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

Authors: Hongqian Lu; Chaoqun Guo; Yue Hu; Wuneng Zhou; Shihao Yan
Journal: Complexity
 (2020)

---

## Abstract

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 ofN. 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.


---

## Body

## 1. 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 [5–7]. 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 event-triggered 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 cyber-attacked 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]. 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]. The methods mentioned above are all aiming to deal with the quadratic integral term such as “−∫t−τMtx˙tWx˙tds”, 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 −∫t−τMtx˙tWx˙tds≤−∫t−τMtx˙tWx˙tds”, where WN = diag{W, 3W, …, (2N + 1)W}, ΨN=1/τM∫t−τMtℒNs−τM/τMxsds, N ≥ 0, and ℒN 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 event-triggered cyber-attacked 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 event-triggered mechanism to the comprehensive delay-dependent 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 cyber-attacked 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 orderN. 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 Section2. 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.Notations 1.
Prob{X} means probability of event X occurring. ∥·∥2 denotes the Euclidean vector norm. ℕ means the natural number. For any matrix A, He(A) = A + AT. ℰ⋅ means the mathematical expectation. S+N expresses the set of positive definite matrices of ℝn×n, ℝn×n denotes the set of all n × n matrices.


## 2. Definitions and Problem Formulation

The considered state space model of NCS is described as follows:(1)x˙t=Axt+But,where xt∈ℝn is the state of the system. ut∈ℝp 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 periodic-triggered 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. The schematic diagram of distributed event-triggered NCS with cyberattacks is shown in Figure1.Figure 1
The schematic diagram of distributed event-triggered NCS with cyberattacks.Define thatt0h, t1h, t2h, … as release times, which means that the sampled states at t0h, t1h, t2h, … satisfy event-triggering condition and can be sent to the transmission channel. System (1) should be described as(2)x˙t=Axt+Butkh.Due tou(t) = Kx(t), we have(3)x˙t=Axt+BKxtkh,where xtkh=x1tk1hTx2tk2hT⋯xntknhTT and K = diag{K1, K2, …, Kn}.In fact, there exists a time delay during the process of signal transmission. Assume thatτkl∈0,τ¯ is the transmission delay, where τ¯ is a positive scalar, l = 1, 2, …, n. The released state xltklh arrives at the actuator at the time tklh+τkl. Next, we will establish the system model with network transmission delay. Define that(4)ϕkl=minα|tklh+τkl+αh≥tk+1lh+τk+1l,α=0,1,2,….Let(5)Iαl=tklh+τkl+α−1h,tklh+τkl+αh,α=1,2,…,ϕkl−1,Iϕkl=tklh+τkl+ϕkl−1h,tk+1lh+τk+1l.Then,(6)tklh+τkl,tk+1lh+τk+1l=∪α=1ϕklIαl.Fort∈tklh+τkl,tk+1lh+τk+1l, define that(7)τlt=t−tklh,t∈I1l,t−tklh−h,t∈I2l,⋮⋮t−tklh−ϕkl−1h,t∈Iϕkl,eklt=0,t∈I1l,xtklh−xtklh+h,t∈I2l,⋮⋮xtklh−xtklh+ϕkl−1h,t∈Iϕkl.Apparently,0<τkl≤τlt≤τ¯+h. Let τM=τ¯+h, then 0<τkl≤τlt≤τM.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 time-varying parameterσk(t), σkt=diagσk1t,σk2t,…,σknt. Set l = 1, 2, …, n and(8)σklt=σkαl,t∈Iαl,α=1,2,…,ϕkl,where(9)σkα+1l=σ¯+ekαlTekαlε+ekαlTekαlσkαl−σ¯,(10)ekαl=xltklh−xltklh+α−1h,α=1,2,…,ϕkl,known constant ɛ > 0, σkα=diagσkα1,σkα2,…,σkαn, σ¯ is the upper bound of σkαl, and σ¯∈0,1, 0<σk1l<σ¯. Next, we put forward the following improved static distributed event-triggered scheme which contains the time-varying parameter σk(t) for t ∈ [tkh + τk, tk+1h + τk+1):(11)ekTtΛekt≤σktxTt−τtΛxt−τt,where xt−τt=x1t−τ1tTx2t−τ2tT⋯xnt−τntTT and Λ is the symmetric positive definite matrix satisfying Λ = diag{Λ1, Λ2, …, Λn}.Usingek(t) and τ(t), for t ∈ [tkh + τk, tk+1h + τk+1), we rewrite u(t) as(12)ut=BKxt−τt+BKekt.Consider cyberattacks launched by adversaries whose aim is to attack the controller. Thus,u(t) can be described as follows:(13)ut=1−βtBKxt−τt+ekt+βtfxt,where the variable β(t) satisfies the Bernoulli distribution. Probβt=1=β¯, Probβt=0=1−β¯, 0≤β¯≤1. 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(14)x˙t=Axt+1−βtBKxt−τt+BKekt+βtfxt,xt=ϕt,t∈−τM,0,where the function Φ(t) is continuous on [−τM, 0]. Note that(15)0<τk≤τt≤τM,d1≤τ˙t≤d2,where scalars d1 < 0 and d2 > 0.To facilitate the analysis, some assumptions and lemmas are given as follows.Assumption 1.
The nonlinear functionf(x(t)) which determines stochastic cyberattacks satisfies the following condition:(16)fxt2≤Gxt2,where G is a known constant matrix.Lemma 1.
(see [32]). Suppose that symmetric positive matrices W1,W2∈ℝn. For ϵ = 0, 1, if there exist the symmetric matrices X1,X2∈ℝn and Y1,Y2∈ℝn such that the following inequality holds:(17)W100W2−ϵX1Y1Y1T0−1−ϵ0Y2Y2T0≥0,then, for all ϵ ∈ (0, 1),(18)1ϵW10011−ϵW2≥W100W2+ϵ0Y1Y1TX2+1−ϵX1Y2Y2T0,is true.
The definitions about Legendre polynomials and the properties of polynomials matrix will be presented as below.Definition 1.
For anyu ∈ [0, 1], f,q∈ℕ, the “shifted” Legendre polynomial is(19)Lfu=−1f∑q=0fρqfuq,where ρqf=−1qfqf+qq and the binomial coefficient fq=f!/f−q!q!.
Correspondingly, the polynomial matrixLN is described as(20)LNu≔L0uIn,L1uIn,…,LNuInT,where n∈ℕ, N∈ℕ. Due to that, the Legendre polynomials have the orthogonality property. Thus, for any symmetric positive definite matrix W, the equation(21)∫01LNuW−1LNTudu=W¯N−1,is true, where W¯N=diagW,3W,…,2N+1W. The evaluation values of the polynomial matrix boundaries LN0 and LN1 are shown as follows:(22)LN0=In−In⋮−1NIn≔χ¯N,(23)LN1=InIn⋮In≔χN.
Next, we give the derivative about the Legendre polynomials matrix which will be employed in the proof process of system stability:(24)dduLNu=ϒ¯NLNu=ϒNLN−1u,(25)dduuLNu=LNu+ΞNLNu,where ϒ¯N=ϒN0nN+1,n, ϒN = υN ⊗ In, and ΞN = êN ⊗ In. The matrices υN∈ℝN+1×N and e^N∈ℝN+1×N+1 are defined as(26)υNf,q=0,iff≥q,2f−11−−1f+q,iff<q,e^Nf,q=0,iff>q,f,iff=q,2f−1,ifk<q.Lemma 2.
(see [33]) (Bessel–Legendre inequality). For x∈ℒ2a,b⟶ℝn, any N∈ℕ and W∈S+N, the inequality(27)∫abxTsWxsds≥b−aΨNTW¯NΨN,holds, where(28)ΨN=1b−a∫abLNs−ab−axsds,W¯N=diagW,3W,…,2N+1W.Remark 1.
For−∫abxTsWxsds, 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 event-triggered mechanism, the less conservative stability criterion of cyber-attacked NCS (14) is obtained via Bessel–Legendre inequalities. The main results are shown in Theorem 1.Theorem 1.
GivenN∈ℕ, scalar ɛ > 0, 0≤β¯≤1, if there exist matrix PN∈S+2N+3n, matrices Q1,Q2,W∈S+n, and Y1,Y2∈ℝN+1n×N+1n for all τt,τ˙t∈ℋ making the following inequality:(29)ΘN′=Θ¯NΠNTτM−τtτMY10ΠNT0τtτMY2TτMΓ1TτMβ¯1−β¯Γ2T∗−τM−τtτMW¯N000∗∗−τtτMW¯N00∗∗∗W−2In0∗∗∗∗W−2In≤0,true; then, system (14) is stable, where(30)ℋ=Co0,0,0,d2,τM,0,τM,d1,Θ¯N=HeE1,N+τ˙tE2,NTPNΦN+E3,N−ΠNTΛNΠN,ΠN=χN−ϒ¯N0−χ¯N00000−ϒ¯NχN−χ¯N00,W¯N=diagW,3W,…,2N+1W,E1,N=A001−β¯BK01−β¯BKβ¯InχN−ϒ¯N0−χ¯N00000−ϒ¯NχN−χ¯N00,E2,N=00000000−ΞN+ϒ¯N0χ¯N00000ΞN−χ¯N000,ΦN=In0000000τtInN0000000τM−τtInN0000,ΛN=W¯N00W¯N+τtτM0Y1Y1TW¯N+τM−τtτMW¯NY2Y2T0,E3,N=diagQ1+GTG,0,0,1−τ˙tQ2−Q1+σktΛ,−Q2,−Λ,−In,Γ1=A001−β¯BK01−β¯BKβ¯In,Γ2=000BK0BK−In,and the unit matrix InN∈ℝN+1n×N+1n and In∈ℝn×n.Proof.
Establish a Lyapunov–Krasovskii functional with respect to Legendre polynomials matrix as(31)VNxt,x˙t,t=V1Nxt,t+V2Nxt,t+V3Nxt,x˙t,t,where(32)V1Nxt,t=ξNTtPNξNt,V2Nxt,t=∫t−τttxTsQ1xsds+∫t−τMt−τtxTsQ2xsds,V3Nxt,x˙t,t=τM∫t−τMt∫θtx˙TsWx˙sdsdθ,where(33)ξNTt=xTtτtΨ1,NTtτM−τtΨ2,NTt,Ψ1,Nt=1τt∫t−τttLNs−t+τtτtxsds,Ψ2,Nt=1τM−τt∫t−τMt−τtLNs−t+τMτM−τtxsds.
Before obtaining the derivative ofV1N, define that(34)ηNTt=xTtΨ1,NTtΨ1,NTtxTt−τtxTt−τMekTtfTxt,then,(35)ξNTt=ηNTtΦNT,where ΦN is defined in Theorem 1. Obviously,(36)ξ˙NTt=x˙TtddtτtΨ1,NTtddtτM−τtΨ2,NTt.
Among them,(37)x˙t=Axt+1−βtBKxt−τt+BKekt+βtfxt,Ψ1,Nt=1τt∫t−τttLNs−t+τtτtxsds,Ψ2,Nt=1τM−τt∫t−τMt−τtLNs−t+τMτM−τtxsds.
Setλ=s−t+τt/τt, then s = λτ(t) + t − τ(t), and rewriting Ψ1,N(t) as(38)Ψ1,Nt=∫01LNλxsλdλ,then,(39)ddtτtΨ1,Nt=τ˙tΨ1,Nt+τt∫01LNλx˙sλλτ˙t+1−τ˙tdλ=τ˙tΨ1,Nt+τ˙tτt∫01λLNλx˙sλdλ+1−τ˙tτt∫01LNλx˙sλdλ.
Using the subsection integration method and employing (23), (25), we can get(40)τt∫01λLNλx˙sλdλ=∫01λLNλdxsλ=LN1xt−∫01xsλddλλLNλdλ=LN1xt−Ψ1,Nt−ΞNΨ1,Nt=χNxt−Ψ1,Nt−ΞNΨ1,Nt.
Correspondingly, forτt∫01LNλx˙sλdλ, using the subsection integration and employing (20), (24), we obtain(41)τt∫01LNλx˙sλdλ=∫01LNλdxsλ=LN1xt−LN0xt−τt−∫01xsλddλLNλdλ=χNxt−χ¯Nxt−τt−ϒ¯NΨ1,Nt.
Thus,(42)ddtτtΨ1,Nt=τ˙tΨ1,Nt+τ˙tχNxt−Ψ1,Nt−ΞNΨ1Nt+1−τ˙tχNxt−χ¯Nxt−τt−ϒ¯Ψ1,Nt.
Obviously, by integral calculation,d/dtτtΨ1,Nt in ξ˙Nt is derived formed by the elements in ηN(t), which can facilitate the realization of inequality in Theorem 1. Next, we deal with d/dtτM−τtΨ2,Nt. Set u=s−t+τM/τM−τt and s = uτM − uτ(t) + t − τM, then we rewrite Ψ2,N(t) as(43)Ψ2,Nt=∫01LNuxsudu.
Then,(44)ddtτM−τtΨ2,Nt=−τ˙tΨ2,Nt+τM−τt∫01LNux˙su−uτ˙t+1du=−τ˙tΨ2,Nt−τ˙tτM−τt∫01uLNux˙sudu+τM−τt∫01LNux˙sudu,where(45)τM−τt∫01uLNux˙sudu=∫01uLNudxsu=LN1xt−τt−Ψ2,Nt−ΞNΨ2,Nt=χNxt−τt−Ψ2,Nt−ΞNΨ2,Nt,τM−τt∫01LNux˙sudu=∫01LNudxsu=LN1xt−τt−LN0xt−τM−ϒ¯NΨ2,Nt=χNxt−τt−χ¯Nxt−τM−ϒ¯NΨ2,Nt.
Thus,d/dtτM−τtΨ2,Nt becomes(46)ddtτM−τtΨ2,Nt=−τ˙tΨ2,Nt−τ˙tχNxt−τt−Ψ2,Nt−ΞNΨ2,Nt+χNxt−τt−χ¯Nxt−τM−ϒ¯NΨ2,Nt.
Combiningẋ(t) and (42), (46), we can obtain(47)ℰξ˙Nt=E1,N+τ˙tE2,NηNt,where E1,N and E2,N are defined in Theorem 1.
According to equations (35) and (47), we get(48)ℰV˙1N=2ξNTtPNℰξ˙Nt=ηTtHeE1,N+τ˙tE2,NTPNΦNηNt.
Next, we deal withℰV˙2Nt. According to Assumption 1, the following inequality is true:(49)xTtGTGxt−fTxtfxt≥0.
Add (49) and term ekTtΛekt−ekTtΛekt on V˙2N, then(50)ℰV˙2N≤xTtQ1xt+1−τ˙txTt−τtQ2−Q1xt−τt−xTt−τMQ2xt−τM+ekTtΛekt−ekTtΛekt+xTtGTGxt−fTxtfxt≤ηNTtE3,NηNt,where the matrix E3,N is defined in Theorem 1:(51)V˙3N=τM2x˙TtWx˙t−τM∫t−τMtx˙TsWx˙sds.
Due to that,ẋ(t) can be written as(52)x˙t=Ω1+β¯−βtΩ2,where(53)Ω1=Axt+1−β¯BKxt−τt+BKekt+β¯fxt,Ω2=β¯−βtBKxt−τt+BKekt−fxt.
Employingℰβ¯−βt=0, ℰβ¯−βt2=β¯1−β¯, then we get(54)ℰτM2x˙TtWx˙t=τM2Ω1TWΩ1+τM2β¯1−β¯Ω2TWΩ2.
Using Lemma2, we have(55)τM∫t−τMtx˙TsWx˙sds≥Ψ¯1,NΨ¯2,NTτMτtW¯N00τMτM−τtW¯NΨ¯1,NΨ¯2,N,where(56)Ψ¯1,N=1τt∫t−τttLNs−t+τtτtx˙sds,Ψ¯2,N=1τM−τt∫t−τMt−τtLNs−t+τMτM−τtx˙sds.
Employing the subsection integration method, we obtain(57)τtΨ¯1,NτM−τtΨ¯2,N=χNxt−χ¯Nxt−τt−ϒ¯NΨ1,NχNxt−τt−χ¯Nxt−τM−ϒ¯NΨ2,N=ΠNηNt,where(58)ΠN=χN−ϒ¯N0−χ¯N0000ϒ¯NχN−χ¯N0.
Thus,(59)−τM∫t−τMtx˙TsWx˙sds≤−ηNTtΠNTτMτtW¯N00τMτM−τtW¯NΠNηNt.
Next, according to Lemma1, 2 and choosing X1=WN−Y1WN−1Y1T and X2=WN−Y2TWN−1Y2 [33], we get the following inequality:(60)−τM∫t−τMtx˙TsWx˙sds≤−ηNTtΠNTΛNΠNηNt+ηNTtΠNTΛN′ΠNηNt,where(61)ΛN=W¯N00W¯N+τtτM0Y1Y1TW¯N+τM−τtτMW¯NY2Y2T0,ΛN′=τM−τtτMY1W¯N−1Y1T00τtτMY2TW¯N−1Y2.
Then, using equations (54) and (60), we obtain ℰV˙3N as follows:(62)ℰV˙3N=ηNTtΠNTΛN′ΠN−ΠNTΛNΠNηNt+τM2Ω1TWΩ1+τM2β¯1−β¯Ω2TWΩ2.
Combining (48), (50) and (62), ℰV˙N satisfies that(63)ℰV˙N≤ηNTtHeE1,N+τ˙tE2,NTPNΦN+E3,N+ΠNTΛN′ΠN−ΠNTΛNΠNηNt+τM2Ω1TWΩ1+τM2β¯1−β¯Ω2TWΩ2=ηNTtΘNηNt,where(64)ΘN=Θ¯N+ΠNTΛN′ΠN+τM2Γ1TWΓ1+τM2β¯1−β¯Γ2TWΓ2,Θ¯N=HeE1,N+τ˙tE2,NTPNΦN+E3,N−ΠNTΛNΠN,Γ1=A001−β¯BK01−β¯BKβ¯In,Γ2=000BK0BK−In.
Due to(65)In−WW−1In−W>0,(66)−W−1≤−2In−W.
By Schur complement, if ΘN ≤ 0 for all τt,τ˙t∈ℋ=Co0,0,0,d2,τM,0,τM,d1, then ΘN′≤0, which guarantees ℰV˙N≤0 for all τt,τ˙t∈ℋ. The proof is completed.Remark 2.
Θ
N
′ which is defined in Theorem 1 is multi-affine on τ(t) and τ˙t, where τt,τ˙t∈ℋ=Co0,0,0,d2,τM,0,τM,d1. 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, τM] × [d1, d2]. Because of the impossible situations that τ˙t is negative when τ(t) = 0 and τ˙t is positive when τ(t) = τM, the vertices (0, d1) and (τM, d2) will never be reached at any time.


## 4. Stabilization Analysis

This part will design a controller for system (14) under the distributed event-triggered mechanism. Specific results are in Theorem 2.Theorem 2.
GivenN∈ℕ, scalar ɛ > 0, 0≤β¯≤1, system (14) is stable if there exist P^1,N∈S+n, P^2,N∈S+N+1n, P^3,N∈S+N+1n, matrices Q1,Q2,W∈S+n, and Y1,Y2∈ℝN+1n×N+1n such that the following inequality:(67)Θ^N′=Θ¯N′ΠNTτM−τtτMY10ΠNT0τtτMY2TτMΓ1TτMβ¯1−β¯Γ2T∗−τM−τtτMW¯N000∗∗−τtτMW¯N00∗∗∗W−2In0∗∗∗∗W−2In≤0,is true for all τt,τ˙t∈ℋ, where(68)ℋ=Co0,0,0,d2,τM,0,τM,d1,Θ¯N′=HeE1,N′+τ˙tE2,NTPNΦN+E3,N−ΠNTΛNΠN,E1,N′=A00−1−β¯P^1,N−10−1−β¯P^1,N−1β¯InχN−ϒ¯N0−χ¯N00000−ϒ¯NχN−χ¯N00,Γ^1=A001−β¯P^1,N−2In01−β¯P^1,N−2Inβ¯In,Γ^2=000P^1,N−2In0P^1,N−2In−In,PN=P^1,N000P^2,N000P^3,N,and other terms are defined in Theorem 1.Proof.
From (29) in Theorem 1, we can find the existence of nonlinear terms such as E1,NTPNΦN. Divide the matrix PN into block matrix as(69)PN=P^1,N000P^2,N000P^3,N,where P^1,N∈S+n, P^2,N∈S+N+1n, and P^3,N∈S+N+1n.
The concrete expressions of the nonlinear terms becomeKTBTP^1,N and P^1,NBK in E1,NPN. To eliminate the nonlinear terms, set K=−B−1P^1,N−1. Then, KTBTP^1,N and P^1,NBK become −In.
Moreover,BK will be replaced by −P^1,N−1. Due to(70)In−P^1,NP^1,N−1In−P^1,N≥0,(71)−P^1,N−1≤P^1,N−2In.
We replaceBK with P^1,N−2In, then we obtain(72)Γ^1=A001−β¯P^1,N−2In01−β¯P^1,N−2Inβ¯In,Γ^2=000P^1,N−2In0P^1,N−2In−In.
Thus, we get(73)Θ^N=Θ¯N′+ΠNTΛN′ΠN++τM2Γ1TWΓ1+τM2β¯1−β¯Γ2TWΓ2,where(74)Θ¯N′=HeE1,N′+τ˙tE2,NTPNΦN+E3,N−ΠNTΛNΠN.
By Schur complement, ifΘ^N≤0, then Θ^N′≤0 in (67) is true. Due to Θ^N′≥ΘN′, according to Theorem 1, system (14) is stable for any delay in allowable delay set ℋ. Thus, the proof ends.
To check the inequalities in Theorem1 and Theorem 2, Algorithm 1 is presented.Algorithm 1: The algorithm for solving linear matrix inequality.
(1)
Set the system model parametersA and B, the probability parameter β¯, and N = 0. Assume the upper bound of the improved event-triggered scheme σ¯ = 0.3, sampling period h = 0.36, ɛ = 0.01, σk11=0.01, and σk12=0.01.
(2)
Set theσkt satisfying equations (8)–(10). Use LMI toolbox in MATLAB to construct the linear matrix inequality (29) or (67).
(3)
Set the initial conditionτM and the constrained condition τ(t), τ˙t.
(4)
Adjust the value ofτM to calculate whether the solution of LMI exists.
(5)
If the feasible solution does not exist, stop the calculation and obtain the maximum value ofτM, the controller gain K, and the triggering parameter Λ, else go to step 4 and increase the value of τM.
(6)
Stop.


## 5. Numerical Examples

Example 1.
Consider the following distributed event-triggered delay-dependent NCS with cyberattacks:(75)A=−0.211.6001.55−1.90000−0.720.40000.25−0.56,B=−0.500001.80000−1.200000.5.
Taken = 2, xt=x1tTx2tTT, σ¯=0.3, σk11=0.01, σk12=0.01, ɛ = 0.01, h = 0.36, β¯=0.5, 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 Figure2, we can see that the system states achieve stability in the fourth second. Figure 3 shows the time-varying parameter σk1t and the release situation of x1(t). Figure 4 shows the time-varying parameter σk2t and the release instants of x2(t). From Figures 3 and 4, σk1t and σk2t 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 x1(t), only 47% of the sampled data is sent. For state x1(t), there is 39% of the sampled data being sent. Thus, the proposed event-triggered scheme shortens the system dynamic process and reduces the transmission burden.
Next, seth = 0.25. Under the improved distributed event-triggered scheme with time-varying parameter σk(t), the release instants and release interval of states x1(t) and x2(t) are given in Figures 5 and 6, respectively. If we replace σk(t) with the constant σ, the corresponding release instants and release interval of states x1(t) and x2(t) are shown in Figures 7 and 8, respectively. Based on the improved distributed event-triggered scheme, transmission frequency at the initial times is higher than that under the general distributed event-triggered strategy.Table 1
The control parametersK and Λ.
t
0
1
…
20
Λ
10.1
0.04
0.13
0.62
0.04
12.3
0.64
0.78
0.13
0.64
21.9
0.28
0.62
0.78
0.28
6.50
1.47
0.01
0.17
0.33
0.01
6.22
0.12
0.04
0.17
0.12
2.90
0.40
0.33
0.04
0.40
17.3
…
0.29
0.03
0.11
0.17
0.03
9.58
0.78
−
2
0.11
0.78
11.2
0
0.17
−
2
0
1
K
−
1.87
0
0
0
0
0.34
0
0
0
0
−
0.91
0
0
0
0
−
0.44
T
−
0.71
0
0
0
0
−
0.14
0
0
0
0
−
2.10
0
0
0
0
1.09
T
…
−
2.34
0
0
0
0
0.47
0
0
0
0
0
1.00
0
0
0
0.97
TFigure 2
State trajectoriesx(t) of the cyber-attacked networked control system.Figure 3
The parameterσk1t and the release instants of x1(t).Figure 4
The parameterσk2t and the release instants of x2(t).Figure 5
The release situation of statex1(t) under the improved distributed event-triggered scheme.Figure 6
The release instants and release interval of statex1(t) under the distributed event-triggered scheme with constant parameter σ.Figure 7
The release instants and release interval of statex2(t) under the improved distributed event-triggered scheme.Figure 8
The release instants and release interval of statex2(t) under the distributed event-triggered scheme with constant parameter σ.Example 2.
Take the same system parameters in Example1. Set different N, get the corresponding upper bounds of network time delay with τt,τ˙t∈ℋ. Table 2 shows the comparison of the upper bounds of delay with other papers. If N = 0, then τM = 2.2161, and it is bigger than other papers. If N = 1, then τM = 2.3521. If N = 2, then τM = 2.9552. Apparently, τM increases with N increasing, which signifies conservativeness of stability criterion decreasing.Table 2
Delay upper boundτM for different τ˙td2=−d1=d˙t.
Method\d˙t
0.1
0.2
0.3
0.8
N = 2
2.9552
2.6306
2.2011
2.1174
N = 1
2.3521
1.9055
1.6732
1.4095
N = 0
2.2161
1.7773
1.4554
1.0161
[34]
0.7522
0.7197
0.6496
0.4768
[35]
2.1403
1.0653
0.8671
0.7644


## 6. Conclusions

The 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. The 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.



---

*Source: 1583286-2020-03-25.xml*