# Soft Sensing Modeling of the SMB Chromatographic Separation Process Based on the Adaptive Neural Fuzzy Inference System

Authors: Dan Wang; Jie-Sheng Wang; Shao-Yan Wang; Shou-Jiang Li; Zhen Yan; Wei-Zhen Sun
Journal: Journal of Sensors
 (2019)

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## Abstract

Simulated moving bed (SMB) chromatographic separation technology is a new adsorption separation technology with strong separation ability. Based on the principle of the adaptive neural fuzzy inference system (ANFIS), a soft sensing modeling method was proposed for realizing the prediction of the purity of the extract and raffinate components in the SMB chromatographic separation process. The input data space of the established soft sensor model is divided, and the premise parameters are determined by utilizing the meshing partition method, subtractive clustering algorithm, and fuzzyC-means (FCM) clustering algorithm. The gradient, Kalman, Kaczmarz, and PseudoInv algorithms were used to optimize the conclusion parameters of ANFIS soft sensor models so as to predict the purity of the extract and raffinate components in the SMB chromatographic separation process. The simulation results indicate that the proposed ANFIS soft sensor models can effectively predict the key economic and technical indicators of the SMB chromatographic separation process.


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## Body

## 1. Introduction

SMB chromatographic separation technology is a new separation technology developed on the basis of traditional fixed bed adsorption operation and true moving bed (TMB) chromatographic separation technology [1]. As the main modern adsorption separation technology, SMB chromatography has been utilized more and more in the complex mixture separation process, such as petrochemical, fine chemical, biopharmaceutical, and food processing. SMB chromatography technology is the cutting-edge technology in separation science, which preserves the high separation rate, low energy consumption, and low material consumption of the chromatogram. It also introduces the continuous, countercurrent, rectification, reflux, and other mechanisms of moving bed technology. Compared with the existing chemical separation technology (distillation, extraction, single-column chromatography), this technology can achieve automatic continuous separation, which can increase separation capacity and improve product yield, yield, and efficiency. It is the key technology for the chemical separation process and can also reuse the stationary phase and mobile phase to reduce cost and energy consumption [2]. In the cyclical operation of continuous production, the SMB system fully reflects its nonlinear, nonideal, and nonequilibrium characteristics. This multidegree-of-freedom system finds it difficult to make the important performance indicators, such as product purity, yield, and mobile phase consumption, in periodicity optimal states [3, 4].The adaptive neural fuzzy inference system is an artificial intelligence inference technology with the advantages of fuzzy logic and neural network [5], which is a good nonlinear mapping technique. ANFIS can construct a clear input-output mapping structure, which provides a powerful learning and decision-making process for target systems including nonlinear systems and has been successfully applied in the prediction of short-term electricity prices [6], clay expansion capacity [7], wind speed profile estimation [8], and the thermal error compensation on the CNC machine [9]. The ANFIS fuzzy neural network model was used to predict the cutting speed in WEDM [10]. The ANFIS model was used to predict the density of ionic liquids at various temperatures [11]. The particle swarm optimization (PSO) algorithm was used to obtain appropriate subtractive clustering (SC) parameter settings, and an adaptive ANFIS model was built to predict the business success rate [12]. ANFIS and artificial neural network (ANN) models were established for predicting the drying characteristics of potato, garlic, and cantaloupe in convection hot air dryers [13]. The polynomial equation of the bank section by utilizing the hybrid ANFIS was used to stabilize the design and implementation of the river section [14]. A data-driven model of the debutanizer based on ANFIS was established for realizing the real-time composition monitoring [15]. ANFIS was applied to develop an accurate temperature-dependent intelligent model for correlating the loading capacity of amino acid salt solutions for a variety of amino acids [16]. The combination of the ANFIS and CFD method provides the nondiscrete domain in various dimensions and makes a smart tool to locally predict multiphase flow [17]. The ANFIS prediction model was established for the response during the stainless steel 202 turning operation [18].In this paper, an ANFIS-based soft modeling method for the SMB chromatography separation process was proposed, which lays a foundation for the SMB system to achieve the periodic optimum stable working condition. The structure of the paper is described as follows. Section2 introduces the SMB chromatographic separation technique and the structure of the established soft sensor model. Three input data space partitioning methods and four conclusion parameter optimization algorithms of ANFIS are introduced in Section 3. Section 4 describes the experimental simulation and result analysis. Finally, the conclusion is listed in the last part.


## 2. SMB Chromatography Separation Technology and Soft Sensor Modeling

### 2.1. SMB Chromatography Separation Technology

SMB chromatographic separation technology simulates the reverse flow of a stationary phase adsorbent by continuously switching the positions of each feed and discharge port. The basic principle of typical SMB is described as follows. The several chromatographic columns are connected to form a loop. The relative countercurrent of the stationary phase and the mobile phase is simulated by orderly moving the eluent inlet, the extract outlet, the feed liquid inlet, and the raffinate outlet in the direction of the mobile phase so as to achieve the separation of two components [19]. Taking the separation of two components as an example, the working principle is briefly explained. The working principle of the typical SMB chromatographic separation process is shown in Figure 1 [20, 21]. It is assumed that two components to be separated are A and B, respectively. The adsorption capacity of component A is stronger than that of component B. D is the desorbent, E is the extract, F is the feed, and R is the raffinate. The entire bed can be divided into four zones (respectively, referred to as I, II, III, and IV or 1, 2, 3, and 4) depending on the location and function of the liquid inlet and outlet. Each zone performs a specific function.Figure 1
Working principle of the SMB cinematographic separation process.Zone I. The strong adsorption component A was resolved from the stationary phase adsorbent by using desorbent D. The fresh desorbent enters from the bottom of zone I and contacts with the adsorbent, and component A is rinsed out and then discharged from the extract outlet at the top of the zone to obtain an extract.Zone II. This is the analytical zone of weakly adsorbed component B. In the liquid entering zone II from zone I (including A and D), since component A has a stronger adsorption capacity than component B, component B in zone II is continuously displaced by component A and enters zone III along with the flowing liquid and the fresh feed (A and B).Zone III. Strongly adsorbed component A is adsorbed by the adsorbent in this zone. B is discharged from the raffinate outlet through a liquid flowing downward to obtain the raffinate.Zone IV. Its function is to achieve regeneration of eluent D and reduce the amount of eluent D. Since the concentration of D in zone IV is much higher than the concentration of D flowing into zone IV from the zone III region (because the region is zone I in the last switching cycle, eluent D is adsorbed by the adsorbent after being flushed with A, so its concentration is higher). Therefore, the adsorption equilibrium will be reestablished and eluent D will be resolved to achieve regeneration. The regenerated eluent D and the fresh eluent D flow into zone I together. In this way, the reverse flow of the stationary phase adsorbent is simulated by continuously switching the liquid inlet and outlet position in the switching direction shown in Figure 1, thereby achieving an effect equivalent to TMB.The feeding and discharging ports divide the SMB flow system into different zones. For the SMB with three zones, the functions of each zone are described as follows.Zone I. Elution zone: it desorbs the adsorbed substances on the stationary phase and flows out from the E port to complete the regeneration process of the stationary phase.Zone II. Distillation zone: continuous elution with the mobile phase makes the fast and slow components further separated in the chromatographic column.Zone III. Adsorption zone: the raw materials from the inlet flow into this zone and are adsorbed and separated, so that fast components flow out of the outlet (R port) of this zone and slow components are adsorbed in the stationary phase.


### 2.2. Soft Sensor Modeling for the SMB Chromatography Separation Process

In the SMB chromatography adsorption separation process, the purity of the product components in the extraction solution and raffinate is more important. However, the online measurement of the component purity as a quality index is difficult to achieve, and there are many factors affecting the product component purity in the SMB separation process. Therefore, the soft sensor modeling method for predicting the component purity in the SMB chromatography separation process has theoretical significance and application value [22]. The basic idea of soft sensing technology is to combine the automatic control theory with the knowledge of the production process. According to some optimal criteria, because some important variables are difficult to measure or temporarily unable to measure, the computer technology is used to select other variables that are easy to measure and to infer or estimate by constructing a mathematical relationship. The soft sensor model can be expressed as
(1)X^=fd,u,y,X∗,t,where X^ is the dominant variable estimated value, d is the measurable disturbance, u is the control input variable, y is the measurable output variable, and X∗ is the off-line sampled value or the analytically calculated value of the estimated variable. The optimal estimate of X can be obtained by constructing the relationship between these variables. Equation (1) reflects the relationship between the dominant variable X and the input variable u, the auxiliary variable y, and the measurable disturbance d2, while the off-line sample value X∗ is often used for self-correction of the model. The structure of the soft sensor model for the SMB chromatographic separation process established in this paper is shown in Figure 2.Figure 2
Structure of the soft sensor model.The auxiliary variables in the soft sensor model are selected from the measurement variables that can be provided in the control system based on the technique mechanism of the production process. Based on the technique mechanism and prior knowledge of the SMB chromatography separation process, the following variables are selected as the auxiliary variables of the soft sensor model.(1)
Flow rate of the injection pump (F pump) of the raw material liquid inlet, whose unit is ml/min
(2)
Flow rate of the flushing pump (D pump) of the flushing fluid inlet, whose unit is ml/min
(3)
Valve switching time, whose unit is minSelect the following variables as the output variable of the established soft sensor model.(1)
The purity of the target in the effluent of the E port. If there is impurity at the E port, this purity is smaller than 1
(2)
The purity of the impurity in the outlet effluent of the R port. If there is target material at the R port, this purity is smaller than 1
(3)
Divide the mass of the target in the E port by the target injection quality so as to obtain the yield of the target at the E port
(4)
Divide the mass of the impurity in the R port by the impurity injection quality so as to obtain the yield of the impurity at the R portThe auxiliary variables are used as inputs, and the purity of the target in the E port and the purity of the impurities in the R port are output variables. ANFIS is used to fit the nonlinear relationship among them so as to establish a predictive model of the corresponding economic and technical indicators. Based on a SMB chromatographic separation process, a soft sensor modeling method on the purity and yield of E and R ports is proposed by utilizing ANFIS. The historical data of the SMB chromatographic separation process were collected, and 1000 sets of historical data with uniformity and representative shown in Table1 are selected as the modeling data. Before carrying out the soft sensor modeling with these data, it is necessary to perform data normalization shown in Equation (2) in order to eliminate the influence of variable unit on modeling accuracy.
(2)y=x−vminvmax−vmin,where x is the initial value of the data set, y is the data after x being normalized, vmax is the maximum in the data set, and vmin is the minimum in the data set.Table 1
Data of the SMB chromatography separation process.
Number
Feed concentration (mg/ml)
Flow rate of F pump (ml/min)
Flow rate of D pump (ml/min)
Switching time (min)
Purity of E port (%)
Purity of E port by simulated calculation (%)
Recovery of R port (%)
Recovery of R port by simulated calculation (%)
1
5.6
0.1
1.0
15.5
95
99.99
71
69.6
2
10
0.1
0.828
17
98.1
99.99
90
88.3
3
10
0.1
0.7
19
98.2
99.99
97.2
99.9
4
10
0.1
1.0
14.5
96.9
99.99
83
84.6
5
10
0.1
0.7
19.5
96
99.99
91
93.2
6
14.3
0.2
0.6
19
98.2
99.99
83
83.1
7
5.3
0.3
0.6
19.5
97.5
99.99
80
81
8
10
0.2
1.0
14.5
96.6
99.99
43
44.3
9
10
0.2
0.4
29
98.5
99.99
64
65.2
…
…
…
…
…
…
…
…
…
1000
10.4
0.2
0.7
19.5
95.7
99.99
48
46.1


## 2.1. SMB Chromatography Separation Technology

SMB chromatographic separation technology simulates the reverse flow of a stationary phase adsorbent by continuously switching the positions of each feed and discharge port. The basic principle of typical SMB is described as follows. The several chromatographic columns are connected to form a loop. The relative countercurrent of the stationary phase and the mobile phase is simulated by orderly moving the eluent inlet, the extract outlet, the feed liquid inlet, and the raffinate outlet in the direction of the mobile phase so as to achieve the separation of two components [19]. Taking the separation of two components as an example, the working principle is briefly explained. The working principle of the typical SMB chromatographic separation process is shown in Figure 1 [20, 21]. It is assumed that two components to be separated are A and B, respectively. The adsorption capacity of component A is stronger than that of component B. D is the desorbent, E is the extract, F is the feed, and R is the raffinate. The entire bed can be divided into four zones (respectively, referred to as I, II, III, and IV or 1, 2, 3, and 4) depending on the location and function of the liquid inlet and outlet. Each zone performs a specific function.Figure 1
Working principle of the SMB cinematographic separation process.Zone I. The strong adsorption component A was resolved from the stationary phase adsorbent by using desorbent D. The fresh desorbent enters from the bottom of zone I and contacts with the adsorbent, and component A is rinsed out and then discharged from the extract outlet at the top of the zone to obtain an extract.Zone II. This is the analytical zone of weakly adsorbed component B. In the liquid entering zone II from zone I (including A and D), since component A has a stronger adsorption capacity than component B, component B in zone II is continuously displaced by component A and enters zone III along with the flowing liquid and the fresh feed (A and B).Zone III. Strongly adsorbed component A is adsorbed by the adsorbent in this zone. B is discharged from the raffinate outlet through a liquid flowing downward to obtain the raffinate.Zone IV. Its function is to achieve regeneration of eluent D and reduce the amount of eluent D. Since the concentration of D in zone IV is much higher than the concentration of D flowing into zone IV from the zone III region (because the region is zone I in the last switching cycle, eluent D is adsorbed by the adsorbent after being flushed with A, so its concentration is higher). Therefore, the adsorption equilibrium will be reestablished and eluent D will be resolved to achieve regeneration. The regenerated eluent D and the fresh eluent D flow into zone I together. In this way, the reverse flow of the stationary phase adsorbent is simulated by continuously switching the liquid inlet and outlet position in the switching direction shown in Figure 1, thereby achieving an effect equivalent to TMB.The feeding and discharging ports divide the SMB flow system into different zones. For the SMB with three zones, the functions of each zone are described as follows.Zone I. Elution zone: it desorbs the adsorbed substances on the stationary phase and flows out from the E port to complete the regeneration process of the stationary phase.Zone II. Distillation zone: continuous elution with the mobile phase makes the fast and slow components further separated in the chromatographic column.Zone III. Adsorption zone: the raw materials from the inlet flow into this zone and are adsorbed and separated, so that fast components flow out of the outlet (R port) of this zone and slow components are adsorbed in the stationary phase.


## 2.2. Soft Sensor Modeling for the SMB Chromatography Separation Process

In the SMB chromatography adsorption separation process, the purity of the product components in the extraction solution and raffinate is more important. However, the online measurement of the component purity as a quality index is difficult to achieve, and there are many factors affecting the product component purity in the SMB separation process. Therefore, the soft sensor modeling method for predicting the component purity in the SMB chromatography separation process has theoretical significance and application value [22]. The basic idea of soft sensing technology is to combine the automatic control theory with the knowledge of the production process. According to some optimal criteria, because some important variables are difficult to measure or temporarily unable to measure, the computer technology is used to select other variables that are easy to measure and to infer or estimate by constructing a mathematical relationship. The soft sensor model can be expressed as
(1)X^=fd,u,y,X∗,t,where X^ is the dominant variable estimated value, d is the measurable disturbance, u is the control input variable, y is the measurable output variable, and X∗ is the off-line sampled value or the analytically calculated value of the estimated variable. The optimal estimate of X can be obtained by constructing the relationship between these variables. Equation (1) reflects the relationship between the dominant variable X and the input variable u, the auxiliary variable y, and the measurable disturbance d2, while the off-line sample value X∗ is often used for self-correction of the model. The structure of the soft sensor model for the SMB chromatographic separation process established in this paper is shown in Figure 2.Figure 2
Structure of the soft sensor model.The auxiliary variables in the soft sensor model are selected from the measurement variables that can be provided in the control system based on the technique mechanism of the production process. Based on the technique mechanism and prior knowledge of the SMB chromatography separation process, the following variables are selected as the auxiliary variables of the soft sensor model.(1)
Flow rate of the injection pump (F pump) of the raw material liquid inlet, whose unit is ml/min
(2)
Flow rate of the flushing pump (D pump) of the flushing fluid inlet, whose unit is ml/min
(3)
Valve switching time, whose unit is minSelect the following variables as the output variable of the established soft sensor model.(1)
The purity of the target in the effluent of the E port. If there is impurity at the E port, this purity is smaller than 1
(2)
The purity of the impurity in the outlet effluent of the R port. If there is target material at the R port, this purity is smaller than 1
(3)
Divide the mass of the target in the E port by the target injection quality so as to obtain the yield of the target at the E port
(4)
Divide the mass of the impurity in the R port by the impurity injection quality so as to obtain the yield of the impurity at the R portThe auxiliary variables are used as inputs, and the purity of the target in the E port and the purity of the impurities in the R port are output variables. ANFIS is used to fit the nonlinear relationship among them so as to establish a predictive model of the corresponding economic and technical indicators. Based on a SMB chromatographic separation process, a soft sensor modeling method on the purity and yield of E and R ports is proposed by utilizing ANFIS. The historical data of the SMB chromatographic separation process were collected, and 1000 sets of historical data with uniformity and representative shown in Table1 are selected as the modeling data. Before carrying out the soft sensor modeling with these data, it is necessary to perform data normalization shown in Equation (2) in order to eliminate the influence of variable unit on modeling accuracy.
(2)y=x−vminvmax−vmin,where x is the initial value of the data set, y is the data after x being normalized, vmax is the maximum in the data set, and vmin is the minimum in the data set.Table 1
Data of the SMB chromatography separation process.
Number
Feed concentration (mg/ml)
Flow rate of F pump (ml/min)
Flow rate of D pump (ml/min)
Switching time (min)
Purity of E port (%)
Purity of E port by simulated calculation (%)
Recovery of R port (%)
Recovery of R port by simulated calculation (%)
1
5.6
0.1
1.0
15.5
95
99.99
71
69.6
2
10
0.1
0.828
17
98.1
99.99
90
88.3
3
10
0.1
0.7
19
98.2
99.99
97.2
99.9
4
10
0.1
1.0
14.5
96.9
99.99
83
84.6
5
10
0.1
0.7
19.5
96
99.99
91
93.2
6
14.3
0.2
0.6
19
98.2
99.99
83
83.1
7
5.3
0.3
0.6
19.5
97.5
99.99
80
81
8
10
0.2
1.0
14.5
96.6
99.99
43
44.3
9
10
0.2
0.4
29
98.5
99.99
64
65.2
…
…
…
…
…
…
…
…
…
1000
10.4
0.2
0.7
19.5
95.7
99.99
48
46.1


## 3. ANFIS and Training Algorithms

### 3.1. Basic Principles of ANFIS Based on Mesh Generation

In the early 1990s, Jang proposed an ANFIS based on the T-S model [5], which is a new type of neural network structure combining fuzzy theory. It uses a neural network to realize three basic processes of fuzzification part, fuzzy inference part, and defuzzification in fuzzy control so that the established system has the advantages of both theories and realizes the adaptive adjustment on the research object [23]. The typical structure of the ANFIS is shown in Figure 3. The prototype ANFIS adopts the meshing partition method to linearly divide the input space so as to determine the number of fuzzy rules [24].Figure 3
Structure diagram of typical ANFIS.Assume that the fuzzy inference system under consideration has two inputsx and y and a single output f. For the first-order Sugeno fuzzy model, the general rule set with two if-then rules is represented as follows.Rule 1.
Ifx is A1 and y is B1, then
(3)f1=p1x+q1y+r1.Rule 2.
Ifx is A2 and y is B2, then
(4)f2=p2x+q2y+r2.Nodes in the same layer have the same function. Here, the output of thejth node in the layer l is Ol,j.Layer 1.
This layer node obscures the input signal.(5)O1,j=μAjx,j=1,2orO1,j=μBj−2x,j=3,4,where x or y is the input node and O1,j is the membership of the fuzzy set A (A1,A2,B1, or B2). Here, the membership function of A takes a bell-shaped function, which can be represented as
(6)μAx=11+x−cj/aj2bj,where {aj, bj, cj} is called the antecedent parameters.Layer 2.
Multiply all input signals passed in the previous layer as the output of this layer.(7)O2,j=μAjxμBjy,j=1,2.Layer 3.
The output in this layer is the normalized excitation intensity. The principle is that thejth node calculates the ratio of the excitation strength of the jth rule to the sum of the excitation strengths of all the rules.
(8)O3,j=w¯j=wjw1+w2,j=1,2.Layer 4.
All nodes in this layer are used as adaptive nodes with the node functions.(9)O4,j=w¯jfj=w¯jpjx+qjy+rj,j=1,2,where w¯j is the normalized excitation intensity from the third layer and {pj, qj, rj} is the conclusion parameters or consequent parameters.Layer 5.
The last layer sums all the signals to calculate the system output.(10)O5,1=∑w¯jfj=∑jwjfj∑jwj,j=1,2.It can be seen from the structure of ANFIS that the system has two adaptive layers (layer1 and layer 4). The first layer has three adjustable antecedent parameters associated with the input membership functions. Layer 4 has three adjustable consequent parameters associated with the first-order polynomial. The linear combination of the consequent parameters can be used to obtain the output of ANFIS, which can be expressed as
(11)f=w¯1f1+w¯2f2=w¯1p1x+q1y+r1+w¯2p2x+q2y+r2=w¯1xp1+w¯1yq1+w¯1r1+w¯2xp2+w¯2xq2+w¯2r2=Aθ,where the elements of the column vector θ constitute a set of conclusion parameters {p1,q1,r1,p2,q2,r2}. If there are t groups of input and output data pairs and given the antecedent parameters, the order of the matrix A,θ,f is t×6, 6×1, and t×1, respectively. In general, the number of sample data is much larger than the number of unknown parameters, that is to say, t>>6. The least squares method (LSM) is used to obtain the best estimate of the conclusion parameter vector θ^ in the sense of minimizing Aθ−f2, namely,
(12)θ^=ATA−1ATf.Then, based on the identification result of the conclusion parametersθ^, the estimated output f^ of ANFIS is calculated by
(13)f^=Aθ^.The root mean square error (RMSE) under the current antecedent and conclusion parameters can be calculated by Equation (14).
(14)RMSE=∑k=1tf^k−fk2t.


### 3.2. Antecedent Parameter Determination Algorithm of ANFIS

In essence, the prototype structure of ANFIS uses the mesh generation method to divide the input space by default. If the input is complex nonlinear data, it will inevitably lead to the exponential growth of the number of fuzzy rules, which will inevitably bring about dimensionality disasters. In addition, the linear division of the grid partition method cannot accurately reflect the spatial distribution of input data. Therefore, the subtraction clustering and fuzzyC-means clustering algorithms are utilized to realize the segmentation of data space and the determination of the antecedent parameters.


#### 3.2.1. Subtractive Clustering Algorithm

In a certain group of data, each data point is regarded as a candidate point of the cluster center, and an independent and fast single-time clustering algorithm that can calculate the number of clusters and the class centers is named as the subtractive clustering method [25]. According to the subtractive clustering method, there is a simple linear relationship between the number of data points and the calculation amount, which is not necessarily related to the dimension of the research object. Assuming that the n data points x1,x2,⋯,xc in the M-dimensional space have been normalized to a hypercube space, the density index at the data point xi is set as
(15)Di=∑j=1nexp−xi−xj2ra/22,where ra is a positive integer representing a neighborhood radius of such data points. It can be seen that a data point with multiple data points in the vicinity has a high density value, and a point outside a certain field has a minimal influence on its density index.In the subtractive clustering algorithm, the class center may be any point. Therefore, the density index of the point and its surrounding points in the neighborhood must be calculated with each point as the class center. The possibility of this point as the class center is analyzed. After the above operations, the first class center is the point with the highest density index selected from it, which is written asxc1 and whose density value is Dc1. The corrected density index of any point xii=1,2,⋯ can be obtained by Eq. (16).
(16)Di=Di−Dc1exp−xi−xc12rb/22,where rb∈Z.Obviously, the closer the data point is toxc1, the smaller the density value and then, the less likely it is a class center. In the neighborhood with radius rb, the density value shows a decreasing trend. In general, in order to prevent the situation of the closer class centers, usually rb>ra, for example, rb=1.5ra. After the density values of all the points are modified, the calculations of other class centers are deduced by the same method, and the density value correction is performed for all the points of the nonclass center. In this paper, the subtractive clustering method is used for pattern decomposition. Suppose the degree of the input and output belonging to a certain kind of center is Fjx and the input set X=X1,X2,⋯,XP contains p input and output data, wherein each point is expressed as
(17)Xk=X1k,X2k,⋯,Xnk,Xn+1k,⋯,Xn+mk,where the input of the kth input and output data pair is represented as X1k,X2k,⋯,Xnk and the corresponding output is Xn+1k,⋯,Xn+mk.The steps of pattern extraction based on the subtractive clustering algorithm are described as follows.Step 1.
There arep vectors vk,k=1,2,⋯,p, and the initial value vk=Xk is defined
Step 2.
Letvk be the reference vector and vl be the comparison vector. Then, calculate the relationship between the two vectors
(18)Fjx=exp−vk−vl2σij2,where k=1,2,⋯,p, l=1,2,⋯,p, and vk−vl2 indicates the Euclidean distance between the two vectors. The variance σ of the Gaussian function is calculated by the mean square error of the performance index
Step 3.
Update the relationship degree between the reference vector and the comparison vector by
(19)Fjx=0,Fjx<ε,Fjx,Other,where ε is a small positive number (ε=0.01)
Step 4.
Calculateωk=ω1k,ω2k,⋯,ωn+mk by Eq. (20).
(20)ωk=∑l=1pFjxvl∑l=1pFjx.
Step 5.
When allωk and vk are the same, the algorithm continues; otherwise, let vk=ωk and go to Step 2
Step 6.
According to the finalvk obtained in the above steps, the number of clusters is equal to the number of convergence vectors. The original data with the same convergence vector is divided into one class with the convergence vector as the center


#### 3.2.2. FCM Clustering Algorithm

The FCM clustering algorithm [26] divides the feature points in the feature space X=x1,x2,⋯,xn into c classes 2≤c≤n, and the cluster center of the ith class is represented by vi, where any feature point xj belongs to the membership degree uij of the ith class, and uij0≤uij≤1 satisfies the following conditions.
(21)∑i=1cuij=1,forj=1,2,⋯,n,0<∑j=1nuij<n,fori=1,2,⋯,c.The objective function of the FCM clustering algorithm is described as(22)JmU,V=∑i=1c∑j=1nuijmxj−vi2,where m>1 is the index weight that affects the degree of fuzzification of the membership matrix. The clustering problem is to find the membership degree matrix U=uijc×n and the category center V=v1,v2,⋯,vc to make Equation (3) minimized. When utilizing the FCM clustering algorithm, a parameter that needs to be determined in advance is the number of classifications of the data set. The clustering validity function used in this paper is described as follows [27].
(23)VpU,c=1n∑k=1nmaxiuik−1K∑i=1c−1∑j=i+1c1n∑k=1nmaxuik,ujk,where
(24)K=∑i=1c−1i.This clustering criterion combines the tightness and separation of fuzzy partitioning. By combining the clustering ISODATA algorithm with the proposed criterion function to obtain the optimal fuzzy partition, the algorithm procedure is described as follows.Step 1.
Specify the maximum number of cluster centerscmax (cmax≤n), the maximum number of iterations T, the index weight m, and the stop threshold ε>0
Step 2.
Initialize the fuzzy clustering centerV0=v10,v20,⋯,vc0 for the number of cluster centers c=2,3,⋯,cmax
Step 3.
For the number of iterationst=1,2,⋯,T, calculate the membership matrix and cluster center according to Eq. (24) and Eq. (25).
(25)uij,t=1/xi−vj21/a−1∑j=1c1/xi−vj21/a−1,(26)vj,t=∑i=1nuij,tmxi∑i=1nuij,tm,where i=1,2,⋯,n and j=1,2,⋯,c. If Vt−Vt−1<ε, go to the next step; otherwise, repeat Step 3
Step 4.
CalculateVpU,c by Equation (23). If c<cmax, go to Step 2; otherwise, stop the iteration and output the optimal cluster number c=cb, which satisfies VpU,cb=minVpU,c,c=2,3,⋯,cmax.


### 3.3. Consequent Parameter Optimization Algorithms of ANFIS

#### 3.3.1. Gradient Algorithm

The gradient descent method, also known as the fastest descent method, is a multidimensional unconstrained optimization problem calculation method based on gradients [28]. Considering an unconstrained optimization problem,
(27)minx∈Rnfx,where fx has a first-order continuous partial derivative. Starting from any point x0∈Rn, the Taylor formula shows that the fx decreases fastest along the direction of the negative gradient. Carry out the first-order Taylor spread on fx at x along p to obtain
(28)fx+λp=fx+λ∇Tfxp+σλ2,where λ is the searching step size and p is a unit vector, that is to say, p=1.Carry out the transposition on Equation (28) to get
(29)fx+λp−fx=∇Tfxp+oλ2λλ.It can be seen from Equation (29) that if ∇Tfxp<0, there are λ¯>0 and 0<λ<λ¯. The term of the square brackets in Equation (29) is negative, and thus, fx+λp<fx, where p is the descending direction of fx at x. Based on the principle of the Cauchy-Schwarz theory, obtain
(30)∇Tfxp≥−∇fx⋅p,(31)p=−∇fx∇fx.Equation (30) is the only condition for Equation (31) to take the equal sign. For the k+1 iteration, get
(32)xk+1=xk+λkpk,where pk=−∇fxk. So, the iteration formula of the steepest descent method can be represented as
(33)xk+1=xk−λk∇fxk,where k is the number of iterations and λ is a constant. The expression of the Jth component in Equation (33) can be described as
(34)xjk+1=xjk−λk∂f∂xj.


#### 3.3.2. Kalman Algorithm

Information fusion is the core of the Kalman algorithm [29]. The probability distribution with position as the variable obeys the Gaussian distribution, which is described as
(35)pix=1σi2πexp−x−x¯22σ2i=1,2.The probability ofx is proportional to p12x=px1px2, and the result of the product also obeys a new Gaussian distribution due to
(36)p12x=exp−x−x¯122δ12−x−x¯222δ22,dp12dx=−x¯12−x¯1σ12+x¯12−x¯2σ12⋅p12x¯12=0.The item in parentheses is 0, so there is(37)x¯12=σ22σ12+σ22x1+σ12σ12+σ22x2.Substitute the mean intopx12 to get
(38)σ122=σ12σ22σ12+σ22.The above description can convert two Gaussian distributions into one Gaussian distribution. When there are multiple Gaussian distributions, they can be merged one by one. So thex^1=x1 and σ^12=σ12 can be replaced with the optimal estimate, that is to say,
(39)x¯2=σ22σ^12+σ22x1+σ12σ^12+σ22x2,x^2=x^1+σ^12σ^12+σ22x2−x^1.There are alsoσ^12=σ12 and σ^22=σ22σ^12/σ^12+σ22. Arrange the equations to obtain a new covariance formula:
(40)σ^22=1−σ^12σ^12+σ22σ^12.Iteratively update the scale factorK by Equation (41).
(41)K=σ^12σ^12+σ22.FeedK into the mean to get
(42)x^2=x^1+Kx2−x^1BringK into the covariance formula to obtain
(43)σ^12=1−Kσ^12.


#### 3.3.3. Kaczmarz Algorithm

The basic principle of the Kaczmarz algorithm is described as follows [30]. Let the map fi from Rn to Rn be defined as
(44)fix=x−x,ai−biai22ai,i=1,2,⋯,m.The mapping fromRm+n to Rn can be defined as
(45)Fb,x=f1∘f2∘⋯∘fmx=f1f2f3⋯fmx⋯.Optionally selectx0=Rn and calculate the iterative sequence xi by utilizing the recursive formula.
(46)xi+1=Fb,xi,i=0,1,2,⋯,where F is defined by Equations (44) and (45).The orthogonal projectionPi from Rn to the Ith plane ai,x=0 can be defined as
(47)pi=I−1ai22aiaiT=δkl−aikailai22.So the mapping shown in Equation (47) is then expressed as
(48)fix=Pix+biai22ai,i=1,2,⋯,m.LetQi=P1P2⋯Pii=1,2,⋯,m, where Q0=I, and matrix R satisfies the following relationship:
(49)Rb=∑i=1mbiai22Qi−1ai.According to the above equations, the expression ofFb,x can be deduced.
(50)Fb,x=Qx+Rb,where Q=Qm, and matrix R depends on matrix A. Based on Equations (49) and (50), the Kaczmarz iterative may be described as
(51)xi=Qix0+∑j=0i−1QjRb.LetQ~=QPRAT, where PRAT is the orthogonal projection from Rn to RAT. Then, define
(52)G=I−Q~−1R.Based on Equation (52), matrix G has the following property:
(53)AGA=A,GAG=G,GA=PRAT,AG=P,where P is a projection along NR to RA and GA=PRAT, which can be replaced by GAT=GA.


#### 3.3.4. PseudoInv Algorithm

The PseudoInv algorithm is implemented using the Moore-Penrose generalized inverse [31]. It is often applied to obtain the least norm least squares solution (least squares method) on the nonuniform linear equations and makes the form of the solution simple. The Moore-Penrose generalized inverse of the matrix is unique in both the real and complex domains and can be obtained by the singular value decomposition. The pseudoinverse of matrix A can be defined as
(54)A+=lima→0ATA+αI−1AT.The actual algorithm for calculating the pseudoinverse adopts Eq. (55).
(55)A+=VD+UT.When the number of columns of matrixA is more than the number of rows, solving the linear equation using pseudoinverse is one of the many possible solutions. Specifically, x=A+y is x2 with the smallest Euclidean norm in all feasible solutions for this equation. When the number of rows in matrix A is more than the number of columns, there may be no solution. In this case, x obtained by pseudoreverse is a solution that minimizes the Euclidean distance Ax−y2 of Ax and y.


## 3.1. Basic Principles of ANFIS Based on Mesh Generation

In the early 1990s, Jang proposed an ANFIS based on the T-S model [5], which is a new type of neural network structure combining fuzzy theory. It uses a neural network to realize three basic processes of fuzzification part, fuzzy inference part, and defuzzification in fuzzy control so that the established system has the advantages of both theories and realizes the adaptive adjustment on the research object [23]. The typical structure of the ANFIS is shown in Figure 3. The prototype ANFIS adopts the meshing partition method to linearly divide the input space so as to determine the number of fuzzy rules [24].Figure 3
Structure diagram of typical ANFIS.Assume that the fuzzy inference system under consideration has two inputsx and y and a single output f. For the first-order Sugeno fuzzy model, the general rule set with two if-then rules is represented as follows.Rule 1.
Ifx is A1 and y is B1, then
(3)f1=p1x+q1y+r1.Rule 2.
Ifx is A2 and y is B2, then
(4)f2=p2x+q2y+r2.Nodes in the same layer have the same function. Here, the output of thejth node in the layer l is Ol,j.Layer 1.
This layer node obscures the input signal.(5)O1,j=μAjx,j=1,2orO1,j=μBj−2x,j=3,4,where x or y is the input node and O1,j is the membership of the fuzzy set A (A1,A2,B1, or B2). Here, the membership function of A takes a bell-shaped function, which can be represented as
(6)μAx=11+x−cj/aj2bj,where {aj, bj, cj} is called the antecedent parameters.Layer 2.
Multiply all input signals passed in the previous layer as the output of this layer.(7)O2,j=μAjxμBjy,j=1,2.Layer 3.
The output in this layer is the normalized excitation intensity. The principle is that thejth node calculates the ratio of the excitation strength of the jth rule to the sum of the excitation strengths of all the rules.
(8)O3,j=w¯j=wjw1+w2,j=1,2.Layer 4.
All nodes in this layer are used as adaptive nodes with the node functions.(9)O4,j=w¯jfj=w¯jpjx+qjy+rj,j=1,2,where w¯j is the normalized excitation intensity from the third layer and {pj, qj, rj} is the conclusion parameters or consequent parameters.Layer 5.
The last layer sums all the signals to calculate the system output.(10)O5,1=∑w¯jfj=∑jwjfj∑jwj,j=1,2.It can be seen from the structure of ANFIS that the system has two adaptive layers (layer1 and layer 4). The first layer has three adjustable antecedent parameters associated with the input membership functions. Layer 4 has three adjustable consequent parameters associated with the first-order polynomial. The linear combination of the consequent parameters can be used to obtain the output of ANFIS, which can be expressed as
(11)f=w¯1f1+w¯2f2=w¯1p1x+q1y+r1+w¯2p2x+q2y+r2=w¯1xp1+w¯1yq1+w¯1r1+w¯2xp2+w¯2xq2+w¯2r2=Aθ,where the elements of the column vector θ constitute a set of conclusion parameters {p1,q1,r1,p2,q2,r2}. If there are t groups of input and output data pairs and given the antecedent parameters, the order of the matrix A,θ,f is t×6, 6×1, and t×1, respectively. In general, the number of sample data is much larger than the number of unknown parameters, that is to say, t>>6. The least squares method (LSM) is used to obtain the best estimate of the conclusion parameter vector θ^ in the sense of minimizing Aθ−f2, namely,
(12)θ^=ATA−1ATf.Then, based on the identification result of the conclusion parametersθ^, the estimated output f^ of ANFIS is calculated by
(13)f^=Aθ^.The root mean square error (RMSE) under the current antecedent and conclusion parameters can be calculated by Equation (14).
(14)RMSE=∑k=1tf^k−fk2t.


## 3.2. Antecedent Parameter Determination Algorithm of ANFIS

In essence, the prototype structure of ANFIS uses the mesh generation method to divide the input space by default. If the input is complex nonlinear data, it will inevitably lead to the exponential growth of the number of fuzzy rules, which will inevitably bring about dimensionality disasters. In addition, the linear division of the grid partition method cannot accurately reflect the spatial distribution of input data. Therefore, the subtraction clustering and fuzzyC-means clustering algorithms are utilized to realize the segmentation of data space and the determination of the antecedent parameters.


### 3.2.1. Subtractive Clustering Algorithm

In a certain group of data, each data point is regarded as a candidate point of the cluster center, and an independent and fast single-time clustering algorithm that can calculate the number of clusters and the class centers is named as the subtractive clustering method [25]. According to the subtractive clustering method, there is a simple linear relationship between the number of data points and the calculation amount, which is not necessarily related to the dimension of the research object. Assuming that the n data points x1,x2,⋯,xc in the M-dimensional space have been normalized to a hypercube space, the density index at the data point xi is set as
(15)Di=∑j=1nexp−xi−xj2ra/22,where ra is a positive integer representing a neighborhood radius of such data points. It can be seen that a data point with multiple data points in the vicinity has a high density value, and a point outside a certain field has a minimal influence on its density index.In the subtractive clustering algorithm, the class center may be any point. Therefore, the density index of the point and its surrounding points in the neighborhood must be calculated with each point as the class center. The possibility of this point as the class center is analyzed. After the above operations, the first class center is the point with the highest density index selected from it, which is written asxc1 and whose density value is Dc1. The corrected density index of any point xii=1,2,⋯ can be obtained by Eq. (16).
(16)Di=Di−Dc1exp−xi−xc12rb/22,where rb∈Z.Obviously, the closer the data point is toxc1, the smaller the density value and then, the less likely it is a class center. In the neighborhood with radius rb, the density value shows a decreasing trend. In general, in order to prevent the situation of the closer class centers, usually rb>ra, for example, rb=1.5ra. After the density values of all the points are modified, the calculations of other class centers are deduced by the same method, and the density value correction is performed for all the points of the nonclass center. In this paper, the subtractive clustering method is used for pattern decomposition. Suppose the degree of the input and output belonging to a certain kind of center is Fjx and the input set X=X1,X2,⋯,XP contains p input and output data, wherein each point is expressed as
(17)Xk=X1k,X2k,⋯,Xnk,Xn+1k,⋯,Xn+mk,where the input of the kth input and output data pair is represented as X1k,X2k,⋯,Xnk and the corresponding output is Xn+1k,⋯,Xn+mk.The steps of pattern extraction based on the subtractive clustering algorithm are described as follows.Step 1.
There arep vectors vk,k=1,2,⋯,p, and the initial value vk=Xk is defined
Step 2.
Letvk be the reference vector and vl be the comparison vector. Then, calculate the relationship between the two vectors
(18)Fjx=exp−vk−vl2σij2,where k=1,2,⋯,p, l=1,2,⋯,p, and vk−vl2 indicates the Euclidean distance between the two vectors. The variance σ of the Gaussian function is calculated by the mean square error of the performance index
Step 3.
Update the relationship degree between the reference vector and the comparison vector by
(19)Fjx=0,Fjx<ε,Fjx,Other,where ε is a small positive number (ε=0.01)
Step 4.
Calculateωk=ω1k,ω2k,⋯,ωn+mk by Eq. (20).
(20)ωk=∑l=1pFjxvl∑l=1pFjx.
Step 5.
When allωk and vk are the same, the algorithm continues; otherwise, let vk=ωk and go to Step 2
Step 6.
According to the finalvk obtained in the above steps, the number of clusters is equal to the number of convergence vectors. The original data with the same convergence vector is divided into one class with the convergence vector as the center


### 3.2.2. FCM Clustering Algorithm

The FCM clustering algorithm [26] divides the feature points in the feature space X=x1,x2,⋯,xn into c classes 2≤c≤n, and the cluster center of the ith class is represented by vi, where any feature point xj belongs to the membership degree uij of the ith class, and uij0≤uij≤1 satisfies the following conditions.
(21)∑i=1cuij=1,forj=1,2,⋯,n,0<∑j=1nuij<n,fori=1,2,⋯,c.The objective function of the FCM clustering algorithm is described as(22)JmU,V=∑i=1c∑j=1nuijmxj−vi2,where m>1 is the index weight that affects the degree of fuzzification of the membership matrix. The clustering problem is to find the membership degree matrix U=uijc×n and the category center V=v1,v2,⋯,vc to make Equation (3) minimized. When utilizing the FCM clustering algorithm, a parameter that needs to be determined in advance is the number of classifications of the data set. The clustering validity function used in this paper is described as follows [27].
(23)VpU,c=1n∑k=1nmaxiuik−1K∑i=1c−1∑j=i+1c1n∑k=1nmaxuik,ujk,where
(24)K=∑i=1c−1i.This clustering criterion combines the tightness and separation of fuzzy partitioning. By combining the clustering ISODATA algorithm with the proposed criterion function to obtain the optimal fuzzy partition, the algorithm procedure is described as follows.Step 1.
Specify the maximum number of cluster centerscmax (cmax≤n), the maximum number of iterations T, the index weight m, and the stop threshold ε>0
Step 2.
Initialize the fuzzy clustering centerV0=v10,v20,⋯,vc0 for the number of cluster centers c=2,3,⋯,cmax
Step 3.
For the number of iterationst=1,2,⋯,T, calculate the membership matrix and cluster center according to Eq. (24) and Eq. (25).
(25)uij,t=1/xi−vj21/a−1∑j=1c1/xi−vj21/a−1,(26)vj,t=∑i=1nuij,tmxi∑i=1nuij,tm,where i=1,2,⋯,n and j=1,2,⋯,c. If Vt−Vt−1<ε, go to the next step; otherwise, repeat Step 3
Step 4.
CalculateVpU,c by Equation (23). If c<cmax, go to Step 2; otherwise, stop the iteration and output the optimal cluster number c=cb, which satisfies VpU,cb=minVpU,c,c=2,3,⋯,cmax.


## 3.2.1. Subtractive Clustering Algorithm

In a certain group of data, each data point is regarded as a candidate point of the cluster center, and an independent and fast single-time clustering algorithm that can calculate the number of clusters and the class centers is named as the subtractive clustering method [25]. According to the subtractive clustering method, there is a simple linear relationship between the number of data points and the calculation amount, which is not necessarily related to the dimension of the research object. Assuming that the n data points x1,x2,⋯,xc in the M-dimensional space have been normalized to a hypercube space, the density index at the data point xi is set as
(15)Di=∑j=1nexp−xi−xj2ra/22,where ra is a positive integer representing a neighborhood radius of such data points. It can be seen that a data point with multiple data points in the vicinity has a high density value, and a point outside a certain field has a minimal influence on its density index.In the subtractive clustering algorithm, the class center may be any point. Therefore, the density index of the point and its surrounding points in the neighborhood must be calculated with each point as the class center. The possibility of this point as the class center is analyzed. After the above operations, the first class center is the point with the highest density index selected from it, which is written asxc1 and whose density value is Dc1. The corrected density index of any point xii=1,2,⋯ can be obtained by Eq. (16).
(16)Di=Di−Dc1exp−xi−xc12rb/22,where rb∈Z.Obviously, the closer the data point is toxc1, the smaller the density value and then, the less likely it is a class center. In the neighborhood with radius rb, the density value shows a decreasing trend. In general, in order to prevent the situation of the closer class centers, usually rb>ra, for example, rb=1.5ra. After the density values of all the points are modified, the calculations of other class centers are deduced by the same method, and the density value correction is performed for all the points of the nonclass center. In this paper, the subtractive clustering method is used for pattern decomposition. Suppose the degree of the input and output belonging to a certain kind of center is Fjx and the input set X=X1,X2,⋯,XP contains p input and output data, wherein each point is expressed as
(17)Xk=X1k,X2k,⋯,Xnk,Xn+1k,⋯,Xn+mk,where the input of the kth input and output data pair is represented as X1k,X2k,⋯,Xnk and the corresponding output is Xn+1k,⋯,Xn+mk.The steps of pattern extraction based on the subtractive clustering algorithm are described as follows.Step 1.
There arep vectors vk,k=1,2,⋯,p, and the initial value vk=Xk is defined
Step 2.
Letvk be the reference vector and vl be the comparison vector. Then, calculate the relationship between the two vectors
(18)Fjx=exp−vk−vl2σij2,where k=1,2,⋯,p, l=1,2,⋯,p, and vk−vl2 indicates the Euclidean distance between the two vectors. The variance σ of the Gaussian function is calculated by the mean square error of the performance index
Step 3.
Update the relationship degree between the reference vector and the comparison vector by
(19)Fjx=0,Fjx<ε,Fjx,Other,where ε is a small positive number (ε=0.01)
Step 4.
Calculateωk=ω1k,ω2k,⋯,ωn+mk by Eq. (20).
(20)ωk=∑l=1pFjxvl∑l=1pFjx.
Step 5.
When allωk and vk are the same, the algorithm continues; otherwise, let vk=ωk and go to Step 2
Step 6.
According to the finalvk obtained in the above steps, the number of clusters is equal to the number of convergence vectors. The original data with the same convergence vector is divided into one class with the convergence vector as the center


## 3.2.2. FCM Clustering Algorithm

The FCM clustering algorithm [26] divides the feature points in the feature space X=x1,x2,⋯,xn into c classes 2≤c≤n, and the cluster center of the ith class is represented by vi, where any feature point xj belongs to the membership degree uij of the ith class, and uij0≤uij≤1 satisfies the following conditions.
(21)∑i=1cuij=1,forj=1,2,⋯,n,0<∑j=1nuij<n,fori=1,2,⋯,c.The objective function of the FCM clustering algorithm is described as(22)JmU,V=∑i=1c∑j=1nuijmxj−vi2,where m>1 is the index weight that affects the degree of fuzzification of the membership matrix. The clustering problem is to find the membership degree matrix U=uijc×n and the category center V=v1,v2,⋯,vc to make Equation (3) minimized. When utilizing the FCM clustering algorithm, a parameter that needs to be determined in advance is the number of classifications of the data set. The clustering validity function used in this paper is described as follows [27].
(23)VpU,c=1n∑k=1nmaxiuik−1K∑i=1c−1∑j=i+1c1n∑k=1nmaxuik,ujk,where
(24)K=∑i=1c−1i.This clustering criterion combines the tightness and separation of fuzzy partitioning. By combining the clustering ISODATA algorithm with the proposed criterion function to obtain the optimal fuzzy partition, the algorithm procedure is described as follows.Step 1.
Specify the maximum number of cluster centerscmax (cmax≤n), the maximum number of iterations T, the index weight m, and the stop threshold ε>0
Step 2.
Initialize the fuzzy clustering centerV0=v10,v20,⋯,vc0 for the number of cluster centers c=2,3,⋯,cmax
Step 3.
For the number of iterationst=1,2,⋯,T, calculate the membership matrix and cluster center according to Eq. (24) and Eq. (25).
(25)uij,t=1/xi−vj21/a−1∑j=1c1/xi−vj21/a−1,(26)vj,t=∑i=1nuij,tmxi∑i=1nuij,tm,where i=1,2,⋯,n and j=1,2,⋯,c. If Vt−Vt−1<ε, go to the next step; otherwise, repeat Step 3
Step 4.
CalculateVpU,c by Equation (23). If c<cmax, go to Step 2; otherwise, stop the iteration and output the optimal cluster number c=cb, which satisfies VpU,cb=minVpU,c,c=2,3,⋯,cmax.


## 3.3. Consequent Parameter Optimization Algorithms of ANFIS

### 3.3.1. Gradient Algorithm

The gradient descent method, also known as the fastest descent method, is a multidimensional unconstrained optimization problem calculation method based on gradients [28]. Considering an unconstrained optimization problem,
(27)minx∈Rnfx,where fx has a first-order continuous partial derivative. Starting from any point x0∈Rn, the Taylor formula shows that the fx decreases fastest along the direction of the negative gradient. Carry out the first-order Taylor spread on fx at x along p to obtain
(28)fx+λp=fx+λ∇Tfxp+σλ2,where λ is the searching step size and p is a unit vector, that is to say, p=1.Carry out the transposition on Equation (28) to get
(29)fx+λp−fx=∇Tfxp+oλ2λλ.It can be seen from Equation (29) that if ∇Tfxp<0, there are λ¯>0 and 0<λ<λ¯. The term of the square brackets in Equation (29) is negative, and thus, fx+λp<fx, where p is the descending direction of fx at x. Based on the principle of the Cauchy-Schwarz theory, obtain
(30)∇Tfxp≥−∇fx⋅p,(31)p=−∇fx∇fx.Equation (30) is the only condition for Equation (31) to take the equal sign. For the k+1 iteration, get
(32)xk+1=xk+λkpk,where pk=−∇fxk. So, the iteration formula of the steepest descent method can be represented as
(33)xk+1=xk−λk∇fxk,where k is the number of iterations and λ is a constant. The expression of the Jth component in Equation (33) can be described as
(34)xjk+1=xjk−λk∂f∂xj.


### 3.3.2. Kalman Algorithm

Information fusion is the core of the Kalman algorithm [29]. The probability distribution with position as the variable obeys the Gaussian distribution, which is described as
(35)pix=1σi2πexp−x−x¯22σ2i=1,2.The probability ofx is proportional to p12x=px1px2, and the result of the product also obeys a new Gaussian distribution due to
(36)p12x=exp−x−x¯122δ12−x−x¯222δ22,dp12dx=−x¯12−x¯1σ12+x¯12−x¯2σ12⋅p12x¯12=0.The item in parentheses is 0, so there is(37)x¯12=σ22σ12+σ22x1+σ12σ12+σ22x2.Substitute the mean intopx12 to get
(38)σ122=σ12σ22σ12+σ22.The above description can convert two Gaussian distributions into one Gaussian distribution. When there are multiple Gaussian distributions, they can be merged one by one. So thex^1=x1 and σ^12=σ12 can be replaced with the optimal estimate, that is to say,
(39)x¯2=σ22σ^12+σ22x1+σ12σ^12+σ22x2,x^2=x^1+σ^12σ^12+σ22x2−x^1.There are alsoσ^12=σ12 and σ^22=σ22σ^12/σ^12+σ22. Arrange the equations to obtain a new covariance formula:
(40)σ^22=1−σ^12σ^12+σ22σ^12.Iteratively update the scale factorK by Equation (41).
(41)K=σ^12σ^12+σ22.FeedK into the mean to get
(42)x^2=x^1+Kx2−x^1BringK into the covariance formula to obtain
(43)σ^12=1−Kσ^12.


### 3.3.3. Kaczmarz Algorithm

The basic principle of the Kaczmarz algorithm is described as follows [30]. Let the map fi from Rn to Rn be defined as
(44)fix=x−x,ai−biai22ai,i=1,2,⋯,m.The mapping fromRm+n to Rn can be defined as
(45)Fb,x=f1∘f2∘⋯∘fmx=f1f2f3⋯fmx⋯.Optionally selectx0=Rn and calculate the iterative sequence xi by utilizing the recursive formula.
(46)xi+1=Fb,xi,i=0,1,2,⋯,where F is defined by Equations (44) and (45).The orthogonal projectionPi from Rn to the Ith plane ai,x=0 can be defined as
(47)pi=I−1ai22aiaiT=δkl−aikailai22.So the mapping shown in Equation (47) is then expressed as
(48)fix=Pix+biai22ai,i=1,2,⋯,m.LetQi=P1P2⋯Pii=1,2,⋯,m, where Q0=I, and matrix R satisfies the following relationship:
(49)Rb=∑i=1mbiai22Qi−1ai.According to the above equations, the expression ofFb,x can be deduced.
(50)Fb,x=Qx+Rb,where Q=Qm, and matrix R depends on matrix A. Based on Equations (49) and (50), the Kaczmarz iterative may be described as
(51)xi=Qix0+∑j=0i−1QjRb.LetQ~=QPRAT, where PRAT is the orthogonal projection from Rn to RAT. Then, define
(52)G=I−Q~−1R.Based on Equation (52), matrix G has the following property:
(53)AGA=A,GAG=G,GA=PRAT,AG=P,where P is a projection along NR to RA and GA=PRAT, which can be replaced by GAT=GA.


### 3.3.4. PseudoInv Algorithm

The PseudoInv algorithm is implemented using the Moore-Penrose generalized inverse [31]. It is often applied to obtain the least norm least squares solution (least squares method) on the nonuniform linear equations and makes the form of the solution simple. The Moore-Penrose generalized inverse of the matrix is unique in both the real and complex domains and can be obtained by the singular value decomposition. The pseudoinverse of matrix A can be defined as
(54)A+=lima→0ATA+αI−1AT.The actual algorithm for calculating the pseudoinverse adopts Eq. (55).
(55)A+=VD+UT.When the number of columns of matrixA is more than the number of rows, solving the linear equation using pseudoinverse is one of the many possible solutions. Specifically, x=A+y is x2 with the smallest Euclidean norm in all feasible solutions for this equation. When the number of rows in matrix A is more than the number of columns, there may be no solution. In this case, x obtained by pseudoreverse is a solution that minimizes the Euclidean distance Ax−y2 of Ax and y.


## 3.3.1. Gradient Algorithm

The gradient descent method, also known as the fastest descent method, is a multidimensional unconstrained optimization problem calculation method based on gradients [28]. Considering an unconstrained optimization problem,
(27)minx∈Rnfx,where fx has a first-order continuous partial derivative. Starting from any point x0∈Rn, the Taylor formula shows that the fx decreases fastest along the direction of the negative gradient. Carry out the first-order Taylor spread on fx at x along p to obtain
(28)fx+λp=fx+λ∇Tfxp+σλ2,where λ is the searching step size and p is a unit vector, that is to say, p=1.Carry out the transposition on Equation (28) to get
(29)fx+λp−fx=∇Tfxp+oλ2λλ.It can be seen from Equation (29) that if ∇Tfxp<0, there are λ¯>0 and 0<λ<λ¯. The term of the square brackets in Equation (29) is negative, and thus, fx+λp<fx, where p is the descending direction of fx at x. Based on the principle of the Cauchy-Schwarz theory, obtain
(30)∇Tfxp≥−∇fx⋅p,(31)p=−∇fx∇fx.Equation (30) is the only condition for Equation (31) to take the equal sign. For the k+1 iteration, get
(32)xk+1=xk+λkpk,where pk=−∇fxk. So, the iteration formula of the steepest descent method can be represented as
(33)xk+1=xk−λk∇fxk,where k is the number of iterations and λ is a constant. The expression of the Jth component in Equation (33) can be described as
(34)xjk+1=xjk−λk∂f∂xj.


## 3.3.2. Kalman Algorithm

Information fusion is the core of the Kalman algorithm [29]. The probability distribution with position as the variable obeys the Gaussian distribution, which is described as
(35)pix=1σi2πexp−x−x¯22σ2i=1,2.The probability ofx is proportional to p12x=px1px2, and the result of the product also obeys a new Gaussian distribution due to
(36)p12x=exp−x−x¯122δ12−x−x¯222δ22,dp12dx=−x¯12−x¯1σ12+x¯12−x¯2σ12⋅p12x¯12=0.The item in parentheses is 0, so there is(37)x¯12=σ22σ12+σ22x1+σ12σ12+σ22x2.Substitute the mean intopx12 to get
(38)σ122=σ12σ22σ12+σ22.The above description can convert two Gaussian distributions into one Gaussian distribution. When there are multiple Gaussian distributions, they can be merged one by one. So thex^1=x1 and σ^12=σ12 can be replaced with the optimal estimate, that is to say,
(39)x¯2=σ22σ^12+σ22x1+σ12σ^12+σ22x2,x^2=x^1+σ^12σ^12+σ22x2−x^1.There are alsoσ^12=σ12 and σ^22=σ22σ^12/σ^12+σ22. Arrange the equations to obtain a new covariance formula:
(40)σ^22=1−σ^12σ^12+σ22σ^12.Iteratively update the scale factorK by Equation (41).
(41)K=σ^12σ^12+σ22.FeedK into the mean to get
(42)x^2=x^1+Kx2−x^1BringK into the covariance formula to obtain
(43)σ^12=1−Kσ^12.


## 3.3.3. Kaczmarz Algorithm

The basic principle of the Kaczmarz algorithm is described as follows [30]. Let the map fi from Rn to Rn be defined as
(44)fix=x−x,ai−biai22ai,i=1,2,⋯,m.The mapping fromRm+n to Rn can be defined as
(45)Fb,x=f1∘f2∘⋯∘fmx=f1f2f3⋯fmx⋯.Optionally selectx0=Rn and calculate the iterative sequence xi by utilizing the recursive formula.
(46)xi+1=Fb,xi,i=0,1,2,⋯,where F is defined by Equations (44) and (45).The orthogonal projectionPi from Rn to the Ith plane ai,x=0 can be defined as
(47)pi=I−1ai22aiaiT=δkl−aikailai22.So the mapping shown in Equation (47) is then expressed as
(48)fix=Pix+biai22ai,i=1,2,⋯,m.LetQi=P1P2⋯Pii=1,2,⋯,m, where Q0=I, and matrix R satisfies the following relationship:
(49)Rb=∑i=1mbiai22Qi−1ai.According to the above equations, the expression ofFb,x can be deduced.
(50)Fb,x=Qx+Rb,where Q=Qm, and matrix R depends on matrix A. Based on Equations (49) and (50), the Kaczmarz iterative may be described as
(51)xi=Qix0+∑j=0i−1QjRb.LetQ~=QPRAT, where PRAT is the orthogonal projection from Rn to RAT. Then, define
(52)G=I−Q~−1R.Based on Equation (52), matrix G has the following property:
(53)AGA=A,GAG=G,GA=PRAT,AG=P,where P is a projection along NR to RA and GA=PRAT, which can be replaced by GAT=GA.


## 3.3.4. PseudoInv Algorithm

The PseudoInv algorithm is implemented using the Moore-Penrose generalized inverse [31]. It is often applied to obtain the least norm least squares solution (least squares method) on the nonuniform linear equations and makes the form of the solution simple. The Moore-Penrose generalized inverse of the matrix is unique in both the real and complex domains and can be obtained by the singular value decomposition. The pseudoinverse of matrix A can be defined as
(54)A+=lima→0ATA+αI−1AT.The actual algorithm for calculating the pseudoinverse adopts Eq. (55).
(55)A+=VD+UT.When the number of columns of matrixA is more than the number of rows, solving the linear equation using pseudoinverse is one of the many possible solutions. Specifically, x=A+y is x2 with the smallest Euclidean norm in all feasible solutions for this equation. When the number of rows in matrix A is more than the number of columns, there may be no solution. In this case, x obtained by pseudoreverse is a solution that minimizes the Euclidean distance Ax−y2 of Ax and y.


## 4. Simulation Experiment and Result Analysis

Based on the technique of the SMB chromatographic separation process, a soft sensor model on the target purity of E and R ports is proposed by utilizing ANFIS. The ANFIS network has three input variables and one output variable. The historical data of the SMB chromatographic separation process was collected, and 1000 sets of historical data with uniformity and representative were selected. Then, the processed data is divided into two parts. The first 800 sets of data are used as training data, and the last 200 sets of data are used to verify the performance of the soft sensor models. According to the reference data, it can be found that the target purity in the E effluent and the yield of the impurity at the R port are correlated, and the impurity purity in the R effluent and the yield of the target at the E port are also correlated. Therefore, in the experimental simulation stage, the target purity in the E effluent and the impurity purity in the R effluent are selected as the output. In order to measure the performance of the predictive models, several performance indicators are defined below, wherey^ is the estimated value and y is the actual value [32]. The model performance indicators are defined as shown in Table 2.Table 2
Definition of model performance indicators.
Root mean square error (RMSE)
RMSE=1/n∑i=1ny∧i−yi21/2
Sum of squared errors (SSE)
SSE=∑i=1ny∧i−yi2
Mean absolute percentage error (MAPE)
MAPE=∑y^i−yi/yi×100/n
Maximum positive error (MPE)
MPE=maxy^−y,0Based on the meshing partition method, subtractive clustering algorithm, and FCM clustering algorithm, the input data space of the SMB chromatographic separation process is divided and the premise parameters are determined. The Sugeno-type ANFIS soft sensor models based on three algorithms are established. Then, these models are optimized by the gradient, Kalman, Kaczmarz, and PseudoInv algorithms to obtain the optimized conclusion parameters so as to achieve the prediction of the target purity of the E port and R port in the SMB chromatographic separation process.


### 4.1. ANFIS Soft Sensor Model Based on Mesh Partition

When the input data space of the soft sensor model is divided by the meshing partition method, the membership function selects the Gaussian function, and the number of membership functions is 5. The membership function curves of the input data of the Sugeno-type ANFIS model based on meshing partition are shown in Figure4. The number of training iterations is 500, and the established soft sensor models are tested by the utilized 200 sets of data with average and representative. The predicted simulation results are shown in Figures 5–8. Figure 5 shows the output contrast of the target purity in the E port effluent in the SMB chromatography process under the gradient, Kalman, Kaczmarz, and PseudoInv algorithms to obtain the optimized conclusion parameters of ANFIS. Figure 6 shows the predictive error comparison curves. Figure 7 shows the output contrast of the impurity purity in the R port effluent in the SMB chromatography process under the gradient, Kalman, Kaczmarz, and PseudoInv algorithms to obtain the optimized conclusion parameters of ANFIS. Figure 8 shows the predictive error comparison curves. Table 3 compares the predicted performance indicators of the established soft sensor models.Figure 4
Input membership function based on the meshing partition method.Figure 5
Effluent purity prediction results in the E port.Figure 6
Effluent purity prediction error in the E port.Figure 7
Prediction results of impurity purity in the R port.Figure 8
Prediction error of impurity purity in the R port.Table 3
Comparison of predictive performance indicators of the ANFIS soft sensor model based on the meshing partition method.
Performance
RMSE
SSE
MAPE
MPE
Purity of the E port
Gradient
3.3668e−04
2.2670e−05
0.0081
0.0026
Kalman
0.0112
0.0252
0.0087
0.1010
Kaczmarz
0.0612
0.7484
0.0337
0.6750
PseudoInv
0.0112
0.0250
0.0087
0.1068
Purity of the R port
Gradient
0.0114
0.0421
0.5512
0.0691
Kalman
0.0145
0.0421
0.5513
0.0900
Kaczmarz
0.0797
1.2699
0.5932
0.3585
PseudoInv
0.0145
0.0418
0.5516
0.0896According to the above simulation results, it can be seen that the input data spatial division and premise parameter determination are realized by the meshing partition method and ANFIS soft sensor models based on the gradient, Kalman, and PseudoInv algorithms for optimizing conclusion parameters have the better prediction results on the key economic and technical indicators of the SMB chromatographic separation process. In order to distinguish the optimization performance of the four algorithms, based on the adopted four performance indicators (RMSE, SSE, MAPE, and MPE), the ANFIS soft sensor model based on the meshing partition method and the gradient algorithm has higher prediction accuracy than the other three algorithms.


### 4.2. ANFIS Soft Sensor Model Based on the Subtractive Clustering Algorithm

When the input data space of the soft sensor model is divided by the subtractive clustering algorithm, the membership function selects the Gaussian function. The membership function curves of the input data of the Sugeno-type ANFIS model based on the subtractive clustering algorithm are shown in Figures9–11. The number of training iterations is 500, and the established soft sensor models are tested by the utilized 200 sets of data with average and representative. The predicted simulation results are shown in Figures 12–15. Figure 12 shows the output contrast of the target purity in the E port effluent in the SMB chromatography process under the gradient, Kalman, Kaczmarz, and PseudoInv algorithms to obtain the optimized conclusion parameters of ANFIS. Figure 13 shows the predictive error comparison curves. Figure 14 shows the output contrast of the impurity purity in the R port effluent in the SMB chromatography process under the gradient, Kalman, Kaczmarz, and PseudoInv algorithms to obtain the optimized conclusion parameters of ANFIS. Figure 15 shows the predictive error comparison curves. Table 4 compares the predicted performance indicators of the established soft sensor models.Figure 9
Membership function of input 1 based on the subtractive clustering method.Figure 10
Membership function of input 2 based on the subtractive clustering method.Figure 11
Membership function of input 3 based on the subtractive clustering method.Figure 12
Effluent purity prediction results in the E port.Figure 13
Effluent purity prediction error in the E port.Figure 14
Prediction results of impurity purity in the R port.Figure 15
Prediction error of impurity purity in the R port.Table 4
Comparison of predictive performance indicators of the ANFIS soft sensor model based on the subtractive clustering method.
Performance
RMSE
SSE
MAPE
MPE
Purity of the E port
Gradient
0.0032
0.0021
0.0103
0.0204
Kalman
0.0068
0.0092
0.0091
0.0434
Kaczmarz
0.0042
0.0035
0.0093
0.0189
PseudoInv
0.0032
0.0020
0.0097
0.0203
Purity of the R port
Gradient
0.0280
0.1567
0.5517
0.1315
Kalman
0.0286
0.1633
0.5514
0.5514
Kaczmarz
0.0386
0.2981
0.5809
0.1624
PseudoInv
0.0280
0.1566
0.5515
0.1313According to the above simulation results, it can be seen that the input data spatial division and premise parameter determination are realized by the subtractive clustering algorithm and ANFIS soft sensor models based on the gradient, Kalman, and PseudoInv algorithms for optimizing conclusion parameters have the better prediction results on the key economic and technical indicators of the SMB chromatographic separation process. In order to distinguish the optimization performance of the four algorithms, based on the adopted four performance indicators (RMSE, SSE, MAPE, and MPE), the ANFIS soft sensor model based on the meshing partition method and the PseudoInv algorithm has higher prediction accuracy than the other three algorithms.


### 4.3. ANFIS Soft Sensor Model Based on the FCM Clustering Algorithm

When the input data space of the soft sensor model is divided by the FCM clustering algorithm, the membership function selects the Gaussian function. The membership function curves of the input data of the Sugeno-type ANFIS model based on the FCM clustering algorithm are shown in Figures16–18. The number of training iterations is 500, and the established soft sensor models are tested by the utilized 200 sets of data with average and representative. The predicted simulation results are shown in Figures 19–22. Figure 19 shows the output contrast of the target purity in the E port effluent in the SMB chromatography process under the gradient, Kalman, Kaczmarz, and PseudoInv algorithms to obtain the optimized conclusion parameters of ANFIS. Figure 20 shows the predictive error comparison curves. Figure 21 shows the output contrast of the impurity purity in the R port effluent in the SMB chromatography process under the gradient, Kalman, Kaczmarz, and PseudoInv algorithms to obtain the optimized conclusion parameters of ANFIS. Figure 22 shows the predictive error comparison curves. Table 5 compares the predicted performance indicators of the established soft sensor models.Figure 16
Membership function of input 1 based on the FCM clustering method.Figure 17
Membership function of input 2 based on the FCM clustering method.Figure 18
Membership function of input 3 based on the FCM clustering method.Figure 19
Effluent purity prediction results in the E port.Figure 20
Effluent purity prediction error in the E port.Figure 21
Prediction results of impurity purity in the R port.Figure 22
Prediction error of impurity purity in the R port.Table 5
Comparison of predictive performance indicators of the ANFIS soft sensor model based on the FCM clustering method.
Performance
RMSE
SSE
MAPE
MPE
Purity of the E port
Gradient
0.0643
0.8281
0.0258
0.8410
Kalman
0.0485
0.4699
0.0258
0.5163
Kaczmarz
0.0517
0.5344
0.0196
0.6431
PseudoInv
0.0440
0.3874
0.0253
0.4297
Purity of the R port
Gradient
0.1245
3.0982
0.5547
0.4398
Kalman
0.087
1.5336
0.5557
0.4262
Kaczmarz
0.1131
2.5603
0.6397
0.5862
PseudoInv
0.0837
1.4009
0.5558
0.4129According to the above simulation results, it can be seen that the input data spatial division and premise parameter determination are realized by utilizing the FCM clustering algorithm and ANFIS soft sensor models based on the Kalman, Kaczmarz, and PseudoInv algorithms for optimizing conclusion parameters have the better prediction results on the key economic and technical indicators of the SMB chromatographic separation process. In order to distinguish the optimization performance of the four algorithms, based on the adopted four performance indicators (RMSE, SSE, MAPE, and MPE), the ANFIS soft sensor model based on the FCM clustering algorithm and the PseudoInv algorithm has higher prediction accuracy than the other three algorithms.


## 4.1. ANFIS Soft Sensor Model Based on Mesh Partition

When the input data space of the soft sensor model is divided by the meshing partition method, the membership function selects the Gaussian function, and the number of membership functions is 5. The membership function curves of the input data of the Sugeno-type ANFIS model based on meshing partition are shown in Figure4. The number of training iterations is 500, and the established soft sensor models are tested by the utilized 200 sets of data with average and representative. The predicted simulation results are shown in Figures 5–8. Figure 5 shows the output contrast of the target purity in the E port effluent in the SMB chromatography process under the gradient, Kalman, Kaczmarz, and PseudoInv algorithms to obtain the optimized conclusion parameters of ANFIS. Figure 6 shows the predictive error comparison curves. Figure 7 shows the output contrast of the impurity purity in the R port effluent in the SMB chromatography process under the gradient, Kalman, Kaczmarz, and PseudoInv algorithms to obtain the optimized conclusion parameters of ANFIS. Figure 8 shows the predictive error comparison curves. Table 3 compares the predicted performance indicators of the established soft sensor models.Figure 4
Input membership function based on the meshing partition method.Figure 5
Effluent purity prediction results in the E port.Figure 6
Effluent purity prediction error in the E port.Figure 7
Prediction results of impurity purity in the R port.Figure 8
Prediction error of impurity purity in the R port.Table 3
Comparison of predictive performance indicators of the ANFIS soft sensor model based on the meshing partition method.
Performance
RMSE
SSE
MAPE
MPE
Purity of the E port
Gradient
3.3668e−04
2.2670e−05
0.0081
0.0026
Kalman
0.0112
0.0252
0.0087
0.1010
Kaczmarz
0.0612
0.7484
0.0337
0.6750
PseudoInv
0.0112
0.0250
0.0087
0.1068
Purity of the R port
Gradient
0.0114
0.0421
0.5512
0.0691
Kalman
0.0145
0.0421
0.5513
0.0900
Kaczmarz
0.0797
1.2699
0.5932
0.3585
PseudoInv
0.0145
0.0418
0.5516
0.0896According to the above simulation results, it can be seen that the input data spatial division and premise parameter determination are realized by the meshing partition method and ANFIS soft sensor models based on the gradient, Kalman, and PseudoInv algorithms for optimizing conclusion parameters have the better prediction results on the key economic and technical indicators of the SMB chromatographic separation process. In order to distinguish the optimization performance of the four algorithms, based on the adopted four performance indicators (RMSE, SSE, MAPE, and MPE), the ANFIS soft sensor model based on the meshing partition method and the gradient algorithm has higher prediction accuracy than the other three algorithms.


## 4.2. ANFIS Soft Sensor Model Based on the Subtractive Clustering Algorithm

When the input data space of the soft sensor model is divided by the subtractive clustering algorithm, the membership function selects the Gaussian function. The membership function curves of the input data of the Sugeno-type ANFIS model based on the subtractive clustering algorithm are shown in Figures9–11. The number of training iterations is 500, and the established soft sensor models are tested by the utilized 200 sets of data with average and representative. The predicted simulation results are shown in Figures 12–15. Figure 12 shows the output contrast of the target purity in the E port effluent in the SMB chromatography process under the gradient, Kalman, Kaczmarz, and PseudoInv algorithms to obtain the optimized conclusion parameters of ANFIS. Figure 13 shows the predictive error comparison curves. Figure 14 shows the output contrast of the impurity purity in the R port effluent in the SMB chromatography process under the gradient, Kalman, Kaczmarz, and PseudoInv algorithms to obtain the optimized conclusion parameters of ANFIS. Figure 15 shows the predictive error comparison curves. Table 4 compares the predicted performance indicators of the established soft sensor models.Figure 9
Membership function of input 1 based on the subtractive clustering method.Figure 10
Membership function of input 2 based on the subtractive clustering method.Figure 11
Membership function of input 3 based on the subtractive clustering method.Figure 12
Effluent purity prediction results in the E port.Figure 13
Effluent purity prediction error in the E port.Figure 14
Prediction results of impurity purity in the R port.Figure 15
Prediction error of impurity purity in the R port.Table 4
Comparison of predictive performance indicators of the ANFIS soft sensor model based on the subtractive clustering method.
Performance
RMSE
SSE
MAPE
MPE
Purity of the E port
Gradient
0.0032
0.0021
0.0103
0.0204
Kalman
0.0068
0.0092
0.0091
0.0434
Kaczmarz
0.0042
0.0035
0.0093
0.0189
PseudoInv
0.0032
0.0020
0.0097
0.0203
Purity of the R port
Gradient
0.0280
0.1567
0.5517
0.1315
Kalman
0.0286
0.1633
0.5514
0.5514
Kaczmarz
0.0386
0.2981
0.5809
0.1624
PseudoInv
0.0280
0.1566
0.5515
0.1313According to the above simulation results, it can be seen that the input data spatial division and premise parameter determination are realized by the subtractive clustering algorithm and ANFIS soft sensor models based on the gradient, Kalman, and PseudoInv algorithms for optimizing conclusion parameters have the better prediction results on the key economic and technical indicators of the SMB chromatographic separation process. In order to distinguish the optimization performance of the four algorithms, based on the adopted four performance indicators (RMSE, SSE, MAPE, and MPE), the ANFIS soft sensor model based on the meshing partition method and the PseudoInv algorithm has higher prediction accuracy than the other three algorithms.


## 4.3. ANFIS Soft Sensor Model Based on the FCM Clustering Algorithm

When the input data space of the soft sensor model is divided by the FCM clustering algorithm, the membership function selects the Gaussian function. The membership function curves of the input data of the Sugeno-type ANFIS model based on the FCM clustering algorithm are shown in Figures16–18. The number of training iterations is 500, and the established soft sensor models are tested by the utilized 200 sets of data with average and representative. The predicted simulation results are shown in Figures 19–22. Figure 19 shows the output contrast of the target purity in the E port effluent in the SMB chromatography process under the gradient, Kalman, Kaczmarz, and PseudoInv algorithms to obtain the optimized conclusion parameters of ANFIS. Figure 20 shows the predictive error comparison curves. Figure 21 shows the output contrast of the impurity purity in the R port effluent in the SMB chromatography process under the gradient, Kalman, Kaczmarz, and PseudoInv algorithms to obtain the optimized conclusion parameters of ANFIS. Figure 22 shows the predictive error comparison curves. Table 5 compares the predicted performance indicators of the established soft sensor models.Figure 16
Membership function of input 1 based on the FCM clustering method.Figure 17
Membership function of input 2 based on the FCM clustering method.Figure 18
Membership function of input 3 based on the FCM clustering method.Figure 19
Effluent purity prediction results in the E port.Figure 20
Effluent purity prediction error in the E port.Figure 21
Prediction results of impurity purity in the R port.Figure 22
Prediction error of impurity purity in the R port.Table 5
Comparison of predictive performance indicators of the ANFIS soft sensor model based on the FCM clustering method.
Performance
RMSE
SSE
MAPE
MPE
Purity of the E port
Gradient
0.0643
0.8281
0.0258
0.8410
Kalman
0.0485
0.4699
0.0258
0.5163
Kaczmarz
0.0517
0.5344
0.0196
0.6431
PseudoInv
0.0440
0.3874
0.0253
0.4297
Purity of the R port
Gradient
0.1245
3.0982
0.5547
0.4398
Kalman
0.087
1.5336
0.5557
0.4262
Kaczmarz
0.1131
2.5603
0.6397
0.5862
PseudoInv
0.0837
1.4009
0.5558
0.4129According to the above simulation results, it can be seen that the input data spatial division and premise parameter determination are realized by utilizing the FCM clustering algorithm and ANFIS soft sensor models based on the Kalman, Kaczmarz, and PseudoInv algorithms for optimizing conclusion parameters have the better prediction results on the key economic and technical indicators of the SMB chromatographic separation process. In order to distinguish the optimization performance of the four algorithms, based on the adopted four performance indicators (RMSE, SSE, MAPE, and MPE), the ANFIS soft sensor model based on the FCM clustering algorithm and the PseudoInv algorithm has higher prediction accuracy than the other three algorithms.


## 5. Conclusions

In this paper, a soft sensing modeling method of the SMB chromatographic separation process based on ANFIS is proposed. Three input data space division and antecedent parameter determination methods are combined with four consequent parameter optimization algorithms to realize the ANFIS soft sensing models on the target purity of the E port and R port in the SMB chromatographic separation process. The simulation results verify that the proposed models can obtain better prediction results on the purity of components, which will supply the foundation of the quality closed-loop control.



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*Source: 1312709-2019-11-13.xml*