Observer-Based PID Security Control for Discrete-Time T-S Fuzzy Systems under Distributed Dynamic Event-Triggered Mechanism

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Introduction
Since the Takagi-Sugeno (T-S) fuzzy model is first proposed in [1], it has received considerable research attention in the past few decades. In practical applications, systems with complex structures usually have nonlinearities, which make analysis and synthesis difficult. As one of the effective methods to deal with nonlinear systems, T-S fuzzy system technology is able to approximate nonlinear systems by a set of local linear systems and a group of if-then rules. erefore, many stability and synchronization analysis methods of linear systems can be easily extended to the study of nonlinear systems. To cope with nonlinearity, the T-S fuzzy model has been extensively studied in the processing of complex nonlinear systems, and a lot of results have been proposed in the literature [2][3][4][5]. For instance, the T-S fuzzy technology has been utilized in [2] to construct a new system model with distributed DETM and multisensor saturations. In [3], by combining the fuzzy control with the sliding mode control, a new fuzzy sliding mode control law is designed to ensure the desired H ∞ performance.
In the industrial processes, proportional-integral-derivative (PID) control is extensively utilized owing to its simple structure, high reliability, and convenience of parameter adjustment [6]. However, with the controlled plants in modern industry becoming more and more complex, traditional PID control mechanisms may not be able to provide satisfactory control performance for complex systems. As such, many researchers have devoted to combined traditional PID control methods with other advanced control schemes to improve PID control performance [7][8][9][10]. It should be pointed out that due to the effectiveness of T-S fuzzy technology in processing nonlinear systems, much effort has been devoted to the research of nonlinear system performance under the framework of the combination of T-S fuzzy control and PID control. For example, in [7], the T-S fuzzy PID control problem is tackled in the framework of time domain, which greatly simplifies the description of the closed-loop system. Based on the T-S fuzzy PID control method, a novel multiloop fuzzy PID-like controller has been designed in [10] to improve the fault tolerance rate of the overall system. In addition, most articles on PID are based on state feedback. In practice, however, due to the existence of economic limitations or technological constraints, the system states might be immeasurable. erefore, the observer-based PID control scheme has important research significance, and a great number of results have been available in the literature [11][12][13][14].
Different from traditional control systems, network control systems (NCSs) have significant advantages in terms of simpler installation, lower cost, and convenient maintainability. In this way, network control systems have been widely applied in various fields such as the Internet, power grids, and transportation networks. However, the communication capacity and computing resources of the network control system are usually limited, and various networkinduced phenomena have inevitably appeared, such as timevarying delays [15][16][17], channel fading [18,19], and actuator saturations [20]. As is well known, if handled improperly, the network-induced phenomenon may cause the decline of the performance of the entire network control system. In this case, various communication protocols have been proposed to reduce the burden of network transmission. Some rather favorable communication protocols are event-triggered protocol (ETP) [21,22], redundant channel transmission protocol (RCTP) [23,24], round-robin (RR) protocol [25,26], weighted try-once-discard (WTOD) protocol [3,27], and stochastic communication protocol (SCP) [28,29]. In particular, under the event-triggered mechanism (ETM), the control information will be transmitted only when the measurement error value exceeds the threshold, thereby greatly reducing the data release rate and the burden on the network. It should be pointed out that the above articles are almost focused on the ETM under the nonlinear continuous systems, while the dynamic event-triggered mechanism (DETM) under the nonlinear discrete systems has not gained sufficient research attention so far despite the great practical significance.
is is one of the research motivations of this article.
Moreover, as network systems become more and more open, networks security has attracted widespread attention. In particular, network attacks have become a major threat to network security, which aimed to affect control performance by destroying/modifying some important data transmitted on the network. In general, cyber-attacks include three main types, mainly, denial of service (DoS) attacks [30][31][32], repeated attacks [33,34], and deception attacks [35][36][37]. Under the replay attacks, the attacker forces the target to receive data repeatedly. DoS attacks primarily intend to hinder the transmission over the communication networks, while deception attacks inject malicious and falsified data into the sensor or control data transmission channels. Among them, deception attacks have been widely studied for their great destructiveness to the system stability. For example, the ISS problems of the discrete-time linear PID control systems under deception attacks have been addressed [11]. Both system state saturation and deception attacks are considered for a class of time-varying stochastic nonlinear systems [35]. Unfortunately, when it comes to the observer-based PID control issues via deception attacks, only a fraction of results have been achieved in existing articles. is is another motivation of our research.
Based on the above discussion, in this article, we are interested in investigating the observer-based PID control for discrete-time T-S fuzzy systems with multisensors under deception attacks and distributed DETM. e main contributions of this study are as follows: (1) in order to reduce the burden on the network, a distributed DETM is introduced. e event generator in the framework of the discrete-time T-S fuzzy system corresponding to each sensor can determine whether to transmit data based on local information; (2) in the presence of distributed DETM, communication delay, and stochastic cyber-attacks, the input-to-state stability and ϱ-security issue have been considered for TCS fuzzy systems in the discrete time domain; (3) a feasible PID control strategy is proposed by considering the effects of distributed DETM and deception attacks.
e remainder of this study is arranged as follows. In Section 2, a controlled model is constructed for the discretetime T-S fuzzy system with DETM and deception attacks. Some sufficient conditions which can guarantee the ISS and ϱ-security of the controlled systems are obtained in Section 3. A numerical example is shown in Section 4. Finally, we conclude this study in Section 5.
Notations: the notation used in this study is fairly standard except where otherwise stated. l 2 [0, ∞) denotes the space of square summable sequences. ‖ · ‖ represents the Euclidean norm of a variable. A function f: R + 0 ⟶ R + 0 is said to be a Q-function if it satisfies f(0) � 0, continuous, and strictly increasing. Furthermore, f is said to be a Q ∞ -function if it is a Q-function and satisfies f(t) ⟶ ∞ when t ⟶ ∞. In addition, a function c: R + 0 × R + 0 ⟶ R + 0 is said to be a P-function if c(·, n) is a Q-function for each fixed n ∈ R + 0 , and for each fixed m ∈ R + 0 , c(m, ·) is decreasing and satisfies c(m, n) ⟶ 0 when n ⟶ ∞.
e maximum eigenvalue and minimum eigenvalue of a matrix is represented by λ max (·) and λ min (·), respectively. sym X { } � X + X T and the symbol inf represent the infimum.

Problem Formulation and Preliminaries
Consider the discrete-time nonlinear systems modeled by the T-S fuzzy model: Plant rule i: if η 1 (k) is g i1 and · · · and η l (k) is g il , then where g ij (i ∈ M ≜ 1, 2, . . . , r { }, j ∈ ≜ 1, 2, . . . , l { }) is the fuzzy set; r represents the number of IF-THEN rules; η(k) � (η 1 (k), η 2 (k), . . . , η l (k)) T denotes the vector consisting of the premise variables; x(k) ∈ R n x and y(k) ∈ R n represent, respectively, the state vector and the measurement output; u(k) ∈ R n u is the control input; τ(k) denotes the discrete-time delay; and φ(l)(l � − τ M , . . . , − 1, 0) are the 2 Discrete Dynamics in Nature and Society initials conditions. ϑ(k) ∈ (l 2 [0, ∞), R n v ) denotes the disturbance input satisfying where ϑ is a given scalar. A i , A τ i , C, D i are the known constant matrices with compatible dimensions and assume the matrix B i is of full column rank, namely, rank(B i ) � n u .
Assumption 1 (See [12]). e discrete-time delay τ(k) satisfies τ m ≤ τ(k) ≤ τ M (k ∈ N + ), where τ m and τ M are the two known scalars, representing the upper bound and the lower bound of τ(k), respectively. By conducting the singleton fuzzifier, center-average defuzzifier, and product fuzzy inference, system (1) can be represented by the following compact form: where the normalized membership functions are given by and g ij (η j (k)) refers to the grade of the membership function of η j (k) in g ij , h i (η(k)) satisfying h i (η(k)) ≥ 0, and r i�1 h i (η(k)) � 1. For simplicity, we use h i to represent h i (η(k)).
From Figure 1, we know that the measurement output y(k) is grouped into y(k) � [y 1 (k), y 2 (k), . . . , y n (k)] T . In order to ease the burden of networked channels, the distributed dynamic event-triggered mechanism is utilized to save system resources. Inspired by [20], each sensor side is equipped with an event-trigger generator to judge whether the measured signal should be transmitted or not. e event generator function for the s th (s ∈ L ≜ 1, 2, . . . , n { }) node is given by where ϕ s (k) ≜ y s (k s t ) − y s (k); k s t denotes the latest triggered time of the s th sensor, and y s (k s t ) is the corresponding measurement output. σ and ε are the two given positive scalars. Obviously, as long as the trigger condition f s (k, ϕ s (k), ζ(k)) > 0 is satisfied, the data will be transmitted to the buffer.
We define the triggering time sequence of the s th (s ∈ L ≜ 1, 2, . . . , n { }) sensor as follows: 0 < k s 0 < k s 1 < · · · < k s t < · · ·. en, the next transmitted instants k s t+1 for the s th sensor can be described as According to (6) and the use of the buffers, for all sensors, we can derive the following event-triggered condition: where ϕ(k) � (ϕ T 1 (k), ϕ T 2 (k)), . . . , ϕ T n (k)) T , n(n > 0) representing the number of sensors, and ζ(k) is an internal dynamical variable satisfying where λ ∈ (0, 1) is a given constant.

Remark 1.
It should be noted that the measured data will be transmitted to the observer when the measured data satisfy the event-triggered condition (7). Moreover, a dynamical variable ζ(k) is introduced in the triggering condition (7) so that each event-trigger generator can determine when transmit data in a dynamic way. In fact, the distributed DETM can be regarded as a static one when σ approaches to infinity.
Since the network communication channel may be attacked by opponents, in this article, we assume that the deception attacks occur randomly in the transmission channel between the sensors and the observer. en, the actual observer input y(k) can be expressed as where y(k) � (y T 1 (k 1 t ), y T 2 (k 2 t ), . . . , y T n (k n t )) T is the real measured data via the distributed event-triggered scheme; ω(k) denotes the deception attacks information launched by the attacker described by where ξ(k) ≠ 0 represents an arbitrary bounded signal satisfying where ξ is a given positive scalar. α(k) ∈ 0, 1 { } is a Bernoulli distributed variable with the following probabilities: where α ∈ [0, 1) is a known constant, and the expectation of Remark 2. As is well known, cyber-attacks are inevitable in the network. Among various attacks, the deception attacks are considered to be the easiest to be sent by an attacker and the most destructive. erefore, in this article, we consider Discrete Dynamics in Nature and Society that the transmission channel between the sensors and the observer is subject to deception attacks. Moreover, in engineering practice, since there exists security protection from the protection institution, the attacks launched by attackers might not be always successful. In this case, the attacks could occur in a random manner. erefore, the randomly occurring deception attacks are described as a Bernoulli sequence in this study, and the attack information ξ(k) sent by the attacker is bounded.
Considering that in industrial practice, not all systems state can be measured directly, then a fuzzy observer is given as follows.
Observer rule i: if η 1 (k) is g i1 and · · · and η l (k) is g il , then where x(k) ∈ R n x is the observer states of x(k); L j denotes the observer gains which will be designed later. en, the fuzzy observer can be constructed as follows: In this study, for the discrete-time T-S fuzzy system (1), we adopt the following observer-based PID control law.
Controller rule i: if η 1 (k) is g i1 and · · · and η l (k) is g il , then where K P i , K I i , K D i are the controller gains which will be designed later; d is a known scalar representing the length of time-windows, and we assume that d ≥ τ M in this study. e considered fuzzy PID control law can be rewritten as follows:  Remark 3. In this study, we consider the observer-based PID controller for discrete-time T-S fuzzy systems. e PID controller consists of three parts. Specifically, the components are the proportional term, integral term, and derivative term, respectively. e gain matrices of the above three terms can be adjusted to meet different requirements in practical systems.
Denoting e(k) ≜ x(k) − x(k), taking (3), (9), (10), and (14) into account, the estimation error system could be described as follows: Now, for simplicity to the following work, we introduce two augmentation vectors: en, based on estimation error system (17) and system (3), we can obtain the closed-loop system of the following form: where It follows from (2) and (11) and one has that Remark 4. Up to now, by considering the influences of timevarying delays and deception attacks, a novel observer-based PID security control model is established for the discretetime T-S fuzzy system under distributed DETM.
Definition 1 (See [11]). System (19) is said to be input-tostate stable (ISS) in mean-square sense if there exist functions κ ∈ QJ and ι ∈ Q ∞ , such that the system dynamic x(k) satisfies the following inequality: Discrete Dynamics in Nature and Society 5 where Definition 2 (See [11]). Denote the desired safety level as ϱ > 0; the closed-loop system (19) is said to be mean-square ϱ-secure if the following inequality holds: Lemma 1 (See [38]). For positive definite matrix Z with suitable dimensions and any real-valued variables x, y, the following inequality holds: where ϵ > 0 is a given constant.

Main Results
We mainly consider the security performance of the controlled plant (1) in this section. First, the sufficient condition is given to guarantee that the closed-loop system (19) is ISS and ϱ-secure in mean-square sense. en, the desired observer gains and PID controller gains are designed.
Discrete Dynamics in Nature and Society In eorem 1, sufficient conditions which can guarantee the ISS and ϱ-secure in mean-square sense of the closed-loop system (19) have been obtained. However, due to the presence of nonlinear terms such as PB i K P j in eorem 1, it cannot be solved by the method of LMIs. erefore, in the next part, we will deal with the nonlinear terms and obtain the observer gains and the controller gains. □ Theorem 2. Let the scalars λ(0 < λ < 1) and σ(σ > 0) satisfy λσ ≥ 1 . For scalar ϱ (ϱ > 0), the closed-loop system (19) is ϱ-secure in mean-square sense if, for all s ∈ L, i, j ∈ M, there exist positive integer n, positive scalars ε, θ, ϵ, c 1 , c 2 , c 3 , positive definite matrices P, Q r (r � 1, 2, . . . , d), R, Y, X, and matrices K P j , K I j , K D j , and L j satisfying Furthermore, the observer gains L j and the PID controller gains K P j , K I j , and K D j are given by Proof. e inequality (27) is equivalent to where 12 Discrete Dynamics in Nature and Society According to the Young inequality, there exists a positive definite matrix X ∈ R n x ×n x satisfying (67) ; applying to the Schur complement, (67) is equivalent to Based on Lemma 2, we can obtain en, by utilizing the Schur complement lemma, we can obtain that (60) can be ensured by (69). So far, the proof is completed. □ Remark 5. It is worth emphasizing that the linearization method adopted in this study can directly calculate the variable Y matrix in the LMI, thereby, quickly obtaining the PID controller gains, which greatly reduces the computational complexity.
Remark 6. Until now, we have addressed the observer-based PID security control issues for the discrete-time T-S fuzzy system with DETM and deception attacks and proposed the desired PID controllers under the designed closed-loop system. Compared with existing articles on the performance of T-S fuzzy systems, this article has two distinctions: (1) the problem solved is new as the security problem for the discrete-time T-S fuzzy system is first constructed under multisensor and deception attacks; (2) the PID control scheme is new in the sense that the influence of discrete-time delay, distributed DETM, and deception attacks on the PID controller are considered at the same time. It is worth mentioning that the main results derived in eorems 1-2 can be utilized to the fuzzy PID control problem for other systems, such as large-scale systems, multiagent systems, and neural networks.

Numerical Examples
For the purpose of illustrating the effectiveness of the designed fuzzy PID controller, we cite a simulation example in this section. Consider system (1) as follows: e model parameters are provided as follows: Assume the upper and lower bounds on the time delay as τ M � 3 and τ m � 1, respectively. e success probabilities of the deception attacks are set to be α � 0.7. Moreover, the event-triggered parameters in (6) and (7) are chosen as Discrete Dynamics in Nature and Society σ � 5, ε � 0.2, and λ � 0.2. e security level of this study is assumed to be ϱ � 0.6, and the length of time-windows is taken as d � 3. e solutions to LMIs (60)-(62) are obtained as follows: erefore, according to eorem 2, the desired PID controller gains and observer gains can be derived as follows: Set the initial states as e disturbance input is assumed to be v(k) � 0.5sin(k), and the deception attacks information launched by adversary is selected as e simulation results are shown in Figures 2-4. To be specific, Figure 2 shows the state trajectories of x(k) for discrete-time fuzzy system (1) under the distributed DETM and the deception attacks with PID control. Obviously, the proposed system (19) with observer-based PID control scheme is ISS and valid. Figures 3-4 and Table 1 show the impact of deception attacks on PID control performance. Specifically, Figures 3 and 4 show the trajectories of control input u(k) and the time instants of successful deception attacks, respectively. Table 1 provides the relationship between the probability of deception attacks and the minimum security level. It is easy to see that as the probability of attack increases, the minimum security level also increases. erefore, we can draw the following conclusion: the security performance of PID control deteriorates when the probability of deception attacks increases.
For the purpose to show the advantages of the DETM adopted in this study compared to the static event-triggered mechanism (SETM), we derive the comparison results shown in  e dynamic triggering instants of three distributed DETM are shown in Figure 5 and the static triggering instants are shown in Figure 6, from which we can know that the number of triggering time of the DETM are 24, 12, and 27 and the number of triggering time of the DETM are 37, 39, and 37. At this point, we can conclude that the triggering time of the DETM is much less than the STEM. Furthermore, we can draw that compared with the SETM, the DETM can save networked resources more effectively.
Remark 7. Based on the above discussion, it can be summarized that (1) under the observer-based PID control law proposed in this study, the closed-loop system (19) can achieve the expected stability; (2) system performance may be destroyed by deception attacks, and the security performance of the system decreases as the probability of the attack increases; (3) compared with the SETM, the DETM has more stringent trigger conditions, so the dynamic eventtriggered mechanism can more effectively reduce pressure on the network bandwidth.
14 Discrete Dynamics in Nature and Society

Conclusion
In this study, by considering DETM and deception attacks, the observer-based PID security control problem has been investigated for the discrete-time T-S fuzzy system. e distributed DETM has been employed to reduce the pressure of network bandwidth. Based on the designed T-S fuzzy model, sufficient conditions for the ISS of the closed-loop system have been derived. Moreover, the desired fuzzy controller gains have been obtained. Finally, the effectiveness of the proposed methods has been demonstrated by a numerical example. In future work, we will continue to pay attention to the observer-based PID control problem for discrete T-S fuzzy systems and combine this framework with various communication protocols, such as redundant channel transmission protocol, round-robin protocol, and stochastic communication protocol.

Data Availability
e data used to support the findings of this study have not been made available because some of our data are still confidential recently.

Conflicts of Interest
e authors declare that they have no conflicts of interest.     16 Discrete Dynamics in Nature and Society