CN107377634A - A kind of hot-strip exports Crown Prediction of Media method - Google Patents
A kind of hot-strip exports Crown Prediction of Media method Download PDFInfo
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- B—PERFORMING OPERATIONS; TRANSPORTING
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- B—PERFORMING OPERATIONS; TRANSPORTING
- B21—MECHANICAL METAL-WORKING WITHOUT ESSENTIALLY REMOVING MATERIAL; PUNCHING METAL
- B21B—ROLLING OF METAL
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Abstract
The hot-strip outlet Crown Prediction of Media method of the present invention includes：Layering Cai Ji be in hot-strip production process strip creation data；Noise reduction process is carried out to creation data；Creation data after noise reduction is divided into training set and test set；Creation data after noise reduction is subjected to dimension-reduction treatment；Input using the normalized matrix after dimensionality reduction as supporting vector machine model, the parameter of supporting vector machine model is optimized using based on the particle swarm optimization algorithm of hybridization；Using best parameter group construction SVMs strip outlet Crown Prediction of Media model；Forecasting model is trained with training set, the Generalization Capability of forecasting model is tested with test set.The forecasting procedure of the present invention determines the optimal parameter of SVMs by Crossbreeding Particle Swarm optimizing, and the precision for the SVMs strip outlet Crown Prediction of Media model for making to be established based on SVMs is improved.Forecasting model is based on a large amount of creation datas, and the collection of creation data is easily operated, and the Generalization Ability of model is stronger.
Description
Technical Field
The invention relates to a hot-rolled strip steel quality control technology, in particular to a hot-rolled strip steel outlet convexity forecasting method.
Background
Strip hot continuous rolling plays a very important role in the steel industry, and approximately half of the total amount of steel in the world comes from hot continuous rolling lines. A hot continuous rolling line includes many precise equipments, a complicated mixing control model and a severe working environment, which all bring difficulties to the improvement of product quality. However, with the development of science and technology, the demand of each department for strip steel is increasing, and the quality of strip steel is also increasing, especially the high requirements for shapes of home appliance steel plates, automobile steel plates, tin-plated steel plates, electrical steel plates and the like are provided. If the section shape of the strip steel is not good, the quality and the service life of a user product are seriously influenced by the appearance of overlarge convexity, local bulge, wedge shape and the like. The hot continuous rolling mill is a dynamic system with nonlinearity, large time lag, multiple variables and strong coupling. There are many factors that affect the strip exit crown, such as: rolling force, roll bending force, roll shape and thermal expansion of the rolls, roll diameter, incoming strip shape, strip width, rolling mill time lag, rolling speed, change in rolling mill rhythm, temperature fluctuation of strips and cooling water, change in rolling mill rolling reduction and the like. It is therefore a difficult task to achieve accurate control of the system. The traditional method is to establish a plate convexity relation model by utilizing a traditional mathematical tool according to a rolling theory and analyze the conditions of deflection, flattening, thermal expansion and the like of a roller in a rolling state. To facilitate modeling, the complexity of the system is simplified, giving many assumptions at the cost of reduced model accuracy. With the increasing requirements of modern manufacturing techniques on the accuracy of strip shapes, the task of improving the accuracy of models or controls becomes very urgent. Therefore, a new method needs to be found to predict and model the rolling mill system more accurately, so as to achieve the purpose of accurately controlling the outlet convexity of the strip steel.
Disclosure of Invention
The embodiment of the invention provides a hot-rolled strip steel outlet convexity prediction method, which optimizes and determines the optimal parameters of a support vector machine through a hybrid particle swarm optimization algorithm, so that the accuracy of a support vector machine strip steel outlet convexity prediction model established based on the support vector machine is improved.
The invention provides a method for forecasting outlet convexity of hot-rolled strip steel, which comprises the following steps:
step 1: respectively acquiring p production data of each strip steel in the production process of the hot rolled strip steel in layers, representing the p production data by using a p-dimensional vector, and dividing the layers according to the steel type, the width of the finish rolled strip steel and the thickness of the finish rolled strip steel;
step 2: noise reduction processing is carried out on the production data of each layer by adopting a statistical 3 sigma principle;
and step 3: dividing the production data subjected to noise reduction into a training set and a test set according to a certain proportion, wherein the consistency of data distribution is kept through set division;
and 4, step 4: forming an observation value matrix by the production data of each layer after noise reduction, and performing standardized transformation and dimension reduction processing on the observation value matrix to obtain a standardized matrix after dimension reduction;
and 5: taking the reduced standardized matrix as the input of the support vector machine model, and optimizing the parameters of the support vector machine model by adopting a particle swarm optimization algorithm based on hybridization;
step 6: constructing a support vector machine strip steel outlet convexity prediction model by adopting the optimized optimal parameter combination;
and 7: training a support vector machine strip steel outlet convexity prediction model by using a training set, and testing the generalization performance of the prediction model by using a test set;
and 8: using a determining coefficient R2And evaluating the overall performance of the support vector machine strip steel outlet convexity prediction model by using the average absolute error MAE, the average absolute percentage error MAPE and the root mean square error RMSE.
The method for forecasting the outlet convexity of the hot-rolled strip steel has the following beneficial effects: the invention adopts an artificial intelligence method to establish a support vector machine strip steel outlet convexity prediction model. The model is based on mass production data, and the acquisition of the production data is easy to operate, so that the popularization capability of the model is strong. In addition, the complex mathematical and physical relationship among variables influencing the outlet convexity of the hot-rolled strip steel is avoided in the model establishing process, and the problems of strong coupling, nonlinearity and the like among all input variables are well solved. By reasonably screening and processing strip steel sample data, the method can effectively forecast the outlet convexity of the hot-rolled strip steel, and lays a foundation for accurate control of the outlet convexity.
Drawings
FIG. 1 is a flow chart of a hot-rolled strip steel outlet convexity prediction method based on a hybrid particle swarm optimization support vector machine;
FIG. 2 is a diagram showing the effect of the production data after being subjected to dimensionality reduction by a principal component analysis method;
FIG. 3 is a diagram of the fitness value and the average fitness variation in the process of optimizing the structure parameters of the support vector machine by the hybrid particle swarm optimization;
FIG. 4 is a diagram of the predicted effect of the strip steel outlet convexity of the model on the training set;
FIG. 5 is a diagram of the predicted effect of the strip outlet convexity of the model on the test set.
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings and examples.
In this embodiment, the final stand rolling data of a 1780 hot continuous rolling mill train is used to predict the strip outlet crown, and the flow of the hot-rolled strip outlet crown prediction method is shown in fig. 1. The forecasting method comprises the following steps:
step 1: p production data of each strip steel in the production process of the hot rolled strip steel are collected in layers and are represented by a p-dimensional vector, and the layers are divided according to the steel type, the width of the finish rolled strip steel and the thickness of the finish rolled strip steel.
In specific implementation, the production data of the last stand of a 1780 hot continuous rolling mill are collected, and the collected production data comprise the inlet temperature T, the outlet temperature T, the inlet thickness H, the outlet thickness H, the strip steel width W and the roll bending force F of each stand in the rolling processBRoll lateral shift S, rolling force FRRoller cooling water flow QmAnd inlet convexity CH. The data acquisition is carried out according to the layers, and the layers are divided according to the steel grades and the specification of the strip steel. The partitioning method is shown in table 1.
Table 1 shows the data acquisition layers in this example.
| Steel grade | Number of samples | Final thickness (mm) | Width of strip steel (mm) |
| SPHC-B | 100 | 4.2 | 1411 |
| SPA-H | 100 | 6.0 | 1199 |
| 65Mn-YT | 100 | 5.0 | 1391 |
| SS400B-1 | 74 | 2.95 | 1332 |
| SS400B-1 | 100 | 3.6 | 1293 |
Step 2: and (4) carrying out noise reduction processing on the production data of each layer by adopting a statistical 3 sigma principle.
In specific implementation, noise data is eliminated by adopting a statistical 3 sigma principle to obtain the production parameters of 474 blocks of steel. Some of the production parameters are shown in table 2.
Table 2 shows the production parameters of a part of the strip.
And step 3: dividing the production data after noise reduction into a training set A and a testing set B according to a certain proportion, wherein the consistency of data distribution is kept through set division.
In specific implementation, 80 percent (380 blocks) of strip steel data of each specification are selected as a training set, and the remaining 20 percent (94 blocks) are selected as a testing set. The training set data is 100 × 80% +74 × 80% +100 × 80% + 380, and the test set data is 100 × 20% + 74% +100 × 20% + 94.
And 4, step 4: and forming an observation value matrix by the production data of each layer after noise reduction, and performing standardized transformation and dimension reduction processing on the observation value matrix to obtain a standardized matrix after dimension reduction.
The step 4 specifically comprises the following steps:
step 4.1: the method comprises the following steps of forming an observed value matrix by the production data of each layer after noise reduction, and carrying out standardized transformation on the observed value matrix to obtain a standardized matrix, wherein the standardized matrix specifically comprises the following steps:
the production data of each strip steel is regarded as a p-dimensional vector X ═ X (X)1,X2,…,Xp) P is the number of production parameters, in this embodiment, p is 10, and after noise reduction, the production data of 474 pieces of strip steel are obtained, and the production data X is (X)1,X2,…,X10) Is expressed as:
after the normalization transformation, a normalization matrix is obtained and expressed as:
the normalized formula is:
wherein,
step 4.2: and performing dimensionality reduction on the standardized matrix by adopting a principal component analysis method. The method specifically comprises the following steps:
step 4.2.1: calculating a correlation coefficient matrix R of the collected production data:
in this embodiment, the matrix elements of the correlation coefficient matrix R are listed in table 3.
Table 3 matrix element list of correlation coefficient matrix R:
wherein,
step 4.2.2: calculating the eigenvalue (lambda) of the correlation coefficient matrix R1,λ2,…,λ10) And corresponding feature vectors, arranged in order of magnitude, lambda1≥λ2≥…λpNot less than 0; respectively calculating corresponding characteristic values lambdaiThe feature vector of (a) is represented as:
ei＝(ei,1,ei,2,…,ei,474),i＝1,2,…,10 (5)
respectively calculating corresponding characteristic values lambdaiThe characteristic vector of (1) is such that | | | ei1 | | |, i.eWherein eijRepresents a vector eiThe jth component of (a). The characteristic values of the correlation coefficient matrix R in the present embodiment are (3.8836, 2.1965, 1.4238, 1.0119, 0.9145, 0.3324, 0.1555, 0.0587, 0.0229, 0.0003)
Step 4.2.3: selecting important principal components, writing a principal component expression, selecting k principal components according to the magnitude of the accumulated contribution rate of each principal component, wherein the contribution rate eta is the proportion of the variance of a certain principal component to all the variances, and is the proportion of a certain eigenvalue of a correlation coefficient matrix R to the total of all eigenvalues, namely:
the cumulative contribution rate is:
and calculating the contribution rate of each principal component and the accumulated contribution rate.
Table 4 main component contribution ratio and cumulative contribution ratio:
| principal component | Characteristic value | Contribution ratio (%) | Cumulative contribution ratio (%) |
| 1 | 3.8836 | 0.3884 | 0.3884 |
| 2 | 2.1965 | 0.2197 | 0.608 |
| 3 | 1.4238 | 0.1424 | 0.7504 |
| 4 | 1.0119 | 0.1012 | 0.8516 |
| 5 | 0.9145 | 0.0915 | 0.943>0.90 |
| 6 | 0.3324 | 0.0332 | 0.9763>0.95 |
| 7 | 0.1555 | 0.0156 | 0.9918 |
| 8 | 0.0587 | 0.0059 | 0.9977 |
| 9 | 0.0229 | 0.0023 | 1 |
| 10 | 0.0003 | 0 | 1 |
The larger the contribution rate is, the stronger the information of the original variable contained in the principal component is, and in the invention, the accumulated contribution rate reaches more than 90%, so that most of the information of the original variable can be ensured. After dimension reduction, the data is changed from 10-dimension data to 5-dimension data, and the principal components are shown in FIG. 2.
And 5: taking the reduced standardized matrix as the input of the support vector machine model, and optimizing the parameters of the support vector machine model by adopting a particle swarm optimization algorithm based on hybridization;
in specific implementation, the production parameters after dimensionality reduction are used as the input of the support vector machine model, and the parameters of the support vector machine model are optimized by adopting a hybrid-based particle swarm optimization algorithm, wherein the parameters comprise a penalty coefficient C, a kernel function parameter sigma and a loss function value of the support vector machine. The optimization step comprises the following steps:
step 5.1: initializing a particle swarm algorithm, and initializing the size of a population, the position and the speed of each particle in the population;
the method specifically comprises the following steps: defining a p-dimensional search space, wherein p is the number of production parameters collected by each strip steel, and a group X consisting of n particles is in the p-dimensional search space (X)1,X2,...,Xn) Wherein the ith particle is represented as a vector X of dimension pi＝[xi1,xi2,…,xip]TRepresenting the position of the ith particle in the p-dimensional search space, and the velocity of the ith particle is Vi＝[Vi1,Vi2,…,Vip]TIndividual extreme value of Pi＝[Pi1,Pi2,…,Pip]TGlobal extremum of the population is Pg＝[Pg1,Pg2,…,Pgp]T. In this example, p is 5.
Step 5.2: calculating the fitness value of each particle in the population; the method specifically comprises the following steps:
step 5.2.1: determining a fitness function, and adopting a Mean Square Error (MSE) between a predicted value and an actual value of the strip steel outlet convexity under a cross validation condition as the fitness function, wherein the fitness function expression is as follows:
wherein,as a prediction of the strip outlet crown, yiThe actual value of the strip steel outlet convexity is obtained;
step 5.2.2: and calculating the fitness value of each particle according to the fitness function.
Step 5.3: comparing the fitness value of each particle with the individual extreme value, and if the fitness value is greater than the individual extreme value, using the fitness value as a new individual extreme value; and comparing the fitness value of each particle with the global extremum, and if the fitness value of each particle is greater than the global extremum, using the fitness value of each particle as a new global extremum.
Step 5.4: updating the position and the speed of the particles according to the new individual extremum and the new global extremum;
wherein, the updating formula of the particle position is as follows:
the updated formula of the particle velocity is:
wherein ω is the inertial weight; d is 1,2, …, p; 1,2, …, n; k is the current iteration number; vidIs the velocity of the particle; c. C1,c2Is an acceleration factor; r is1,r2Is a random number between 0 and 1. In this example, ω is 0.72, c1＝c2The population number is 20, the maximum number of iterations is 100, and the cross-validation method has a score of 5, which is 1.19. The C search range is 0-100, the sigma search range is 0-100, and the search range is 0-1.
Step 5.5: reinitializing the population, selecting a specific number of particles according to the hybridization probability and putting the particles into a hybridization pool, and randomly hybridizing every two parent particles in the pool to generate the same number of child particles to form a new population;
wherein the position of the daughter particle and the velocity of the daughter particle are expressed as:
wherein m isxIs the position of the parent particle, nxIs the position of the daughter particle, mvIs the velocity of the parent particle, nvI is a random number between 0 and 1, which is the velocity of the daughter particles.
Step 5.6: and repeating the step 5.2 to the step 5.4 to update the individual extreme value and the global extreme value, and further updating the position and the speed of the child particle.
Step 5.7: when the number of iterations reaches the set value, the optimization is stopped and the optimization result is output, and the combination of the optimal parameters found in the embodiment is (16.75,0.097, 0.0473).
As shown in fig. 3, a fitness value variation diagram in the optimization process of the parameters of the support vector machine model by using the hybrid-based particle swarm optimization algorithm in this embodiment is shown, c1＝c2The population number is 20 and the maximum number of iterations is 100, 1.19.
Step 6: constructing a support vector machine strip steel outlet convexity prediction model by adopting the optimal parameter combination (C, sigma) found in the step 5, and specifically comprising the following steps:
step 6.1: the collected production data and the actual value of the strip steel outlet convexity form a data setxiFor selected production data, y, affecting the strip outlet crowniDefining a decision plane f (x) w for the actual value of the strip outlet convexityTPhi (x) + b is a prediction model of the strip steel outlet convexity of the support vector machine, and the problem expression of the prediction model is defined as:
wherein phi (x)i) The high-dimensional feature space i is 1, …, m, w is an adjustable weight vector of the decision plane, and b is the offset of the decision plane, i.e. the offset of the decision plane relative to the origin;
C>0, denotes f (x) and yiMaximum deviation between,/ In order to be insensitive to the loss function,
step 6.2 introduction of relaxation variable ξi,Rewriting the problem expression of the forecasting model to obtain the rewritten problem expression:
step 6.3 introduction of Lagrange multiplier α*,μ,μ*The lagrange function is obtained as follows:
step 6.4 let L (w, b, α)*,ξ,ξ*,μ,μ*) For w, b, ξ*The partial derivative is zero:
step 6.5: substituting the Lagrange function into the rewritten problem expression to obtain a dual problem expression:
solving the dual problem to obtain a solution of w, b:
step 6.6: obtaining a support vector machine strip steel outlet convexity prediction model:
wherein,
and 7: training the support vector machine strip steel outlet convexity prediction model constructed in the step 6 by using a training set A, and testing the generalization performance of the support vector machine strip steel outlet convexity prediction model by using a test set B;
and 8: using a determining coefficient R2And evaluating the overall performance of the support vector machine strip steel outlet convexity prediction model by using the average absolute error MAE, the average absolute percentage error MAPE and the root mean square error RMSE. They calculateThe formula is as follows:
table 5 model error calculation results.
| Training set | Test set | |
| MAE | 1.7426 | 2.1042 |
| MAPE(％) | 2.8704 | 3.2767 |
| RMSE | 2.1374 | 1.3018 |
The predictive effect of the predictive model on the training set is shown in fig. 4, and the predictive effect on the test set is shown in fig. 5.
The method for forecasting the outlet convexity of the hot-rolled strip steel has the following beneficial effects: the invention adopts an artificial intelligence method to establish a plate convexity prediction model of hot-rolled strip steel. The model is based on mass production data, and the acquisition of the production data is easy to operate, so that the popularization capability of the model is strong. In addition, the complex mathematical and physical relationship among variables influencing the outlet convexity of the hot-rolled strip steel is avoided in the model establishing process, and the problems of strong coupling, nonlinearity and the like among all input variables are well solved. By reasonably screening and processing strip steel sample data, the method can effectively forecast the outlet convexity of the hot-rolled strip steel, and lays a foundation for accurate control of the convexity.
The above description is only exemplary of the preferred embodiments of the present invention, and is not intended to limit the present invention, and any modifications, equivalents and improvements made within the spirit and principle of the present invention should be included in the scope of the present invention.
Claims (10)
1. A method for forecasting outlet convexity of hot-rolled strip steel is characterized by comprising the following steps:
step 1: respectively acquiring p production data of each strip steel in the production process of the hot rolled strip steel in layers, representing the p production data by using a p-dimensional vector, and dividing the layers according to the steel type, the width of the finish rolled strip steel and the thickness of the finish rolled strip steel;
step 2: noise reduction processing is carried out on the production data of each layer by adopting a statistical 3 sigma principle;
and step 3: dividing the production data subjected to noise reduction into a training set and a test set according to a certain proportion, wherein the consistency of data distribution is kept through set division;
and 4, step 4: forming an observation value matrix by the production data of each layer after noise reduction, and performing standardized transformation and dimension reduction processing on the observation value matrix to obtain a standardized matrix after dimension reduction;
and 5: taking the reduced standardized matrix as the input of the support vector machine model, and optimizing the parameters of the support vector machine model by adopting a particle swarm optimization algorithm based on hybridization;
step 6: constructing a support vector machine strip steel outlet convexity prediction model by adopting the optimized optimal parameter combination;
and 7: training a support vector machine strip steel outlet convexity prediction model by using a training set, and testing the generalization performance of the prediction model by using a test set;
and 8: using a determining coefficient R2And evaluating the overall performance of the support vector machine strip steel outlet convexity prediction model by using the average absolute error MAE, the average absolute percentage error MAPE and the root mean square error RMSE.
2. The method of predicting hot rolled strip exit crown as claimed in claim 1, wherein said step 4 comprises:
step 4.1: forming an observation value matrix by the production data of each layer after noise reduction, and carrying out standardized transformation on the observation value matrix to obtain a standardized matrix;
step 4.2: and performing dimensionality reduction on the standardized matrix by adopting a principal component analysis method.
3. The method for forecasting the outlet convexity of the hot-rolled strip steel as claimed in claim 2, wherein the step 4.1 is specifically as follows:
the production data of each strip steel is regarded as a p-dimensional vector X ═ X (X)1,X2,…,Xp) Obtaining the production data of the N strip steel after noise reduction, wherein the production data X ═ X1,X2,…,Xp) Is expressed as:
<mrow> <mi>X</mi> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>x</mi> <mn>11</mn> </msub> </mtd> <mtd> <msub> <mi>x</mi> <mn>12</mn> </msub> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <msub> <mi>x</mi> <mrow> <mn>1</mn> <mi>p</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mn>21</mn> </msub> </mtd> <mtd> <msub> <mi>x</mi> <mn>22</mn> </msub> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <msub> <mi>x</mi> <mrow> <mn>2</mn> <mi>p</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow> <mi>N</mi> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>x</mi> <mrow> <mi>N</mi> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <msub> <mi>x</mi> <mrow> <mi>N</mi> <mi>p</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> </mrow>
after the normalization transformation, a normalization matrix is obtained and expressed as:
<mrow> <msup> <mi>X</mi> <mi>R</mi> </msup> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>x</mi> <mn>11</mn> <mi>R</mi> </msubsup> </mtd> <mtd> <msubsup> <mi>x</mi> <mn>12</mn> <mi>R</mi> </msubsup> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <msubsup> <mi>x</mi> <mrow> <mn>1</mn> <mi>p</mi> </mrow> <mi>R</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>x</mi> <mn>21</mn> <mi>R</mi> </msubsup> </mtd> <mtd> <msubsup> <mi>x</mi> <mn>22</mn> <mi>R</mi> </msubsup> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <msubsup> <mi>x</mi> <mrow> <mn>2</mn> <mi>p</mi> </mrow> <mi>R</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>x</mi> <mrow> <mi>N</mi> <mn>1</mn> </mrow> <mi>R</mi> </msubsup> </mtd> <mtd> <msubsup> <mi>x</mi> <mrow> <mi>N</mi> <mn>2</mn> </mrow> <mi>R</mi> </msubsup> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <msubsup> <mi>x</mi> <mrow> <mi>N</mi> <mi>p</mi> </mrow> <mi>R</mi> </msubsup> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> </mrow>
the normalized formula is:
<mrow> <msubsup> <mi>x</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> <mi>R</mi> </msubsup> <mo>=</mo> <mfrac> <mrow> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>-</mo> <msub> <mover> <mi>x</mi> <mo>&OverBar;</mo> </mover> <mi>j</mi> </msub> </mrow> <msqrt> <mrow> <mi>var</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>j</mi> </msub> <mo>)</mo> </mrow> </mrow> </msqrt> </mfrac> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>...</mo> <mo>,</mo> <mi>N</mi> <mo>;</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>...</mo> <mo>,</mo> <mi>p</mi> <mo>,</mo> </mrow>
wherein,
4. the method of predicting hot rolled strip exit crown as claimed in claim 3, wherein said step 4.2 comprises:
step 4.2.1: calculating a correlation coefficient matrix R of the collected production data:
<mrow> <mi>R</mi> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>r</mi> <mn>11</mn> </msub> </mtd> <mtd> <msub> <mi>r</mi> <mn>12</mn> </msub> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <msub> <mi>r</mi> <mrow> <mn>1</mn> <mi>p</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>r</mi> <mn>21</mn> </msub> </mtd> <mtd> <msub> <mi>r</mi> <mn>22</mn> </msub> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <msub> <mi>r</mi> <mrow> <mn>2</mn> <mi>p</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <msub> <mi>r</mi> <mrow> <mi>p</mi> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>r</mi> <mrow> <mi>p</mi> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <msub> <mi>r</mi> <mrow> <mi>p</mi> <mi>p</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> </mrow>
wherein,
step 4.2.2: calculating the eigenvalue (lambda) of the correlation coefficient matrix R1,λ2,…λp) And arranged in order of magnitude, λ1≥λ2≥…λpNot less than 0; respectively calculating corresponding characteristic values lambdaiCharacteristic vector e ofi(i-1, 2, …, p), and (ii) ei1 | | |, i.eWherein eijRepresents a vector eiThe jth component of (a);
step 4.2.3: selecting important principal components, writing a principal component expression, selecting k principal components according to the magnitude of the accumulated contribution rate of each principal component, wherein the contribution rate eta is the proportion of the variance of a certain principal component to all the variances, and is the proportion of a certain eigenvalue of a correlation coefficient matrix R to the total of all eigenvalues, namely:
<mrow> <msub> <mi>&eta;</mi> <mi>i</mi> </msub> <mo>=</mo> <mfrac> <msub> <mi>&lambda;</mi> <mi>i</mi> </msub> <mrow> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>p</mi> </munderover> <msub> <mi>&lambda;</mi> <mi>i</mi> </msub> </mrow> </mfrac> </mrow>
the cumulative contribution rate is:
<mfrac> <mrow> <munderover> <mo>&Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>i</mi> </munderover> <msub> <mi>&lambda;</mi> <mi>k</mi> </msub> </mrow> <mrow> <munderover> <mo>&Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>p</mi> </munderover> <msub> <mi>&lambda;</mi> <mi>k</mi> </msub> </mrow> </mfrac>
the larger the contribution rate is, the stronger the information of the original variable contained in the principal component is, and the cumulative contribution rate reaches more than 90%, so that most of the information of the original variable can be guaranteed.
5. The method of predicting hot rolled strip exit crown as claimed in claim 1, wherein said step 5 comprises:
step 5.1: initializing a particle swarm algorithm, and initializing the size of a population, the position and the speed of each particle in the population;
step 5.2: calculating the fitness value of each particle in the population;
step 5.3: comparing the fitness value of each particle with the individual extreme value, and if the fitness value is greater than the individual extreme value, using the fitness value as a new individual extreme value; comparing the fitness value of each particle with the global extremum, and if the fitness value is greater than the global extremum, using the fitness value as a new global extremum;
step 5.4: updating the position and the speed of the particles according to the new individual extremum and the new global extremum;
step 5.5: reinitializing the population, selecting a specific number of particles according to the hybridization probability and putting the particles into a hybridization pool, and randomly hybridizing every two parent particles in the pool to generate the same number of child particles to form a new population;
step 5.6: repeating the step 5.2 to the step 5.4 to update the individual extreme value and the global extreme value, and further updating the position and the speed of the child particle;
step 5.7: and when the iteration times reach a set value, stopping optimization and outputting the optimal parameters of the support vector machine model.
6. The method for forecasting the outlet convexity of the hot-rolled strip steel as claimed in claim 5, wherein the step 5.1 is specifically as follows:
defining a p-dimensional search space, wherein p is the number of production parameters collected by each strip steel, and a group X consisting of n particles is in the p-dimensional search space (X)1,X2,...,Xn) Wherein the ith particle is represented as a vector X of dimension pi＝[xi1,xi2,…,xip]TRepresenting the position of the ith particle in the p-dimensional search space, and the velocity of the ith particle is Vi＝[Vi1,Vi2,…,Vip]TIndividual extreme value of Pi＝[Pi1,Pi2,…,Pip]TGlobal extremum of the population is Pg＝[Pg1,Pg2,…,Pgp]T。
7. The method for forecasting the outlet convexity of the hot-rolled strip steel as claimed in claim 6, wherein the step 5.2 is specifically as follows:
step 5.2.1: determining a fitness function, and adopting a Mean Square Error (MSE) between a predicted value and an actual value of the strip steel outlet convexity under a cross validation condition as the fitness function, wherein the fitness function expression is as follows:
<mrow> <mi>M</mi> <mi>S</mi> <mi>E</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msup> <mrow> <mo>(</mo> <msubsup> <mi>y</mi> <mi>i</mi> <mo>*</mo> </msubsup> <mo>-</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>;</mo> </mrow>
wherein,as a prediction of the strip outlet crown, yiThe actual value of the strip steel outlet convexity is obtained;
step 5.2.2: and calculating the fitness value of each particle according to the fitness function.
8. The method of predicting hot rolled strip exit crown as claimed in claim 6, wherein the updated formula for the positions of the particles in step 5.4 is:
<mrow> <msubsup> <mi>V</mi> <mrow> <mi>i</mi> <mi>d</mi> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>&omega;V</mi> <mrow> <mi>i</mi> <mi>d</mi> </mrow> <mi>k</mi> </msubsup> <mo>+</mo> <msub> <mi>c</mi> <mn>1</mn> </msub> <msub> <mi>r</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <msubsup> <mi>P</mi> <mrow> <mi>i</mi> <mi>d</mi> </mrow> <mi>k</mi> </msubsup> <mo>-</mo> <msubsup> <mi>X</mi> <mrow> <mi>i</mi> <mi>d</mi> </mrow> <mi>k</mi> </msubsup> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>c</mi> <mn>2</mn> </msub> <msub> <mi>r</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <msubsup> <mi>P</mi> <mrow> <mi>g</mi> <mi>d</mi> </mrow> <mi>k</mi> </msubsup> <mo>-</mo> <msubsup> <mi>X</mi> <mrow> <mi>i</mi> <mi>d</mi> </mrow> <mi>k</mi> </msubsup> <mo>)</mo> </mrow> <mo>,</mo> </mrow>
the updated formula of the particle velocity is:
<mrow> <msubsup> <mi>X</mi> <mrow> <mi>i</mi> <mi>d</mi> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>X</mi> <mrow> <mi>i</mi> <mi>d</mi> </mrow> <mi>k</mi> </msubsup> <mo>+</mo> <msubsup> <mi>V</mi> <mrow> <mi>i</mi> <mi>d</mi> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>;</mo> </mrow>
wherein ω is the inertial weight; d is 1,2, …, p; 1,2, …, n; k is the current iteration number; vidIs the velocity of the particle; c. C1,c2Is an acceleration factor; r is1,r2Is a random number between 0 and 1.
9. The method for forecasting the outlet crown of a hot-rolled strip as claimed in claim 6, wherein the positions of the daughter particles and the velocities of the daughter particles in the step 5.5 are expressed as:
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>n</mi> <mi>x</mi> </msub> <mo>=</mo> <mi>i</mi> <mo>*</mo> <msub> <mi>m</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>*</mo> <msub> <mi>m</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>n</mi> <mi>v</mi> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>m</mi> <mi>v</mi> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>m</mi> <mi>v</mi> </msub> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> <mrow> <mo>|</mo> <msub> <mi>m</mi> <mi>v</mi> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>m</mi> <mi>v</mi> </msub> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>|</mo> </mrow> </mfrac> <mo>|</mo> <msub> <mi>m</mi> <mi>v</mi> </msub> <mo>|</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> </mrow>
wherein m isxIs the position of the parent particle, nxIs the position of the daughter particle, mvIs the velocity of the parent particle, nvI is a random number between 0 and 1, which is the velocity of the daughter particles.
10. The method of predicting hot rolled strip exit crown as claimed in claim 1, wherein said step 6 comprises:
step 6.1: the collected production data and the actual value of the strip steel outlet convexity form a data setxiFor selected production data, y, affecting the strip outlet crowniDefining a decision plane f (x) w for the actual value of the strip outlet convexityTPhi (x) + b is a prediction model of the strip steel outlet convexity of the support vector machine, and the problem expression of the prediction model is defined as:
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </mtd> <mtd> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>|</mo> <mo>|</mo> <mi>w</mi> <mo>|</mo> <msup> <mo>|</mo> <mn>2</mn> </msup> <mo>+</mo> <mi>C</mi> <munderover> <mi>&Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <msub> <mi>l</mi> <mi>&epsiv;</mi> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>-</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mfenced>
wherein phi (x)i) The high-dimensional feature space i is 1, …, m, w is an adjustable weight vector of the decision plane, and b is the offset of the decision plane, i.e. the offset of the decision plane relative to the origin;
C>0, is f (x) and yiMaximum deviation between,/ In order to be insensitive to the loss function,
step 6.2 introduction of relaxation variable ξi,Rewriting the problem expression of the forecasting model to obtain the rewritten problem expression:
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <munder> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mrow> <mi>w</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>&xi;</mi> <mo>,</mo> <msup> <mi>&xi;</mi> <mo>*</mo> </msup> </mrow> </munder> </mtd> <mtd> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msup> <mi>w</mi> <mi>T</mi> </msup> <mi>w</mi> <mo>+</mo> <mi>C</mi> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <msub> <mi>&xi;</mi> <mi>i</mi> </msub> <mo>+</mo> <mi>C</mi> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <msubsup> <mi>&xi;</mi> <mi>i</mi> <mo>*</mo> </msubsup> </mrow> </mtd> </mtr> </mtable> </mfenced>
s.t. wTφ(xi)+b-yi≤+ξi,；
<mrow> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo>-</mo> <msup> <mi>w</mi> <mi>T</mi> </msup> <mi>&phi;</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>b</mi> <mo>&le;</mo> <mi>&epsiv;</mi> <mo>+</mo> <msubsup> <mi>&xi;</mi> <mi>i</mi> <mo>*</mo> </msubsup> <mo>,</mo> </mrow>
<mrow> <msub> <mi>&xi;</mi> <mi>i</mi> </msub> <mo>,</mo> <msubsup> <mi>&xi;</mi> <mi>i</mi> <mo>*</mo> </msubsup> <mo>&GreaterEqual;</mo> <mn>0</mn> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>...</mo> <mo>,</mo> <mi>m</mi> <mo>.</mo> </mrow>
step 6.3 introduction of Lagrange multiplier α*,μ,μ*The lagrange function is obtained as follows:
<mrow> <mtable> <mtr> <mtd> <mrow> <mi>L</mi> <mrow> <mo>(</mo> <mrow> <mi>w</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>&alpha;</mi> <mo>,</mo> <msup> <mi>&alpha;</mi> <mo>*</mo> </msup> <mo>,</mo> <mi>&xi;</mi> <mo>,</mo> <msup> <mi>&xi;</mi> <mo>*</mo> </msup> <mo>,</mo> <mi>&mu;</mi> <mo>,</mo> <msup> <mi>&mu;</mi> <mo>*</mo> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msup> <mi>w</mi> <mi>T</mi> </msup> <mi>w</mi> <mo>+</mo> <mi>C</mi> <munderover> <mi>&Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <mrow> <mo>(</mo> <mrow> <msub> <mi>&xi;</mi> <mi>i</mi> </msub> <mo>+</mo> <msubsup> <mi>&xi;</mi> <mi>i</mi> <mo>*</mo> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <munderover> <mi>&Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <msub> <mi>&mu;</mi> <mi>i</mi> </msub> <msub> <mi>&xi;</mi> <mi>i</mi> </msub> <mo>-</mo> <munderover> <mi>&Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <msubsup> <mi>&mu;</mi> <mi>i</mi> <mo>*</mo> </msubsup> <msubsup> <mi>&xi;</mi> <mi>i</mi> <mo>*</mo> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <munderover> <mi>&Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <msub> <mi>&alpha;</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mrow> <msup> <mi>w</mi> <mi>T</mi> </msup> <mi>&phi;</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>b</mi> <mo>-</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo>-</mo> <mi>&epsiv;</mi> <mo>-</mo> <msub> <mi>&xi;</mi> <mi>i</mi> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <munderover> <mi>&Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <msubsup> <mi>&alpha;</mi> <mi>i</mi> <mo>*</mo> </msubsup> <mrow> <mo>(</mo> <mrow> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo>-</mo> <msup> <mi>w</mi> <mi>T</mi> </msup> <mi>&phi;</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>b</mi> <mo>-</mo> <mi>&epsiv;</mi> <mo>-</mo> <msubsup> <mi>&xi;</mi> <mi>i</mi> <mo>*</mo> </msubsup> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>;</mo> </mrow>
step 6.4 let L (w, b, α)*,ξ,ξ*,μ,μ*) For w, b, ξ*Partial derivative of zero
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mfrac> <mrow> <mo>&part;</mo> <mi>L</mi> </mrow> <mrow> <mo>&part;</mo> <mi>w</mi> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> <mo>&RightArrow;</mo> <mi>w</mi> <mo>=</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <mrow> <mo>(</mo> <msubsup> <mi>&alpha;</mi> <mi>i</mi> <mo>*</mo> </msubsup> <mo>-</mo> <msub> <mi>&alpha;</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>x</mi> <mi>i</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mo>&part;</mo> <mi>L</mi> </mrow> <mrow> <mo>&part;</mo> <mi>b</mi> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> <mo>&RightArrow;</mo> <mn>0</mn> <mo>=</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <mrow> <mo>(</mo> <msubsup> <mi>&alpha;</mi> <mi>i</mi> <mo>*</mo> </msubsup> <mo>-</mo> <msub> <mi>&alpha;</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mo>&part;</mo> <mi>L</mi> </mrow> <mrow> <mo>&part;</mo> <msub> <mi>&xi;</mi> <mi>i</mi> </msub> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> <mo>&RightArrow;</mo> <mi>C</mi> <mo>=</mo> <msub> <mi>&alpha;</mi> <mi>i</mi> </msub> <mo>+</mo> <msub> <mi>&mu;</mi> <mi>i</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mo>&part;</mo> <mi>L</mi> </mrow> <mrow> <mo>&part;</mo> <msubsup> <mi>&xi;</mi> <mi>i</mi> <mo>*</mo> </msubsup> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> <mo>&RightArrow;</mo> <mi>C</mi> <mo>=</mo> <msubsup> <mi>&alpha;</mi> <mi>i</mi> <mo>*</mo> </msubsup> <mo>+</mo> <msubsup> <mi>&mu;</mi> <mi>i</mi> <mo>*</mo> </msubsup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>;</mo> </mrow>
Step 6.5: substituting the Lagrange function into the rewritten problem expression to obtain a dual problem expression:
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <munder> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mrow> <msub> <mi>&alpha;</mi> <mi>i</mi> </msub> <mo>,</mo> <msubsup> <mi>&alpha;</mi> <mi>i</mi> <mo>*</mo> </msubsup> </mrow> </munder> </mtd> <mtd> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <munderover> <mo>&Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <msup> <mrow> <mo>(</mo> <msub> <mi>&alpha;</mi> <mi>i</mi> </msub> <mo>-</mo> <msubsup> <mi>&alpha;</mi> <mi>i</mi> <mo>*</mo> </msubsup> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mrow> <mo>(</mo> <msub> <mi>&alpha;</mi> <mi>i</mi> </msub> <mo>-</mo> <msubsup> <mi>&alpha;</mi> <mi>i</mi> <mo>*</mo> </msubsup> <mo>)</mo> </mrow> <mi>Q</mi> <mo>+</mo> <mi>&epsiv;</mi> <mrow> <mo>(</mo> <msub> <mi>&alpha;</mi> <mi>i</mi> </msub> <mo>+</mo> <msubsup> <mi>&alpha;</mi> <mi>i</mi> <mo>*</mo> </msubsup> <mo>)</mo> </mrow> <mo>+</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <msub> <mi>y</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>&alpha;</mi> <mi>i</mi> </msub> <mo>-</mo> <msubsup> <mi>&alpha;</mi> <mi>i</mi> <mo>*</mo> </msubsup> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced>
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>s</mi> <mo>.</mo> <mi>t</mi> <mo>.</mo> </mrow> </mtd> <mtd> <mrow> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <mrow> <mo>(</mo> <msub> <mi>&alpha;</mi> <mi>i</mi> </msub> <mo>-</mo> <msubsup> <mi>&alpha;</mi> <mi>i</mi> <mo>*</mo> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mfenced>
<mrow> <mn>0</mn> <mo>&le;</mo> <msub> <mi>&alpha;</mi> <mi>i</mi> </msub> <mo>,</mo> <msubsup> <mi>&alpha;</mi> <mi>i</mi> <mo>*</mo> </msubsup> <mo>&le;</mo> <mi>C</mi> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>...</mo> <mo>,</mo> <mi>m</mi> <mo>.</mo> <mo>;</mo> </mrow>
solving the dual problem to obtain a solution of w, b
<mrow> <mi>b</mi> <mo>=</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo>+</mo> <mi>&epsiv;</mi> <mo>-</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <mrow> <mo>(</mo> <msubsup> <mi>&alpha;</mi> <mi>i</mi> <mo>*</mo> </msubsup> <mo>-</mo> <msub> <mi>&alpha;</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mi>&phi;</mi> <msup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mi>&phi;</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>j</mi> </msub> <mo>)</mo> </mrow> <mo>.</mo> </mrow>
<mrow> <mi>w</mi> <mo>=</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <mrow> <mo>(</mo> <msubsup> <mi>&alpha;</mi> <mi>i</mi> <mo>*</mo> </msubsup> <mo>-</mo> <msub> <mi>&alpha;</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mi>&phi;</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>j</mi> </msub> <mo>)</mo> </mrow> <mo>.</mo> <mo>;</mo> </mrow>
Step 6.6: obtaining a support vector machine strip steel outlet convexity prediction model:
<mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <mrow> <mo>(</mo> <msubsup> <mi>&alpha;</mi> <mi>i</mi> <mo>*</mo> </msubsup> <mo>-</mo> <msub> <mi>&alpha;</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mi>K</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>x</mi> <mi>j</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>b</mi> <mo>.</mo> </mrow>
wherein,
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Application publication date: 20171124 Assignee: SHANDONG PROVINCE METALLURGICAL ENGINEERING Co.,Ltd. Assignor: Northeastern University Contract record no.: X2021210000025 Denomination of invention: A prediction method of exit crown of hot rolled strip Granted publication date: 20181016 License type: Common License Record date: 20210520 |
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