Sustainable Operation and Management of a Dynamic Supply Chain under the Framework of a Community with a Shared Future for Mankind

Abstract

The values of a community with a shared future for mankind include the views of common interests, sustainable development, and global governance. This article will fully consider introducing the value concept of a community with a shared future into the operation and management of dynamic supply chains. Based on the optimal information fusion mechanism of artificial intelligence, this article aims to examine the operation and management of dynamic supply chains within the framework of a community with a shared future for mankind. The core idea is to consider the common interests among enterprises, establish a global collaborative operation concept for upstream, midstream, and downstream enterprises, and achieve the goal of sustainable development. Firstly, a type of composite dynamic supply chain model is considered, in which the total inventory of each node in the supply chain is further subdivided into raw material inventory and finished product inventory. At the same time, we have considered factors such as the signing of procurement contracts between core enterprises and upstream enterprises, as well as the signing of supply contracts between core enterprises and downstream enterprises. Secondly, the static and dynamic monitoring information of the enterprise has been established. We use steady-state Kalman filtering theory to obtain dynamic reference signals for upstream enterprises, core enterprises, and downstream enterprises. Based on the optimal information fusion processing mechanism of artificial intelligence, the coefficient weighting method is used to obtain the optimal fusion signals of upstream enterprises, core enterprises, and downstream enterprises. Once again, through high-quality switching strategies, enterprises can achieve in-order switching, improve production efficiency, reduce downtime, enhance their competitiveness and responsiveness, and transform the dynamic supply chain, including order switching, into a discrete-time linear switching system for processing. Fourthly, sufficient conditions, robustness analysis results, and inventory control criteria for the solvability of dynamic supply chain ${\mathcal{H}}^{\infty }$ with order switching are provided. Finally, data analysis is conducted using historical order information from three fruit companies to verify the validity and feasibility of the conclusions in this article and to improve the performance of the dynamic supply chain system. The research findings of this article enrich the exploration of the operation and management of dynamic supply chains and the construction of a community with a shared future for mankind.

1. Introduction

During the fight against the COVID-19 pandemic, the medical supply chain of many countries responded quickly [1,2]. Related enterprises in manufacturing countries such as China rapidly increased the production and supply of masks, protective clothing, ventilators, and other materials by dynamically adjusting production plans and supply chain arrangements. These materials were transported to various parts of the world through a global supply chain network to help other countries fight the pandemic together [3,4]. This reflects the positive role of dynamic supply chains in ensuring material supply and promoting international cooperation in the face of global public health crises, making contributions to building a community with a shared future for mankind [5]. When dealing with natural disasters in certain regions, the international supply chain of food, medicine, and other materials can be dynamically adjusted according to the changing needs of affected areas, timely delivering urgently needed materials, alleviating the difficulties of the affected areas, demonstrating cross-border mutual assistance and cooperation, and helping to reduce the impact of disasters and promote the recovery of affected areas. This is also a manifestation of the positive role of dynamic supply chains [6]. In the global electronics industry, the dynamic supply chain of components ensures that enterprises from different countries and regions can timely access necessary resources, jointly promote technological progress and industrial development, and promote interdependence and common prosperity of the global economy, which is of great significance for building a common future for mankind at the economic level [7]. Therefore, we must explore the operation and management of dynamic supply chains under the concept of a community with a shared future for mankind.

1.1. Motivation

The dynamic supply chain is a supply chain model with high flexibility, adaptability, and agility [8]. The dynamic supply chain can quickly respond to changes in market demand, fluctuations in customer orders, and various uncertain factors, such as unexpected events, technological changes, and changes in the competitive landscape. Dynamic supply chains can self-adjust and optimize according to different situations and environments, including adjusting production plans, logistics distribution paths, supplier cooperation relationships, etc. [9]. A dynamic supply chain emphasizes the ability to quickly perceive and respond to changes to ensure competitiveness in a dynamically changing environment. The dynamic supply chain relies on advanced information technology and data-sharing mechanisms to grasp real-time information about various links in the supply chain, to ensure timely decisions and actions. The dynamic supply chain involves close collaboration among various nodes in the supply chain, including suppliers, manufacturers, distributors, retailers, etc., to jointly respond to dynamic changes and achieve maximum overall benefits [10]. The goal of a dynamic supply chain is to better meet customer needs, improve the operational efficiency and efficiency of the supply chain, and enhance the ability of enterprises to respond to risks and changes in a complex and ever-changing business environment.

A community with a shared future for mankind aims to balance the legitimate concerns of other countries while pursuing its interests and promoting the common development of all countries in the pursuit of its own development [11]. The values of a community with a shared future for mankind include the views of common interests, sustainable development, and global governance [12]. The efficient operation of dynamic supply chains helps to promote the smooth development of the global economy and the rational allocation of resources, which is an important support for the goal of economic prosperity and common development in building a community with a shared future for mankind [13]. By optimizing the supply chain, economic connections and cooperation among enterprises can be strengthened, creating better conditions for the common development of humanity. In the face of global challenges such as epidemics and natural disasters, a good dynamic supply chain can more flexibly allocate resources, help humanity cope with crises together, reflect the interdependence and shared responsibility of humanity in dealing with difficulties, and conform to the concept of working together to address global issues emphasized by the community with a shared future for mankind. In the operation of dynamic supply chains, there may be an unequal distribution of benefits, which may conflict with the principles of fairness and equality advocated by the community with a shared future for mankind and require coordination and resolution. Overall, dynamic supply chains can provide a positive impetus for the construction of a community with a shared future for mankind to a certain extent, but they also need to focus on coordination and solving possible problems in the development process to better serve the common development and progress of humanity. The concept of a community with a shared future for mankind can fully guide the operation and sustainable development of enterprises [14]. We can fully consider the common interests between core enterprises and upstream and downstream enterprises throughout the entire industry chain, introduce the concept of global governance into the global coordinated development between enterprises, and provide a new paradigm for green manufacturing and sustainable development of enterprises [15].

Dynamic supply chain control is a concept and method for managing and optimizing the supply chain [16]. Dynamic supply chain control emphasizes the ability to respond to various dynamic changes in real time and flexibly during the operation of the supply chain. These changing factors may include fluctuations in market demand, uncertainty in supply, unexpected events (such as natural disasters, epidemics, etc.), technological changes, etc. Through dynamic supply chain control, enterprises can take a series of measures, such as real-time monitoring and analysis of data in various links of the supply chain, rapid adjustment of production plans, procurement strategies, inventory levels, logistics distribution, etc., to better match supply and demand relationships, improve supply chain efficiency and agility, reduce costs and risks, and enhance supply chain adaptability and competitiveness [17]. Dynamic supply chain control emphasizes the use of advanced information technology and data analysis methods to achieve dynamic monitoring and precise regulation of the supply chain, ensuring that the supply chain can continue to operate stably in complex and ever-changing environments. The core of dynamic supply chain control includes the following aspects [18]: (1) Real-time information acquisition and sharing: timely and accurate acquisition of information from various links in the supply chain, and effective sharing among relevant parties, which is the foundation of dynamic response; (2) Agility and adaptability: ability to quickly and flexibly adjust supply chain strategies, processes, and operations to adapt to constantly changing internal and external environments; (3) Collaborative cooperation: including close collaboration between different departments within the enterprise and upstream and downstream enterprises in the supply chain, forming a joint force to respond to dynamic changes; (4) Data analysis and decision support: using data analysis as an insight into changing trends, to identify problems and opportunities, and provide a basis for scientific decision-making to optimize supply chain operations; (5) Risk management: effectively identify, evaluate, and respond to various risks, ensuring that the supply chain maintains stability and continuous operation in dynamic changes; (6) Continuous optimization capability: continuously improving and perfecting all aspects of the supply chain to enhance overall performance and the ability to respond to dynamic changes.

Artificial intelligence optimal information fusion refers to the fusion of multiple information sources in the field of artificial intelligence to obtain optimal information [19]. Artificial intelligence optimal information fusion is a multimodal information fusion technology aimed at fusing different types of information to improve accuracy and reliability [20]. The optimal information fusion of artificial intelligence has a wide range of applications, such as image recognition, speech recognition, natural language processing, intelligent transportation, intelligent healthcare, and other fields. Artificial intelligence optimal information fusion can improve the accuracy and reliability of information, thereby improving the performance and efficiency of the system [21]. Inspired by it, we can apply it to the operation management and inventory scheduling of dynamic supply chains to improve the resilience and robustness of enterprises.

In summary, we must discuss the operation and management of dynamic supply chains within the framework of a community with a shared future for mankind. It is feasible to explore the control of dynamic supply chains based on the optimal information fusion mechanism of artificial intelligence. The main work of this article will provide new ideas, methods, and directions for the sustainable operation and management of dynamic supply chains.

1.2. Literature Review

In terms of research on the theory and application of a common future for mankind, Qi Xu et al. carried out in-depth research on the construction of a community with a shared future for the oceans, thought about the construction of a community with a future for the oceans by the Chinese Academy of Sciences, discussed the theoretical concept of literature on the construction of a community with a shared future for the oceans, and discussed the theoretical roadmap of a community with a shared future for the oceans, including its theoretical basis and its contribution to the theory of international ocean governance [22]. Chuanliang Wang explored the significance of the concept of a community with a shared future for the ocean in the process of international law formulation [23]. Considering the impact of economic growth in the late period of the COVID-19 pandemic on future sustainable development, green economic recovery can achieve a win–win situation between economic recovery and environmental improvement, and bring about environmentally sustainable economic growth. Xin-Xin Zhao et al. analyzed the opportunities and challenges faced by China’s green economic recovery based on a comparative study of international green economic recovery policy practice experience, and put forward countermeasures and suggestions on how to promote its sustainable development in the post-pandemic era [24]. Zeyu Xing et al. creatively proposed a low-carbon digital economy view from the perspectives of resource flow, digital flow, and energy flow, and studied the synergistic effects of low-carbon digital development by constructing a synergistic model for economic and social development through low-carbon digital development. On the basis of theoretical research, a collaborative model and an evaluation index system for low-carbon digital development and economic and social development were constructed, and empirical analysis was conducted based on data related to regional low-carbon digital development and economic and social development in China from 2014 to 2019 [25]. However, most of the aforementioned theoretical research on the common destiny of humanity has focused on the fields of marine science, environmental science, and ecological science, and there have been no relevant reports on the operation and management of dynamic supply chains so far. On the one hand, we can try to expand the application research of the values of a community with a shared future for mankind in the field of dynamic supply chain and promote the vigorous development of this field. On the other hand, it can provide new ideas and directions for the operation and management of dynamic supply chains at the macro level.

In terms of enterprise operation management and dynamic supply chain control, based on Marx’s capital cycle theory, Shen Youjia et al. elucidated the political and economic principles of industrial chain supply chain activities, explored the inherent logical relationship and abstract coherence between the two, and combined with the reality of the fragility of China’s industrial chain supply chain, analyzed the essence and root causes of the problem and explored the basic path to improve the resilience and security level of the industrial chain supply chain, but did not provide substantive policy recommendations to enhance the resilience and security level of the industrial chain supply chain [26]. Lihua Hu et al. established a green supply chain production and operation model with dual channels from the perspectives of safety scheduling and green production. They also considered the situation where enterprises fail to supply goods due to force majeure. They designed a supply scheduling scheme through ${\mathcal{H}}^{\infty }$ control to ensure the fault-tolerant stability of the dual-channel supply chain in the event of goods supply failure [27]. Lihua Hu et al. established a new mathematical model for complex dynamic supply chains by collecting information on green manufacturing production. They considered the impact of force majeure (such as the COVID-19 pandemic) on the production scale control and resource scheduling optimization of complex dynamic supply chains, providing new ideas for planning production plans and scheduling schemes in dynamic supply chains and improving the robustness of closed-loop dynamic supply chains [28]. In response to the situation where upstream enterprises fail to timely obtain current order information from downstream enterprises, Baolin Zhang et al. proposed a linear combination prediction method that uses historical order information to estimate the order quantity of downstream node enterprises in the supply chain. This method transformed the supply chain system model into a linear uncertainty system with multiple states and provided sufficient conditions for robust stabilization of the supply chain system through state feedback and a design method for state feedback controllers [29]. However, the aforementioned achievements have many shortcomings. On the one hand, current literature on supply chain modeling mostly considers a single product, without taking into account both raw materials and finished products and without considering the ability of enterprises to self-process. Many companies currently possess the ability to self-process, and it is necessary for us to further expand our theoretical research work. In addition, it is unrealistic for enterprises to rely solely on ordering channels to meet their demand for products, especially when emergencies occur, such as traffic interruptions and large-scale closures caused by the COVID-19 epidemic. On the other hand, to ensure product quality and save inventory costs, many agricultural and sideline product enterprises have also established their processing plants. For such enterprises, they have strong self-processing capabilities. Therefore, we must establish a dynamic supply chain model for enterprises with self-processing capabilities, and provide sufficient research on their supply chain control and guidance on their operational management issues. Thirdly, Baolin Zhang et al. proposed an order prediction scheme based on the order information of downstream enterprises for several consecutive order cycles, which provided sufficient conditions for the asymptotic stability of the dynamic supply chain system. However, they were unable to achieve real-time control.

In terms of the principle and application of optimal information fusion in artificial intelligence, Xie Chen et al. provided a scientific and technological perspective for the research of optimal information fusion in artificial intelligence and provided independent insights and inspirations for its future development [30]. A. S. Albahri et al. summarized in detail the application of the optimal information fusion principle of artificial intelligence in the field of healthcare [21]. Based on the principle of optimal information fusion in artificial intelligence, Mohammad G. Zamani et al. studied a multi-model data fusion method for reservoir water quality, which promotes the sustainable development of human life and the social environment [21]. Based on the principle of optimal information fusion in artificial intelligence, Yike Zhao et al. studied the intelligent diagnosis problem of bearings with small and unbalanced samples [31]. Based on the principle of optimal information fusion in artificial intelligence, Pengfei Zhang et al. proposed a new probability distribution information representation model, namely the probability distribution information system. They designed an unsupervised feature selection algorithm based on minimum separability and minimum uncertainty, which can fully integrate multiple possible information, maintain as much information as possible, and minimize information uncertainty [20]. Based on the principle of optimal information fusion in artificial intelligence, Gao Hu et al. proposed a decentralized multi-sensor information fusion method and established a robust local state estimation for fault detection. The traditional optimal data fusion technology was extended to nonlinear systems, and the local state estimation of the fusion subsystem was fused using a traceless transformation within the framework of minimum square error estimation [32]. However, through investigation, it was found that this theory has not yet been applied in the field of operation and management of dynamic supply chain. On the one hand, we can try to expand the application scenarios of the optimal information fusion principle of artificial intelligence and promote the vigorous development of this field. On the other hand, new technologies and methods can be provided at the micro level for the operation and management of dynamic supply chains.

In view of this, this paper will explore the sustainable operation and management of dynamic supply chains via optimal information fusion mechanism of artificial intelligence within the framework of a community with a shared future for mankind.

1.3. Contribution of This Article

Based on the above discussion, this article will examine the operation and management of dynamic supply chains from the perspective of optimal information fusion in artificial intelligence within the framework of a community with a shared future for mankind. Firstly, we established a composite dynamic supply chain model, in which the total inventory of each node in the supply chain is further subdivided into raw material inventory and finished product inventory. At the same time, we consider factors such as the signing of procurement contracts between core enterprises and upstream enterprises, and the signing of supply contracts between core enterprises and downstream enterprises. Secondly, we establish static and dynamic monitoring information for the core enterprise. Using steady-state Kalman filtering theory to obtain dynamic reference signals for upstream enterprises, core enterprises, and downstream enterprises. Based on the optimal information fusion processing mechanism of artificial intelligence, the coefficient weighting method is used to obtain the optimal fusion signals of upstream enterprises, core enterprises, and downstream enterprises. Once again, high-quality switching strategies, help enterprises achieve and do a good job in order switching, improve production efficiency, reduce downtime, enhance their competitiveness and responsiveness, and transform the dynamic supply chain, including order switching, into a discrete-time linear switching system for processing. Fourthly, we provide sufficient conditions, robustness analysis results, and inventory control criteria for the ${\mathcal{H}}^{\infty }$ solvability of dynamic supply chain with order switching. Finally, data analysis is conducted using historical order information from Chongqing Hongjiu Fruit Co., Ltd. (Chongqing, China), Wuhan Golden Orchard Trading Co., Ltd. (Wuhan, China), and Hangzhou Xianfeng Fruit Co., Ltd. (Hangzhou, China) to verify the validity and feasibility of the conclusions drawn in this article and to improve the performance of the dynamic supply chain system. In summary, the innovation and contribution of this article mainly focus on the following five aspects:

(1)

This article introduces the values of a community with a shared future for mankind into the operation and management of dynamic supply chains, providing new tacks and ideas for the operation and management of dynamic supply chains.

(2)

This article establishes an information supply chain model with self-processing capability for the first time, considering raw material inventory and finished product inventory in a hierarchical manner.

(3)

This article establishes for the first time the static and dynamic monitoring information of core enterprises, achieving real-time data collection and providing effective strategies for real-time control of dynamic supply chains, enhancing the robustness of control algorithms.

(4)

This article introduces the principle of artificial intelligence optimal information fusion into the operation and management of dynamic supply chains, overcoming the public problem of traditional optimal control in dynamic supply chains that requires solving the Jacobian matrix. The design concept and process are also simple and easy.

(5)

In the face of the public problem of “ordering less may not be enough to meet the needs of consumers, while ordering more may bring huge losses to themselves” in agricultural product enterprises, this article establishes an order-switching model with market conditions, holidays, weather conditions, and other factors as switching factors for the first time, providing new ideas for the quantitative solution of this open problem.

1.4. Organization

By elaborating on the research motivation and analyzing the literature review, this article aims to discuss the sustainable operation and management of dynamic supply chains within the framework of a community with a shared future for mankind. Firstly, a type of composite dynamic supply chain model is considered. In this dynamic supply chain model, the total inventory of each node in the supply chain is further subdivided into raw material inventory and finished product inventory. Secondly, based on the optimal information fusion mechanism of artificial intelligence, this article aims to examine the operation and management of dynamic supply chains within the framework of a community with a shared future for mankind. Thirdly, based on the fundamental principles of control theory, a sustainable operation mechanism for dynamic supply chains is established. Fourthly, empirical research demonstrates the effectiveness and feasibility of the conclusions drawn in this article. Finally, we summarize and discuss the work presented in this article.

The organizational structure of this article is as follows: in Section 2, we establish a mathematical model of a composite dynamic supply chain. We design the static/dynamic monitoring information and optimal information fusion mechanism for the core enterprise in Section 3. In Section 4, we consider the principle of order switching and establish a mathematical model of a closed-loop dynamic supply chain network. We also provide sufficient conditions, robustness analysis results, and inventory control criteria for the solvability of ${\mathcal{H}}^{\infty }$ in the dynamic supply chain. We complete the empirical research in Section 5. Finally, in Section 6, we summarize the main work of this article and provide policy recommendations.

2. Dynamic Supply Chain Model

This article considers a type of composite dynamic supply chain model, whose local network topology is shown in Figure 1. In this dynamic supply chain model, the total inventory of each node in the supply chain is further subdivided into raw material inventory and finished product inventory. $x_{\mathrm{core}}^1\left(k\right)$ represents the raw material inventory level of the core enterprise considered in this article. $x_{\mathrm{core}}^2\left(k\right)$ represents the finished product inventory level of the core enterprise considered in this article. $x_{\mathrm{up}}^1\left(k\right)$ represents the raw material inventory level of upstream enterprises. $x_{\mathrm{up}}^2\left(k\right)$ represents the finished product inventory level of upstream enterprises. $x_{\mathrm{down}}^1\left(k\right)$ represents the raw material inventory level of downstream enterprises. $x_{\mathrm{down}}^2\left(k\right)$ represents the finished product inventory level of downstream enterprises. $x_{\mathrm{core}}\left(k\right)$ represents the total inventory of the core enterprise considered in this article. $x_{\mathrm{up}}\left(k\right)$ represents the total inventory of upstream enterprises. $x_{\mathrm{down}}\left(k\right)$ represents the total inventory of downstream enterprises. Referring to the supply chain structure established by Baolin Zhang et al. [29], there is a simple relationship as follows:

$$\begin{array}{cc}\hfill x_{\mathrm{up}}\left(k\right)& =x_{\mathrm{up}}^1\left(k\right)+x_{\mathrm{up}}^2\left(k\right),\hfill \\ \hfill x_{\mathrm{core}}\left(k\right)& =x_{\mathrm{core}}^1\left(k\right)+x_{\mathrm{core}}^2\left(k\right),\hfill \\ \hfill x_{\mathrm{down}}\left(k\right)& =x_{\mathrm{down}}^1\left(k\right)+x_{\mathrm{down}}^2\left(k\right).\hfill \end{array}$$

(1)

However, the dynamic supply chain model (1) does not fully consider the actual situation of the enterprise. Raw materials and finished products cannot be compared, and it is not reasonable to directly add the two. Starting from the physical meaning and actual situation, it is also necessary to consider its dimensions and units.

The cycle iteration relationship of the dynamic supply chain is shown in Figure 2. In each iteration cycle, the loss factor of raw materials is ${\alpha }_1$, and the loss factor of finished products is ${\alpha }_2$. Within an iteration cycle, ${\beta }_1$ of raw materials is processed into finished products, and the processing rate is ${\beta }_2$. Therefore, there is

$$\begin{array}{cc}\hfill \left[\begin{array}{c}{x}_{\mathrm{core}}^1(k+1)\\ x_{\mathrm{core}}^2(k+1)\end{array}\right]=& \left[\begin{array}{cc}(1-{\alpha }_1)(1-{\beta }_1)& 0\\ (1-{\alpha }_1){\beta }_1{\beta }_2& (1-{\alpha }_2)\end{array}\right]\left[\begin{array}{c}{x}_{\mathrm{core}}^1\left(k\right)\\ x_{\mathrm{core}}^2\left(k\right)\end{array}\right].\hfill \end{array}$$

(2)

And it can be abbreviated as

$$\begin{array}{cc}\hfill x_{\mathrm{core}}(k+1)& =A{x}_{\mathrm{core}}\left(k\right),\hfill \end{array}$$

(3)

where $A=\left[\begin{array}{cc}(1-{\alpha }_1)(1-{\beta }_1)& 0\\ (1-{\alpha }_1){\beta }_1{\beta }_2& (1-{\alpha }_2)\end{array}\right]\in R^{2\times 2}$ is a constant coefficient matrix, and $x_{\mathrm{core}}\left(k\right)=\left[\begin{array}{c}{x}_{\mathrm{core}}^1\left(k\right)\\ x_{\mathrm{core}}^2\left(k\right)\end{array}\right]\in R^2$ is the inventory status of the core enterprise.

At the same time, it is also necessary to consider factors such as the signing of procurement contracts between core enterprises and upstream enterprises, and the signing of supply contracts between core enterprises and downstream enterprises. $u_{\mathrm{purchase}}^1\left(k\right)$ is the purchase quantity of raw materials. $u_{\mathrm{purchase}}^2\left(k\right)$ is the purchase quantity of finished products. Let the core enterprise’s procurement quantity from upstream enterprises be $u_{\mathrm{purchase}}\left(k\right)={[u_{\mathrm{purchase}}^1\left(k\right),u_{\mathrm{purchase}}^2\left(k\right)]}^T$. Let the supply volume of the core enterprise to downstream enterprises be $u_{\mathrm{supply}}\left(k\right)={[u_{\mathrm{supply}}^1\left(k\right),u_{\mathrm{supply}}^2\left(k\right)]}^T$, where $u_{\mathrm{supply}}^1\left(k\right)$ is the supply of raw materials and $u_{\mathrm{supply}}^2\left(k\right)$ is the quantity of finished products supplied. The supply and purchase quantities we mentioned here are normalized, which is easy to understand in the field of information science. At the same time as signing supply contracts with downstream enterprises, we need to consider our company’s inventory and actual production capacity. As a result, we have signed contracts with upstream enterprises. At the same time as purchasing quantity, it is also necessary to consider the inventory level and actual economic strength of the core enterprise. Therefore, there is

$$\begin{array}{cc}\hfill x_{\mathrm{core}}(k+1)=& A{x}_{\mathrm{core}}\left(k\right)+u_{\mathrm{purchase}}\left(k\right)-u_{\mathrm{supply}}\left(k\right)\hfill \\ \hfill =& A{x}_{\mathrm{core}}\left(k\right)+[I_{2\times 2},-I_{2\times 2}]\left[\begin{array}{c}{u}_{\mathrm{purchase}}\left(k\right)\\ u_{\mathrm{supply}}\left(k\right)\end{array}\right]\hfill \\ \hfill \triangleq & A{x}_{\mathrm{core}}\left(k\right)+Bu\left(k\right),\hfill \end{array}$$

(4)

where $A\in R^{2\times 2},B\in R^{2\times 4}$ is a constant coefficient matrix, and $u\left(k\right)\in R^4$ represents the order quantity and procurement quantity of the core enterprise, which is also the dynamic output feedback adjustment signal of this article. The dynamic supply chain model considered in this article is a discrete dynamic system model. Therefore, the supply chain considered in this article is a dynamic model.

Remark 1. 

Common supply chain models do not differentiate between considering raw material inventory and finished product inventory [27,28], which is often unrealistic. On the one hand, in high-end agricultural food processing enterprises, these enterprises will purchase and sell raw materials as inventory; on the other hand, they will purchase and sell finished products, such as the fruit enterprise surveyed in this article. Baolin Zhang et al. also proposed a supply chain that distinguishes between raw material inventory and finished product inventory, but their model is very rough and does not consider indicators such as loss rate and processing factor [29]. Starting from the physical meaning and practical situation, it is also necessary to consider its dimensions and units. Therefore, the supply chain model considered in this article is more detailed and comprehensive.

3. Artificial Intelligence Optimal Information Fusion

This article aims to establish dynamic monitoring information for core enterprises. Upstream enterprises can monitor the inventory dynamics of core enterprises in real-time and grasp their production scale in real-time. Downstream enterprises can monitor the inventory dynamics of core enterprises in real-time and adjust their order scale in real-time. Core enterprises can monitor their inventory dynamics in real-time, adjust their procurement and supply scales in real-time, and ensure the maximization of enterprise benefits.

Firstly, it is easy to establish static monitoring information for the core enterprise. The static monitoring information of upstream enterprises on core enterprises is as follows:

$$\begin{array}{cc}\hfill y_{\mathrm{up}}\left(k\right)& =C_{\mathrm{up}}{x}_{\mathrm{core}}\left(k\right),\hfill \\ \hfill z_{\mathrm{up}}\left(k\right)& =D_{\mathrm{up}}{x}_{\mathrm{core}}\left(k\right)+v_{\mathrm{up}}\left(k\right).\hfill \end{array}$$

(5)

The static monitoring information of the core enterprise itself is

$$\begin{array}{cc}\hfill y_{\mathrm{core}}\left(k\right)& =C_{\mathrm{core}}{x}_{\mathrm{core}}\left(k\right),\hfill \\ \hfill z_{\mathrm{core}}\left(k\right)& =D_{\mathrm{core}}{x}_{\mathrm{core}}\left(k\right)+v_{\mathrm{core}}\left(k\right).\hfill \end{array}$$

(6)

The static monitoring information of downstream enterprises on core enterprises is

$$\begin{array}{cc}\hfill y_{\mathrm{down}}\left(k\right)& =C_{\mathrm{down}}{x}_{\mathrm{core}}\left(k\right),\hfill \\ \hfill z_{\mathrm{down}}\left(k\right)& =D_{\mathrm{down}}{x}_{\mathrm{core}}\left(k\right)+v_{\mathrm{down}}\left(k\right).\hfill \end{array}$$

(7)

In static monitoring information systems (5)–(7), $y_{\mathrm{up}}\left(k\right),y_{\mathrm{core}}\left(k\right),y_{\mathrm{down}}\left(k\right)\in R^2$ represent the monitoring information of upstream enterprises, core enterprises in this article, and downstream enterprises on the core enterprises in this article, respectively. $z_{\mathrm{up}}\left(k\right),z_{\mathrm{core}}\left(k\right),z_{\mathrm{down}}\left(k\right)\in R^2$ represent the monitoring output information of upstream enterprises, core enterprises in this article, and downstream enterprises on the core enterprises in this article, respectively. $v_{\mathrm{up}}\left(k\right),v_{\mathrm{core}}\left(k\right),v_{\mathrm{down}}\left(k\right)\in R^2$ are all independent Gaussian noise with a mean of 0 and a variance of R. $C_{\mathrm{up}},C_{\mathrm{core}},C_{\mathrm{down}}\in R^{2\times 2}$ are reversible constant coefficient matrices, and $D_{\mathrm{up}},D_{\mathrm{core}},D_{\mathrm{down}}\in R^{2\times 2}$ are constant coefficient matrices.

Secondly, in order to accurately reflect the status information of the core enterprise’s supply chain, this article adopts the dynamic information monitoring method of upstream enterprises, core enterprises, and downstream enterprises to grasp the inventory status of the core enterprise. Considering the discontinuity and periodic iteration of information in dynamic supply chains, this paper first establishes a common control theory filtering model [33] and uses steady-state Kalman filtering theory to obtain dynamic signals of upstream enterprises, core enterprises, and downstream enterprises. Based on the optimal information fusion processing mechanism of artificial intelligence, the coefficient weighting method is used to obtain the optimal fusion signals of upstream enterprises, core enterprises, and downstream enterprises. As shown in Figure 3, the output information of the optimal information fusion processing mechanism of artificial intelligence is used as the dynamic inventory scheduling quantity of the core enterprise, and then the dynamic inventory of the core enterprise is adjusted to provide satisfactory results for enterprise operation. Based on static monitoring information (5)–(7), a dynamic monitoring information model consisting of output equations and measurement equations can be further constructed. The dynamic monitoring information of upstream enterprises on core enterprises is

$$\begin{array}{cc}\hfill y_{\mathrm{up}}(k+1)=& {\mathrm{Y}}_{\mathrm{up}}{y}_{\mathrm{up}}\left(k\right)+{\Gamma }_{\mathrm{up}}u\left(k\right),\hfill \\ \hfill z_{\mathrm{up}}\left(k\right)=& {\Lambda }_{\mathrm{up}}{y}_{\mathrm{up}}\left(k\right)+v_{\mathrm{up}}\left(k\right).\hfill \end{array}$$

(8)

The dynamic monitoring information of the core enterprise itself is

$$\begin{array}{cc}\hfill y_{\mathrm{core}}(k+1)=& {\mathrm{Y}}_{\mathrm{core}}{y}_{\mathrm{core}}\left(k\right)+{\Gamma }_{\mathrm{core}}u\left(k\right),\hfill \\ \hfill z_{\mathrm{core}}\left(k\right)=& {\Lambda }_{\mathrm{core}}{y}_{\mathrm{core}}\left(k\right)+v_{\mathrm{core}}\left(k\right).\hfill \end{array}$$

(9)

The dynamic monitoring information of downstream enterprises on core enterprises is

$$\begin{array}{cc}\hfill y_{\mathrm{down}}(k+1)=& {\mathrm{Y}}_{\mathrm{down}}{y}_{\mathrm{down}}\left(k\right)+{\Gamma }_{\mathrm{down}}u\left(k\right),\hfill \\ \hfill z_{\mathrm{down}}\left(k\right)=& {\Lambda }_{\mathrm{down}}{y}_{\mathrm{down}}\left(k\right)+v_{\mathrm{down}}\left(k\right).\hfill \end{array}$$

(10)

In dynamic monitoring information systems (8)–(10), $y_{\mathrm{up}}\left(k\right),y_{\mathrm{core}}\left(k\right),y_{\mathrm{down}}\left(k\right)\in R^2$ represent the monitoring information status of upstream enterprises, core enterprises in this article, and downstream enterprises on the core enterprises in this article, respectively. $z_{\mathrm{up}}\left(k\right),z_{\mathrm{core}}\left(k\right),z_{\mathrm{down}}\left(k\right)\in R^2$ represent the monitoring output information of upstream enterprises, core enterprises in this article, and downstream enterprises on the core enterprises in this article, respectively. ${\mathrm{Y}}_{\mathrm{up}}=C_{\mathrm{up}}A{C}_{\mathrm{up}}^{-1},{\mathrm{Y}}_{\mathrm{core}}=C_{\mathrm{core}}A{C}_{\mathrm{core}}^{-1},{\mathrm{Y}}_{\mathrm{down}}=C_{\mathrm{down}}A{C}_{\mathrm{down}}^{-1},{\Gamma }_{\mathrm{up}}=C_{\mathrm{up}}B,{\Gamma }_{\mathrm{core}}=C_{\mathrm{core}}B,{\Gamma }_{\mathrm{down}}=C_{\mathrm{down}}B,{\Lambda }_{\mathrm{up}}=D_{\mathrm{up}}{C}_{\mathrm{up}}^{-1},{\Lambda }_{\mathrm{core}}=D_{\mathrm{core}}{C}_{\mathrm{core}}^{-1},{\Lambda }_{\mathrm{down}}=D_{\mathrm{down}}{C}_{\mathrm{down}}^{-1}$ are constant coefficient matrices.

Lemma 1 

([34]). According to the steady-state Kalman filtering theory, the estimators of dynamic monitoring information systems (8)–(10) can be designed as follows:

$$\begin{array}{cc}\hfill {\widehat{y}}_{\mathrm{up}}\left(k\right)=& (I_n-{\Phi }_{\mathrm{up}}{\Lambda }_{\mathrm{up}}){\mathrm{Y}}_{\mathrm{up}}{\widehat{y}}_{\mathrm{up}}\left(k\right)+{\Phi }_{\mathrm{up}}{z}_{\mathrm{up}}\left(k\right),\hfill \\ \hfill {\Phi }_{\mathrm{up}}=& {\Psi }_{\mathrm{up}}{\Lambda }_{\mathrm{up}}^{\mathrm{T}}{\left[{\Lambda }_{\mathrm{up}}{\Psi }_{\mathrm{up}}{\Lambda }_{\mathrm{up}}^{\mathrm{T}}+{\Omega }_{\mathrm{up}}\right]}^{-1},\hfill \\ \hfill {\Xi }_{\mathrm{up}}=& \left[I_n-{\Phi }_{\mathrm{up}}{\Lambda }_{\mathrm{up}}\right]{\Psi }_{\mathrm{up}},\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill {\widehat{y}}_{\mathrm{core}}\left(k\right)=& (I_n-{\Phi }_{\mathrm{core}}{\Lambda }_{\mathrm{core}}){\mathrm{Y}}_{\mathrm{core}}{\widehat{y}}_{\mathrm{core}}\left(k\right)+{\Phi }_{\mathrm{core}}{z}_{\mathrm{core}}\left(k\right),\hfill \\ \hfill {\Phi }_{\mathrm{core}}=& {\Psi }_{\mathrm{core}}{\Lambda }_{\mathrm{core}}^{\mathrm{T}}{\left[{\Lambda }_{\mathrm{core}}{\Psi }_{\mathrm{core}}{\Lambda }_{\mathrm{core}}^{\mathrm{T}}+{\Omega }_{\mathrm{core}}\right]}^{-1},\hfill \\ \hfill {\Xi }_{\mathrm{core}}=& \left[I_n-{\Phi }_{\mathrm{core}}{\Lambda }_{\mathrm{core}}\right]{\Psi }_{\mathrm{core}},\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill {\widehat{y}}_{\mathrm{down}}\left(k\right)=& (I_n-{\Phi }_{\mathrm{down}}{\Lambda }_{\mathrm{down}}){\mathrm{Y}}_{\mathrm{down}}{\widehat{y}}_{\mathrm{down}}\left(k\right)+{\Phi }_{\mathrm{down}}{z}_{\mathrm{down}}\left(k\right),\hfill \\ \hfill {\Phi }_{\mathrm{down}}=& {\Psi }_{\mathrm{down}}{\Lambda }_{\mathrm{down}}^{\mathrm{T}}{\left[{\Lambda }_{\mathrm{down}}{\Psi }_{\mathrm{down}}{\Lambda }_{\mathrm{down}}^{\mathrm{T}}+{\Omega }_{\mathrm{down}}\right]}^{-1},\hfill \\ \hfill {\Xi }_{\mathrm{down}}=& \left[I_n-{\Phi }_{\mathrm{down}}{\Lambda }_{\mathrm{down}}\right]{\Psi }_{\mathrm{down}},\hfill \end{array}$$

(13)

where ${\Phi }_{\mathrm{up}},{\Phi }_{\mathrm{core}},and{\Phi }_{\mathrm{down}}$ are the coefficient matrices for steady-state filtering, and ${\Xi }_{\mathrm{up}},{\Xi }_{\mathrm{core}},and{\Xi }_{\mathrm{down}}$ are the prediction error variance matrices for steady-state filtering. If matrix Q is the variance matrix of $u\left(k\right)$ in the dynamic monitoring information systems (8)–(10) [35], then ${\Psi }_{\mathrm{up}},{\Psi }_{\mathrm{core}},and{\Psi }_{\mathrm{down}}$ are the unique positive definite solutions of the Riccati equations as follows:

$$\begin{array}{cc}\hfill {\Psi }_{\mathrm{up}}=& {\mathrm{Y}}_{\mathrm{up}}\left[{\Psi }_{\mathrm{up}}-{\Psi }_{\mathrm{up}}{\Lambda }_{\mathrm{up}}^{\mathrm{T}}{\left({\Lambda }_{\mathrm{up}}{\Psi }_{\mathrm{up}}{\Lambda }_{\mathrm{up}}^{\mathrm{T}}+{\Omega }_{\mathrm{up}}\right)}^{-1}{\Lambda }_{\mathrm{up}}{\Psi }_{\mathrm{up}}\right]{\mathrm{Y}}_{\mathrm{up}}^{\mathrm{T}}+{\Gamma }_{\mathrm{up}}Q{\Gamma }_{\mathrm{up}}^{\mathrm{T}},\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill {\Psi }_{\mathrm{core}}=& {\mathrm{Y}}_{\mathrm{core}}\left[{\Psi }_{\mathrm{core}}-{\Psi }_{\mathrm{core}}{\Lambda }_{\mathrm{core}}^{\mathrm{T}}{\left({\Lambda }_{\mathrm{core}}{\Psi }_{\mathrm{core}}{\Lambda }_{\mathrm{core}}^{\mathrm{T}}+{\Omega }_{\mathrm{core}}\right)}^{-1}{\Lambda }_{\mathrm{core}}{\Psi }_{\mathrm{core}}\right]{\mathrm{Y}}_{\mathrm{core}}^{\mathrm{T}}+{\Gamma }_{\mathrm{core}}Q{\Gamma }_{\mathrm{core}}^{\mathrm{T}},\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill {\Psi }_{\mathrm{down}}=& {\mathrm{Y}}_{\mathrm{down}}\left[{\Psi }_{\mathrm{down}}-{\Psi }_{\mathrm{down}}{\Lambda }_{\mathrm{down}}^{\mathrm{T}}{\left({\Lambda }_{\mathrm{down}}{\Psi }_{\mathrm{down}}{\Lambda }_{\mathrm{down}}^{\mathrm{T}}+{\Omega }_{\mathrm{down}}\right)}^{-1}{\Lambda }_{\mathrm{down}}{\Psi }_{\mathrm{down}}\right]{\mathrm{Y}}_{\mathrm{down}}^{\mathrm{T}}+{\Gamma }_{\mathrm{down}}Q{\Gamma }_{\mathrm{down}}^{\mathrm{T}}.\hfill \end{array}$$

(16)

Lemma 2 

([34]). Without considering the correlation between the states of various dynamic monitoring information systems (8)–(10), according to the trajectory-weighted optimal fusion criterion of the variance matrix, the optimal fusion of the state information of each dynamic monitoring information system is as follows:

$$\begin{array}{c}\hfill \widehat{y}\left(k\right)=a_{\mathrm{up}}{\widehat{y}}_{\mathrm{up}}\left(k\right)+a_{\mathrm{core}}{\widehat{y}}_{\mathrm{core}}\left(k\right)+a_{\mathrm{down}}{\widehat{y}}_{\mathrm{down}}\left(k\right),\end{array}$$

(17)

where $a_{\mathrm{up}},a_{\mathrm{core}},a_{\mathrm{down}}$ are optimal fusion coefficients, Tr represents the trace of the matrix, and there are

$$\begin{array}{cc}\hfill a_{\mathrm{up}}=& {\displaystyle \frac{1}{Tr\left({\Xi }_{\mathrm{up}}\right)}}{\left({\displaystyle \frac{1}{Tr\left({\Xi }_{\mathrm{up}}\right)+Tr\left({\Xi }_{\mathrm{core}}\right)+Tr\left({\Xi }_{\mathrm{down}}\right)}}\right)}^{-1},\hfill \\ \hfill a_{\mathrm{core}}=& {\displaystyle \frac{1}{Tr\left({\Xi }_{\mathrm{core}}\right)}}{\left({\displaystyle \frac{1}{Tr\left({\Xi }_{\mathrm{up}}\right)+Tr\left({\Xi }_{\mathrm{core}}\right)+Tr\left({\Xi }_{\mathrm{down}}\right)}}\right)}^{-1},\hfill \\ \hfill a_{\mathrm{down}}=& {\displaystyle \frac{1}{Tr\left({\Xi }_{\mathrm{down}}\right)}}{\left({\displaystyle \frac{1}{Tr\left({\Xi }_{\mathrm{up}}\right)+Tr\left({\Xi }_{\mathrm{core}}\right)+Tr\left({\Xi }_{\mathrm{down}}\right)}}\right)}^{-1}.\hfill \end{array}$$

The optimal information fusion mechanism for artificial intelligence in this article includes a dynamic monitoring information mechanism, which provides the possibility for demand forecasting in dynamic supply chains.

4. ${\mathcal{H}}^{\mathbf{\infty }}$ Control of Dynamic Supply Chain

The model construction of a composite dynamic supply chain network was discussed earlier, and the steady-state Kalman filtering method was used to establish static/dynamic monitoring information for the core enterprise. The artificial intelligence optimal information fusion mechanism for the core enterprise was established. This section will discuss the ${\mathcal{H}}^{\infty }$ control of dynamic supply chain networks. Firstly, considering the order-switching principle of the supply chain, the dynamic supply chain network is transformed into a linear discrete switching system model, and the ${\mathcal{H}}^{\infty }$ control of the dynamic supply chain network is transformed into a ${\mathcal{H}}^{\infty }$ control of a linear discrete switching system with disturbances. Secondly, it provides sufficient conditions for the ${\mathcal{H}}^{\infty }$ solvability of a dynamic supply chain. Once again, we consider the ${\mathcal{H}}^{\infty }$ robustness of dynamic supply chains. Finally, the ${\mathcal{H}}^{\infty }$ control criteria for the dynamic supply chain are provided to ensure the stability of the dynamic supply chain. A dynamic inventory scheduling algorithm is designed to satisfy the ${\mathcal{H}}^{\infty }$ performance of the closed-loop dynamic supply chain network.

4.1. Order Switching

Order switching refers to efficiently switching from the current order to the next order during the production process to reduce production downtime and improve production efficiency. As with pre-switching preparation, order preparation involves preparing relevant information, drawings, process flow, etc., for the next order in advance to ensure that production personnel are familiar with the requirements of the new order. Material preparation is the preparation of raw materials, components, and other materials required for the next order in advance, ensuring the smooth flow of the production line. If the equipment is adjusted, clean and maintain the production equipment before order switching to ensure that the equipment is in good condition and avoid switching delays caused by equipment failures. According to the requirements of the new order, adjust the equipment parameters and process flow to ensure that the equipment operates in accordance with the requirements of the new order. Clear order switching during team collaboration and handover communication between production teams can ensure accurate information transmission. Ensure close cooperation between different departments, coordinating material supply, equipment adjustment, and personnel allocation to ensure smooth order zero switching. By implementing a high-quality switching strategy $\sigma \left(k\right)$, enterprises can effectively switch orders, improve production efficiency, reduce downtime, and enhance their competitiveness and responsiveness.

The optimal information fusion problem for inventory adjustment in dynamic supply chain networks is considered. By using the steady-state Kalman filtering theory in Lemma 1 and the optimal information fusion criterion in Lemma 2, the optimal fusion output $\widehat{y}\left(k\right)$ of the dynamic monitoring information of the upstream enterprise, the core enterprise in this paper, and the downstream enterprise can be obtained, which serves as the dynamic output feedback regulation signal of the dynamic supply chain system (4).

$$\begin{array}{c}\hfill u\left(k\right)=K_{\sigma \left(k\right)}\widehat{y}\left(k\right),\end{array}$$

(18)

where $K_{\sigma \left(k\right)}$ is the adjustment gain, $\sigma \left(k\right)\in \mathcal{S}=\{1,2,\cdots ,s\}$, and s is the number of order switching states in the dynamic supply chain. In order to control the switching states s with different distributions, this paper introduces the method of switching control.

Remark 2. 

Binchuan County, China, is known as the “world-class hometown of fruits”. With abundant light and heat, drought, and little rain, Binchuan County has unique advantages in fruit cultivation. When we conducted research in Binchuan County, China, we found that the majority of agricultural products are based on the “Fruit Base–Distributor–Seller” model. Compared to other categories of goods, agricultural products (fruits) have shorter shelf life and greater storage difficulties. The technical content of their ordering is also relatively high, and they often repeatedly tug between ordering less than enough and selling more, and ordering more with high losses. Ordering more leads to greater losses. So, does ordering less or not ordering less mean there is no loss or less loss? Of course not, not ordering or underordering can be classified as hidden losses, wasting the sales potential that could have been produced, and insufficient sales silently devour sales growth space. Therefore, one can adjust their business strategy through high-quality switching strategies $\sigma \left(k\right)$.

The dynamic output feedback regulation signal of the dynamic supply chain system (4) can be obtained from the optimal information fusion (17) and the dynamic output feedback regulation signal (18) as follows:

$$\begin{array}{c}\hfill u\left(k\right)=K_{\sigma \left(k\right)}(a_{\mathrm{up}}{\widehat{y}}_{\mathrm{up}}\left(k\right)+a_{\mathrm{core}}{\widehat{y}}_{\mathrm{core}}\left(k\right)+a_{\mathrm{down}}{\widehat{y}}_{\mathrm{down}}\left(k\right)).\end{array}$$

(19)

By substituting the dynamic output feedback regulation signal (19) into the state equation of the dynamic supply chain system (4), it can be obtained that

$$\begin{array}{c}\hfill x_{\mathrm{core}}(k+1)=A{x}_{\mathrm{core}}\left(k\right)+B{K}_{\sigma \left(k\right)}(a_{\mathrm{up}}{\widehat{y}}_{\mathrm{up}}\left(k\right)+a_{\mathrm{core}}{\widehat{y}}_{\mathrm{core}}\left(k\right)+a_{\mathrm{down}}{\widehat{y}}_{\mathrm{down}}\left(k\right)).\end{array}$$

(20)

According to the static monitoring output information (5)–(7) of the dynamic supply chain system (4), it can be inferred that the estimators (11)–(13) of the dynamic monitoring information (8)–(10) can be further written as

$$\begin{array}{c}\hfill {\widehat{y}}_{\mathrm{up}}\left(k\right)=\left(I_n-{\Phi }_{\mathrm{up}}{\Lambda }_{\mathrm{up}}\right){\mathrm{Y}}_{\mathrm{up}}{\widehat{y}}_{\mathrm{up}}(k-1)+{\Phi }_{\mathrm{up}}{D}_{\mathrm{up}}x\left(k\right)+{\Phi }_{\mathrm{up}}{v}_{\mathrm{up}}\left(k\right),\end{array}$$

(21)

$$\begin{array}{c}\hfill {\widehat{y}}_{\mathrm{core}}\left(k\right)=\left(I_n-{\Phi }_{\mathrm{core}}{\Lambda }_{\mathrm{core}}\right){\mathrm{Y}}_{\mathrm{core}}{\widehat{y}}_{\mathrm{core}}(k-1)+{\Phi }_{\mathrm{core}}{D}_{\mathrm{core}}x\left(k\right)+{\Phi }_{\mathrm{core}}{v}_{\mathrm{core}}\left(k\right),\end{array}$$

(22)

$$\begin{array}{cc}\hfill {\widehat{y}}_{\mathrm{down}}\left(k\right)=& \left(I_n-{\Phi }_{\mathrm{down}}{\Lambda }_{\mathrm{down}}\right){\mathrm{Y}}_{\mathrm{down}}{\widehat{y}}_{\mathrm{down}}(k-1)+{\Phi }_{\mathrm{down}}{D}_{\mathrm{down}}x\left(k\right)+{\Phi }_{\mathrm{down}}{v}_{\mathrm{down}}\left(k\right).\hfill \end{array}$$

(23)

By inputting the output feedback regulation signal (19) and dynamic monitoring estimator (21)–(23) into the dynamic supply chain system (4), it can be obtained that

$$\begin{array}{cc}\hfill x_{\mathrm{core}}(k+1)=& A{x}_{\mathrm{core}}\left(k\right)+B{K}_{\sigma \left(k\right)}\{a_{\mathrm{up}}[\left(I_n-{\Phi }_{\mathrm{up}}{\Lambda }_{\mathrm{up}}\right){\mathrm{Y}}_{\mathrm{up}}{\widehat{y}}_{\mathrm{up}}(k-1)+{\Phi }_{\mathrm{up}}{D}_{\mathrm{up}}x\left(k\right)+{\Phi }_{\mathrm{up}}{v}_{\mathrm{up}}\left(k\right)]\hfill \\ \hfill & +a_{\mathrm{core}}[\left(I_n-{\Phi }_{\mathrm{core}}{\Lambda }_{\mathrm{core}}\right){\mathrm{Y}}_{\mathrm{core}}{\widehat{y}}_{\mathrm{core}}(k-1)+{\Phi }_{\mathrm{core}}{D}_{\mathrm{core}}x\left(k\right)+{\Phi }_{\mathrm{core}}{v}_{\mathrm{core}}\left(k\right)]\hfill \\ \hfill & +a_{\mathrm{down}}[\left(I_n-{\Phi }_{\mathrm{down}}{\Lambda }_{\mathrm{down}}\right){\mathrm{Y}}_{\mathrm{down}}{\widehat{y}}_{\mathrm{down}}(k-1)+{\Phi }_{\mathrm{down}}{D}_{\mathrm{down}}x\left(k\right)+{\Phi }_{\mathrm{down}}{v}_{\mathrm{down}}\left(k\right)]\}.\hfill \\ \hfill =& A{x}_{\mathrm{core}}\left(k\right)+B{K}_{\sigma \left(k\right)}(a_{\mathrm{up}}{\Phi }_{\mathrm{up}}{D}_{\mathrm{up}}+a_{\mathrm{core}}{\Phi }_{\mathrm{core}}{D}_{\mathrm{core}}+a_{\mathrm{down}}{\Phi }_{\mathrm{down}}{D}_{\mathrm{down}})x\left(k\right)\hfill \\ \hfill & +B{K}_{\sigma \left(k\right)}[a_{\mathrm{up}}\left(I_n-{\Phi }_{\mathrm{up}}{\Lambda }_{\mathrm{up}}\right){\mathrm{Y}}_{\mathrm{up}}{\widehat{y}}_{\mathrm{up}}(k-1)+a_{\mathrm{core}}\left(I_n-{\Phi }_{\mathrm{core}}{\Lambda }_{\mathrm{core}}\right){\mathrm{Y}}_{\mathrm{core}}{\widehat{y}}_{\mathrm{core}}(k-1)\hfill \\ \hfill & +a_{\mathrm{down}}\left(I_n-{\Phi }_{\mathrm{down}}{\Lambda }_{\mathrm{down}}\right){\mathrm{Y}}_{\mathrm{down}}{\widehat{y}}_{\mathrm{down}}(k-1)]\hfill \\ \hfill & +B{K}_{\sigma \left(k\right)}[a_{\mathrm{up}}{\Phi }_{\mathrm{up}}{v}_{\mathrm{up}}\left(k\right)+a_{\mathrm{core}}{\Phi }_{\mathrm{core}}{v}_{\mathrm{core}}\left(k\right)+a_{\mathrm{down}}{\Phi }_{\mathrm{down}}{v}_{\mathrm{down}}\left(k\right)].\hfill \end{array}$$

(24)

Let the augmented state vector be $\overline{x}\left(k\right)=[x_{\mathrm{core}}^{\mathrm{T}}\left(k\right),{\widehat{y}}_{\mathrm{up}}^{\mathrm{T}}(k-1),{\widehat{y}}_{\mathrm{core}}^{\mathrm{T}}(k-1),{\widehat{y}}_{\mathrm{down}}^{\mathrm{T}}(k-1)]\in R^8$, then the state equation of the closed-loop dynamic supply chain system is

$$\begin{array}{c}\hfill \overline{x}(k+1)=\left(\overline{A}+\overline{B}{K}_{\sigma \left(k\right)}L\right)\overline{x}\left(k\right)+{\overline{B}}_{\sigma \left(k\right)}v\left(k\right),\end{array}$$

(25)

where

$$\overline{A}=\left[\begin{array}{cccc}A& 0& 0& 0\\ {\Phi }_{\mathrm{up}}{D}_{\mathrm{up}}& \left(I_n-{\Phi }_{\mathrm{up}}{\Lambda }_{\mathrm{up}}\right){\mathrm{Y}}_{\mathrm{up}}& 0& 0\\ {\Phi }_{\mathrm{core}}{D}_{\mathrm{core}}& 0& \left(I_n-{\Phi }_{\mathrm{core}}{\Lambda }_{\mathrm{core}}\right){\mathrm{Y}}_{\mathrm{core}}& 0\\ {\Phi }_{\mathrm{down}}{D}_{\mathrm{down}}& 0& 0& \left(I_n-{\Phi }_{\mathrm{down}}{\Lambda }_{\mathrm{down}}\right){\mathrm{Y}}_{\mathrm{down}}\end{array}\right],$$

$$\overline{B}=\left[\begin{array}{c}B\\ 0\\ 0\\ 0\end{array}\right],L={\left[\begin{array}{c}{a}_{\mathrm{up}}{D}_{\mathrm{up}}^{\mathrm{T}}{\Phi }_{\mathrm{up}}^{\mathrm{T}}+a_{\mathrm{core}}{D}_{\mathrm{core}}^{\mathrm{T}}{\Phi }_{\mathrm{core}}^{\mathrm{T}}+a_{\mathrm{down}}{D}_{\mathrm{down}}^{\mathrm{T}}{\Phi }_{\mathrm{down}}^{\mathrm{T}}\\ a_{\mathrm{up}}{\mathrm{Y}}_{\mathrm{up}}^{\mathrm{T}}{\left(I_n-{\Phi }_{\mathrm{up}}{\Lambda }_{\mathrm{up}}\right)}^{\mathrm{T}}\\ a_{\mathrm{core}}{\mathrm{Y}}_{\mathrm{core}}^{\mathrm{T}}{\left(I_n-{\Phi }_{\mathrm{core}}{\Lambda }_{\mathrm{core}}\right)}^{\mathrm{T}}\\ a_{\mathrm{down}}{\mathrm{Y}}_{\mathrm{down}}^{\mathrm{T}}{\left(I_n-{\Phi }_{\mathrm{down}}{\Lambda }_{\mathrm{down}}\right)}^{\mathrm{T}}\end{array}\right]}^{\mathrm{T}},$$

$${\overline{B}}_{\sigma \left(k\right)}v\left(k\right)\triangleq \left[\begin{array}{ccc}B{K}_{\sigma \left(k\right)}{a}_{\mathrm{up}}{\Phi }_{\mathrm{up}}& B{K}_{\sigma \left(k\right)}{a}_{\mathrm{core}}{\Phi }_{\mathrm{core}}& B{K}_{\sigma \left(k\right)}{a}_{\mathrm{down}}{\Phi }_{\mathrm{down}}\\ {\Phi }_{\mathrm{up}}& 0& 0\\ 0& {\Phi }_{\mathrm{core}}& 0\\ 0& 0& {\Phi }_{\mathrm{down}}\end{array}\right]\left[\begin{array}{c}{v}_{\mathrm{up}}\left(k\right)\\ v_{\mathrm{core}}\left(k\right)\\ v_{\mathrm{down}}\left(k\right)\end{array}\right].$$

For the convenience of discussion, the closed-loop dynamic supply chain system (25) is written in the following form:

$$\begin{array}{c}\hfill \overline{x}(k+1)={\overline{A}}_{\sigma \left(k\right)}\overline{x}\left(k\right)+{\overline{B}}_{\sigma \left(k\right)}v\left(k\right),\end{array}$$

(26)

where ${\overline{A}}_{\sigma \left(k\right)}=\overline{A}+\overline{B}{K}_{\sigma \left(k\right)}L$, $\sigma \left(k\right)\in \mathcal{S}$ is the switching signal. When $\sigma \left(k\right)=i$, it indicates that the closed-loop dynamic supply chain system switches to the i-th subsystem, and $v\left(k\right)$ is the interference signal of the linear closed-loop dynamic supply chain system (26). From the expression, it can be seen that the dimensions of Equations (25) and (26) are higher than those of the original dynamic supply chain model (4), which is also the cost we pay to solve the problem. While obtaining superior control performance, we need to pay higher computational costs, which is in line with the opposing and unified views of Marxist theoretical philosophy.

In order to study the ${\mathcal{H}}^{\infty }$ performance of a linear dynamic supply chain system (26), the following assumption is given for the observable output of a closed-loop dynamic supply chain system.

Assumption A1. 

If ${\overline{z}}_i\left(k\right)$ is the observable output of the i-th subsystem of the closed-loop dynamic supply chain system (26), then

$$\begin{array}{c}\hfill \overline{z}\left(k\right)={\overline{C}}_{\sigma \left(k\right)}\overline{x}\left(k\right),\end{array}$$

(27)

where

$${\overline{C}}_{\sigma \left(k\right)}=\left[\begin{array}{cccc}{C}_{\sigma \left(k\right)}& 0& 0& 0\\ 0& I_8& 0& 0\\ 0& 0& I_8& 0\\ 0& 0& 0& I_8\end{array}\right].$$

Definition 1. 

For any $i\in \mathcal{S}$, the index function $\xi \left(k\right)={\left[{\xi }_1\left(k\right),{\xi }_2\left(k\right),\cdots ,{\xi }_s\left(k\right)\right]}^{\mathrm{T}}$ is defined as follows:

$${\xi }_i\left(k\right)=\left\{\begin{array}{c}1,\mathit{when}\mathit{the}\mathit{system}\mathit{switched}\mathit{to}\mathit{the}i\mathit{-}\mathit{th}\mathit{subsystem}\mathit{at}\mathit{that}\mathit{time}K,\hfill \\ 0,\mathit{other}\mathit{situations}.\hfill \end{array}\right.$$

According to Definition 1, a closed-loop dynamic supply chain system (26) can be described as

$$\left\{\begin{array}{cc}\hfill \overline{x}(k+1)& =\sum _{i=1}^s{\xi }_i\left(k\right)\left({\overline{A}}_i\overline{x}\left(k\right)+{\overline{B}}_{i}v\left(k\right)\right),\hfill \\ \hfill \overline{z}\left(k\right)& =\sum _{i=1}^s{\xi }_i\left(k\right){\overline{C}}_i\overline{x}\left(k\right),\hfill \end{array}\right.$$

(28)

where ${\overline{A}}_i$ is the system matrix of the i-th subsystem of the closed-loop dynamic supply chain system.

4.2. Sufficient Conditions for ${\mathcal{H}}^{\infty }$ Solvability in Dynamic Supply Chains

The solvability of ${\mathcal{H}}^{\infty }$ control is an important concept in control theory [36]. In this section, we provide the definition and sufficient conditions for ${\mathcal{H}}^{\infty }$ solvability of dynamic supply chain systems (26)–(28).

Definition 2 

(${\mathcal{H}}^{\mathbf{\infty }}$Solvability of the Dynamic Supply Chain). The dynamic supply chain system (26)–(28) has an ${\mathcal{H}}^{\infty }$ norm bound γ stability under any switching of $\sigma \left(k\right)$; that is, it simultaneously satisfies the following conditions:

(1) 

When $v\left(k\right)=0$, the dynamic supply chain system (26)–(28) is asymptotically stable under any switching of $\sigma \left(k\right)$;

(2) 

Under zero initial conditions, with the action of arbitrary switching of $\sigma \left(k\right)$, the tuned output $\overline{z}\left(k\right)$ satisfies

$$\begin{array}{c}\hfill \sum _{k=0}^{\infty }{∥\overline{z}\left(k\right)∥}^2<{\gamma }^2\sum _{k=0}^{\infty }{∥v\left(k\right)∥}^2.\end{array}$$

(29)

The dynamic supply chain system (26)–(28) is ${\mathcal{H}}^{\infty }$ solvable under any switching of $\sigma \left(k\right)$.

A necessary condition for the ${\mathcal{H}}^{\infty }$ solvability of a dynamic supply chain system (26)–(28) under any switching law $\sigma \left(k\right)$ is that each subsystem is ${\mathcal{H}}^{\infty }$ solvable. This is obvious and easy to understand. In fact, if the j-th subsystem is not ${\mathcal{H}}^{\infty }$ solvable, and if $i\equiv j$ is set, then the dynamic supply chain system (26)–(28) is clearly not ${\mathcal{H}}^{\infty }$ solvable. However, the solvability of each subsystem ${\mathcal{H}}^{\infty }$ is not a sufficient condition for the ${\mathcal{H}}^{\infty }$ solvability of dynamic supply chain systems (26)–(28). This is because even if each subsystem is stable (condition 1 for ${\mathcal{H}}^{\infty }$ solvability), the dynamic supply chain system (26)–(28) may be unstable under a special switching signal $\sigma \left(k\right)$. Furthermore, even under the action of a switching signal $\sigma \left(k\right)$, the dynamic supply chain system (26)–(28) is stable and does not necessarily have an ${\mathcal{H}}^{\infty }$ norm bound (condition 2 for ${\mathcal{H}}^{\infty }$ solvability). Therefore, it is evident that the dynamic supply chain system (26)–(28) is not ${\mathcal{H}}^{\infty }$ solvable. Therefore, in order to obtain the solvability of the ${\mathcal{H}}^{\infty }$ problem in the dynamic supply chain system (26)–(28), in addition to each subsystem being ${\mathcal{H}}^{\infty }$ solvable, the dynamic supply chain system (26)–(28) must also satisfy certain conditions. The main purpose of this section is to find sufficient conditions for the ${\mathcal{H}}^{\infty }$ solvability of dynamic supply chain systems (26)–(28) under the action of any switching law $\sigma \left(k\right)$.

Theorem 1 

(Sufficient Conditions for ${\mathcal{H}}^{\mathbf{\infty }}$Solvability in the Dynamic Supply Chain). In the case where each subsystem of the dynamic supply chain system (26)–(28) is ${\mathcal{H}}^{\infty }$ solvable, each subsystem is asymptotically stable and has an ${\mathcal{H}}^{\infty }$ performance bound ${\gamma }_j$. If the inequality

$$\begin{array}{c}\hfill {\mathit{U}}_j=\left[\begin{array}{cc}{\overline{A}}_j^{\mathrm{T}}\mathit{Q}{\overline{A}}_j-\mathit{Q}+{\overline{C}}_j^{\mathrm{T}}{\overline{C}}_j& {\overline{A}}_j^{\mathrm{T}}\mathit{Q}{\overline{B}}_j\\ {\overline{B}}_j^{\mathrm{T}}\mathit{Q}{\overline{A}}_j& {\overline{B}}_j^{\mathrm{T}}\mathit{Q}{\overline{B}}_j-{\gamma }_j^{2}I\end{array}\right]<0,j\in \mathcal{S}\end{array}$$

(30)

has a common positive definite solution $\mathit{Q}$, then the dynamic supply chain system (26)–(28) is ${\mathcal{H}}^{\infty }$ solvable under any switching law $\sigma \left(k\right)$, and its ${\mathcal{H}}^{\infty }$ performance bound $\gamma =\underset{j\in \mathcal{S}}{max}\left|{\gamma }_j\right|$.

Proof. 

Under condition (30), constructing a Lyapunov function $V\left(k\right)={\overline{x}}^{\top }\left(k\right)\mathit{Q}\overline{x}\left(k\right)$, it can be concluded that the dynamic supply chain system (26)–(28) is asymptotically stable. To prove that under zero initial conditions, the dynamic supply chain system (26)–(28) has

$$\sum _{k=0}^{\infty }{∥\overline{z}\left(k\right)∥}^2<{\gamma }^2\sum _{k=0}^{\infty }{∥v\left(k\right)∥}^2,\gamma =\underset{j\in \mathcal{S}}{max}\left\{{\gamma }_j\right\}.$$

Assuming $\overline{x}\left(0\right)=0$, introduce

$$\begin{array}{c}\hfill W\left(\overline{z}\left(k\right),v\left(k\right)\right)={\gamma }_j^2{v}^{\mathrm{T}}\left(k\right)v\left(k\right)-{\overline{z}}^{\mathrm{T}}\left(k\right)\overline{z}\left(k\right).\end{array}$$

(31)

If there is

$$\begin{array}{c}\hfill \Delta V\left(k\right)=V(k+1)-V\left(k\right)<W\left(\overline{z}\left(k\right),v\left(k\right)\right),\end{array}$$

(32)

summing k from 0 to ∞ on both sides of inequality (32) yields

$$\begin{array}{cc}\hfill 0& <V(\infty )\hfill \\ \hfill & =V(\infty )-V\left(0\right)\hfill \\ \hfill & =\sum _{k=0}^{\infty }\Delta V\left(k\right)\hfill \\ \hfill & <\sum _{k=0}^{\infty }W\left(\overline{z}\left(k\right),v\left(k\right)\right)\hfill \\ \hfill & =\sum _{k=0}^{\infty }\left({\gamma }_j^2{v}^{\mathrm{T}}\left(k\right)v\left(k\right)-{\overline{z}}^{\mathrm{T}}\left(k\right)\overline{z}\left(k\right)\right)\hfill \\ \hfill & ⩽\sum _{k=0}^{\infty }\left({\gamma }^2{v}^{\mathrm{T}}\left(k\right)v\left(k\right)-{\overline{z}}^{\mathrm{T}}\left(k\right)\overline{z}\left(k\right)\right);\hfill \end{array}$$

(33)

that is, ${\displaystyle \sum _{k=0}^{\infty }}{∥\overline{z}\left(k\right)∥}^2<{\gamma }^2\sum _{k=0}^{\infty }{∥v\left(k\right)∥}^2$.

Let us prove inequality (32) below. Let ${\eta }^{\mathrm{T}}\left(k\right)={[\overline{x}\left(k\right),v^{\mathrm{T}}\left(k\right)]}^{\mathrm{T}}$,

$$\begin{array}{cc}\hfill \Delta V\left(k\right)=& V(k+1)-V\left(k\right)\hfill \\ \hfill =& {\overline{x}}^{\mathrm{T}}(k+1)\mathit{Q}{\overline{x}}^{\mathrm{T}}(k+1)-{\overline{x}}^{\mathrm{T}}\left(k\right)\mathit{Q}{\overline{x}}^{\mathrm{T}}\left(k\right)\hfill \\ \hfill =& \left({\overline{x}}^{\mathrm{T}}\left(k\right){\overline{A}}_j^{\mathrm{T}}+v^{\mathrm{T}}\left(k\right){\overline{B}}_j^{\mathrm{T}}\right)\mathit{Q}\left({\overline{A}}_j\overline{x}\left(k\right)+{\overline{B}}_{j}v\left(k\right)\right)-{\overline{x}}^{\mathrm{T}}\left(k\right)\mathit{Q}{\overline{x}}^{\mathrm{T}}\left(k\right)\hfill \\ \hfill =& {\eta }^{\mathrm{T}}\left(k\right)\left[\begin{array}{cc}{\overline{A}}_j^{\mathrm{T}}\mathit{Q}{\overline{A}}_j-\mathit{Q}& {\overline{A}}_j^{\mathrm{T}}\mathit{Q}{\overline{B}}_j\\ {\overline{B}}_j^{\mathrm{T}}\mathit{Q}{\overline{A}}_j& {\overline{B}}_j^{\mathrm{T}}\mathit{Q}{\overline{B}}_j\end{array}\right]\eta \left(k\right),\hfill \\ \hfill -W\left(\overline{z}\left(k\right),v\left(k\right)\right)=& {\overline{z}}^{\mathrm{T}}\left(k\right)\overline{z}\left(k\right)-{\gamma }_j^2{v}^{\mathrm{T}}\left(k\right)v\left(k\right)\hfill \\ \hfill =& {\eta }^{\mathrm{T}}\left(k\right)\left[\begin{array}{c}{\overline{C}}_j^{\mathrm{T}}\\ 0\end{array}\right]\left[\begin{array}{cc}{\overline{C}}_j& 0\end{array}\right]\eta \left(k\right)-{\gamma }_j^2{v}^{\mathrm{T}}\left(k\right)v\left(k\right)\hfill \\ \hfill =& {\eta }^{\mathrm{T}}\left(k\right)\left[\begin{array}{cc}{\overline{C}}_j^{\mathrm{T}}{\overline{C}}_j& 0\\ 0& -{\gamma }_j^{2}I\end{array}\right]\eta \left(k\right).\hfill \end{array}$$

Therefore,

$$\begin{array}{c}\hfill \Delta V\left(k\right)-W\left(\overline{z}\left(k\right),v\left(k\right)\right)={\eta }^{\mathrm{T}}\left(k\right){\mathit{U}}_j\eta \left(k\right).\end{array}$$

(34)

According to the inequality (30) with a common solution $\mathit{Q}$, it can be inferred that ${\mathit{U}}_j<0$. Therefore, $\Delta V\left(k\right)<W\left(\overline{z}\left(k\right),v\left(k\right)\right)$; that is, $\sum _{k=0}^{\infty }{∥\overline{z}\left(k\right)∥}^2<{\gamma }^2\sum _{k=0}^{\infty }{∥v\left(k\right)∥}^2$ holds. □

Remark 3. 

Through the ${\mathcal{H}}^{\infty }$ control principle, Lihua Hu and Tao Fan studied the production planning and scheduling of dynamic supply chains [28] and also studied the ${\mathcal{H}}^{\infty }$ fault-tolerant control of complex supply chain systems with dual-channels [27]. However, they did not include sufficient discussion on the solvability of the ${\mathcal{H}}^{\infty }$ problem, and our current research has found that this problem seriously restricts the promotion and application of its results. The above results in this article just make up for this deficiency.

4.3. ${\mathcal{H}}^{\infty }$ Robustness of Dynamic Supply Chains

The ${\mathcal{H}}^{\infty }$ robustness of the dynamic supply chain system studied in this article can be described as follows: if the dynamic supply chain system (26)–(28) is internally stable ($v\left(k\right)=0$), and under zero initial conditions, the non-zero interference signal $v\left(k\right)$ with limited energy satisfies $J=\parallel \overline{z}{\left(k\right)\parallel }_2^2-{\gamma }^2{∥v\left(k\right)∥}_2^2<0$, then the dynamic supply chain system (26)–(28) is said to be robust stable and has an ${\mathcal{H}}^{\infty }$ disturbance attenuation degree $\gamma$.

Lemma 3 

(Matrix Inverse Lemma [37]).$A,B,C,D$ are matrices with appropriate dimensions, and if A and D are not singular, then

$$\begin{array}{c}\hfill {\left(A+B{D}^{-1}C\right)}^{-1}=A^{-1}-A^{-1}B{\left(C{A}^{-1}B+D\right)}^{-1}C{A}^{-1}\end{array}$$

holds.

Lemma 4 

(Improving Schur’s Complement Lemma [38]).Given matrices P and G with appropriate dimensions, where $P>0$, the condition $I-G^{T}PG>0$ is equivalent to $P^{-1}-G{G}^T>0$.

This section will combine Lemmas 3 and 4 and analyze the ${\mathcal{H}}^{\infty }$ robustness of dynamic supply chain systems (26)–(28) using the common quadratic Lyapunov function method and the piecewise quadratic Lyapunov function method.

Theorem 2 

(Common Quadratic Lyapunov Function). For dynamic supply chain systems (26)–(28), if $\forall i\in \mathcal{S}$, there exists a positive definite matrix P that satisfies

$$\begin{array}{c}\hfill P^{-1}-{\displaystyle \frac{1}{{\gamma }^2}}{\overline{B}}_i{\overline{B}}_i^{\top }>0,\end{array}$$

(35)

$$\begin{array}{c}\hfill {\overline{A}}_i^{\top }{\left(P^{-1}-{\displaystyle \frac{1}{{\gamma }^2}}{\overline{B}}_i{\overline{B}}_i^T\right)}^{-1}{\overline{A}}_i-P+{\overline{C}}_i^T{\overline{C}}_i<0,\end{array}$$

(36)

then the dynamic supply chain system (26)–(28) is robust and stable for any switching $\sigma \left(k\right)$, and has an ${\mathcal{H}}^{\infty }$ disturbance attenuation degree γ.

Proof. 

Firstly, we demonstrate the robust stability of the dynamic supply chain system (26)–(28). Let us assume that $\sigma \left(k\right)=i$ and the dynamic supply chain system (26)–(28) switches to the i-th subsystem at time k, taking the Lyapunov function as $V\left(\overline{x}\left(k\right)\right)={\overline{x}}^{\top }\left(k\right)P\overline{x}\left(k\right)$. If the dynamic supply chain system (26)–(28) is internally stable, we need to find the difference negative definite of $V\left(\overline{x}\left(k\right)\right)$ when $v\left(k\right)=0$; that is,

$$\begin{array}{c}\hfill \Delta V\left(x\left(k\right)\right)=x^{\top }\left(k\right)\left(A_i^{\top }P{A}_i-P\right)x\left(k\right)<0.\end{array}$$

(37)

According to condition (36) of Theorem 2, it can be inferred that

$$\begin{array}{c}\hfill {\overline{A}}_i^{\top }{\left(P^{-1}-{\displaystyle \frac{1}{{\gamma }^2}}{\overline{B}}_i{\overline{B}}_i^T\right)}^{-1}{\overline{A}}_i-P+{\overline{C}}_i^T{\overline{C}}_i<0.\end{array}$$

(38)

Applying Lemma 3 can yield the following:

$$\begin{array}{c}\hfill {\overline{A}}_i^{\top }P{\overline{A}}_i-P+{\overline{C}}_i^{\top }{\overline{C}}_i-{\overline{A}}_i^{\mathrm{T}}P{\overline{B}}_i{\left({\overline{B}}_i^{\top }P{\overline{B}}_i-{\gamma }^{2}I\right)}^{-1}{\overline{B}}_i^{\top }P{\overline{A}}_i<0.\end{array}$$

(39)

According to Formula (35), there is $P^{-1}-{\displaystyle \frac{1}{{\gamma }^2}}{\overline{B}}_i{\overline{B}}_i^{\top }>0$. According to Lemma 4, it can be inferred that ${\overline{B}}_i^{\top }P{\overline{B}}_i-{\gamma }^{2}I<0$. From Formula (39) and ${\overline{C}}_i^{\top }{\overline{C}}_i⩾0$, it can be inferred that

$$\begin{array}{c}\hfill {\overline{A}}_i^{\top }P{\overline{A}}_i-P<0.\end{array}$$

(40)

According to Formula (40), there is $\Delta V\left(x\right(k\left)\right)<0$, that is, the dynamic supply chain system (26)–(28) is robust and stable.

Secondly, it is proven that the dynamic supply chain system (26)–(28) has an ${\mathcal{H}}^{\infty }$ disturbance attenuation degree $\gamma$. Let

$$\begin{array}{c}\hfill J_{\sigma \left(k\right)}={\overline{z}}^{\mathrm{T}}\left(k\right)\overline{z}\left(k\right)-{\gamma }^2{v}^{\top }\left(k\right)v\left(k\right).\end{array}$$

(41)

We can reorganize and obtain

$$\begin{array}{c}\hfill J_{\sigma \left(k\right)}={\left[\begin{array}{c}v\left(k\right)\\ \overline{x}\left(k\right)\end{array}\right]}^{\top }\left[\begin{array}{cc}{\overline{B}}_{\sigma \left(k\right)}^{\top }P{\overline{B}}_{\sigma \left(k\right)}-{\gamma }^{2}I& {\overline{B}}_{\sigma \left(k\right)}^{\top }P{\overline{A}}_{\sigma \left(k\right)}\\ {\overline{A}}_{\sigma \left(k\right)}^{\top }P{\overline{B}}_{\sigma \left(k\right)}& {\overline{A}}_{\sigma \left(k\right)}^{\top }P{\overline{A}}_{\sigma \left(k\right)}-P+{\overline{C}}_{\sigma \left(k\right)}^{\top }{\overline{C}}_{\sigma \left(k\right)}\end{array}\right]\left[\begin{array}{c}v\left(k\right)\\ \overline{x}\left(k\right)\end{array}\right]-\Delta V\left(x\left(k\right)\right).\end{array}$$

(42)

Based on the previous assumption that $\sigma \left(k\right)=i$, and according to Schur’s complement lemma, (39) can be concluded that

$$\begin{array}{c}\hfill \left[\begin{array}{cc}{\overline{B}}_i^{\top }P{\overline{B}}_i-{\gamma }^{2}I& {\overline{B}}_i^{\top }P{\overline{A}}_i\\ {\overline{A}}_i^{\top }P{\overline{B}}_i& {\overline{A}}_i^{\top }P{\overline{A}}_i-P+{\overline{C}}_i^{\top }{\overline{C}}_i\end{array}\right]<0,\forall i\in \mathcal{S}.\end{array}$$

(43)

Because $J=\sum _{k=0}^{\infty }{J}_{\sigma \left(k\right)}$ and $V\left(x\right(0\left)\right)=0$, when inequality (43) has $J<0$, and the dynamic supply chain system (26)–(28) has an ${\mathcal{H}}^{\infty }$ disturbance attenuation degree $\gamma$, Theorem 2 holds. □

Corollary 1. 

$\forall i\in \mathcal{S}$ if there is a positive definite matrix P such that the LMI,

$$\begin{array}{c}\hfill \left[\begin{array}{cc}{\displaystyle \frac{1}{{\gamma }^2}}{\overline{B}}_i^{\top }P{\overline{B}}_i-I& {\displaystyle \frac{1}{\gamma }}{\overline{B}}_i^{\top }P{\overline{A}}_i\\ {\displaystyle \frac{1}{\gamma }}{\overline{A}}_i^{\top }P{\overline{B}}_i& {\overline{A}}_i^{\top }P{\overline{A}}_i-P+{\overline{C}}_i^{\mathrm{T}}{\overline{C}}_i\end{array}\right]<0,\end{array}$$

(44)

is solvable, then the dynamic supply chain system (26)–(28) is robustly stable for any switching $\sigma \left(k\right)$ and has an ${\mathcal{H}}^{\infty }$ disturbance decay degree γ.

Proof. 

Applying Lemma 4 and Schur’s complement lemma can prove the validity of Corollary 1. □

Secondly, the existence of a common quadratic Lyapunov function in the dynamic supply chain system (26)–(28) indicates that the dynamic supply chain system (26)–(28) is quadratic stable, and the quadratic Lyapunov function can be regarded as a special case of piecewise quadratic Lyapunov function. If there is a common piecewise quadratic Lyapunov function in the dynamic supply chain system (26)–(28), it can be ensured that the dynamic supply chain system (26)–(28) is so-called “piecewise quadratic stability”, and piecewise quadratic stability is a weaker condition than quadratic stability. Therefore, using the piecewise quadratic Lyapunov function method can theoretically reduce the conservatism of the conclusion of Theorem 2. There may be two alternative quadratic Lyapunov functions:

$$\begin{array}{c}\hfill V_1\left(\overline{x}\left(k\right)\right)={\overline{x}}^{\top }\left(k\right)P_1\overline{x}\left(k\right);\\ \hfill V_2\left(\overline{x}\left(k\right)\right)={\overline{x}}^{\top }\left(k\right)P_2\overline{x}\left(k\right),\end{array}$$

available. The common Lyapunov function of the dynamic supply chain system (26)–(28) is taken as

$$\begin{array}{c}\hfill V\left(\overline{x}\left(k\right)\right)=min\left\{{\overline{x}}^{\top }\left(k\right)P_1\overline{x}\left(k\right),{\overline{x}}^{\top }\left(k\right)P_2\overline{x}\left(k\right)\right\};\end{array}$$

(45)

that is, the common Lyapunov function of the dynamic supply chain system (26)–(28) is composed of two piecewise quadratic Lyapunov functions, and the following results are obtained:

Theorem 3 

(Piecewise Quadratic Lyapunov Function). $\forall i\in \mathcal{S}$, there exist positive definite matrices $P_1,P_2$, and appropriate non-negative constants ${\mu }_{i,1},{\mu }_{i,2}$, such that

$$\begin{array}{c}\hfill P_1^{-1}-{\displaystyle \frac{1}{{\gamma }^2}}{\overline{B}}_i{\overline{B}}_i^{\mathrm{T}}>0,\\ \hfill P_2^{-1}-{\displaystyle \frac{1}{{\gamma }^2}}{\overline{B}}_i{\overline{B}}_i^{\mathrm{T}}>0,\\ \hfill {\overline{A}}_i^{\mathrm{T}}{\left(P_1^{-1}-{\displaystyle \frac{1}{{\gamma }^2}}{\overline{B}}_i{\overline{B}}_i^{\mathrm{T}}\right)}^{-1}{\overline{A}}_i-P_1+{\overline{C}}_i^{\mathrm{T}}{\overline{C}}_i+{\mu }_{i,1}\left(P_2-P_1\right)<0,\\ \hfill {\overline{A}}_i^{\mathrm{T}}{\left(P_2^{-1}-{\displaystyle \frac{1}{{\gamma }^2}}{\overline{B}}_i{\overline{B}}_i^{\mathrm{T}}\right)}^{-1}{\overline{A}}_i-P_2+{\overline{C}}_i^{\mathrm{T}}{\overline{C}}_i+{\mu }_{i,1}\left(P_1-P_2\right)<0,\end{array}$$

then the dynamic supply chain system (26)–(28) is robust and stable for any switching $\sigma \left(k\right)$, and has an ${\mathcal{H}}^{\infty }$ disturbance attenuation degree γ.

Proof. 

Referring to the proof process of Theorem 2 and considering the piecewise case of Lyapunov function (45), Theorem 3 can be proven. □

Corollary 2. 

$\forall i\in \mathcal{S}$ if there is a positive definite matrix P, such that the LMIs

$$\begin{array}{c}\hfill U_{1,i}=\left[\begin{array}{cc}{\displaystyle \frac{1}{{\gamma }^2}}{\overline{B}}_i^{\top }{P}_1{\overline{B}}_i-I& {\displaystyle \frac{1}{\gamma }}{\overline{B}}_i^{\top }{P}_1{\overline{A}}_i\\ {\displaystyle \frac{1}{\gamma }}{\overline{A}}_i^{\top }{P}_1{\overline{B}}_i& {\overline{A}}_i^{\top }{P}_1{\overline{A}}_i-P_1+{\overline{C}}_i^{\mathrm{T}}{\overline{C}}_i+{\mu }_{i,1}\left(P_2-P_1\right)\end{array}\right]<0,\end{array}$$

(46)

$$\begin{array}{c}\hfill U_{2,i}=\left[\begin{array}{cc}{\displaystyle \frac{1}{{\gamma }^2}}{\overline{B}}_i^{\top }{P}_2{\overline{B}}_i-I& {\displaystyle \frac{1}{\gamma }}{\overline{B}}_i^{\top }{P}_2{\overline{A}}_i\\ {\displaystyle \frac{1}{\gamma }}{\overline{A}}_i^{\top }{P}_2{\overline{B}}_i& {\overline{A}}_i^{\top }{P}_2{\overline{A}}_i-P_2+{\overline{C}}_i^{\mathrm{T}}{\overline{C}}_i+{\mu }_{i,2}\left(P_1-P_2\right)\end{array}\right]<0,\end{array}$$

(47)

are solvable, then the dynamic supply chain system (26)–(28) is robustly stable for any switching $\sigma \left(k\right)$ and has an ${\mathcal{H}}^{\infty }$ disturbance decay degree γ.

Proof. 

Applying Lemma 4 and Schur’s complement lemma, it can be proven from Theorem 3 that Corollary 2 holds. □

Remark 4. 

The conclusions of Theorems 2 and 3 are sufficient conditions and both have conservatism. For Theorem 3, if more alternative quadratic Lyapunov functions are used to form piecewise quadratic Lyapunov functions, it can theoretically reduce conservatism, but the computational complexity of the analysis process will increase. Simply put, while reducing conservatism, it inevitably increases computational complexity. The conflicting parties are mutually conditional and coexist in a unified entity, where they transform into each other under certain conditions.

4.4. ${\mathcal{H}}^{\infty }$ Control Criteria for Dynamic Supply Chains

In the case of any order switching $\sigma \left(k\right)$, design a dynamic output feedback regulation signal (18) so that the closed-loop dynamic supply chain system (26)–(28) is internally stable, and under zero initial conditions, for a non-zero interference signal $v\left(k\right)$ with limited energy, $J=\parallel \overline{z}{\left(k\right)\parallel }_2^2-{\gamma }^2{∥v\left(k\right)∥}_2^2<0$ is satisfied.

Definition 3. 

Given a ${\mathcal{H}}^{\infty }$ performance metric $\gamma >0$, the following conditions are met:

(1) 

When the initial state of the closed-loop dynamic supply chain system (26)–(28) is $\overline{x}\left(0\right)=0$,

$$\begin{array}{c}\hfill ∥\overline{z}\left(k\right)∥<\gamma ∥v\left(k\right)∥.\end{array}$$

(48)

(2) 

When $v\left(k\right)\equiv 0$, the closed-loop dynamic supply chain system (26)–(28) is asymptotically stable.

At this point, the closed-loop dynamic supply chain system (26)–(28) is robust and stable under the action of the dynamic output feedback regulation signal (18), and has an ${\mathcal{H}}^{\infty }$ disturbance attenuation degree γ. Therefore, the closed-loop dynamic supply chain system (26)–(28) satisfies the ${\mathcal{H}}^{\infty }$ control criterion, where the norm is defined by the inner product of the discrete ${\mathcal{L}}_2[0,+\infty )$ space.

Theorem 4. 

If there are s symmetric positive definite matrices $G_1,G_2,\cdots ,G_s$, and the constant $\gamma >0$, the following matrix inequality holds:

$$\begin{array}{c}\hfill \left[\begin{array}{cc}{\overline{A}}_i^{\mathrm{T}}{G}_j{\overline{A}}_i-G_i+{\gamma }^{-1}{\overline{C}}_i^{\mathrm{T}}{\overline{C}}_i& {\overline{A}}_i^{\mathrm{T}}{G}_j{\overline{B}}_i\\ {\overline{B}}_i^{\mathrm{T}}{G}_j{\overline{A}}_i& {\overline{B}}_i^{\mathrm{T}}{G}_j{\overline{B}}_i-\gamma I\end{array}\right]<0,i,j\in \mathcal{S}.\end{array}$$

(49)

Then, there exists a dynamic output feedback regulation signal (18) that satisfies the ${\mathcal{H}}^{\infty }$ control criterion in the closed-loop dynamic supply chain system (26)–(28).

Proof. 

Construct Lyapunov functions for closed-loop dynamic supply chain systems (26)–(28) as follows:

$$\begin{array}{c}\hfill V\left(k,\overline{x}\left(k\right)\right)={\overline{x}}^{\mathrm{T}}\left(k\right)G\left({\xi }_k\right)\overline{x}\left(k\right)={\overline{x}}^{\mathrm{T}}\left(k\right)\left(\sum _{i=1}^s{\xi }_i\left(k\right)G_i\right)\overline{x}\left(k\right);\end{array}$$

(50)

then, the difference of $V(k,\overline{x}\left(k\right))$ is

$$\begin{array}{cc}\hfill \Delta V& =V\left(k+1,\overline{x}(k+1)\right)-V\left(k,\overline{x}\left(k\right)\right)\hfill \\ \hfill & ={\overline{x}}^{\mathrm{T}}(k+1)G\left({\xi }_{k+1}\right)\overline{x}(k+1)-{\overline{x}}^{\mathrm{T}}\left(k\right)G\left({\xi }_k\right)\overline{x}\left(k\right).\hfill \end{array}$$

(51)

Considering that the switching signal $\sigma \left(k\right)$ of the closed-loop dynamic supply chain system (26)–(28) is arbitrary, when the discrete system randomly switches to the i-th subsystem at time k, the performance index function in the closed-loop dynamic supply chain system (28) is equivalent to the following form:

$$\left\{\begin{array}{cc}\hfill {\xi }_i\left(k\right)& =1,{\xi }_{l\ne i}\left(k\right)=0,\hfill \\ \hfill {\xi }_j(k+1)& =1,{\xi }_{l\ne j}(k+1)=0.\hfill \end{array}\right.$$

(52)

According to Equation (52), the difference Equation (51) of the closed-loop dynamic supply chain system (28) is rewritten as follows:

$$\begin{array}{cc}\hfill \Delta V=& {\overline{x}}^{\mathrm{T}}\left(k\right){\overline{A}}_i^{\mathrm{T}}{G}_j{\overline{A}}_i\overline{x}\left(k\right)-{\overline{x}}^{\mathrm{T}}\left(k\right)G_i\overline{x}\left(k\right)+\left({\overline{x}}^{\mathrm{T}}\left(k\right){\overline{A}}_i^{\mathrm{T}}{G}_j{\overline{B}}_{i}v\left(k\right)+v^{\mathrm{T}}\left(k\right){\overline{B}}_i^{\mathrm{T}}{G}_j{A}_i\overline{x}\left(k\right)\right)\hfill \\ \hfill & +v^{\mathrm{T}}\left(k\right){\overline{B}}_i^{\mathrm{T}}{G}_j{\overline{B}}_{i}v\left(k\right).\hfill \end{array}$$

(53)

According to the closed-loop dynamic supply chain system (26)–(28) and inequality (48), for a constant $\gamma >0$, let

$$\begin{array}{c}\hfill W\left(\overline{z}\left(k\right),v\left(k\right)\right)=\gamma v^{\mathrm{T}}\left(k\right)v\left(k\right)-{\gamma }^{-1}{\overline{x}}^{\mathrm{T}}\left(k\right){\overline{C}}_i^{\mathrm{T}}{\overline{C}}_i\overline{x}\left(k\right).\end{array}$$

(54)

From Equations (53) and (54), it can be concluded that

$$\Delta V-W={\left[\begin{array}{c}\overline{x}\left(k\right)\\ v\left(k\right)\end{array}\right]}^{\mathrm{T}}\left[\begin{array}{cc}{\overline{A}}_i^{\mathrm{T}}{G}_j{\overline{A}}_i-G_i+{\gamma }^{-1}{\overline{C}}_i^{\mathrm{T}}{\overline{C}}_i& {\overline{A}}_i^{\mathrm{T}}{G}_j{\overline{B}}_i\\ {\overline{B}}_i^{\mathrm{T}}{G}_j{\overline{A}}_i& {\overline{B}}_i^{\mathrm{T}}{G}_j{\overline{B}}_i-\gamma I\end{array}\right]\left[\begin{array}{c}\overline{x}\left(k\right)\\ v\left(k\right)\end{array}\right].$$

According to inequality (49), it can be inferred that

$$\begin{array}{c}\hfill \Delta V\left(k,\overline{x}\left(k\right)\right)-W\left(\overline{z}\left(k\right),v\left(k\right)\right)<0.\end{array}$$

(55)

To make the closed-loop dynamic supply chain system (26)–(28) satisfy the ${\mathcal{H}}^{\infty }$ control, according to condition (1) in Definition 3, it is necessary to prove that the following equation holds:

$$\begin{array}{c}\hfill \sum _{k=0}^{\infty }{\overline{z}}^{\mathrm{T}}\left(k\right)\overline{z}\left(k\right)-{\gamma }^2\sum _{k=0}^{\infty }{v}^{\mathrm{T}}\left(k\right)v\left(k\right)<0.\end{array}$$

(56)

When the initial state of the closed-loop dynamic supply chain system (26)–(28) is $\overline{x}\left(0\right)=0$, the iteration of inequality (55) yields

$$\begin{array}{c}\hfill \sum _{k=0}^{\infty }\Delta V\left(k,\overline{x}\left(k\right)\right)-\sum _{k=0}^{\infty }W\left(\overline{z}\left(k\right),v\left(k\right)\right)=V(\infty )-\sum _{k=0}^{\infty }\gamma v^{\mathrm{T}}\left(k\right)v\left(k\right)+\sum _{k=0}^{\infty }{\gamma }^{-1}{\overline{z}}^{\mathrm{T}}\left(k\right)\overline{z}\left(k\right)<0.\end{array}$$

(57)

According to the condition $V(\infty )>0$, so there is

$$\begin{array}{c}\hfill \sum _{k=0}^{\infty }\gamma v^{\mathrm{T}}\left(k\right)v\left(k\right)-\sum _{k=0}^{\infty }{\gamma }^{-1}{\overline{z}}^{\mathrm{T}}\left(k\right)\overline{z}\left(k\right)>0.\end{array}$$

(58)

The multiplication of the left and right sides of the above inequality by $\gamma$ proves that inequality (56) holds. The closed-loop system (28) satisfies condition (1) in Definition 3.

When the interference $v\left(k\right)\equiv 0$ in the closed-loop dynamic supply chain system (26)–(28), it can be obtained from Equation (54) that

$$\begin{array}{c}\hfill W\left(\overline{z}\left(k\right),v\left(k\right)\right)=-{\gamma }^{-1}\overline{x}{\left(k\right)}^{\mathrm{T}}{\overline{C}}_i^{\mathrm{T}}{\overline{C}}_i\overline{x}\left(k\right).\end{array}$$

Due to the constant $\gamma >0$, according to inequality (55), it can be inferred that $\Delta V\left(k,\overline{x}\left(k\right)\right)<0$. Therefore, when $v\left(k\right)\equiv 0$, according to Lyapunov stability theory, system (28) is asymptotically stable; that is, condition (2) in Definition 3 holds. □

Theorem 5. 

For the dynamic supply chain system (26)–(28), if there is a positive definite matrix $P_i$ and a matrix $K_i$ that satisfy the conditions,

$$\begin{array}{c}\hfill P_j^{-1}-{\displaystyle \frac{1}{{\gamma }^2}}{\overline{B}}_i{\overline{B}}_i^{\mathrm{T}}>0,\end{array}$$

(59)

$$\begin{array}{c}\hfill {\left(\overline{A}+\overline{B}{K}_{i}L\right)}^{\mathrm{T}}{\left(P_j^{-1}-{\displaystyle \frac{1}{{\gamma }^2}}{\overline{B}}_i{\overline{B}}_i^{\mathrm{T}}\right)}^{-1}\left(\overline{A}+\overline{B}{K}_{i}L\right)-P_i+{\overline{C}}_i^{\mathrm{T}}{\overline{C}}_i<0,\end{array}$$

(60)

for $\forall i,j\in \mathcal{S}$, then the closed-loop dynamic supply chain system (26)–(28) is robust and stable for any switching $\sigma \left(k\right)$ under the action of the dynamic output feedback regulation signal (18), and has an ${\mathcal{H}}^{\infty }$ disturbance attenuation degree γ.

Proof. 

Firstly, we demonstrate the robust stability of the dynamic supply chain system (26)–(28). The Lyapunov function of the dynamic supply chain system (26)–(28) is taken as $V\left(\overline{x}\left(k\right)\right)={\overline{x}}^{\mathrm{T}}\left(k\right)P_{\sigma \left(k\right)}\overline{x}\left(k\right)$. In the proof process, it can be assumed that $\sigma \left(k\right)=i$ and $\sigma (k+1)=j$. To make the internal robust stability of the dynamic supply chain system (26)–(28), it is required that the difference of $V\left(\overline{x}\left(k\right)\right)$ is negatively definite when $v\left(k\right)=0$; that is, $\forall i,j\in \mathcal{S}$ satisfies

$$\begin{array}{c}\hfill \Delta V\left(\overline{x}\left(k\right)\right)={\overline{x}}^{\mathrm{T}}\left(k\right)\left({\left(\overline{A}+\overline{B}{K}_{i}L\right)}^{\mathrm{T}}{P}_j\left(\overline{A}+\overline{B}{K}_{i}L\right)-P_i\right)\overline{x}\left(k\right)<0.\end{array}$$

(61)

From inequality (59), it can be inferred that $P_j^{-1}-{\displaystyle \frac{1}{{\gamma }^2}}{\overline{B}}_i{\overline{B}}_i^{\mathrm{T}}>0$, and from inequality (60), it can be inferred that

$$\begin{array}{c}\hfill {\left(\overline{A}+\overline{B}{K}_{i}L\right)}^{\mathrm{T}}{\left(P_j^{-1}-{\displaystyle \frac{1}{{\gamma }^2}}{\overline{B}}_i{\overline{B}}_i^{\mathrm{T}}\right)}^{-1}\left(\overline{A}+\overline{B}{K}_{i}L\right)-P_i+{\overline{C}}_i^{\mathrm{T}}{\overline{C}}_i<0.\end{array}$$

(62)

According to matrix inverse Lemma 3, there is

$$\begin{array}{c}\hfill {\left(P_j^{-1}-{\displaystyle \frac{1}{{\gamma }^2}}{\overline{B}}_i{\overline{B}}_i^{\mathrm{T}}\right)}^{-1}=P_j-P_j{\overline{B}}_i{\left({\overline{B}}_i^{\mathrm{T}}{P}_j{\overline{B}}_i-{\gamma }^{2}I\right)}^{-1}{\overline{B}}_i^{\top }{P}_j.\end{array}$$

(63)

Substituting into inequality (62) yields

$$\begin{array}{cc}\hfill & {\left(\overline{A}+\overline{B}{K}_{i}L\right)}^{\mathrm{T}}{P}_j\left(\overline{A}+\overline{B}{K}_{i}L\right)-P_i+{\overline{C}}_i^{\mathrm{T}}{\overline{C}}_i-{\left(\overline{A}+\overline{B}{K}_{i}L\right)}^{\mathrm{T}}{P}_j{\overline{B}}_i{\left({\overline{B}}_i^{\mathrm{T}}{P}_j{\overline{B}}_i-{\gamma }^{2}I\right)}^{-1}{\overline{B}}_i^{\top }{P}_j\left(\overline{A}+\overline{B}{K}_{i}L\right)<0.\hfill \end{array}$$

(64)

From inequality (59), $P_j^{-1}-{\displaystyle \frac{1}{{\gamma }^2}}{\overline{B}}_i{\overline{B}}_i^{\mathrm{T}}>0$ can be obtained. If Schur’s lemma is applied again and ${\overline{B}}_i^T{P}_j{\overline{B}}_i-{\gamma }^{2}I<0$ is obtained, then from inequality (64), there is

$$\begin{array}{c}\hfill {\left(\overline{A}+\overline{B}{K}_{i}L\right)}^{\mathrm{T}}{P}_j\left(\overline{A}+\overline{B}{K}_{i}L\right)-P_i<0.\end{array}$$

(65)

That is to say, $\Delta V\left(\overline{x}\left(k\right)\right)<0$, the dynamic supply chain system (26)–(28) is robust and stable.

Then, it is proved that the dynamic supply chain system (26)–(28) has an ${\mathcal{H}}^{\infty }$ disturbance attenuation degree $\gamma$ under the action of any switching law $\sigma \left(k\right)$, which satisfies $J=\parallel \overline{z}{\left(k\right)\parallel }_2^2-{\gamma }^2{\parallel v\left(k\right)\parallel }_2^2<0$. Let $J\left(k\right)={\overline{z}}^{\mathrm{T}}\left(k\right)\overline{z}\left(k\right)-{\gamma }^2{v}^{\mathrm{T}}\left(k\right)v\left(k\right)$, there is

$$\begin{array}{c}\hfill J\left(k\right)={\overline{z}}^{\mathrm{T}}\left(k\right)\overline{z}\left(k\right)-{\gamma }^2{v}^{\mathrm{T}}\left(k\right)v\left(k\right)+\Delta V\left(x\left(k\right)\right)-\Delta V\left(x\left(k\right)\right).\end{array}$$

(66)

Let ${\overline{A}}_i=\overline{A}+\overline{B}{K}_{i}L$, there is

$$J\left(k\right)={\left[\begin{array}{c}\overline{x}\left(k\right)\\ v\left(k\right)\end{array}\right]}^{\top }\left[\begin{array}{cc}{\overline{A}}_i^{\top }{P}_j{\overline{A}}_i-P_i+{\overline{C}}_i^{\top }{\overline{C}}_i& {\overline{A}}_i^{\top }{P}_j{\overline{B}}_i\\ {\overline{B}}_i^{\top }{P}_j{\overline{A}}_i& {\overline{B}}_i^{\top }{P}_j{\overline{B}}_i-{\gamma }^{2}I\end{array}\right]\left[\begin{array}{c}\overline{x}\left(k\right)\\ v\left(k\right)\end{array}\right]-\Delta V\left(\overline{x}\left(k\right)\right).$$

According to the inequality (64), there is

$$\begin{array}{c}\hfill {\overline{A}}_i^{\mathrm{T}}{P}_j{\overline{A}}_i-P_i+{\overline{C}}_i^{\mathrm{T}}{\overline{C}}_i-{\overline{A}}_i^{\mathrm{T}}{P}_j{\overline{B}}_i{\left({\overline{B}}_i^{\mathrm{T}}{P}_j{\overline{B}}_i-{\gamma }^{2}I\right)}^{-1}{\overline{B}}_i^{\top }{P}_j{\overline{A}}_i<0.\end{array}$$

(67)

According to Schur’s complement lemma, the inequality (67) can be rewritten as

$$\begin{array}{c}\hfill \left[\begin{array}{cc}{\overline{A}}_i^{\mathrm{T}}{P}_j{\overline{A}}_i-P_i+{\overline{C}}_i^{\mathrm{T}}{\overline{C}}_i& {\overline{A}}_i^{\mathrm{T}}{P}_j{\overline{B}}_i\\ {\overline{B}}_i^{\top }{P}_j{\overline{A}}_i& {\overline{B}}_i^{\mathrm{T}}{P}_j{\overline{B}}_i-{\gamma }^{2}I\end{array}\right]<0.\end{array}$$

(68)

Under zero initial conditions, $V\left(\overline{x}\left(0\right)\right)=0$, then $J=\parallel \overline{z}{\left(k\right)\parallel }_2^2-{\gamma }^2{\parallel v\left(k\right)\parallel }_2^2=\sum _{z=0}^{\infty }J\left(k\right)<0$, Theorem 5 holds. □

Theorem 5 can be transformed into a directly solvable LMI problem by replacing variables and applying Schur’s lemma multiple times.

Theorem 6. 

For dynamic supply chain systems (26)–(28), given a constant $\gamma >0$, if there is a positive definite matrix $Q_i$ and any matrix $Y_i$, the following LMI

$$\begin{array}{c}\hfill \left[\begin{array}{cccc}-Q_j& 0& \left(\overline{A}{Q}_i+\overline{B}{Y}_i\right)& {\displaystyle \frac{1}{\gamma }}{\overline{B}}_i\\ 0& -I& {\overline{C}}_i{Q}_i& 0\\ {\left(\overline{A}{Q}_i+\overline{B}{Y}_i\right)}^{\mathrm{T}}& {\left({\overline{C}}_i{Q}_i\right)}^{\mathrm{T}}& -Q_i& 0\\ {\displaystyle \frac{1}{\gamma }}{\overline{B}}_i^T& 0& 0& -I\end{array}\right]<0\forall i,j\in \mathcal{S}\end{array}$$

(69)

holds. Then, there exists an adjustment gain $K_i=-Y_i{Q}_i^{-1}$, and under the action of the dynamic output feedback adjustment signal (18), the closed-loop dynamic supply chain system (26)–(28) is robustly stable and has an ${\mathcal{H}}^{\infty }$ disturbance attenuation degree γ for any switching $\sigma \left(k\right)$.

Proof. 

Referring to the result of Theorem 5, further matrix transformation is performed, and the result is obvious. □

Remark 5. 

This article mainly studies the scheduling of the core enterprise shown in Figure 1. If readers understand the main work of this article, the scheduling of original manufacturing enterprises or consumer terminal enterprises can also be easily solved. In order to save space, we will not elaborate on it again. It is easy to generalize the conclusion of this article to such situations.

Remark 6. 

This article introduces the values of a community with a shared future for mankind into the sustainable operation and management of dynamic supply chains for the first time, which is the characteristic and innovation of this work. However, the main scientific tools and technical methods used in this article are artificial intelligence optimal information fusion and control theory. Therefore, mathematical description and theoretical analysis are inevitable.

5. Empirical Research

To verify Theorem 1 in this article, which is a sufficient condition for the ${\mathcal{H}}^{\infty }$ solvability of dynamic supply chain, we conducted data analysis using Chongqing Hongjiu Fruit Co., Ltd. as an example. Chongqing Hongjiu Fruit Co., Ltd. is a limited company with fruit as its main business. It is a multi-brand fresh fruit group listed on the main board of the Hong Kong Stock Exchange, focusing on the full industry chain operation of imported fruits and high-quality domestic fruits. Through an “end-to-end” advanced digital supply chain, it directly delivers delicious and high-quality fresh fruit products from global orchards to national fruit retail terminals. It adopts an “orchard distributor retail terminal” model. Through preliminary market research and literature review, its operational parameters are

$${\alpha }_1=0.0346,{\alpha }_2=0.0172,{\beta }_1=0.5694,{\beta }_2=0.9851.$$

It should be considered that enterprise orders are controlled by two modes of switching, namely imported fruits and high-quality domestic fruits, with interference suppression set as

$${\gamma }_1=0.1280,{\gamma }_2=0.2452.$$

Let the positive definite matrix be as follows:

$$\mathit{Q}=\mathrm{diag}\left\{\left[\begin{array}{ccc}0.4545& 0.5215& -0.0703\\ 0.5215& 2.9590& -1.1287\\ -0.0703& -1.1287& 0.5872\end{array}\right]\right\}.$$

By using Theorem 1 for simple calculations, the following can be concluded:

$$max\left(\lambda \left({\mathit{U}}_1\right)\right)=-9.8989,max\left(\lambda \left({\mathit{U}}_2\right)\right)=-8.5223.$$

It can be seen that the eigenvalues are far from the imaginary axis and have a large negative real part, which also indicates that the algorithm proposed in this paper can be fully applied to the operation and management of Chongqing Hongjiu Fruit Co., Ltd. Taking the 2019 enterprise operation data as an example, its dynamic response is shown in Figure 4. It can be seen that the data results perfectly match the theoretical analysis.

To verify Theorems 2 and 3 and Corollaries 1 and 2 of this article, namely the ${\mathcal{H}}^{\infty }$ robustness of the dynamic supply chain, we conducted data analysis using Wuhan Golden Orchard Trading Co., Ltd. as an example. Wuhan Golden Orchard Trade Co., Ltd. is an integrated service provider focusing on fruit planting, wholesale, and distribution. It has its own direct sales department, nearly 10,000 mu of green fruit planting base in the Chinese mainland, and has established a long-term stable direct procurement system with overseas fruit bases. Wuhan Golden Orchard Trading Co., Ltd. mainly produces imported fruits, domestically produced high-quality fruits, and independently developed Hezhong Organic Seedless Black Tea (referred to as Hezhong Black Tea). Through preliminary market research and literature review, its operational parameters are

$${\alpha }_1=0.0215,{\alpha }_2=0.0287,{\beta }_1=0.4572,{\beta }_2=0.8964.$$

Considering that enterprise orders are controlled by two modes of switching, namely high-quality imported fruit (black prickly durian) and high-quality domestic fruit (Xinjiang Korla fragrant pear), the interference suppression is as follows:

$${\gamma }_1=0.0128,{\gamma }_2=0.0253,\gamma =max\{{\gamma }_1,{\gamma }_2\}.$$

Let the positive definite matrices be as follows:

$$P_1=\mathrm{diag}\left\{\left[\begin{array}{cc}0.8432\hfill & 0.0427\hfill \\ 0.0427\hfill & 0.9256\hfill \end{array}\right]\right\},P_2=\mathrm{diag}\left\{\left[\begin{array}{cc}0.7828\hfill & -0.4446\hfill \\ -0.4446\hfill & 0.8564\hfill \end{array}\right]\right\}.$$

Based on the simple calculation of Corollary 2, the following can be concluded:

$$max\left(\lambda \left(U_{1,i}\right)\right)=-8.1425,max\left(\lambda \left(U_{2,i}\right)\right)=-7.4668.$$

It can be seen that the eigenvalues are far from the imaginary axis and have a large negative real part, which also indicates that the algorithm proposed in this paper can be fully applied to the operation and management of Wuhan Golden Orchard Trading Co., Ltd. Taking the 2021 enterprise operation data as an example, its dynamic response is shown in Figure 5. It can be seen that the data results perfectly match the theoretical analysis.

To verify Theorems 4–6, which are the ${\mathcal{H}}^{\infty }$ control criteria for a dynamic supply chain, we conducted data analysis using Hangzhou Xianfeng Fruit Co., Ltd. as an example. Hangzhou Xianfeng Fruit Co., Ltd. is a global enterprise integrating new retail, smart cold chain logistics, and supply chain B2B platform, and is one of the largest fruit chain enterprises in the Chinese mainland. The company has successively won honors such as the National Key Leading Enterprise in Agricultural Industrialization, the Zhejiang Province Agricultural Leading Enterprise, and the China Famous Trademark. At present, Hangzhou Xianfeng Fruit Co., Ltd. has 23 modern cold-chain storage centers with a total area of 480,000 square meters. More than 200 product experts are stationed in more than 300 planting bases around the world, monitoring the planting and picking process to achieve supply chain stability and reliability. Hangzhou Xianfeng Fruit Co., Ltd. actively promotes industry standardization and refinement through its own standardized development. It establishes a stable interest linkage mechanism through the “Base + Farmers” model upstream of the industry, directly driving 23,000 farmers to start businesses. In 2018, the cumulative purchase amount was CNY 1681 million, driving fruit farmers to increase their income by CNY 46 million. Through standardized and branded operations, it has driven fruit farmers to achieve an income increase of over CNY 1 billion. In the downstream of the industry, through the chain franchise model, we aim to deepen mass entrepreneurship and innovation, drive innovation and entrepreneurship among social entities, solve employment problems for more than 10,000 people, and actively build a globally leading digital fruit ecosystem. Through preliminary market research and literature review, its operational parameters are

$${\alpha }_1=0.0346,{\alpha }_2=0.0172,{\beta }_1=0.5694,{\beta }_2=0.9851.$$

It should be considered that enterprise orders are controlled by two modes of switching, namely high-quality imported fruits and high-quality domestic fruits, with interference suppression set as

$${\gamma }_1=0.1280,{\gamma }_2=0.2452.$$

Through Theorem 6, a simple calculation can be obtained:

$$\begin{array}{c}\hfill K_1=\left[-0.3594,0.3497\right],K_2=\left[-0.1877,0.4186\right].\end{array}$$

(70)

Taking the 2023 enterprise operation data as an example, combined with the adjustment gain (70) and dynamic output feedback adjustment information (18), the dynamic response is shown in Figure 6, which also indicates that the algorithm proposed in this paper can be fully applied to the operation management of Hangzhou Xianfeng Fruit Co., Ltd. It can be seen that the data results perfectly match the theoretical analysis.

Remark 7. 

The manuscript has provided a detailed introduction to the core characteristics and basic information of three empirical research companies. However, considering the confidentiality of business operations, the companies request to conceal their names, hoping that readers can understand. If readers are interested in this, they are welcome to discuss the matter with the authors.

6. Conclusions and Inspiration

6.1. Key Findings

To summarize the main work of this article, firstly, we established a dynamic supply chain model in Section 2. Secondly, we designed the optimal information fusion mechanism for artificial intelligence in Section 3. Thirdly, we introduced an order-switching mechanism in the dynamic supply chain model in Section 4.1 of the manuscript. Finally, sustainable operation and management mechanisms for dynamic supply chains are presented in Section 4.2, Section 4.3 and Section 4.4.

This article established a composite dynamic supply chain network model, which is a good addition to the development of composite dynamic supply chain network models. On the one hand, this article considered the raw material inventory and finished product inventory of enterprises hierarchically. It did not assume that enterprises only have a single inventory, nor did it simply assume that the raw material inventory and finished product inventory have a simple algebraic superposition relationship. This has broad guiding significance for the operation and production of agricultural product enterprises. On the other hand, current large enterprises can often process products, which is fully reflected in the resilience and superiority of such enterprises when the COVID-19 pandemic occurred. At the same time, many agricultural enterprises now operate in groups and possess such capabilities [39], such as COFCO Corporation Co., Ltd. (Beijing, China), New Hope Group Co., Ltd. (Chengdu, China), Beidahuang Group Co., Ltd Ltd. (Harbin, China), and so on.

This article developed a dynamic supply chain operation and management mechanism based on the optimal information fusion processing mechanism of artificial intelligence within the framework of a community with a shared future for mankind. On the one hand, the values of a community with a shared future for mankind emphasize the interdependent concept of common interests. This article also fully explored the information flow and logistics of the “upstream enterprise–core enterprise–downstream enterprise” model, grasping the common values and interests of relevant enterprises in an industrial chain and designing static monitoring information of upstream enterprises towards core enterprises, core enterprises themselves, and downstream enterprises towards core enterprises. On the other hand, the values of a community with a shared future for mankind emphasize the concept of sustainable development. This article fully explored the dynamic characteristics of enterprise supply chains in the current social context, focusing on green production and sustainable development of enterprises and designing dynamic monitoring information and estimators that can achieve real-time monitoring and operation management of the supply chain, enhancing the robustness of control algorithms. Thirdly, the values of a community with a shared future for mankind emphasize the concept of global governance. Inspired by this, we introduced this concept into the operation and management of dynamic supply chains, emphasizing the global operation concept between core enterprises and upstream enterprises and between core enterprises and downstream enterprises. We also established the optimal information fusion mechanism for dynamic monitoring information systems, injecting new ideas and concepts into enterprise operation and management, and opening up a new situation for the development of human society.

This article considered the situation of order switching. On the one hand, enterprises can set switching thresholds based on different factors (such as their own economic situation, seasonal weather changes, economic policy changes, etc.). Enterprises can adjust purchase and supply orders in real-time to improve the robustness of their operations. On the other hand, order interruption caused by force majeure natural disasters (such as the COVID-19 epidemic) can be avoided, which will cause great losses to enterprises. We can formulate and adjust the operation strategy as soon as possible according to the real-time dynamic information of the supply chain. The third aspect is that “ordering less may not be enough to meet the needs of consumers, while ordering more may bring huge losses to themselves” has always been an open international challenge in the operation and management of agricultural product enterprises. This article provides a preliminary quantitative analysis basis for this challenge, and we can further consider various factors to model order switching clearly, pointing out the direction for subsequent research on dynamic supply chains.

This article investigated the ${\mathcal{H}}^{\infty }$ control of a dynamic supply chain network composed of multiple independent order-switching subsystems. On the one hand, this article discussed the sufficient conditions for the ${\mathcal{H}}^{\infty }$ solvability of dynamic supply chain networks. In each subsystem, it was ${\mathcal{H}}^{\infty }$-solvable, and a sufficient condition was given for the dynamic supply chain network to be ${\mathcal{H}}^{\infty }$-solvable under any switching action; that is, the Riccati inequality of each subsystem has a common positive definite solution. Although the ${\mathcal{H}}^{\infty }$ gain of the dynamic supply chain network is greater than or equal to the ${\mathcal{H}}^{\infty }$ gain of each subsystem, the results are also effective. On the other hand, this article investigated the ${\mathcal{H}}^{\infty }$ performance analysis problem of dynamic supply chain networks. Under arbitrary switching conditions, the robust stability and disturbance suppression performance of dynamic supply chain systems were analyzed using the common quadratic Lyapunov function method and the piecewise quadratic Lyapunov function method, respectively. Thirdly, this article investigated the ${\mathcal{H}}^{\infty }$ control problem of dynamic supply chain networks. In the case of arbitrary switching laws, the robust stability and disturbance suppression performance of a switching dynamic supply chain network with supply chain scheduling were analyzed using the multiple Lyapunov function method. Based on the LMI-solving algorithm, a dynamic supply chain scheduling scheme was proposed. The main conclusions obtained are applicable to the analysis and design of robust ${\mathcal{H}}^{\infty }$ feedback control for order-switching dynamic supply chain systems.

6.2. Policy Recommendations

Starting from the perspective of optimal information fusion in artificial intelligence and within the framework of a community with a shared future for mankind, this article conducted quantitative research and in-depth exploration of the operation and management of dynamic supply chains. The empirical research of Section 5 conclusions indicate that the operational efficiency of enterprises can be fully improved. This article is the first to introduce the values of a community with a shared future for mankind into the sustainable operation and management process of dynamic supply chains. This was the core idea of this article, and also the characteristic and innovation of its work. The main scientific tools and technical methods used in this article were artificial intelligence optimal information fusion and control theory. Therefore, our first policy suggestion is to manage dynamic supply chains by introducing modern information science. The community with a shared future for mankind includes the development and extension of Marxist theory. Therefore, our second policy suggestion is to apply Marxist principles to guide the operation and management of dynamic supply chains. The community with a shared future for mankind was first proposed by China and is also the current concept of peaceful development in China. Therefore, the third policy suggestion provided in this article is to apply China’s current development concept to guide the operation and management of dynamic supply chains. Specifically, this article proposes the following policy recommendations:

Firstly, we can manage dynamic supply chains by introducing modern information science. For example, real-time monitoring and early warning. This article used real-time dynamic monitoring information technology to track key indicators of various links in the supply chain, such as inventory levels and order status. The order status here refers to the order quantity and purchase quantity of the core enterprise considered in this article, logistics progress, etc., and sets up early warning mechanisms to detect anomalies promptly. For the dynamic adjustment of demand forecasting, enterprises continuously update demand forecasting models based on new data and market changes, improve prediction accuracy, and adjust production and procurement plans accordingly. This is the basis for establishing order switching in this article. Agile production requires enterprises to have the ability to quickly adjust production processes, capacity, and product combinations to cope with market demand fluctuations and changes. This is the starting point and landing point for establishing dynamic monitoring information in this article. The organic collaboration between upstream and downstream enterprises is fully reflected in the optimal information fusion mechanism of artificial intelligence in this article, which involves close cooperation, information sharing, collaborative response to supply fluctuations and risks, and joint development of emergency plans. Dynamic inventory management adopts dynamic inventory strategies, such as dynamic setting of safety stock, supplier managed inventory, etc., and flexibly adjusts inventory levels according to actual situations. In terms of logistics optimization, real-time optimization of logistics paths, transportation methods, etc., should be introduced to improve logistics efficiency and respond to changes. Regular supply chain risk assessments should be conducted to develop targeted response measures to prevent and resolve risks and improve the robustness of the supply chain network. Multiple emergency plans should be developed in advance, including response plans for different emergencies, to ensure that the supply chain can quickly recover in abnormal situations and improve the resilience of the supply chain network. Data-driven decision-making, relying on big data analysis and intelligent algorithms to assist decision-making, makes supply chain decision-making more scientific, reasonable, and timely. As a result of cross-departmental collaboration, the raw materials and cost inventory considered in this article are fully reflected, promoting efficient collaboration and information sharing among departments such as procurement, production, sales, logistics, etc., in order to expect various links of the enterprise to respond quickly to changes.

Secondly, we can use Marxist principles to guide the operation and management of dynamic supply chains. From the research results of this article, it can be seen that there is a high degree of correlation and fit between dynamic supply chain and Marxist theory. From the perspective of dialectical materialism, both emphasize change and development. Marxism emphasizes that things are constantly evolving and changing, which is in line with the need for dynamic supply chains to constantly adapt to market, technological, and other dynamic changes. Both require a developmental perspective to view problems and respond to challenges. Both have a relationship between material and consciousness, and the construction and operation of dynamic supply chains are based on the understanding and grasp of material factors such as actual market demand and resources, as well as the guidance of correct concepts and strategies at the level of consciousness. From the perspective of social reproduction theory, there exists a circular relationship of production, distribution, exchange, and consumption. The dynamic supply chain involves product production, distribution, exchange, and other links. There is a certain connection between the dynamic supply chain and Marxist theory on the process of social reproduction, both focusing on the systematic and coherent nature of economic activities. However, it should be pointed out that Marxist theory mainly profoundly reveals the overall laws and essence of socio-economic development, while dynamic supply chain is a specific concept in modern management and economics. They are not directly one-to-one correspondences, but the basic principles and methodology of Marxism can provide a macro perspective and analytical framework for analyzing and understanding the operation and development of dynamic supply chains in the socio-economic context.

Finally, we can use China’s current development philosophy to guide the operation and management of dynamic supply chains. From the research results of this article, it can be seen that there is a high degree of correlation and compatibility between dynamic supply chains and 654 China’s current development concepts. The series of governance and development concepts currently proposed by China include the new development concept and the community with a shared future for mankind. On the one hand, in the economic field, emphasizing innovative development and coordinated development is the main theme of China’s current socio-economic development. A dynamic supply chain reflects flexible adaptation and dynamic adjustment to market changes and various factors, which is in line with the concept of innovative development. By continuously innovating and optimizing supply chain management models, efficiency and competitiveness can be improved. At the same time, building an efficient dynamic supply chain also helps to promote coordinated economic development, and strengthen collaboration between different industries and regions, which also meets the requirements of coordinated development. On the other hand, under the advocacy of building a community with a shared future for mankind, dynamic supply chains also play a positive role in promoting international economic cooperation and exchanges, reflecting the concept of open and shared development, and helping to achieve common development and prosperity. It should be pointed out that this correlation is a reflection and response at a more macro level of ideas and development requirements.