Improved Distributed Event-Triggered Control for Networked Control System under Random Cyberattacks via Bessel–Legendre Inequalities

The stability problem of networked control system (NCS) with cyberattacks and processing delay is considered under an event-triggered scheme. An improved distributed event-triggered mechanism is proposed, which optimizes the performance of system dynamics and decreases the network transmission load simultaneously. By means of Bessel–Legendre inequality method and constructing an active Lyapunov–Krasovskii functional, a series of larger upper bounds of delay are obtained corresponding to the order of N . It is worth mentioning that the upper bound increases with N , which means that the conservatism of the stability criterion lowers. Finally, a distributed event-triggered controller is designed. The validity of the results is veriﬁed by numerical examples.


Introduction
In a practical NCS, it is inevitable that there exist some problems such as network-induced delay, perturbation, and packet dropouts. As for NCS with time delay, a large number of researchers have done many relevant investigations. Filtering for discrete NCS with random time delays has been studied in [1]. A fault detection filter has been designed in [2]. Control problem for delay-dependent NCS with actuator faults has been researched in [3,4]. Recently, researchers began to pay a significant attention to cyberattacks, which can cause a serious network security issues [5][6][7]. Cyberattacks often lead to instability of system and deterioration of performance. In fact, cyberattacks include three forms: denial of service [8,9], replay attacks [10], and deception attacks [11]. Kalman filtering for nonlinear systems with denial of service attack has been studied in [12]. For stochastic system with deception attacks, distributed filtering problem has been investigated in [13]. Performance analysis of NCS under replay attacks has been given in [14]. Although many research works on cyberattack problems have been conducted in the literature, cyberattack issue has not been fully addressed for various NCS, which serves as the main motivation of this work.
With the development of information technology, an ever-increasing amount of data needs to be sent through networks. Unfortunately, the bandwidth of the network channel is subject to limited resources. For the sake of reducing the burden of network transmission, it is preferred to utilize the event-based rules, which can release the data to controller only when the sampled state satisfies the event-based rule. Event-triggered scheme overcomes the shortcoming of traditional periodic-triggered scheme and derives extensive application [15,16]. An eventtriggered controller was designed via a delay system method in [17]. Event-triggered real-time scheduling method has been researched in [18]. For the stochastic Markovian jumping system, event-triggered state estimation has been considered in [19]. Event-triggered consensus control for multiagent systems has referred to in [20]. Recently, some modified event-triggered schemes have been proposed to adapt different system demands. Distributed event-triggered mechanism has been proposed in [21] for estimation of wireless sensor network system. Adaptive event-triggered mechanism has attracted comprehensive attention [22]. To stochastic state estimation problem, a deterministic event-triggered scheme has been proposed in [23]. In addition, in order to further reduce the release times, a dynamic event-triggered mechanism was put forward by introducing a dynamic variable [24]. Nowadays, people are not only committed to saving network transmission resources but also devoting to enhance the system dynamics behavior. For instance, a new static event-triggered scheme has been raised to accelerate the dynamic process by constructing a time-varying parameter in triggering rule [25]. Here, we will establish an improved distributed event-triggered scheme for cyberattacked networked control system.
is can not only reduce the load of the network communication but also enhance the property of system dynamics, which has never been tackled in the literature.
It is significant to guarantee that the system is stable within a certain range of delay. In order to reduce the conservatism of the upper bound of system delay, we usually expect the upper bound as large as possible. Lyapunov-Krasovskii functional method as a powerful method has two main matters to research to further increase the upper bound of time delay: constructing an appropriate Lyapunov-Krasovskii functional and estimating the derivative of the functional. In terms of more accurate estimation of derivative, over the past few years, many approaches have been proposed by researchers. For example, the model transformation method was employed in [26]. Free weighting matrix approach was researched in [27]. Later, researchers began to pay attention to Jensen inequality to derive the better upper bound of delay [28]. Wirtinger-based inequality has been utilized to estimate the derivative of Lyapunov-Krasovskii functional in [29]. In addition, the auxiliary function-based integral inequality has been introduced to various systems [30]. e methods mentioned above are all aiming to deal with the quadratic integral term such as "− x(t)ds", which can be concluded in the derivative of Lyapunov-Krasovskii functional. Recently, a method called Bessel-Legendre inequality method was proposed in [31], which read as − x(s)ds, N ≥ 0, and L N is the "shifted" Legendre polynomial matrix. e criterion of stability is related to the order N, and its conservatism will decrease as N grows. Now, a suitable Lyapunov-Krasovskii functional will be established, and a larger delay upper bound of the event-triggered cyberattacked NCS will be got by means of Bessel-Legendre inequality.
Consequently, the following questions on the comprehensive NCS under cyberattacks will be addressed: (1) In order to save network transmission resources and improve dynamic property, we expect that suitable triggering scheme can realize that more triggers at the initial times and less triggers at the period tend to stable. How to devise an improved distributed eventtriggered mechanism to the comprehensive delaydependent NCS under cyberattacks for achieving above expectation? (2) Whether can we apply the Bessel-Legendre inequality approach to the investigation of stability for the system in this article? How to establish a powerful Lyapunov-Krasovskii functional applicable to the Bessel-Legendre inequality method? (3) Under the improved distributed event-triggered scheme, is it possible to design an effective controller to the NCS?
Motivated by the aforementioned challenges, the major contributions are listed as follows: (1) A more practical model of the networked control system subject to cyberattacks and time delay is constructed. A novel distributed event-triggered scheme is established for the comprehensive system researched in this paper, which can not only accelerate the system dynamics but also reduce communication burden. (2) For the analysis of cyberattacked NCS, not alike previous researches, this paper constructs a Lyapunov-Krasovskii functional with respect to Legendre polynomials and applies Bessel-Legendre inequality approach to acquire a less conservative stability condition, which is related to the order N. When N increases, the upper bound of delay increases. (3) An effective controller is devised. e remainder of this article is organized as follows: definitions and problem formulation are described in Section 2. In addition, the proposed improved distributed event-triggered scheme is also given in Section 2. Section 3 gives the specific stability analysis process. A controller is designed in Section 4. In Section 5, numerical examples are shown to illustrate the effectiveness of the proposed method. At last, conclusions are described in Section 6.

Definitions and Problem Formulation
e considered state space model of NCS is described as follows: where x(t) ∈ R n is the state of the system. u(t) ∈ R p is the control input. A and B are the known constant matrices. It is well known that the event-triggered scheme can compensate the shortcoming of traditional periodic-triggered scheme. For example, under the traditional periodictriggered scheme, some unnecessary signals can be sent to the channel, which places a burden on the limited bandwidth. Consider that the system has multiple sensors. In order to not only reduce the network transmission burden but also improve the system dynamics, a novel distributed event-triggered scheme will be introduced. e schematic diagram of distributed event-triggered NCS with cyberattacks is shown in Figure 1.
Define that t 0 h, t 1 h, t 2 h, . . . as release times, which means that the sampled states at t 0 h, t 1 h, t 2 h, . . . satisfy eventtriggering condition and can be sent to the transmission channel. System (1) should be described as where x(t k h) � x 1 (t 1 k h) T x 2 (t 2 k h) T · · · x n (t n k h) T T and K � diag{K 1 , K 2 , . . ., K n }.
In fact, there exists a time delay during the process of signal transmission. Assume that τ l k ∈ (0, τ) is the transmission delay, where τ is a positive scalar, l � 1, 2, . . ., n. e released state x l (t l k h) arrives at the actuator at the time t l k h + τ l k . Next, we will establish the system model with network transmission delay. Define that Let en, In order to shorten the system dynamic process, we expect that there are more packets transmitted at the initial times and the triggering frequency lowers when the system gets close to the steady state. us, we introduce the timevarying parameter where Next, we put forward the following improved static distributed eventtriggered scheme which contains the time-varying param-  where Using e k (t) and Consider cyberattacks launched by adversaries whose aim is to attack the controller. us, u(t) can be described as follows: where the variable β(t) satisfies the Bernoulli distribution.
When β(t) � 1, cyberattacks occur. When β(t) � 0, the released signals will be sent through network without cyberattacks. Nonlinear function f(x(t)) denotes the cyberattack characteristics. Next, with the novel distributed event-triggered mechanism, the complete model of cyber-attacked NCS with time delay is where the function Φ(t) is continuous on [− τ M , 0]. Note that where scalars d 1 < 0 and d 2 > 0.
To facilitate the analysis, some assumptions and lemmas are given as follows.

Assumption 1.
e nonlinear function f(x(t)) which determines stochastic cyberattacks satisfies the following condition: where G is a known constant matrix.
The definitions about Legendre polynomials and the properties of polynomials matrix will be presented as below. where Correspondingly, the polynomial matrix L N is described as where n ∈ N, N ∈ N. Due to that, the Legendre polynomials have the orthogonality property. us, for any symmetric positive definite matrix W, the equation e evaluation values of the polynomial matrix boundaries L N (0) and L N (1) are shown as follows: Next, we give the derivative about the Legendre polynomials matrix which will be employed in the proof process of system stability: where Lemma 2 (see [33]) (Bessel-Legendre inequality). For holds, where Bessel-Legendre inequality can estimate a tighter upper bound than other methods. In addition, the obtained upper bound can be as tight as possible along with N approaching to infinity. Consequently, the stability criterion to be obtained next will be less conservative, and the effectiveness will be verified in final example.

Stability Analysis
With the improved distributed event-triggered mechanism, the less conservative stability criterion of cyber-attacked NCS (14) is obtained via Bessel-Legendre inequalities. e main results are shown in eorem 1.
) ∈ H making the following inequality: Complexity true; then, system (14) is stable, where and the unit matrix I nN ∈ R (N+1)n×(N+1)n and I n ∈ R n×n .

Due to
It is deserved to mention that the solution of LMI (29) with allowable delay set H has lower conservatism than that with allowable delay set [0, . Because of the impossible situations that _ τ(t) is negative when τ(t) � 0 and _ τ(t) is positive when τ(t) � τ M , the vertices (0, d 1 ) and (τ M , d 2 ) will never be reached at any time.
Proof. From (29) in eorem 1, we can find the existence of nonlinear terms such as E T 1,N P N Φ N . Divide the matrix P N into block matrix as where P 1,N ∈ S n + , P 2,N ∈ S (N+1)n + , and P 3,N ∈ S (N+1)n + . e concrete expressions of the nonlinear terms become K T B T P 1,N and P 1,N BK in E 1,N P N . To eliminate the nonlinear terms, set K � − B − 1 P − 1 1,N .
en, K T B T P 1,N and P 1,N BK become − I n .
Moreover, BK will be replaced by − P − 1 1,N . Due to We replace BK with P 1,N − 2I n , then we obtain us, we get where By Schur complement, if Θ N ≤ 0, then Θ N ′ ≤ 0 in (67) is true. Due to Θ N ′ ≥ Θ N ′ , according to eorem 1, system (14) is stable for any delay in allowable delay set H. us, the proof ends.
To check the inequalities in eorem 1 and eorem 2, Algorithm 1 is presented.
From Figure 2, we can see that the system states achieve stability in the fourth second. Figure 3 shows the timevarying parameter σ 1 k (t) and the release situation of x 1 (t). Figure 4 shows the time-varying parameter σ 2 k (t) and the release instants of x 2 (t). From Figures 3 and 4, σ 1 k (t) and σ 2 k (t) vary from 0.01 to 0.3. e release frequency of the x(t) is higher at the beginning times than other times. When the states tend to be stable, the release times gradually decrease. For state x 1 (t), only 47% of the sampled data is sent. For state x 1 (t), there is 39% of the sampled data being sent. us, the proposed event-triggered scheme shortens the system dynamic process and reduces the transmission burden.
Next, set h � 0.25. Under the improved distributed eventtriggered scheme with time-varying parameter σ k (t), the release instants and release interval of states x 1 (t) and x 2 (t) are given in Figures 5 and 6, respectively. If we replace σ k (t) with the constant σ, the corresponding release instants and (1) Set the system model parameters A and B, the probability parameter β, and N � 0. Assume the upper bound of the improved eventtriggered scheme σ � 0.3, sampling period h � 0.36, ε � 0.01, σ 1 k1 � 0.01, and σ 2 k1 � 0.01. (2) Set the σ kt satisfying equations (8)- (10). Use LMI toolbox in MATLAB to construct the linear matrix inequality (29) or (67).     Example 2. Take the same system parameters in Example 1. Set different N, get the corresponding upper bounds of network time delay with (τ(t), _ τ(t)) ∈ H. Table 2 shows the comparison of the upper bounds of delay with other papers. If N � 0, then τ M � 2.2161, and it is bigger than other papers.      If N � 1, then τ M � 2.3521. If N � 2, then τ M � 2.9552. Apparently, τ M increases with N increasing, which signifies conservativeness of stability criterion decreasing.

Conclusions
e distributed event-triggered control problem for NCS with time delay was investigated in this paper. We considered a scenario where NCS may suffer from deception cyberattacks. e distributed event-triggered scheme is improved by introducing a time-varying parameter in triggered strategy to achieve that there is a higher frequency of release at the beginning times than other times. To obtain a less conservative stability criterion, the Bessel-Legendre inequality method was applied. In addition, a Lyapunov-Krasovskii functional-related Legendre polynomial was constructed. Consequently, we get a larger upper bound on the time delay. At last, a distributed event-triggered controller was also designed.
Future research directions include, but not limited to, filter research of NCS with multiple cyberattacks and time delay based on novel distribute event-triggered strategy, distributed event-triggered stabilization of cyber-attacked NCS with serious uncertainties, and further improvement of an event-triggered scheme for various specific requirements.

Data Availability
All data generated or analyzed during this study are included in this article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.