# Exponential Synchronization of Complex Dynamical Networks via a Novel Sampled-Data Control

Authors: Haixia Liu; Tianbo Wang
Journal: Complexity
 (2022)

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## Abstract

This paper investigates the exponential synchronization of complex dynamical networks based on the sampled-data control method. The sampled-data control means that the control input remains unchanged for a long time after each sampling, which can reduce the sampling number. By using the stability theory of the dynamical systems, this paper provides a novel sampling controller and estimates the bound of the sampling interval. Finally, a numerical example is given to demonstrate the effectiveness of the proposed design technique.


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## Body

## 1. Introduction

Recently, complex networks have received a great deal of attention from researchers in several fields such as industrial networks, financial networks, biology, and power systems [1–4]. Complex networks are usually composed of lots of interconnected nodes, in which each node can adjust its behavior according to the received information from its neighbor nodes, so complex networks can be used to describe some complex phenomena in science and engineering fields. As a result, there are some existing papers [5–8] to study the complex networks from diverse facets.Synchronization is one of the important issues concerning complex networks, which means that the collective behaviour that all the states of the network converge towards a predefined target trajectory along with the time evolution. In fact, synchronization is a difficult problem because of the different initial values and various dynamical structures. Thus, there are many papers [9–15] to investigate the synchronization of complex networks by using different theories and methods. Furthermore, many kinds of synchronization of complex networks have been investigated, which include asymptotic synchronization [16], finite-time synchronization [17], and fixed-time synchronization [18].With the rapid development of computer hardware technology, sampled-data control has been applied widely for its convenience and good control efficiency. Sampled-data control only requires the system states information at the sampling instants, which can greatly reduce the computation and the number of transmitted information. Therefore, there are many results reported in the literature [19–25]. For example, for the periodic sampling case, the authors presented some sufficient conditions of sampled-data synchronization criteria for the complex dynamical networks with time-varying coupling delay by constructing a suitable augmented Lyapunov function, and with the help of introduced integral inequalities, and employing the convex combination technique in [22]. By employing a time-dependent Lyapunov functional and the sampled-data control, the authors of [23] investigated the synchronization control problem for chaotic neural networks subject to actuator saturation and provided a sampled-data controller to regionally synchronize the drive neural networks and response neural networks. For the stochastic sampling case, the authors of [24] studied the stabilization problem for a class of sampled-data systems under noisy sampling interval by introducing a Vandermonde matrix and Kronecker product operation and provided the corresponding stabilization controller. By using the reciprocally convex matrix inequality, proper integral inequalities, and the linear convex combination method, the authors of [25] investigated the dissipative analysis and quantized sampled-data control design issues for T-S fuzzy networked control system under stochastic cyberattacks and obtained a new quantitative sample data controller to ensure that the system is asymptotically stable and dissipative.Motivated by the above discussion, we will consider the exponential synchronization control problem for complex dynamical networks in this paper. The main contributions of our paper are as follows: (1) we consider the exponential synchronization of complex networks by using the sampled-data control; (2) the presented controller is novel; (3) the upper bound of the sampling interval is estimated.The rest of this paper is organized as follows: in Section2, the model description and preliminary results are presented. In Section 3, the sampled-data controller guaranteed the complex network to be exponential synchronization is derived. In Section 4, a numerical example is provided to illustrate the electiveness of our obtained results. Finally, this paper is ended with a conclusion in Section 5.Throughout this paper, the following notations are used.R+=0,+∞ denotes the set of the positive real number. Rn and Rn×m, respectively, denote the n-dimensional Euclidean space and the set of n×m real matrices. For a given vector x=x1,x2,…,xnT∈Rn,x=∑i=1nxi2 denotes its norm. The notation X ≥ Y (respectively, X > Y), where X, Y are symmetric matrices, means that X≥Y is a symmetric semidefinite matrix (respectively, positive definite matrix). For a given matrix A, AT denotes its transpose, and A denotes its norm defined as A=λmaxATA.


## 2. Problem Formulation and Preliminaries

In this paper, we consider the following complex networks composed ofN identical nodes with linear couplings. The ith node is an n-dimensional dynamical system, whose state equation is(1)x˙it=Axit+Bfxit+∑j=1NcijΓxjt+uit,i=1,2,…,N,where N is the number of coupled nodes. xit=(xi1t,xi1t,…,xint)T∈Rn denotes the state vector, A∈Rn×n and B∈Rn×n are known constant real matrices, f⋅∈Rn is a continuous differential vector function with f0=0. C=cij∈RN×N is the outer-coupling matrix, where cij is defined as follows: if there exists a connection between node i with node j, then cij>0; otherwise, cij=0. In addition, the elements of matrix C satisfy(2)cii=−∑j=1,j≠iNcij,i=1,2,…,N,Γ∈Rn×n is the innercoupling matrix, and uit∈Rn is the control input to be designed in the sequel.Assume that the target node is(3)s˙t=Ast+Bfst,where st=s1t,s2t,…,snt∈Rn may be an equilibrium point, a periodic orbit, or a chaotic orbit of dynamical systems.Definition 1.
Complex network (1) is said to be exponential synchronization with target node (3) if there exist scalars κ≥0 and θ>0 such that(4)xit−st≤κe−θt,i=1,2,…,N,for t≥0 and any initial conditions.
Next, for the nodei, we let the sequence of sampling data instants be t0,t1,t2,… and correspond a sequence of control updates uit0,uit1,uit2,…. Between the adjacent control updates, the value of input uit is held constant in a zero-order hold fashion. That is,(5)uit=uitk,∀t∈tk,tk+1.
Thus, the controllers are piecewise constant between any sampling intervaltk,tk+1 for k=0,1,2,….
In order to ensure that complex network (1) exponentially synchronizes with target node (3), we intend to use the following feedback controller:(6)uit=−μe−μt−tkxitk−stk,t∈tk,tk+1,to control the i th node, where μ>0 is a scalar to be determined.
Lettingyit=xit−st, while t∈tk,tk+1, we obtain the closed-loop system(7)y˙it=Ayit+Bfxit−fst+∑j=1NcijΓyjt−μe−μt−tkyitk,i=1,2,…,N.
Defineeit=yit−e−μt−tkyitk, then the above system can be rewritten as(8)y˙it=Ayit+Bfxit−fst+∑j=1NcijΓyjt−μyit+μeit,i=1,2,…,N,t∈tk,tk+1.
Denote byyt=y1Tt,y2Tt,…,yNTtT,et=e1Tt,e2Tt,…,eNTtT,ϕit=fxit−fst,ϕt=ϕ1Tt,ϕ2Tt,…,ϕNTtT. Following the above notations, system (8) can be written as the compact form(9)y˙t=IN⊗A−μInN+C⊗Γyt+IN⊗Bϕt+μet=A˜yt+IN⊗Bϕt+μet,for t∈tk,tk+1, where A˜=IN⊗A−μIn+C⊗Γ
Before presenting our main results, we give the following assumptions and lemmas.Assumption 1.
Assume that there exists a constantL≥0 such that(10)fξ1t−fξ2t≤Lξ1t−ξ2t,for any ξ1t,ξ2t∈R, and t≥t0.Assumption 2.
The complex network is connection. That is, there does not exist isolated node. Intuitively, it is a necessary condition for the synchronization of complex network.Lemma 1.
[26]. The Kronecker product ⊗ has the following properties:(1)
A+B⊗C=A⊗C+B⊗C,C⊗A+B=C⊗A+C⊗B(2)
A⊗BT=AT⊗BT(3)
A⊗B−1=A−1⊗B−1(4)
A⊗CB⊗D=AB⊗CD(5)
A⊗B=A⋅Bwhere A,B,C,D are matrices with appropriate dimensions.Lemma 2.
[27]. Consider a differentiable function νt:R+⟶R+ satisfying the following inequality:(11)ν˙t≤−aνt+b∫tktνϑdϑ,∀t∈tk,tk+1,where 0<tk+1−tk≤τM<+∞, a and b are positive reals satisfying bτM<a. Then, the function νt exponentially converges to zero; that is,(12)vt≤vt0e−θt−t0,with θ=a−bτMe−aτM.


## 3. Main Results

In this section, we will present several updated conditions of sampling data such that complex network (1) synchronizes with the target node (3).Theorem 1.
Suppose that Assumptions1 and 2 hold. If there exist a positive definite symmetric matrix P∈Rn×n, scalars α>0,β>0 and μ>0 such that(13)PB+BTP−βIn<0,(14)IN⊗PA+ATP−2μP+αIn+C⊗PΓ+ΓTP<0,and(15)α>βL,then complex network (1) exponentially synchronizes with target node (3) under the action of the controller (6).Proof.
Choose the following Lyapunov functional as(16)Vt=yTtIN⊗Pyt.
Fort∈tk,tk+1, the derivative of Vt with respect to the trajectories of system (9) is(17)V˙t=2yTtIN⊗PA˜yt+IN⊗Bϕt+μet=yTtA˜yt+yTtIN⊗PB+BTPϕt+2μyTtIN⊗Pet,where A˜=IN⊗PA+ATP−2μP+C⊗PΓ+ΓTP. It follows from Assumption 1 and inequality (13) that(18)yTtIN⊗PB+BTPϕt≤yt⋅PB+BTP⋅ϕt≤βyt⋅ϕt≤βLyt2.
In view of (14) and (15), we have(19)V˙t≤yTtA˜+βL⋅Inyt+2μyTtIN⊗Pet≤βL−αyTtyt+2μyt⋅P⋅et≤βL−αλmaxPVt+2μyt⋅P⋅et.
Since(20)e˙t=y˙t+μe−μt−tkytk=A˜yt+IN⊗Bϕt+μet+μe−μt−tkytk=A˜yt+IN⊗Bϕt+μyt.
Integrating fromtk to t on the above equation and using etk=0, one gets(21)et=∫tktA˜yϑ+IN⊗Bϕϑ+μyϑdϑ.
Thus, we obtain(22)et≤∫tktA˜yϑ+IN⊗Bϕϑ+μyϑdϑ≤∫tktA˜⋅yϑ+B⋅ϕϑ+μ⋅yϑdϑ≤A˜+LB+μ∫tktyϑdϑ.
Substituting (22) into (19), then(23)V˙t≤βL−αλPmaxVt+2μyt⋅P⋅A˜+LB+μ∫tktyϑdϑ≤βL−αλPmaxVt+2μλminPVt⋅P⋅A˜+LB+μ∫tktVϑdϑ≤−2g1Vt+2g2Vt∫tktVϑdϑ,where g1=α−βL/2λmaxP and g2=2μ/λminP⋅P⋅A˜+LB+μ which leads to(24)dVtdt≤−g1Vt+g2∫tktVϑdϑ.
It follows from Lemma2 that(25)Vt≤Vt0e−2ρt−t0,with ρ=g1−g2τMe−g1τM>0. By the fact λminPyt2≤Vt, we get(26)yt≤Vt0λminPe−ρt−t0,
which shows that complex network (1) exponentially synchronizes with target node (3) under the action of the controller (6). This completes the proof.Remark 1.
It follows fromρ=g1−g2τMe−g1τM>0 that(27)τM<g1g2=α−βL2λmaxP⋅λminPμPA˜+LB+μ≤α−βL2μPA˜+LB+μ,which gives the upper bound of the sampling data interval. That is, only if the sampling period is not bigger than the bound τM, then complex network (1) can exponentially synchronize with the target node (3) under controller (6). Moreover, if we take the sampling period T∈0,τM as a constant, then it becomes a zero-order hold control with a fixed sampling period.Remark 2.
Although inequalities (13)∼(15) are nonlinear on variables, if we take μ beforehand, then these inequalities are linear matrix inequalities. Furthermore, by solving(28)minμ,subject to9,10,11,one can obtain the most optical μ.
In this paper, the graph of the considered complex network is undirect, so the out-coupling matrixC is real symmetric. According to the properties of a real symmetric matrix, there must exist an orthogonal matrix U such that UTCU=diagλ1,λ2,…,λN, where λi1≤i≤N are the eigenvalues of matrix C. By this fact, we have the following results.Theorem 2.
Suppose that Assumptions1 and 2 hold and the graph is undirect. If there exist a positive definite symmetric matrix P∈Rn×n, scalars α>0,β>0, and μ>0 such that(29)PB+BTP−βIn<0,PA+ATP−2μP+αIn+λiPΓ+ΓTP<0,i=1,2,…,N,and(30)α>βL,then complex network (1) exponentially synchronizes with target node (3) under the action of controller (6).Proof.
Lift multiplyUT⊗In and right multiply UT⊗In on both sides of inequality (14), one can obtain inequality (29). The proof is completed.
It is noticed that we also can use controller (6) to control a part of nodes in complex network (1), which is called the pining control and has been usually used to investigate the synchronization of complex networks in the literature such as [28–30]. Without loss of generality, we only control the former l1≤l≤N nodes with index i=1,2,…,l. Otherwise, we can rearrange the index of nodes.Theorem 3.
Suppose that Assumptions1 and 2 hold. If there exist a positive definite symmetric matrix P∈Rn×n, scalars α>0,β>0 and μ>0 such that(31)PB+BTP−βIn<0,IN⊗PA+ATP+αIn−2μ¯⊗P+C⊗PΓ+ΓTP<0,and(32)α>βL,then complex network (1) exponentially synchronizes with target node (3) under the pinning controller(33)uit=−μe−μt−tkxitk−stk,t∈tk,tk+1,1≤i≤l≤N,where μ¯=diagμ,…,μ︸l,0,…,0︸N−l.Proof.
Substituting (33) into (1), we yield(34)y˙t=IN⊗A−μ¯⊗In+C⊗Γyt+In⊗Bϕt+μ¯⊗Inet=A˜yt+IN⊗Bϕt+μ¯⊗Inet,where A¯=IN⊗A−μ¯⊗In+C⊗Γ. Similar to the proof of Theorem 1, we can obtain the above results. This completes the proof.Remark 3.
It is similar to Remark1, we obtain the length of the sampling interval(35)τ¯M≤α−βL2μPA¯+LB+μ.
In addition, (14) holds if (32) holds because of μ^⊗P>0, where μ^=diag0,…,0︸l,μ,…,μ︸N−l But the converse does not hold, which shows that the complex network can achieve synchronization by using the pinning control, then it can achieve synchronization by controlling all the nodes.


## 4. A Numerical Example

In this section, we provide the following example to show the effectiveness of our proposed method.It is well known that Chua’s dynamical system is described by(36)s˙1t=10s2t−s1t−f1s1t,s˙2t=s1t−s2t+s3t,s˙3t=−15s2t−0.0385s3t,where f1s1t=bs1t+0.5a−bs1t+1−s1t−1, a and b are two constants. Furthermore, it has been shown that system (36) possesses a chaotic behavior if a and b are appropriately chosen. In this example, we choose a=−1.14 and b=−0.714, then the phase diagram of this system is shown in Figure 1 with the initial condition s0=−500,400,−300T.Figure 1
State trajectories of the Chua’s system with the given initial conditions0.On the other hand, this system can be rewritten as(37)s˙t=Ast+Bfst,where st=s1t,s2t,s3tT,A=−101001−110−15−0.0385,B=I3,fst=−10f1s1t,0,0T.Since(38)fst−fs^t=10f1s1t−f1s^t≤10b+a−b⋅s1t−s^t≤10b+a−b⋅st−s^t,for any st,s^t∈R3,fst satisfies Assumption 1 as L=11.4.Consider complex network (1) with five nodes N=5 and take Chua’s dynamical system (37) as the target node. Assume that the outer-coupling matrix is(39)C=−520122−610301−302100−212321−8,and the inner-coupling matrix is Γ=diag0.2,0.5,0.7, respectively. By the LMI toolbox inMatlab, we obtain the feasible solutions of inequalities (13)–(15) as follows:(40)α=5.0296,β=0.4101,P=0.10550.0252−0.01430.02520.1284−0.0386−0.0143−0.03860.1574.According to Theorem1, we know that complex network (1) can synchronize with target node s˙t=Ast+fst under the action of controller (6). Taking the initial condition(41)y0=−20,30,−40,−40,−40,−80,100,70,−40,50,−50,−75,−30,−60,−140T,the state trajectories of the error system (9) with the given parameters are shown in Figure 2, which shows that all the nodes synchronize well with each other, and our proposed method is effective.Figure 2
The error trajectories of system (9) with each node being the Chua’s system i=1,2,…,5.
(a)(b)(c)


## 5. Conclusions

This paper has investigated the exponential synchronization of complex network based on sampled-data control. By constructing a novel sampling-data controller, the complex networks can exponentially synchronize with the target node. Moreover, we have estimated the bound of the sampling intervals. A numerical example has shown that our method is effective. In the future, we will continue to study the synchronization of complex networks with multiple influencing factors, find out the quantity relationship of the synchronization with the complex network parameters, and provide some more effective synchronization controllers. Specially, we will consider how to apply the existing results and methods of complex networks to solve practical problems.



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