Research Article  Open Access
Improved Distributed EventTriggered Control for Networked Control System under Random Cyberattacks via Bessel–Legendre Inequalities
Abstract
The stability problem of networked control system (NCS) with cyberattacks and processing delay is considered under an eventtriggered scheme. An improved distributed eventtriggered 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 eventtriggered controller is designed. The validity of the results is verified by numerical examples.
1. Introduction
In a practical NCS, it is inevitable that there exist some problems such as networkinduced 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 delaydependent 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 [5–7]. 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 everincreasing 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 eventbased rules, which can release the data to controller only when the sampled state satisfies the eventbased rule. Eventtriggered scheme overcomes the shortcoming of traditional periodictriggered scheme and derives extensive application [15, 16]. An eventtriggered controller was designed via a delay system method in [17]. Eventtriggered realtime scheduling method has been researched in [18]. For the stochastic Markovian jumping system, eventtriggered state estimation has been considered in [19]. Eventtriggered consensus control for multiagent systems has referred to in [20]. Recently, some modified eventtriggered schemes have been proposed to adapt different system demands. Distributed eventtriggered mechanism has been proposed in [21] for estimation of wireless sensor network system. Adaptive eventtriggered mechanism has attracted comprehensive attention [22]. To stochastic state estimation problem, a deterministic eventtriggered scheme has been proposed in [23]. In addition, in order to further reduce the release times, a dynamic eventtriggered 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 eventtriggered scheme has been raised to accelerate the dynamic process by constructing a timevarying parameter in triggering rule [25]. Here, we will establish an improved distributed eventtriggered scheme for cyberattacked networked control system. This 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]. Wirtingerbased inequality has been utilized to estimate the derivative of Lyapunov–Krasovskii functional in [29]. In addition, the auxiliary functionbased integral inequality has been introduced to various systems [30]. The methods mentioned above are all aiming to deal with the quadratic integral term such as “”, 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 ”, where = diag{, , …, (2N + 1)}, , N ≥ 0, and is the “shifted” Legendre polynomial matrix. The 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 eventtriggered 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 eventtriggered 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 eventtriggered 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.
The remainder of this article is organized as follows: definitions and problem formulation are described in Section 2. In addition, the proposed improved distributed eventtriggered 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.
Notations 1. Prob{X} means probability of event X occurring. ∥·∥_{2} denotes the Euclidean vector norm. means the natural number. For any matrix A, He(A) = A + A^{T}. means the mathematical expectation. expresses the set of positive definite matrices of , denotes the set of all n × n matrices.
2. Definitions and Problem Formulation
The considered state space model of NCS is described as follows:where is the state of the system. is the control input. A and B are the known constant matrices.
It is well known that the eventtriggered scheme can compensate the shortcoming of traditional periodictriggered 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 eventtriggered scheme will be introduced. The schematic diagram of distributed eventtriggered 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
Due to u(t) = Kx(t), we havewhere and K = diag{K_{1}, K_{2}, …, K_{n}}.
In fact, there exists a time delay during the process of signal transmission. Assume that is the transmission delay, where is a positive scalar, l = 1, 2, …, n. The released state arrives at the actuator at the time . Next, we will establish the system model with network transmission delay. Define that
Let
Then,
For , define that
Apparently, . Let , then .
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. Thus, we introduce the timevarying parameter σ_{k}(t), . Set l = 1, 2, …, n andwhereknown constant ɛ > 0, , is the upper bound of , and , . Next, we put forward the following improved static distributed eventtriggered scheme which contains the timevarying parameter σ_{k}(t) for t ∈ [t_{k}h + τ_{k}, t_{k+1}h + τ_{k+1}):where and Λ is the symmetric positive definite matrix satisfying Λ = diag{Λ_{1}, Λ_{2}, …, Λ_{n}}.
Using e_{k}(t) and τ(t), for t ∈ [t_{k}h + τ_{k}, t_{k+1}h + τ_{k+1}), we rewrite u(t) as
Consider cyberattacks launched by adversaries whose aim is to attack the controller. Thus, u(t) can be described as follows:where the variable β(t) satisfies the Bernoulli distribution. Prob, Prob, . 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 eventtriggered mechanism, the complete model of cyberattacked NCS with time delay iswhere the function Φ(t) is continuous on [−τ_{M}, 0]. Note thatwhere scalars d_{1} < 0 and d_{2} > 0.
To facilitate the analysis, some assumptions and lemmas are given as follows.
Assumption 1. The nonlinear function f(x(t)) which determines stochastic cyberattacks satisfies the following condition:where is a known constant matrix.
Lemma 1. (see [32]). Suppose that symmetric positive matrices . For ϵ = 0, 1, if there exist the symmetric matrices and such that the following inequality holds:then, for all ϵ ∈ (0, 1),is true.
The definitions about Legendre polynomials and the properties of polynomials matrix will be presented as below.
Definition 1. For any u ∈ [0, 1], , the “shifted” Legendre polynomial iswhere and the binomial coefficient .
Correspondingly, the polynomial matrix is described aswhere , . Due to that, the Legendre polynomials have the orthogonality property. Thus, for any symmetric positive definite matrix , the equationis true, where . The evaluation values of the polynomial matrix boundaries and 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 , ϒ_{N} = υ_{N} ⊗ I_{n}, and Ξ_{N} = ê_{N} ⊗ I_{n}. The matrices and are defined as
Lemma 2. (see [33]) (Bessel–Legendre inequality). For , any and , the inequalityholds, where
Remark 1. For , 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.
3. Stability Analysis
With the improved distributed eventtriggered mechanism, the less conservative stability criterion of cyberattacked NCS (14) is obtained via Bessel–Legendre inequalities. The main results are shown in Theorem 1.
Theorem 1. Given , scalar ɛ > 0, , if there exist matrix , matrices , and for all making the following inequality:true; then, system (14) is stable, whereand the unit matrix and .
Proof. Establish a Lyapunov–Krasovskii functional with respect to Legendre polynomials matrix aswherewhereBefore obtaining the derivative of V_{1N}, define thatthen,where Φ_{N} is defined in Theorem 1. Obviously,Among them,Set , then s = λτ(t) + t − τ(t), and rewriting Ψ_{1,N}(t) asthen,Using the subsection integration method and employing (23), (25), we can getCorrespondingly, for , using the subsection integration and employing (20), (24), we obtainThus,Obviously, by integral calculation, in is derived formed by the elements in η_{N}(t), which can facilitate the realization of inequality in Theorem 1. Next, we deal with . Set and s = uτ_{M} − uτ(t) + t − τ_{M}, then we rewrite Ψ_{2,N}(t) asThen,whereThus, becomesCombining ẋ(t) and (42), (46), we can obtainwhere E_{1,N} and E_{2,N} are defined in Theorem 1.
According to equations (35) and (47), we getNext, we deal with . According to Assumption 1, the following inequality is true:Add (49) and term on , thenwhere the matrix E_{3,N} is defined in Theorem 1:Due to that, ẋ(t) can be written aswhereEmploying , , then we getUsing Lemma 2, we havewhereEmploying the subsection integration method, we obtainwhereThus,Next, according to Lemma 1, 2 and choosing and [33], we get the following inequality:whereThen, using equations (54) and (60), we obtain as follows:Combining (48), (50) and (62), satisfies thatwhereDue toBy Schur complement, if Θ_{N} ≤ 0 for all , then , which guarantees for all . The proof is completed.
Remark 2. which is defined in Theorem 1 is multiaffine on τ(t) and , where . It is deserved to mention that the solution of LMI (29) with allowable delay set has lower conservatism than that with allowable delay set [0, τ_{M}] × [d_{1}, d_{2}]. Because of the impossible situations that is negative when τ(t) = 0 and is positive when τ(t) = τ_{M}, the vertices (0, d_{1}) and (τ_{M}, d_{2}) will never be reached at any time.
4. Stabilization Analysis
This part will design a controller for system (14) under the distributed eventtriggered mechanism. Specific results are in Theorem 2.
Theorem 2. Given , scalar ɛ > 0, , system (14) is stable if there exist , , , matrices , and such that the following inequality:is true for all , whereand other terms are defined in Theorem 1.
Proof. From (29) in Theorem 1, we can find the existence of nonlinear terms such as . Divide the matrix P_{N} into block matrix aswhere , , and .
The concrete expressions of the nonlinear terms become and in E_{1,N}P_{N}. To eliminate the nonlinear terms, set . Then, and become −I_{n}.
Moreover, BK will be replaced by . Due toWe replace BK with , then we obtainThus, we getwhereBy Schur complement, if , then in (67) is true. Due to , according to Theorem 1, system (14) is stable for any delay in allowable delay set . Thus, the proof ends.
To check the inequalities in Theorem 1 and Theorem 2, Algorithm 1 is presented.

5. Numerical Examples
Example 1. Consider the following distributed eventtriggered delaydependent NCS with cyberattacks:Take n = 2, , , , , ɛ = 0.01, h = 0.36, , and N = 0. The designed controller gain K and triggering parameter Λ are shown in Table 1. The state trajectories are shown in Figure 2.
From Figure 2, we can see that the system states achieve stability in the fourth second. Figure 3 shows the timevarying parameter and the release situation of x^{1}(t). Figure 4 shows the timevarying parameter and the release instants of x^{2}(t). From Figures 3 and 4, and vary from 0.01 to 0.3. The 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. Thus, the proposed eventtriggered scheme shortens the system dynamic process and reduces the transmission burden.
Next, set h = 0.25. Under the improved distributed eventtriggered scheme with timevarying 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 release interval of states x^{1}(t) and x^{2}(t) are shown in Figures 7 and 8, respectively. Based on the improved distributed eventtriggered scheme, transmission frequency at the initial times is higher than that under the general distributed eventtriggered strategy.
