# New Stability Criteria for Event-Triggered Nonlinear Networked Control System with Time Delay

Authors: Hongqian Lu; Yue Hu; Chaoqun Guo; Wuneng Zhou
Journal: Complexity
 (2019)

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## Abstract

This note focuses on the stability and stabilization problem of nonlinear networked control system with time delay. To alleviate the burden of transformation channel and shorten the dynamic process simultaneously, an improved event-triggered scheme is proposed. This paper employs an improved time delay method to enhance the performance and reduce the delay upper bound conservatism. Less conservative stability criteria related to the orderN are derived by establishing an augmented Lyapunov-Krasovskii functional manufactured for the use of Bessel-Legendre inequality. In addition, an event-triggered controller is designed for nonlinear networked control system with time delay. At last, numerical examples are proposed to verify the effectiveness of the new method.


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## Body

## 1. Introduction

In recent years, the networked control has been applied to the practical control process [1]. Compared with the point to point control method, the networked control system has better reliability and can reduce power requirements, operation cost [2, 3]. There are different kinds of networked control systems, such as centralized networked control system [4], decentralized networked control system [5], distributed networked control system [6], and wireless networked control system [7]. In practice, the performance of networked control system is always influenced by the uncertainties and disturbance and nonlinear factors. Nonlinear networked control system’s asymptotic behavior has been researched in [8]. Stability of nonlinear networked control system has been studied in [9]. The literature [10, 11] have investigated robust stability of nonlinear networked control system with uncertainties. This paper also takes into account nonlinearity to augment the performance of networked control system.For traditional networked control system [12, 13], transmitting all sampled packets into the network is not always necessary from the point of view of the limited network channel resource under the time-triggered scheme. Thus, the event-triggered scheme (ETS) is proposed to reduce the burden of channel. Under the event-triggered condition, stochastic stability of nonlinear system was studied in [14]. The fault detection issue for nonlinear discrete-time networked systems was discussed in [15]. In recent years, researchers have improved event-triggered scheme continuously to adjust to various networked environments. A periodic ETS was proposed to overcome the shortcoming needing extra hardware to check triggering condition instantaneously [16]. For wireless sensor networks, the decentralized ETS was developed to better save the channel resource [17]. In order to shorten the dynamic process, an improved static ETS was researched in [18] which can increase the frequency of transmission at initial times. Furthermore, dynamic ETS was proposed by introducing a dynamic variable in triggering condition [19]. In this paper, we will put forward a new improved static ETS for the nonlinear networked control system with time delay to decrease the burden of channel and improve system dynamics.At the same time, time delay problem has been widely investigated in the practical control system [20–24]. Variable time delay problems appear in control system [25–27]. For example, the stability problems of delay neural networks was studied in [28]. Based on the stochastic process, the random delay was researched in [29, 30]. Robust H∞ stability of time delay system was researched widely [31–33]. Reference [34] studied a network-induced delay to deal with the network transmission delay problem. The distributed delay was developed for a class of neural network control system. However, in this improved event-triggered networked control system, the time delay problems still have a lot of room for improvement [35–38].For the sake of reducing the conservatism of stability criterion of time delay systems, a series of technical approaches have been proposed. Before listing these approaches, we state that the conservatism of stability criteria mainly results from the estimation gap of the integral terms expressed as∫t-d2tx˙T(s)Rx˙(s)ds in the derivative of Lyapunov-Krasovskii functional. To study stability of system, model transformation approach was used in [39]. The stability criteria obtained by model transformation approach have large conservatism. To decrease the conservatism of stability criteria, a free weighting matrix method which can remedy the drawback of model transformation was proposed in [40]. However, free weighting matrix method will increase the decision variables, which makes the computation complex. To overcome this point and better estimate the integral terms, the Jensen inequality was used widely [41]. Afterwards, Wirtinger-based inequality which is considered a tighter method than Jensen inequality for estimation of the integral term was developed and employed in various systems [42]. In recent years, many researchers have improved the Wirtinger-based inequality approach, such as free-matrix-based integral inequality [43], auxiliary function-based integral inequality [44]. In this note, we will make use of a new integral inequality called Bessel-Legendre inequality together with reciprocally convex combination lemma to research the stability of nonlinear networked control system with time delay under improved event-triggered scheme.As is well known, there is a quadratic integral term∫t-d2t∫θtx˙T(s)Rx˙(s)dsdθ in Lyapunov-Krasovskii functional, which means that there will be a term ∫t-d2tx˙T(s)Rx˙(s)ds in the derivative of Lyapunov-Krasovskii functional. We apply the Bessel-Legendre inequality to estimate ∫t-d2tx˙T(s)Rx˙(s)ds and obtain that (1)∫t-d2tx˙TsRx˙sds≥d2ϕNTR¯NϕN,where ϕN=(1/d2)∫-d20LN((s+d2)/d2)x(s)ds, R¯N=diag(R,3R,…,(2N+1)R) and LN is Legendre polynomial matrix. This inequality provides a tighter bound on this specific term, which makes the obtained stability condition less conservative. In addition, we will construct an appropriate Lyapunov-Krasovskii functional manufactured for the use of Bessle-Legendre inequality.The main contributions of this paper are summarized as follows. Firstly, an improved ETS is put forward for nonlinear networked control system in this paper to reduce transmission load of channel by decreasing the number of signal transmission. The triggering parameter in this improved scheme is time-varying to achieve the situation that transmission frequency at the beginning instants is higher than at the other times, which can shorten the dynamic process of system effectively. Secondly, less conservative stability criteria subject to the orderN are obtained by employing the Bessel-Legendre inequality and introducing a Legendre-based Lyapunov-Krasovskii functional. Conservatism will be reduced with the increase of the N. Furthermore, a controller is designed for event-triggered nonlinear networked control system with time delay.The rest of the paper is summarized as follows. Section2 gives the considered nonlinear networked control system and puts forward an improved ETS. In Section 3, less conservative stability criteria are derived via Bessel-Legrendre inequality method. Section 4 designs a controller for the system in this paper. To verify the effectiveness of results, numerical examples are shown in Section 5. Finally, conclusions are summarized in Section 6.Notations: In this paper, symbolT denotes the transpose. The Rn denotes the n-dimensional Euclidean space. Rn×m is the set of all n×m matrices. The set Sn(S+n) means the set of the symmetric (positive definite) matrices of Rn×n. Furthermore, He(A)=A+AT. For matrices A∈Ru×v and B∈Rm×n, their Kronecker product is a matrix in Rum×vn denoted as A⊗B. The N denotes non-negative integer. In denotes the identity matrix with n×n dimensions. InN denotes the identity matrix with n(N+1)×n(N+1) dimensions.


## 2. Problem Formulation

In this paper, it is assumed that the networked control system has nonlinear function which satisfies the Lipschitz condition and the system state is fully observable. Thus, in this section, we establish the system model as(2)x˙t=Axt+Aτfxt+But,zt=Cxt,xt=ϕt,t∈-d2,0,where x(t)∈Rn is the state; z(t)∈Rq is the controlled output; u(t)∈Rm denotes the control input; A, Aτ, B, C are real constant matrices; ϕ(t) denotes the initial condition function; d2 is a positive scalar. Furthermore, f(x(t)) is nonlinear function and satisfies (3)fx-fy≤Fx-y,where F is a known constant matrix.In the nonlinear networked control system, there exists a phenomenon of transmitting some unnecessary sampling data during the transmission from the sensor to the controller. In order to improve the networked control system transmission performance, this paper will propose an improved event-triggered mechanism to reduce the load of the network transmission. Next, we will build an event-triggered generator for the nonlinear networked control system. It is assumed that the sampling sequence isSs={0,h,2h,…,nh}. Suppose tkh is the current released time and tk+1h is the next released time. In addition, tk+1h=tkh+nh, where nh is the release interval of the transmitted data.Although the data is released attkh, it will arrive at actuator at tkh+τ^k instant resulting from the existence of time delay τ^k∈[0,τ¯), k∈{0,1,2,···}, scalar τ¯>0.Next, based on [18, 35] and the diagram of event-triggered networked control system in Figure 1, a network time delay model for the nonlinear networked control system can be constructed. Suppose that (4)ρk=min⁡j∣tkh+τ^k+jh≥tk+1h+τ^k+1,j=0,1,2,….Figure 1
Schematic of networked control system with ETS.The intervaltkh+τ^k,tk+1h+τ^k+1 can be rewritten as (5)tkh+τ^k,tk+1h+τ^k+1=⋃j=1ρkΠj,where(6)Πj=tkh+τ^k+j-1h,tkh+τ^k+jh,j=1,2,…,ρk-1,Πρk=tkh+ρk-1h+τ^k,tk+1h+τ^k+1.(7)dt=t-tkh,t∈Π1t-tkh-h,t∈Π2⋮⋮t-tkh-ρk-1h,t∈Πρk(8)ekt=0,t∈Π1xtkh-xtkh+h,t∈Π2⋮⋮xtkh-xtkh+ρk-1h,t∈Πρkwhere 0<τ^k≤d(t)≤τ¯+h. We set the d2=τ¯+h, then 0<τ^k≤d(t)≤d2.For thet∈tkh+τ^k,tk+1h+τ^k+1, the event-triggered condition is(9)ekTtΩekt≤σktxTt-dtΩxt-dt,Ω ∈ S + n, σk(t) is a time-varying function as(10)σkt=σkj,t∈Πj,j=1,2,…,ρk,where(11)σkj+1=σ¯+ekjTekjλ+ekjTekjσkj-σ¯,(12)ekj=xtkh-xtkh+j-1h,j=1,2,…,ρk,where known constantsσ¯∈[0,1], λ>0, 0<σk1<σ¯, and σk1 is the initial value of σkj. It is easy to see that the σk(t) is monotonically increasing and has an upper bound σ¯.Remark 1.
The sampled data will be transmitted when condition (9) is not satisfied. It is noticed that the σk(t) in the improved event-triggered scheme (9) is time-varying and meets (10), which can increase the triggering time at the initial times to optimize the dynamic process of the system in this paper.
Therefore, according to formulae (2)-(9), we have(13)utkh=Kxtkh=Kekt+Kxt-dt,where t∈[tkh+τ^k,tk+1h+τ^k+1), K∈Rm×n represents networked controller gain, then the system model can be rewritten as follows:(14)x˙t=Axt+AτFxt+BKxt-dt+BKekt,zt=Cxt,xt=ϕt,-d2,0,where(15)0<τ^k≤dt≤d2,dm≤d˙t≤dM,dm, dM are known constants, dm<0, dM>0.
For analyzing the stability and stabilization problem of nonlinear networked control system conveniently, we will give a definition and some lemmas.Definition 2 (see [24]).
For given scalarsi, j∈N, the Legendre polynomial considered over the interval μ∈[0,1] is (16)Liμ=-1i∑j=0ipjiμj,where pji=-1jiji+jj, and kl means k!/(k-l)!l!.
Define that(17)LNμ≔L0μIn,…,LNμInT is a polynomial matrix with (N+1)n×n dimensions, where the integers N≥0, n>0.Lemma 3 (see [28], reciprocally convex inequality).
Let integern>0 and R1, R2 be in S+n. If there exist X1, X2 in Sn and Y1, Y2 in Rn×n such that (18)R100R2-αX1Y1Y1T0-1-α0Y2Y2TX2≥0holds for α=0,1, then the following inequality (19)1αR10011-αR2≥R100R2+1-αX1Y2Y2T0+α0Y1Y1TX2holds for all α∈0,1.Lemma 4 (see [3]).
S 1, S2 and S3 are given constant matrices, where S1T=S1, S2=S2T. If and only if (20)S1S3S3T-S2<0or-S2S3S3TS1<0,then we have S1+S3TS2-1S3<0.Lemma 5 (see [42]).
For any matrixR∈S+n, integer N≥0, time functions a, b, a<b, and a function x in L2([a,b])→Rn), the inequality (21)∫abxTsRxsds≥b-aϕNTR¯NϕNholds, where (22)ϕN=1b-a∫abLNs-ab-axsdsR¯N=diagR,3R,…,2N+1R.Lemma 6 (see [29]).
For any given positive matricesL>0, Y>0, if the following inequality holds (23)L-YY-1L-Y>0,then we have (24)LY-1L>2L-Y.Remark 7.
For the networked control system, we have designed an improved event-triggered generator in this section. Under the nonlinear function and improved event-triggered condition, the time delay problem will be reconsidered. Based on above preliminaries, we will give the related stability analysis in the next section.


## 3. Stability Criteria

In this section, let us investigate the stability problem of nonlinear networked control system. Compared with the previous networked control system, we will employ the Bessel-Legendre inequality method to reduce the delay upper bound conservatism of the nonlinear networked control system with time delay. At first, the relevant properties of the Legendre polynomials will be introduced. For any given matrixR∈S+n, it holds that (25)∫01LNμR-1LNTμdμ=R¯N-1,where(26)R¯N=diagR,3R,…,2N+1RLN1=InIn⋮In≔1N,LN0=In-In⋮-1NIn≔1¯N.According to the properties of the orthogonal polynomials, the Legendre polynomials satisfy(27)dLNμdμ=Γ¯NLNμ=ΓNLN-1μ,dμLNμdμ=LNμ+ΘNLNμ,where Γ¯N=[ΓN0n(N+1),n],ΓN=γN⊗In and ΘN=θN⊗In, matrices γN∈R(N+1)×N and θN∈R(N+1)×(N+1) are defined as(28)γNk,i=0ifk≥i,2k-11--1i+kifk<i,θNk,i=0ifk≥i,kifk=i,2k-11--1i+kifk<i.Theorem 8.
For givenN∈N, the system (14) is stable if there exist any matrices PN∈S+(2N+3)n, Q1∈S+n, Q2∈S+n, R∈S+n, Ω∈S+n, and Y1∈R(N+2)n×(N+2)n, Y2∈R(N+2)n×(N+2)n such that the following inequality holds for all (d(t),d˙(t))∈H:(29)Ξ=ΞN0dt,d˙tWNTY1WNTY2∗-d2-dtd2R~N+10∗∗-dtd2R~N+1≤0,where(30)H=Co0,0,0,dM,d2,0,d2,dm,HN=00000001¯N00Γ¯N-ΘN00-1¯N000ΘN,JN=A+AτFBKBK0001N-1¯N00-Γ¯N001N0-1¯N0-Γ¯N,WN=1N+1-1¯N+100-ΓN+1001N+10-1¯N+10-ΓN+1,ΛN=R¯N+100R¯N+1+d2-dtd2R¯N+1Y2Y2T0+dtd20Y1Y1TR¯N+1,Y1=d2-dtd2Y10,Y2=0dtd2Y2T,FN=A+AτFBKBK000,R~N+1=diagR,3R,…,2N+1R,2N+3R,(31)GNdt=In000000000dtInN000000d2-dtInN,ΣNd˙t=diagQ1,-1-d˙tQ1-Q2+σktΩ,-Ω,-Q2,0,0,ΞN0dt,d˙t=HeGNTdtPNJN+d˙tHN+ΣNd˙t+d22FNTRFN-WNTΛNWN.Proof.
Now we choose the Lyapunov-Krasovskii functional candidate to investigate nonlinear networked control system as(32)Vxt,x˙t=V1xt,x˙t+V2xt,x˙t+V3xt,x˙t,(33)V1xt,x˙t=x~TtPNx~t,V2xt,x˙t=∫t-dttxTsQ1xsds+∫t-d2t-dtxTsQ2xsds,V3xt,x˙t=d2∫t-d2t∫θtx˙TsRx˙sdsdθ, where (34)x~Tt=xTtdtϕ1,NTtd2-dtϕ2,NTt,ϕ1,Nt=1dt∫t-dttLNs-t+dtdtxsds,ϕ2,Nt=1d2-dt∫t-d2t-dtLNs-t+d2d2-dtxsds.
Taking the derivative ofV(x(t),x˙(t)), we have (35)V˙xt,x˙t=V˙1xt,x˙t+V˙2xt,x˙t+V˙3xt,x˙t,V˙1xt,x˙t=Hex~TtPNx~˙t,V˙2xt,x˙t=xTtQ1xt-1-d˙txTt-dtQ1-Q2xt-dt-xTt-d2Q2xt-d2,V˙3xt,x˙t=d22x˙TtRx˙t-d2∫t-d2tx˙sRx˙sds.
Now, we define the(36)ξTt=xTtxTt-dtekTtxTt-d2ϕ1,NTtΦ2,NTt,then due to the networked control system model (29), the derivative of x~(t) can be represented by x~˙(t)=(JN+d˙(t)HN)ξ(t), where matrices JN and HN are defined in (31). On the one hand, let us consider (37)ϕN=1b-a∫abLNs-ab-axsds,where a and b are time functions. Now we set s(μ)=(b-a)μ+a to get (38)ϕN=∫01LNμxsμdμ,and(39)db-aϕNdt=a˙ψ1,N+b˙-a˙ψ2,N+ϕN,where(40)ψ1,N=b-a∫01LNμx˙sμdμ,ψ2,N=b-a∫01μLNμx˙sμdμ.Applying integration by parts, we obtain(41)ψ1,N=1Nxb-1¯Nxa-Γ¯NϕN,ψ2,N=1Nxb-ΘNϕN-ϕN.Adding these equations into (39), for all t∈R+, (42)db-aϕNdt=a˙1Nxb-1¯Nxa-Γ¯NϕN+b˙-a˙1Nxb-ΘNϕN-ϕN+ϕN=b˙1Nxb-a˙1¯Nxa-a˙Γ¯N+b˙-a˙ΘNϕN.
Now, let us analyze two cases that(a,b)=(t-d(t),t) and (a,b)=(t-d2,t-d(t)). (43)x~˙=x˙Ttddtϕ1,NTdtdd2-dtϕ2,NTdt,x˙t=At+Adxt-dt,then we have (44)ddtϕ1,Ndt=1Nxt-1-d˙t1¯Nxt-dt-1-d˙tΓ¯N+d˙tΘNϕ1,N,dd2-dtϕ2,Ndt=1-d˙t1Nxt-dt-1¯Nxt-d2-Γ¯N-d˙tΘNϕ2,N.
On the other hand, for the function ofV3(x(t),x˙(t)), we have (45)∫t-d2tx˙TsRx˙sds≥ϕ~1,N+1ϕ~2,N+1TdtR~N+100d2-dtR~N+1ϕ~1,N+1ϕ~2,N+1,where(46)ϕ~1,N+1=1dt∫t-dttLN+1s-t+dtdtx˙sds,ϕ~2,N+1=1d2-dt∫t-d2t-dtLs-t+d2d2-dtx˙sds.
Due to equation (27), we obtain (47)dtϕ~1,N+1d2-dtϕ2,N+1=1N+1xt-1¯N+1xt-dt-ΓN+1ϕ1,N1N+1xt-dt-1¯N+1xt-dt-ΓN+1ϕ2,N=WNξN,where WNξN is defined in (31).
According to Lemma5 and Bessel-Legendre inequality, we have (48)d2∫t-d2tx˙TsRx˙sds≥ξTtWNTd2dtR~N+100d2d2-dtR~N+1WNξt.
In addition, for anyY1, Y2∈R(N+2)n×(N+2)n, consider the matrix (49)ΛN0=d2-dtd2Y1R~N+1-1Y1T00dtd2Y2TR~N+1-1Y2.
By applying Lemma3, we have(50)d2∫t-d2tx˙TsRx˙sds≥ξTtWNTΛN-ΛN0WNξt,where ΛN is defined in (31).
In addition, let us consider the following event-triggered condition of the networked control system(51)ekTtΩekt≤σktxTt-dtΩxt-dt.
Let us add (51) into the derivative of V(x(t),x˙(t)), then, we have(52)V˙xt,x˙t=HeξTtGNTdtPNx~˙t+ξTtd22FNTRFN+ΣNd˙tξt-d2∫t-d2tx˙TsRx˙sds,where GN, FN, ΣN(d˙(t)) are given in (31).
By the integral inequality method (50), the derivative of Lyapunov-Krasovskii can be rewritten as(53)V˙xt,x˙t≤ξTtHeGNTdtPNJN+d˙tHN+ΣNd˙t+d22FNTRFN-WNTΞNdt-ΞN0dtWNξt=ξTtΨN0dt,d˙t+WNTΞN0dtWNξt.
Notice that theΨN0(d(t),d˙(t))+WNTΞN0(d(t))WN is multi-affine about d(t) and d˙(t), where (d(t),d˙(t))∈H. Therefore, by the Schur’s complement, if the matrix Ξ<0, then V˙(x(t),x˙(t))<0 for (d(t),d˙(t))∈H, the system is stable. This proof is completed.Remark 9.
Notice that LMI (29) is considered satisfying (d(t),d˙(t))∈H, where (54)H=Co0,0,0,dM,d2,0,d2,dm.That is because the vertices (0,dm), (d2,dM) are impossible to reach. In another words, at the lower bound of time delay 0, the derivative of time delay d˙(t) can not be negative; at the upper bound of time delay d2, the derivative of time delay d˙(t) should be non-positive. Thus, we choose the allowable delay set as H.Remark 10.
In this section, the stability problem of the nonlinear networked control system has been discussed. We employed the Bessel-Legendre inequality method to improve the Lyapunov-Krasovskii functional and obtained the delay upper bound. For theV1(x(t),x˙(t)), we take the terms ϕ1,N(t) and ϕ2,N(t) into x~(t) to consider the delay dependent condition. Furthermore, the V2(x(t),x˙(t)) will be zoomed by this integral method.


## 4. Stabilization of Networked Control System

In this section, we will deal with the stabilization problem of the event-triggered nonlinear networked control system with time delay. In order to more effectively control the system state and achieve a stable, fast and accurate networked control system, the state feedback controller will be designed. Next, the relevant stabilization theorem is given as follows.Theorem 11.
For given integer scalarN≥0, scalar ϵ>0, system (14) with the feedback controller gain K=-ϵB-1P^1-1 is stable if there exist matrices P^1∈S+n, P^2∈S+(N+1)n, P^3∈S+(N+1)n, Q1, Q2, R∈S+n, Ω∈S+n, and Y1, Y2∈R(N+2)n×(N+2)n, such that(55)Ξ=Ξ~N0dt,d˙tWNTY1WNTY2d22F^NT-WNTΛN∗-d2-dtd2R~N+1000∗∗-dtd2R~N+100∗∗∗-2In+R0∗∗∗∗ΛN≤0holds for all (d(t),d˙(t))∈H, where(56)H=Co0,0,0,dM,d2,0,d2,dm,F^N=A+AτFϵ-2In+P^1ϵ-2In+P^1000,WN=1N+1-1¯N+100-ΓN+1001N+10-1¯N+10-ΓN+1,ΛN=R¯N+100R¯N+1+d2-dtd2R¯N+1Y2Y2T0+dtd20Y1Y1TR¯N+1,R~N+1=diagR,3R,…,2N+1R,2N+3R,GNdt=In000000000dtInN000000d2-dtInN,ΣNd˙t=diagQ1,-1-d˙tQ1-Q2+σktΩ,-Ω,-Q2,0,0,Ξ~N0dt,d˙t=HeGNTdtΞ¯N0d˙t+ΣNd˙t,Ξ¯N0d˙t=P^1A+AτF-ϵIn-ϵIn000P^21N-P^21¯N00-P^2Γ¯N00P^31N0-P^31¯N0-P^3Γ¯N+d˙t0000000P^21¯N00P^2Γ¯N-P^2ΘN00-P^31¯N000P^3ΘN,Y1=d2-dtd2Y10,Y2=0dtd2Y2T,Proof.
According to Theorem8 and Lemma 4, we have obtained the LMI as follows:(57)Ξ=Ξ^N0dt,d˙tWNTY1WNTY2d22FNT∗-d2-dtd2R~N+100∗∗-dtd2R~N+10∗∗∗-R-1≤0, where Ξ^N0(d(t),d˙(t))=He(GNTPN(JN+d˙(t)HN))+ΣN(d˙(t))+WNTΛNWN.
For the symmetric matrixPN, we set(58)PN=P^1000P^2000P^3,where P^1∈S+n, P^2∈S+(N+1)n and P^3∈S+(N+1)n. Then we have(59)GNTΞ¯N0dt,d˙t=GNTPNJN+d˙tHN=GNTP^1000P^2000P^3A+AτFBKBK0001N-1¯N00-Γ¯N001N0-1¯N0-Γ¯N+d˙t00000001¯N00Γ¯N-ΘN00-1¯N000ΘN.At the same time, we set the controller gain K=-ϵB-1P^1-1, then we obtain(60)GNTΞ¯N0dt,d˙t=GNTP^1A+AτF-ϵIn-ϵIn000P^21N-P^21¯N00-P^2Γ¯N00P^31N0-P^31¯N0-P^3Γ¯N+d˙t0000000P^21¯N00P^2Γ¯N-P^2ΘN00-P^31¯N000P^3ΘN.
Thanks to Lemma6, we replace -R-1 with -2In+R; on the other hand, define(61)FN=A+AτFϵ-2In+P^1000.
Combining equations (58), (59), (60) and (61), we obtain the following inequality:(62)Ξ^N0dt,d˙tWNTY1WNTY2d22FNT∗-d2-dtd2R~N+100∗∗-dtd2R~N+10∗∗∗-2In+R≤0.
In addition, apply the Schur’s complement to (62) again. Then, linear matrix inequality is obtained in Theorem 11. This proof is completed.


## 5. Numerical Examples

Example 1.
In order to understand the applicability of the system more clearly, and demonstrate the effectiveness of Bessel-Legendre inequalities method, we give Example1. First of all, let us consider the system model as follows: (63)x˙t=Axt+AτFxt+But, and the system relevant parameters are given as (64)A=-200-0.9,Aτ=-10-1-1,B=-10-0.5-1,F=-10-0.5-1.
For testifying the less conservatism of the obtained stability condition in this paper, the comparison with other paper’s results about the upper bound of delay is given in Table1. From Table 1, for N=0, d˙(t) is taken different values as 0.1, 0.3, 0.5, 0.9, respectively, then the obtained upper bound of time delays are 2.3544, 1.1498, 1.0755, 1.0526, respectively, which are all larger than the values in [13, 45, 46] for the corresponding values of d˙(t). Obviously, for this nonlinear networked control system, the conservatism of the stability criteria derived by using Bessel-Legendre inequality method in this paper has been greatly reduced. Furthermore, the obtained stability criteria are related to the order N. As we can see in Table 1, the upper bound of time delay increases with N increasing. In other words, the larger N, the lower conservatism.Table 1
Delay upper boundd2 for different d˙(t)(dM=-dm=d˙(t)).
Method\d˙(t) 0.1 0.3 0.5 0.9 Theorem8, N=3 4.7526 3.01780 3.0472 2.9856  Theorem8, N=2 3.2546 2.3136 2.1797 2.0732  Theorem8, N=1 2.8572 2.6095 2.1472 2.0153  Theorem8, N=0 2.3544 1.1498 1.0755 1.0526  Theorem 1 of [45] 2.1403 1.0653 0.8671 0.7644  Theorem 1 of [46] 0.7522 0.7197 0.6496 0.4768  Theorem 1 of [13] 1.2890 1.0126 0.9775 0.6324Next, we propose Example2 to investigate the triggering performance of nonlinear networked control system under the improved event-triggered scheme.Example 2.
This example concerns the parameters of nonlinear networked control system as follows:(65)A=00.50000-30-10010.50-30.7,Aτ=00.1000010-10010.50-31,B=00.10-0.01,F=50000-0.30002.0-0.66670.66670-2.70.1-0.8.Set σk1=0.01, σ¯=0.6, h=0.22, d˙(t)=0.1. Under the improved event-triggered scheme, the obtained controller gains K and triggering parameters Ω at t=0, t=1,…,t=30 are shown in Table 2.Table 2
The controller gainsK and triggering parameters Ω at different times.
t 0 1 … 30 Ω 4.57 1.06 4.46 0.49 1.06 6.19 0.14 0.95 4.46 0.14 9.79 3.70 0.49 0.95 3.70 9.65 4.47 0.15 1.67 1.33 0.15 7.62 0.13 0.95 1.67 0.13 8.89 0.40 1.33 0.95 0.40 7.23 … 9.77 0.13 0.22 0.36 0.13 5.14 0.20 0.94 0.22 0.20 6.94 0.18 0.36 0.94 0.18 5.53   K 6.2667 5.6134 2.0162 4.8761 T 4.4035 5.1434 6.1441 6.5962 T … 3.3367 2.4931 5.0829 8.9023 TIn this improved event-triggered scheme, the parameterσk(t) is time-varying and satisfies (10), which can change the release rate at different times while saving the network transmission resource. When h=0.22, λ=0.01, the variation of σk(t) and release instants are shown in Figure 2. In Figure 2, there are only 29 sampled signals which takes 21.3% of the whole sampled signals need to be sent out when t∈[0,30]. In addition, from Figure 2, we can see that the parameter σk(t) varies from 0.01 to 0.4 and the frequency of triggering in initial times is higher than other times, which can shorten the dynamic process.Figure 2
The variation ofσk(t) and release instants, h=0.22.To further illustrate the advantage of our improved static event-triggered scheme, we compare it with the general static event-triggered scheme in which the parameterσk(t) is time invariant. We set h=0.12, then the results are shown in Figures 3 and 4. Obviously, the release times in initial times in Figure 4 are more than those in Figure 3 and there are more packets that can be transmitted in initial times.Figure 3
Improved static event-based release interval and release instants, h=0.12.Figure 4
Static event-based release interval and release instants, h=0.12.The state response of the event-triggered nonlinear networked control system is shown in Figure5. At 3 seconds, the system achieves steady.Figure 5
State response of the event-trigger nonlinear networked control system.


## 6. Conclusions

In this paper, we discussed the stability and stabilization problem of the event-triggered nonlinear networked control system with time varying delay. Firstly, on one hand, compared with the previous investigation of networked control system, this paper utilizes the Bessel-Legendre inequality method to reduce the conservatism of the system delay upper bound. On the other hand, by designing an appropriate state feedback controller, the stabilization problem of the nonlinear networked control system has been solved. Secondly, an improved ETS was put forward to reduce transmission load and make the system has a better dynamic process. Finally, two simulation examples have been shown to verify the effectiveness of the improved time delay method and ETS. Further study can be concentrated on the discrete time matters for the networked control system and finite time issues for nonlinear networked control switch system under an improved ETS.



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*Source: 1648576-2019-06-10.xml*