r)) {V V}_{k k} + + {B B}_{n no} ((x x,, r r)) {ω ω}_{n no} (({V V}_{k k})) - - - - - - ((88))$

其中，in,

C₀(x,r)＝[B₁(x,r),B₂(x,r),…,B_n-1(x,r)]，其中，C₀(x,r)为1×(n-1)维基函数的变换向量，B_n-1(x,r)为第n-1个基函数；V_k＝[ω₁(u_k),ω₂(u_k),…,ω_n-1(u_k)]^T，其中，V_k为(n-1)×1维权值向量，ω_n-1(u_k)为第n-1个基函数对应的权值；[ω₁(u_k),ω₂(u_k),…,ω_n-1(u_k)]^T为[ω₁(u_k),ω₂(u_k),…,ω_n-1(u_k)]的转置矩阵；ω_n(V_k)为第n个权值对应的权值；B_n(x,r)为第n个基函数。C ₀ (x,r)=[B ₁ (x,r), B ₂ (x,r),…,B _n-1 (x,r)], where C ₀ (x,r) is 1× (n-1) The transformation vector of the wiki function, B _n-1 (x, r) is the n-1th basis function; V _k ＝[ω ₁ (u _k ),ω ₂ (u _k ),…,ω _n-1 (u _k )] ^T , where V _k is the (n-1)×1-dimensional weight vector, and ω _n-1 (u _k ) is the weight corresponding to the n-1th basis function; [ω ₁ (u _k ),ω ₂ (u _k ),…,ω _n-1 (u _k )] ^T is [ω ₁ (u _k ),ω ₂ (u _k ),…,ω _n-1 (u _k ) ] transpose matrix; ω _n (V _k ) is the weight corresponding to the nth weight; B _n (x, r) is the nth basis function.

3、加入权值的动态变化部分构成基于平方根B样条模型的三维输出PDF动态模型3. The dynamic change part of the weight is added to form a three-dimensional output PDF dynamic model based on the square root B-spline model

上述设计的三维输出PDF平方根模型没有涉及权值变化，很多情况下，输出PDF与输入之间是动态关系。一般假设V_k与控制输入u_k之间是线性动态相关的，这里假设权值的动态变化部分表示为：The 3D output PDF square root model designed above does not involve weight changes, and in many cases, the relationship between the output PDF and the input is dynamic. It is generally assumed that there is a linear dynamic relationship between V _k and the control input u _k . Here, it is assumed that the dynamic change part of the weight is expressed as:

V_k＝AV_k-1+Bu_k-1 (9)V _k = A V _k-1 + Bu _k-1 (9)

式(9)中，A为表示系统动态关系的(n-1)×(n-1)维参数矩阵,B为表示系统动态关系的(n-1)×1维参数矩阵；V_k-1为k-1时刻对应的n-1维权值向量，u_k-1为k-1时刻对应的控制量。In formula (9), A is a (n-1)×(n-1)-dimensional parameter matrix representing the dynamic relationship of the system, and B is a (n-1)×1-dimensional parameter matrix representing the dynamic relationship of the system; V _k-1 is the n-1 dimension weight vector corresponding to k-1 time, u _k-1 is the control amount corresponding to k-1 time.

于是，基于平方根B样条模型的三维输出PDF动态模型表示为：Therefore, the dynamic model of the 3D output PDF based on the square root B-spline model is expressed as:

V_k＝AV_k-1+Bu_k-1 (10)V _k = A V _k-1 + Bu _k-1 (10)

$\sqrt{γ γ ((x x,, r r,, {u u}_{k k}))} = = {C C}_{00} ((x x,, r r)) {V V}_{k k} + + {B B}_{n no} ((x x,, r r)) {ω ω}_{n no} (({V V}_{k k})) - - - - - - ((1111))$

4、三维输出PDF动态模型权值之间的动态解耦；4. Dynamic decoupling between the weights of the 3D output PDF dynamic model;

由(10)、(11)式得出基函数对应的权值之间是非线性关系，为了解决这个问题，需做如下变换，以实现权值的动态解耦。From equations (10) and (11), it can be concluded that the weights corresponding to the basis functions have a nonlinear relationship. In order to solve this problem, the following transformation is required to realize the dynamic decoupling of the weights.

将式(11)两边同乘以[C₀ ^T(x，r)B_n(x，r)]^T可得：Multiply both sides of formula (11) by [C ₀ ^T (x, r)B _n (x, r)] ^T to get:

$[\begin{matrix} {C C}_{00}^{T T} ((x x,, r r)) \\ {B B}_{n no} ((x x,, r r)) \end{matrix}] \sqrt{γ γ ((x x,, r r,, {u u}_{k k}))} = = [\begin{matrix} {C C}_{00}^{T T} ((x x,, r r)) {C C}_{00} ((x x,, r r)) & {C C}_{00}^{T T} ((x x,, r r)) {B B}_{n no} ((x x,, r r)) \\ {B B}_{n no} ((x x,, r r)) {C C}_{00} ((x x,, r r)) & {B B}_{n no}^{22} ((x x,, r r)) \end{matrix}] [\begin{matrix} {V V}_{k k} \\ {ω ω}_{n no} (({V V}_{k k})) \end{matrix}] - - - - - - ((1212))$

取x∈[a₁,b₁]r∈[a₂,b₂]为随机变量取值范围，上式两边积分可得：Taking x∈[a ₁ ,b ₁ ]r∈[a ₂ ,b ₂ ] as the value range of the random variable, the integral on both sides of the above formula can be obtained:

$[\begin{matrix} {C C}_{11} \\ {C C}_{22} \end{matrix}] = = Q Q [\begin{matrix} {V V}_{k k} \\ {ω ω}_{n no} (({V V}_{k k})) \end{matrix}] - - - - - - ((1313))$

其中，in,

当取得的B样条正交时，式(13)中Q矩阵的逆总是存在，式(13)表示为：When the obtained B-splines are orthogonal, the inverse of the Q matrix in formula (13) always exists, and formula (13) is expressed as:

$[\begin{matrix} {V V}_{k k} \\ {ω ω}_{n no} (({V V}_{k k})) \end{matrix}] = = {Q Q}^{- - 11} [\begin{matrix} {C C}_{11} \\ {C C}_{22} \end{matrix}] - - - - - - ((1414))$

式(14)实现了实现ω_n(V_k)和V_k的动态解耦。Equation (14) realizes the dynamic decoupling of ω _n (V _k ) and V _k .

4、三维输出PDF动态模型满足自然约束条件推导如下：4. The 3D output PDF dynamic model satisfies the natural constraints and is deduced as follows:

因为γ(x,r,u_k)为输出概率密度函数，则在其定义域范围内应满足积分为一的约束：Since γ(x,r,u _k ) is the output probability density function, it should satisfy the constraint that the integral is one within its domain:

${&Integral; &Integral;}_{{a a}_{11}}^{{b b}_{22}} {&Integral; &Integral;}_{{a a}_{11}}^{{b b}_{11}} γ γ ((x x,, r r,, {u u}_{k k})) dxdr wxya = = {&Integral; &Integral;}_{{a a}_{22}}^{{b b}_{22}} {&Integral; &Integral;}_{{a a}_{11}}^{{b b}_{11}} {((\sqrt{γ γ ((x x,, r r,, {u u}_{k k}))}))}^{22} dxdr wxya = = 11 - - - - - - ((1515))$

根据式(15)推出：According to formula (15), it is deduced that:

V_k ^T∑₀V_k+2∑₁V_kω_n(u_k)+∑₂ω_n ²(u_k)＝1 (16)V _k ^T ∑ ₀ V _k +2∑ ₁ V _k ω _n (u _k )+∑ ₂ ω _n ² (u _k )＝1 (16)

解式(16)可得Solve equation (16) to get

${ω ω}_{n no} (({u u}_{k k})) = = \frac{11}{{Σ Σ}_{22}} ((- - {Σ Σ}_{11} &PlusMinus; &PlusMinus; \sqrt{{V V}_{k k}^{T T} {Σ Σ}_{11}^{T T} {Σ Σ}_{11} {V V}_{k k} - - {V V}_{k k}^{T T} {Σ Σ}_{00} {Σ Σ}_{22} {V V}_{k k} - - {Σ Σ}_{22}})) - - - - - - ((1717))$

由上式得出ω_n(u_k)与V_k之间是非线性关系，并且ω_n(u_k)能用其它n-1个自由权值表示，记ω_n(u_k)＝h(V_k)。为了保证上式有解，应该满足下式：It can be concluded from the above formula that there is a nonlinear relationship between ω _n (u _k ) and V _k , and ω _n (u _k ) can be expressed by other n-1 free weights, record ω _n (u _k )=h(V _k ). In order to ensure that the above formula has a solution, the following formula should be satisfied:

V_k ^T∑₁ ^T∑₁ ^TV_k-V_k ^T∑₀∑₂V_k-∑₂≥0 (18)V _k ^T ∑ ₁ ^T ∑ ₁ ^T V _k -V _k ^T ∑ ₀ ∑ ₂ V _k - ∑ ₂ ≥0 (18)

化简式(18)可得如下非线性约束Simplify Equation (18) to obtain the following nonlinear constraints

||V_k||_∑≤1 (19)||V _k || _∑ ≤1 (19)

其中，V_k为(n-1)×1维权值向量；||V_k||_∑＝V_k ^T∑V_k，表示V_k的∑范数；∑为∑₀、∑₁和∑₂的变换向量。Among them, V _k is a (n-1)×1-dimensional weight vector; ||V _k || _∑ ＝V _k ^T ∑V _k , indicating the ∑ norm of V _k ; ∑ is the transformation vector of ∑ ₀ , ∑ ₁ and ∑ ₂ .

计算求得的系统权值只要满足式(19)所具备的条件，就能满足输出PDF在其定义域内积分为1的自然约束条件。As long as the calculated system weight satisfies the conditions of formula (19), it can satisfy the natural constraint that the output PDF integrates to 1 within its domain of definition.

通过上述推导得到三维输出PDF动态模型满足自然约束具备的条件为：Through the above derivation, the conditions for the 3D output PDF dynamic model to satisfy the natural constraints are as follows:

||V_k||_∑≤1||V _k || _∑ ≤1

5、三维输出PDF的输入输出模型5. Input and output model of 3D output PDF

上述(9)式表示的权值动态关系在实际系统中不容易得到，于是，需要对建立的基于平方根B样条模型的三维输出PDF动态模型进行一下变换，将(10)(11)式表示的动态模型表示成输入输出形式。令The weight dynamic relationship represented by the above formula (9) is not easy to obtain in the actual system, so it is necessary to transform the established 3D output PDF dynamic model based on the square root B-spline model, and express the formulas (10) and (11) The dynamic model of is expressed in the form of input and output. make

$f f ((x x,, r r,, {u u}_{k k})) = = \sqrt{γ γ ((x x,, r r,, {u u}_{k k}))} - - {B B}_{n no} ((x x,, r r)) h h (({V V}_{k k})) = = {C C}_{00} ((x x,, r r)) {V V}_{k k} - - - - - - ((2020))$

其中，f(x,r,u_k)为具有输出特性的等价输出函数。对权值动态方程(10)引入位移算子z^-1，将式(20)改写为：Among them, f(x,r,u _k ) is an equivalent output function with output characteristics. Introduce the displacement operator z ^-1 to the weight dynamic equation (10), and rewrite the equation (20) as:

f(x,r,u_k)＝C₀(x,r)(I-Az^-1)^-1Bu_k-1 (21)f(x,r,u _k )＝C ₀ (x,r)(I-Az ^-1 ) ^-1 Bu _k-1 (21)

然后根据(I-Az^-1)^-1B的展开式将上式化简，即得到三维输出PDF的输入输出模型：Then, according to the expansion of (I-Az ^-1 ) ^-1 B, the above formula is simplified, that is, the input and output model of the three-dimensional output PDF is obtained:

$f f ((x x,, r r,, {u u}_{k k})) = = {Σ Σ}_{i i = = 11}^{n no - - 11} {a a}_{i i} f f ((x x,, r r,, {u u}_{k k - - i i})) + + {Σ Σ}_{j j = = 00}^{n no - - 22} {C C}_{00} ((x x,, r r)) {D D.}_{j j} {u u}_{k k - - j j - - 11} - - - - - - ((22 twenty two))$

其中，I为(n-1)×(n-1)为单位矩阵；n为选取的基函数个数；f(x,r,u_k-i)为k-i时刻对应的等价输出函数；u_k-i为k-i时刻对应的控制作用；u_k-j-1为k-j-1时刻对应的控制作用；a_i为k-i时刻对应的f(x,r,u_k-i)的系数，D_j＝[d_j1,…,d_ji,…,d_j(n-1)]^T为需要辨识的参数，d_ji为与C₀(x,r)中的项相对应的系数。Among them, I is (n-1)×(n-1) as the unit matrix; n is the number of selected basis functions; f(x,r,u _ki ) is the equivalent output function corresponding to the moment ki; u _ki is The control action corresponding to time ki; u _kj-1 is the control action corresponding to time kj-1; a _i is the coefficient of f(x,r,u _ki ) corresponding to time ki, D _j =[d _j1 ,…,d _ji ,…,d _j(n-1) ] ^T is the parameter to be identified, and d _ji is the coefficient corresponding to the term in C ₀ (x,r).

D_j和a_i都为(22)式中的未知量，根据(22)式的形式采用递归最小二乘算法辨识三维输出PDF的输入输出模型中的未知参数。辨识过程如下：Both D _j and a _i are unknown quantities in (22), and the recursive least squares algorithm is used to identify the unknown parameters in the input-output model of the 3D output PDF according to the form of (22). The identification process is as follows:

定义definition

θ＝[a₁,…,a_n-1,d₀₁,…,d_0(n-1),d₁₁,…，d_1(n-1),…,d_(n-2)1,…,d_(n-2)(n-1)]^T (23)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-1C₀₁(x,r),…,u_k-1C_0(n-1)(x,r),…,φ(x,r,k)=[f(x,r,k-1),…,f(x,r,k-n+1),u _k-1 C ₀₁ (x,r),…, u _k-1 C _0(n-1) (x,r),…,

u_k-n+1C₀₁(x,r),…,u_0(k-n+1)C_0(n-1)(x,r)]^T (24)u _k-n+1 C ₀₁ (x,r),…,u _0(k-n+1) C _0(n-1) (x,r)] ^T (24)

其中，θ为需要辨识的权值；φ(x,r,k)为已知量，当实际系统中采集到的输入输出数据确定时。Among them, θ is the weight value that needs to be identified; φ(x, r, k) is a known quantity, when the input and output data collected in the actual system are determined.

取x∈[a₁,b₁]r∈[a₂,b₂]为随机变量取值范围，在其定义域范围内分别选择N_x和N_r个采样点来组成f(x_i,r_j,k)：Take x∈[a ₁ ,b ₁ ]r∈[a ₂ ,b ₂ ] as the value range of the random variable, and select N _x and N _r sampling points in its domain to form f(x _i ,r _j ,k):

f(x_i,r_j,k)＝θ^Tφ(x_i,r_j,k) (25)f(x _i ,r _j ,k)=θ ^T φ(x _i ,r _j ,k) (25)

x_i,r_j为X轴和R轴的采样点，i＝1,2,…,N_x，j＝1,2,…,N_r。x _i , r _j are the sampling points of the X axis and the R axis, i=1, 2,..., N _x , j=1, 2,..., N _r .

递归最小二乘算法定义如下：The recursive least squares algorithm is defined as follows:

$θ θ ((i i + + 11,, j j + + 11)) = = θ θ ((i i,, j j)) + + \frac{P P ((i i,, j j)) φ φ (({x x}_{i i},, {r r}_{j j},, k k)) ϵ ϵ ((i i,, j j))}{11 + + {φ φ}^{T T} (({x x}_{i i},, {r r}_{j j},, k k)) P P ((i i,, j j)) φ φ (({x x}_{i i},, {r r}_{j j},, k k))} - - - - - - ((2626))$

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

$P P ((i i + + 11,, j j + + 11)) = = ((I I - - \frac{P P ((i i,, j j)) φ φ (({x x}_{i i},, {r r}_{j j},, k k)) {φ φ}^{T T} (({x x}_{i i},, {r r}_{j j},, k k))}{11 + + {φ φ}^{T T} (({x x}_{i i},, {r r}_{j j},, k k)) P P ((i i,, j j)) φ φ (({x x}_{i i},, {r r}_{j j},, k k))})) P P ((i i,, j j)) - - - - - - ((2828))$

其中，P(i,j)为辨识参数时所需变换矩阵；ε(i,j)为辨识误差；P(1,1)＝10^3-6I_n(n-1)为初始矩阵，其中，I_n(n-1)为n×(n-1)为单位矩阵；θ(1,1)＝θ₀为初始权值向量。Among them, P(i,j) is the transformation matrix required for identifying parameters; ε(i,j) is the identification error; P(1,1)=10 ^3-6 I _n(n-1) is the initial matrix, where , I _n(n-1) is n×(n-1) is the identity matrix; θ(1,1)=θ ₀ is the initial weight vector.

三维输出PDF的输入输出模型参数辨识步骤如下：The steps to identify the input and output model parameters of the 3D output PDF are as follows:

(1)选择合适的二维B样条基函数，计算基函数B_i(x,r)(i＝1,2,…,n)的值。(1) Select an appropriate two-dimensional B-spline basis function, and calculate the value of the basis function B _i (x, r) (i=1, 2, . . . , n).

(2)在采样时刻k(k≥n)，收集系统控制输入{u_k-1,…,u_k-n+1}和定义域范围内采样点处γ(x_i,r_j,u_k-1),…,γ(x_i,r_j,u_k-n+1)的值；(2) At the sampling time k(k≥n), collect system control input {u _k-1 ,…,u _k-n+1 } and γ( _xi ,r _j ,u _{k -1} ),…,γ(x _i ,r _j ,u _k-n+1 );

(3)按式(14)计算h(V_k-1),…,h(V_k-n-1)值，按定义计算f(x_i,r_j,u_k-1),…,f(x_i,r_j,u_k-n+1)和φ(x_i,r_j,k)的值；(3) Calculate h(V _k-1 ),…,h(V _kn-1 ) values according to formula (14), and calculate f(x _i ,r _j ,u _k-1 ),…,f(x _i ,r _j ,u _k-n+1 ) and φ(x _i ,r _j ,k);

(4)根据式(26)-(28)，估计参数θ，记θ(N_x,N_r)为采样时刻k的估计值；(4) According to equations (26)-(28), estimate the parameter θ, denote θ(N _x , N _r ) as the estimated value of sampling time k;

(5)如果k小于N，则k增加1，转向第二步。(5) If k is less than N, increase k by 1 and turn to the second step.

辨识后，θ为已知量，则得到三维输出PDF的输入输出模型：After identification, θ is a known quantity, and the input-output model of the three-dimensional output PDF is obtained:

6、瞬时优化跟踪控制器设计6. Instantaneous optimal tracking controller design

根据上面建立的三维输出PDF的输入输出模型，即可进行相应的控制器设计。设计控制器的目的是选择合适的控制输入使系统实际的输出PDF分布形状尽可能的逼近期望PDF分布形状，考虑建立的模型，假设模型的权值满足约束条件，于是选择平方根二次型性能指标：According to the input and output model of the three-dimensional output PDF established above, the corresponding controller can be designed. The purpose of designing the controller is to select the appropriate control input to make the actual output PDF distribution shape of the system as close as possible to the expected PDF distribution shape. Considering the established model, assuming that the weight of the model satisfies the constraints, the square root quadratic performance index is selected :

$J J = = {&Integral; &Integral;}_{{a a}_{11}}^{{b b}_{22}} {&Integral; &Integral;}_{{a a}_{11}}^{{b b}_{11}} {((\sqrt{γ γ ((x x,, r r,, {u u}_{k k + + 11}))} - - \sqrt{g g ((x x,, r r))}))}^{22} dxdr wxya + + {Ru Ru}_{k k}^{22} - - - - - - ((2929))$

根据式(20)(22)上式改写为：According to formula (20) (22), the above formula is rewritten as:

$J J (({u u}_{k k})) = = {&Integral; &Integral;}_{{a a}_{22}}^{{b b}_{22}} {&Integral; &Integral;}_{{a a}_{11}}^{{b b}_{11}} {(({C C}_{00} ((x x,, r r)) {D D.}_{00} {u u}_{k k} + + \overset{^^}{g g} ((x x,, r r))))}^{22} dxdr wxya + + {Ru Ru}_{k k}^{22} - - - - - - ((3030))$

其中：in:

$\overset{^^}{g g} ((x x,, r r)) = = {Σ Σ}_{i i = = 22}^{n no - - 11} (({a a}_{i i} f f ((x x,, r r,, k k - - i i + + 11)) + + {C C}_{00} ((x x,, r r)) {D D.}_{i i - - 11} {u u}_{k k - - i i + + 11})) + + {a a}_{11} f f ((x x,, r r,, k k)) + + {B B}_{n no} ((x x,, r r)) h h (({V V}_{k k})) - - \sqrt{g g ((x x,, r r))}$

展开式(30)得：Expansion (30) gets:

$J J (({u u}_{k k})) = = {u u}_{k k}^{22} {&Integral; &Integral;}_{{a a}_{22}}^{{b b}_{22}} {&Integral; &Integral;}_{{a a}_{11}}^{{b b}_{11}} {(({C C}_{00} ((x x,, r r)) {D D.}_{00}))}^{22} dxdr wxya + + {22 u u}_{k k} {&Integral; &Integral;}_{{a a}_{22}}^{{b b}_{22}} {&Integral; &Integral;}_{{a a}_{11}}^{{b b}_{11}} {C C}_{00} ((x x,, r r)) {D D.}_{00} \overset{^^}{g g} ((x x,, r r)) dxdr wxya + + {&Integral; &Integral;}_{{a a}_{22}}^{{b b}_{22}} {&Integral; &Integral;}_{{a a}_{11}}^{{b b}_{11}} \overset{^^}{g g} {((x x,, r r))}^{22} dxdr wxya + + {Ru Ru}_{k k}^{22} - - - - - - ((3131))$

采用优化控制的算法，对u_k求偏导，令则得到：Using the algorithm of optimal control, the partial derivative of u _k is obtained, so that then get:

${u u}_{k k} = = - - \frac{{&Integral; &Integral;}_{{a a}_{22}}^{{b b}_{22}} {&Integral; &Integral;}_{{a a}_{11}}^{{b b}_{11}} (({C C}_{00} ((x x,, r r)) {D D.}_{00} \overset{^^}{g g} ((x x,, r r)))) dxdr wxya}{{&Integral; &Integral;}_{{a a}_{22}}^{{b b}_{22}} {&Integral; &Integral;}_{{a a}_{11}}^{{b b}_{11}} {(({C C}_{00} ((x x,, r r)) {D D.}_{00}))}^{22} dxdr wxya + + R R} - - - - - - ((3232))$

自此，在本发明中，一种三维输出概率密度函数的动态建模与控制器设计方法完毕。Since then, in the present invention, a method for dynamic modeling and controller design of a three-dimensional output probability density function has been completed.

实施例如下：Examples are as follows:

由于实验条件有限，不易得到实际系统的输入和输出数据，假设系统的动态向量A、B已知，下面构造基于平方根B样条模型的三维输出PDF动态模型为：Due to the limited experimental conditions, it is difficult to obtain the input and output data of the actual system. Assuming that the dynamic vectors A and B of the system are known, the three-dimensional output PDF dynamic model based on the square root B-spline model is constructed as follows:

V_k＝AV_k-1+Bu_k-1 V _k ＝AV _k-1 +Bu _k-1

$\sqrt{γ γ ((x x,, r r,, {u u}_{k k}))} = = {C C}_{00} ((x x,, r r)) {V V}_{k k} + + {B B}_{n no} ((x x,, r r)) h h (({V V}_{k k}))$

其中in

$A A = = (\begin{matrix} 0.991 0.991 & 00 & 00 & 00 & 00 \\ 00 & 0.994 0.994 & 00 & 00 & 00 \\ 0.704 0.704 & 0.056 0.056 & 0.887 0.887 & 00 & 00 \\ 00 & 00 & 00 & 0.947 0.947 & 00 \\ 00 & 00 & 00 & 0.057 0.057 & 0.976 0.976 \end{matrix});;$

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

根据二维B样条基函数的定义，这里选择一维B样条基函数构造二维B样条函数。假设x,r的取值范围为x∈[0,1]r∈[0,1]，x和r轴上的基函数定义如下：According to the definition of the two-dimensional B-spline basis function, the one-dimensional B-spline basis function is selected here to construct the two-dimensional B-spline function. Assuming that the value range of x and r is x∈[0,1]r∈[0,1], the basis functions on the x and r axes are defined as follows:

(1)X轴上的基函数：(1) Basis functions on the X-axis:

B₁(x)＝xI_x1+(2-x)I_x2 B ₁ (x)=xI _x1 +(2-x)I _x2

B₂(x)＝(x-1)I_x2+(3-x)I_x3 B ₂ (x)=(x-1)I _x2 +(3-x)I _x3

B₃(x)＝(x-2)I_x3+(4-x)I_x4 B ₃ (x)=(x-2)I _x3 +(4-x)I _x4

其中 in

(2)R轴上的基函数(2) Basis functions on the R axis

${B B}_{11} ((r r)) = = \frac{11}{22} {r r}^{22} {I I}_{r r 11} + + ((- - {r r}^{22} + + 33 r r - - \frac{33}{22})) {I I}_{r r 22} + + \frac{11}{22} {((r r - - 33))}^{22} {I I}_{r r 33}$

${B B}_{22} ((r r)) = = \frac{11}{22} {((r r - - 11))}^{22} {I I}_{r r 22} + + ((- - {r r}^{22} + + 55 r r - - \frac{1111}{22})) {I I}_{r r 33} + + \frac{11}{22} {((r r - - 44))}^{22} {I I}_{r r 44}$

其中 in

控制器设计需要的已知量为：The known quantities required for controller design are:

初始权值V₀＝[0.688 2.129 1.551 0.166 0.792]^T；Initial weight V ₀ =[0.688 2.129 1.551 0.166 0.792] ^T ;

输入约束因子为R＝0.0005；The input constraint factor is R=0.0005;

控制作用取值范围为u∈[0,1]；The value range of the control action is u∈[0,1];

初始控制作用为u₀＝0.3；The initial control effect is u ₀ =0.3;

期望输出PDF对应的控制输入为u＝0.65；The control input corresponding to the expected output PDF is u=0.65;

这里选的二维B样条基函数如图1所示。图2，图3分别给出了三维线性系统的初始输出PDF分布和期望输出PDF分布图像，图4为跟踪给定输出分布的系统输出PDF响应曲面，最后时刻期望输出PDF与控制输出PDF的跟踪误差如图5所示，图6给出的是控制过程中控制输入作用的响应曲线，由图得到控制输入能够收敛并接近于期望输入。图7为控制过程中性能指标的变化曲线。The two-dimensional B-spline basis function selected here is shown in Figure 1. Figure 2 and Figure 3 show the initial output PDF distribution and expected output PDF distribution images of the three-dimensional linear system respectively, and Figure 4 shows the system output PDF response surface tracking a given output distribution, and the tracking of the expected output PDF and the control output PDF at the last moment The error is shown in Figure 5, and Figure 6 shows the response curve of the control input during the control process. From the figure, the control input can converge and be close to the desired input. Figure 7 is the change curve of the performance index in the control process.

Claims

1. a dynamic modeling and controller design method of three-dimensional output probability density function, it is characterized in that, the method comprises the following steps:

Step 1: Construct a 3D output PDF dynamic model based on the square root B-spline model;

The three-dimensional output PDF dynamic model based on the square root B-spline model comprises the following steps:

Step S1: constructing an instantaneous square root B-spline model of the three-dimensional output PDF according to the two-dimensional B-spline function;

The two-dimensional B-spline function is represented by the tensor product of two one-dimensional B-spline functions as follows:

{B B}_{j j,, i i} ((x x,, r r)) = = {B B}_{{j j}_{x x},, {i i}_{x x}} ((x x)) {B B}_{{j j}_{r r},, {i i}_{r r}} ((r r))

in, It is obtained by the following recursive formula:

{B B}_{11,, {i i}_{x x}} ((x x)) = = \{\begin{matrix} 11,, & x x &Element; &Element; [[{x x}_{{i i}_{x x}},, {x x}_{{i i}_{x x} + + 11})) \\ 00,, & x x &NotElement; &NotElement; [[{x x}_{{i i}_{x x}},, {x x}_{{i i}_{x x} + + 11})) \end{matrix}

{B B}_{{j j}_{x x},, {i i}_{x x}} ((x x)) = = \frac{x x - - {x x}_{{i i}_{x x}}}{{x x}_{{i i}_{x x} + + {j j}_{x x} - - 11} - - {x x}_{{i i}_{x x}}} {B B}_{{j j}_{x x} - - 11,, {i i}_{x x}} ((x x)) + + \frac{{x x}_{{i i}_{x x} + + {j j}_{x x}} - - x x}{{x x}_{{i i}_{x x} + + {j j}_{x x}} - - {x x}_{{i i}_{x x} + + 11}} {B B}_{{j j}_{x x} - - 11,, {i i}_{x x} + + 11} ((x x))

It is obtained by the following recursive formula:

{B B}_{11,, {i i}_{r r}} ((r r)) = = \{\begin{matrix} 11,, & r r &Element; &Element; [[{r r}_{{i i}_{r r}},, {r r}_{{i i}_{r r} + + 11})) \\ 00,, & r r &NotElement; &NotElement; [[{r r}_{{i i}_{r r}},, {r r}_{{i i}_{r r} + + 11})) \end{matrix}

{B B}_{{j j}_{r r},, {i i}_{r r}} ((r r)) = = \frac{r r - - {r r}_{{i i}_{r r}}}{{r r}_{{i i}_{r r} + + {j j}_{r r} - - 11} - - {r r}_{{i i}_{r r}}} {B B}_{{j j}_{r r} - - 11,, {i i}_{r r}} ((r r)) + + \frac{{r r}_{{i i}_{r r} + + {j j}_{r r}} - - r r}{{r r}_{{i i}_{r r} + + {j j}_{r r}} - - {r r}_{{i i}_{r r} + + 11}} {B B}_{{j j}_{r r} - - 11,, {i i}_{r r} + + 11} ((r r))

Among them, B _j,i (x, r) is a two-dimensional B-spline basis function; is a one-dimensional B-spline basis function; is a one-dimensional B-spline basis function; x, r are variables defined in space, x∈[a ₁ ,b ₁ ], r∈[a ₂ ,b ₂ ]; a ₁ is the X-axis setting interval The lower limit value of the setting; b ₁ is the upper limit value set in the X-axis setting interval; [a ₁ ,b ₁ ] is an interval including a ₁ and b ₁ ; a ₂ is in the R-axis setting interval The lower limit value set; b ₂ is the upper limit value set in the R-axis setting interval; [a ₂ ,b ₂ ] is an interval including a ₂ and b ₂ ; j represents the order of the two-dimensional B-spline i represents the number of two-dimensional B-spline basis functions; j _x is the order of the basis functions selected on the X-axis; i _x is the number of basis functions selected on the X-axis; j _r is the order of the basis functions selected on the R-axis The order of the basis function; i _r is the number of basis functions selected on the R axis;

is the i _xth B-spline function of order 1; is the i _xth B-spline function of order j _x -1; is the i _x +1th B-spline basis function of order j _x -1; is the node value, where i _x ＝-j _x +1,…,m _x +j _x , j _x >1, and

x_{j_{x} + 1} < . . . < a_{1} = x_{0} < . . . < x_{m_{x} + 1} = b_{1} < . . . < x_{m_{x} + j_{x}};

m _x is the number of effective nodes in the interval [a ₁ , b ₁ ], j _x -1 is the number of outer nodes on the left and right sides of the interval; to include but does not include an interval of

is the i _rth B-spline function of order 1; is the i _rth B-spline function of order j _r -1; is the i _r +1th B-spline basis function of order j _r -1; is the node value, where i _r ＝-j _r +1,…,m _r +j _r , j _r >1, and

r_{j_{r} + 1} < . . . < a_{2} = r_{0} < . . . < r_{m_{r} + 1} = b_{2} < . . . < r_{m_{r} + j_{r}};

m _r is the number of effective nodes in the interval [a ₂ , b ₂ ], j _r -1 is the number of outer nodes on the left and right sides of the interval; to include but does not include an interval of

The two-dimensional B-spline function B _j,i (x,r), omitting the order j of the B-spline, that is, the two-dimensional B-spline function B _j,i (x,r) is recorded as B _i (x,r );

The instantaneous square root B-spline model of the three-dimensional output PDF based on the two-dimensional square root B-spline function is:

\sqrt{γ γ ((x x,, r r,, {u u}_{k k}))} = = {C C}_{00} ((x x,, r r)) {V V}_{k k} + + {B B}_{n no} ((x x,, r r)) {ω ω}_{n no} (({V V}_{k k}))

in,

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

n is the number of selected two-dimensional B-spline functions, and k is the sampling time;

C ₀ (x,r)=[B ₁ (x,r), B ₂ (x,r),…,B _n-1 (x,r)], where C ₀ (x,r) is 1× (n-1) wiki function transformation vector;

B _i (x, r) is a two-dimensional B-spline function;

V _k ＝[ω ₁ (u _k ),ω ₂ (u _k ),…,ω _n-1 (u _k )] ^T , where V _k is the (n-1)×1-dimensional weight vector corresponding to time k ;

ω _i (u _k ) is the weight dependent on u _k , and u _k is the corresponding control effect at time k;

B _n (x, r) is a two-dimensional B-spline function;

ω _n (V _k ) is the weight corresponding to the nth basis function;

Step S2: On the basis of step S1, adding the dynamic change part of the weight value to obtain a three-dimensional output PDF dynamic model based on the square root B-spline model;

Assume that the dynamic part of the added weight is:

V _k ＝AV _k-1 +Bu _k-1

Among them, A is a (n-1)×(n-1)-dimensional parameter matrix representing the dynamic relationship of the system, B is a (n-1)×1-dimensional parameter matrix representing the dynamic relationship of the system; V _k-1 is k-1 The n-1 dimension weight vector corresponding to the moment, u _k-1 is the control amount corresponding to the k-1 moment;

The 3D output PDF dynamic model based on the square root B-spline model is:

V _k ＝AV _k-1 +Bu _k-1

\sqrt{γ γ ((x x,, r r,, {u u}_{k k}))} = = {C C}_{00} ((x x,, r r)) {V V}_{k k} + + {B B}_{n no} ((x x,, r r)) {ω ω}_{n no} (({V V}_{k k}))

Step S3: There is a nonlinear relationship between the weight vector V _k of the system and the weight corresponding to the nth basis function, and the dynamic decoupling between the weights of the three-dimensional output PDF dynamic model is obtained through analysis;

The decoupling formula between the weights of the three-dimensional output PDF dynamic model is:

[\begin{matrix} {V V}_{k k} \\ {ω ω}_{n no} (({V V}_{k k})) \end{matrix}] = = {Q Q}^{- - 11} [\begin{matrix} {C C}_{11} \\ {C C}_{22} \end{matrix}]

in,

Among them, C ₁ is the integral of the root mean square of the output probability density function and the product of the basis function transformation matrix at time k;

Among them, C ₂ is the integral of the root mean square of the output probability density function at time k and the nth two-dimensional basis function B _n (x, r);

Q = [\begin{matrix} Σ_{0} & {Σ_{1}}^{T} \\ Σ_{1} & Σ_{2} \end{matrix}],

Among them, Q is the transformation form of Σ ₀ , Σ ₁ , and Σ _2. When the basis function is selected and the input and output data of the actual system are known, Q is a known quantity;

Among them, Σ ₀ is the integral of the square value of the basis function transformation vector C ₀ (x, r) in its domain;

Among them, Σ ₁ is the integral of the product of the basis function transformation vector C ₀ (x, r) and the nth basis function within its domain of definition;

Among them, Σ ₂ is the integral of the square of the nth basis function B _n (x, r) in its domain;

a ₁ is the lower limit value set in the X-axis setting interval; b ₁ is the upper limit value set in the X-axis setting interval;

a ₂ is the lower limit value set in the R-axis setting interval; b ₂ is the upper limit value set in the R-axis setting interval;

Σ ₁ ^T is the transposition matrix of Σ ₁ ;

Step S4: Analyze the three-dimensional output PDF dynamic model to meet the conditions of natural constraints:

||V _k || _Σ ≤ 1

Where, ||V _k || _Σ = V _k ^T ΣV _k ,

Σ = Σ_{0} - \frac{{Σ_{1}}^{T} Σ_{1}}{Σ_{2}};

In the above two formulas, V _k is the n-1 dimension weight vector corresponding to time k; V _k ^T is the transposition matrix of V _k ; Σ is the transformation vector of Σ ₀ , Σ ₁ and Σ ₂ ;

Step 2: Use the input and output data collected in the actual system to establish the input and output model of the three-dimensional output PDF through the recursive least squares algorithm;

The input and output model of the established 3D output PDF is:

f f ((x x,, r r,, {u u}_{k k})) = = {Σ Σ}_{i i = = 11}^{n no - - 11} {a a}_{i i} f f ((x x,, r r,, {u u}_{k k - - i i})) + + {Σ Σ}_{j j = = 00}^{n no - - 22} {C C}_{00} ((x x,, r r)) {D D.}_{j j} {u u}_{k k - - j j - - 11}

in,

f (x, r, u_{k}) = \sqrt{γ (x, r, u_{k})} - B_{no} (x, r) ω_{no} (V_{k});

In the above two formulas, f(x,r,u _k ) is the transformed form of the output probability density function corresponding to time k; a _i is the coefficient of f(x,r,u _ki ) corresponding to time ki; f(x, r,u _ki ) is the transformed form of the output probability density function corresponding to time ki; u _ki is the control action corresponding to time ki; u _kj-1 is the control action corresponding to time kj-1; D _j ＝[d _j1 ,… ,d _ji ,…,d _j(n-1) ] ^T is the parameter to be identified; d _ji is the coefficient corresponding to the item in C ₀ (x,r);

Step 3: Select the instantaneous square root performance index to design the controller, and design the control quantity of the controller by optimizing the instantaneous square root performance index, so as to realize the shape of the system output PDF distribution tracking the given distribution and output the shape of the PDF distribution;

The instantaneous square root performance index of the selection is:

J J = = {&Integral; &Integral;}_{{a a}_{22}}^{{b b}_{22}} {&Integral; &Integral;}_{{a a}_{11}}^{{b b}_{11}} {((\sqrt{γ γ ((x x,, r r,, {u u}_{k k + + 11}))} - - \sqrt{g g ((x x,, r r))}))}^{22} dxdr wxya + + {Ru Ru}_{k k}^{22}

Among them, J is the instantaneous square root performance index value; γ(x,r,u _k+1 ) is the three-dimensional output probability density function; g(x,r) is the given three-dimensional output PDF distribution function; R is the constraint constant of the control effect ; a ₁ is the lower limit value set in the X-axis setting interval; b ₁ is the upper limit value set in the X-axis setting interval; a ₂ is the lower limit value set in the R-axis setting interval; b ₂ is the upper limit value set in the R-axis setting interval;

The control quantity obtained by optimizing the instantaneous square root performance index is as follows:

{u u}_{k k} = = - - \frac{{&Integral; &Integral;}_{{a a}_{22}}^{{b b}_{22}} {&Integral; &Integral;}_{{a a}_{11}}^{{b b}_{11}} (({C C}_{00} ((x x,, r r)) {D D.}_{00} \overset{^^}{g g} ((x x,, r r)))) dxdr wxya}{{&Integral; &Integral;}_{{a a}_{22}}^{{b b}_{22}} {&Integral; &Integral;}_{{a a}_{11}}^{{b b}_{11}} {(({C C}_{00} ((x x,, r r)) {D D.}_{00}))}^{22} dxdr wxya + + R R}

in,

\overset{^^}{g g} ((x x,, r r)) = = {Σ Σ}_{i i = = 22}^{n no - - 11} (({a a}_{i i} f f ((x x,, r r,, k k - - i i + + 11)) + + {C C}_{00} ((x x,, r r)) {D D.}_{i i - - 11} {u u}_{k k - - i i + + 11})) + + {a a}_{11} f f ((x x,, r r,, k k)) + + {B B}_{n no} ((x x,, r r)) {ω ω}_{n no} (({V V}_{k k})) - - \sqrt{g g ((x x,, r r))};;

in, is the transformation form of known quantities and parameters; f(x,r,k-i+1) is the transformation form of the output probability density function corresponding to k-i+1 moment; D ₀ is the identified parameter value;

The control quantity u _k is passed to In f(x,r,k),f(x,r,k-1),…,f(x,r,k-n+2),ω _n (V _k ) and u _k-1 ,u _{k -2} ,...,u _k-n+2 values are adjusted to realize the system output PDF distribution shape tracking the given output PDF distribution shape.