# Further Results on Sampled-Data Synchronization for Complex Dynamical Networks with Time-Varying Coupling Delay

Authors: Hea-Min Lee; Wookyong Kwon; Sangmoon Lee; Dongyeop Kang
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This paper deals with the sampled-data synchronization problem for complex dynamical networks (CDNs) with time-varying coupling delay. To get improved results, two-sided free-weighting stabilization method is utilized with a novel looped functional taking the information of the present sampled states and next sampled states, which can more precisely account for the sawtooth shape of the sampling delay. Also, the quadratic generalized free-weighting matrix inequality (QGFWMI), which provides additional degree of freedom (DoF), is utilized to calculate the upper limit of the integral term. Based on the novel looped functional and QGFWMI, improved conditions of stability are derived from forms of linear matrix inequalities (LMIs). The numerical examples show the validity and effectiveness.


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## Body

## 1. Introduction

Complex dynamical networks (CDNs) are an attempt to model a set of interconnected dynamic properties of nodes with specific contents. For example, there are human interaction networks, ad hoc networks, secure communications, harmonic oscillations, biological systems, and chaotic systems, financial systems, social networks, and neural networks. CDNs are faced with the problems of expressing structural complexity and connection diversity at the same time. Furthermore, the dynamic characteristics of the network make it difficult to provide a solution to the real world because modeling should be done with the node’s insufficient information from the network. Nevertheless, CDNs have attracted lots of attention in various fields of engineering [1–4]. Especially, the problem of synchronization has been focused by many researchers [5–7], as the synchronization of CDNs is a fundamental phenomenon. In nature, complex networks in the synchronization encounter time delay in biological and physical networks, because of the limited speed of network transmission, traffic jams, and signal propagation. The time delay is a source of degradation synchronization performance and instability, and thus complex networks with time-varying delay are of importance and generality [8, 9].The design of control has been developed including pinning control [10], impulsive control [11], hybrid control [12], fuzzy adaptive output feedback control [13], and sampled-data control [5] to accomplish stable synchronization. Among these methods, sampled-data control for the synchronization of CDNs has been studied extensively with the development of digital communication since sampled-data offers many benefits in modern control systems. The advantages of sampled-data control are as follows: Firstly, the sampled-data control is more realistic than continuous control in that it can be implemented in practical systems. Secondly, in the case of the signal in the form of pulse data, the information is supplied immediately with a small outlay. Lastly, the control system for better performance is generally achieved by a sampled-data control. For continuous systems, the differentiator not only improves the existing noise but also generate additional noise. In the sampled-data system, the differential operation can be implemented without increasing noise problem. For that reason, sampled-data control was used because of these benefits: practicality, immediacy, economics, and accuracy. In sampled-data control systems, it is the main issue to design controllers that can get larger sampling interval. Increasing maximum sampling interval is very important because it not only enlarges the stable region but also improves performance when considered with other aspects: H∞ [14] and dissipativity [15].Several criteria for CDNs with time delay are developed to derive stability conditions on sampled-data intervals. Sampled-data signals which are discontinuous at every sampling time can be treated as continuous time-varying delayed signals. In [5], the problem of sampled-data synchronization control for a class of general complex networks with time-varying coupling delay is firstly handled using Jensen’s inequality found in the input delay approach. The time-dependent Lyapunov functional and convex combination techniques are used in [16] to derive a less conservative condition for the sampled-data synchronization. The synchronization in memory neural networks with time-varying delays was studied in [2, 17, 18]. In [17], a sampled-data feedback controller was proposed by using the Lyapunov function theory and Jensen’s inequality method to guarantee the synchronization of memristive Bidirectional Associative Memory (BAM) neural networks with leakage and two additive time-varying delays. The authors in [18] obtained less conservative results by constructing a Lyapunov function and using the stochastic differential inclusions and some inequality techniques. Recently, Wirtinger’s inequality is used in [19, 20]. Also, the augmented Lyapunov function approach and Lyapunov function with triple integral have been reported in the literature [15, 21, 22]. In [23], a looped-functional-based approach was proposed for the stability analysis of linear impulsive systems. This approach easily formulates sampling interval result for discrete time stability using a continuous time’s approach. In [24], a new looped-functional for stability analysis was proposed. This functional entirely uses the information on both interval tk and tk+1, which improved stability condition. However, there are still more rooms for improvement, which motivates our research. To consider the information of sampling time at tk and tk+1, the modified looped-functional is proposed by using the augmented vector for two-sided sampling time.In this paper, enhanced results on sampled-data synchronization criteria and controller design are given for the complex dynamical networks with time-varying coupling delay. The stability and stabilization criteria are presented in forms of linear matrix inequalities (LMIs). The superiority of the proposed scheme is shown through numerical examples. The main contribution of this note is summarized as follows:(i)
Free-weighing matrices at time sequencetk and tk+1 are separately employed with an additional scalar parameter in consideration of system dynamics in CDN satisfying convexity.(ii)
In order to fully consider the information of sawtooth shape sampling pattern attk and tk+1, novel looped functional is employed with augmented vectors which become zero by constructing the vector crossly aligned at each sampling time tk or tk+1. Namely, the dimension of the LMI variable extends from R2×2 (2 × 2 dimensional Euclidean space) to R4×4. Therefore, augmented vectors provide an increased degrees of freedom (DoF) and improved results.(iii)
QGFMI is firstly applied to sampled-data synchronization. QGFMI estimates the upper limit of the integral term more tightly. Thus it contributes to deriving a less conservative result.


## 2. Preliminaries

CDNs composed of N identical coupled nodes with n-dimensional dynamics are described as follows:(1)x˙it=fxit+c∑j=1NGijDxjt+c∑j=1NGijAxjt-τt+uit,fori=1,2,…,Nwhere f(·) is the continuous vector-valued nonlinear function, Gij is the outer coupling matrix from node i to j with weight, A and D are the inner coupling matrix, xit is the state variable of node i, uit is input variable e of node i, and c>0 is the coupling strength. Gij is defined as follows:(2)Gij>0ifthereisconnectionbetweennodeiandjGij=0ifthereisnoconnectionbetweennodeiandjThe diagonal elements of Gij are denoted as Gij=-∑i=1,j=iNGij for i=1,2,…,N. The bounded time-varying delay τ(t) satisfies(3)0≤τt≤τM,-μ≤τ˙t≤μ,where τM and μ are positive known constants. Without loss of generality, the nonlinear function f· is assumed to satisfy a sector-bounded condition as(4)fx-fy-W1x-yTfx-fy-W2x-y≤0,where W1 and W2 are matrices with appropriate dimensions. Let rt be an unforced isolated node, r˙t=f(r(t)), then the error dynamics of each dynamical system is derived as (5)e˙it=geit,rt+c∑j=1NGijDxjt+c∑j=1NGijAxjt-τt+uit,fori=1,2,…,N,where eit=xi-r(t) and geit,rt=f(xit)-f(rt). Utilizing Kronecker product, the whole CDN is represented as (6)e˙it=g^et,rt+cG⊗Det+cG⊗Aet-τt+utwhere(7)et=e1te2t…eNt,et-τt=e1t-τte2t-τt…eNt-τt,g^et,rt=ge1t,rtge2t,rt…geNt,rt,ut=u1tu2t…uNt.For the given system, our objective is to design a sampled-data controller which makes the error systems converge to zero. When considering that the measurement sensor generates a signal, discretized sampled signals are only sent to the controller through the network in system topology, and the control input can be utilized using zero order function. The control input signals using sampler are generated with a sequence of hold time 0=t0<t1<…<tk<… where limk→∞⁡tk=∞ andk is a positive integer. The sampling intervals are represented as(8)tk+1-tk≤hk≤hMwhere hM is the maximum sampling instant. Then, the sampled-data controller is designed as (9)ut=utk=Ketk,fortk≤t≤tk+1.where k=diag⁡{K1,K2,…,KN} is the control gain matrix and etk is the sampled error signal. The closed-loop error systems with sampled-data control are rewritten as (10)e˙it=g^et,rt+cG⊗Det+cG⊗Aet-τt+KutkTo develop main results, useful lemmas are introduced.Lemma 1 (Wirtinger’s inequalities; [25]).
For given a matrixR>0 and a continuous differentiable function ω(t) in [a,b], the following inequality holds:(11)∫abωTsRωsds≥1b-aη1TRη1+3b-aη2TRη2where η1=∫abωsds and η2=∫abωsds-2/b-a∫ab∫sbωududs.Lemma 2 (reciprocally convex combination method; [26]}).
For a given scalar0<θ<1, vectors ω1, ω2, and matrices M>0,N satisfying MNNM>0, then the following holds:(12)1θω1TMω1+11-θω2TMω2≤ω1ω2TMNNMω1ω2.Lemma 3 (QGFMI).
Given matricesX,Y and positive semidefinite matrixR, the inequality is given by (13)∫abωTsRωsds≥-η0η3Tb-aXR-1XTXI0∗b-a3YR-1YT+sym⁡Y-I2Iη0η3where η0 is any vector and η3=[∫abωTsds,1/b-a∫ab∫sbω(u)Tduds]T, and ω(t) is a differentiable function which is a continuous on [a,b].Proof.
The proof of Lemma3 is omitted as it is similar to that of [27].Remark 4.
The QGFMI is used to calculate the upper limit of integral term in the derivative of Lyapunov-Krasovskii function, which increases freedom to choose a free-selectable vector [27]. Furthermore, the new free-weighting matrix plays a vital role in filling in the diagonal element and corresponding augmented vector provides additional flexibility.Remark 5.
The control technique using the sample value data in (9) can be applied to systems such as event-triggered communication as in [28, 29].


## 3. Main Results

The matricesIi=R11n×n for i=1,2,…,11 are denoted to represent matrices composed of (n-1) zero elements matrices with ith identity matrix. For example, I1=[I0000000000]Tand I5=[0000I000000]T. Moreover, the following are declared:(14)ζt=et,e˙t,etk,1t-tk∫tkteTαdα,etk+1,1tk+1-t∫ttk+1eTαdα,et-τt,et-τM,1τt∫t-τtteTαdα,1τM-τt∫t-τMt-τteTαdα,g^Tet,st,κ1=I1,I3+I5,I4+I6,I2,κ2=I1-I3,I1-I4,κ3=I5-I1,I5-I6,κ4=I1-I7,I1+I7-I9,I7-I8,I7+I8-I10,η1=I1-I3,t-tkI5,t-tkI3,t-tkI4,η2=I5-I1,tk+1-tI3,tk+1-tI5,tk+1-tI6,η1d=I2,I5,I3,I1,η2d=-I2,-I3,-I5,-I1,S1t=t-tkS10-tk+1-tS11S12S12TS13∈R4n×4n,S10∈Rn×n,S11∈Rn×n,S12∈Rn×3n,S13∈R3n×3n,S1d=S10-S110n×n0n×n0n×n,R^=R00R.Theorem 6.
For a given scalar parameterhM and a gain matrix K, if there exist a positive scalar δ, symmetric matrices P>0, Q1>0, Q2>0, R>0, T1>0, and T2>0, any matrices G1, G2, G3, G4, S1(t), S2, X1, X2, Y1, Y2, and M^, and a scalar δ>0 satisfying the inequalities (15)Π1,ht=0κ1X2κ3Y2∗-hMT20∗0-3hMT2≤0(16)Π1,ht=hκ1X1κ2Y1∗-hMT10∗0-3hMT1≤0(17)R^M^M^R^≥0where(18)Π1,ht=sym⁡I1PI2T+I1Q1I1T-1-μI7Q1I7T+I1Q2I1T-I8Q2I8T+τMI2RI2T-κ4R^M^M^R^κ4T+sym⁡η1S1tη2dT+η1dS1tη2T+η1S1dη2T+sym⁡I1-I3t-tkI4S2-I2T-I1T+I2I1S2I5T-I1Ttk+1-tI6+tk+1-tI2T1I2T+t-tkI2T2I2T+sym⁡κ1X1I0κ2T+κ2Y1-I2Iκ2T+sym⁡κ1X2I0κ3T+κ1Y2-I2Iκ3T-δI1I11W~1W~2∗2II1TI11T+t-tksym⁡I1G1+I2G2·cC⊗DI1T+cC⊗AI7T+KI3T-I2T+I11T+tk+1-tsym⁡I1G3+I2G4·cC⊗DI1T+cC⊗AI7T+KI3T-I2T+I11Tand W1~=I⊗W1W2+(W1W2⊗I), W2~=-I⊗(W1T+W2T), then CDNs are asymptotically stable.Proof.
Construct the following Lyapunov-Krasovskii function fort∈[tk,tk+1):(19)Vxt=V1t+V2t+V3t+Vl1t+Vl2t+Vl3t+Vl4t,where(20)V1t=eTtPet,V2t=∫t-τtteTαQ1eαdα+∫t-τMteTαQ2eαdα,V3t=∫t-τMt∫βte˙TαRe˙αdαdβ,Vl1t=2et-etkt-tketk+1t-tketk∫tkteαdαTS1etk+1-ettk+1-tetktk+1-tetk+1∫ttk+1eαdα,Vl2t=2et-etk∫tkteαdαTS2etk+1-et∫ttk+1eαdα,Vl3t=tk+1-t∫tkte˙TαT1e˙αdα,Vl4t=-t-tk∫ttk+1e˙TαT2e˙αdα.Differentiating each LKF in equation (20) provides(21)V˙xt=V˙1t+V˙2t+V˙3t+V˙l1t+V˙l2t+V˙l3t+V˙l4t,where(22)V˙1t=2etPe˙t=sym⁡I1TPI2,V˙2t=eTtQ1et-1-μeTt-τtQ1et-τt+eTtQ2et-eTt-τMQ2et-τM=I1Q1I1T-1-μI7Q1I7T+I1Q2I1T-I8Q2I8T,V˙3t=τMe˙TtRe˙t-∫t-τMte˙TαRe˙αdα=τMI2RI2T-∫t-τMte˙TαRe˙αdα,V˙l1t=2et-etkt-tketk+1t-tketk∫tkteαdαTS1t-e˙t-etk-etk+1-et+2e˙tetk+1etketTS1tetk+1-ettk+1-tetktk+1-tetk+1∫ttk+1eαdα+2et-etkt-tketk+1t-tketk∫tkteαdαTS1detk+1-ettk+1-tetktk+1-tetk+1∫ttk+1eαdα=sym⁡η1S1tη2dT+η1dS1tη2T+η1S1dη2T,V˙l2t=2et-etk∫tkteαdαTS2-e˙t-et+2e˙tetTS2etk+1-et∫ttk+1eαdα=sym⁡I2-I3t-tkI4S2-I2T-I1T+I2I1S2I5T-I1Ttk+1-tI6T,V˙l3t=tk+1-te˙TtT1et˙-∫tkte˙TαT1e˙αdα=tk+1-tI2T1I2T-∫tkte˙TαT1e˙αdα,V˙l4t=t-tke˙TtT2e˙t-∫ttk+1e˙TαT2e˙αdα=t-tkI2T2I2T-∫ttk+1e˙TαT2e˙αdα.Separating the integral in V˙2 for two sides and applying Lemmas 1 and 2, we have(23)-∫t-τMte˙TαRe˙αdα=-∫t-τtte˙TαRe˙αdα-∫t-τMt-τte˙TαRe˙αdα≤-ζTtΣR^M^M^R^ΣTζt=-ζTtκ4R^M^M^R^κ4Tζtwhere Σ=I1-I6I1+I6-I8I1-I7I1+I7-I9, R^=R00R.
Using Lemma3, the upper bound of each integrals in V˙l3t, V˙l4t is estimated as follows:(24)-∫tkte˙TαT1e˙αdα≤ζTtκ1Tκ2TTt-tkX1T1-1X1TX110∗t-tk3Y1T1-1Y1T+sym⁡Y1-I2Iκ1Tκ2Tζt,(25)-∫ttk+1e˙TαT2e˙αdα≤ζTtκ1Tκ3TTtk+1-tX2T2-1X2TX210∗tk+1-t3Y2T2-1Y2T+sym⁡Y2-I2Iκ1Tκ3Tζt,Then, the following is satisfied with the given nonlinear function as(26)geit,sit-W1eitTgeit,sit-W2eit≤0.Equation (13) is equivalent to (27)-δetg^et,rtTW1~W2~∗2Ietg^et,rt≥0where W1~=I⊗W1W2+(W1W2⊗I), W2~=-I⊗(W1T+W2T), and δ is a constant scalar. Considering the dynamic equations (6) with auxiliary matrices G1, G2, G3, G4, the following holds:(28)2t-tkG1+tk+1-tG2I1+t-tkG3+tk+1-tG4I2·cC⊗DI1+cC⊗AI7+KI3-I2+I11=0,Summing up (21), (24), (25), (27), and (28), the derivative of LKF is expressed as (29)V˙xt=ζTtΛhtζtwhere (30)Λht=Π1,ht+t-tkΠ2+tk+1-tΠ3Π1,ht=sym⁡I1PI2T+I1Q1I1T-1-μI7Q1I7T+I1Q2I1T-I8Q2I8T+τMI2RI2T-κ4R^M^M^R^κ4T+sym⁡η1S1tη2dT+η1dS1tη2T+η1S1dtη2T+sym⁡I1-I3t-tkI4S2-I2T-I1T+I2I1S2I5T-I1Ttk+1-tI6T+tk+1-tI2T1I2T+t-tkI2T2I2T+sym⁡κ1X1I0κ2T+κ2Y1-I2Iκ2T+sym⁡κ1X2I0κ3T+κ1Y2-I2Iκ3T-δI1I11W~1W~2∗2II1TI11T+sym⁡I1G1+I2G2·cC⊗DI1T+cC⊗AI7T+KI3T-I2T+I11TΠ2=κ1X1T1-1X1Tκ1T+13κ2Y1T1-1Y1Tκ2TΠ3=κ1X2T2-1X2Tκ1T+13κ3Y2T2-1Y2Tκ3TTherefore, the conditions ζT(t)Λh(t)ζ(t)<0 implies that V˙<0 for t∈tk,tk+1. From the convex combination method, Λh(t)<0 for ht∈0,hM is equivalent to the conditions in Theorem 6 using Schur complements, and thus the synchronization error system is asymptotically stable for given sampling time hM. This is the end of the proof.Remark 7.
The constructed Lyapunov functionals include novel looped functionals, which areVl1, Vl2,Vl3, and Vl4. Those functionals satisfy Vl1=Vl2=Vl3=0 at every sampling instances tk,tk+1 so it is looped functional as defined in [23].Remark 8.
The additional consideration oftk+1 in Lyapunov-Krasovskii functional (LKF) results in deriving less conservative results, which reflect more information for sampling time. The idea is motivated by the two-sided looped-functional [24], which is applied to CDNs with time-varying coupling delay to achieve more stable synchronization between two or more nodes. In the stability analysis of the sampled-data system using looped-functional, it is important to take into account the sampled-time-dependent vectors which become zero at each end point of each sampling time. However, in [24], the product term between the vectors xt-x(tk) and xtk+1-x(t) and the product term between x(tk) and x(tk+1) are separately considered. In this paper, more generalized results are derived by defining the vector as η which provides each cross terms at the same time.Remark 9.
The novelty of the proposed looped functional is in the formation ofVl1. In Vl1, the variables are chosen as η1, η2 and S(t) is composed of sampled-time-dependent matrix. The dimension of the LMI variable S1(t) is extended to R4×4. Therefore, augmented vectors provide an increased degrees of freedom (DoF) and improved results compared with the existing results. Herein, the DoF is the number of independent variables or equations that must be specified to solve the problem uniquely. It extends the range of the solution and provides less conservative conditions.Remark 10.
In Theorem6, a sufficient condition for the synchronization is derived in terms of LMIs which is obtained by constructing new looped functional. The results are sufficient conditions, which imply that there is still room for further improvement. Some approaches to reducing the conservatism are available. The conservativeness will be reduced by augmented vector or segmenting formulas. Also, new Lyapunov functions such as Lyapunov-Krasovskii or discontinuous Lyapunov [30] may play an essential role in the further reduction of the conservativeness.Based on Theorem6, the following corollary is constructed for the stabilization problem.Corollary 11.
For given scalarsα,0≤β≤1, and hM, if there exist a positive scalar δ, symmetric matrices P>0, Q1>0, Q2>0, R>0, T1>0, and T2>0, and any matrices S1(t), S2, X1, X2, Y1, Y2, M^, Ha, Hb, Ga=diag⁡Ga1,Ga2,…,Gan, Gb=diag⁡Gb1,Gb2,…,Gbn,(31)Π^1,ht=0κ1X2κ3Y2∗-hMT20∗0-3hMT2≤0,(32)Π^1,ht=hκ1X1κ2Y1∗-hMT10∗0-3hMT1≤0,(33)R^M^M^R^≥0,where(34)Π^1,ht=sym⁡I1P1I2T+I1Q1I1T-1-μI7Q1I7T+I1Q2I1T-I8Q2I8T+τMI2RI2T-κ4R^M^M^R^κ4T+sym⁡η1S1tη2dT+η1dS1tη2T+η1S1dtη2T+sym⁡I1-I3t-tkI4S2-I2T-I1T+I2I1S2I5T-I1Ttk+1-tI6T+tk+1-tI2T1I2T+t-tkI2T2I2T+sym⁡κ1X1I0κ2T+κ2Y1-I2Iκ2T+sym⁡κ1X2I0κ3T+κ1Y2-I2Iκ3T-δI1I11W~1W~2∗2II1TI11T+t-tksym⁡β1I1+αI2·GaI11T+cGaC⊗DI1T+cGaC⊗AI7T+HaI3T-GI2T+tk+1-tsym⁡β2I1+αI2·GaI11T+cGbC⊗DI1T+cGbC⊗AI7T+HbI3T-GbI2T,then the control gain is given by h=Ga-1Ha which stabilizes the error dynamics.Proof.
Substituting the variablesG1, G2, G3, and G4 in the zero equation (28) into β1Ga, β2Gb, αGa, and αGb, respectively, the following are obtained:(35)2t-tkβ1I1+αI2·cGaC⊗DI1T+cGaC⊗AI7T+HaI3T-GaI2T+GaI11T=0,(36)2tk+1-tβ2I1+αI2·cGbC⊗DI1T+cGbC⊗AI7T+HbI3T-GbI2T+GbI11T=0.where Ha=GaK and Hb=GbK. The other part of the proof is omitted for brevity as it is similar to that of Theorem 6.


## 4. Numerical Examples

In this section, two examples are revisited from literature [5].Example 12.
Let us consider the CDNs composed of 3 nodes with the following matrices and parameters:(37)A=1001,D=0000,G=-1010-11112,c=0.5,τM=0.5,μ=0.25.The nonlinear function is defined as (38)fxit=-0.5xi1+tanh⁡0.2xi1+0.2xi10.95xi2-tanh⁡0.75xi2,satisfying the condition in (3) by the following matrices:(39)W1=-0.50.200.95,W2=-0.30.200.2.For α=1.2, hM=2.01, the corresponding gain matrices are given by(40)K1=0.1117-0.1514-0.0761-0.8361,K2=0.1117-0.1514-0.0761-0.8361,K3=0.2682-0.1716-0.0949-0.8335.K are designed using the toolbox YALMIP 3.0 and SeDuMi 1.3 of MATLAB. The time-varying delay is chosen as τ(t)=0.25+0.1sin2(5t). The initial state is x0=[4,-3,-2,1,2,-5]T. The state trajectories of error system and the sampled-data input signals are, respectively, presented in Figures 1 and 2, which show stable convergence. Figure 1 shows the trajectories of error dynamics which is controlled by using the gain matrices at the simulation time from 0 to 30 seconds. The input signals (u1, u2, and u3) converge, respectively, to 0 in Figure 2.Figure 1
The state trajectories of the system (38) with c=0.5, τM=0.5, and μ=0.25.Figure 2
The input signals of the system (38) with c=0.5, τM=0.5, and μ=0.25.Example 13.
Consider Chua’s circuit composed of 4 nodes. The dynamics of Chua’s circuit is represented as(41)x˙1t=ϵ1-x1t+x2t-hx1tx˙2t=-x1t-x2t+x3tx˙3t=-ϵ2x2twhere hx1t=m2x1t+0.5(m1+m2)(x1t+1-x1t-1) with the parameters ϵ1=10, ϵ2=14.87, m1=-1.27, and m2=-0.68. Then, the nonlinear function is denoted as(42)fxit=-ϵ11+m2ϵ101-110-ϵ20xit+-ϵ1m1+m2xi1t+1-xi1t-100From (4), the matrices are given as(43)W1=2.71001-100-14.870,W2=-3.21001-110-14.870.For given systems, the inner coupling matrix A, the outer coupling D, and parameters are chosen as(44)A=IN,D=0n,G=-31111-21011-20100-1,c=0.5,τM=0.5,μ=0.1.The controller gain of Example 13 was calculated using toolbox YALMIP 3.0 and SeDuMi 1.3 of MATLAB in the same manner of Example 12. The corresponding gain with α=0.7, hM=0.2138 is given as (45)K1=-6.7314-4.13561.0795-0.0834-1.1048-0.78956.67117.6457-2.1652,K2=-7.7827-3.56181.0721-0.1367-2.0333-0.55216.64587.3950-3.0248,K3=-7.7821-3.56791.0678-0.1372-2.0340-0.55086.63467.4183-3.0214,K4=-7.8447-3.40301.1874-0.0468-2.3492-0.59906.41077.2538-3.3484.The trajectories of error dynamics and the sampled-data input signals are represented in Figures 3 and 4, respectively.Figure 3
The error trajectories of system (41) with c=0.5, τM=0.5, and μ=0.1.Figure 4
The sampled-data input signals of system (41) with c=0.5, τM=0.5, and μ=0.1.The maximum allowable sampling time is computed; the result using Corollary11 is compared with existing results in Table 1. The results show that Corollary 11 provides more considerable maximum sampling period. It means that the synchronized error system guarantees stability in the broader sampling region. The last line of Table 1 represents a number of decision variables (NoV). It can be expressed as the product of the number of node and dimension of the node, which is defined as ι=N×n. Our method has higher complexity due to a larger NoV than ones of other methods. However, the proposed method is efficient on offline work with a more considerable allowable maximum sampling time.Table 1
The allowable maximum sampling time for Examples12 and 13.
ref. [5] ref. [31] ref. [16] ref. [19] Corollary11 h M 0.5409 0.5573 0.9016 1.3756 1.9789  h M 0.0790 0.0793 0.1607 0.1833 0.2138  Nov 3 ι 2 + ( 4 + n )  ι + 1 3.5 ι 2 + 3.5  ι + 1 11.5 ι 2 + 3.5  ι + 1 7 ι 2 + ( 4 + n )  ι + 1 34 ι 2 + 5   ι + 1


## 5. Conclusions

This paper provides the new stabilization criteria to increase the maximum sampling interval for the synchronization of CDN with time-varying coupling delay. Novel two-sided looped functional and QGFWI are presented to obtain the enhanced results. The two-sided stabilization method is formulated by additional free matrices for present and next sampling time. The proposed looped functional, which vanishes at current sampling time and the next one, is constructed by using the augmented states to consider the information of sawtooth shape sampling pattern. Finally, simulation results show that the proposed synchronization approach provides a larger maximum sampling interval than one of the existing results. In other words, the proposed method contributes to extending the stable region and deriving a less conservative result. Also, the effectiveness of the proposed sampled-data control scheme has been demonstrated by numerical examples.In future work, practical situations will be considered in sampled-data control for CDNs. For example, the proposed scheme could be applied to the synchronization of CDN systems with heterogeneous time-varying delay or with asynchronous and aperiodic sampling characteristics.



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