J. Cent. South Univ. (2019) 26: 63-74

DOI: https://doi.org/10.1007/s11771-019-3982-1

Modeling of goethite iron precipitation process based on time-delay fuzzy gray cognitive network

CHEN Ning(陈宁), ZHOU Jia-qi(周佳琪), PENG Jun-jie(彭俊洁),GUI Wei-hua(桂卫华), DAI Jia-yang(戴佳阳)

School of Information Science and Engineering, Central South University, Changsha 410083, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Abstract: The goethite iron precipitation process consists of several continuous reactors and involves a series of complex chemical reactions, such as oxidation reaction, hydrolysis reaction and neutralization reaction. It is hard to accurately establish a mathematical model of the process featured by strong nonlinearity, uncertainty and time-delay. A modeling method based on time-delay fuzzy gray cognitive network (T-FGCN) for the goethite iron precipitation process was proposed in this paper. On the basis of the process mechanism, experts’ practical experience and historical data, the T-FGCN model of the goethite iron precipitation system was established and the weights were studied by using the nonlinear hebbian learning (NHL) algorithm with terminal constraints. By analyzing the system in uncertain environment of varying degrees, in the environment of high uncertainty, the T-FGCN can accurately simulate industrial systems with large time-delay and uncertainty and the simulated system can converge to steady state with zero gray scale or a small one.

Key words: time-delay fuzzy gray cognitive network (T-FGCN); iron precipitation process; nonlinear Hebbian learning

Cite this article as: CHEN Ning, ZHOU Jia-qi, PENG Jun-jie, GUI Wei-hua, DAI Jia-yang. Modeling of goethite iron precipitation process based on time-delay fuzzy gray cognitive network [J]. Journal of Central South University, 2019, 26(1): 63–74. DOI: https://doi.org/10.1007/s11771-019-3982-1.

1 Introduction

The iron precipitation method by leaching zinc sulfide in acid solution to recycle elemental sulfur is one of the best zinc hydrometallurgy methods and also a process of obtaining zinc sulphate which is adopted by most zinc smelting enterprises [1–3]. Rich iron in the zinc sulfide concentrate makes the zinc sulfate solution rich in large amounts of ferrous ions, leading to low purity of finished zinc ingots and extra consumption of raw material and other issues. Therefore, iron removal is an important process in zinc hydrometallurgy.

The iron precipitation process is a process with complex physical and chemical reactions which is cascaded by multi-reactors, and is of multi-reaction with coupling, nonlinearity, high uncertainty and large time-delay. The main chemical reactions involved in the reaction process are oxidation reaction, hydrolysis reaction and neutralization reaction which are coupled with each other and the change of the ions’ concentration in the solution is nonlinear. Because of the complexity, confinement and high uncertainty inside reactors, the nonlinear relationship between the production target and the process parameters as well as other related information is hard to be described and estimated. Besides, workers regularly sample to obtain the concentration of each ion in the solution, and then compare the acquired data with the historical data to adjust the input of control variables, which is of large time-delay and high uncertainty. Therefore, it is of great realistic meaning to propose a new modeling method based on experts’ practical experience, historical data and process mechanism which can accurately reflect the dynamic change of the iron removal process, to guide the practical application and realize the efficient use of resources and save energy.

Fuzzy cognitive map (FCM) is a method that can realize the communication between raw data and researchers. It has the characteristics of formal description, numerical reasoning and the fuzzy information expression. ANNINOU et al [4] used FCM to present a mathematical model for Parkinson’s syndrome by analyzing the factors that are likely to make people get Parkinson's syndrome, which provided a very interesting research direction. In Ref. [5], the relationship between personality and emotion was analyzed, and a mathematical model based on FCM was established. This model was of great significance to develop humanized systems. OBIEDAT et al [6] used the method combining kinetic system with neural network to analyze social ecosystem and established the FCM model of social ecosystem. The results showed that this method can provide suitable suggestions for decision makers. However, there are different degrees of time-delay in the actual systems. Many researchers have done some researches about the systems with time-delay [7–9]. There also exists such a problem that the traditional FCM modeling method can not reflect the dynamic changes of the system very well when it comes to the system with large time-delay. In Ref. [10], on the basis of professional knowledge and process analysis, the definition of different time delay that reflect different degrees of influence on the weight value was made, a T-FCM modeling method was proposed and applied to the medical decision support system. The results did indicate that this method can provide good advice for medical work, but it requires a high degree of understanding of the system and depends on experts’ practical experience too much. NEOCLEOUS et al [11] studied the factors affecting the political economy in Cyprus and analyzed the time-delay and time constant between various factors and results. Based on historical data and professional knowledge, the FCM model related to the change of time delay and time constant was established for the political economy dynamics of Cyprus. The results suggested that the model can provide constructive advice for the political and economic construction, but this method also has too much dependence on the professional experience. PARK et al [12] added the concept of time-delay into the FCM model to describe the existence of the systems with time-delay, this method can simulate the actual system more accurately, but required a higher degree of understanding of the system and specific group of time-delay. BOURGANI et al [13] adopted timed fuzzy cognitive map (T-FCM) to help evaluate or dispute doctors’ decisions on estimating the health status of both mother and fetus during delivery, but it’s difficult to incorporate knowledge, experience and literature into a system that infers a decision comprehensively. LEE et al [14] proposed a learning rule for temporal FCMs involving post- and pre-delay time by extending Oja’s learning rule and applied it to solve a time-delayed chemical plant control problem. But this method depends on experts’ experience too much when we define time delay between nodes. ZHANG et al [15] firstly analyzed the factors affecting the trust of virtual enterprises (VE) and defined the strength affected by time lag between each node, the fuzzy cognitive time maps (FCTM) model of the trust of VE was established. Finally, a swift trust model was used to validate the effectiveness of FCTM as an example. But the weights between each node were affected by time lag which was hard to define, the FTCM modeling method might be not very accurate [15].

The traditional FCM achieves system modeling through off-line simulation, and the accuracy of the system model is influenced by the accuracy of professional knowledge. Fuzzy cognitive networks (FCN) [16–18] derived from FCM can realize the close connection between the established model and actual system and is of the function of putting excellent control and decision-making on the actual system [19]. The FCN framework includes a fuzzy cognitive map, an update mechanism based on system feedback and the storage of knowledge gained throughout the operation. So FCN can be completely divorced from experts’ practical experience.

Gray system theory (GST) has made great contributions in the establishment of the system model, which uses the method of probability theory and mathematical statistics to obtain the new data sequence with weakened randomness and enhanced regularity [20]. Since the new sequence not only reflects the change of the original sequence but also eliminates the volatility, the gray system theory can solve system problems with some unknown parameters. JI [21, 22] proposed an unbiased gray prediction model, which greatly eliminates the inherent bias of the traditional model. MA et al [23] proposed a novel time-delayed polynomial gray prediction model which is abbreviated as TDPGM(1, 1), and established a gray prediction model of the consumption of natural gas from 2014 to 2020 and verified the accuracy of the model. DING et al [24] proposed a novel gray multivariable model that is of high accuracy and good adaptability to predict the carbon dioxide emissions from 2014 to 2020. The result proved that the model can provide a solid basis for formulating environmental policies and energy consumption plans.

In view of the superior performance of FCN and gray system theory in system modeling, the combined applications of the two are of great significance in the modeling of complex systems with the characteristics of coupling, nonlinearity and high uncertainty. However, the accuracy of a model can not be separated from the setting of some parameters, and FCN is no exception. Similar to parameter identification of the neural network, weight learning uses a certain method to update the relationship between the various nodes, and constantly adjust the initial weight of the system to improve the accuracy of FCN. Hebbian learning method is one of the commonly used methods of learning weights, firstly proposed by Hebbian in 1949. PAPAGEORGIOU et al [25] presented the nonlinear Hebbian learning method for nonlinear systems, which extended Hebbian learning method by modifying the weight update formula. A LASSOFCM learning method was proposed, which can learn large-scale FCM systems without prior experts’ practical experience and large amounts of data, but it is difficult to transfer the FCM learning problem into the problem of sparse signal reconstruction [26]. NATARAJAN et al [27] analyzed some factors affecting the yield of sugarcane in India, such as the soil and climate factors, and established the FCM model. The parameters of the model were identified through data-driven non-linear Hebbian learning method (DDNHL) and genetic algorithm. We have studied nonlinear Hebbian learning algorithm with terminal constraints before, which is of high convergence rate and depends little on the given intial value to train model’s weights [28, 29]. So we adpot it to train the weights of the established model.

In this paper, goethite iron precipitation process was studied. Based on the fuzzy cognitive map and the gray system theory, a time-delay fuzzy gray cognitive network model of the iron precipitation system is established by taking the coupling characteristics between each reactor in the series structure, the nonlinear characteristics of the ions concentration inside the reactors, the high uncertainty of the control amounts and the large time-delay of the reaction into consideration. The nonlinear Hebbian learning method with terminal constraints is used to calculate the weights of the fuzzy gray cognitive network model of the iron precipitation process. Finally, the simulations were carried out under different uncertainties. The results proved that T-FGCN modeling method can accurately simulate industrial systems with time-delay in the environment with a high degree of uncertainty.

2 Flow of goethite iron precipitation process

The objective of goethite iron precipitation process is the zinc sulfate solution which can be obtained by leaching zinc concentrate in acid solution. The reaction needs to be carried out at temperature ranging from 65 to 82 °C and pH value ranging from 3.0 to 4.0.

The schematic of goethite iron precipitation process is shown in Figure 1. The process consists of five reactors in series structure which are placed from the higher place to lower place and the reaction process requires constant addition of calcine and oxygen. The solution flows through each reactor in turn and then is firstly delivered to 1# reactor which is subjected to a sufficient chemical reaction under the stirring of the continuous stirred tank reactor to achieve the effect of the initial iron removal. Then the solution will flow to 2#, 3#, 4#, 5# in turn to achieve iron removal again. In this process, it is required that the concentration of ferrous ions, ferric ions and pH value in each reactor should be kept within a certain range, as well as the outlet concentration of ferrous ions in the solution flowing from 5# reactor must be less than 1 g/L and the pH value in every reactor must kept within range from 3.0 to 4.0.

Figure 1 Schematic of goethite iron precipitation process

The reactions of the gas, solid and liquid phases are involved in each reactor during the goethite iron precipitation process. Three very important chemical reactions are as follows.

Oxidation reaction:

              (1)

Neutralization reaction:

                    (2)

Hydrolysis reaction:

                 (3)

The purpose of the oxidation reaction is to oxidize the ferrous ions in the solution into ferric ions; the neutralization reaction aims to ensure a certain reaction condition, that is, the pH value of the solution should be kept within a certain range; the hydrolysis reaction is intended to hydrolyze the ferric ions in the solution to produce goethite precipitation.

In the chemical reaction process, the chemical reactions have different reaction time, and the change of the reactants can not lead to the change of the product immediately. There is a period of certain reaction time, that is called time-delay, and different chemical reactions are of different time-delay. In the goethite iron precipitation process, the oxidation rate of ferrous ions is mainly affected by dissolved oxygen concentration, the concentration of itself and pH value of the solution. In the whole progress, the oxidation rate is rather fast, the hydrolysis rate of ferric ions depends on their own concentration, so the hydrolysis rate is relatively slow. Thus, there are different time-delays in the three chemical reactions involved in the goethite iron precipitation process, leading the nodes in the FCN model to update asynchronously in the iron precipitation process. Meanwhile, the inlet concentration of the ions in the reactor is obtained by sampling and is of a certain degree of uncertainty. Thus, the traditional modeling method based on FCN can not accurately simulate the dynamic changes of the actual systems. It is necessary to work out a new modeling method based on FCN for complex system like goethite iron precipitation process by taking full advantage of the process mechanism, experts’ practical experience and historical data.

3 Modeling of goethite iron precipitation process based on T-FGCN

3.1 Fuzzy gray cognitive network

An FGCN model provides an intuitive, detailed abstract modeling method to inaccurately describe gray relationship between fuzzy concept nodes by representing unstructured knowledge on the basis of FCN.

A nonlinear system can be expressed as a directed graph with nodes, weights and system feedback. The directed graphs with different initial values of nodes and different weights correspond to different systems. Nodes’ set in directed graph is expressed as C={C₁, C₂, …, C_n} and each node represents a characteristic of the system, such as variable, state, event, target etc. FGCN modeling method can make FGCN model and the actual system interact more closely through introducing the concept of control nodes, stable nodes, output nodes and intermediate nodes. The control nodes represent the control variables of the system; the stable nodes represent the node that affects other nodes but will not be affected by other nodes; the output nodes represent the output of the system; the intermediate nodes represent the other nodes besides the above three nodes. The state values of nodes in FGCN are variable, which reflect the values of the nodes (The values are fuzzy or determined). The state values of nodes are gray numbers, which can be expressed as follows.

        (4)

where n is the total number of nodes in FGCN model; can be gray numbers or white numbers

is used to represent the state value of node at time k, and its value is in range [0,1], which is converted from the actual value of the system. W_ij is used to represent the degree of causal impact between node C_i and node C_j and the value of W_ij is in range [–1, 1]. If w_ij>0, then the state of the result node C_j changes in proportion to the state value of the cause node C_i; If w_ij<0 instead, the state of the result node is inversely proportional to that of the cause node; If w_ij=0, the state of the result node is not associated with that of the cause node.

The edge between the two nodes represents the causal relationship between the nodes. Since the FGCN is a combination of the neural network and gray system theory, the value of the arc with weight between the two nodes is expressed as a gray number weight.

      (5)

where i represents cause nodes; j represents result nodes.

FGCN model is a dynamic system containing feedback, where each node represents the behavior and event of the system. A node can affect other nodes and also be affected by other nodes. Thus, the update equation for the state value of the nodes is as follows.

             (6)

In particular, the state values of the stable nodes should be gray numbers based on the actual measured values of the system.

                         (7)

where represent the gray state value vector at time k, k–1 respectively;represents actual measured values of the system of the stable node or output nodes, which are measured online or given in advance; ε represents amplitude and f represents transfer function which is used to convert the state values to interval [0, 1] with assurance.

According to the three chemical reactions and analysis of the iron precipitation process, the reaction process mainly involves the changes of the concentration of ferrous iron, ferric ions and hydrogen ions, as well as the addition of oxygen and calcine. The concentrations of ferrous ions, ferric ions and hydrogen ions interact with each other and are affected by oxygen and calcine, so the nodes representing the ferrous ions, ferric ions and hydrogen ions are regarded as the output nodes. Oxygen and calcine are control variables, and they remain constant throughout the iron precipitation process, the nodes representing them are not only regarded as control nodes but also stable nodes. Therefore, the FGCN model of the 1# reactor is established as an example.

C₁, C₂, C₃, C₄, C₅ represent oxygen, calcine,ferrous ions, ferric ions and hydrogen ions respectively. Oxygen and calcine are not affected by other nodes and itself as they are control variables, there is no arc pointing to the two nodes. The other nodes interact with each other, so there are arcs between the nodes.

Figure 2 FGCN model of 1# reactor in goethite iron precipitation process

3.2 Update mechanism of nodes in T-FGCN

There is a problem that the output nodes in the FGCN model of 1# reactor can not update synchronously. Therefore, it is necessary to deeply analyze the relationship that the state values of nodes change over time.

Different time-delay can be obtained by analyzing the change of molar mass in the oxidation, hydrolysis and neutralization reactions involved in the goethite iron precipitation process. Assuming that the time required for consuming 4 mol ferrous ions in the oxidation reaction is t, the reaction reaches equilibrium at the same time. According to conservation of mass, consuming 4 mol ferrous ions needs consuming 4 mol hydrogen ions and 1 mol oxygen in the oxidation reaction (1) and 4 mol ferric ions are generated simultaneously. 4 mol ferric ions generated by the oxidation reaction will produce 12 mol hydrogen ions in the event of complete reaction in the neutralization reaction (3). 8 mol hydrogen ions remain in the solution by calculating the 4 mol hydrogen ions consumed in the oxidation reaction and 12 mol hydrogen ions generated in the neutralization reaction. Consuming the 8 mol hydrogen ions remaining in the solution needs 4 mol calcine. Above all, consuming 4 mol oxygen takes time 4t. consuming 4 mol calcine takes time t, and generating 4 mol ferric ions takes time t, generating 4 mol hydrogen ions takes time 0.5t. We define the ratio of the time required for consuming or generating same molar mass of the materials as the time-delay between nodes. Because generating the same 4 mol hydrogen ions as the other reactants is 0.5t, we define 0.5t as one time period. For example, the time required for node C₅ representing hydrogen ions to change from node C₃ representing ferrous ions is 4 time periods, the time required for node C₃ to change from node C₅ is 1 time period instead. That is to say, the time-delay are 4 time periods and 1 time period, respectively. We define a matrix, where each element represents the time required for a node in the system model to cause another node to change, that is the time-delay between nodes.

                   (8)

where n represents the number of nodes in system model; Γ_ij represents the time required for the node C_i to cause node C_j to change, i, j=1, 2, 3, …, n.

Combined with the established FGCN model of the 1# reactor, the time-delay matrix in the goethite iron precipitation process system model is shown as follow.

                      (9)

There is no time-delay between the node and itself, that is, the time-delay is 0. The stable nodes in the system will not be affected by other nodes, so the time-delay between the other nodes and stable nodes is 0 too.

The relationship between the dynamic behavior of the system is stored in the network structure of the fuzzy cognitive map and the causal relationship between nodes. In the model, the state value of a node at a certain time is affected by the state value of the node itself at previous time and that of the nodes having causal relationship with it. When there are different time-delay between nodes, the state value of a node at time k is not only related to that of itself at time k–1, but also to the state values at time k–1, k–2, …, k–m of the nodes having casual relationship with it. So the update formula of the state value of a node is as follows.

    (10)

where n represents the total number of system nodes; represent the gray state value of node C_j at time k, k–1 respectively; represents the gray weight between node C_i and node C_j at time m, Γ represents the time-delay matrix of the system; Γ_ij represents the time-delay between node C_i and node C_j.

Therefore, the T-FGCN model of the 1# reactor is established by adding the time-delay between the nodes in the FGCN model of the goethite iron precipitation process.

3.3 Nonlinear Habbian learning algorithm with terminal constraints

The state value of a node obtained by each iteration can be regarded as control quantity to be applied to the original system, and a real-time state value can be obtained to be used as a learning target. The nonlinear Hebbian learning algorithm with terminal constraints introduces the actual feedback of the system as constraint, and views the error of the model predictive value and the actual measured value of the system as a criterion of adjusting the weight. The error directly reflects the difference between the predicted value and the actual value and it can achieve directional correction of the weights. The algorithm not only improves convergence rate, but also solves the problem that traditional NHL algorithm without supervision strongly depends on initial value. In this paper, we use the nonlinear Hebbian learning algorithm with terminal constraints to train the weights of the T-FGCN model.

represents the gray state value of nodes in T-FGCN model, represents gray weights between two nodes at time k. The weights’ update formula of NHL algorithm with terminal constraints is as follow.

     (11)

where represent the gray weights between node C_i and node C_j at time k+1, k respectively; γ, η, κ represent decay rate, learning rate and correction rate respectively, 0<γ, η, κ<0.1; represents the gray scale of node C_i and its formula is shown as follows.

                 (12)

The two termination criteria for the nonlinear Hebbian algorithm with terminal constraints based on the T-FGCN model are as follows.

Criterion 1:

            (13)

whererepresent the white numbers calculated by T-FGCN at time k, k–1 respectively and equalization whitening is used in the whitening process; F₁ represents the absolute value of the error between the values of the white numbers calculated by T-FGCN at time k, k–1 respectively. When it is less than 0.002, the system is considered to be in equilibrium.

Criterion 2:

                             (14)

where represents the corresponding white number of When errors of all nodes are less than the given threshold, the iteration can be terminated, generally e=0.001.

The flow of the nonlinear Hebbian learning algorithm with terminal constraints based on T-FGCN is as follows:

Step 1: Set initial state value and the initial weight matrix according to experts’ practical experience and historical data;

Step 2: Calculate the gray state value of each node at the next moment according to Eqs. (7) and (10);

Step 3: According to Eq. (12), calculate the errors between the calculated state values of the nodes and state values of the nodes in the actual system; calculate the gray errors between the nodes at current time and last time;

Step 4: Judge the errors whether satisfy Eqs. (13) and (14). If not, go to step5; store the final weight matrix otherwise;

Step 5: Update the weight according to Eq. (11) and go to step 2.

4 Simulation and analysis

It can be seen from the T-FGCN model of the 1# reactor shown in Figure 3 that the state variables are the concentration of ferrous ions, ferric ions and hydrogen ions, and they must be kept within the defined range, that is, between the upper and lower bounds. The control ranges of the three kinds of ions are shown as Eq.(15).

                    (15)

According to the requirements of the actual process, the ranges of the outlet concentration of ferrous ions, ferric ions and pH value are 6–10.5 g/L, 0.6–1.4 g/L and 3.0–4.0, respectively. The units of the data are processed uniformly, and then normalized by S-curve function, and the control target the model is shown as Eq. (16).

                  (16)

It is found that the amounts of the addition of oxygen and calcine are constant throughout the process, but have an effect on the concentration of the other three ions. According to the experts’ practical experience and historical data, the initial weight matrix of 1# reactor of T-FGCN model is obtained as follow.

Figure 3 T-FGCN model of 1# reactor in goethite iron process

     (17)

According to the measured data of the actual process, the inlet concentrations of ferrous ions, ferric ions of the 1# reactor are 11.4 g/L, 2.05 g/L respectively, and the pH value is 2.5. Meantime, the corresponding amount of oxygen added is 19.38 m³/h and the amount of calcine added is 0.265 t/h. After a period of sufficient reaction, the concentration of the ferrous ions, ferric ions are 7.8 g/L, 0.93 g/L, respectively, and the pH value is 3.4. The initial state values and steady state values of the model are as follows after unit being processed uniformly and normalized by S-curve function.

    (18)

  (19)

4.1 Simulation and analysis of FCN model

According to Eqs. (17) and (19), the nonlinear Hebbain learning algorithm with terminal constraint is used to train the weights of FCN model, and the final weight matrix of FCN model is as follow.

   (20)

Substituting the final weight matrix into the FCN model, the system will be in equilibrium after several iterations. When the system is in equilibrium, the state values of each node are shown in Table 1.

Table 1 Simulation results of FCN model

It can be seen from Table 1 that FCN model can accurately describe the actual conditions of the system through adding the system feedback, and the state of each node can reach their equilibrium point and the value of each node is kept within the range shown in Eq. (16).

4.2 Simulation and analysis of FGCN model

FCN model can reflect the system characteristics but requires the measured data to be absolutely accurate, so it can not distinguish and deal with the uncertainty in the measured data. FGCN model can deal with the complex systems with uncertainty by introducing gray system theory.

The gray scale values of initial state values are chosen to be 0, 0.2 and 0.4, respectively, and then corresponding vectors of initial state values are as follows.

                   (21)

                    (22)

                   (23)

The initial weight matrix given by the expert is considered to be of gray uncertainty, so the initial weight matrix of the system shown as Eq. (17) are extended to be as follows in the case of gray scale values being 0, 0.2 and 0.4.

 (24)

 (25)

 (26)

The nonlinear Hebbain learning algorithm with terminal constraint is used to train the weights of FGCN model. Substituting the obtained weight matrix into the model, the steady state values of the nodes are obtained.

According to the comparison between Tables 1 and 2, it can be seen that the steady state values of the FGCN model are exactly the same as that of the FCN model when the initial values are whitening values, the FCN model does not have the parameters to judge the uncertainty but the FGCN model adds additional information with gray scale to judge the uncertainty of the results.

According to the corresponding whitening values of the gray steady state values in Tables 2–4, it can be seen that FGCN model can converge to a precise equilibrium point and the control nodes reach the target range though the initial value of the system has a certain degree of gray uncertainty. It can be seen that the output results are of small gray scale or zero gray scale when the initial states values are of small gray uncertainty. It can be concluded that outputs results are of small gray scale or gray scale of zero when the initial states values are of great gray uncertainty, which reduce the uncertainty of the system to a great extent.

Table 2 Simulation results of FGCN model with gray scale of initial state value being 0

Table 3 Simulation results of FGCN model with gray scale of initial state value being 0.2

Table 4 Simulation results of FGCN model with gray scale of initial state value being 0.4

4.3 Simulation and analysis of T-FGCN model

FGCN model neglects the problem that the update of nodes’ state values are not synchronous, because the actual systems mostly have time-delay. Therefore, the simple FGCN modeling method can not simulate the actual system accurately. T-FGCN model adds the concept of time-delay between system nodes based on the FGCN model, and can simulate the system with time-delay and uncertainty.

The final weight matrix of T-FGCN model can be obtained by nonlinear Habbian learning algorithm with terminal constraints. Substituting the weight matrix into the T-FGCN model, we get the steady state values of each node.

According to the comparison between Tables 2 and 5, Tables 3 and 6, Tables 4 and 7 respectively, it can be seen that steady state values of T-FGCN are closer to the steady state values of the actual system than that of FGCN model when the gray scales are 0, 0.2 and 0.4 respectively, and the values are within the range shown as Eq. (16).

According to Tables 5 and 6, it can be seen that the whitening values of output nodes are very close to steady state values of the actual system when the system has small gray uncertainty, and the output values of the T-FGCN model are of very small gray scale. From Table 7, we can conclude that when the system has great gray uncertainty, the output values of the T-FGCN model are also of very small gray scale. Above all, the T-FGCN modeling method can describe the system with time-delay and big gray scale accurately and it can achieve excellent outputs.

In summary, T-FGCN modeling method can model systems with time-delay and uncertainty accurately, reflecting the dynamic characteristics of the systems, and the outputs of the system can converge to more accurate balanced state. Besides, the gray scale is used as the evaluation criterion of the output reliability, which increases the flexibility and applicability of the T-FGCN modeling method.

Table 5 Simulation results of T-FGCN model with gray scale of initial state value being 0

Table 6 Simulation results of T-FGCN model with gray scale of initial state value being 0.2

Table 7 Simulation results of T-FGCN model with gray scale of initial state value being 0.4

5 Conclusions

T-FGCN modeling method combining the neural network and gray system theory can accurately simulate the industrial systems with time-delay and uncertainty. The T-FGCN model of actual systems after being trained by the nonlinear Hebbian algorithm with terminal constraints can reflect system characteristics in the case of data with uncertainty and information with errors. The results show that T-FGCN modeling method can accurately reflect the dynamic characteristics of the system with a certain degree of gray scale, and the outputs of the system can converge to the steady state that satisfies the control range.

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(Edited by HE Yun-bin)

中文导读

基于时滞模糊灰色认知网络的铁矿沉铁过程建模方法

摘要：针铁矿沉铁过程是由多个连续反应器级联，并且包含氧化反应、还原反应以及中和反应等一系列复杂化学反应的复杂过程，具有强非线性、不确定性及大时滞性等特点，难以建立精确的数学模型。本文提出了一种基于T-FGCN（Time-delay Fuzzy Gray Cognitive Network，T-FGCN）的针铁矿沉铁过程的建模方法。根据过程机理、专家经验和历史数据，建立针铁矿沉铁系统的T-FGCN模型，利用带终端约束的非线性Hebbian学习算法（Nonlinear Hebbian Learning,NHL）对模型权值进行学习。通过在不同程度上的不确定性环境下对系统进行分析，结果表明，T-FGCN建模方法能在不确定性高的环境下对具有大时滞的工业系统进行较为精确的模拟，系统稳定状态值能收敛到一个灰度为零或者灰度很小的灰数平衡点。

关键词：沉铁过程；模糊灰色认知网络；时滞；非线性Hebbian学习

Foundation item: Project(61673399) supported by the National Natural Science Foundation of China; Project(2017JJ2329) supported by the Natural Science Foundation of Hunan Province, China; Project(2018zzts550) supported by the Fundamental Research Funds for Central Universities, China

Received date: 2017-12-18; Accepted date: 2018-05-02

Corresponding author: CHEN Ning, PhD, Professor; Tel: +86-13875915950; E-mail: ningchen@csu.edu.cn; ORCID: 0000-0001-8384- 2948