Penicillin fermentation process penicillin concentration modeling method based on dual-network recursive interval type-2 fuzzy neural network

Abstract

A penicillin concentration modeling method in a penicillin fermentation process based on a dual-network recursive interval type-2 fuzzy neural network belongs to the field of intelligent control and industrial fault diagnosis and prediction maintenance, and comprises the following steps: step 1, establishing the dual-network recursive interval type-2 fuzzy neural network based on singular value decomposition total least square optimization, generating a network and a state evaluation network for the output; 2, optimizing a network structure by adopting a rule generation algorithm with online learning capability; and step 3, optimizing parameters of the dual-network recursion interval type-2 fuzzy neural network model based on the overall least square optimization of singular value decomposition by adopting a hybrid method of a non-dominated sorting genetic algorithm II and the overall least square optimization of singular value decomposition, and carrying out penicillin concentration modeling in the penicillin fermentation process. According to the method, the penicillin concentration can be quickly and accurately predicted.

Description

Technical Field

[0001] The present invention belongs to the fields of intelligent control and bioprocess engineering, and particularly relates to a method for modeling the penicillin concentration in the penicillin fermentation process based on a dual-network recursive interval type-2 fuzzy neural network. Background Art

[0002] The penicillin fermentation process is one of the classic applications with important practical significance in the field of bioengineering. During this process, the production of penicillin involves complex biochemical reactions and process control, so it is crucial to accurately model and optimize it. The penicillin fermentation process is usually divided into two main stages: the first stage is the growth stage of penicillin, during which penicillin bacteria consume a large amount of nutrients in the substrate and proliferate rapidly, but hardly produce penicillin; the second stage is the synthesis stage of penicillin, which is the key link in industrial production. The core goal of industrial production is to promote the synthesis of penicillin by preventing substrate inhibition, controlling the fermentation environment, inhibiting product autolysis, etc., so as to improve production efficiency. In order to accurately simulate and optimize the penicillin fermentation process, the Pensim fermentation simulation platform is widely used. Pensim is a powerful simulation tool that can accurately simulate various biochemical reactions and parameter changes during fermentation. It can provide rich process variables and output data, providing a reliable basis for the optimization design of researchers and engineers in experiments and production. The output variables of the Pensim fermentation simulation platform include six: the substrate concentration in the fermenter, the biomass concentration, the penicillin product concentration, the dissolved oxygen concentration, the carbon dioxide concentration in the tank, and the volume of the fermenter reactor. Among them, the penicillin product concentration is the most concerned in the industrial production process because it not only directly determines the fermentation rate, but also significantly affects the production cost and overall economic benefits.

[0003] However, traditional fuzzy neural networks and interval type-2 fuzzy neural networks usually adopt a feedforward network structure and lack the memory ability to process historical information, which limits the performance of the model in the concentration identification and prediction accuracy of penicillin products, resulting in problems such as difficult prediction, slow calculation speed, and poor accuracy of the penicillin concentration in the penicillin fermentation process. Summary of the Invention

[0004] In view of the above deficiencies of the existing technology, the present invention provides a method for modeling the penicillin concentration in the penicillin fermentation process based on a dual-network recursive interval type-2 fuzzy neural network, aiming to solve the problems of difficult prediction, slow calculation speed, and poor accuracy of the penicillin concentration in the penicillin fermentation process.

[0005] The present invention is realized by the following method:

[0006] A method for modeling the penicillin concentration in the penicillin fermentation process based on a dual-network recursive interval type-2 fuzzy neural network model, comprising the following steps:

[0007] Step 1: Establish a dual-network recursive interval type-2 fuzzy neural network model based on total least squares optimization with singular value decomposition for penicillin concentration prediction in the penicillin fermentation process. The model consists of two Takagi-Sugeno type interval type-2 fuzzy neural networks, namely an output generation network and a state evaluation network. Both networks are composed of six layers. Among them, after the output of the state evaluation network is processed by the membership function, the result is fed back to the output generation network;

[0008] Step 2: Adopt a variability adaptive clustering algorithm. The algorithm adaptively clusters by dynamically splitting the clustering centers to adapt to the data distribution and uses variance as the splitting criterion, thereby realizing adaptive clustering, that is, optimizing the network structure using a rule generation algorithm with online learning ability;

[0009] Step 3: Adopt a hybrid method of non-dominated sorting genetic algorithm II and total least squares with singular value decomposition to optimize the parameters of the dual-network recursive interval type-2 fuzzy neural network model based on total least squares optimization with singular value decomposition. Use the non-dominated sorting genetic algorithm II to optimize the antecedent parameters of the rules and the total least squares method with singular value decomposition to optimize the consequent parameters of the rules.

[0010] Furthermore, the output generation network is divided into an input layer, a membership function layer, a rule layer, a defuzzification layer, a recursive layer, and an output layer; the state evaluation network is divided into an input layer, a membership function layer, a rule layer, a defuzzification layer, a state output layer, and a membership function mapping layer.

[0011] Furthermore, Step 1 specifically includes the following steps:

[0012] Step 1.1: The first layer is the input layer. In the input layer, the input variables x = [x 1 , x 2 ,..., x I are received. This layer has I nodes, and each node represents an input. The function of this layer is to directly transfer the input signal to the next layer for fuzzy calculation, so no processing or calculation is performed in this layer. The output formula of this layer is as follows:

[0013] y0 i (1) = y (1) (x i (t)) = x i (t)

[0014] Step 1.2: The second layer is the membership function layer. The membership function layer is also called the fuzzification layer. Each node in this layer uses a Gaussian membership function to fuzzify the input, and each node represents the calculation of a membership function. The formula is as follows:

[0015]

[0016] Among them, a ij (1 ≤ i ≤ I, 1 ≤ j ≤ J) is the mean value of the j-th fuzzy rule for the i-th input variable, and s ij are the standard deviations of the upper and lower membership functions. a ij 、 and s ij are the antecedent parameters of the fuzzy rules. The function of this layer is to calculate the values of the input variables on the membership functions, and the output formula is as follows:

[0017]

[0018] Step 1.3: The third layer is the rule layer. The rule layer has J nodes, and each node represents a rule. The output of this layer is as follows:

[0019]

[0020] Among them,

[0021]

[0022] Step 1.4: The fourth layer is the defuzzification layer. The defuzzification layer has two nodes corresponding to the lower output and the upper output respectively, and the output is a defuzzified set [y l , y r , which mainly completes the defuzzification operation of the interval type-2 fuzzy neural network. The output formula of this layer is as follows:

[0023] yo (4) = y (4) (yo j (3) ) = [y l , y r

[0024] Among them,

[0025]

[0026] Step 1.5: The fifth layer is the state output layer in the state evaluation network. There is only one node in the state output layer. Using the Begian-Melek-Mendel (BMM) defuzzification method, the weight coefficient q (0 ≤ q ≤ 1) can be used to change the ratio of y l and y r to improve the accuracy of the network. The output yo s (5) can be expressed as follows:

[0027] ​

[0028] Step 1.6: The sixth layer is the membership function mapping layer in the state evaluation network. Through this membership function mapping layer, the membership degree of each input variable under a specific membership function can be determined more accurately, thereby improving the overall performance and accuracy of the system. The membership function here can be crisp or fuzzy. Using a crisp membership function, the state of the network becomes a discrete state, and using a fuzzy membership function, the state of the network becomes a continuous state. Assuming g(x) is the membership function, the output of this layer is as follows:

[0029] yo s1 (6) = g 1 (yo s (5) )

[0030] yo s2 (6) = g 2 (yo s (5) );

[0031] Step 1.7: The fifth layer is the recurrent layer in the output generation network. In this recurrent layer, it can accept the output feedback yo s1 (6) and yo s2 (6) . This process not only helps the system generate the current output but also effectively integrates and remembers the historical information of past observations. Through this feedback mechanism, the system can dynamically adjust its parameters, optimize the membership degree calculation of input variables, and improve the overall performance and accuracy. In addition, the output generation network can use the historical information provided by the state evaluation network to more accurately predict and respond to future input changes, ensuring the stability and reliability of the system in a complex environment. This design enables the system to have good memory and adaptive capabilities and can continuously optimize its operating state. The output of this layer is as follows:

[0032]

[0033] Step 1.8: The sixth layer is the output layer in the output generation network. The output of this output layer is the output of the entire model, and the output of this layer is as follows:

[0034]

[0035] Furthermore, the specific steps of the said Step 2 include the following steps:

[0036] Step 2.1: Initialization. Start from the average value of all data samples and set an initial clustering center. Specifically, calculate the average value of all samples to initialize the first clustering center. The initialization formula is as follows:

[0037]

[0038] where Clusters is the clustering center matrix, DataNum is the number of samples, and Data is the input data matrix;

[0039] Step 2.2: Sample assignment. Calculate the distance from each sample to all clustering centers and assign the sample to the nearest clustering center. The specific steps are as follows:

[0040] (i) Calculate the Euclidean distance from each sample to all clustering centers.

[0041] Distace(i,j)=||Data(i,:)-Clusters(j,:)||

[0042] where i represents the sample index and j represents the clustering center index.

[0043] (ii) Assign the sample to the nearest clustering center.

[0044] result(i,:)=[Data(i,:),index]

[0045] where index is the index of the clustering center with the minimum distance.

[0046] Step 2.3: Variance calculation and clustering center splitting. For each clustering center, calculate the variance of its internal samples as the basis for splitting. The specific steps are as follows:

[0047] (i) Calculate the variance: For each clustering center, calculate the variance of its internal samples:

[0048]

[0049] where Point represents the set of samples belonging to the i-th clustering center, and PointNum is the number of samples in the clustering center. The algorithm selects the clustering center with the largest variance for splitting. If the maximum variance is greater than the set threshold Vth, perform a splitting operation on this clustering center and split it into two new clustering centers;

[0050] (ii) Select the maximum variance: Select the clustering center with the largest variance for splitting:

[0051] [maxVariance,SplitPos]=max(VarianceSum)

[0052] Among them, VarianceSum is the sum of variances of all cluster centers;

[0053] (iii) Cluster center splitting: If the maximum variance is greater than the set threshold Vth, perform a splitting operation on this cluster center and split it into two new cluster centers:

[0054] Clusters(SplitPos,:) = Clusters(SplitPos,:) - ε

[0055] Clusters(ClusterNum + 1,:) = Clusters(SplitPos,:) + ε

[0056] Among them, ε is a small positive number used to generate new cluster centers;

[0057] Step 2.4: Iterative update. After each split, the algorithm enters the iterative update stage to recalculate each cluster center. The specific steps are as follows:

[0058] (i) Sample reallocation: Using the minimum distance principle, allocate samples to the new cluster centers and calculate the distances from each sample to all cluster centers:

[0059] CurrentDistace(i,j) = ||Data(i,:) - Clusters(j,:)||

[0060] (ii) Update cluster centers: Update the positions of the cluster centers according to the sample allocation situation:

[0061]

[0062] Among them, quan is an indicator matrix used to represent which cluster a sample belongs to;

[0063] (iii) RAF algorithm convergence judgment: Judge whether the position changes of all cluster centers are less than the set threshold ε:

[0064] ||OldClusters - NewClusters|| < ε

[0065] If the condition is met, the algorithm terminates; otherwise, return to Step 2 to continue the iteration.

[0066] Furthermore, the said Step 3 includes Step 3.1 antecedent parameter optimization and Step 3.2 consequent parameter optimization;

[0067] Step 3.1: The said antecedent parameter optimization includes the following steps:

[0068] Step 3.1.1: Initialize the population. In NSGA-II, first, randomly generate an initial population P 0 , with a population size of N p ; each individual consists of multiple genes, and these genes correspond to the antecedent parameters a ij , and s ij :

[0069] The mean vector of the antecedent membership function is expressed as follows:

[0070]

[0071] The standard deviation vector of the membership function is expressed as follows:

[0072]

[0073] N p The N solution vectors are expressed as follows:

[0074]

[0075] The initialized population is expressed as:

[0076]

[0077] where k = 1, 2,..., N p ;

[0078] Step 3.1.2: Calculate the objective function. For each individual in the population, decode its genes and map them to the corresponding model; then, use the decoded parameters to train and validate the model, and calculate the mean square error, smoothness, and complexity of the objective function;

[0079]

[0080] f 3 (t) = p(t) + r(t)

[0081] where N is the number of training samples, y(t) is the true output value, is the output of the model, p(t) represents the number of model parameters, and r(t) represents the number of fuzzy rules;

[0082] Step 3.1.3: Non-dominated sorting and crowding distance calculation. Perform non-dominated sorting on the calculated objective function values and calculate the crowding distance. Non-dominated sorting is one of the core steps of the NSGA-II algorithm; its purpose is to divide the population into several levels, and the individuals in each level are not dominated by the individuals in other levels; for each individual x i in the population, calculate the number of individuals n that it dominatesi and the set S of individuals dominated by it i ; If for two individuals x i and x j , the following is satisfied:

[0083]

[0084] and

[0085]

[0086] then x is said to dominate x i ; j ;

[0087] Let F i be the non-dominated front at the i-th layer, and its crowding degree calculation steps are as follows:

[0088] (ii) Initialize the crowding degree d of each individual i = 0;

[0089] (ii) For each objective function f k :

[0090] Sort the individuals in this layer according to the objective function value f k ;

[0091] For the sorted individuals, set the crowding degree of the boundary individuals to infinity;

[0092] For non-boundary individuals, calculate their crowding degree according to the difference in objective function values of adjacent individuals:

[0093]

[0094] where f k max and f k min are the maximum and minimum values of the k-th objective function in this layer respectively;

[0095] Step 3.1.4: Selection, crossover, and mutation. In the evolutionary process of each generation, use selection, crossover, and mutation operations to generate a new offspring population;

[0096] Step 3.1.5: Population update. Merge the current population and the generated offspring population, and perform non-dominated sorting. Select the first Popsize individuals from the merged population as the next generation population according to the non-dominated rank and crowding degree;

[0097] Step 3.1.6: Repeat steps 3.1.4 to 3.1.6 until the predetermined maximum number of iterations MaxGenerations is reached;

[0098] Step 3.2: The consequent parameter optimization includes the following steps:

[0099] The output of the dual-network recursive interval type-2 fuzzy neural network model optimized by total least squares based on singular value decomposition and the consequent parameter w ij have a linear relationship. The consequent parameters can be optimized by the total least squares method based on singular value decomposition. For the known N training data T t =(x(t), y(t)), the network output is:

[0100]

[0101] q ∈ (0, 1), p' t =[x 0 , x t =[x 0 , x 1 ,..., x I , x 0 =1, w j =[w 0j , w 1j ,..., w Ij , where X is the consequent parameter and also the parameter to be optimized, X = [w 1 , w 2 ,..., w J T , denotes the Kronecker product; for N groups of training data, the loss function can be defined as E(X) as follows:

[0102] E(X)=||Q - BX||

[0103] where,

[0104] Q = [y 1 , y 2 ,..., y N T

[0105]

[0106] X = [w 1 , w 2 ,..., w J T

[0107] The basic idea of using total least squares is to use the error matrices ΔB and ΔQ to perturb B and Q, which makes the loss function become:

[0108] E(X)=||(Q + ΔQ)-(B + ΔB)X||​​​

[0109] In the absence of explicit error matrices ΔB and ΔQ, the process of using singular value decomposition to implement total least squares automatically determines these errors by processing an augmented matrix; first, construct the augmented matrix A:

[0110] A = [B | -Q]

[0111] Perform singular value decomposition on the augmented matrix A:

[0112] A = UΣV T

[0113] where U and V are orthogonal matrices, Σ is a diagonal matrix, and the diagonal elements are the singular values of A; then identify the smallest singular value and its corresponding singular vector. In the result of singular value decomposition, the smallest singular value usually corresponds to the direction of noise or error in the system; the right singular vector v corresponding to the smallest singular value min is the last column of V; next, process the right singular vector, and divide the right singular vector v corresponding to the smallest singular value min into two parts:

[0114]

[0115] Finally, correct matrices B and Q, which can be updated as follows:

[0116]

[0117] After correction, the error function becomes as follows according to formulas (37) and (41):

[0118]

[0119] The goal is to make the error function E(X) as small as possible. Next, the least squares estimation based on singular value decomposition will be given; first, construct the augmented matrix H:

[0120]

[0121] Perform singular value decomposition on the augmented matrix H:

[0122] H = SVD T

[0123] where S and D are orthogonal matrices, V is a diagonal matrix, and the diagonal elements are the singular values of H; determine the effective rank p of H, and calculate the (p + 1)×(p + 1) matrix M (p) :

[0124]

[0125] where,

[0126]

[0127] Here, v(i,k) is the element in the i-th row and k-th column of the unitary matrix D, and σ j is the singular value; then find the inverse matrix M (p) of M -(p) , and obtain the optimal least-squares approximation solution of formula (47) from formula (52):

[0128]

[0129] Compared with the prior art, the beneficial effects of the present invention are as follows: First, the structure learning algorithm of the model has the self-evolution characteristic, which means that it can automatically adjust and optimize the network structure (including the number of rules) and related parameters according to the training data samples; Second, the dual-network recursive interval type-2 fuzzy neural network has the ability to process historical information and can effectively process time-series data. The dual-network structure enhances the modeling ability of the model for dynamic and complex systems through the recursive mechanism and can more accurately capture the time-varying characteristics of the system. The interval type-2 fuzzy logic improves the robustness and noise resistance of the model in complex environments by dealing with parameter uncertainties. Furthermore, in terms of the parameter learning of the model, the overall least-squares method combining the non-dominated sorting genetic algorithm II and the singular value decomposition has significant advantages in optimizing the antecedent parameters and consequent parameters. As an efficient multi-objective optimization algorithm, the non-dominated sorting genetic algorithm II finds the optimal solution set through non-dominated sorting, can optimize multiple objective functions simultaneously when dealing with complex problems, effectively balance the accuracy, smoothness and complexity of the model, and thus achieve more reliable fault diagnosis and prediction in practical applications. The overall least-squares method of singular value decomposition reduces the computational complexity through singular value decomposition, effectively deals with the multicollinearity problem in high-dimensional data, and thus improves the accuracy of parameter estimation. The combined application of these two methods not only improves the optimization effect of the model, but also improves the computational efficiency, ensuring the robustness and accuracy of the final model when dealing with complex systems. The present invention can ensure the accuracy of penicillin concentration prediction in the penicillin fermentation process. BRIEF DESCRIPTION OF THE DRAWINGS

[0130] Figure 1 is the structural diagram of the dual-network recursive interval type-2 fuzzy neural network optimized based on the overall least-squares of singular value decomposition;

[0131] Figure 2 is the training diagram of penicillin concentration in the penicillin fermentation process;

[0132] Figure 3 is the training error diagram of penicillin concentration in the penicillin fermentation process;

[0133] Figure 4 is the test diagram of penicillin concentration in the penicillin fermentation process;

[0134] Figure 5 It is a graph of the test error of penicillin concentration in the penicillin fermentation process;

[0135] Figure 6 It is a graph of the coefficient of determination in the training process;

[0136] Figure 7 It is a graph of the coefficient of determination in the test process. Detailed implementation manners

[0137] The present invention will be further described below in conjunction with the accompanying drawings and embodiments.

[0138] The concept, specific structure and technical effects generated of the present disclosure will be clearly and completely described below with reference to the accompanying drawings, so as to fully understand the purpose, solution and effects of the present invention. It should be noted that, without conflict, the embodiments in the present application and the features in the embodiments may be combined with each other.

[0139] In this embodiment, the penicillin concentration in the penicillin fermentation process is predicted.

[0140] First, 400 groups of data obtained from the Pensim simulation platform of the penicillin fermentation process are used as the data source for the simulation experiment and input samples based on the double-network recursive interval type-2 fuzzy neural network optimized by total least squares based on singular value decomposition, and the output of the fuzzy neural network is calculated according to step 1; then, the structure of the double-network recursive interval type-2 fuzzy neural network optimized by total least squares based on singular value decomposition is adjusted according to step 2; next, the parameters of the double-network recursive interval type-2 fuzzy neural network optimized by total least squares based on singular value decomposition are adjusted according to step 3; finally, the trained double-network recursive interval type-2 fuzzy neural network optimized by total least squares based on singular value decomposition is used to predict the penicillin concentration in the penicillin process.

[0141] To verify the effectiveness of the inventive method, under normal conditions set on the penicillin fermentation simulation platform Pensim, the simulation time is set to 300 h, and 400 groups of data of the required auxiliary variables and dominant variables are extracted from the generated simulation data as the training sample data for model establishment. In addition, it is set to run for 300 h under normal conditions, and 400 groups of data are selected at equal time intervals starting from the 0th moment for testing and prediction. After the training process of 400 groups of data, 3 rules are generated.

[0142] Figure 1 Shows the structure diagram of the double-network recursive interval type-2 fuzzy neural network optimized by total least squares based on singular value decomposition, Figure 2 Shows the training graph of the penicillin concentration in the penicillin fermentation process, Figure 3 Shows the training error graph, such as Figure 2 、3 As shown, the training curve of the dual-network recursive interval type-2 fuzzy neural network model optimized by total least squares based on singular value decomposition is basically fitted with the penicillin concentration data curve measured by the simulation platform, indicating that the model has good accuracy and fitting ability in predicting penicillin concentration. Figure 4 The penicillin concentration test chart of the penicillin fermentation process is shown, Figure 5 The test error chart is shown, as Figure 4 、 5 As shown, the high degree of coincidence between the output and the true value curve indicates that the model can effectively process unseen data sets and successfully predict the change trend of penicillin concentration. This accurate prediction ability indicates that the model not only successfully captures the internal laws of the data during the training process, but also effectively applies this regularity to new data. Figure 6 The coefficient of determination chart during the training process is shown, Figure 7 The coefficient of determination chart during the test process is shown, as Figure 6 、 7 As shown, the coefficient of determination on the training set is 0.997, indicating that the model fits the training data very well and can almost explain all variances. The coefficient of determination on the test set is 0.989. Although slightly lower than the training set, it is still very close to 1, indicating that the model also has strong generalization ability on new data. This high consistency indicates that the model performs excellently in capturing data patterns and avoids obvious overfitting problems, with good stability and reliability.

[0143] Only some exemplary embodiments of the present invention are described by way of illustration above. Undoubtedly, for those of ordinary skill in the art, the described embodiments can be modified in various different ways without departing from the spirit and scope of the present invention. Therefore, the above drawings and descriptions are illustrative in nature and should not be construed as limiting the protection scope of the claims of the present invention.

Claims

1. A penicillin concentration modeling method for penicillin fermentation process based on a dual-network recursive interval type-2 fuzzy neural network model, characterized in that: The following steps are involved: Step 1: Establish a dual-network recursive interval type-2 fuzzy neural network model based on total least squares optimization of singular value decomposition for penicillin concentration prediction in penicillin fermentation process, the model consists of two Takagi-Sugeno type interval type-2 fuzzy neural networks, namely an output generation network and a state evaluation network, both networks are composed of six levels, wherein the output of the state evaluation network is processed by a membership function and the result is fed back to the output generation network; Step 2: Adopting a variability adaptive clustering algorithm, the algorithm dynamically splits the cluster center to adapt to the distribution of data and uses variance as a splitting criterion to achieve adaptive clustering, that is, using a rule generation algorithm with online learning capability to optimize the network structure; Step 3: A hybrid method of non-dominated sorting genetic algorithm II and total least squares of singular value decomposition is used to optimize the parameters of the dual-network recursive interval type-2 fuzzy neural network model based on total least squares optimization of singular value decomposition. The non-dominated sorting genetic algorithm II is used to optimize the rule antecedent parameters, and the total least squares method of singular value decomposition is used to optimize the rule consequent parameters.

2. The penicillin concentration modeling method of penicillin fermentation process based on dual network recursive interval type-2 fuzzy neural network model according to claim 1 is characterized in that: The output generation network is divided into an input layer, a membership function layer, a rule layer, a reduction layer, a recursive layer and an output layer; the state evaluation network is divided into an input layer, a membership function layer, a rule layer, a reduction layer, a state output layer and a membership function mapping layer.

3. The penicillin concentration modeling method for penicillin fermentation process based on dual network recursive interval type-2 fuzzy neural network model according to claim 2 is characterized in that: The step 1 specifically comprises the following steps: Step 1.1: The first layer is the input layer, which accepts input variables x = [x1, x2, ..., x I ], this layer has I nodes, each node represents an input, the role of this layer is to pass the input signal directly to the next layer for fuzzy calculation, so no processing or calculation is performed in this layer. The output formula of this layer is as follows: I i (1) =and (1) (x i (t))=x i (t) Step 1.2: The second layer is the membership function layer, which is also called the fuzzification layer. Each node in this layer uses a Gaussian membership function to fuzzify the input, and each node represents the calculation of a membership function. The formula is as follows: Among them, a ij (1≤i≤I,1≤j≤J) is the mean of the jth fuzzy rule for the ith input variable, and s ij are the upper and lower membership function standard deviations. a ij , and s ij is the antecedent parameter of the fuzzy rule. The function of this layer is to calculate the value of the input variable on the membership function, and the output formula is as follows: Step 1.3: The third layer is the rule layer, which has J nodes, each of which represents a rule. The output of this layer is as follows: in, Step 1.4: The fourth layer is a reduction layer, which has two nodes corresponding to the lower output and the upper output respectively, and the output is a reduction set [y l ,y r ], mainly completes the reduction operation of interval type-2 fuzzy neural network. The output formula of this layer is as follows: I (4) =and (4) (I j (3) )=[and l ,and r ] in, Step 1.5: The fifth layer is the state output layer in the state evaluation network. There is only one node in the state output layer. The Begian-Melek-Mendel (BMM) defuzzification method is used. The weight coefficient q (0≤q≤1) can be used to change y l and r The ratio of to improve the accuracy of the network. Output yo s (5) It can be expressed as follows: Step 1.6: The sixth layer is the membership function mapping layer in the state assessment network. Through the membership function mapping layer, the degree of membership of each input variable under a specific membership function can be determined more accurately, thereby improving the overall performance and accuracy of the system. The membership function here can be clear or fuzzy. Using a clear membership function, the state of the network becomes a discrete state, and using a fuzzy membership function, the state of the network becomes a continuous state. Assuming g(x) is the membership function, the output of this layer is as follows: I s1 (6) =g1(I s (5) ) I s2 (6) =g2(I s (5) ); Step 1.7: The fifth layer is a recurrent layer in the output generation network, in which the output feedback yo from the state evaluation network can be received s1 (6) and yo s2 (6) This process not only helps the system generate the current output, but also effectively integrates and remembers the historical information of past observations; through this feedback mechanism, the system can dynamically adjust its parameters, optimize the membership calculation of the input variables, and improve the overall performance and accuracy; in addition, the output generation network uses the historical information provided by the state evaluation network to more accurately predict and respond to future input changes, ensuring the stability and reliability of the system in complex environments. This design enables the system to have good memory and adaptive capabilities, and can continuously optimize its operating status. The output of this layer is as follows: Step 1.8: The sixth layer is the output layer in the output generation network. The output layer is the output of the entire model. The output of this layer is as follows:

4. The penicillin concentration modeling method for penicillin fermentation process based on dual network recursive interval type-2 fuzzy neural network model according to claim 3 is characterized in that: The step 2 specifically includes the following steps: Step 2.1: Initialization, starting from the average value of all data samples, set an initial cluster center. Specifically, initialize the first cluster center by calculating the average value of all samples. The initialization formula is as follows: Among them, Clusters is the cluster center matrix, DataNum is the number of samples, and Data is the input data matrix; Step 2.2: Sample allocation, calculate the distance from each sample to all cluster centers, and assign the sample to the cluster center closest to it. The specific steps are as follows: (i) Calculate the Euclidean distance from each sample to all cluster centers. Distace(i,j)=||Data(i,:)-Clusters(j,:)|| Among them, i represents the sample index and j represents the cluster center index. (ii) Assign the sample to the cluster center that is closest to it. result(i,:)=[Data(i,:),index] Among them, index is the index of the cluster center with the smallest distance. Step 2.3: Variance calculation and cluster center splitting. For each cluster center, calculate the variance of its internal samples as the basis for splitting. The specific steps are as follows: (i) Calculate variance: For each cluster center, calculate the variance of its internal samples: Where Point represents the sample set belonging to the i-th cluster center, and PointNum is the number of samples in the cluster center. The algorithm selects the cluster center with the largest variance for splitting. If the maximum variance is greater than the set threshold Vth, the cluster center is split into two new cluster centers. (ii) Select the maximum variance: Select the cluster center with the largest variance for splitting: [maxVariance,SplitPos]=max(VarianceSum) Among them, VarianceSum is the sum of the variances of all cluster centers; (iii) Cluster center splitting: If the maximum variance is greater than the set threshold Vth, the cluster center is split into two new cluster centers: Clusters(SplitPos,:)=Clusters(SplitPos,:)-ε Clusters(ClusterNum+1,:)=Clusters(SplitPos,:)+ε Among them, ε is a small positive number used to generate new cluster centers; Step 2.4: Iterative update: After each split, the algorithm enters the iterative update phase and recalculates each cluster center. The specific steps are as follows: (i) Sample redistribution: Use the minimum distance principle to assign samples to new cluster centers and calculate the distance from each sample to all cluster centers: CurrentDistace(i,j)=||Data(i,:)-Clusters(j,:)|| (ii) Update the cluster center: Update the location of the cluster center according to the sample distribution: Among them, quan is an indicator matrix used to indicate which cluster the sample belongs to; (iii) RAF algorithm convergence judgment: Determine whether the position change of all cluster centers is less than the set threshold ε: ||OldClusters-NewClusters||<ε If the conditions are met, the algorithm terminates; otherwise, it returns to Step 2 and continues to iterate.

5. The method for modeling penicillin concentration in penicillin fermentation process based on dual network recursive interval type-2 fuzzy neural network model according to claim 4, characterized in that: The step 3 includes step 3.1 antecedent parameter optimization and step 3.2 subsequent parameter optimization; Step 3.1: The prior parameter optimization includes the following steps: Step 3.1.1: Initialize the population. In NSGA-II, first, randomly generate an initial population P0 with a population size of N. p ; Each individual consists of multiple genes, which correspond to the regular antecedent parameters a of the model ij , and s ij : The mean vector of the rule antecedent membership function is expressed as follows: The standard deviation vector of the membership function is expressed as follows: N p The solution vector is expressed as follows: The initial population is expressed as: Where k = 1, 2, ..., N p ; Step 3.1.2: Objective function calculation. For each individual in the population, decode its genes and map them to the corresponding model. Then, use the decoded parameters to train and verify the model, and calculate the mean square error, smoothness, and complexity of the objective function. f3(t)=p(t)+r(t) Where N is the number of training samples, y(t) is the true output value, is the output of the model, p(t) represents the number of model parameters, and r(t) represents the number of fuzzy rules; Step 3.1.3: Non-dominated sorting and crowding calculation. Perform non-dominated sorting on the calculated objective function value and calculate the crowding distance. Non-dominated sorting is one of the core steps of the NSGA-II algorithm; its purpose is to divide the population into several levels, and the individuals in each level are not dominated by the individuals in other levels; for each individual x in the population i , calculate its number of dominant individuals n i and the set of individuals S dominated by it i ; If for two individuals x i and x j ,satisfy: and Then x i dominatex j ; Let F i is the i-th non-dominated layer, and its congestion calculation steps are as follows: (i) Initialize the crowding degree d of each individual i =0; (ii) For each objective function f k : The individuals in this layer are divided into k Sorting; For the sorted individuals, let the crowding degree of the border individuals be infinite; For non-boundary individuals, their crowding degree is calculated based on the difference in the objective function values ​​of adjacent individuals: Among them, f k max and f k min are the maximum and minimum values ​​of the kth objective function in this layer respectively; Step 3.1.4: Selection, crossover and mutation. In each generation of evolution, selection, crossover and mutation operations are used to generate new offspring populations. Step 3.1.5: Population update, merge the current population and the generated offspring population, and perform non-dominated sorting. According to the non-dominated level and crowding degree, select the first Popsize individuals from the merged population as the next generation population; Step 3.1.6: Repeat steps 3.1.4 to 3.1.6 until the predetermined maximum number of iterations MaxGenerations is reached; Step 3.2: The subsequent parameter optimization includes the following steps: Output and rule consequent parameter w of dual network recursive interval type-2 fuzzy neural network model based on total least squares optimization of singular value decomposition ij The linear relationship between the subsequent parameters can be optimized by the total least squares method of singular value decomposition. For the known N training data T t =(x(t),y(t)), the network output is: q∈(0,1),p' t =[x0,x t ]=[x0,x1,...,x I ],x0=1,w j =[w 0j ,w 1j ,...,w Ij ], where X is the subsequent parameter, which is also the parameter that needs to be optimized, X = [w1, w2, ..., w J ] T , Represents the Kronecker product; for N sets of training data, the loss function can be defined as E(X) as follows: E(X)=||Q-BX|| in, Q=[y1,y2,...,y N ] T X=[w1,w2,...,w J ] T The basic idea of ​​using total least squares is to use the error matrices ΔB and ΔQ to interfere with B and Q, which makes the loss function become: E(X)=||(Q+ΔQ)-(B+ΔB)X|| In the absence of explicit error matrices ΔB and ΔQ, the process of implementing total least squares using singular value decomposition is to automatically determine these errors by processing an augmented matrix; first construct the augmented matrix A: A=[B|-Q] Perform singular value decomposition on the augmented matrix A: A=U∑V T Where U and V are orthogonal matrices, ∑ is a diagonal matrix, where the diagonal is the singular value of A; then identify the minimum singular value and its corresponding singular vector. In the result of singular value decomposition, the smallest singular value usually corresponds to the direction of noise or error in the system; the right singular vector v corresponding to the minimum singular value min is the last column of V; next, process the right singular vector and convert the right singular vector v corresponding to the minimum singular value min It is divided into two parts: Finally, the correction matrices B and Q can be updated as follows: After correction, the error function (37) (41) becomes as follows: The goal is to make the error function E(X) as small as possible. Next, the least squares estimate based on singular value decomposition will be given; first, the augmented matrix H is constructed: Perform singular value decomposition on the augmented matrix H: H=SVD T Where S and D are orthogonal matrices, V is a diagonal matrix with the singular values ​​of H on the diagonal; determine the effective rank p of H and calculate the (p+1)×(p+1) matrix M (p) : in, Here, v(i,k) is the element in the i-th row and k-th column of the unitary matrix D, σ j are singular values; Then ask M (p) The inverse matrix M -(p) , and the optimal least squares approximate solution of formula (47) is obtained by formula (52):

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