*Article* **Identification and Analysis of Factors Influencing Green Growth of Manufacturing Enterprises Based on DEMATEL Method—Wooden Flooring Manufacturing Companies as a Case**

**Wei Li and Xia Wu \***

College of Applied Science and Technology, Beijing Union University, Beijing 100012, China **\*** Correspondence: yykjtwuxia@buu.edu.cn

**Abstract:** It is significant to scientifically identify what factors influence the green growth of manufacturing enterprises and analyze the relationship among these factors, thus promoting green growth. Firstly, the corresponding conceptual model is designed; then, the DEMATEL method and steps used to identify the influencing factors are introduced; finally, the DEMATEL method is adopted to empirically analyze wooden flooring manufacturing companies so as to identify influencing factors of their green growth. According to the results, there are six reason factors, namely environmental standard constraints, green market demand, market competition, green technology advancement, upstream and downstream synergy of green industrial chain, and policy support, which provide the most important external support to enterprises' green growth and main driving power to wooden flooring manufacturing ones.

**Keywords:** green growth; influencing factor; factor identification; DEMATEL method; manufacturing enterprises

#### **1. Introduction**

Crucial to the industrial economy, manufacturing is a typical and fundamental symbol of comprehensive national strength and international status. Over the past 40 years since the reform and opening-up of China, China's manufacturing industry has made great progress, now with more than 30% of the world's total manufacturing output, a comprehensive industrial system with all sectors, and a complete industrial chain [1], However, despite the huge volume, China is still not a manufacturing powerhouse [2] subject to scale expansion [3]; therefore, transformation is urgent to upgrade China's manufacturing industry.

To pursue a higher quality of industrial development, green growth has become an important symbol [4]. In 2015, the Chinese government officially put forward the "Made in China 2025" Initiative, in which green growth was a guideline for strategic implementation, emphasizing that sustainable economic development should be coordinated with the natural environment towards fully green manufacturing [5]. In particular, manufacturing enterprises are required to shift from a traditional development model that was at the expense of the environment to foster green transformation and upgrading to realize green growth [6,7].

"Green growth" refers to the process that manufacturing enterprises grow stronger through green strategies and green behaviors, fewer pollutant residues, less consumption of resources and energy, and more environmentally friendly, safe, and healthy products, together with ever-increasing green competitiveness. In particular, the leading concept is green development throughout the production and management practices of manufacturing enterprises relying on relevant technological and management innovations, featured by less environmental pollution and higher resource efficiency [8,9].

**Citation:** Li, W.; Wu, X. Identification and Analysis of Factors Influencing Green Growth of Manufacturing Enterprises Based on DEMATEL Method—Wooden Flooring Manufacturing Companies as a Case. *Processes* **2022**, *10*, 2594. https:// doi.org/10.3390/pr10122594

Academic Editors: Hideki Kita and Jochen Strube

Received: 9 October 2022 Accepted: 22 November 2022 Published: 5 December 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

terprises'

It is clear that green growth essentially emphasizes sustainable development to realize both economic growth and environmental protection [10]. Enterprise behaviors should follow the basic green premise and corresponding requirements under green constraints, which is necessary for their survival and development. " "

Such green growth of manufacturing enterprises is a "systematic project" with a wide range, rich contents, and many influencing factors [11]. As a result, it is pragmatic to identify and refine these key factors.

This paper aims to effectively identify the influencing factors of manufacturing enterprises' green growth, then further define and utilize key ones to promote green growth. Major contributions made in this paper include a conceptual model of the factors influencing the green growth of manufacturing enterprises and a method to further explore the relevant dynamic mechanisms and key influencing factors identified through the DE-MATEL method and verify the effectiveness of the DEMATEL method. Finally, this paper concludes with six verified key influencing factors. In particular, the introduction of the "industrial chain" factor and the proposed "collaboration within the industrial chain" factor have improved comprehensiveness in analyzing green growth dynamics and provided more perspectives apart from previous individual enterprises. " " " "

#### **2. The Research Objectives and Method**

#### *2.1. Setting up a Conceptual Model of Influencing Factors*

Enterprise management pursues more values and shareholders' maximum benefits. That is, green behaviors cannot sustain without more business values or better financial performance [12]; therefore, based on rational human assumption, manufacturing enterprises driven by their interests (including short-term and long-term gains) will be more willing to adopt green behaviors with various "green motivation" factors that add values. In this sense, effective identification of external factors that can improve such "green behavior willingness" [13] is necessary; thus, a conceptual model of influencing factors is designed as shown in Figure 1. ' " " " "

**Figure 1.** Conceptual model of the main factors influencing green growth.

The model logic is that due to external influencing factors, manufacturing enterprises have a stronger willingness to green behaviors driven by their interests. Then these green behaviors realize green benefits and green growth in different ways. That is, "external influencing factors → manufacturing enterprises → stronger willingness to green behaviors → green behaviors → more revenues → green growth."

Here are definitions of several relevant concepts in Figure 1.

" → → → → → " (1) Green behavior willingness, with manufacturing enterprises as the subject. It refers to enterprises' subjective readiness for green behaviors under comprehensive influencing factors, which is crucial to connect external and internal factors.

(2) Green input, mainly including capital input and technical staff input. The former one reflects emphasis and implementation of environmental protection, while the latter is key to ensuring deliveries and green competitive advantages.

(3) Green behaviors, namely a series of environmentally friendly actions. There are mainly four categories: Firstly, green product development, that is, relying on green technologies to develop new green products with less resource or energy consumption, emissions, and pollution. Secondly, green process improvement, including clean production and end-of-pipe management, that is, better new technologies, processes, and equipment are adopted to save energy and control pollution. Next is the research and development of green technologies so as to improve green competitiveness in pollution prevention and control. Finally, stronger green management. Through building up green corporate culture, staff training, and a green management system, employees can have and implement this concept with stronger green awareness and consciously fulfill their environmental responsibilities.

(4) Green benefits, namely, more environmental and economic benefits as a result of green behaviors. Specifically, environmental benefits are obvious in saving energy and reducing consumption, and further enhancing green image indirectly. However, economic benefits refer to revenue and profit increases, along with fewer costs and expenses.

#### *2.2. Methods and Steps to Identify Influencing Factors*

This paper adopts the widely recognized DEMATEL method to identify influencing factors of manufacturing enterprises' green growth. Compared to methods such as structural equations, linear regression analysis, and system dynamics, the DEMATEL method can not only analyze the influence relationship between individual factors, but also show corresponding specific influence levels [14]. Simply, this method is powerful in simplifying intricate relationships. Firstly, the direct impact matrix is established by judging the logical relationships between factors in the system with the help of professional expertise and rich experience. The influence level of each factor on other factors and the degree of being influenced are analyzed using this matrix, thus calculating the centrality and the reason degree of each factor [15]. This helps to identify key influencing factors for system optimization decisions.

The major steps are as follows:

Firstly, selecting out influencing factors of green growth. As numerous relevant factors constrain and interact with each other, it is neither practical nor necessary to examine them one by one. Instead, "literature reference+ expert consultation" is more feasible to conduct preliminary screening and form the "alternative sets" {*f*1, *f*2, ··· , *fn*}, so as to have more sensible analysis and decisions from theoretical or practical perspectives.

Next is to determine the relationship between these factors through comparisons between each other. A panel of experts was invited to score each group from 0 and 4 according to "influence level".

Corresponding figures are shown in Table 1.

**Table 1.** Judgment basis of impact degree.


Thirdly, direct impact matrix. Based on scores, direct impact matrix A of these influencing factors can be set up as *A* = - *aij n*×*n* , *aij* representing the influence level of factor *f<sup>i</sup>* on factor *f<sup>j</sup>* .

The fourth step is to normalize the direct impact matrix so as to obtain the normalized influence matrix B: *B* = - *bij n*×*n* .

$$b\_{\rm ij} = a\_{\rm ij} \times \frac{1}{\max\_{1 \le i \le n} \left( \sum\_{j=1}^{n} a\_{\rm ij} \right)} \text{ ( $i, j = 1, 2, \dots, n$ )}\tag{1}$$

Then, the next step is to calculate the comprehensive impact matrix T according to the formula *T* = *B*(*I* − *B*) −1 , where I is the unit matrix. That is, *T* = *B*(*I* − *B*) <sup>−</sup><sup>1</sup> = [*tij*] *nn* , *tij* indicates the level of direct and indirect influence of factor *f<sup>i</sup>* on factor *f<sup>j</sup>* .

The sixth step is to calculate corresponding levels of influence and being influenced. According to the comprehensive impact matrix T, the relationship between each influencing factor is determined, specifically the influencing level *D<sup>i</sup>* and the level being influenced *F<sup>i</sup>* . The calculation formula is:

$$D\_i = \sum\_{j=1}^{n} t\_{ij}(i = 1, 2, \dots, n) \tag{2}$$

$$F\_i = \sum\_{i=1}^{n} t\_{ij}(i = 1, 2, \dots, n) \tag{3}$$

*Di* is a row-wise sum of the elements in T, which represents the comprehensive value of the influence level of *X<sup>i</sup>* on other factors.

*Fi* is a column sum of the elements in T, which represents the comprehensive value of how much *X<sup>i</sup>* is influenced by other factors.

Finally, the centrality and reason degree of each factor are calculated. The formulas for the centrality *H<sup>i</sup>* and the reason degree *J<sup>i</sup>* are as follows:

$$H\_{\mathbf{i}} = D\_{\mathbf{i}} + F\_{\mathbf{i}}(\mathbf{i} = 1, 2, \dots, n) \tag{4}$$

$$J\_i = D\_i - F\_i(i = 1, 2, \dots, n) \tag{5}$$

The centrality *H<sup>i</sup>* is the total sum of *D<sup>i</sup>* and *F<sup>i</sup>* , which indicates the position of the factor *Xi* in the system. A larger *H<sup>i</sup>* indicates that *X<sup>i</sup>* has a higher position in the system and *X<sup>i</sup>* plays a larger role [16].

The reason degree *J<sup>i</sup>* is the difference between *D<sup>i</sup>* and *F<sup>i</sup>* . It indicates how the influence is realized among influencing factors, literally whether a factor is to influence or to be influenced. If *J<sup>i</sup>* > 0, it is called a reason factor, indicating that factor *X<sup>i</sup>* has a strong influence on other factors; but if *J<sup>i</sup>* < 0, it is called a result factor, indicating that factor *X<sup>i</sup>* is strongly influenced by other factors [17].

In summary, the DEMATEL method not only helps to identify key influencing factors, but also provides a preliminary analysis of the interaction mechanism among these factors according to centrality and reason degree, thus providing a reference for exploring the green growth mechanism of manufacturing enterprises.

#### **3. Empirical Analysis of Wooden Flooring Manufacturing Enterprises as a Case**

Wooden flooring manufacturing is a typical traditional type. This industry in China grew rapidly from the 1980s, with an average production scale of 400 million m<sup>2</sup> for many years and an annual output value of nearly 100 billion RMB, making China the world's largest producer and consumer of wooden flooring [18]; however, along with rapid expansion, this industry also faces some environmental problems that cannot be ignored or yet to be fundamentally solved, especially low resource utilization, high unit energy consumption, harmful emissions, free formaldehyde residues [19]. The environmental pressure and utilization chain are shown in Figure 2.

Concerning increasing pressure on resources and the environment, the consumer demand for green wooden flooring products is growing. Faced with both challenges and opportunities brought by "greenness", enterprises urgently need to identify and grasp key influencing factors of green growth so as to empower their sustainable growth [20].

#### *3.1. The Main Factors Influencing Green Growth of Wooden Flooring Manufacturing Enterprises*

Berry and Rondinelli [21] held that government, customer, employee, and competitor pressure are driving enterprises' shift to proactive environmental management. Bansal and Roth [22] proposed competitiveness, legitimacy, and ecological responsibility as three main elements leading to the ecological responsiveness of enterprises. Zhu [23] identified through factor analysis that corporate awareness of laws and regulations, environmental strategies, supply chain pressure, market demand, and the cost of green activities are the major factors of pressure/motivation and practice of green supply chain management in companies, also pointed out that green supply chain management had become an effective means for companies to improve their competitiveness. Hao et al. [24] used the "entropy decision model" to investigate 30 enterprises and concluded that expected benefits, environmental regulations, ecological environment, cluster network characteristics, and corporate social responsibility are key factors affecting green behavior decisions of enterprises in resource-based industrial clusters. Jiang [25] found through an empirical study that demand pressure, competitive pressure, policy opportunities, demand opportunities, and competitive opportunities all contribute to green performance. Furthermore, Zeng [18] also found through an empirical study that command-and-control policy instruments in environmental regulation, international market pull in demand-pull factors, and ISO14001 certified firms in supply-side factors are all key factors influencing enterprises to engage in green innovations.

**Figure 2.** Environmental pressure from manufacturing to consumption.

Based on relevant literature and characteristics, together with many rounds of discussions by the CGE, a prepared set of factors influencing the green growth of wooden flooring manufacturing enterprises was formed as follows.

" " (1) Policies, basically government policy support and environmental standard constraints. The former mainly promotes enterprises to adopt green behaviors and implement green growth through favorable, subsidy, and incentive policies; the latter sets relevant environmental standards to constrain and regulate enterprises' behaviors [26].

' (2) Industry, such as green market demand, market competition, green technology advancement, and local support. Among them, green market demand is a necessary precondition for green growth, which is mainly reflected comprehensively by population, purchasing power, and purchasing desire. Moreover, market competition mainly focuses on the status of competition between various industrial chains, with wooden flooring manufacturing enterprises as the core. Green technology advancement is crucial to support green innovations and a decisive factor for quality "greenness" [27]. In terms of local support, namely industrial support and other "hardware and software" in the area where an enterprise is located, it provides protection to enterprises' green growth. Specifically, such support includes public service facilities, production factors' trading market, logistics' supporting network, local economy, and government–industry–university–research cooperation and innovation, along with industrial information environment.

'

"

"

(3) Industrial chain. This mainly refers to the green synergy of enterprises in the industry chain, which is represented by their cooperation level, the "green requirements" for wooden flooring manufacturing enterprises, and response levels of upstream and downstream enterprises.

(4) Green behavior willingness. It demonstrates how strong enterprises' readiness are to adopt green behaviors, which is the key link to transforming the "external factors" of green growth into "internal ones".

(5) Green input. This shows the quantity and quality of input resources and generates economic and environmental benefits through the process of "green input—green output", which is mainly reflected by the green input intensity (such as upgrading green products, manufacturing processes, production equipment, end-of-pipe pollution control, etc.) and the number of technicians.

(6) Green management level. Led by the sustainable development idea, environmental protection is integrated into the whole process of enterprise production and operation so as to control pollution, save resources, shape the green image, and finally achieve sustainable growth of enterprises embodied in "green" comprehensive management capacity. To define such a level, symbol factors mainly refer to green strategies and their implementation, higher green quality of products, and enterprises' green images.

(7) Green output. This refers to outcomes of enterprises' green governance and green R&D through green inputs; for example, the number of patent applications, especially invention patents that can reflect the comprehensive strength of enterprise scientific research, which can be used as an important indicator reflecting green outputs.

Building a Direct Impact Matrix.Firstly, the major influencing factors of wooden flooring manufacturing enterprises' green growth are named and listed in Table 2.


**Table 2.** The main factors impacting the green growth of wooden flooring manufacturing companies.

Next, a panel of nine professionals in business growth and green development was established, including four university professors, two researchers from research institutions, two senior consultants from consulting organizations, and one top executive from a wooden flooring manufacturing enterprise. They were invited to evaluate the above 14 influencing factors and score them according to Table 1, thus forming the quantitative relationship values between these factors. Each set of quantified influence relationship values indicates the direct effect of an influencing factor on another.

Finally, the experts' scores are averaged and rounded to form a direct impact matrix *A* ′ . Then *A* ′ is sent back to experts for confirmation and correction so as to gain direct impact matrix A, as shown in Table 3.


**Table 3.** The direct impact matrix A.

#### *3.2. Calculations and Results*

3.2.1. The Comprehensive Influence Matrix

Based on matrix A, the normalized influence matrix B is calculated according to Equation (1). Then, according to *T* = *B*(*I* − *B*) <sup>−</sup><sup>1</sup> = [*tij*] *nn* , the comprehensive influence matrix T is obtained, as shown in Table 4.

**Table 4.** The comprehensive influence matrix T.


3.2.2. The Levels of Influence, Being Influenced, the Reason Degree, and Centrality

According to Equations (2)–(5), the levels of influence, being influenced, reason degree, and centrality of each factor are calculated, respectively [18], as shown in Table 5.


**Table 5.** *D<sup>i</sup>* , *F<sup>i</sup>* , *H<sup>i</sup>* , *J<sup>i</sup>* of each index.

Based on Table 5, factors' positions in the plane coordinate system were marked to form a diagram of their integrated influence relationship. In this figure, centrality is the horizontal coordinate, reason degree is the vertical coordinate, the intersection of the horizontal and vertical coordinates is [k,0], and the distances from k to the maximum and minimum values of centrality are equal—see Figure 3. Table 5, factors' positions —

**Figure 3.** Schematic diagram of comprehensive impacting factors.

#### *3.3. Analysis of Results*

−

By calculating factors in comprehensive influence matrix T, the levels of influence, being influenced, reason degree, and centrality of each factor are derived. Through further analysis of the results, the following conclusions were obtained.

−

'

(1) In terms of reason degree, each factor has positive and negative values, which indicates that how each factor influences wooden flooring manufacturing enterprises' green growth is complicated. Among them, *f*1−*f*<sup>7</sup> are positive or reason factors and *f*8−*f*<sup>14</sup> are negative or result factors.

Reason factors (reason level greater than 0) are Green Market Demand *f*<sup>3</sup> > Environmental Standard Constraints *f*<sup>2</sup> > Government Policy Support *f*1, Market Competition *f*<sup>4</sup> > Upstream and Downstream Green Synergy *f*<sup>7</sup> > Green Technology Advancement *f*<sup>5</sup> > Local Support *f*<sup>6</sup> according to importance.

Result factors (reason level lower than 0) can contribute to green growth through the influence exerted by reason factors.

(2) Concerning centrality, Green Behavior Willingness *f*<sup>8</sup> demonstrates the largest value. Other influencing factors with a centrality level greater than 4 are Product Green Quality *f*12, Green Input Intensity *f*9, and Green Strategy Formulation and Implementation *f*11, which should be the focus of corporate management.

(3) According to Figure 3, Environmental Standard Constraints *f*2, Green Market Demand *f*3, Market Competition *f*4, Green Technology Advancement *f*5, Upstream and Downstream Green Synergy *f*<sup>7</sup> in the first quadrant and are Driving Factors, with the greatest influence and most critical role in promoting the green growth of wooden flooring manufacturing enterprises [28–30].

Government Policy Support *f*<sup>1</sup> and Local Support *f*<sup>6</sup> in the second quadrant are called Voluntariness, which plays a supportive role in the model. Specifically, Local Support *f*<sup>6</sup> has the lowest centrality value, indicating that this factor has little influence on wooden flooring manufacturing enterprises' green growth, which is the same as its reason level. As result, this factor can be excluded from the analysis. The reason level of Government Policy Support *f*<sup>1</sup> is higher with a certain centrality level, which will promote green growth.

Located in the fourth quadrant, Green Behavior Willingness *f*8, Green Input Intensity *f*9, Number of Technical Staff *f*10, Green Strategy Formulation and Implementation *f*11, Product Green Quality *f*12, Corporate Green Image *f*13, and Number of Patent Applications *f*<sup>14</sup> are called Core Problems. They are key elements vulnerable to other factors' influence, which are involved in different ways in enterprise production and operation to promote green growth. Among them, Green Behavior Willingness *f*<sup>8</sup> is crucial to connect enterprises' external motivating factors and internal influencing factors by transforming external motivation into internal actions.

To sum up, six factors, namely environmental standard constraints, green market demand, market competition, green technology advancement, upstream and downstream green synergy, and government policy support, work together to enhance enterprises' willingness to conduct green behaviors, then generating green benefits to promote green growth of wooden flooring manufacturing enterprises. Therefore, driven by the ultimate goal of profit maximization, the above six factors provide the most important external support for the green growth of enterprises, especially the key driving force for wooden flooring manufacturing ones.

#### **4. Conclusions**

The green growth of manufacturing enterprises is certainly affected by interactions and joint influence of multiple factors, which is very complex and challenging to analyze the corresponding relationship.

This study uses the DEMATEL method to identify factors influencing the green growth of wooden flooring manufacturing enterprises, concluding with six factors, namely environmental standard constraints, green market demand, market competition, green technology advancement, upstream and downstream green synergy, together with government policy support as reason factors. They are the most important external support for enterprises' green growth, particularly major driving factors for wooden flooring manufacturing ones.

The DEMATEL method is relatively easy to operate and can generate clear and straightforward outcomes; however, there are also limitations, such as the subjective part of experts' scores and relatively few samples. In future studies, a larger scope and more samples can be utilized to obtain more reliable data. Apart from the DEMATEL method, fuzzy sets theory and the interpretative structural modeling method (ISM) can also be used to further analyze factors influencing the green growth of manufacturing enterprises [31].

**Author Contributions:** Conceptualization, W.L. and X.W.; methodology, W.L.; software, W.L.; validation, W.L. and X.W.; formal analysis, X.W.; investigation, W.L.; re-sources, W.L.; data curation, X.W.; writing—original draft preparation, W.L.; writing—review and editing, X.W.; visualization, W.L. and X.W.; supervision, X.W.; project administration, W.L.; funding acquisition, W.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Research Project of Beijing Union University "Research on financial support for green technology innovation in manufacturing enterprises in Beijing, Tianjin and Hebei" (grant number: SK30202102).

**Data Availability Statement:** The data used to support findings in this study are available from corresponding authors upon request.

**Acknowledgments:** Appreciation to BUU for funding support. Thanks to the expert panel members for their professional ratings. Thanks to the reviewers for their rigorous review comments. We also thank the editors for their hard work in revising and typesetting.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Yanqin Wang 1,2,3, Weijian Ren 3,4, \*, Zhuoqun Liu 5 , Jing Li <sup>1</sup> and Duo Zhang 3,4**


**Abstract:** Continuous stirring tank reactors are widely used in the chemical production process, which is always accompanied by nonlinearity, time delay, and uncertainty. Considering the characteristic of the actual reaction of the continuous stirring tank reactors, the fault detection problem is studied in terms of the T-S fuzzy model. Through a fault detection filter performance analysis, the sufficient condition for the filtering error dynamics is obtained, which meets the exponential stability in the mean square sense and the given performance requirements. The design of the fault detection filter is transformed into one that settles the convex optimization issue of linear matrix inequality. Numerical analysis shows the effectiveness of this scheme.

**Keywords:** continuous stirring reactors; fault detection; T-S fuzzy model; channel fading

#### **1. Introduction**

Continuous stirring tank reactors (CSTR) are the most widely used chemical reactors in chemical production [1]. The CSTR reaction process is an important chemical production process, and the complexity and risk of its operation are determined by the nonlinearity, time delay, and uncertainty of the reaction process. With the development of chemical equipment being geared towards integration and larger scales, the importance of fault detection (FD) for the reaction process has increased and the technology used in its performance is continuously being improved [2]. The nonlinear dynamic equation of CSTR can be established according to the equilibrium formula of reaction materials. However, in the actual production process, most of the systems are uncertain nonlinear systems, and the uncertainty is represented by model error, parameter perturbation, and unknown disturbance, which increases the complexity and difficulty of FD.

As is well known, the task of FD is to check whether there is a fault in the system and to determine the time of the fault occurrence [3]. During the past several decades, the technology for detecting faults has already been widely adopted in industrial processes and has gradually become a significant method of enhancing both system security and reliability [4–11]. For linear systems, the FD issue has been discussed since the 1970s, and several applicable FD methods have been developed [12–16]. Nevertheless, numerous industrial systems exhibit inherent nonlinearity. Nonlinearity is known to be a primary factor that impacts system performance. The existence of nonlinearity raises the system complexity, which simultaneously brings significant challenges to the issue of system analysis and synthesis. Note that these problems can no longer be solved by using the former FD approaches for linear systems. So far, the problem of FD for nonlinear systems has not been discussed enough [17–19].

**Citation:** Wang, Y.; Ren, W.; Liu, Z.; Li, J.; Zhang, D. T-S Fuzzy Model-Based Fault Detection for Continuous Stirring Tank Reactor. *Processes* **2021**, *9*, 2127. https://doi.org/10.3390 /pr9122127

Received: 19 October 2021 Accepted: 19 November 2021 Published: 25 November 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

On the other hand, the fuzzy set theory has been proven to be a powerful method in dealing with nonlinear systems, and a considerable number of reports have been published on it [20,21]. More particularly, a substantial amount of attention has been paid to the Takagi–Sugeno (T-S) fuzzy model for the reason that it can approach any smooth nonlinear system reaching an arbitrarily designated accuracy inside any compact set. This approach has been employed in numerous fields, e.g., electrical controlling, quantitative modeling, signal processing and pattern recognition, intelligent decision-making, and robot investigation [22,23]. Compared with the extensive research on controller and filter design problems with regard to the T-S fuzzy system, the corresponding FD problem has not been investigated thoroughly [24].

The channel fading phenomenon unavoidably occurs in systems linked through wireless and shared connections. As is known, the fading effect is one of the major features of wireless transmission. Diffraction, reflection, and scattering seriously affect signal power, which results in fading or attenuation. Some scholars have paid attention to the problem of channel fading, and some works have emerged. For instance, [25] studied the filtering problem of linear systems subject to channel fading. An event-based state-feedback controller is designed in [26] for interval type-2 fuzzy systems over fading channels. Nevertheless, despite the large number of research findings about filtering and control issues in the case of channel fading [27], the FD problem still has not received enough attention.

Inspired by the aforementioned statements, this paper is devoted to dealing with the FD issue in CSTR with regard to parameter uncertainty and channel fading within a networked environment and in terms of the T-S fuzzy model. We are to realize the FD by carrying out the fuzzy FD filtering, which presents a residual signal in order to obtain the estimate of the fault signal. The primary principle is to decrease the error between the residual and the fault to the minimum. Distinct from other published results in previous papers, the highlights of this paper are as follows: (1) the issue discussed is novel in view of the fact that this paper represents the first of a few endeavors to settle the H∞ fault detection issue against parameter uncertainties, channel fading, and delays for the CSTR reaction process; (2) the considered system is comprehensive and reflects the reality of the CSTR reaction process, which involves the Takagi-Sugeno fuzzy model, parameter uncertainties, time delay, and channel fading; and (3) a specific fault detection scheme is proposed, which ensures that CSTR fuzzy systems achieve exponential stability in the mean square and H∞ performance.

The rest of this paper is organized as follows. The T-S fuzzy model of CSTR is established in Section 2. The performance of an FD dynamic system is analyzed in Section 3. A fuzzy FD filter is designed in Section 4. Section 5 presents a numerical example. A conclusion is given in Section 6.

#### **2. Model of CSTR**

The material enters CSTR at a certain concentration and temperature for exothermic reaction. The operational goal is to continuously adjust the coolant temperature to make the product concentration and reactor temperature meet the production requirements, as shown in Figure 1. Based on the law of energy conservation and the principle of chemical dynamics, the dimensionless mechanism model of the CSTR system is as follows [1]:

$$\dot{\mathbf{x}}\_1(t) = D\_\mathbf{d} \left[ 1 - \mathbf{x}\_1(t) \right] \exp\left[ \frac{\mathbf{x}\_2(t)}{1 + \mathbf{x}\_2(t) / \gamma\_0} \right] - \frac{1}{\lambda} \mathbf{x}\_1(t) + \left( \frac{1}{\lambda} - 1 \right) \mathbf{x}\_1(t - d(t))$$

$$\begin{split} \dot{\mathbf{x}}\_2(t) &= H D\_\mathbf{d} \left[ 1 - \mathbf{x}\_1(t) \right] \exp\left[ \frac{\mathbf{x}\_2(t)}{1 + \mathbf{x}\_2(t) / \gamma\_0} \right] - \left( \frac{1}{\lambda} + \beta \right) \mathbf{x}\_2(t) + \left( \frac{1}{\lambda} - 1 \right) \mathbf{x}\_2(t - d(t)) + \zeta \mathbf{w}(t) \\ &+ \delta f(t) \end{split}$$

where *<sup>x</sup>*1(*t*) = *<sup>C</sup>*0−*C<sup>a</sup>* (*t*) *C*0 and *<sup>x</sup>*2(*t*) = *<sup>γ</sup>*0(*T<sup>a</sup>* (*t*)−*T*0) *T*0 represent the dimensionless product concentration and reactor temperature, respectively.

−

−

+

=

**Figure 1.** Schematic of CSTR.

 *dt*() The symbols in the formula are explained as follows: *λ*, *Dα*, *γ*0, *H*, *β*, *T*<sup>0</sup> are dimensionless system parameters, *ζ* is the disturbance coefficient, *w*(*t*) is the external disturbance, *d*(*t*) is the term of variable time delay. In this paper, the T-S fuzzy model is adopted in order to approach the mechanism model. The reactor temperature, which is easier to measure online, is chosen as the precursor variable, and the linear processing is carried out near each steady-state equilibrium point. Then, considering the parameter uncertainty, the T-S fuzzy model is obtained, which is expressed as follows:

 

 

 = − − + + − − +

+

−

=

 

 Plant Rule *i*: IF *θ*1(*k*) is *Mi*<sup>1</sup> , *θ*2(*k*) is *Mi*<sup>2</sup> , . . . . . . *θp*(*k*) is *Mip* , then

$$\begin{cases} \begin{aligned} \mathbf{x}(k+1) &= (A\_i + \Delta A\_i)\mathbf{x}(k) + (A\_{di} + \Delta A\_{di})\mathbf{x}(k - d(k)) + D\_{\text{li}}\mathbf{w}(k) + \mathbf{G}\_{\text{if}}f(k) \\ \mathbf{y}(k) &= \mathbf{C}\_i \mathbf{x}(k) + D\_{\text{li}}\mathbf{w}(k) \\ \mathbf{x}(k) &= \boldsymbol{\psi}(k), \,\forall k \in \left[ -\overline{d}, 0 \right] \end{aligned} \end{cases} \tag{1}$$

 = − = *fk*() ) where *r* is the IF-THEN rule number; *Mij* is the fuzzy set; *θ*(*k*) = [*θ*1(*k*), *θ*2(*k*), · · · , *θp*(*k*)] is the premise variable vector; *<sup>x</sup>*(*k*) <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* is the state vector; *<sup>y</sup>*(*k*) <sup>∈</sup> <sup>R</sup>*<sup>m</sup>* is the measurement output; *<sup>w</sup>*(*k*) <sup>∈</sup> <sup>R</sup>*<sup>q</sup>* is the disturbance input; *<sup>f</sup>*(*k*) <sup>∈</sup> <sup>R</sup>*<sup>l</sup>* is the fault signal; *w*(*k*) and *f*(*k*) belong to *<sup>l</sup>*2[0, <sup>∞</sup>); 0 ≤ *<sup>d</sup>*(*k*) ≤ *<sup>d</sup>* represents time delay; system matrices *<sup>A</sup><sup>i</sup>* , *C<sup>i</sup>* , *D*1*<sup>i</sup>* , *D*2*<sup>i</sup>* , and *G<sup>i</sup>* are given real-valued matrices with appropriate dimensions; *ψ*(*k*), *k* ∈ h −*d*, 0 i is the given initial state and satisfies sup*k*∈[−*d*,0] E n k*ψ*(*i*)k 2 o < ∞; ∆*A<sup>i</sup>* and ∆*Adi* represent norm-bounded parameter uncertainties, which satisfy the following formula:

$$\begin{bmatrix} \Delta A\_i & \Delta A\_{di} \end{bmatrix} = H\_i F(k) \begin{bmatrix} E\_d & E\_d \end{bmatrix} \tag{2}$$

−

 where *F*(*k*) is the unknown matrix that satisfies *F T* (*k*)*F*(*k*) ≤ *I*, and *H<sup>i</sup>* , *Ea*, *E<sup>d</sup>* stand for known matrices with appropriate dimensions.

For the T-S fuzzy system (1), the defuzzified output is denoted as follows:

$$\begin{cases} \mathbf{x}(k+1) = \sum\_{i=1}^{r} h\_i(\theta(k)) [(A\_i + \Delta A\_i)\mathbf{x}(k) + (A\_{di} + \Delta A\_{di})\mathbf{x}(k - d(k)) \\ \qquad + D\_{1i}\mathbf{w}(k) + G\_i f(k)] \\ y(k) = \sum\_{i=1}^{r} h\_i(\theta(k)) [\mathbf{C}\_i \mathbf{x}(k) + D\_{2i}\mathbf{w}(k)] \\ \mathbf{x}(k) = \boldsymbol{\psi}(k), \forall k \in \mathbb{Z}^- \end{cases} \tag{3}$$

where the fuzzy basis functions are described as

$$h\_i(\theta(k)) = \frac{\theta\_i(\theta(k))}{\sum\_{i=1}^r \theta\_i(\theta(k))}$$

with *ϑi*(*θ*(*k*)) = *p* ∏ *j*=1 *Mij*(*θj*(*k*)), *ϑi*(*θ*(*k*)) ≥ 0, *i* = 1, 2, · · · ,*r*, *r* ∑ *i*=1 *ϑi*(*θ*(*k*)) > 0, *Mij*(*θj*(*k*)) denoting the membership of *θj*(*k*) in *Mij*, understandably.

> *hi*(*θ*(*k*)) ≥ 0, *i* = 1, 2, · · · ,*r*, *r* ∑ *hi*(*θ*(*k*)) = 1

For simplicity, we denote *h<sup>i</sup>* = *hi*(*θ*(*k*)).

Considering that the fading phenomenon occurs in the transmission process of the measurement signal from the sensor to the FD filter, based on the *L*th-order rice fading model, the measurement signal obtained by the fault detection filter is expressed in the following form:

$$y\_f(k) = \sum\_{s=0}^{\ell} \beta\_s(k) y(k-s) + E\_y \mathfrak{f}(k) \tag{4}$$

*i*=1

where ℓ is a given positive scalar and *β s k* (*<sup>s</sup>* <sup>=</sup> 0, 1, · · · , <sup>ℓ</sup>) represent the channel coefficients, and they are mutually independent. Moreover, *β s k* own the probability density function over the interval [0, 1], which has the expectation *β<sup>s</sup>* and variance *β*e<sup>∗</sup> *s* . *<sup>ξ</sup><sup>k</sup>* <sup>∈</sup> *<sup>l</sup>*2([0, <sup>∞</sup>); <sup>R</sup>*m*) stands for external noise and *E<sup>y</sup>* denotes a given real-valued matrix with a proper dimension.

**Remark 1.** *In this paper, channel fadings are characterized* via *the improved Lth-order Rice model. Such a model has been extensively utilized in fields of signal processing and remote control due to its capacity to describe both channel fadings and random time-delays at the same time. Differing from the conventional model of channel fadings, in model (4), the channel coefficients are described by random variables obeying an arbitrary probabilistic distribution over the interval [0, 1]. Note that the consideration of channel fadings increases the complexity of acquiring the FD filter*.

Taking into account the physical object described by (1) and (2), an FD filter is constructed with the following expression:

Filter Rule *i*: IF *θ*1(*k*) is *Mi*<sup>1</sup> , *θ*2(*k*) is *Mi*<sup>2</sup> , . . . . . . *θp*(*k*) is *Mip*, then

$$\begin{cases}
\pounds(k+1) = A\_{fi}\pounds(k) + B\_{fi}y\_f(k) \\
r(k) = \mathcal{C}\_{fi}\pounds(k) + D\_{fi}y\_f(k)
\end{cases} \tag{5}$$

where *<sup>x</sup>*ˆ(*k*) <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* denotes the state vector of the filter, *<sup>r</sup>*(*k*) <sup>∈</sup> <sup>R</sup>*<sup>l</sup>* represents the residual signal being compatible with the fault signal *f*(*k*), *Af i*, *Bf i*, *Cf i*, and *Df i* are appropriately dimensioned filter gains to be decided. Therefore, the whole fuzzy fault detection filter is constructed in the following formulation:

$$\begin{cases} \mathfrak{x}(k+1) = \sum\_{i=1}^{r} h\_{i} [A\_{fi}\mathfrak{x}(k) + B\_{fi}y\_{f}(k)] \\\ r(k) = \sum\_{i=1}^{r} h\_{i} [\mathbb{C}\_{fi}\mathfrak{x}(k) + D\_{fi}y\_{f}(k)]. \end{cases} \tag{6}$$

In what follows, we denote

$$\sum\_{a\_1, a\_2, \dots, a\_s = 1}^r h\_{a\_1} h\_{a\_2} \dotsm h\_{a\_s} = \sum\_{a\_1 = 1}^r h\_{a\_1} \sum\_{a\_2 = 1}^r h\_{a\_2} \dotsm \sum\_{a\_s = 1}^r h\_{a\_s}, \forall s \ge 1$$

$$\eta(k) = \left[ \mathbf{x}^T(k) \ \mathbf{\hat{x}}^T(k) \right]^T, \ v(k) = \left[ \mathbf{w}^T(k) \ \mathbf{\hat{y}}^T(k) \ f^T(k) \right]^T, \ \mathbf{\tilde{r}}(k) = r(k) - f(k),$$

$$\eta^\*(k) = \left[ \eta^T(k-1) \ \eta^T(k-2) \ \cdots \ \eta^T(k-\ell) \right]^T, \ \mathbf{\tilde{v}}(k) = \left[ \ \ \ \ \mathbf{v}^T(k) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$$

By (3) and (6), the following FD dynamic system can be obtained:

$$\begin{cases} \begin{aligned} \eta(k+1) &= \sum\_{i,j=1}^{r} h\_{i} h\_{j} \Big[ (\overline{A}\_{ij} + \Delta \overline{A}\_{ij} + \widetilde{\beta}\_{0}(k) \widehat{A}\_{ij}) \eta(k) + (\overline{A}\_{di} + \Delta \overline{A}\_{di}) \eta(k - d(k)) \\ &+ (\overline{\Lambda}\_{l} A\_{ij}^{\*} + \widetilde{\Lambda}\_{l}(k) A\_{ij}^{\*}) \eta^{\*}(k) + (\overline{B}\_{li} + \widetilde{\beta}\_{0}(k) \widehat{B}\_{lj}) \eta(k) \\ &+ (\overline{\Lambda}\_{l} B\_{ij}^{\*} + \widetilde{\Lambda}\_{l}(k) B\_{ij}^{\*}) \eta^{\*}(k) \end{aligned} &(7) \\ \begin{aligned} \overline{\tau}(k) &= \sum\_{i,j=1}^{r} h\_{i} h\_{j} [(\overline{C}\_{ij} + \widetilde{\beta}\_{0}(k) \widehat{C}\_{ij}) \eta(k) + (\overline{\Lambda}\_{l} C\_{ij}^{\*} + \widetilde{\Lambda}\_{l}(k) C\_{ij}^{\*}) \eta^{\*}(k) \\ &+ (\overline{D}\_{lj} + \widetilde{\beta}\_{0}(k) \widehat{D}\_{lj}) v(k) + (\overline{\Lambda}\_{l} D\_{lj}^{\*} + \widetilde{\Lambda}\_{l}(k) D\_{lj}^{\*}) v^{\*}(k) \end{aligned} &(8) \end{cases}$$

where *Aij* = *A<sup>i</sup>* 0 *<sup>B</sup>f jC<sup>i</sup> <sup>A</sup>f j* , ∆*A<sup>i</sup>* = ∆*A<sup>i</sup>* 0 0*<sup>i</sup>* 0 , *A*ˆ *ij* = 0 0 *Bf jC<sup>i</sup>* 0 , *Bij* = *D*1*<sup>i</sup>* 0 *G<sup>i</sup> β*0 *Bf jD*2*<sup>i</sup> Bf jE<sup>y</sup>* 0 , *B*ˆ *ij* = 0 0 0 *<sup>B</sup>f jD*2*<sup>i</sup>* 0 0 , *Adi* = *Adi* 0 0 0 , ∆*Adi* = ∆*Adi* 0 0 0 , *Cij* = - *<sup>β</sup>*0*Df jC<sup>i</sup> <sup>C</sup>f j* , *C*ˆ *ij* = - *Df jC<sup>i</sup>* 0 , *Dij* = - *β*0*Df jD*2*<sup>i</sup> E<sup>y</sup>* −*I* , *D*ˆ *ij* = - *Df jD*2*<sup>i</sup>* 0 0 , *A* ∗ *ij* <sup>=</sup> diag{*A*<sup>ˆ</sup> *ij*, · · · , *<sup>A</sup>*<sup>ˆ</sup> *ij* | {z } } ℓ , *B* ∗ *ij* <sup>=</sup> diag{*B*<sup>ˆ</sup> *ij*, · · · , *<sup>B</sup>*<sup>ˆ</sup> *ij* | {z } } ℓ , Λ*<sup>l</sup>* = [*β*<sup>1</sup> *I*, · · · , *β<sup>l</sup> I*], Λe *l* (*k*) = [*β*e <sup>1</sup> *I*, · · · , *β*e *l I*], *C* ∗ *ij* <sup>=</sup> diag{*C*<sup>ˆ</sup> *ij*, · · · , *<sup>C</sup>*<sup>ˆ</sup> *ij* | {z } } ℓ , *D* ∗ *ij* <sup>=</sup> diag{*D*<sup>ˆ</sup> *ij*, · · · , *<sup>D</sup>*<sup>ˆ</sup> *ij* | {z } } ℓ , <sup>e</sup>*αm*(*k*) = *<sup>α</sup>m*(*k*) <sup>−</sup> *<sup>α</sup>m*, <sup>E</sup>{e*αm*(*k*)} <sup>=</sup> 0, <sup>E</sup> n e*α* 2 *<sup>m</sup>*(*k*) o = *αm*(1 − *αm*), E n *β*e2 *s* (*k*) o = *β*e<sup>∗</sup> *s* , *β*e*s*(*k*) = *βs*(*k*) − *β<sup>s</sup>* (*s* = 0, 1, . . . , *l*).

**Definition 1.** *With the FD dynamic system (7) and each initial condition ψ*, *in the situation of v*ˆ(*k*) = 0, *system (7) is said to be exponentially mean-square stable if there are constants δ* > 0 *and* 0 < *κ* < 1, *which achieve the following [28].*

$$\mathbb{E}\left\{\left\|\eta(k)\right\|^2\right\} \le \delta \kappa^k \sup\_{i \in \mathbb{Z}^-} \mathbb{E}\left\{\left\|\psi(i)\right\|^2\right\}, \forall k \ge 0.$$

*Thus, the ideal FD filter is designed via the following steps*:

*Step (1) Introduce a residual signal. With system (2), a fuzzy FD filter expressed as (5) is designed to produce a residual signal r*(*k*). *Then, the filter is devised to guarantee that the whole FD system (6) achieves exponential stability in the mean square and the following H*∞ *performance under the zero-initial condition*:

$$\sum\_{k=0}^{\infty} \mathbb{E}\left\{ \left\| \overline{r}(k) \right\|^2 \right\} \le \gamma^2 \sum\_{k=0}^{\infty} \left\| \vartheta(k) \right\|^2 \tag{8}$$

*where <sup>v</sup>*ˆ(*k*) <sup>6</sup><sup>=</sup> <sup>0</sup> *and <sup>γ</sup>* <sup>&</sup>gt; <sup>0</sup> *are made as small as possible in the feasibility of (8)*.

*Step (2) Establish a residual evaluation stage containing an evaluation function J*(*k*) *and a threshold J*th *as follows [29]:*

$$J(k) = \left\{ \sum\_{k=s-L}^{k=s} r^T(k)r(k) \right\}^{\frac{1}{2}}, \; f\_{\text{th}} = \sup\_{w \in l\_2f = 0} \mathbb{E}\{f(k)\} \tag{9}$$

*where L is the length of the finite evaluating time horizon. Based on (9), whether a fault occurs is detected according to the rule below:*

$$J(k) > J\_{\rm th} \to \text{ fault occurs and alarm}$$

*J*(*k*) ≤ *J*th → no fault occurs.

#### **3. Performance Analysis of an FD Dynamic System**

In this part, we are concerned with the performance analysis of the FD filter for the T-S fuzzy system, as stated previously. Before proceeding, we present several useful lemmas:

**Lemma 1.** *(Schur Complement) Given constant matrices X* = *X*<sup>11</sup> *X*<sup>12</sup> *<sup>X</sup>*<sup>21</sup> *<sup>X</sup>*<sup>22</sup> , *where X*<sup>11</sup> *is r* × *r*, *the following three conditions are equivalent*:


**Lemma 2.** *(S-procedure) Given matrix E* = *E T , M and N are real matrices with suitable dimensions, and F satisfies F <sup>T</sup><sup>F</sup>* <sup>≤</sup> *I, then the sufficient condition for <sup>E</sup>* <sup>+</sup> *MFN* <sup>+</sup> *<sup>N</sup>T<sup>F</sup> <sup>T</sup>M<sup>T</sup>* < 0 *is that there is a positive number, so that*

$$E + \mu M M^T + \mu^{-1} N^T N < 0 \text{ or } \Pi = \begin{bmatrix} E & \mu M & N^T \\ \mu M^T & -\mu I & 0 \\ N & 0 & -\mu I \end{bmatrix} < 0.$$

**Lemma 3.** *For any real matrices <sup>X</sup>ij*, *<sup>i</sup>*, *<sup>j</sup>* <sup>=</sup> 1, 2, · · · ,*<sup>r</sup> and* <sup>Λ</sup> <sup>&</sup>gt; <sup>0</sup> *with proper dimensions, one has [30].*

$$\sum\_{i=1}^{r} \sum\_{j=1}^{r} \sum\_{k=1}^{r} \sum\_{l=1}^{r} h\_{i} h\_{j} h\_{k} h\_{l} X\_{ij}^{T} \Delta X\_{kl} \le \sum\_{i=1}^{r} \sum\_{j=1}^{r} h\_{i} h\_{j} X\_{ij}^{T} \Delta X\_{ij} \tag{10}$$

*The following analysis outcome provides a theoretical basis for the subsequent discussion.*

**Theorem 1.** *For the fuzzy CSTR system (2) with known filter parameters and a specified H*∞ *performance γ* > 0. *The fuzzy FD system (6) becomes exponentially stable in the mean square with a disturbance attenuation level γ if there are positive definite matrices P* > 0 and *Q* > 0 *satisfying*

$$
\Pi \Pi\_{\ddot{\imath}i}^T \widetilde{P} \Pi\_{\dddot{\imath}i} + \widetilde{\Gamma} \Pi\_{\dddot{\imath}i}^T \widetilde{P} \Upsilon\_{\dddot{\imath}i} + \overline{P}\_{\dddot{\imath}i} < 0 \tag{11}
$$

$$\mathcal{D}(\overline{P}\_{\overline{i}i} + \overline{P}\_{\overline{j}\overline{i}}) + (\Pi\_{\overline{i}\overline{j}} + \Pi\_{\overline{j}i})^T \widetilde{P} (\Pi\_{\overline{i}\overline{j}} + \Pi\_{\overline{j}i}) + (\widetilde{\Pi}\_{\overline{i}\overline{j}} + \widetilde{\Pi}\_{\overline{j}i})^T \not{P} (\widetilde{\Pi}\_{\overline{i}\overline{j}} + \widetilde{\Pi}\_{\overline{j}i}) < 0 \tag{12}$$

*where*

Π*ij* = " *Aij* + ∆*A Adi* + ∆*Adi* Λ*<sup>l</sup> A* ∗ *ij Bij* Λ*lB* ∗ *ij Cij* 0 Λ*lC* ∗ *ij Dij* Λ*lD*<sup>∗</sup> *ij* # , Πe *ij* = h Πe *T* <sup>1</sup>*ij* <sup>Π</sup><sup>e</sup> *<sup>T</sup>* <sup>2</sup>*ij* <sup>Π</sup><sup>e</sup> *<sup>T</sup>* <sup>3</sup>*ij* <sup>Π</sup><sup>e</sup> *<sup>T</sup>* <sup>4</sup>*ij* <sup>i</sup>*<sup>T</sup>* , Πe <sup>1</sup>*ij* = - *β*ˇ*A*ˆ *ij* 0 0 *β*ˇ*B*ˆ *ij* 0 , <sup>Π</sup><sup>e</sup> <sup>2</sup>*ij* = h 0 0 *β*ˇ *l A* ∗ *ij* <sup>0</sup> *<sup>β</sup>*<sup>ˇ</sup> *lB* ∗ *ij* i , Πe <sup>3</sup>*ij* = - *β*ˇ*C*ˆ *ij* 0 0 *β*ˇ*D*ˆ *ij* 0 , <sup>Π</sup><sup>e</sup> <sup>4</sup>*ij* = h 0 0 *β*ˇ *lC* ∗ *ij* <sup>0</sup> *<sup>β</sup>*<sup>ˇ</sup> *lD*<sup>∗</sup> *ij* i , *<sup>P</sup>*´ <sup>=</sup> diag{*P*, *<sup>P</sup>*<sup>ℓ</sup> , *<sup>I</sup>*, *<sup>I</sup>*},*<sup>R</sup>* <sup>=</sup> *<sup>I</sup>*ℓ+<sup>2</sup> <sup>⊗</sup> *<sup>P</sup>*, *<sup>P</sup>is* <sup>=</sup> diag *Q*, −*Q*, −*γ* 2 *I* , *Q* = −*P* + (*d* + 1)*Q* + ℓ ∑ *l*=1 *Rl* , *<sup>P</sup>*<sup>e</sup> <sup>=</sup> diag{*P*, *<sup>I</sup>*},*R<sup>l</sup>* <sup>=</sup> diag{*R*1, · · · , *<sup>R</sup>*ℓ}, *<sup>β</sup>*<sup>ˇ</sup> <sup>=</sup> q *β*e∗ 0 *I*, *β*ˇ *<sup>l</sup>* <sup>=</sup> diagq *β*e∗ 1 *I*, · · · , q *β*e∗ ℓ *I* , Λe∗ *<sup>l</sup>* <sup>=</sup> diag<sup>n</sup> *β*e∗ 1 , · · · , *β*e<sup>∗</sup> *l* o ,*β*e∗ *<sup>l</sup>* = E n *β*e2 *l* (*k*) o , E n Λe *T l* (*k*)*P*Λ<sup>e</sup> *l* (*k*) o <sup>=</sup> diag<sup>n</sup> *β*e∗ 1 *P*, · · · , *β*e<sup>∗</sup> *l P* o , <sup>Λ</sup>e<sup>∗</sup> *<sup>l</sup>* ⊗ *P*.

**Proof.** For simplicity, denote *η*ˆ(*k*) = - *η T* (*k*) *η T* (*k* − *d*(*k*)) *η* ∗*T* (*k*) *v T* (*k*) *v* ∗*T* (*k*) *T* . With the dynamic system (7), define the following Lyapunov function:

$$V(k) = \sum\_{i=1}^{4} V\_i(k) \tag{13}$$

where

$$\begin{aligned} V\_1(k) &= \eta^T(k) P \eta(k), \; V\_2(k) = \sum\_{i=k-d(k)}^{k-1} \eta^T(i) Q \eta(i), \\ V\_3(k) &= \sum\_{n=-\overline{d}+1}^0 \sum\_{i=k+n}^{k-1} \eta^T(i) Q \eta(i), \; V\_4(k) = \sum\_{l=1}^\ell \sum\_{i=k-l}^{k-1} \eta^T(i) R\_l \eta(i), \end{aligned}$$

where *P* > 0 and *Q* > 0 denote unknown matrices yet to be decided. By (7), one has

<sup>E</sup>{∆*V*1(*k*)} <sup>=</sup> <sup>E</sup> *η T* (*k* + 1)*Pη*(*k* + 1) − *η T* (*k*)*Pη*(*k*) <sup>=</sup> <sup>E</sup>{ *r* ∑ *i*,*j*,*s*,*t*=1 *hihjhsh<sup>t</sup>* [*η T* (*k*)((*Aij* + ∆*Ai*) *T P*(*Ast* + ∆*As*) + *β*e<sup>∗</sup> <sup>0</sup>*A*<sup>ˆ</sup> *<sup>T</sup> ijPA*<sup>ˆ</sup> *st* − *P*)*η*(*k*) +2*η T* (*k*)(*Aij* + ∆*Ai*) *T P*Λ*<sup>l</sup> A* ∗ *stη* ∗ (*k*) + 2*η T* (*k*)(*Aij* + ∆*Ai*) *T PBstv*(*k*) +2*β*e<sup>∗</sup> 0 *η T* (*k*)*A*ˆ *<sup>T</sup> ijPB*<sup>ˆ</sup> *stv*(*k*) + 2*η T* (*k*)(*Aij* + ∆*Ai*) *T P*Λ*lB* ∗ *stv* ∗ (*k*) +2*η T* (*k*)(*Aij* + ∆*Ai*) *T <sup>P</sup>*(*Ads* + <sup>∆</sup>*Ads*)*η*(*<sup>k</sup>* − *<sup>d</sup>*(*k*)) +*η* ∗*T* (*k*)*A* ∗*T ij* Λ *T <sup>l</sup> P*Λ*<sup>l</sup> A* ∗ *stη* ∗ (*k*) + *η* ∗*T* (*k*)*A* ∗*T ij* (Λe<sup>∗</sup> *<sup>l</sup>* ⊗ *P*)*A* ∗ *stη* ∗ (*k*) +2*η* ∗*T* (*k*)*A* ∗*T ij* Λ *T <sup>l</sup> PBstv*(*k*) + 2*η* ∗*T* (*k*)*A* ∗*T ij* Λ *T <sup>l</sup> P*Λ*lB* ∗ *stv* ∗ (*k*) +2*η* ∗*T* (*k*)*A* ∗*T ij* (Λe<sup>∗</sup> *<sup>l</sup>* ⊗ *P*)*B* ∗ *stv* ∗ (*k*) + 2*η* ∗*T* (*k*)*A* ∗*T ij* Λ *T <sup>l</sup> <sup>P</sup>*(*Ads* + <sup>∆</sup>*Ads*)*η*(*<sup>k</sup>* − *<sup>d</sup>*(*k*)) +*v T* (*k*)*B T ijPBstv*(*k*) + *β*e<sup>∗</sup> 0 *v T* (*k*)*B*ˆ*<sup>T</sup> ijPB*<sup>ˆ</sup> *stv*(*k*) + 2*v T* (*k*)*B T ijP*Λ*lB* ∗ *stv* ∗ (*k*), +2*v T* (*k*)*B T ijP*(*Ads* + <sup>∆</sup>*Ads*)*η*(*<sup>k</sup>* − *<sup>d</sup>*(*k*)) + *<sup>v</sup>* ∗*T* (*k*)*B* ∗*T ij* Λ *T <sup>l</sup> P*Λ*lB* ∗ *stv* ∗ (*k*) +*v* ∗*T* (*k*)*B* ∗*T ij* (Λe<sup>∗</sup> *<sup>l</sup>* ⊗ *P*)*B* ∗ *stv* ∗ (*k*) + 2*v* ∗*T* (*k*)*B* ∗*T ij* Λ *T <sup>l</sup> <sup>P</sup>*(*Ads* + <sup>∆</sup>*Ads*)*η*(*<sup>k</sup>* − *<sup>d</sup>*(*k*)) +*η T* (*<sup>k</sup>* − *<sup>d</sup>*(*k*)(*Adi* + <sup>∆</sup>*Adi*)*P*(*Ads* + <sup>∆</sup>*Ads*)*η*(*<sup>k</sup>* − *<sup>d</sup>*(*k*))], (14)

$$\begin{aligned} \mathbb{E}\{\Delta V\_2(k)\} &= \mathbb{E}\{V\_2(k+1) - V\_2(k)\} \\ &\le [\eta^T(k)Q\eta(k) - \eta^T(k - d(k))Q\eta(k - d(k)) + \sum\_{i=k-\overline{d}+1}^{k} \eta^T(i)Q\eta(i)], \end{aligned} \tag{15}$$

$$\begin{aligned} \mathbb{E}\{\Delta V\_3(k)\} &= \mathbb{E}\{V\_3(k+1) - V\_3(k)\} \\ &\le \mathbb{E}\left[\overline{d}\eta^T(k)Q\eta(k) - \sum\_{i=k-\overline{d}+1}^k \eta^T(i)Q\eta(i)\right], \end{aligned} \tag{16}$$

$$\begin{aligned} &\mathbb{E}\{\Delta V\_{4}(k)\} = \mathbb{E}\{V\_{4}(k+1) - V\_{4}(k)\} \\ &= \sum\_{l=1}^{\ell} \left\{ \sum\_{i=k+1-l}^{k} \eta^{T}(i)\mathbb{R}\_{l}\eta(i) - \sum\_{i=k-l}^{k-1} \eta^{T}(i)\mathbb{R}\_{l}\eta(i) \right\} \\ &= \sum\_{l=1}^{\ell} \left\{ \eta^{T}(k)\mathbb{R}\_{l}\eta(k) - \eta^{T}(k-l)\mathbb{R}\_{l}\eta(k-l) \right\}. \end{aligned} \tag{17}$$

In the next stage, firstly, we are to verify the exponential stability of the FD dynamic system (7) with *v*ˆ(*k*) = 0. By (14)–(17) and Lemma 1, we acquire the following:

<sup>E</sup>{∆*V*1(*k*)|*v*ˆ(*k*) = <sup>0</sup>} <sup>≤</sup> <sup>E</sup> *r* ∑ *i*,*j*,*s*,*t*=1 *hihjhsh<sup>t</sup>* [*η T* (*k*)((*Aij* + ∆*Ai*) *T P*(*Ast* + ∆*As*) + *β*e<sup>∗</sup> <sup>0</sup>*A*<sup>ˆ</sup> *<sup>T</sup> ijPA*<sup>ˆ</sup> *st* − *P*)*η*(*k*) +2*η T* (*k*)(*Aij* + ∆*Ai*) *T P*Λ*<sup>l</sup> A* ∗ *stη* ∗ (*k*) + 2*η T* (*k*)(*Aij* + ∆*Ai*) *T P*Λ*lB* ∗ *stv* ∗ (*k*) +2*η T* (*k*)(*Aij* + ∆*Ai*) *T <sup>P</sup>*(*Ads* + <sup>∆</sup>*Ads*)*η*(*<sup>k</sup>* − *<sup>d</sup>*(*k*)) + *<sup>η</sup>* ∗*T* (*k*)*A* ∗*T ij* Λ *T <sup>l</sup> P*Λ*<sup>l</sup> A* ∗ *stη* ∗ (*k*) +*η* ∗*T* (*k*)*A* ∗*T ij* (Λe<sup>∗</sup> *<sup>l</sup>* ⊗ *P*)*A* ∗ *stη* ∗ (*k*) + 2*η* ∗*T* (*k*)*A* ∗*T ij* Λ *T <sup>l</sup> <sup>P</sup>*(*Ads* + <sup>∆</sup>*Ads*)*η*(*<sup>k</sup>* − *<sup>d</sup>*(*k*)) +*η T* (*<sup>k</sup>* − *<sup>d</sup>*(*k*)(*Adi* + <sup>∆</sup>*Adi*)*P*(*Ads* + <sup>∆</sup>*Ads*)*η*(*<sup>k</sup>* − *<sup>d</sup>*(*k*))]. (18)

Denote A*ij* = [*Aij* + ∆*AAdi* + ∆*Adi*Λ*<sup>l</sup> A* ∗ *ij*], Ae *ij* <sup>=</sup> diag<sup>n</sup> *β*ˇ*A*ˆ *ij*, 0, *β*ˇ *l A* ∗ *ij*o , then

$$\begin{split} & \quad \mathbb{E}\{\boldsymbol{\Delta}\boldsymbol{V}(k)|\boldsymbol{\theta}(k) = 0\} \\ & \leq \quad \mathbb{E}\{\sum\limits\_{i,j,s,t=1}^{r} h\_{i}h\_{j}h\_{s}h\_{t}\boldsymbol{\eta}^{T}(k)(\mathbf{A}\_{ij}^{-T}P\mathbf{A}\_{st} + \widetilde{\mathbf{A}}\_{ij}^{T}\boldsymbol{P}\widetilde{\mathbf{A}}\_{st} + \boldsymbol{\hat{P}}\_{is})\boldsymbol{\eta}(k)\} \\ & \leq \quad \sum\limits\_{i,j=1}^{r} h\_{i}h\_{j}\boldsymbol{\eta}^{T}(k)\mathbf{A}\_{i}^{T}P\mathbf{A}\_{ij} + \widetilde{\mathbf{A}}\_{ij}^{T}P\widetilde{\mathbf{A}}\_{ij} + \boldsymbol{\hat{P}}\_{ii})\boldsymbol{\eta}(k) \\ & \leq \quad \sum\limits\_{i=1}^{r} h\_{ij}\boldsymbol{\frac{2}{\gamma}}\boldsymbol{\eta}^{T}(k)\mathbf{A}\_{ii}^{-T}P\mathbf{A}\_{ii} + \widetilde{\mathbf{A}}\_{ii}^{T}P\widetilde{\mathbf{A}}\_{ii} + \boldsymbol{\hat{P}}\_{ii})\boldsymbol{\eta}(k) \\ & \quad + \frac{1}{2}\sum\limits\_{i,j=1, i$$

where *P*ˆ *is* = diag{*Q*, −*Q*, −*Rl*}, *Q* = −*P* + (*d* + 1)*Q* + ℓ ∑ *l*=1 *Rl* .

By Theorem 1, we have Ω < 0. Furthermore, according to the method used in the proof in reference [31], it is observed that system (7) reaches exponential stability. Next, the *H*∞ performance of fuzzy dynamic system (7) is analyzed. Suppose zero initial conditions and construct the exponential function as follows:

$$\begin{split} J(n) &= \mathbb{E} \sum\_{k=0}^{n} \left[ \overline{r}^T(k)\overline{r}(k) - \gamma^2 \widehat{v}^T(k)\widehat{v}(k) \right] \\ \leq \mathbb{E} \sum\_{k=0}^{n} \left[ \overline{r}^T(k)\overline{r}(k) - \gamma^2 v^T(k)v(k) - \gamma^2 v^{\*T}(k)v^\*(k) + \Delta V(k) \right]. \end{split} \tag{20}$$

It can be deduced from (7) that

*r T* (*k*)*r*(*k*) = *r* ∑ *i*,*j*,*s*,*t*=1 *hihjhsh<sup>t</sup>* [*η T* (*k*)(*C T ijCst* + *β*e<sup>∗</sup> 0*C*ˆ*<sup>T</sup> ijC*ˆ *st*)*η*(*k*) +2*η T* (*k*)(*C T ij*Λ*lC* ∗ *st*)*η* ∗ (*k*) + 2*η T* (*k*)(*C T ijDst* + *β*e<sup>∗</sup> 0*C*ˆ*<sup>T</sup> ijD*ˆ *st*)*v*(*k*) +2*η T* (*k*)(*C T ij*Λ*lD*<sup>∗</sup> *st*)*v* ∗ (*k*) + *η* ∗*T* (*k*)(*C* ∗*T ij* Λ *T <sup>l</sup>* Λ*lC* ∗ *st* + <sup>Λ</sup>e<sup>∗</sup> *l C* ∗*T ij C* ∗ *st*)*η* ∗ (*k*) +2*η* ∗*T* (*k*)((Λ*lC* ∗ *ij*) *T Dst*)*v*(*k*) + 2*η* ∗*T* (*k*)(*C* ∗*T ij* Λ *T <sup>l</sup>* Λ*lD*<sup>∗</sup> *st* + <sup>Λ</sup>e<sup>∗</sup> *l C* ∗*T ij D*<sup>∗</sup> *st*)*v* ∗ (*k*) +*v T* (*k*)(*D T ijDst* + *β*e<sup>∗</sup> <sup>0</sup>*D*<sup>ˆ</sup> *<sup>T</sup> ijD*ˆ *st*)*v*(*k*) + 2*v T* (*k*)(*D T ij*Λ*lD*<sup>∗</sup> *st*)*v* ∗ (*k*) +*v* ∗*T* (*k*)(*D*∗*<sup>T</sup> ij* Λ *T <sup>l</sup>* Λ*lD*<sup>∗</sup> *st* + <sup>Λ</sup>e<sup>∗</sup> *<sup>l</sup> <sup>D</sup>*∗*<sup>T</sup> ij D*<sup>∗</sup> *st*)*v* ∗ (*k*)]. (21)

Denote

$$
\boldsymbol{\eta}(k) = \begin{bmatrix} \eta^T(k) & \eta^T(k - d(k)) & \eta^{\*T}(k) \end{bmatrix}^T,\\
\boldsymbol{\tilde{\eta}}(k) = \begin{bmatrix} \boldsymbol{\eta}^T(k) & \boldsymbol{\upsilon}^T(k) & \boldsymbol{\upsilon}^{\*T}(k) \end{bmatrix}^T.
$$

By (19) and (20) and Lemma 1, we have

$$\begin{split} J(n) &\leq \mathbb{E} \left\{ \sum\_{k=0}^{n} \sum\_{i,j,s,t=1}^{r} h\_{i} h\_{j} h\_{s} h\_{t} \widetilde{\eta}^{T}(k) (\Pi\_{ij}^{T} \widetilde{P} \Pi\_{tl} + \widetilde{\Pi}\_{ij}^{T} \widetilde{P} \widetilde{\Pi}\_{lj} + \overline{P}\_{is}) \widetilde{\eta}(k) \right\} \\ &\leq \sum\_{i,j=1}^{r} h\_{i} h\_{j} \widetilde{\eta}^{T}(k) (\Pi\_{ij}^{T} \widetilde{P} \Pi\_{ij} + \widetilde{\Pi}\_{ij}^{T} \widetilde{P} \widetilde{\Pi}\_{lj} + \overline{P}\_{ii}) \widetilde{\eta}(k) \\ &\leq \sum\_{i=1}^{r} h\_{i}^{2} \widetilde{\eta}^{T}(k) (\Pi\_{ii}^{T} \widetilde{P} \Pi\_{il} + \widetilde{\Pi}\_{ii}^{T} \widetilde{P} \widetilde{\Pi}\_{il} + \overline{P}\_{ii}) \widetilde{\eta}(k) \\ &+ \frac{1}{2} \sum\_{i,j=1, i$$

With Theorem 1, *<sup>J</sup>*(*n*) <sup>≤</sup> 0, then (8) is obtained, and the proof is complete.

#### **4. Fuzzy FD Filter Design**

In this section, on the basis of the previous analysis, the fuzzy FD filter design problem will be settled by the subsequent theorem.

**Theorem 2.** *Consider the fuzzy dynamic system (7) and make γ* > 0 *a known scalar. If there are matrices P* > 0*, Q* > 0*, Xand K satisfying the following linear matrix inequality (LMI)*:

$$
\begin{bmatrix}
\overline{\Gamma}\_1 & \* & \* \\
 M\_i^T & -\varepsilon I & \* \\
 \varepsilon N & 0 & -\varepsilon I
\end{bmatrix} < 0 \tag{23}
$$

$$
\begin{bmatrix}
\overline{\Gamma}\_2 & \* & \* \\
 M\_i^T + M\_j^T & -\varepsilon I & \* \\
 \varepsilon N & 0 & -\varepsilon I
\end{bmatrix} < 0 \tag{24}
$$

*then the FD filter in the form of (6) exists with the following*:

Γ<sup>1</sup> = *Pii* ∗ Z1*ii* −*P* , Γ<sup>2</sup> = 2(*Pii* + *Pjj*) ∗ Z1*ij* + Z1*ji* −*P* , Z1*ij* = *PA*ˆ <sup>0</sup>*<sup>i</sup>* + *XjR*ˆ <sup>1</sup>*<sup>i</sup> PAdi* <sup>Λ</sup>*<sup>l</sup>* <sup>⊗</sup> (*XjR*<sup>ˆ</sup> <sup>2</sup>*i*) *PB*ˆ <sup>0</sup>*<sup>i</sup>* + *XjR*ˆ <sup>3</sup>*<sup>i</sup>* <sup>Λ</sup>*<sup>l</sup>* <sup>⊗</sup> (*XjR*<sup>ˆ</sup> 4*i*) *KjR*ˆ <sup>1</sup>*<sup>i</sup>* <sup>0</sup> <sup>Λ</sup>*<sup>l</sup>* <sup>⊗</sup> (*KjR*<sup>ˆ</sup> <sup>2</sup>*i*) *D*<sup>0</sup> + *KjR*ˆ <sup>3</sup>*<sup>i</sup>* <sup>Λ</sup>*<sup>l</sup>* <sup>⊗</sup> (*KjR*<sup>ˆ</sup> 4*i*) *β*ˇ*XjR*ˆ <sup>2</sup>*<sup>i</sup>* 0 0 *β*ˇ*XjR*ˆ <sup>4</sup>*<sup>i</sup>* 0 0 0 *β*ˇ *<sup>l</sup>* <sup>⊗</sup> (*XjR*<sup>ˆ</sup> <sup>2</sup>*i*) 0 *β*ˇ *<sup>l</sup>* <sup>⊗</sup> (*XjR*<sup>ˆ</sup> 4*i*) *β*ˇ*KjR*ˆ <sup>2</sup>*<sup>i</sup>* 0 0 *β*ˇ*KjR*ˆ <sup>4</sup>*<sup>i</sup>* 0 0 0 *β*ˇ *<sup>l</sup>* <sup>⊗</sup> (*KjR*<sup>ˆ</sup> <sup>2</sup>*i*) 0 *β*ˇ *<sup>l</sup>* <sup>⊗</sup> (*KjR*<sup>ˆ</sup> 4*i*) , *M<sup>i</sup>* = h 0 0 0 0 0 *H T <sup>i</sup> <sup>P</sup>* 0 0 0 0 0 <sup>i</sup>*<sup>T</sup>* , *N* = - *<sup>E</sup><sup>a</sup> <sup>E</sup><sup>d</sup>* 0 0 0 0 0 0 0 0 0 , *E*ˆ = [ 0 *I* ] *T* , *D*<sup>0</sup> = - 0 −*I* ,*A*ˆ <sup>0</sup>*<sup>i</sup>* = *A<sup>i</sup>* 0 0 0 , *B*ˆ <sup>0</sup>*<sup>i</sup>* = *D*1*<sup>i</sup> G<sup>i</sup>* 0 0 , *R*ˆ <sup>1</sup>*<sup>i</sup>* = 0 *I C<sup>i</sup>* 0 , *R*ˆ <sup>2</sup>*<sup>i</sup>* = 0 0 *D*2*<sup>i</sup>* 0 

*If P, Q, X<sup>j</sup> and K<sup>j</sup> are feasible solutions to (23) and (24), then the FD filter gains of (5) are computed* via *the following formula:*

$$[A\_{f\circ}B\_{f\circ}] = \left(\triangle^T P\triangle\right)^{-1}\triangle^T \mathcal{X}\_{\circ} \; [\mathcal{C}\_{f\circ}D\_{f\circ}] = \mathcal{K}\_{\circ}.$$

**Proof.** For the purpose of avoiding splitting the matrix *P*, *Qm*, and *R<sup>l</sup>* , the parameters in Theorem 1 are rewritten as follows:

$$
\overline{A}\_{\text{i}\text{j}} = \hat{A}\_{0\text{i}} + \hat{E}L\_{\text{j}}\hat{R}\_{1\text{i}\text{\prime}} \\
\overline{B}\_{\text{i}\text{j}} = \hat{B}\_{0\text{i}} + \hat{E}L\_{\text{j}}\hat{R}\_{2\text{i}\text{\prime}} \\
\overline{C}\_{\text{i}\text{\prime}} = K\_{\text{j}}\hat{R}\_{1\text{i}\text{\prime}} \\
\overline{D}\_{\text{i}\text{\prime}} = \overline{D}\_{0} + K\_{\text{j}}\hat{R}\_{2\text{i}\text{\prime}}
$$

where *L<sup>j</sup>* = [*Af j Bf j*], *K<sup>j</sup>* = [*Cf j Df j*].

Then, according to Lemma 1, (11) and (12) are rewritten as follows:

$$
\begin{bmatrix}
\overline{P}\_{\text{ii}} & \* \\
\widetilde{Z}\_{\text{ii}} & -\overline{P}^{-1}
\end{bmatrix} < 0 \tag{25}
$$

$$
\begin{bmatrix}
\mathbf{2}(\overline{P}\_{\overline{i}i} + \overline{P}\_{\overline{j}j}) & \* \\
\widetilde{\mathbf{Z}}\_{\overline{i}j} + \widetilde{\mathbf{Z}}\_{\overline{j}i} & -\overline{P}^{-1}
\end{bmatrix} < 0 \tag{26}
$$

where 1 <sup>≤</sup> *<sup>i</sup>* <sup>&</sup>lt; *<sup>j</sup>* <sup>≤</sup> *<sup>r</sup>* (*i*, *<sup>j</sup>* <sup>∈</sup> *<sup>R</sup>*).

Ze *ij* = *A*ˆ <sup>0</sup>*<sup>i</sup>* + *EL*ˆ *jR*ˆ <sup>1</sup>*<sup>i</sup>* <sup>+</sup> <sup>∆</sup>*A<sup>i</sup> <sup>A</sup>di* <sup>+</sup> <sup>∆</sup>*Adi* <sup>Λ</sup>*<sup>l</sup>* <sup>⊗</sup> (*XjR*<sup>ˆ</sup> <sup>2</sup>*i*) *B*ˆ <sup>0</sup>*<sup>i</sup>* + *EL*ˆ *jR*ˆ <sup>3</sup>*<sup>i</sup>* <sup>Λ</sup>*<sup>l</sup>* <sup>⊗</sup> (*EL*<sup>ˆ</sup> *jR*ˆ 4*i*) *KjR*ˆ <sup>1</sup>*<sup>i</sup>* <sup>0</sup> <sup>Λ</sup>*<sup>l</sup>* <sup>⊗</sup> (*KjR*<sup>ˆ</sup> <sup>2</sup>*i*) *D*<sup>0</sup> + *KjR*ˆ <sup>3</sup>*<sup>i</sup>* <sup>Λ</sup>*<sup>l</sup>* <sup>⊗</sup> (*KjR*<sup>ˆ</sup> 4*i*) *β*ˇ*EL*ˆ *jR*ˆ <sup>2</sup>*<sup>i</sup>* 0 0 *β*ˇ*EL*ˆ *jR*ˆ <sup>4</sup>*<sup>i</sup>* 0 0 0 *β*ˇ *<sup>l</sup>* <sup>⊗</sup> (*EL*<sup>ˆ</sup> *jR*ˆ <sup>2</sup>*i*) 0 *β*ˇ *<sup>l</sup>* <sup>⊗</sup> (*EL*<sup>ˆ</sup> *jR*ˆ 4*i*) *β*ˇ*KjR*ˆ <sup>2</sup>*<sup>i</sup>* 0 0 *β*ˇ*KjR*ˆ <sup>4</sup>*<sup>i</sup>* 0 0 0 *β*ˇ *<sup>l</sup>* <sup>⊗</sup> (*KjR*<sup>ˆ</sup> <sup>2</sup>*i*) 0 *β*ˇ *<sup>l</sup>* <sup>⊗</sup> (*KjR*<sup>ˆ</sup> 4*i*) 

Pre- and post-multiply inequalities (25) and (26) by diag *I*, *P* , respectively, and denote *X<sup>j</sup>* = *PEL*ˆ *j* , one acquires the following:

$$
\Gamma\_1 = \begin{bmatrix}
\overline{P}\_{ii} & \* \\
\overline{Z}\_{ii} & -\overline{P}
\end{bmatrix} < 0 \tag{27}
$$

$$
\Gamma\_2 = \begin{bmatrix}
\mathfrak{L} (\overline{P}\_{\bar{i}i} + \overline{P}\_{\bar{j}j}) & \* \\
\overline{\mathcal{Z}}\_{i\bar{j}} + \overline{\mathcal{Z}}\_{\bar{j}i} & -\overline{P}
\end{bmatrix} < 0 \tag{28}
$$

where Z*ij* = *PA*ˆ <sup>0</sup>*<sup>i</sup>* + *XjR*ˆ <sup>1</sup>*<sup>i</sup>* <sup>+</sup> *<sup>P</sup>*∆*A<sup>i</sup> PAdi* <sup>+</sup> *<sup>P</sup>*∆*Adi* <sup>Λ</sup>*<sup>l</sup>* <sup>⊗</sup> (*XjR*<sup>ˆ</sup> <sup>2</sup>*i*) *PB*ˆ <sup>0</sup>*<sup>i</sup>* + *XjR*ˆ <sup>3</sup>*<sup>i</sup>* <sup>Λ</sup>*<sup>l</sup>* <sup>⊗</sup> (*XjR*<sup>ˆ</sup> 4*i*) *KjR*ˆ <sup>1</sup>*<sup>i</sup>* <sup>0</sup> <sup>Λ</sup>*<sup>l</sup>* <sup>⊗</sup> (*KjR*<sup>ˆ</sup> <sup>2</sup>*i*) *D*<sup>0</sup> + *KjR*ˆ <sup>3</sup>*<sup>i</sup>* <sup>Λ</sup>*<sup>l</sup>* <sup>⊗</sup> (*KjR*<sup>ˆ</sup> 4*i*) *β*ˇ*XjR*ˆ <sup>2</sup>*<sup>i</sup>* 0 0 *β*ˇ*XjR*ˆ <sup>4</sup>*<sup>i</sup>* 0 0 0 *β*ˇ *<sup>l</sup>* <sup>⊗</sup> (*XjR*<sup>ˆ</sup> <sup>2</sup>*i*) 0 *β*ˇ *<sup>l</sup>* <sup>⊗</sup> (*XjR*<sup>ˆ</sup> 4*i*) *β*ˇ*KjR*ˆ <sup>2</sup>*<sup>i</sup>* 0 0 *β*ˇ*KjR*ˆ <sup>4</sup>*<sup>i</sup>* 0 0 0 *β*ˇ *<sup>l</sup>* <sup>⊗</sup> (*KjR*<sup>ˆ</sup> <sup>2</sup>*i*) 0 *β*ˇ *<sup>l</sup>* <sup>⊗</sup> (*KjR*<sup>ˆ</sup> 4*i*) 

According to the expression of the uncertainty parameters, we have

∆*A<sup>i</sup>* = *HiF*(*k*)*Ea*, ∆*Adi* = *HiF*(*k*)*E<sup>d</sup>* , *H<sup>i</sup>* = - *H<sup>T</sup> i* 0 *T* , *E<sup>a</sup>* = - *E<sup>a</sup>* 0 , *E<sup>d</sup>* = - *E<sup>d</sup>* 0 

Equations (27) and (28) can be rewritten as follows:

$$
\overline{\Gamma}\_1 + M F(k) N + N^T F^T(k) M\_l^T < 0 \tag{29}
$$

.

$$
\overline{\Gamma}\_2 + (M\_i + M\_j)F(k)N + N^T F^T(k)(M\_i + M\_j)^T < 0 \tag{30}
$$

where 1 <sup>≤</sup> *<sup>i</sup>* <sup>&</sup>lt; *<sup>j</sup>* <sup>≤</sup> *<sup>r</sup>* (*i*, *<sup>j</sup>* <sup>∈</sup> *<sup>R</sup>*); the parameters therein are defined in Theorem 2. In accordance with the S-procedure in Lemma 2, (23) and (24) are obtained, and the proof is now complete.

**Remark 2.** *Until now, the H*∞ *fault detection filter design has been accomplished for the CSTR reaction process subject to parameter uncertainties, channel fadings, and delays. The main results of this paper are thus highlighted as follows. In Section 3, Lemmas 1–3 lay a necessary foundation for later analysis and design, and Theorem 1 realizes the performance analysis (exponential stability in the mean square of the error dynamics of the fault detection filter and the H*∞ *disturbance rejection level of the residual filtering error against external disturbances). In Section 4, the fault detection filter design is fulfilled in Theorem 2, the gain expression of the desired fault detection filter is acquired by virtue of the feasible solution to certain LMIs. More specifically, Theorem 2 contains all the system parameters such as delay bound, parameters in the parameter uncertainties, and statistical characteristics of the channel coefficient*.

**Remark 3.** *The main work of this paper is further emphasized as follows: (1) constructing a fuzzy T-S model to reflect the CSTR reaction process on the basis of the dimensionless mechanism model; (2) the channel fading phenomenon is considered in the transmission process of CSTR measurement signal from the sensor to the FD filter, which is characterized by the improved Lth-order Rice fading model by reflecting the actual situation of signal transmission more accurately; and (3) a reinforced stochastic analysis technique is implemented in order to conform to the H*∞ *performance of the fault detection filter concerning the CSTR fuzzy systems, except for the constraint of exponential stability in the mean square*.

#### **5. Numerical Example**

The chosen CSTR system parameters are the following: *γ*<sup>0</sup> = 20, *H* = 8, *β* = 1, *D<sup>α</sup>* = 0.072, and *λ* = 0.8. Let *d* = 5, *D*<sup>11</sup> = *D*<sup>12</sup> = *D*<sup>13</sup> = - 0 1 *<sup>T</sup>* . In the reaction, the CSTR system has three equilibrium points: *x*ˆ<sup>01</sup> = - 0.1440 0.8862 *<sup>T</sup>* , *x*ˆ<sup>02</sup> = - 0.4472 2.7520 *<sup>T</sup>* , *x*ˆ<sup>03</sup> = - 0.7646 4.7052 *<sup>T</sup>* , the following T-S fuzzy rules are then employed to expand near the three equilibrium points.

Rule 1: If *x*2(*k*) is small (*x*2(*k*) is about 0.8862), then

$$\mathbf{x}(k+1) = (A\_1 + \Delta A\_1)\mathbf{x}(k) + (A\_{d1} + \Delta A\_{d1})\mathbf{x}(k - d(k)) + D\_{11}\mathbf{w}(k) + G\_1 f(k);$$

Rule 2: If *x*2(*k*) is medium (*x*2(*k*) is about 2.7520), then

$$\mathbf{x}(k+1) = (A\_2 + \Delta A\_2)\mathbf{x}(k) + (A\_{d2} + \Delta A\_{d2})\mathbf{x}(k - d(k)) + D\_{12}\mathbf{w}(k) + G\_2f(k)\mathbf{y}$$

Rule 3: If *x*2(*k*) is large (*x*2(*k*) is about 4.7052), then

$$\mathbf{x}(k+1) = (A\_3 + \Delta A\_3)\mathbf{x}(k) + (A\_{d3} + \Delta A\_{d3})\mathbf{x}(k - d(k)) + D\_{13}w(k) + G\_3f(k).$$

Here, *x*(*k*) and *x*(*k* − *d*(*k*)) are the set of differences between the temperature state value and the corresponding equilibrium point temperature value. According to the selected parameters, there are

$$\begin{aligned} A\_{1} &= \begin{bmatrix} 0.0418 & 0.0132 \\ 0.0346 & -0.0194 \end{bmatrix}, A\_{2} = \begin{bmatrix} 0.0590 & 0.0346 \\ -0.0472 & 0.0515 \end{bmatrix}, A\_{3} = \begin{bmatrix} 0.0498 & -0.0167 \\ 0.0983 & 0.0758 \end{bmatrix}, \\\ A\_{4} &= A\_{42} = A\_{43} = \text{diag}\{0.25, 0.25\}, F(k) = \sin(0.6k), \mathcal{C}\_{2} = \begin{bmatrix} -0.79 & 0.65 \end{bmatrix}, \\\ H\_{1} &= H\_{2} = H\_{3} = \begin{bmatrix} 0.2 \\ 0.01 \end{bmatrix}, E\_{4} = \begin{bmatrix} 0 & 0.15 \end{bmatrix}, E\_{5} = \begin{bmatrix} 0 & 0.2 \end{bmatrix}, \mathcal{G}\_{1} = \begin{bmatrix} 0.21 \\ -0.14 \end{bmatrix}, \mathcal{G}\_{3} = \begin{bmatrix} -0.81 & 0.65 \end{bmatrix}, \\\ \mathcal{G}\_{2} &= \begin{bmatrix} 0.20 \\ -0.12 \end{bmatrix}, \mathcal{G}\_{3} = \begin{bmatrix} 0.19 \\ -0.15 \end{bmatrix}, \mathcal{G}\_{2} = \begin{bmatrix} 0.19 \\ -0.15 \end{bmatrix}, \mathcal{D}\_{1} = \begin{bmatrix} 0.2 & 0.65 \end{bmatrix}. \end{aligned}$$

The membership functions are shown in Figure 2.

**Figure 2.** Membership function.

= 3

=

 −

= 2 The order of the fading model is ℓ = 2, the probability quality function of the channel coefficient is as follows:

$$\begin{array}{c} f(\beta\_0) = 0.0005(e^{9.89\beta\_0} - 1), \ 0 \le \beta\_0 \le 1\\ f(\beta\_1) = \begin{cases} 10\beta\_1, & 0 \le \beta\_1 \le 0.20\\ -2.50(\beta\_1 - 1), & 0.20 < \beta\_1 \le 1 \end{cases} \\ f(\beta\_2) = 8.5017e^{-8.5\beta\_2}, \ 0 \le \beta\_2 \le 1 \end{array}$$

 − = = The mathematical expectations *β<sup>s</sup>* (*s* = 0, 1, 2) are acquired as 0.8991, 0.4000, and 0.1174, the variance (*β* e *s*) 2 are 0.0133, 0.0467, and 0.01364, respectively. In terms of the above parameters and using the LMI toolbox in the Matlab software, the gains of the FD filter can be calculated by solving the feasible solution to matrix inequalities (23) and (24). The obtained gains of the fault detection filter (5) are shown in Table 1.


**Table 1.** The computed gains of the fault detection filter.

= 2 − − − − − The initial state is taken as *x*(0) = - 0.9 0.9 *T* , noise *w*(*k*) = ( 0.2rand(1, 1), 30 ≤ *k* ≤ 130 , and the fault signal *f*(*k*) is chosen as follows:

 − − − − 0, else

$$f(k) = \begin{cases} 1, & \text{50} \le k \le 100 \\ 0, & \text{else} \end{cases}$$

= 

 = = Figure 3 plots measurement curves, in which the dashed line denotes the ideal measurement output, and the solid line represents the signal actually received by the fault detection filter. It can be seen that the amplitude change of the received signal is more intense than that of the ideal measurement, which validates that channel fadings may lead to the signal distortion (signal missing and delays). Additionally, the occurrence and existence of faults cause the abnormal values of the measurement signals. Figures 4 and 5 describe the residual signal curves with and without noise, respectively. We notice that the residual signal curve without noise is smoother than the one with noise, and the influence of both faults and channel fadings on the residual signal is obvious, which is in accordance with Equation (5). In terms of Equation (9), Figures 6 and 7 reflect the evolution of the

detection.

detection.

#### 200 0 sup ( ) ( ) *<sup>T</sup> th f J r h r h* 200 0 sup ( ) ( ) *<sup>T</sup> th f J r h r h*

of faults cause the abnormal values of the measurement signals. Figures 4 and 5 describe the residual signal curves with and without noise, respectively. We notice that the residual signal curve without noise is smoother than the one with noise, and the influence of both faults and channel fadings on the residual signal is obvious, which is in accordance with Equation (5). In terms of Equation (9), Figures 6 and 7 reflect the evolution of the residual evaluation function curves with and without noise, respectively. It is shown that there are more fluctuations in the residual evaluation function with noise than those without noise. In Figure 6 (or 7), the dashed line and the solid line depict the residual evaluation function with and without faults, respectively. It is also illustrated that the value of the residual evaluation function increases due to the existence of faults, which lay a basis for the fault

of faults cause the abnormal values of the measurement signals. Figures 4 and 5 describe the residual signal curves with and without noise, respectively. We notice that the residual signal curve without noise is smoother than the one with noise, and the influence of both faults and channel fadings on the residual signal is obvious, which is in accordance with Equation (5). In terms of Equation (9), Figures 6 and 7 reflect the evolution of the residual evaluation function curves with and without noise, respectively. It is shown that there are more fluctuations in the residual evaluation function with noise than those without noise. In Figure 6 (or 7), the dashed line and the solid line depict the residual evaluation function with and without faults, respectively. It is also illustrated that the value of the residual evaluation function increases due to the existence of faults, which lay a basis for the fault

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*Processes* **2021**, *9*, x FOR PEER REVIEW 14 of 17

residual evaluation function curves with and without noise, respectively. It is shown that there are more fluctuations in the residual evaluation function with noise than those without noise. In Figure 6 (or Figure 7), the dashed line and the solid line depict the residual evaluation function with and without faults, respectively. It is also illustrated that the value of the residual evaluation function increases due to the existence of faults, which lay a basis for the fault detection. Assuming the threshold 0 *h* , after 200 fault-free simulation runs, the average threshold is then 0.2622 *th <sup>J</sup>* . It can be recognized from Figure 6 that 2.519 (59) (60) 2.772 *th J J J* , i.e., the fault is detected in step 10, after it occurs. It can be concluded that the residual can not only reflect the fault in time, but also detect the fault in the case of disturbance. Assuming the threshold 0 *h* , after 200 fault-free simulation runs, the average threshold is then 0.2622 *th <sup>J</sup>* . It can be recognized from Figure 6 that 2.519 (59) (60) 2.772 *th J J J* , i.e., the fault is detected in step 10, after it occurs. It can be concluded that the residual can not only reflect the fault in time, but also detect the fault in the case of disturbance.

**Figure 3.** Ideal and practical measurement outputs. **Figure 3.** Ideal and practical measurement outputs. **Figure 3.** Ideal and practical measurement outputs.

**Figure 4.** Residual signal with noises. **Figure 4.** Residual signal with noises. **Figure 4.** Residual signal with noise.

**Figure 5.** Residual signal without noises. **Figure 5.** Residual signal without noise.

**Figure 6.** Residual evaluation function with noises.

**Figure 7.** Residual evaluation function without noises.

**6. Conclusions** 

**Figure 5.** Residual signal without noises.

**Figure 5.** Residual signal without noises.

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*Processes* **2021**, *9*, x FOR PEER REVIEW 15 of 17

**Figure 6.** Residual evaluation function with noises. **Figure 6.** Residual evaluation function with noise. **Figure 6.** Residual evaluation function with noises.

**Figure 7.** Residual evaluation function without noises. **Figure 7.** Residual evaluation function without noises. **Figure 7.** Residual evaluation function without noise.

**6. Conclusions 6. Conclusions**  Assuming the threshold *Jth* = sup*f*=<sup>0</sup> s 200 ∑ *h*=0 *r <sup>T</sup>*(*h*)*r*(*h*), after 200 fault-free simulation runs, the average threshold is then *Jth* = 0.2622. It can be recognized from Figure 6 that 2.519 = *J*(59) < *Jth* < *J*(60) = 2.772, i.e., the fault is detected in step 10, after it occurs. It can be concluded that the residual can not only reflect the fault in time, but also detect the fault in the case of disturbance.

#### **6. Conclusions**

In this paper, the FD issue for CSTR with respect to time delay, uncertainty parameters, and channel fadings was investigated in terms of the T-S fuzzy model. Norm-bounded uncertainties were adopted to describe parameter imprecision caused by modelling errors. The phenomenon of channel fadings was considered while the measurement output signal was transmitted from the sensor to the FD filter, which was then reflected with an improved *L*-th Rice fadings model. The performance constraints to be met by the constructed fault detection filter were both the exponential stability in the mean square of the filtering error system and the *H*∞ disturbance rejection level of the residual filtering error in resistance to external disturbances. With the help of the Lyapunov stability theory and reinforced stochastic analysis techniques, the analysis of the performance and the design of the fault

detection filter were carried out for the CSTR. As a result, a sufficient condition was put forward, ensuring the existence of a satisfactory FD filter. Simultaneously, a direct expression was acquired from the FD filter in accordance with the feasible solution to a specified LMI, which is solved conveniently via the standard Matlab software. Lastly, a simulation example demonstrated that faults can be reflected and detected in time under circumstances of disturbance by choosing the thresholds appropriately, which validates the effectiveness and the correctness of the developed FD strategy for CSTR in this paper. For subsequent research topics, we would like to deal with fault estimation, fault prognosis, and related issues therein [32].

**Author Contributions:** Model construction, system performance analysis, filter design, and writing original draft preparation, Y.W.; investigation, D.Z.; writing—review and editing, Z.L.; supervision and project administration, W.R.; numerical simulation, J.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Starting Fund for Doctoral Research of Daqing Normal University, grant number 19ZR09, and the Natural Science Foundation of Heilongjiang Province of China, grant number LH2021F006.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** Authors declare no conflict of interest.

#### **References**

