CN111695229B - Novel distributed non-Gaussian process monitoring method based on GA-ICA - Google Patents

Novel distributed non-Gaussian process monitoring method based on GA-ICA Download PDF

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CN111695229B
CN111695229B CN201910229753.2A CN201910229753A CN111695229B CN 111695229 B CN111695229 B CN 111695229B CN 201910229753 A CN201910229753 A CN 201910229753A CN 111695229 B CN111695229 B CN 111695229B
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唐俊苗
童楚东
史旭华
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Ningbo University
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Abstract

The invention discloses a novel distributed non-Gaussian process monitoring method based on GA-ICA, which aims to solve an ICA model by utilizing a decimal genetic algorithm and deduce the implementation process of a plurality of ICA algorithms, and can implement distributed non-Gaussian process monitoring on the basis. In the process of implementing multi-block modeling, the method firstly utilizes a decimal genetic algorithm to solve the separation vectors and independent components corresponding to all the measured variables, and then separates the independent components corresponding to each variable sub-block according to the uniqueness of each variable sub-block. Therefore, the method comprehensively considers the integrity of all measured variables and the local characteristics of each variable sub-block when implementing multi-block modeling, and is a brand-new decentralized non-Gaussian process monitoring method. In addition, the advantages of the method of the present invention will be demonstrated in the present embodiment, thereby illustrating that the method of the present invention is a more preferred decentralized non-gaussian process monitoring method.

Description

Novel distributed non-Gaussian process monitoring method based on GA-ICA
Technical Field
The invention relates to a data-driven process monitoring method, in particular to a novel distributed non-Gaussian process monitoring method based on GA-ICA.
Background
Under industry "big data" hot flashes, modern industrial processes are moving towards digital management. Because the production process object can store and measure massive data on line, the data contains information which can embody the running state of the production process, and the monitoring of the running state of the production process by using the sampling data is popular among more students. In fact, a great deal of human and material research is put into the process monitoring method taking fault detection as a core task, both in academia and industry. In the field of data-driven process monitoring research, statistical process monitoring is the most studied approach, with principal component analysis (Principal Component Analysis, PCA) and independent component analysis (Independent Component Analysis, ICA) being the most dominant implementation techniques. Compared with the PCA algorithm, the ICA algorithm can mine out hidden non-Gaussian component information in the data, and can reveal the essence of the object, so that the method is more suitable for monitoring the non-Gaussian process object. In addition, it can be found in the existing scientific literature and patent information that the fault detection effect of the ICA method is generally no worse than that of the PCA method.
When the traditional ICA method is applied to non-Gaussian process monitoring, an ICA model is built, namely, all the measurement variables corresponding to the production units are taken as a whole. In practical application, considering that modern industrial processes gradually move to large-scale development, the production process is fully automatically controlled, and comprises a plurality of production units which are mutually connected and restrained, and different production units are treated differently. Therefore, modern industrial processes prefer distributed process monitoring method technology for decentralized monitoring of individual production units. However, due to the mutual coupling between the individual production units, again, the integrity must be taken into account. In this regard, decentralized monitoring should take into account the integrity in addition to local characteristics when building a process monitoring model.
Although the multi-block PCA (MBPCA) algorithm has long been proposed and applied to decentralized process monitoring, the multi-block ICA (MBICA) algorithm has not been proposed. This is mainly because the solution process of the ICA model typically uses the fastgica iterative algorithm to solve for the separation vector that maximizes non-gaussian by the principle of gradient descent. Therefore, the ICA based on the fastca algorithm cannot be directly extended to the MBICA algorithm. Since non-gaussian maximization is an optimization problem, intelligent optimization algorithms, such as genetic (Genetic Algorithm, abbreviated GA) algorithms, can be used in addition to gradient descent. The classical GA algorithm requires binary coding, and conversion of decimal into binary increases the computational effort. Therefore, it is still to be questionable to solve the ICA model directly using the GA algorithm.
Disclosure of Invention
The main technical problems to be solved by the invention are as follows: and solving an ICA model by using a decimal genetic algorithm, deducing the implementation process of a plurality of ICA algorithms, and carrying out distributed non-Gaussian process monitoring on the basis. Specifically, the method of the invention firstly utilizes the GA-ICA algorithm to solve the independent components corresponding to all the measured variables, then carries out block processing on all the measured variables of the process object, and finally carries out distributed non-Gaussian process monitoring by utilizing the independent components.
The technical scheme adopted for solving the technical problems is as follows: a novel distributed non-Gaussian process monitoring method based on GA-ICA comprises the following steps:
step (1): under the normal operation state of the production process object, n sample data x are collected 1 ，x 2 ，…，x n Composition of training data matrix x= [ X ] 1 ，x 2 ，…，x n ]∈R m×n And calculates the mean vector μ= (x) 1 +x 2 +…+x n ) N and standard deviation vector delta epsilon R m×1 ：
Wherein, the ". Is a vector (x i - μ) and (x) i - μ) corresponding element multiplication, x i ∈R m×1 Represents the i-th sample data, n is the number of training sample data, m is the number of measurement variables, i=1, 2, …, n, R is the real number set, R m×n Representing a matrix of real numbers in the m x n dimension.
Step (2): according to the formulaFor sample data x 1 ，x 2 ，…，x n Respectively performing normalization processing to obtain matrix +.>Wherein (1)>Representing vector (x) i Mu) is divided by the corresponding element in the standard deviation vector delta,/and>is a normalized data vector.
Step (3): determining the measurement variables related to each production unit according to the composition structure of the production process object, dividing the m measurement variables into B variable sub-blocks, and dividing the matrix according to the B variable sub-blocksSelecting corresponding row vectors to respectively form B sub-block matrixes +.>
It is noted that due to the interrelation between the production units, there is a possibility that the measurement variables involved in the respective production units overlap. Thus, B variable sub-block matricesThe sum of the numbers of all row vectors is m or more.
Step (4): setting the number of the separation vectors as D, and solving a matrix by utilizing a decimal Genetic (GA) algorithmD separation vectors w of (2) 1 ，w 2 ，…，w D D mixing vectors p 1 ，p 2 ，…，p D And D independent component vectors t 1 ，t 2 ，…，t D In which the parameter D should be smaller than the total number m of measured variables, D.epsilon. [ m/4, m/2 can be set in the usual case]. The specific implementation process is as follows:
step (4.1): initializing d=1 and u=0, and randomly generating a real matrix W of dimension n×m over the interval [ -1,1], where N is the population number of the GA algorithm, where N is required to be set to an even number greater than m.
Step (4.2): according to the formulaSolving the matrix F epsilon R N×n And calculates the mean vector f= (F) of n column vectors in the matrix F 1 +F 2 +…+F n ) N, wherein F i Representing the ith column vector in matrix F.
Step (4.3): finding the maximum value f of the elements in the mean vector f max And records the position v where the maximum is located in the mean vector f.
Step (4.4): judging whether the condition U < U is satisfied? Wherein U is the maximum iteration number of the genetic algorithm, and is recommended to be 5m or more and is not more than 50m or less, if yes, after u=u+1 is set, the step (4.5) is skipped; if not, the upsilon line vector W (upsilon) in the real number matrix W is recorded as the d separation vector W d And jumps to step (4.13).
Step (4.5): in turn according to the formula gamma k ＝f k /(f 1 +f 2 +…+f N ) And formula (VI)Calculating the selection probability gamma k And cumulative probability->Where k=1, 2, …, N, f k Representing the kth element in the mean vector f.
Step (4.6): randomly generating a number s in the interval (0, 1), thenFind out the meeting conditionMinimum value +.>And the kth row vector in the real matrix W is retained.
Step (4.7): repeating the step (4.6) for N times, thereby forming a matrix V epsilon R by the reserved N row vectors N×m And initializes j=1.
Step (4.8): after randomly generating a number c in the interval (0, 1), judging whether the condition is satisfied:wherein->Representing the crossover probability, suggesting a value +.>If yes, the j-th row vector V in the matrix V is calculated according to the following formula (2) j And the j+1th row vector V j+1 Performing a crossover operation:
in the above, θ j Is a number randomly generated on the interval (0, 1); if not, the j-th row vector V in the matrix V j And the j+1th row vector V j+1 Remain unchanged.
Step (4.9): judging whether the condition is satisfied: j < N-1? If yes, after j=j+2, returning to the step (4.8); if not, a matrix V after the crossover operation is obtained, and k=1 is initialized.
Step (4.10): after randomly generating a number g in the interval (0, 1), judging whether the condition is satisfied:wherein the method comprises the steps ofRepresenting the probability of variation, suggesting a value +.>If yes, the k-th row vector V in the matrix V is calculated according to the following formula (3) k The mutation operation is performed on any one element V (k, epsilon):
in the above formula, epsilon represents that the arbitrarily selected element V (k, epsilon) is represented by the vector V k The position of r k Function h (u) =r for a number randomly generated over interval (0, 1) k (1-u/U) 2 The method comprises the steps of carrying out a first treatment on the surface of the If not, the kth row vector V (k) in the matrix V is kept unchanged.
Step (4.11): judging whether the condition is satisfied: k < N? If yes, after setting k=k+1, returning to the step (4.10); if not, a matrix V after the mutation operation is obtained.
Step (4.12): according to formula V N Update of row vector V of last row in matrix V =w (V) N Wherein W (V) represents a V-th row vector in the real matrix W, and after setting the real matrix w=v, the step (4.2) is returned.
Step (4.13): judging whether the condition is satisfied: d < D? If yes, sequentially according to the formula And->Respectively calculating the d independent component vector t d Hybrid vector p d And update matrix->D=d+1 and u=0 are reset and are within the interval [ -1,1]Randomly generating an Nxm b Returning to the step (4.2) after the real matrix W of the dimension; if it isIf not, all D separation vectors w are obtained 1 ，w 2 ，…，w D D mixing vectors p 1 ，p 2 ，…，p D And D independent component vectors t 1 ，t 2 ，…，t D Wherein the upper reference T indicates a transpose of the matrix or vector.
Step (5): according to D separation vectors w 1 ，w 2 ，…，w D Sequentially for B variable sub-block matricesThe extraction of the independent components is implemented, and the specific implementation process is as follows:
step (5.1): initializing d=1.
Step (5.2): separating vector w from d-th variable sub-block according to B-th variable sub-blocks in step (3) d Selecting corresponding elements to respectively form separation vectors w corresponding to the B variable sub-blocks d (1)，w d (2)，…，w d (B)。
Step (5.2): for all values of B epsilon {1,2, …, B }, judging whether the conditions are satisfied or not respectively:if not, then the formula +.>And->Calculating independent component vectors t corresponding to the b-th variable sub-block respectively d (b) And a load vector a d (b) Then, jumping to the step (5.3); if yes, the separation vector w corresponding to the b variable sub-block is obtained 1 (b)，w 2 (b)，…，w d (b) Form a separation matrix W b And respectively forming an independent component matrix T by all independent component vectors and all load vectors corresponding to the calculated b variable sub-block b And a load matrix P b After that, the process goes to step (5.4). Wherein b represents the b-th variable sub-A block.
Step (5.3): according to the formulaUpdating the b-th variable sub-block matrix +.>After that, the process goes to step (5.5).
Step (5.4): reserving an independent component matrix T corresponding to the b-th variable sub-block b Separation matrix W b And a load matrix P b 。
Step (5.5): judging whether the condition is satisfied: d < D? If yes, after d=d+1, returning to the step (5.2); if not, obtaining separation matrixes W corresponding to all B variable sub-blocks 1 ，W 2 ，…，W B Load matrix P 1 ，P 2 ，…，P B 。
Step (6): firstly, according to a formula lambda b ＝T b T b T (n-1) calculation of covariance matrix Λ 1 ，Λ 2 ，…，Λ B Then according to formula Q b ＝diag(T b T Λ b -1 T b ) Calculating and monitoring a statistical index vector Q 1 ，Q 2 ，…，Q B Wherein diag () represents an operation of converting an element of a matrix diagonal into a vector.
Step (7): respectively for monitoring statistical index vector Q 1 ，Q 2 ，…，Q B The elements in the variable sub-blocks are arranged in descending order according to the size, and the n/100 th maximum value is respectively used as the upper limit CL of the monitoring statistical index of each variable sub-block 1 ，CL 2 ，…，CL B 。
The steps (1) to (7) are offline modeling stages of the method, and after the offline modeling stages are completed, online process monitoring can be implemented, and specific implementation processes are as follows.
Step (8): collecting sample data x epsilon R at new sampling time m×1 And according to the formulaThe normalization of x is performed to obtain a vector +.>
Step (9): from the B variable sub-blocks in step (3), the vector can be correspondingly dividedDivided into B sub-vectors
Step (10): firstly according to the formulaRespectively calculating each subvector +.>Corresponding independent component vector y 1 ，y 2 ，…，y B Then according to formula->Calculating monitoring index->
Step (11): judging whether the condition is satisfied or not for all values of the index b:if yes, the production process object at the current sampling moment is normally operated, and the step (8) is returned to monitor the sample data at the next moment continuously; if not, the production process object at the current sampling moment enters an abnormal operation state.
Compared with the traditional method, the method has the advantages that:
in the process of implementing multi-block modeling, the method firstly utilizes a decimal genetic algorithm to solve the separation vectors and independent components corresponding to all the measured variables, and then separates the independent components corresponding to each variable sub-block according to the uniqueness of each variable sub-block. Therefore, the method comprehensively considers the integrity of all measured variables and the local characteristics of each variable sub-block when implementing multi-block modeling, and is a brand-new decentralized non-Gaussian process monitoring method. In addition, the advantages of the method of the present invention will be demonstrated in the present embodiment, thereby illustrating that the method of the present invention is a more preferred decentralized non-gaussian process monitoring method.
Drawings
FIG. 1 is a flow chart of the method of the present invention.
Fig. 2 is a diagram showing the construction of TE process objects.
FIG. 3 is a comparative diagram of TE process fault monitoring details.
Detailed Description
The process according to the invention is described in detail below with reference to the drawings and to the specific examples.
The invention discloses a novel distributed non-Gaussian process monitoring method based on GA-ICA, and the implementation process and the superiority of the method relative to the prior method are described below by combining a specific industrial process example.
The application object is from an experiment of Tenn-Issmann (TE) chemical process in the United states, and the prototype is a practical technological process in an Issmann chemical production workshop. At present, the TE process is widely used as a standard experimental platform for fault detection research due to the complexity of the flow. The entire TE process includes 22 measured variables, 12 manipulated variables, and 19 component measured variables. The TE process object may simulate a variety of different fault types, such as material inlet temperature step changes, cooling water fault changes, and the like. To monitor the process, 33 process variables as shown in Table 1 were selected. Because of the short sampling interval, the TE process samples data with unavoidable sequence autocorrelation, and the detailed description of the steps of the invention is provided below in connection with the TE process.
Table 1: TE process monitor variables.
Firstly, as shown in fig. 1, the implementation flow of the method of the present invention builds a non-gaussian process monitoring model by using n=960 sample data sampled under the normal working condition of the TE process, and the method comprises the following steps:
step (1): in the normal operation state of the production process object, n=960 sample data x are collected 1 ，x 2 ，…，x 960 Form training data matrix X and calculate mean vector μ= (X) 1 +x 2 +…+x 960 ) 960 and standard deviation vector delta epsilon R 33×1 。
Step (2): according to the formulaFor sample data x 1 ，x 2 ，…，x 960 Respectively performing normalization processing to obtain matrix +.>
Step (3): according to the composition structure of the TE process object as shown in fig. 2, it comprises: the reactor, condenser, compressor, gas-liquid separator, and stripper were used to determine the measured variables involved in each production unit, thereby dividing the m measured variables into b=5 variable sub-blocks, the details of which are shown in table 2, and from the matrix based on the 5 variable sub-blocksSelecting corresponding row vectors to respectively form 5 sub-block matrixes +.>
TABLE 2 5 variable sub-blocks of TE process object
Production unit name: variable number in variable sub-block:
reactor for producing a catalyst 1，2，3，6，7，8，9，21，23，24，25，32
Condenser 17，33
Gas-liquid separator 4，10，15，16，18，19，26，28，29
Stripping tower 10，11，12，13，14，17，22，28，30，31
Compressor 5，20，27
Step (4): setting the number of the separation vectors to be D=9, and solving a matrix by using a decimal Genetic (GA) algorithmD=9 separation vectors w 1 ，w 2 ，…，w 9 9 mixing vectors p 1 ，p 2 ，…，p 9 And 9 independent component vectors t 1 ，t 2 ，…，t 9 。
Step (5): according to 9 separation vectors w 1 ，w 2 ，…，w 9 Sequentially for 5 variable sub-block matricesExtraction of the individual components is performed.
Step (6): firstly, according to a formula lambda b ＝T b T b T (n-1) calculation of covariance matrix Λ 1 ，Λ 2 ，…，Λ 5 Then according to formula Q b ＝diag(T b T Λ b -1 T b ) Calculating and monitoring a statistical index vector Q 1 ，Q 2 ，…，Q 5 Where diag () represents an operation of converting an element of a diagonal of a matrix into a vector, and the upper reference T represents a transpose of the matrix or the vector.
Step (7): respectively for monitoring statistical index vector Q 1 ，Q 2 ，…，Q 5 The elements in the variable sub-blocks are arranged in descending order according to the size, and the n/100 th maximum value is respectively used as the upper limit CL of the monitoring statistical index of each variable sub-block 1 ，CL 2 ，…，CL 5 。
The offline modeling phase is completed, and then online monitoring is performed. On-line fault detection is implemented by using 960 sampling data under the TE process fault condition, wherein the first 160 sampling data are normal, and the last 800 data are collected from the fault working condition.
Step (8): collecting sample data x epsilon R at new sampling time 33×1 And according to the formulaThe normalization of x is performed to obtain a vector +.>
Step (9): from the b=5 variable sub-blocks in step (3), the vector can be correspondingly setDivided into 5 sub-vectors
Step (10): firstly according to the formulaRespectively calculating each subvector +.>Corresponding independent component vector y 1 ，y 2 ，…，y 5 Then according to formula->Calculating monitoring index->
Step (11): judging whether the condition is satisfied or not for all values of the index b:if yes, the production process object at the current sampling moment is normally operated, and the step (8) is returned to monitor the sample data at the next moment continuously; if not, the production process object at the current sampling moment enters an abnormal operation state.
As shown in FIG. 3, the inventive method is in detail compared with the ICA-based conventional non-Gaussian process monitoring method in monitoring TE process faults. As is apparent from the comparison in fig. 3, the method of the present invention is significantly superior to the conventional method in failure detection success rate. Thus, it can be said that the method of the present invention has more reliable process monitoring performance.
The above embodiments are merely illustrative of specific implementations of the invention and are not intended to limit the invention. Any modification made to the present invention that comes within the spirit of the present invention and the scope of the appended claims falls within the scope of the present invention.

Claims (2)

1. The novel distributed non-Gaussian process monitoring method based on the GA-ICA is characterized by comprising the following steps of:
the offline modeling stage includes steps (1) to (7):
step (1): under the normal operation state of the production process object, n sample data x are collected 1 ，x 2 ，…，x n Composition of training data matrix x= [ X ] 1 ，x 2 ，…，x n ]∈R m×n And calculates the mean vector μ= (x) 1 +x 2 +…+x n ) N and standard deviation vector delta epsilon R m ×1 ：
Wherein, the ". Is a vector (x i - μ) and (x) i - μ) corresponding element multiplication, x i ∈R m×1 Represents the i-th sample data, m is the total number of measured variables, i=1, 2, …, n, R is the real set, R m×n Representing a real matrix in m x n dimensions;
step (2): according to the formulaFor sample data x 1 ，x 2 ，…，x n Respectively performing normalization processing to obtain matrix +.>Wherein (1)>Representing vector (x) i Mu) is divided by the corresponding element in the standard deviation vector delta,is a normalized data vector;
step (3): according to production processDetermining the measurement variables related to each production unit by the composition structure of the process object, dividing the m measurement variables into B variable sub-blocks, and dividing the matrix according to the B variable sub-blocksSelecting corresponding row vectors to respectively form B sub-block matrixes +.>
Step (4): setting the number of the separation vectors as D, and solving a matrix by utilizing a decimal genetic algorithmD separation vectors w of (2) 1 ，w 2 ，…，w D D mixing vectors p 1 ，p 2 ，…，p D And D independent component vectors t 1 ，t 2 ，…，t D ；
Step (5): according to D separation vectors w 1 ，w 2 ，…，w D Sequentially for B variable sub-block matricesThe extraction of the independent components is implemented, and the specific implementation process is as follows:
step (5.1): initializing d=1;
step (5.2): separating vector w from d-th variable sub-block according to B-th variable sub-blocks in step (3) d Selecting corresponding elements to respectively form separation vectors w corresponding to the B variable sub-blocks d (1)，w d (2)，…，w d (B)；
Step (5.2): for all values of B epsilon {1,2, …, B }, judging whether the conditions are satisfied or not respectively:if not, then the formula +.>And->Calculating independent component vectors t corresponding to the b-th variable sub-block respectively d (b) And a load vector a d (b) Then, jumping to the step (5.3); if yes, the separation vector w corresponding to the b variable sub-block is obtained 1 (b)，w 2 (b)，…，w d (b) Form a separation matrix W b And respectively forming an independent component matrix T by all independent component vectors and all load vectors corresponding to the calculated b variable sub-block b And a load matrix P b Then, jumping to the step (5.4);
step (5.3): according to the formulaUpdating the b-th variable sub-block matrix +.>Then, jumping to the step (5.5);
step (5.4): reserving an independent component matrix T corresponding to the b-th variable sub-block b Separation matrix W b And a load matrix P b ；
Step (5.5): judging whether the condition is satisfied: d < D? If yes, after d=d+1, returning to the step (5.2); if not, obtaining separation matrixes W corresponding to all B variable sub-blocks 1 ，W 2 ，…，W B Load matrix P 1 ，P 2 ，…，P B ；
Step (6): firstly, according to a formula lambda b ＝T b T b T (n-1) calculation of covariance matrix Λ 1 ，Λ 1 ，…，Λ B Then according to formula Q b ＝diag(T b T Λ b -1 T b ) Calculating and monitoring a statistical index vector Q 1 ，Q 2 ，…，Q B Wherein diag () tableIllustrating the operation of converting elements of a diagonal of a matrix into vectors, the reference T above indicates the transpose of the matrix or vector;
step (7): respectively for monitoring statistical index vector Q 1 ，Q 2 ，…，Q B The elements in the variable sub-blocks are arranged in descending order according to the size, and the n/100 th maximum value is respectively used as the upper limit CL of the monitoring statistical index of each variable sub-block 1 ，CL 2 ，…，CL B ；
The online monitoring stage performs the following steps:
step (8): collecting sample data x epsilon R at new sampling time m×1 And according to the formulaThe normalization of x is performed to obtain a vector +.>
Step (9): from the B variable sub-blocks in step (3), the vector can be correspondingly dividedDivided into B sub-vectors
Step (10): firstly according to the formulaRespectively calculating each subvector +.>Corresponding independent component vector y 1 ，y 2 ，…，y B Then according to formula->Calculating monitoring index->
Step (11): judging whether the condition is satisfied or not for all values of the index b:if yes, the production process object at the current sampling moment is normally operated, and the step (8) is returned to monitor the sample data at the next moment continuously; if not, the production process object at the current sampling moment enters an abnormal operation state.
2. The novel distributed non-gaussian process monitoring method based on GA-ICA according to claim 1, wherein the detailed implementation process of step (4) is specifically:
step (4.1): setting the number D of the separation vectors and the maximum iteration number U of the genetic algorithm, initializing d=1 and u=0, and randomly generating an N x m-dimensional real matrix W on the interval [ -1,1], wherein N is the population number;
step (4.2): according to the formulaSolving the matrix F epsilon R N×n And calculates the mean vector f= (F) of n column vectors in the matrix F 1 +F 2 +…+F n ) N, wherein F i Representing the ith column vector in matrix F;
step (4.3): finding the maximum value f of the elements in the mean vector f max And recording the position v of the maximum value in the mean vector f;
step (4.4): judging whether the condition U < U is satisfied? If yes, after u=u+1 is set, jumping to the step (4.5); if not, the upsilon line vector W (upsilon) in the real number matrix W is recorded as the d separation vector W d And jumps to step (4.13);
step (4.5): in turn according to the formula gamma k ＝f k /(f 1 +f 2 +…+f N ) And formula (VI)Calculating the selection probability gamma k And cumulative probability->Where k=1, 2, …, N, f k Represents the kth element in the mean vector f;
step (4.6): randomly generating a number s in the interval (0, 1), thenFind out the meeting conditionMinimum value +.>And reserving the kth row vector in the real matrix W;
step (4.7): repeating the step (4.6) for N times, thereby forming a matrix V epsilon R by the reserved N row vectors N×m And initializing j=1;
step (4.8): after randomly generating a number c in the interval (0, 1), judging whether the condition is satisfied:wherein->Representing the crossover probability; if yes, the j-th row vector V in the matrix V is calculated according to the following formula (2) j And the j+1th row vector V j+1 Performing a crossover operation:
in the above, θ j Randomly generated one for interval (0, 1)A number; if not, the j-th row vector V in the matrix V j And the j+1th row vector V j+1 Remain unchanged;
step (4.9): judging whether the condition is satisfied: j < N-1? If yes, after j=j+2, returning to the step (4.8); if not, obtaining a matrix V after the cross operation is implemented, and initializing k=1;
step (4.10): after randomly generating a number g in the interval (0, 1), judging whether the condition is satisfied:wherein->Representing the probability of variation, suggesting a value +.>If yes, the k-th row vector V in the matrix V is calculated according to the following formula (3) k The mutation operation is performed on any one element V (k, epsilon):
in the above formula, epsilon represents that the arbitrarily selected element V (k, epsilon) is represented by the vector V k The position of r k Function h (u) =r for a number randomly generated over interval (0, 1) k (1-u/U) 2 The method comprises the steps of carrying out a first treatment on the surface of the If not, keeping the k-th row vector V (k) in the matrix V unchanged;
step (4.11): judging whether the condition is satisfied: k < N? If yes, after setting k=k+1, returning to the step (4.10); if not, obtaining a matrix V after the mutation operation is implemented;
step (4.12): according to formula V N Update of row vector V of last row in matrix V =w (V) N Wherein W (V) represents a V-th row vector in the real matrix W, and after setting the real matrix w=v, returning to the step (4.2);
step (4.13): judging whether the condition is satisfied: d < D? If it isThen in turn according to the formula And->Respectively calculating the d independent component vector t d Hybrid vector p d And update matrix->D=d+1 and u=0 are reset and are within the interval [ -1,1]Randomly generating an Nxm b Returning to the step (4.2) after the real matrix W of the dimension; if not, all D separation vectors w are obtained 1 ，w 2 ，…，w D D mixing vectors p 1 ，p 2 ，…，p D And D independent component vectors t 1 ，t 2 ，…，t D 。
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