CN103336427B - A kind of dynamic modeling of three-dimensional output probability density function and controller design method - Google Patents
A kind of dynamic modeling of three-dimensional output probability density function and controller design method Download PDFInfo
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Abstract
The invention discloses dynamic modeling and the controller design method of a kind of three-dimensional output probability density function in the theoretical field of stochastic distribution control.The method comprises the following steps: step 1: the three-dimensional built based on square root B-spline model exports PDF dynamic model; Step 2: utilize the inputoutput data collected in real system to set up the three-dimensional input/output model exporting PDF by recursive least squares; Step 3: select instantaneous square root performance index CONTROLLER DESIGN, by the controlled quentity controlled variable of optimization instantaneous square root performance index CONTROLLER DESIGN, the system that realizes exports the shape that PDF distribution shape tracing preset exports PDF distribution; The present invention devises optimization routine controller, by optimizing square root performance index, realizes exporting the tracking of PDF distribution to given output PDF.The present invention has enriched three-dimensional output PDF control theory, provides new method for having the three-dimensional industrial process exporting distribution character.
Description
Technical Field
The invention belongs to the field of random distribution control theory, and particularly relates to a dynamic modeling and controller design method of a three-dimensional output probability density function.
Background
The stochastic system control theory is one of the important branches of control theory and application, mainly because most industrial processes are interfered by stochastic signals, and the stochastic control theory of the system is formed aiming at the practical problem. Early research results focused on statistical characteristics of control system variables, the control targets were mean and variance of the system, and in the existing methods, most of the random variables in the random system were assumed to obey gaussian distribution, however, this assumption did not meet practical applications, such as fiber length distribution in the paper making process, grain particle distribution in grain processing, and boiler flame temperature distribution. The general random system distribution is represented by an output PDF (probability density function) probability density function, when a random variable is a Gaussian variable, the output PDF of the system can be controlled by controlling the mean value and the variance of the system, for a system with the random variable not meeting the Gaussian distribution, the mean value and the variance of the system cannot contain all information of the system, and the control of the mean value and the variance of the system cannot control the output PDF of the system. For such systems, wang macro professor proposed in 1998 a method of directly controlling the system output PDF shape, i.e., output PDF control. This type of approach directly designs the controller to have the system output PDF distribution shape track the given PDF distribution shape.
The output PDF can be approximated through a B spline neural network, so that the decoupling of the original complex coupling system is realized to a certain extent, and the decoupled output PDF control is called SDC (statistical dependence control) random system distribution control. The method breaks through the limitation of random control research, and converts the traditional random control problem into a method for establishing a simple and reasonable model and designing a high-efficiency and proper control algorithm. The method converts the partial differential equation which implicitly describes the dynamic characteristics of the system into the decoupled state space model description, and finally achieves the purpose that the dynamic behavior of the random distribution system can be described by using a more accurate model similarly to a determination system. SDC is more realistic than the conventional stochastic control theory, and therefore, the theory is integrated into a new industrial field and provides strong vitality to the industrial field.
A relatively perfect theoretical system has been established for a random distribution system with two-dimensional characteristics. For example, in the aspect of system modeling, a linear B-spline model, a rational B-spline model, a square root B-spline model, a rational square root B-spline model, an input/output ARMAX model, a neural network PDF model and the like are established. In the aspect of controller design, an instantaneous optimal tracking control algorithm, an optimal tracking control algorithm, a model reference adaptive control algorithm, a prediction control algorithm, a structured controller algorithm, an iterative learning control algorithm and the like are realized. In recent years, researchers have done a lot of work on robust control of random distribution control, minimum entropy control, fault diagnosis, filter design, and the like.
In summary, great progress has been made in the control of two-dimensional output distribution, but a class of three-dimensional output distribution problems also exist in the actual industrial process, such as three-dimensional temperature field representing the flame temperature of the boiler, three-dimensional distribution of material concentration in the coal-fired circulating fluidized bed boiler of the power plant, and the like. The three-dimensional distribution is closely related to the operation condition of the industrial process, and the method has important application value in the aspects of improving the production efficiency, reducing the environmental pollution and the like of the whole industrial process. With the rapid development of computer image processing and sensor technology, the detection of three-dimensional output distribution conditions has been rapidly developed. However, the method of obtaining the output distribution of the system by online measurement using advanced technology is complicated to implement and requires expensive equipment.
The invention relates to a three-dimensional output PDF control problem which is an important component of SDC theory, however, the research on the Modeling and control problems of the three-dimensional PDF is still incomplete, the article 'Modeling and control soft magnetic temperature distribution using basic reliability mapping' carries out static Modeling and controller design on the three-dimensional output PDF, selects a two-dimensional B-spline basis function and establishes a static model of the three-dimensional output PDF through a least square algorithm, optimizes secondary performance indexes, obtains local optimal control input of a system by a gradient method, and obtains a reasonable result through computer simulation research. So far, studies on the control of three-dimensional output distribution of dynamic processes are rarely published, but no study report on the control of three-dimensional output distribution is directly reported.
In order to further improve the three-dimensional output PDF control theory and enable the realization of the three-dimensional random distribution control problem to be possible, the invention firstly establishes an instantaneous square root B spline model of the three-dimensional output PDF, adds a 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, realizes the dynamic decoupling between the weights and analyzes the condition that the three-dimensional output PDF dynamic model meets the natural constraint; then, a three-dimensional output PDF input and output model is established through a recursive least square algorithm according to system input and output data; and finally, selecting an instantaneous square root performance index, and designing a conventional optimal controller. The invention perfects the PDF control theory of three-dimensional output and provides a new method and thought for controlling the three-dimensional output distribution problem.
Disclosure of Invention
The invention provides a dynamic modeling and controller design method of a three-dimensional output probability density function, aiming at the needs of three-dimensional output PDF theory to be perfected and the needs of an actual industrial process.
A dynamic modeling and controller design method of a three-dimensional output probability density function comprises the following steps:
step 1: constructing a three-dimensional output PDF dynamic model based on a square root B spline model;
the method for constructing 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 a two-dimensional B spline function;
the two-dimensional B-spline function is represented by the tensor product of the two one-dimensional B-spline functions as follows:
wherein,the following recursion formula yields:
the following recursion formula yields:
wherein, Bj,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 and r are respectively variables defined in space, and x belongs to [ a ]1,b1]、r∈[a2,b2]；a1Setting a lower limit value in a section for the X axis; b1Setting an upper limit value in a section for the X axis; [ a ] A1,b1]To comprise a1And b1An interval of (a); a is2Setting a lower limit value in a range for the R axis; b2Setting an upper limit value in a section for the R axis; [ a ] A2,b2]To comprise a2And b2An interval of (a); j represents the order of a two-dimensional B-spline; i represents the number of two-dimensional B-spline basis functions; j is a function ofxThe order of the selected basis function on the X axis; i.e. ixThe number of basis functions selected on the X axis; j is a function ofrThe order of the selected basis function on the R axis; i.e. irThe number of basis functions selected on the R axis;
is 1 st order ixB spline functions;is jx1 st order ixB spline functions;is jx1 st order ix+ 1B-spline basis functions;is a node value and hasmxIs the interval [ a1,b1]Number of valid nodes in, jx-1 is the number of outer nodes on the left and right sides of the interval;to compriseBut do not compriseAn interval of (a);
is 1 st order irB spline functions;is jr1 st order irB spline functions;is jr1 st order ir+ 1B-spline basis functions;is a node value and hasmrIs the interval [ a2,b2]Number of valid nodes in, jr-1 is the number of outer nodes on the left and right sides of the interval;to compriseBut do not compriseAn interval of (a);
two-dimensional B spline function Bj,i(x, r), omitting the order j of the B-spline, i.e. the two-dimensional B-spline function Bj,i(x, r) is denoted as Bi(x,r)；
The instantaneous square root B-spline model of the three-dimensional output PDF obtained based on the two-dimensional square root B-spline function is as follows:
wherein,
γ(x,r,uk) Outputting a probability density function in three dimensions;
n is the number of the selected two-dimensional B-spline functions, and k is the sampling time;
C0(x,r)＝[B1(x,r),B2(x,r),…,Bn-1(x,r)]wherein, C0(x, r) is a 1 × (n-1) dimensional basis function transform vector;
Bi(x, r) is a two-dimensional B-spline function;
Vk＝[ω1(uk),ω2(uk),…,ωn-1(uk)]Twherein V iskThe vector is a (n-1) multiplied by 1-dimensional weight value vector corresponding to the k time;
ωi(uk) To be dependent on ukWeight of ukThe control action corresponding to the k moment.
Bn(x, r) is a two-dimensional B-spline function;
ωn(Vk) The weight value corresponding to the nth basis function.
Step S2: on the basis of the step S1, adding a dynamic change part of the weight to obtain a three-dimensional output PDF dynamic model based on a square root B spline model;
the weight dynamic part assumed to be added is:
Vk＝AVk-1+Buk-1
wherein, A is an (n-1) x (n-1) dimensional parameter matrix representing the dynamic relationship of the system, and B is an (n-1) x 1 dimensional parameter matrix representing the dynamic relationship of the system; vk-1Is a corresponding n-1 dimensional weight value vector u at the k-1 timek-1The control quantity corresponding to the k-1 moment;
the three-dimensional output PDF dynamic model based on the square root B-spline model is as follows:
step S3: weight vector V of systemkThe weight corresponding to the nth basis function has a nonlinear relation, and dynamic decoupling between the three-dimensional output PDF dynamic model weights is obtained through analysis;
the decoupling formula between the three-dimensional output PDF dynamic model weights is as follows:
wherein,
wherein, C1Outputting the integral of the product of the probability density function root mean square and the basis function transformation matrix for the moment k;
wherein, C2Outputting a probability density function root mean square and an nth two-dimensional basis function B for k timen(x, r);
Therein, sigma0Transforming vector C for basis function0(x, r) is the integral of the squared value of (x, r) within its domain of definition;
therein, sigma1Transforming vector C for basis function0(x, r) the integral of the product of the nth basis function over its domain of definition;
therein, sigma2Is the nth basis function Bn(x, r) the integral of the square over its domain of definition;
a1setting a lower limit value in a section for the X axis; b1Setting an upper limit value in a section for the X axis;
a2setting a lower limit value in a range for the R axis; b2Setting an upper limit value in a section for the R axis;
∑1 Tis sigma1The transposed matrix of (2).
Step S4: the conditions for analyzing the three-dimensional output PDF dynamic model to meet the natural constraint are as follows:
||Vk||∑≤1
wherein, | | Vk||∑＝Vk T∑Vk，
In the above two formulae, VkIs an n-1 dimensional weight value vector corresponding to the k moment; vk TIs a VkThe transposed matrix of (2); sigma is sigma0、∑1Sum Σ2The transformed vector of (2);
step 2: establishing an input/output model of three-dimensional output PDF by using input/output data acquired in an actual system through a recursive least square algorithm;
the built input and output model of the three-dimensional output PDF is as follows:
wherein, <math>
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in the above two formulae, f (x, r, u)k) The transformation form of the output probability density function corresponding to the k moment; a isiF (x, r, u) corresponding to the k-i timek-i) The coefficient of (a); f (x, r, u)k-i) The transformation form of the output probability density function corresponding to the k-i moment; u. ofk-iThe control function corresponding to the k-i moment; u. ofk-j-1The corresponding control action at the moment k-j-1 is taken; dj＝[dj1,…,dji,…,dj(n-1)]TThe parameters to be identified; djiIs and C0The terms in (x, r) correspond to coefficients.
And step 3: and selecting an instantaneous square root performance index design controller, and realizing that the system output PDF distribution shape tracks the shape of the given distribution output PDF distribution by optimizing the control quantity of the instantaneous square root performance index design controller.
The selected transient square root performance indicator is:
wherein J is an instantaneous square root performance indicator value; gamma (x, r, u)k+1) Outputting a probability density function in three dimensions; g (x, r) is given three-dimensional outputA PDF distribution function; r is a constraint constant of control action; a is1Setting a lower limit value in a section for the X axis; b1Setting an upper limit value in a section for the X axis; a is2Setting a lower limit value in a range for the R axis; b2And setting an upper limit value set in the interval for the R axis.
The control quantity obtained by optimizing the instantaneous square root performance index is as follows:
wherein,
wherein,in the form of a transformation of known quantities and parameters; f (x, r, k-i +1) is a transformation form of an output probability density function corresponding to the k-i +1 moment; d0Is the identified parameter value.
The control quantity ukIs formed byWhere f (x, r, k), f (x, r, k-1), … f (x, r, k-n +2), ωn(Vk) And uk-1,uk-2,…,uk-n+2And adjusting the value to realize that the shape of the system output PDF distribution tracks the shape of the given output PDF distribution.
The invention has the beneficial effects that: 1. the invention guarantees the constraint condition that the system output PDF is more than 1, and analyzes the constraint condition that the weight value should meet when the integral of the system output PDF in the definition domain is 1; 2. according to the method, a dynamic model of three-dimensional output PDF is established according to a square root B spline model, then the established dynamic model based on the square root B spline function is transformed, and an input and output model of a system is established according to collected input and output data; 3. the invention designs a conventional optimization controller, obtains the control function of the system by optimizing the square root performance index, and realizes the tracking of the output PDF distribution shape to the 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 the three-dimensional output distribution characteristic.
Drawings
FIG. 1 is a two-dimensional B-spline image;
FIG. 2 is an initial PDF distribution for a three-dimensional dynamic system;
FIG. 3 is a given output PDF distribution for a three-dimensional dynamical system;
FIG. 4 is an output PDF response surface of a three-dimensional dynamical system;
FIG. 5 is a diagram of the last moment control output PDF versus a given PDF tracking error;
FIG. 6 is a response curve of a control amount in a control process;
FIG. 7 is a graph of performance index variation during control;
fig. 8 is an overall flow chart of the present invention.
Detailed Description
For the purpose of promoting an understanding of the invention, reference will now be made in detail to the embodiments of the invention illustrated in the accompanying drawings. It should be emphasized that the following description is merely exemplary in nature and is not intended to limit the scope of the invention or its application.
For practical industrial process needs, the output PDF control theory is applied to a system with three-dimensional distribution characteristics in order to simplify the complexity caused by establishing a system model and controller design by using a mechanistic approach. The invention provides a dynamic modeling and controller design method of a three-dimensional output probability density function. For enabling tracking of the overall output PDF distribution shape.
The invention comprises the following steps:
firstly, constructing an instantaneous square root B spline model of three-dimensional output PDF, adding a dynamic change part of a 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, realizing dynamic decoupling between the weights of the three-dimensional output PDF dynamic model, and analyzing that the three-dimensional output PDF dynamic model meets the conditions of natural constraints;
secondly, in order to design a controller conveniently on the basis of the first step, the dynamic model is transformed, and an input/output model of three-dimensional output PDF is established by using input/output data acquired in an actual system through a recursive least square algorithm;
and thirdly, selecting an instantaneous square root performance index to design a controller on the basis of the step two, and achieving the purpose that the distribution shape of the system output PDF tracks the distribution shape of the given distribution output PDF by adjusting the control function obtained by the controller.
The method comprises the following specific steps:
1. two-dimensional B-spline function representation method
The two-dimensional B-spline function is represented by the tensor product of two one-dimensional B-splines:
wherein,the calculation formula is obtained by the following recursion formulaTo:
the following recursion formula yields:
wherein, Bj,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 and r are respectively variables defined in space, and x belongs to [ a ]1,b1]、r∈[a2,b2]；a1A lower limit value set in a set interval; b1Setting an upper limit value in a set interval; [ a ] A1,b1]To comprise a1And b1An interval of (a); a is2A lower limit value set in a set interval; b2Setting an upper limit value in a set interval; [ a ] A2,b2]To comprise a2And b2An interval of (a); j represents the order of a two-dimensional B-spline; i represents the number of two-dimensional B-spline basis functions; j is a function ofxThe order of the selected basis function on the X axis; i.e. ixThe number of basis functions selected on the X axis; j is a function ofrThe order of the selected basis function on the R axis; i.e. irThe number of basis functions selected on the R axis;
is 1 st order ixB spline functions;is jx1 st order ixB spline functions;is jx1 st order ix+ 1B-spline basis functions;is a node value and hasmxIs the interval [ a1,b1]Number of valid nodes in, jx-1 is the number of outer nodes on the left and right sides of the interval;to compriseAndan interval of (a);
is 1 st order irB spline functions;is jr1 st order irB spline functions;is jr1 st order ir+ 1B-spline basis functions;is a node value and hasmrIs the interval [ a2,b2]Number of valid nodes in, jr-1 is the number of outer nodes on the left and right sides of the interval;to compriseAndan interval of (a);
2. constructing transient square root model of three-dimensional output PDF
The square root model is to approximate the square root of the system output PDF by using a two-dimensional B-spline function so as to ensure that the output PDF of the stochastic system is non-negative in the control process.
The discrete form of the instantaneous square root model of the three-dimensional output PDF is represented as:
wherein,
γ(x,r,uk) Is an output probability density function;
n is the number of the selected two-dimensional B-spline functions, and k is the sampling time;
Bi(x, r) is a two-dimensional B-spline function, wherein the order j of the B-spline is omitted;
ωi(uk) To be dependent on ukThe weight of (2); u. ofkThe control action corresponding to the k moment.
e0Is the approximate error of the system;
normally ignore e0. The instantaneous square root model of the three-dimensional output PDF is then expressed as:
for a given three-dimensional output PDF function, equation (7) is unique, taking x ∈ [ a ]1,b1]r∈[a2,b2]For the range of random variable values, the transient square root model of the three-dimensional output PDF is further expressed according to equation (7) as:
wherein,
C0(x,r)＝[B1(x,r),B2(x,r),…,Bn-1(x,r)]wherein, C0(x, r) is a transformation vector of 1 × (n-1) -dimensional basis function, Bn-1(x, r) is the (n-1) th basis function; vk＝[ω1(uk),ω2(uk),…,ωn-1(uk)]TWherein V iskIs a (n-1) × 1-dimensional weight vector, ωn-1(uk) The weight value corresponding to the (n-1) th basis function; [ omega ]1(uk),ω2(uk),…,ωn-1(uk)]TIs [ omega ]1(uk),ω2(uk),…,ωn-1(uk)]The transposed matrix of (2); omegan(Vk) The weight value corresponding to the nth weight value; b isn(x, r) is the nth basis function.
3. The dynamic change part added with the weight forms a three-dimensional output PDF dynamic model based on a square root B spline model
The three-dimensional output PDF square root model designed above does not involve weight change, and in many cases, the output PDF and the input are in a dynamic relationship. General assumption VkAnd a control input ukThe linear dynamic correlation is adopted, and the dynamic change part of the weight value is assumed to be expressed as:
Vk＝AVk-1+Buk-1 (9)
in the formula (9), A is an (n-1) x (n-1) dimensional parameter matrix representing the dynamic relationship of the system, and B is an (n-1) x 1 dimensional parameter matrix representing the dynamic relationship of the system; vk-1Is a corresponding n-1 dimensional weight value vector u at the k-1 timek-1Is the control quantity corresponding to the time k-1.
Thus, the three-dimensional output PDF dynamical model based on the square root B-spline model is represented as:
Vk＝AVk-1+Buk-1 (10)
4. dynamically decoupling the three-dimensional output PDF dynamic model weights;
the nonlinear relation between the weights corresponding to the basis functions is obtained by the formulas (10) and (11), and in order to solve the problem, the following transformation is needed to realize the dynamic decoupling of the weights.
Multiplying both sides of formula (11) by [ C0 T(x，r)Bn(x，r)]TThe following can be obtained:
take x ∈ [ a ]1,b1]r∈[a2,b2]For the random variable value range, the integral on both sides of the above equation can be obtained:
wherein,
wherein, C1Outputting the integral of the product of the probability density function root mean square and the basis function transformation matrix for the moment k;
wherein, C2Outputting a probability density function root mean square and an nth two-dimensional basis function B for k timen(x, r);
Therein, sigma0Transforming vector C for basis function0The square value of (x, r) is inIt defines the integral within the domain;
therein, sigma1Transforming vector C for basis function0(x, r) the integral of the product of the nth basis function over its domain of definition;
therein, sigma2Is the nth basis function Bn(x, r) the integral of the square over its domain of definition;
a1setting a lower limit value in a section for the X axis; b1Setting an upper limit value in a section for the X axis;
a2setting a lower limit value in a range for the R axis; b2Setting an upper limit value in a section for the R axis;
∑1 Tis sigma1The transposed matrix of (2).
When the acquired B-splines are orthogonal, the inverse of the Q matrix in equation (13) always exists, equation (13) is expressed as:
the formula (14) realizes the realization of omegan(Vk) And VkDynamic decoupling of (3).
4. The derivation that the three-dimensional output PDF dynamic model meets the natural constraint condition is as follows:
because of γ (x, r, u)k) To output the probability density function, the constraint of integral to one should be satisfied within its domain:
derived from equation (15):
Vk T∑0Vk+2∑1Vkωn(uk)+∑2ωn 2(uk)＝1 (16)
can be obtained by solving formula (16)
From the above formula, ωn(uk) And VkIs a non-linear relationship between, and ωn(uk) Can be represented by other n-1 free weight values, and is recorded as omegan(uk)＝h(Vk). To ensure that the above equation has a solution, the following equation should be satisfied:
Vk T∑1 T∑1 TVk-Vk T∑0∑2Vk-∑2≥0 (18)
simplifying the equation (18) yields the following nonlinear constraints
||Vk||∑≤1 (19)
Wherein, VkIs a (n-1) x 1-dimensional weight vector; i Vk||∑＝Vk T∑VkIs denoted by VkSigma norm of (d);sigma is sigma0、∑1Sum Σ2The transformed vector of (2).
The system weight obtained by calculation satisfies the natural constraint condition that the integral of the output PDF in the domain of its definition is 1, as long as the condition of expression (19) is satisfied.
The conditions that the three-dimensional output PDF dynamic model obtained by the derivation meets the natural constraint are as follows:
||Vk||∑≤1
5. input/output model of three-dimensional output PDF
Since the weight dynamics relationship expressed by the above expression (9) is not easily obtained in an actual system, it is necessary to perform a down-conversion on the established three-dimensional output PDF dynamic model based on the square root B-spline model to express the dynamic model expressed by the expressions (10) and (11) in an input/output format. Order to
Wherein, f (x, r, u)k) Is an equivalent output function with output characteristics. Introducing a displacement operator z into the weight dynamic equation (10)-1The formula (20) is rewritten as:
f(x,r,uk)＝C0(x,r)(I-Az-1)-1Buk-1 (21)
then according to (I-Az)-1)-1And B, simplifying the above formula by using an expansion formula to obtain an input and output model of the three-dimensional output PDF:
wherein I is (n-1) x (n-1) is an identity matrix; n is the number of selected basis functions; f (x, r, u)k-i) Equivalent output functions corresponding to the k-i moments; u. ofk-iThe control function corresponding to the k-i moment; u. ofk-j-1The corresponding control action at the moment k-j-1 is taken; a isiF (x, r, u) corresponding to the k-i timek-i) Coefficient of (D)j＝[dj1,…,dji,…,dj(n-1)]TFor the parameters to be identified, djiIs and C0The terms in (x, r) correspond to coefficients.
DjAnd aiAre all unknown quantities in the formula (22), and the unknown parameters in the input-output model of the three-dimensional output PDF are identified by adopting a recursive least square algorithm according to the form of the formula (22). The identification process is as follows:
definition of
θ＝[a1,…,an-1,d01,…,d0(n-1),d11,…，d1(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),uk-1C01(x,r),…,uk-1C0(n-1)(x,r),…,
uk-n+1C01(x,r),…,u0(k-n+1)C0(n-1)(x,r)]T (24)
Wherein, theta is a weight value to be identified; phi (x, r, k) is a known quantity, when the input and output data collected in the actual system is determined.
Take x ∈ [ a ]1,b1]r∈[a2,b2]Selecting N in the range of the definition domain for the value range of the random variablexAnd NrA number of sampling points to form f (x)i,rj,k)：
f(xi,rj,k)＝θTφ(xi,rj,k) (25)
xi,rjSample points for the X and R axes, i ═ 1,2, …, Nx，j＝1,2,…,Nr。
The recursive least squares algorithm is defined as follows:
(i,j)＝f(xi,rj,k)-θT(i,j)φ(xi,rj,k) (27)
wherein, P (i, j) is a transformation matrix required when identifying parameters; (i, j) is the recognition error; p (1,1) ═ 103-6In(n-1)Is an initial matrix, wherein In(n-1)Is n (n-1) as an identity matrix; θ (1,1) ═ θ0Is the initial weight vector.
The identification steps of the input and output model parameters of the three-dimensional output PDF are as follows:
(1) selecting proper two-dimensional B-spline basis function and calculating basis function Bi(x, r) (i ═ 1,2, …, n).
(2) At sampling time k (k ≧ n), the collection system control input { u ≧ nk-1,…,uk-n+1And gamma (x) at sampling points within the domaini,rj,uk-1),…,γ(xi,rj,uk-n+1) A value of (d);
(3) calculating h (V) according to equation (14)k-1),…,h(Vk-n-1) Value, by definition, f (x)i,rj,uk-1),…,f(xi,rj,uk-n+1) And phi (x)i,rjThe value of k);
(4) according to the expressions (26) to (28), the parameter theta is estimated, and theta (N) is recordedx,Nr) Is an estimated value of sampling time k;
(5) if k is less than N, k is increased by 1 and the process goes to the second step.
After identification, if θ is a known quantity, an input-output model of the three-dimensional output PDF is obtained:
6. instantaneous optimization tracking controller design
And according to the input and output model of the three-dimensional output PDF established above, corresponding controller design can be carried out. The purpose of designing the controller is to select a proper control input to enable the actual output PDF distribution shape of the system to be close to the expected PDF distribution shape as much as possible, the established model is considered, the weight of the model is assumed to meet the constraint condition, and then the square root quadratic form performance index is selected:
the above formula is rewritten as follows from formulas (20) and (22):
wherein:
the expansion formula (30) is as follows:
using an algorithm for optimal control, for ukCalculating a deviation guide, orderThen the following results are obtained:
therefore, the dynamic modeling and controller design method of the three-dimensional output probability density function is completed in the invention.
The examples are as follows:
because experimental conditions are limited, input and output data of an actual system are not easy to obtain, and assuming that the dynamic vector A, B of the system is known, the following three-dimensional output PDF dynamic model based on a square root B-spline model is constructed as follows:
Vk＝AVk-1+Buk-1
wherein
B＝[0.0209 0.0448 0.0246 0.0292 0.0305]T；
According to the definition of the two-dimensional B-spline base function, the one-dimensional B-spline base function is selected to construct the two-dimensional B-spline function. Assuming that x, r has a value range of x ∈ [0,1] r ∈ [0,1], the basis functions on the x and r axes are defined as follows:
(1) basis function on the X-axis:
B1(x)＝xIx1+(2-x)Ix2
B2(x)＝(x-1)Ix2+(3-x)Ix3
B3(x)＝(x-2)Ix3+(4-x)Ix4
wherein
(2) Basis function on the R-axis
Wherein
Known quantities required for controller design are:
initial weight V0＝[0.688 2.129 1.551 0.166 0.792]T；
The input constraint factor is R0.0005;
the value range of the control action is u belongs to [0,1 ];
the initial control action is u0＝0.3；
The control input corresponding to the desired output PDF is u ═ 0.65;
the two-dimensional B-spline basis function chosen here is shown in FIG. 1. Fig. 2, fig. 3 show the initial output PDF distribution and the expected output PDF distribution images of a three-dimensional linear system, respectively, fig. 4 shows the system output PDF response curved surface for tracking the given output distribution, the tracking error between the final expected output PDF and the control output PDF is shown in fig. 5, fig. 6 shows the response curve of the control input action in the control process, and the control input can be converged and close to the expected input by the graph. Fig. 7 is a graph showing the variation of the performance index during the control process.
Claims (1)
1. A dynamic modeling and controller design method of a three-dimensional output probability density function is characterized by comprising the following steps:
step 1: constructing a three-dimensional output PDF dynamic model based on a square root B spline model;
the method for constructing 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 a two-dimensional B spline function;
the two-dimensional B-spline function is represented by the tensor product of the two one-dimensional B-spline functions as follows:
wherein,the following recursion formula yields:
is obtained by the following recursion formula：
Wherein, Bj,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 and r are respectively variables defined in space, and x belongs to [ a ]1,b1]、r∈[a2,b2]；a1Setting a lower limit value in a section for the X axis; b1Setting an upper limit value in a section for the X axis; [ a ] A1,b1]To comprise a1And b1An interval of (a); a is2Setting a lower limit value in a range for the R axis; b2Setting an upper limit value in a section for the R axis; [ a ] A2,b2]To comprise a2And b2An interval of (a); j represents the order of a two-dimensional B-spline; i represents the number of two-dimensional B-spline basis functions; j is a function ofxThe order of the selected basis function on the X axis; i.e. ixThe number of basis functions selected on the X axis; j is a function ofrThe order of the selected basis function on the R axis; i.e. irThe number of basis functions selected on the R axis;
is 1 st order ixB spline functionCounting;is jx1 st order ixB spline functions;is jx1 st order ix+ 1B-spline basis functions;is a node value, where ix＝-jx+1,…,mx+jx,jx>1, and is provided with mxIs the interval [ a1,b1]Number of valid nodes in, jx-1 is the number of outer nodes on the left and right sides of the interval;to compriseBut do not compriseAn interval of (a);
is 1 st order irB spline functions;is jr1 st order irB spline functions;is jr1 st order ir+ 1B-spline basis functions;is a node value, where ir＝-jr+1,…,mr+jr,jr>1, and is provided with mrIs the interval [ a2,b2]Number of valid nodes in, jr-1 is the number of outer nodes on the left and right sides of the interval;to compriseBut do not compriseAn interval of (a);
two-dimensional B spline function Bj,i(x, r), omitting the order j of the B-spline, i.e. the two-dimensional B-spline function Bj,i(x, r) is denoted as Bi(x,r)；
The instantaneous square root B-spline model of the three-dimensional output PDF obtained based on the two-dimensional square root B-spline function is as follows:
wherein,
γ(x,r,uk) Outputting a probability density function in three dimensions;
n is the number of the selected two-dimensional B-spline functions, and k is the sampling time;
C0(x,r)＝[B1(x,r),B2(x,r),…,Bn-1(x,r)]wherein, C0(x, r) is a 1 × (n-1) dimensional basis function transform vector;
Bi(x, r) is a two-dimensional B-spline function;
Vk＝[ω1(uk),ω2(uk),…,ωn-1(uk)]Twherein V iskThe vector is a (n-1) multiplied by 1-dimensional weight value vector corresponding to the k time;
ωi(uk) To be dependent on ukWeight of ukThe control function corresponding to the k moment;
Bn(x, r) is a two-dimensional B-spline function;
ωn(Vk) The weight value corresponding to the nth basis function;
step S2: on the basis of the step S1, adding a dynamic change part of the weight to obtain a three-dimensional output PDF dynamic model based on a square root B spline model;
the weight dynamic part assumed to be added is:
Vk＝AVk-1+Buk-1
wherein, A is an (n-1) x (n-1) dimensional parameter matrix representing the dynamic relationship of the system, and B is an (n-1) x 1 dimensional parameter matrix representing the dynamic relationship of the system; vk-1Is a corresponding n-1 dimensional weight value vector u at the k-1 timek-1The control quantity corresponding to the k-1 moment;
the three-dimensional output PDF dynamic model based on the square root B-spline model is as follows:
Vk＝AVk-1+Buk-1
step S3: weight vector V of systemkThe weight corresponding to the nth basis function has a nonlinear relation, and dynamic decoupling between the three-dimensional output PDF dynamic model weights is obtained through analysis;
the decoupling formula between the three-dimensional output PDF dynamic model weights is as follows:
wherein,
wherein, C1Outputting the integral of the product of the probability density function root mean square and the basis function transformation matrix for the moment k;
wherein, C2Outputting a probability density function root mean square and an nth two-dimensional basis function B for k timen(x, r);
wherein, sigma0Transforming vector C for basis function0(x, r) is the integral of the squared value of (x, r) within its domain of definition;
wherein, sigma1Transforming vector C for basis function0(x, r) the integral of the product of the nth basis function over its domain of definition;
wherein, sigma2Is the nth basis function Bn(x, r) the integral of the square over its domain of definition;
a1setting a lower limit value in a section for the X axis; b1Setting an upper limit value in a section for the X axis;
a2setting a lower limit value in a range for the R axis; b2Setting an upper limit value in a section for the R axis;
Σ1 Tis sigma1The transposed matrix of (2);
step S4: the conditions for analyzing the three-dimensional output PDF dynamic model to meet the natural constraint are as follows:
||Vk||Σ≤1
wherein, | | Vk||Σ＝Vk TΣVk， <math>
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In the above two formulae, VkIs an n-1 dimensional weight value vector corresponding to the k moment; vk TIs a VkThe transposed matrix of (2); sigma is0、Σ1Sum-sigma2The transformed vector of (2);
step 2: establishing an input/output model of three-dimensional output PDF by using input/output data acquired in an actual system through a recursive least square algorithm;
the built input and output model of the three-dimensional output PDF is as follows:
wherein, <math>
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in the above two formulae, f (x, r, u)k) The transformation form of the output probability density function corresponding to the k moment; a isiF (x, r, u) corresponding to the k-i timek-i) The coefficient of (a); f (x, r, u)k-i) The transformation form of the output probability density function corresponding to the k-i moment; u. ofk-iThe control function corresponding to the k-i moment; u. ofk-j-1The corresponding control action at the moment k-j-1 is taken; dj＝[dj1,…,dji,…,dj(n-1)]TThe parameters to be identified; djiIs and C0Coefficients corresponding to the terms in (x, r);
and step 3: selecting an instantaneous square root performance index design controller, and realizing that the system output PDF distribution shape tracks the shape of the given distribution output PDF distribution by optimizing the control quantity of the instantaneous square root performance index design controller;
the selected transient square root performance indicator is:
wherein J is an instantaneous square root performance indicator value; gamma (x, r, u)k+1) Outputting a probability density function in three dimensions; g (x, r) is a given three-dimensional output PDF distribution function; r is a constraint constant of control action; a is1Setting a lower limit value in a section for the X axis; b1Setting an upper limit value in a section for the X axis; a is2Setting a lower limit value in a range for the R axis; b2Setting an upper limit value in a section for the R axis;
the control quantity obtained by optimizing the instantaneous square root performance index is as follows:
wherein,
wherein,in the form of a transformation of known quantities and parameters; f (x, r, k-i +1) is a transformation form of an output probability density function corresponding to the k-i +1 moment; d0Is the identified parameter value;
the control quantity ukIs formed byWhere f (x, r, k), f (x, r, k-1), …, f (x, r, k-n +2), ωn(Vk) And uk-1,uk-2,…,uk-n+2And adjusting the value to realize that the shape of the system output PDF distribution tracks the shape of the given output PDF distribution.
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