# Tracking Control of the Dynamic Input-Output Economic System Based on Data Fusion

Authors: Yirun Chen; Wensheng Dai
Journal: Security and Communication Networks
 (2022)

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## Abstract

In recent years, under the background of stable economic operation, people's research on economic systems has become increasingly popular. The dynamic input-output model reflects the change and development process of the input-output relationship of the economic system over a period of time. The main purpose of tracking control is to design a suitable controller so that the output of the control system can track the output of the reference system as much as possible. In the economic system, data is an important factor. Based on this, this paper mainly studies the tracking control of dynamic input-output economic system based on data fusion. This research takes the data fusion of the dynamic input-output economic system as the starting point and takes the optimal control and tracking of the economic system as the research object of this research. Based on data fusion technology, a new dynamic input-output economic system tracking and control is proposed. This paper studies the finite-time optimal tracking control of linear systems. Through numerical examples, comparing the finite-time and infinite-time optimal control simulation results, it is proved that the algorithm can achieve good tracking control. Experimental data shows that the optimal and suboptimal performance indicators for a limited time are 0.7729412 and 1.5687310, respectively. Therefore, compared with the infinite-time optimal control, the performance loss and the final tracking error of the suboptimal control proposed in this study are reduced.


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## Body

## 1. Introduction

Up to the present position, although input-output technology has only a history of more than half a century, it has made great development in depth and breadth [1, 2]. A data fusion produced by the intersection, integration, and extension of big data technology with other disciplines has become a new research content in the field of data mining [3, 4]. Input outputs were determined by using a finite mixed regression model to determine their contribution to the probability of attending and the probability of attending more. Sociodemographic and socioeconomic characteristics, participation in other cultural activities, ticket prices, and theatre supply are considered to varying degrees [5]. Data fusion-related technologies and algorithms have effective and extensive theoretical foundations and application prospects in many fields and have attracted wide attention at home and abroad. The tracking control of the dynamic input-output economic system not only needs to design the controller to meet the stability of the dynamic input-output economic system, but also requires the system to finally realize the output tracking of the system under the action of a tracking controller with a given performance reference system output [6]. Therefore, from the point of view of data fusion, it has important practical significance to study the tracking control method of dynamic input-output economic system.Data fusion technology refers to the information processing technology that uses computers to automatically analyze and synthesize several observational information obtained in time series under certain criteria to complete the required decision-making and evaluation tasks. Data fusion technology includes the collection, transmission, synthesis, filtering, correlation, and synthesis of useful information given by various information sources, in order to assist people in situation/environment determination, planning, detection, verification, and diagnosis [7, 8].Regarding the research on the input and output economic system, many scholars have explored it from multiple angles. For example, Acemoglu studied the microscopic origin of macroeconomic tail risk [9]; from the perspective of input and output, Wu studied the current situation of China's economic energy use [10]; Wiebe studied the global circular economy scenario under the multiregional input-output framework [11]; Hung analyzed the links between economic opening construction activities through multiregional input-output [12]; He took the Australian economic system and waste management as the research object and studied the input-output relationship between the two [13]. These scholars' research on input and output can control the economy to a certain extent. However, most of these researches are analyzed in a static environment, and their practicability is not very high, and it is not even applicable to a dynamic environment. It can be seen that there are not many studies on the input and output economic system, but the method of data fusion is used; there are not many researches on the tracking control of economic systems. Therefore, the research in this article helps to enrich the theoretical system in this field.The output in input-output technology refers to the result of a system performing an activity process. For example, the result of production activities in a production system is the products (material products and services) produced by various departments in the system. Input-output technology can be used not only in economic systems, but also in other systems, such as military systems, ecosystems, and so on. These systems also have inputs and outputs in the process of carrying out various activities. For example, in military activities, various materials such as ammunition, gasoline, etc. will be input, as well as the results achieved by military activities.The purpose of this paper is to study the tracking control of a dynamic input-output economic system based on data fusion. This article first analyzes the dynamic input-output model and builds a continuous dynamic input-output model. The relationship between input and output in the reproduction process of each period is studied with continuous time variables. The output of each department and the final net product are functions of time and are a system of first-order linear ordinary differential equations. Then, this article summarizes the data fusion theory and the meaning of the amount of information and puts forward the information fusion theory of the economic system and the data fusion control method of the economic system. This article briefly introduces the Solow growth model, Ramsey-Cass-Koopmans model, Diamond model, and models with credit friction. In this paper, an output feedback tracking controller is designed to make the dynamic input-output economic system asymptotically stable when the secondary performance index function is satisfied. In this paper, numerical examples are used to compare the finite-time and infinite-time optimal control simulation results, and it is proved that the algorithm can achieve good tracking control. The data shows that the optimal and suboptimal performance metrics in finite time are 0.7729412 and 1.5687310, respectively.


## 2. Dynamic Input-Output Economic System Based on Data Fusion

### 2.1. Dynamic Input-Output Model

The input-output analysis method is a practical economic analysis method. Through the establishment of related models, it can deeply analyze the interdependence between various industrial sectors in production activities. Through the input-output model, we can clearly see the direct and indirect connections, production consumption, and distribution between the various industrial sectors of the national economy in the production process. Its research object can be either the input-output relationship of various sectors of the national economy, or a certain economic entity (such as a certain city or a certain enterprise). When the research object is a certain region, it reflects the internal relations between industrial sectors; when the research object is a certain industry, it reflects the supply-demand connection between different products subdivided in the industry; when the research object is a certain enterprise, it reflects the internal relations of the enterprise the relationship between departments or production links [14, 15].There are many types of input-output tables, and different forms of models can be established according to different needs. For example, according to the different units at the time of statistics, it is divided into physical input-output tables and value-based input-output tables; according to the different representation structures, it is divided into symmetrical input-output tables and UV-type input-output tables; according to product the substitutability is different, divided into the competition type table and the noncompetition type table, etc., and there are different types of input-output tables classified according to the different geographical scopes involved in the statistics, the different time spans, and the different fields of investigation.The data in the input-output table is generally divided into three parts, distributed in the first, second, and third quadrants.The first quadrant is the core of the input-output table and the key to input-output analysis. The data in this quadrant forms ann×n matrix. Each data in the matrix has two meanings, which can reflect a certain industry sector. In the production process, the value of products or services consumed by each industrial sector (column direction) can reflect the value of products or services used by each industrial sector (row direction). The analysis of the data in this quadrant can reflect the value flow process between different industrial sectors.The second quadrant and the first quadrant sharen industrial sectors in the vertical direction (that is, the same main column), but the object column is composed of columns such as consumption, investment, and export. The data in this quadrant reflects the value and category of products or services produced by various industrial sectors in the final consumption process, that is, the final use situation.The third quadrant and the first quadrant share the horizontaln industrial sectors (that is, the same object column). However, the main column consists of value-added items such as fixed asset depreciation, labor compensation, and net income. The data in this quadrant reflects the increase in industrial value of each industry sector in production activities and the different sources of these increases. In simple terms, the basic form of the input-output table is shown in Table 1 (department is represented by D).Table 1
Basic form of input-output table.
Intermediate useEnd useTotal outputD1D2...DnConsumptionInvestExitImportTotalIntermediate inputD1Xi1Xi2...XinCIYE1YM1Y1X1D2X21X22...X2NCIYE2YM2Y2X2..............................DnXn1Xn2...XnnCnInYEnYMnnXnInitial investmentDepreciationD1D2...Dn——————Labor remunerationW1W2...Wn——————Tax and profitV1V2...Vn——————TotalF1F2...Fn——————Total investmentX1X2...Xn—————The dynamic input-output model is developed on the basis of the static model, and it is a process of change. By studying the interrelationship between the reproduction process in several periods and the reproduction process in each period, it reflects the connection between the various sectors of the national economy in different periods and their quantitative dependence. In this respect, the dynamic input-output model can better reflect the actual situation of economic development.


### 2.2. Continuous Dynamic Input-Output Model

The continuous dynamic input-output model is represented by differential equations. Among them, this research mainly focuses on the expression method of the open model, in which the model includes some factors including household consumption as the final product of dynamic input and output, which is used as the exogenous variable of the model. The final product is the symmetry of the “intermediate product,” which is produced in a certain period and is no longer processed in the same period and is available for final consumption and use. It is an important indicator for examining the results of economic work from the perspective of the input-output balance of the national economy, and it is also the basis for calculating the gross national product. Final products are strictly defined as “products for living consumption,” and in practical applications, final products are divided into individual final consumer goods and collective consumer goods. The calculation method is as follows:(1)yi=ΔFi+y∼i.Here,ΔFi represents the quantity of productive investment products used by department i, and y∼i represents the final net product. Divide ΔFi according to the production department, as shown in(2)ΔFi=∑j=1nΔFij,i=1,2,...,n.Here,ΔFij represents the number of newly added sector i products in sector j as productive investment products. To express the total output of products in year t, as shown in(3)xit=∑j=1naijxjt+yit,i=1,2,...,n.Combining the two, the total output of products in yeart can be obtained, as shown in(4)xit=∑j=1naijxjt+∑j=1nΔFijt+yit,i=1,2,...,n,(5)xit=∑j=1naijxjt−∑j=1nΔFijt=yit,i=1,2,...,n.Here, the direct consumption coefficientaij is the basis for calculating the complete consumption coefficient. The direct consumption coefficient A represents the technical and economic relations between the various sectors of the national economy. The complete consumption coefficient matrix is denoted by B, and its calculation method is shown in(6)B=I−A−1−I.SubstitutingΔFij into it, the relationship between the two is shown in (7)ΔFijt=Fijt+1−FijtΔt⋅Δt=ΔFijtΔt≈dFijtdt⋅Δt,(8)dFijtdt=dFijtdxjt⋅dxjtdt.Among them, the definition of this formula is shown in(9)bijt=dFijtdxjt.Among them,bij represents the investment coefficient (capital coefficient) in year t, which represents the ratio of the marginal growth of output to the marginal growth of productive investment. When △t = 1, the total product model of year t is shown in(10)xit−∑j=1naijxjt−∑j=1nbijtdxjtdt=yit,i=1,2,...,n.Its matrix form expression method is shown in(11)xt−Axt−Bx.t=y∼t.In the formula,B=bijn×n is the investment coefficient matrix. Under normal circumstances, the A and B arrays are constant. Therefore, the dynamic input-output economic system model is highly hypothetical. In practical applications, the departmental elements in the model can also be expanded to include various content in the final demand field. For example, the resident consumption factor is expressed as the resident department, the government consumption factor is expressed as the government department, and the import and export are expressed for the foreign trade department, include these parts in the model, and expand the content of bij to get a dynamic closed loop model, as shown in(12)xt−Axt−Bx.t=0.


### 2.3. Basic Way of Data Fusion

The so-called data fusion is the association, correlation, and integration of data and data obtained from single and multiple data sources to obtain accurate location and identity estimates, as well as the situation, characteristics, attributes, trends, and threats and their importance, with a data processing process for comprehensive and timely evaluation; this process is a continuous refinement process for its estimation, evaluation, and evaluation of additional data source needs [16, 17]. It is also a process of continuous self-correction in the data processing process to obtain improved results. The basic idea of data fusion is mainly shown in Figure 1.Figure 1
The idea of data fusion.Data fusion theory studies the optimal fusion of related data, unrelated data, definite data, uncertain data, and nonlinear data and applies it to control problems [18–20]. Data fusion methods include centralized and sequential methods. In the centralized method, the nodes of data fusion are on the central processing unit, so all the fusion processes are carried out on one central processing unit. However, this structure has many disadvantages, such as the observed data that needs to be transmitted to the central processing unit, so a high data bandwidth is required. In addition, the time required to transmit data from different data sources to the central processing unit is different, and the data needs to be synchronized. Therefore, although the centralized structure is a perfect structure in theory, it is subject to various constraints in practice. In the sequence structure, the processing for each data source node is carried out separately, the processing result is sent to the fusion node, and each fusion node is responsible for processing the received information of other nodes. This structure provides a more flexible configuration, there can be only one fusion node, or there can be multiple intermediate fusion nodes.


#### 2.3.1. Centralized Method

Data fusion solution is relatively simple, only needs to concentrate on processing all the collected data, and then directly calculates the total amount of data. Assuming datayi∧,y2∧,...,yn∧, the estimated quantity x represents a data set that can be fused, and Ri represents a symmetric matrix. P represents the fusion data volume of the data. Then, the calculation method of the total data volume of n data is shown in(13)P=HTR−1H=∑i=1nHiTRi−1Hi.If the fused data (set) has a nonsingular information about the estimated quantity, it is called the fused data (set), which means thatHiTRi−1Hi is nonsingular. Calculating the optimal fusion method of x, as shown in(14)x∧=P−1∑i=1nHiTRi−1yi.


#### 2.3.2. Sequential Way

Sequential mode accumulates data one by one. Assuming datayi∧,y2∧,...,yn∧, the same estimated quantity x represents a data set that can be fused, and P1=HT1R1−1H1 is positive definite; then x∧=P1−1H1TR1−1y1. Among them, P1∗ represents the data volume of the first to i-th data, xi∗ represents the corresponding optimal fusion, and the calculation methods for P1∗ and x∧i∗ are shown in(15)P1∗=P∗i−1+HTiRi−1Hi,(16)x∧i∗=Pi∗−1P∗i−1x∧i−1+∗HTiRi−1yi.In the formula,i sequentially from 2 to n, when i = n, there is Pn∗  = P, xn∗=x∧.


### 2.4. Data Fusion Modeling of the Dynamic Input-Output Economic System

Input-output data is the basis of system tracking and control. The dynamic input-output economic control system, the state equation, and output equation are shown in(17)xk+1=Axk+Buk,yk=Cxk,x0=x0.Then, the model expression of the dynamic input-output economic system is shown in(18)xk=Axk+Bxk+1−xk+yk.In the formula,x(k) represents the output vector of multisectoral products, y(k) represents the output vector of consumer goods (final products), A represents the input-output (consumption coefficient) matrix, and B is the investment (capital) coefficient matrix. The input-output relationship of each part is shown in Table 2.Table 2
The input-output relationship of the dynamic input-output economic system model.
Cyclek − 1CyclekCyclek + 1Outputx (k − 1)x (k)x (k + 1)Consumables inputAx (k-1)Ax (k)Ax (k + 1)Fixed asset investmentBx (k − 1)Bx (k)Bx (k + 1)Fixed asset formationBx (k)Bx (k + 1)Bx (k + 2)Add fixed assetsB[x (k) − x (k − 1)]B[x (k + 1) − x (k1)]B[x (k + 2) − x (k + 1)]Consumptiony (k − 1)y (k)y (k + 1)


## 2.1. Dynamic Input-Output Model

The input-output analysis method is a practical economic analysis method. Through the establishment of related models, it can deeply analyze the interdependence between various industrial sectors in production activities. Through the input-output model, we can clearly see the direct and indirect connections, production consumption, and distribution between the various industrial sectors of the national economy in the production process. Its research object can be either the input-output relationship of various sectors of the national economy, or a certain economic entity (such as a certain city or a certain enterprise). When the research object is a certain region, it reflects the internal relations between industrial sectors; when the research object is a certain industry, it reflects the supply-demand connection between different products subdivided in the industry; when the research object is a certain enterprise, it reflects the internal relations of the enterprise the relationship between departments or production links [14, 15].There are many types of input-output tables, and different forms of models can be established according to different needs. For example, according to the different units at the time of statistics, it is divided into physical input-output tables and value-based input-output tables; according to the different representation structures, it is divided into symmetrical input-output tables and UV-type input-output tables; according to product the substitutability is different, divided into the competition type table and the noncompetition type table, etc., and there are different types of input-output tables classified according to the different geographical scopes involved in the statistics, the different time spans, and the different fields of investigation.The data in the input-output table is generally divided into three parts, distributed in the first, second, and third quadrants.The first quadrant is the core of the input-output table and the key to input-output analysis. The data in this quadrant forms ann×n matrix. Each data in the matrix has two meanings, which can reflect a certain industry sector. In the production process, the value of products or services consumed by each industrial sector (column direction) can reflect the value of products or services used by each industrial sector (row direction). The analysis of the data in this quadrant can reflect the value flow process between different industrial sectors.The second quadrant and the first quadrant sharen industrial sectors in the vertical direction (that is, the same main column), but the object column is composed of columns such as consumption, investment, and export. The data in this quadrant reflects the value and category of products or services produced by various industrial sectors in the final consumption process, that is, the final use situation.The third quadrant and the first quadrant share the horizontaln industrial sectors (that is, the same object column). However, the main column consists of value-added items such as fixed asset depreciation, labor compensation, and net income. The data in this quadrant reflects the increase in industrial value of each industry sector in production activities and the different sources of these increases. In simple terms, the basic form of the input-output table is shown in Table 1 (department is represented by D).Table 1
Basic form of input-output table.
Intermediate useEnd useTotal outputD1D2...DnConsumptionInvestExitImportTotalIntermediate inputD1Xi1Xi2...XinCIYE1YM1Y1X1D2X21X22...X2NCIYE2YM2Y2X2..............................DnXn1Xn2...XnnCnInYEnYMnnXnInitial investmentDepreciationD1D2...Dn——————Labor remunerationW1W2...Wn——————Tax and profitV1V2...Vn——————TotalF1F2...Fn——————Total investmentX1X2...Xn—————The dynamic input-output model is developed on the basis of the static model, and it is a process of change. By studying the interrelationship between the reproduction process in several periods and the reproduction process in each period, it reflects the connection between the various sectors of the national economy in different periods and their quantitative dependence. In this respect, the dynamic input-output model can better reflect the actual situation of economic development.


## 2.2. Continuous Dynamic Input-Output Model

The continuous dynamic input-output model is represented by differential equations. Among them, this research mainly focuses on the expression method of the open model, in which the model includes some factors including household consumption as the final product of dynamic input and output, which is used as the exogenous variable of the model. The final product is the symmetry of the “intermediate product,” which is produced in a certain period and is no longer processed in the same period and is available for final consumption and use. It is an important indicator for examining the results of economic work from the perspective of the input-output balance of the national economy, and it is also the basis for calculating the gross national product. Final products are strictly defined as “products for living consumption,” and in practical applications, final products are divided into individual final consumer goods and collective consumer goods. The calculation method is as follows:(1)yi=ΔFi+y∼i.Here,ΔFi represents the quantity of productive investment products used by department i, and y∼i represents the final net product. Divide ΔFi according to the production department, as shown in(2)ΔFi=∑j=1nΔFij,i=1,2,...,n.Here,ΔFij represents the number of newly added sector i products in sector j as productive investment products. To express the total output of products in year t, as shown in(3)xit=∑j=1naijxjt+yit,i=1,2,...,n.Combining the two, the total output of products in yeart can be obtained, as shown in(4)xit=∑j=1naijxjt+∑j=1nΔFijt+yit,i=1,2,...,n,(5)xit=∑j=1naijxjt−∑j=1nΔFijt=yit,i=1,2,...,n.Here, the direct consumption coefficientaij is the basis for calculating the complete consumption coefficient. The direct consumption coefficient A represents the technical and economic relations between the various sectors of the national economy. The complete consumption coefficient matrix is denoted by B, and its calculation method is shown in(6)B=I−A−1−I.SubstitutingΔFij into it, the relationship between the two is shown in (7)ΔFijt=Fijt+1−FijtΔt⋅Δt=ΔFijtΔt≈dFijtdt⋅Δt,(8)dFijtdt=dFijtdxjt⋅dxjtdt.Among them, the definition of this formula is shown in(9)bijt=dFijtdxjt.Among them,bij represents the investment coefficient (capital coefficient) in year t, which represents the ratio of the marginal growth of output to the marginal growth of productive investment. When △t = 1, the total product model of year t is shown in(10)xit−∑j=1naijxjt−∑j=1nbijtdxjtdt=yit,i=1,2,...,n.Its matrix form expression method is shown in(11)xt−Axt−Bx.t=y∼t.In the formula,B=bijn×n is the investment coefficient matrix. Under normal circumstances, the A and B arrays are constant. Therefore, the dynamic input-output economic system model is highly hypothetical. In practical applications, the departmental elements in the model can also be expanded to include various content in the final demand field. For example, the resident consumption factor is expressed as the resident department, the government consumption factor is expressed as the government department, and the import and export are expressed for the foreign trade department, include these parts in the model, and expand the content of bij to get a dynamic closed loop model, as shown in(12)xt−Axt−Bx.t=0.


## 2.3. Basic Way of Data Fusion

The so-called data fusion is the association, correlation, and integration of data and data obtained from single and multiple data sources to obtain accurate location and identity estimates, as well as the situation, characteristics, attributes, trends, and threats and their importance, with a data processing process for comprehensive and timely evaluation; this process is a continuous refinement process for its estimation, evaluation, and evaluation of additional data source needs [16, 17]. It is also a process of continuous self-correction in the data processing process to obtain improved results. The basic idea of data fusion is mainly shown in Figure 1.Figure 1
The idea of data fusion.Data fusion theory studies the optimal fusion of related data, unrelated data, definite data, uncertain data, and nonlinear data and applies it to control problems [18–20]. Data fusion methods include centralized and sequential methods. In the centralized method, the nodes of data fusion are on the central processing unit, so all the fusion processes are carried out on one central processing unit. However, this structure has many disadvantages, such as the observed data that needs to be transmitted to the central processing unit, so a high data bandwidth is required. In addition, the time required to transmit data from different data sources to the central processing unit is different, and the data needs to be synchronized. Therefore, although the centralized structure is a perfect structure in theory, it is subject to various constraints in practice. In the sequence structure, the processing for each data source node is carried out separately, the processing result is sent to the fusion node, and each fusion node is responsible for processing the received information of other nodes. This structure provides a more flexible configuration, there can be only one fusion node, or there can be multiple intermediate fusion nodes.


### 2.3.1. Centralized Method

Data fusion solution is relatively simple, only needs to concentrate on processing all the collected data, and then directly calculates the total amount of data. Assuming datayi∧,y2∧,...,yn∧, the estimated quantity x represents a data set that can be fused, and Ri represents a symmetric matrix. P represents the fusion data volume of the data. Then, the calculation method of the total data volume of n data is shown in(13)P=HTR−1H=∑i=1nHiTRi−1Hi.If the fused data (set) has a nonsingular information about the estimated quantity, it is called the fused data (set), which means thatHiTRi−1Hi is nonsingular. Calculating the optimal fusion method of x, as shown in(14)x∧=P−1∑i=1nHiTRi−1yi.


### 2.3.2. Sequential Way

Sequential mode accumulates data one by one. Assuming datayi∧,y2∧,...,yn∧, the same estimated quantity x represents a data set that can be fused, and P1=HT1R1−1H1 is positive definite; then x∧=P1−1H1TR1−1y1. Among them, P1∗ represents the data volume of the first to i-th data, xi∗ represents the corresponding optimal fusion, and the calculation methods for P1∗ and x∧i∗ are shown in(15)P1∗=P∗i−1+HTiRi−1Hi,(16)x∧i∗=Pi∗−1P∗i−1x∧i−1+∗HTiRi−1yi.In the formula,i sequentially from 2 to n, when i = n, there is Pn∗  = P, xn∗=x∧.


## 2.3.1. Centralized Method

Data fusion solution is relatively simple, only needs to concentrate on processing all the collected data, and then directly calculates the total amount of data. Assuming datayi∧,y2∧,...,yn∧, the estimated quantity x represents a data set that can be fused, and Ri represents a symmetric matrix. P represents the fusion data volume of the data. Then, the calculation method of the total data volume of n data is shown in(13)P=HTR−1H=∑i=1nHiTRi−1Hi.If the fused data (set) has a nonsingular information about the estimated quantity, it is called the fused data (set), which means thatHiTRi−1Hi is nonsingular. Calculating the optimal fusion method of x, as shown in(14)x∧=P−1∑i=1nHiTRi−1yi.


## 2.3.2. Sequential Way

Sequential mode accumulates data one by one. Assuming datayi∧,y2∧,...,yn∧, the same estimated quantity x represents a data set that can be fused, and P1=HT1R1−1H1 is positive definite; then x∧=P1−1H1TR1−1y1. Among them, P1∗ represents the data volume of the first to i-th data, xi∗ represents the corresponding optimal fusion, and the calculation methods for P1∗ and x∧i∗ are shown in(15)P1∗=P∗i−1+HTiRi−1Hi,(16)x∧i∗=Pi∗−1P∗i−1x∧i−1+∗HTiRi−1yi.In the formula,i sequentially from 2 to n, when i = n, there is Pn∗  = P, xn∗=x∧.


## 2.4. Data Fusion Modeling of the Dynamic Input-Output Economic System

Input-output data is the basis of system tracking and control. The dynamic input-output economic control system, the state equation, and output equation are shown in(17)xk+1=Axk+Buk,yk=Cxk,x0=x0.Then, the model expression of the dynamic input-output economic system is shown in(18)xk=Axk+Bxk+1−xk+yk.In the formula,x(k) represents the output vector of multisectoral products, y(k) represents the output vector of consumer goods (final products), A represents the input-output (consumption coefficient) matrix, and B is the investment (capital) coefficient matrix. The input-output relationship of each part is shown in Table 2.Table 2
The input-output relationship of the dynamic input-output economic system model.
Cyclek − 1CyclekCyclek + 1Outputx (k − 1)x (k)x (k + 1)Consumables inputAx (k-1)Ax (k)Ax (k + 1)Fixed asset investmentBx (k − 1)Bx (k)Bx (k + 1)Fixed asset formationBx (k)Bx (k + 1)Bx (k + 2)Add fixed assetsB[x (k) − x (k − 1)]B[x (k + 1) − x (k1)]B[x (k + 2) − x (k + 1)]Consumptiony (k − 1)y (k)y (k + 1)


## 3. Tracking Control of the Dynamic Input-Output Economic System

### 3.1. System Model Conversion and Solution

#### 3.1.1. Model Conversion

In a dynamic input-output economic system based on data fusion, under a pure market mechanism, the consumptiony(k) is determined by the wage level and product price level, and the output x(k) is determined by the profit. Profit is also determined by the level of wages and the price level of products, and the relationship between them can be shown in Figure 2.Figure 2
The relationship between supply and demand under the market mechanism.As shown in Figure2, using a pure market mechanism, under certain conditions (such as no monopoly, adequate data exchange, etc.), system resources can be automatically adjusted to the optimal allocation point. However, as far as the current market economy is concerned, this mechanism has inherent shortcomings. The system completely relies on automatic adjustment of resource allocation to balance supply and demand. Once the system deviates from the balance point of supply and demand for some reason, it may take a long time to adjust itself back to the balance point. At the same time, if prices fluctuate greatly, they will cause greater psychological reactions and speculative behaviors, which will form an unfavorable factor for the automatic adjustment of the system. The closed loop system includes two control loops.The first closed loop is composed of a linear control system and a virtual model, wherew(k) is the given input signal of the closed loop, and y(k) is the output signal of the closed loop. When the first closed loop becomes asymptotically stable, the output signal y(k) of the closed loop gradually tracks the signal w(k). The second closed loop control loop is composed of PIDNN compensator and actual model. Among them, y(k) is the input signal of the closed loop, and y∧(k) is the output signal of the closed loop. When the first closed loop becomes asymptotically stable, the output signal of the closed loop gradually tracks the given signal y(k). Therefore, when the two closed loops are stable at the same time, the output signal of the actual controlled object will gradually track the given signal w(k).


#### 3.1.2. Solution Method

The dynamic input-output model is generally solved by the reverse iteration method; using the inverse iteration method to solve the eigenvalue problem, a series of eigenvalues and corresponding eigenvectors from low to high can be obtained in turn. For the dynamic topology optimization problem of the specified frequency, it can avoid solving the frequency before the specified frequency and directly calculate the specified frequency, and the main process is shown in Figure3.Figure 3
Dynamic input-output model solution and operation process.


### 3.2. Input-Output Data Fusion

#### 3.2.1. Data Collection

According to the established corresponding relationship model, first set up the cost center group, then specify its corresponding development unit, and set the cost allocation motivation and the target index of the collection; the system records the setting content, performs the operation, and records the log [21, 22]. The workload of data collection is large and the process is complicated. In order to prevent the creation of a collection log from misoperation, the collection operation can be reexecuted in batches, effectively avoiding duplication of workload. The realization of data collection in the dynamic input-output economic system is shown in Figure 4.Figure 4
Data collection.Since data collection involves basic settings such as cost center grouping and development unit correspondence, if the basic settings change, the data collection linkage mode will respond to the collection data according to the new basic settings and minimize manual workload.At the same time, due to the huge amount of data collected at a cost, the detailed data collected each time reaches a million level. Conventional data processing methods are adopted, which require a long processing time and high risk of errors.


#### 3.2.2. Data Integration

Since other professional departments have different requirements on the statistical caliber, calculation accuracy, and collection period of their respective operating systems, they must consider establishing a reasonable verification mechanism when extracting these system data, control the data quality from the source to ensure accuracy and completeness, and ensure that the evaluation results of the system are correct and credible. Data processing and integration are carried out on the middleware layer, and the integrated data is released through the standard interface of the middleware layer. There is a virtual data service layer on the middle layer, which is connected with various data sources of the data layer through JDBC, FILE adapter, application adapter, etc. and maps various data entities in the data source into virtual data of the middleware. The tables in the virtual data layer only have metadata and do not store actual production data. On the virtual data layer, users can define data mapping relationships by using a visual graphical interface and perform data processing and integration. These data processing logics are generally stored in files or databases.


### 3.3. Traceability of the Dynamic Input-Output Economic System

In the research of dynamic systems, numerical calculation methods are usually used to describe the orbit of the system, and the equalization path with errors is the “pseudo-orbit.” Pseudo-orbit tracking research has been an indispensable part of dynamic system research. Scholars have defined various pseudo-orbit tracking according to their research needs, for example, average tracking, asymptotic average tracking, limit tracking, traversal tracking, speciﬁcation-ness, almost speciﬁcation-ness.


#### 3.3.1. System Model

(1) Solow Growth Model. The Solow growth model analyzes economic growth from different perspectives (factors), provides a reference for the analysis and understanding of various economic growth theories, and also provides a template for the establishment of new models. Among them, the form of the production function is shown in(19)Yt=FKt,AtLt.Among them, Y, K,L, A, t, respectively, represent output, capital, labor efficiency, knowledge (technical development level), and time. Since only capital is an endogenous variable in the Solow growth model, economic behavior can be described by analyzing changes in capital.(2) Ramsey-Cass-Koopmans Model. The economy described by the Ramsey-Cass-Koopmans model is as follows: the market is perfect, and there are only manufacturers and families in the market. Homogeneous firms rent capital, hire labor, and produce and sell products in a competitive market (factors, products); households are homogeneous and constant in number, with an indefinite life, and households mainly supply labor and hold capital (only use for saving and consumption). The savings rate of this model changes endogenously, and capital changes are subject to the interaction of households and firms to maximize their respective interests.(3) Credit Model. The model is based on the Diamond model as the basic framework. In each period, the new generation of potential entrepreneurs (homogeneous) has a unit of labor endowment. In the model, the productivity shock is caused by endogenous changes in the credit composition of heterogeneous projects. If the credit portfolio does not change, the investment dynamics will be the same as the standard neoclassical growth model. Due to credit friction, these projects are not necessarily the most productive projects. The increase in the borrower’s net assets not only eases borrowing restrictions, but may also lead to a shift in the credit structure to more productive projects. Credit friction may lead to capital deepening effects. Therefore, the rate of return may move procyclically.


#### 3.3.2. Model Control Methods

(1) No Model Control. Modern industrial processes have greatly increased the complexity of its manufacturing technology, production equipment, and production environment. These all exhibit complex features such as strong nonlinearity, multivariable coupling, diverse operating conditions, and uncertain complex structures and unknowns parameters, so system modeling based on physical and chemical conditions becomes increasingly difficult. In this case, data-based control theories and methods have received more and more attention.Generally speaking, data-based control methods can be divided into two categories: indirect methods and direct methods.(2) The Direct Method. It uses the measured input/output data to design the controller directly without modeling the system or constructing an equivalent model before the system is operated. However, this data-based control method requires some prior knowledge of system characteristics. There are some typical examples that can be directly used for data-based control methods, such as iterative learning control (ILC), model-free control (MFC), virtual reference feedback tuning method (VRFT), and so on. The IAC method requires the system to repeatedly perform the same operation task within a limited time interval. The MFC method is suitable for nonlinear systems with continuous partial derivative control inputs. This control method only requires some basic system prior knowledge as training. The VRFT method is usually used to control discrete-time output systems. The controller design problem is transformed into a parameter identification problem with the help of a virtual reference signal, and it is assumed that the control structure is known.The basis of model-free control is the use of basic algebraic differentiation. With the help of online parameter identification methods, estimation techniques have become very important. As long as the system dynamics estimation deviation is small enough, the tracking control of the system can be realized. Since the tracking control does not rely on the accurate model of the system dynamics, it only completes the control design based on the system input and the derivative estimation of the system output, which is called model-free control. However, it is not difficult to find that the core part of the control method is the accurate estimation of the system dynamics and the estimation of the high-order derivative of the system output signal, which is a challenging task.(3) Decoupling Method. Decoupling control is to select a suitable decoupling controller, so that the generalized system composed of the controller and the controlled object is no longer a multiple-input multiple-output system, but a generalized system composed of multiple single-input single-output systems. At this time, the output variables of each independent single-input single-output system are fully controlled only by the input variables of the independent system. In decoupling control, the degree of decoupling a coupled system sometimes affects the control result. Generally speaking, the better the decoupling effect, the easier the control; the worse the decoupling effect, the more difficult the control. Therefore, how to remove the influence of coupling on the system to the greatest extent is a question worth considering.For a multi-input and multioutput coupled system, the method is to treat the coupling of other control loops to a loop as a disturbance and then design a disturbance observer on this loop to estimate and eliminate the disturbance, so as to achieve the purpose of decoupling. Obviously, when the coupling between the loops is strong, this method has the disadvantage of incomplete decoupling. Moreover, the disturbance observer used in this method is generally an extended state observer, so this method is generally suitable for second-order systems. If the controlled object is a high-order system, the selection of the parameters of the extended state observer is a tricky problem.


#### 3.3.3. Controllability

The controllability problem is a theoretical prerequisite to ensure that the dynamic input-output economic system can be widely used. It can restrict the stability of multiagents under certain external input conditions and then effectively control the target. In fact, in the case of a single external input, a topological structure can be found. When any node is selected as the leader node, the system can be controlled.


#### 3.3.4. Consistency

For the first-order system, the realization of consistency is the stable convergence of the dynamic input-output economic system state. The second-order system often includes position information and speed information. The result of achieving consistency is that the position state is synchronized, and the speed state is stable and consistent. At this time, the external performance of the dynamic input-output economic system is a dynamic synchronous movement. Dichotomous consistency is an extended discussion of consistency under the symbolic diagram. Since the connection between input and output economic systems in the symbol diagram is mixed and weighted, when the system achieves dichotomous consistency, the state of the system is divided into two groups, and they show the characteristics of the same modulus and opposite signs.For a dynamic input-output economic system, the controllability not only is related to Jordan corresponding to the repeated characteristic value, but also depends on the number of externally coupled inputs. A large number of simulation examples show that the greater the number of external inputs is, the simpler the controllability of the system is achieved. In other words, the fewer the number of external inputs, the more difficult it is for the system to achieve controllability. However, in actual engineering applications, the greater the number of externally coupled inputs, the greater the production cost; when the number of externally coupled inputs is less, the production cost can be reduced well.


## 3.1. System Model Conversion and Solution

### 3.1.1. Model Conversion

In a dynamic input-output economic system based on data fusion, under a pure market mechanism, the consumptiony(k) is determined by the wage level and product price level, and the output x(k) is determined by the profit. Profit is also determined by the level of wages and the price level of products, and the relationship between them can be shown in Figure 2.Figure 2
The relationship between supply and demand under the market mechanism.As shown in Figure2, using a pure market mechanism, under certain conditions (such as no monopoly, adequate data exchange, etc.), system resources can be automatically adjusted to the optimal allocation point. However, as far as the current market economy is concerned, this mechanism has inherent shortcomings. The system completely relies on automatic adjustment of resource allocation to balance supply and demand. Once the system deviates from the balance point of supply and demand for some reason, it may take a long time to adjust itself back to the balance point. At the same time, if prices fluctuate greatly, they will cause greater psychological reactions and speculative behaviors, which will form an unfavorable factor for the automatic adjustment of the system. The closed loop system includes two control loops.The first closed loop is composed of a linear control system and a virtual model, wherew(k) is the given input signal of the closed loop, and y(k) is the output signal of the closed loop. When the first closed loop becomes asymptotically stable, the output signal y(k) of the closed loop gradually tracks the signal w(k). The second closed loop control loop is composed of PIDNN compensator and actual model. Among them, y(k) is the input signal of the closed loop, and y∧(k) is the output signal of the closed loop. When the first closed loop becomes asymptotically stable, the output signal of the closed loop gradually tracks the given signal y(k). Therefore, when the two closed loops are stable at the same time, the output signal of the actual controlled object will gradually track the given signal w(k).


### 3.1.2. Solution Method

The dynamic input-output model is generally solved by the reverse iteration method; using the inverse iteration method to solve the eigenvalue problem, a series of eigenvalues and corresponding eigenvectors from low to high can be obtained in turn. For the dynamic topology optimization problem of the specified frequency, it can avoid solving the frequency before the specified frequency and directly calculate the specified frequency, and the main process is shown in Figure3.Figure 3
Dynamic input-output model solution and operation process.


## 3.1.1. Model Conversion

In a dynamic input-output economic system based on data fusion, under a pure market mechanism, the consumptiony(k) is determined by the wage level and product price level, and the output x(k) is determined by the profit. Profit is also determined by the level of wages and the price level of products, and the relationship between them can be shown in Figure 2.Figure 2
The relationship between supply and demand under the market mechanism.As shown in Figure2, using a pure market mechanism, under certain conditions (such as no monopoly, adequate data exchange, etc.), system resources can be automatically adjusted to the optimal allocation point. However, as far as the current market economy is concerned, this mechanism has inherent shortcomings. The system completely relies on automatic adjustment of resource allocation to balance supply and demand. Once the system deviates from the balance point of supply and demand for some reason, it may take a long time to adjust itself back to the balance point. At the same time, if prices fluctuate greatly, they will cause greater psychological reactions and speculative behaviors, which will form an unfavorable factor for the automatic adjustment of the system. The closed loop system includes two control loops.The first closed loop is composed of a linear control system and a virtual model, wherew(k) is the given input signal of the closed loop, and y(k) is the output signal of the closed loop. When the first closed loop becomes asymptotically stable, the output signal y(k) of the closed loop gradually tracks the signal w(k). The second closed loop control loop is composed of PIDNN compensator and actual model. Among them, y(k) is the input signal of the closed loop, and y∧(k) is the output signal of the closed loop. When the first closed loop becomes asymptotically stable, the output signal of the closed loop gradually tracks the given signal y(k). Therefore, when the two closed loops are stable at the same time, the output signal of the actual controlled object will gradually track the given signal w(k).


## 3.1.2. Solution Method

The dynamic input-output model is generally solved by the reverse iteration method; using the inverse iteration method to solve the eigenvalue problem, a series of eigenvalues and corresponding eigenvectors from low to high can be obtained in turn. For the dynamic topology optimization problem of the specified frequency, it can avoid solving the frequency before the specified frequency and directly calculate the specified frequency, and the main process is shown in Figure3.Figure 3
Dynamic input-output model solution and operation process.


## 3.2. Input-Output Data Fusion

### 3.2.1. Data Collection

According to the established corresponding relationship model, first set up the cost center group, then specify its corresponding development unit, and set the cost allocation motivation and the target index of the collection; the system records the setting content, performs the operation, and records the log [21, 22]. The workload of data collection is large and the process is complicated. In order to prevent the creation of a collection log from misoperation, the collection operation can be reexecuted in batches, effectively avoiding duplication of workload. The realization of data collection in the dynamic input-output economic system is shown in Figure 4.Figure 4
Data collection.Since data collection involves basic settings such as cost center grouping and development unit correspondence, if the basic settings change, the data collection linkage mode will respond to the collection data according to the new basic settings and minimize manual workload.At the same time, due to the huge amount of data collected at a cost, the detailed data collected each time reaches a million level. Conventional data processing methods are adopted, which require a long processing time and high risk of errors.


### 3.2.2. Data Integration

Since other professional departments have different requirements on the statistical caliber, calculation accuracy, and collection period of their respective operating systems, they must consider establishing a reasonable verification mechanism when extracting these system data, control the data quality from the source to ensure accuracy and completeness, and ensure that the evaluation results of the system are correct and credible. Data processing and integration are carried out on the middleware layer, and the integrated data is released through the standard interface of the middleware layer. There is a virtual data service layer on the middle layer, which is connected with various data sources of the data layer through JDBC, FILE adapter, application adapter, etc. and maps various data entities in the data source into virtual data of the middleware. The tables in the virtual data layer only have metadata and do not store actual production data. On the virtual data layer, users can define data mapping relationships by using a visual graphical interface and perform data processing and integration. These data processing logics are generally stored in files or databases.


## 3.2.1. Data Collection

According to the established corresponding relationship model, first set up the cost center group, then specify its corresponding development unit, and set the cost allocation motivation and the target index of the collection; the system records the setting content, performs the operation, and records the log [21, 22]. The workload of data collection is large and the process is complicated. In order to prevent the creation of a collection log from misoperation, the collection operation can be reexecuted in batches, effectively avoiding duplication of workload. The realization of data collection in the dynamic input-output economic system is shown in Figure 4.Figure 4
Data collection.Since data collection involves basic settings such as cost center grouping and development unit correspondence, if the basic settings change, the data collection linkage mode will respond to the collection data according to the new basic settings and minimize manual workload.At the same time, due to the huge amount of data collected at a cost, the detailed data collected each time reaches a million level. Conventional data processing methods are adopted, which require a long processing time and high risk of errors.


## 3.2.2. Data Integration

Since other professional departments have different requirements on the statistical caliber, calculation accuracy, and collection period of their respective operating systems, they must consider establishing a reasonable verification mechanism when extracting these system data, control the data quality from the source to ensure accuracy and completeness, and ensure that the evaluation results of the system are correct and credible. Data processing and integration are carried out on the middleware layer, and the integrated data is released through the standard interface of the middleware layer. There is a virtual data service layer on the middle layer, which is connected with various data sources of the data layer through JDBC, FILE adapter, application adapter, etc. and maps various data entities in the data source into virtual data of the middleware. The tables in the virtual data layer only have metadata and do not store actual production data. On the virtual data layer, users can define data mapping relationships by using a visual graphical interface and perform data processing and integration. These data processing logics are generally stored in files or databases.


## 3.3. Traceability of the Dynamic Input-Output Economic System

In the research of dynamic systems, numerical calculation methods are usually used to describe the orbit of the system, and the equalization path with errors is the “pseudo-orbit.” Pseudo-orbit tracking research has been an indispensable part of dynamic system research. Scholars have defined various pseudo-orbit tracking according to their research needs, for example, average tracking, asymptotic average tracking, limit tracking, traversal tracking, speciﬁcation-ness, almost speciﬁcation-ness.


### 3.3.1. System Model

(1) Solow Growth Model. The Solow growth model analyzes economic growth from different perspectives (factors), provides a reference for the analysis and understanding of various economic growth theories, and also provides a template for the establishment of new models. Among them, the form of the production function is shown in(19)Yt=FKt,AtLt.Among them, Y, K,L, A, t, respectively, represent output, capital, labor efficiency, knowledge (technical development level), and time. Since only capital is an endogenous variable in the Solow growth model, economic behavior can be described by analyzing changes in capital.(2) Ramsey-Cass-Koopmans Model. The economy described by the Ramsey-Cass-Koopmans model is as follows: the market is perfect, and there are only manufacturers and families in the market. Homogeneous firms rent capital, hire labor, and produce and sell products in a competitive market (factors, products); households are homogeneous and constant in number, with an indefinite life, and households mainly supply labor and hold capital (only use for saving and consumption). The savings rate of this model changes endogenously, and capital changes are subject to the interaction of households and firms to maximize their respective interests.(3) Credit Model. The model is based on the Diamond model as the basic framework. In each period, the new generation of potential entrepreneurs (homogeneous) has a unit of labor endowment. In the model, the productivity shock is caused by endogenous changes in the credit composition of heterogeneous projects. If the credit portfolio does not change, the investment dynamics will be the same as the standard neoclassical growth model. Due to credit friction, these projects are not necessarily the most productive projects. The increase in the borrower’s net assets not only eases borrowing restrictions, but may also lead to a shift in the credit structure to more productive projects. Credit friction may lead to capital deepening effects. Therefore, the rate of return may move procyclically.


### 3.3.2. Model Control Methods

(1) No Model Control. Modern industrial processes have greatly increased the complexity of its manufacturing technology, production equipment, and production environment. These all exhibit complex features such as strong nonlinearity, multivariable coupling, diverse operating conditions, and uncertain complex structures and unknowns parameters, so system modeling based on physical and chemical conditions becomes increasingly difficult. In this case, data-based control theories and methods have received more and more attention.Generally speaking, data-based control methods can be divided into two categories: indirect methods and direct methods.(2) The Direct Method. It uses the measured input/output data to design the controller directly without modeling the system or constructing an equivalent model before the system is operated. However, this data-based control method requires some prior knowledge of system characteristics. There are some typical examples that can be directly used for data-based control methods, such as iterative learning control (ILC), model-free control (MFC), virtual reference feedback tuning method (VRFT), and so on. The IAC method requires the system to repeatedly perform the same operation task within a limited time interval. The MFC method is suitable for nonlinear systems with continuous partial derivative control inputs. This control method only requires some basic system prior knowledge as training. The VRFT method is usually used to control discrete-time output systems. The controller design problem is transformed into a parameter identification problem with the help of a virtual reference signal, and it is assumed that the control structure is known.The basis of model-free control is the use of basic algebraic differentiation. With the help of online parameter identification methods, estimation techniques have become very important. As long as the system dynamics estimation deviation is small enough, the tracking control of the system can be realized. Since the tracking control does not rely on the accurate model of the system dynamics, it only completes the control design based on the system input and the derivative estimation of the system output, which is called model-free control. However, it is not difficult to find that the core part of the control method is the accurate estimation of the system dynamics and the estimation of the high-order derivative of the system output signal, which is a challenging task.(3) Decoupling Method. Decoupling control is to select a suitable decoupling controller, so that the generalized system composed of the controller and the controlled object is no longer a multiple-input multiple-output system, but a generalized system composed of multiple single-input single-output systems. At this time, the output variables of each independent single-input single-output system are fully controlled only by the input variables of the independent system. In decoupling control, the degree of decoupling a coupled system sometimes affects the control result. Generally speaking, the better the decoupling effect, the easier the control; the worse the decoupling effect, the more difficult the control. Therefore, how to remove the influence of coupling on the system to the greatest extent is a question worth considering.For a multi-input and multioutput coupled system, the method is to treat the coupling of other control loops to a loop as a disturbance and then design a disturbance observer on this loop to estimate and eliminate the disturbance, so as to achieve the purpose of decoupling. Obviously, when the coupling between the loops is strong, this method has the disadvantage of incomplete decoupling. Moreover, the disturbance observer used in this method is generally an extended state observer, so this method is generally suitable for second-order systems. If the controlled object is a high-order system, the selection of the parameters of the extended state observer is a tricky problem.


### 3.3.3. Controllability

The controllability problem is a theoretical prerequisite to ensure that the dynamic input-output economic system can be widely used. It can restrict the stability of multiagents under certain external input conditions and then effectively control the target. In fact, in the case of a single external input, a topological structure can be found. When any node is selected as the leader node, the system can be controlled.


### 3.3.4. Consistency

For the first-order system, the realization of consistency is the stable convergence of the dynamic input-output economic system state. The second-order system often includes position information and speed information. The result of achieving consistency is that the position state is synchronized, and the speed state is stable and consistent. At this time, the external performance of the dynamic input-output economic system is a dynamic synchronous movement. Dichotomous consistency is an extended discussion of consistency under the symbolic diagram. Since the connection between input and output economic systems in the symbol diagram is mixed and weighted, when the system achieves dichotomous consistency, the state of the system is divided into two groups, and they show the characteristics of the same modulus and opposite signs.For a dynamic input-output economic system, the controllability not only is related to Jordan corresponding to the repeated characteristic value, but also depends on the number of externally coupled inputs. A large number of simulation examples show that the greater the number of external inputs is, the simpler the controllability of the system is achieved. In other words, the fewer the number of external inputs, the more difficult it is for the system to achieve controllability. However, in actual engineering applications, the greater the number of externally coupled inputs, the greater the production cost; when the number of externally coupled inputs is less, the production cost can be reduced well.


## 3.3.1. System Model

(1) Solow Growth Model. The Solow growth model analyzes economic growth from different perspectives (factors), provides a reference for the analysis and understanding of various economic growth theories, and also provides a template for the establishment of new models. Among them, the form of the production function is shown in(19)Yt=FKt,AtLt.Among them, Y, K,L, A, t, respectively, represent output, capital, labor efficiency, knowledge (technical development level), and time. Since only capital is an endogenous variable in the Solow growth model, economic behavior can be described by analyzing changes in capital.(2) Ramsey-Cass-Koopmans Model. The economy described by the Ramsey-Cass-Koopmans model is as follows: the market is perfect, and there are only manufacturers and families in the market. Homogeneous firms rent capital, hire labor, and produce and sell products in a competitive market (factors, products); households are homogeneous and constant in number, with an indefinite life, and households mainly supply labor and hold capital (only use for saving and consumption). The savings rate of this model changes endogenously, and capital changes are subject to the interaction of households and firms to maximize their respective interests.(3) Credit Model. The model is based on the Diamond model as the basic framework. In each period, the new generation of potential entrepreneurs (homogeneous) has a unit of labor endowment. In the model, the productivity shock is caused by endogenous changes in the credit composition of heterogeneous projects. If the credit portfolio does not change, the investment dynamics will be the same as the standard neoclassical growth model. Due to credit friction, these projects are not necessarily the most productive projects. The increase in the borrower’s net assets not only eases borrowing restrictions, but may also lead to a shift in the credit structure to more productive projects. Credit friction may lead to capital deepening effects. Therefore, the rate of return may move procyclically.


## 3.3.2. Model Control Methods

(1) No Model Control. Modern industrial processes have greatly increased the complexity of its manufacturing technology, production equipment, and production environment. These all exhibit complex features such as strong nonlinearity, multivariable coupling, diverse operating conditions, and uncertain complex structures and unknowns parameters, so system modeling based on physical and chemical conditions becomes increasingly difficult. In this case, data-based control theories and methods have received more and more attention.Generally speaking, data-based control methods can be divided into two categories: indirect methods and direct methods.(2) The Direct Method. It uses the measured input/output data to design the controller directly without modeling the system or constructing an equivalent model before the system is operated. However, this data-based control method requires some prior knowledge of system characteristics. There are some typical examples that can be directly used for data-based control methods, such as iterative learning control (ILC), model-free control (MFC), virtual reference feedback tuning method (VRFT), and so on. The IAC method requires the system to repeatedly perform the same operation task within a limited time interval. The MFC method is suitable for nonlinear systems with continuous partial derivative control inputs. This control method only requires some basic system prior knowledge as training. The VRFT method is usually used to control discrete-time output systems. The controller design problem is transformed into a parameter identification problem with the help of a virtual reference signal, and it is assumed that the control structure is known.The basis of model-free control is the use of basic algebraic differentiation. With the help of online parameter identification methods, estimation techniques have become very important. As long as the system dynamics estimation deviation is small enough, the tracking control of the system can be realized. Since the tracking control does not rely on the accurate model of the system dynamics, it only completes the control design based on the system input and the derivative estimation of the system output, which is called model-free control. However, it is not difficult to find that the core part of the control method is the accurate estimation of the system dynamics and the estimation of the high-order derivative of the system output signal, which is a challenging task.(3) Decoupling Method. Decoupling control is to select a suitable decoupling controller, so that the generalized system composed of the controller and the controlled object is no longer a multiple-input multiple-output system, but a generalized system composed of multiple single-input single-output systems. At this time, the output variables of each independent single-input single-output system are fully controlled only by the input variables of the independent system. In decoupling control, the degree of decoupling a coupled system sometimes affects the control result. Generally speaking, the better the decoupling effect, the easier the control; the worse the decoupling effect, the more difficult the control. Therefore, how to remove the influence of coupling on the system to the greatest extent is a question worth considering.For a multi-input and multioutput coupled system, the method is to treat the coupling of other control loops to a loop as a disturbance and then design a disturbance observer on this loop to estimate and eliminate the disturbance, so as to achieve the purpose of decoupling. Obviously, when the coupling between the loops is strong, this method has the disadvantage of incomplete decoupling. Moreover, the disturbance observer used in this method is generally an extended state observer, so this method is generally suitable for second-order systems. If the controlled object is a high-order system, the selection of the parameters of the extended state observer is a tricky problem.


## 3.3.3. Controllability

The controllability problem is a theoretical prerequisite to ensure that the dynamic input-output economic system can be widely used. It can restrict the stability of multiagents under certain external input conditions and then effectively control the target. In fact, in the case of a single external input, a topological structure can be found. When any node is selected as the leader node, the system can be controlled.


## 3.3.4. Consistency

For the first-order system, the realization of consistency is the stable convergence of the dynamic input-output economic system state. The second-order system often includes position information and speed information. The result of achieving consistency is that the position state is synchronized, and the speed state is stable and consistent. At this time, the external performance of the dynamic input-output economic system is a dynamic synchronous movement. Dichotomous consistency is an extended discussion of consistency under the symbolic diagram. Since the connection between input and output economic systems in the symbol diagram is mixed and weighted, when the system achieves dichotomous consistency, the state of the system is divided into two groups, and they show the characteristics of the same modulus and opposite signs.For a dynamic input-output economic system, the controllability not only is related to Jordan corresponding to the repeated characteristic value, but also depends on the number of externally coupled inputs. A large number of simulation examples show that the greater the number of external inputs is, the simpler the controllability of the system is achieved. In other words, the fewer the number of external inputs, the more difficult it is for the system to achieve controllability. However, in actual engineering applications, the greater the number of externally coupled inputs, the greater the production cost; when the number of externally coupled inputs is less, the production cost can be reduced well.


## 4. Tracking Control Results of the Dynamic Input-Output Economic System Based on Data Fusion

### 4.1. Changes in Capital Output Rate

The capital output rate is the amount of capital output per unit. Regarding the prediction of long-term changes in the capital output rate, there are hardly any rules in economic theory. Using the Chanary multicountry model to predict the capital output rate, the results are shown in Figure5.Figure 5
Changes in the rate of capital output in the Chanary’s multicountry model.It can be seen from Figure5 that the most obvious changes in capital output rate with income growth are agriculture and extractive industries, which decrease with economic development, and the magnitude of the change is greater. Infrastructure and service industries increase with economic development, and increase of the magnitude is small. This trend can be used as a basis for forecasting changes in China's capital output rate.


### 4.2. Limited Time Optimal Output Tracking Control

Aiming at the problem of finite-time optimal output tracking control, the research-selected finite-time performance index is shown in(20)J=12eTtfQtf+12∫t0tfeTtQet+uTtfRutdt.Among them,Qf and R represent a symmetric positive definite weight matrix, and Q represents a symmetric positive semidefinite weight matrix. e(t) is the tracking error between the reference output and the actual output of the dynamic input-output economic system.The parameter definition of the performance index of the dynamic input-output economic systemxk+1=Axk+Bukyk=Cxk,x0=x0, as shown in (21)A=01−0.12,B=01,C=10.Among them,Q = 1, R = 1, and Qf  = 1. Taking t0  = 0, tf  = 8s, then the initial value of the state variable is set as shown in (22)x0=10T.Among them, take the correction indexa = 0.01, compare the optimal control and the suboptimal control, and the result is shown in Figure 6.Figure 6
Control curve and tracking error curve of optimal control and suboptimal control.
(a)(b)It can be seen from Figure6 that the “overshoot” of the finite-time suboptimal tracking control is smaller than the finite time at 0∼5s, and the “overshoot” is larger than the finite time at 5∼8s, which compensates for part of the tracking error.


### 4.3. Optimal Control Comparison Results

Using MATLAB software to simulate the optimal control and suboptimal control proposed in this study and calculate the performance under the three methods, the results are shown in Table3.Table 3
Performance indicators of different control schemes.
Serial numberControl planPerformance1Limited time optimal control0.77294122Infinite-time optimal control1.65701423Limited time suboptimal control1.5687310It can be known from the data in Table3 that the optimal performance index and the suboptimal performance index value in a limited time are 0.7729412 and 1.5687310, respectively. Therefore, compared with the infinite-time optimal control, the performance loss and the final tracking error of the suboptimal control proposed in this study are reduced.


### 4.4. Stability of Input-Output Economic System Based on Data Fusion

The initial value of the given system isx(o) = [1, −1], the state delay is d(t) = 0.51 + 0.51sin(t), and the response result of the state is shown in Figure 7.Figure 7
The stability of the input-output economic system based on data fusion.It can be seen from Figure7 that the stability results presented in this paper are less conservative, especially in the case of a large time lag.


### 4.5. Quantitative Tracking Control Simulation Triggered by State-Based Events

When quantizing the signal, the uniform quantizer will find the set element closest to the signal value in the static uniformly distributed quantization set according to the size of the signal value and use this element as the output of the quantizer. This strategy is to quantify the signal value. The uniform quantizer divides the quantization interval into multiple intervals of uniform size and takes the middle value of the interval as the quantization value in each interval, and the upper bound of the quantization error is the same in all intervals. The uniform quantizer has the advantages of simple structure and easy implementation, but its quantization parameters are uniformly distributed and have nothing to do with the parameters of the system, so it has a greater impact on the stability of the system; in addition, due to the nature of the quantization series, there will be some quantization intervals that have little effect on the stability of the system, and the quantization areas divided in these quantization intervals will cause unnecessary calculations. Regarding the use of the sector-bound method to process the log quantizer, the expression of the log quantizer is shown in(23)fv=1+Δv.Among them, △ represents the quantization error; then the logarithmizer is shown in Figure8.Figure 8
Log quantizer.Among them,u1=1−δv, u2=v, u3=1+δv, and δ=1−ρ/1+ρ. ρ represents the quantization density. The transmission reduction is measured by the transmission rate, where the transmission rate = the total number of event triggers/total step length. First, study the influence of quantization density and trigger parameters on the system transmission rate and performance optimization. The results are shown in Table 4. When the quantization density is 0.6 and the trigger parameters are 0.1, 0.3, 0.4, and 0.5, the transmission rates are, respectively, 0.53, 0.48, 0.47, and 0.46; when the quantization density is 0.7 and the trigger parameters are 0.1, 0.3, 0.4, and 0.5, the transmission rate is 0.53, 0.50, 0.46, and 0.46, respectively.Table 4
Transmission rate and trace (M) under different quantization densities and trigger parameters.
Trigger parameter0.60.70.8Transfer rateTrace (M)Transfer rateTrace (M)Transfer rateTrace (M)0.10.531.29640.531.24840.571.20130.30.481.43470.501.39240.531.32150.40.474.56930.461.51800.491.44720.50.462.10330.461.81250.461.67490.60.365.19780.353.99830.393.2109It can be seen from Table4 that as the quantization density increases, the transmission rate will decrease, and trace (M) will increase. When the quantization density increases, trace (M) will also increase. Assuming that the quantization density is 0.8, the trigger parameters are 0.1 and 0.6. The output trajectory of the system and the output trajectory of the reference system are shown in Figure 9.Figure 9
The output of the system and the reference system under different trigger parameters.It can be seen from Figure9 that the system can reach a balanced state under the action of the state feedback tracking controller. At this time, the system can stably track the reference system. In addition, the smaller the trigger parameter under the same quantization density, the smaller the system's. The smaller the tracking error, the better the tracking effect.


### 4.6. Tracking Control of Total Output and Total Income

For the tracking problem of dynamic input and output economic system, the data fusion tracking strategy proposed in this research can be used to obtain the data of total output and total income, respectively. The results are shown in Figure10.Figure 10
Total output and total income.It can be seen from Figure10 that the use of data fusion method to track and control the dynamic input-output economic system can play the role of the system in tracking the total income data according to the total output data and achieve the purpose of controlling the balance of payments. This shows that the use of data fusion to track and control the system can achieve the purpose of tracking input and output values. However, in the tracking control process, the actual total output growth rate has a certain lag compared with the actual total income growth rate, so there will be errors in the tracking.


## 4.1. Changes in Capital Output Rate

The capital output rate is the amount of capital output per unit. Regarding the prediction of long-term changes in the capital output rate, there are hardly any rules in economic theory. Using the Chanary multicountry model to predict the capital output rate, the results are shown in Figure5.Figure 5
Changes in the rate of capital output in the Chanary’s multicountry model.It can be seen from Figure5 that the most obvious changes in capital output rate with income growth are agriculture and extractive industries, which decrease with economic development, and the magnitude of the change is greater. Infrastructure and service industries increase with economic development, and increase of the magnitude is small. This trend can be used as a basis for forecasting changes in China's capital output rate.


## 4.2. Limited Time Optimal Output Tracking Control

Aiming at the problem of finite-time optimal output tracking control, the research-selected finite-time performance index is shown in(20)J=12eTtfQtf+12∫t0tfeTtQet+uTtfRutdt.Among them,Qf and R represent a symmetric positive definite weight matrix, and Q represents a symmetric positive semidefinite weight matrix. e(t) is the tracking error between the reference output and the actual output of the dynamic input-output economic system.The parameter definition of the performance index of the dynamic input-output economic systemxk+1=Axk+Bukyk=Cxk,x0=x0, as shown in (21)A=01−0.12,B=01,C=10.Among them,Q = 1, R = 1, and Qf  = 1. Taking t0  = 0, tf  = 8s, then the initial value of the state variable is set as shown in (22)x0=10T.Among them, take the correction indexa = 0.01, compare the optimal control and the suboptimal control, and the result is shown in Figure 6.Figure 6
Control curve and tracking error curve of optimal control and suboptimal control.
(a)(b)It can be seen from Figure6 that the “overshoot” of the finite-time suboptimal tracking control is smaller than the finite time at 0∼5s, and the “overshoot” is larger than the finite time at 5∼8s, which compensates for part of the tracking error.


## 4.3. Optimal Control Comparison Results

Using MATLAB software to simulate the optimal control and suboptimal control proposed in this study and calculate the performance under the three methods, the results are shown in Table3.Table 3
Performance indicators of different control schemes.
Serial numberControl planPerformance1Limited time optimal control0.77294122Infinite-time optimal control1.65701423Limited time suboptimal control1.5687310It can be known from the data in Table3 that the optimal performance index and the suboptimal performance index value in a limited time are 0.7729412 and 1.5687310, respectively. Therefore, compared with the infinite-time optimal control, the performance loss and the final tracking error of the suboptimal control proposed in this study are reduced.


## 4.4. Stability of Input-Output Economic System Based on Data Fusion

The initial value of the given system isx(o) = [1, −1], the state delay is d(t) = 0.51 + 0.51sin(t), and the response result of the state is shown in Figure 7.Figure 7
The stability of the input-output economic system based on data fusion.It can be seen from Figure7 that the stability results presented in this paper are less conservative, especially in the case of a large time lag.


## 4.5. Quantitative Tracking Control Simulation Triggered by State-Based Events

When quantizing the signal, the uniform quantizer will find the set element closest to the signal value in the static uniformly distributed quantization set according to the size of the signal value and use this element as the output of the quantizer. This strategy is to quantify the signal value. The uniform quantizer divides the quantization interval into multiple intervals of uniform size and takes the middle value of the interval as the quantization value in each interval, and the upper bound of the quantization error is the same in all intervals. The uniform quantizer has the advantages of simple structure and easy implementation, but its quantization parameters are uniformly distributed and have nothing to do with the parameters of the system, so it has a greater impact on the stability of the system; in addition, due to the nature of the quantization series, there will be some quantization intervals that have little effect on the stability of the system, and the quantization areas divided in these quantization intervals will cause unnecessary calculations. Regarding the use of the sector-bound method to process the log quantizer, the expression of the log quantizer is shown in(23)fv=1+Δv.Among them, △ represents the quantization error; then the logarithmizer is shown in Figure8.Figure 8
Log quantizer.Among them,u1=1−δv, u2=v, u3=1+δv, and δ=1−ρ/1+ρ. ρ represents the quantization density. The transmission reduction is measured by the transmission rate, where the transmission rate = the total number of event triggers/total step length. First, study the influence of quantization density and trigger parameters on the system transmission rate and performance optimization. The results are shown in Table 4. When the quantization density is 0.6 and the trigger parameters are 0.1, 0.3, 0.4, and 0.5, the transmission rates are, respectively, 0.53, 0.48, 0.47, and 0.46; when the quantization density is 0.7 and the trigger parameters are 0.1, 0.3, 0.4, and 0.5, the transmission rate is 0.53, 0.50, 0.46, and 0.46, respectively.Table 4
Transmission rate and trace (M) under different quantization densities and trigger parameters.
Trigger parameter0.60.70.8Transfer rateTrace (M)Transfer rateTrace (M)Transfer rateTrace (M)0.10.531.29640.531.24840.571.20130.30.481.43470.501.39240.531.32150.40.474.56930.461.51800.491.44720.50.462.10330.461.81250.461.67490.60.365.19780.353.99830.393.2109It can be seen from Table4 that as the quantization density increases, the transmission rate will decrease, and trace (M) will increase. When the quantization density increases, trace (M) will also increase. Assuming that the quantization density is 0.8, the trigger parameters are 0.1 and 0.6. The output trajectory of the system and the output trajectory of the reference system are shown in Figure 9.Figure 9
The output of the system and the reference system under different trigger parameters.It can be seen from Figure9 that the system can reach a balanced state under the action of the state feedback tracking controller. At this time, the system can stably track the reference system. In addition, the smaller the trigger parameter under the same quantization density, the smaller the system's. The smaller the tracking error, the better the tracking effect.


## 4.6. Tracking Control of Total Output and Total Income

For the tracking problem of dynamic input and output economic system, the data fusion tracking strategy proposed in this research can be used to obtain the data of total output and total income, respectively. The results are shown in Figure10.Figure 10
Total output and total income.It can be seen from Figure10 that the use of data fusion method to track and control the dynamic input-output economic system can play the role of the system in tracking the total income data according to the total output data and achieve the purpose of controlling the balance of payments. This shows that the use of data fusion to track and control the system can achieve the purpose of tracking input and output values. However, in the tracking control process, the actual total output growth rate has a certain lag compared with the actual total income growth rate, so there will be errors in the tracking.


## 5. Conclusions

The input-output analysis is based on the input-output table. The traditional input-output table uses flow as the research object, which reflects the economic connection between the production and use of products in various sectors of the national economy. However, in actual economic operation, the economic system is an open system, which will inevitably be affected by the external environment. Natural resources, material capital, human capital, and other factors have also exerted a certain influence on the operation and development of the economy, and traditional input-output technologies cannot analyze them well. Therefore, when there are certain parameter perturbations and uncertainties in the economic system, it is necessary to incorporate the tracking control of the economic system into the research purpose. On this basis, with the development of computer technology, data fusion is gradually applied to the tracking control of the system, combining input and output with optimization methods. The innovative tracking control method of dynamic input-output economic systems is a future research in this field.



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*Source: 1461977-2022-03-21.xml*