# Fixed-Time Synchronization for Different Dimensional Complex Network Systems with Unknown Parameters via Adaptive Control

Authors: Yude Ji; Yunli Gong; Shan Su; Xiaoxue Bai
Journal: Complexity
 (2021)

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## Abstract

This article is related to the issue of fixed-time synchronization of different dimensional complex network systems with unknown parameters. Two suitable adaptive controllers and dynamic parameter estimations are proposed such that the complex network driving and response systems can be synchronized in the settling time. Based on fixed-time control theory and Lyapunov functional method, novel sufficient conditions are provided to guarantee the synchronization within the fixed times, and the settling times are explicitly evaluated, which are independent of the initial synchronization errors. Finally, a numerical example is given to illustrate the effectiveness of the proposed control algorithms.


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## Body

## 1. Introduction

Complex network systems (CNSs) usually comprise a family of coupled interconnected nodes, in which each node represents a dynamical system. Over the past decades, the field of CNS [1–6] has been gaining wide attention due to its broad applications in nature and human society, such as World Wide Web, metabolic pathways, ecological networks, food webs, electrical power grids, and so on. The synchronization of CNS has been intensively investigated due to its important applications in secure communication [7, 8], image encryption [9, 10], automatic control [11], information processing [12], and other fields. Synchronization refers to a process that all the nodes seek to adjust a certain property of their motion to a common behavior as time evolves. Generally speaking, there are two typical synchronizations of CNS including inner synchronization, i.e. synchronization of all the nodes inside every network, and outer synchronization between whole complex networks simultaneously. In recent years, many synchronization patterns have been proposed to the CNS, for instance, exponential synchronization [13–15], cluster synchronization [16, 17], phase synchronization [18], robust synchronization [19], generalized synchronization [20], projective synchronization [21, 22], complete synchronization [23], antisynchronization [24], etc. There are several control approaches for the synchronization of CNS, including sliding mode control [25], intermittent control and impulsive control [26, 27], pinning adaptive control [28, 29], optimal control [30], fuzzy control [31] and so on.In the analysis of synchronization problems, the convergence rate of the proposed controller has been considered as a practically important issue, which leads to the investigation of finite-time synchronization [32–39]. On the one hand, compared with the infinite-time synchronization, finite-time synchronization essentially requires high-precision convergence; furthermore, the states of CNS remain completely identical after some finite time, which is called the settling time. On the other hand, a critical issue of finite-time synchronization is that the settling time heavily depends upon the initial synchronization error of considered systems. Therefore, the modification of initial synchronization errors will lead to the different settling times. However, the initial conditions of many practical systems might be random, which leads to the inaccessibility of settling time. To overcome the defect of finite-time control, the concept of fixed-time stability was firstly proposed by Polyakov [40]. Different from the finite-time method, the settling time of fixed-time technique can be estimated offline. Namely, the stability of system can be stabilized in a predefined time by properly selecting control gains [41, 42]. Until now, many relevant results concerning fixed-time synchronization of CNS have been published in [43–48]. The fixed-time controller for delayed memristor-based recurrent neural networks is discussed in [43]. The fixed-time synchronization of CNS with nonidentical nodes and stochastic noise perturbations was discussed in [44]. Specially, regardless of the initial conditions, the settling time of fixed-time synchronization for hybrid coupled networks can be adjusted to the desired values in [45]. In addition, based on a new fast fixed-time control method and delay feedback control, the fixed-time synchronization for the delayed neural networks and delayed Cohen–Grossberg neural networks was investigated in [46, 47], respectively. The fixed-time cluster synchronization for CNS via pinning control is considered in [48]. By designing new Lyapunov function and comparison systems, the fixed-time synchronization of CNS with impulsive effects via nonchattering control was investigated in [49]. However, most of the aforementioned results just presented either were more specific to synchronization of some systems or did not analyze the effects of unknown parameters for systems.From the practical point of view, uncertainties in the plant models may arise due to parameter variation, neglected dynamics, unmodeled dynamics, and either unknown or approximately known parameters. Therefore, it is important and necessary to study the synchronization in such systems with unknown parameters. In recent years, some results of synchronization for CNS with unknown parameters have been obtained. In [50], the adaptive synchronization was studied for the response system contains unknown uncertain nonlinearities and unknown dead zones in neural networks. In [51], based on the invariant principle of functional differential equations and parameter identification, the rigorous adaptive feedback scheme is derived to achieve synchronization of two coupled neural networks with time-varying delay. But in real application, the parameters of the CNS may be mismatched. Paper [52] proved that for the general complex networks with identification of the topological structure and unknown parameters, synchronization can be achieved by using adaptive control approach. The adaptive control, as presented in [51, 52], evaluates the unknown parameters to achieve synchronization of two networks. The adaptive finite-time synchronization of different dimensional coupled chaotic (or hyperchaotic) systems with unknown parameters has been discussed in [53, 54]. In [55], the Lyapunov stability theory and the nonlinear feedback control were used to get the robust finite-time synchronization for the chaotic systems with different orders. However, this paper does not analyze the effects of finite time adaptive feedback control on unknown parameters identification of two chaotic neural networks. Recently, the generalized outer synchronization between two nondissipatively coupled CNSs with different time-varying coupling delays was investigated in [56]. Based on new concepts of finite-time passivity, two models of delayed multiweighted CNS with different dimensional nodes were presented in [57]. The finite-time synchronization problem of all parameters estimation between two different dimensional fractional-order complex dynamical networks was proposed in [58]. By using fractional adaptive laws, the fractional-order nonlinear systems with actuator faults whose parametric uncertainties and parameters are fully unknown were investigated in [59, 60], respectively. These works have provided important theoretical and application values than the previous model for single driving and single response system.It is worthwhile to emphasize that all results above only considered the same dimensional CNS, while the different dimensional CNSs are very universal. Actually, we can observe that both circulatory and respiratory systems behave in synchronous way, but their models are essentially different and they have different dimensions. So, the theoretical research of synchronization for strictly different dynamical systems is very important. Although many finite-time synchronization schemes have been developed in the literature for various CNSs, to the best of the authors’ knowledge, few published papers consider the fixed-time synchronization and identification of unknown parameters for the different dimensional CNSs. So, it is meaningful to consider fixed-time synchronization of all parameters’ estimation for different dimensional CNSs. This paper aims to solve this challenging problem.The remainder of this paper is organized as follows. The preliminary results and problem formulation are given in Section2. The main results are derived in Section 3. The numerical simulation is provided to verify the results in Section 4. The conclusion is drawn in Section 5.Throughout the paper, the following notations will be used. Letℝn denote n-dimension real space and ℝ+ denote 1-dimension positive real space, and In is the identity matrix with n dimensions. For any x∈ℝn, let x=xTx1/2, and superscript T denotes the transpose of a vector or matrix. For a matrix A∈ℝn×n∈ℝn×m, A−1 denotes the inverse matrix of A (the right inverse matrix of A) and A⊗B represents the Kronecker product of matrices of A and B. For any vector st=s1,s2,…,snT∈ℝn,stμ=s1μ,s2μ,…,snμT, signst=signs1,signs2,…,signsnT, and sigμst=signs1s1μ,signs2s2μ,…,signsnsnμT, where sign⋅ is the standard sign function.


## 2. Preliminary Results and Problem Formulation

In order to facilitate the synchronization control of different dimensional CNSs with unknown parameters, some definitions and lemmas are presented in the section.For the following driving system:(1)x˙=Fxt,x0=x0,where xt∈ℝn represents the system state, F:ℝn⟶ℝn is a continuous function on an open neighbourhood D⊆ℝn of the origin with F0=0.The controlled response system is given by(2)y˙=Gyt+ut,where yt∈ℝm represents the system state, ut∈ℝm represents the control input, and G:ℝm⟶ℝm is a continuous function.Remark 1.
Whenn=m, the driving and response systems (1) and (2) have the same dimension. There are a lot of relevant works about the problem of synchronization. When n≠m, the generalized synchronization of systems (1) and (2) with different dimensional is studied in [53–55]. There are few results about the finite-time and fixed-time synchronization of systems (1) and (2) with different dimensions. Therefore, it is essential to give some theoretical results. In addition, it should be noticed that lots of well-known chaotic systems could be written in the form of (1) and (2), such as, Lorenz system, Chen system, Rössler system, Duffing system, hyperchaotic Lorenz system, and hyperchaotic Rössler system.
The definition of fixed-time generalized synchronization is given as follows.Definition 1.
Systems (1) and (2) are said to be in finite-time generalized synchronization by adding a suitable designed controller to response system (2), if there exist two continuously differentiable functions ϕ:ℝn⟶ℝr,φ:ℝm⟶ℝr and there exists a number T^e0≥0 such that(3)limt⟶T^e0et=0,et=0,for allt>T^e0,where et=ϕx−φy∈ℝr represents the synchronization error of different dimensional systems (1) and (2) and ⋅ denotes the Euclidean norm.
The time(4)Te0=infT^e0≥0:et=0,for allt>T^e0,is called the settling time of synchronization, which is dependent of the initial value e0. Furthermore, if there exists a fixed-time T≥0 such that Te0≤T for any e0∈ℝr, systems (1) and (2) are said to be in fixed-time generalized synchronization.Remark 2.
In Definition1, the functions ϕ and φ are two arbitrary continuously differentiable functions; according to the different form of functions ϕ and φ, the generalized synchronization can also be reduced to the projective synchronization, so the research about synchronization has more extensive meaning than previous results. If m=n,ϕx=x,φy=βy, where β is a scalar, then the generalized synchronization will be reduced to the projective synchronization and β denotes the projective coefficient. Especially, if β=1, then the generalized synchronization will be reduced to the complete synchronization; if β=−1, then the generalized synchronization will be reduced to the projective antisynchronization.Remark 3.
Ifm=n,ϕx≡0,φy=y, the equation in Definition 1 is reduced to the following form:(5)limt⟶T^y0yt=0,yt=0,for allt>T^y0.
In this case, system (2) is said to be globally finite-time asymptotically stabilized to the origin.
The following lemmas are useful for our main results.Lemma 1.
(see [61]). Let a1,a2,…,aN≥0 and 0<p≤1,q>1. Then, the following two inequalities hold:(6)∑i=1Nαip≥∑i=1Nαip,∑i=1Nαiq≥N1−q∑i=1Nαiq.Lemma 2.
(see [62]). Consider a scalar system(7)y˙=−αym/n−βyp/q,y0=y0,where α>0 and β>0 and m,n,p,q are positive odd integers satisfying m>n and p<q. Then, the equilibrium of system (7) is fixed-time stable and the settling time is(8)T=nαm−n+qβq−p.In this paper, the fixed time and adaptive controller are designed to realize the synchronization between the driving network systems and response ones with unknown parameters in the settling timeT, i.e., limt⟶Tet=0.


## 3. Main Results

In this section, the sufficient conditions of fixed-time synchronization for two different dimensional CNSs are obtained via adaptive control methods.We consider the following driving dynamical networks consisting ofN identical nodes:(9)xi.t=Fixit+fixitΦi+∑j=1Naijxjt,i=1,2,…,N,where xit=xi1t,xi2t,…,xintT∈ℝn denotes the state vector of ith node, Fi:ℝn⟶ℝn,fi:ℝn⟶ℝn×ℝp are continuous functions, and Φi=Φi1,Φi2,⋯ΦipT∈ℝp represents the vector of unknown parameters. A=aij∈ℝN×N denotes the weight configuration matrix. If there is a connection between nodes i and ji≠j, then aij=aji>0; otherwise, aij=aji=0. The diagonal elements of matrices A are defined as(10)aii=−∑j=1,j≠iNaij.For simplicity, driving system (9) is inferred to be in the following form:(11)x˙t=Fxt+fxtΦ+A⊗Inxt,where xt=x1Tt,x2Tt,…,xNTtT,Fxt=F1Tx1t,F2Tx2t,…,FNTxNtT, fxt=diagf1x1t,f2x2t,…,fNxNt,Φ=Φ1T,Φ2T,…,ΦNTT.Correspondingly, the response system can be given as follows:(12)yi.t=Giyit+giyitΨi+∑j=1Naijyjt+uit,i=1,2,…,N.Similarly, response system (12) is inferred to be in the following form:(13)y˙t=Gyt+gytΨ+A⊗Imyt+ut,where yt=y1Tt,y2Tt,…,yNTtT,yit=yi1t,yi2t,…,yimtT∈ℝm denotes the response state vector of the ith node, Gyt=G1Ty1t,G2Ty2t,…,GNTyNtT,Gi:ℝm⟶ℝm, gyt=diagg1y1t,g2y2t,…,gNyNt, gi:ℝm⟶ℝm×ℝq are continuous functions, Ψ=Ψ1T,Ψ2T,…,ΨNTT,Ψi=Ψi1,Ψi2,⋯ΨiqT∈ℝq represent the vector of unknown parameters, and ut=u1Tt,u2Tt,…,uNTtT,uit∈ℝm denotes the control input of the ith node.Consider the driving system (9) and response system (12) with unknown parameters; we define the synchronization error as et=e1Tt,e2Tt,…,eNTtT,eit=ϕxit−φyit, where ϕ=ϕ1,ϕ1,…,ϕrT:ℝn⟶ℝr and φ=φ1,φ1,…,φrT:ℝm⟶ℝr are both continuously differentiable functions; then, it is well known that the fixed-time synchronization problem between driving system (9) and response system (12) can be transformed to the equivalent problem of the fixed-time stabilization for the following error system:(14)e˙it=JϕxiFixi+∑j=1Naijxjt+fixiΦi−JφyiGiyi+∑j=1Naijyjt+giyiΨi+uit,where(15)Jϕxi=∂ϕ1xi∂xi1∂ϕ1xi∂xi2⋯∂ϕ1xi∂xin∂ϕ2xi∂xi1∂ϕ2xi∂xi2⋯∂ϕ2xi∂xin⋮⋮⋱⋮∂ϕrxi∂xi1∂ϕrxi∂xi2⋯∂ϕrxi∂xin,Jφyi=∂φ1yi∂yi1∂φ1yi∂yi2⋯∂φ1yi∂yim∂φ2yi∂yi1∂φ2yi∂yi2⋯∂φ2yi∂yim⋮⋮⋱⋮∂φryi∂yi1∂φryi∂yi2⋯∂φryi∂yim,denote the Jacobin matrices of functions ϕxi and φyi, respectively.For simplicity, error system (14) is inferred to be in the following form:(16)e˙t=JϕxFxt+A⊗Inxt+fxtΦ−JφyGyxt+A⊗Imyt+gyxtΨ+ut,where Jϕx=diagJϕx1,Jϕx2,…,JϕxN and Jφy=diagJφy1,Jφy2,…,JφyN.Assumption 1.
Letr≤minm,n and matrix Jφ be row full rank.
Now, two novel fixed-time synchronization control methodologies are given in the following two theorems.Theorem 1.
Under Assumption1, for error system (14) of driving system (9) and response system (12) with unknown parameters, the controller ut is designed as(17)uit=−Giyi−∑j=1Naijyjt−giyiΨ^i+Jφ−1yiJϕxiFixi+∑j=1Naijxjt+fixiΦ^i+α1⋅sigm1/n1ei+β1⋅sigp1/q1ei,where Φ^i, Ψ^i denote the estimations of Φi and Ψi, α1>0,β1>0 and m1,n1,p1,q1 are all positive odd integers satisfying m1>n1,p1<q1. And the following adaptive update laws are proposed to estimate the unknown parameters Φ^i and Ψ^i:(18)Φ^˙i=JϕxifixiTeit−α1⋅sigm1/n1Φ˜i−β1⋅sigp1/q1Φ˜i,(19)Ψ^˙i=−JφyigiyiTeit−α1⋅sigm1/n1Ψ˜i−β1⋅sigp1/q1Ψ˜i,where Φ˜i=Φ^i−Φi, Ψ˜i=Ψ^i−Ψi. Then, systems (9) and (12) are fixed-time synchronized with the setting time T1=1/2m1+n1/2n1α1N1n1−m1/2n12n1/m1−n1+1/2p1+q1/2q1β12q1/q1−p1, where N1=Nr+p+q.Proof.
Substituting controller (17) into the error system (14) yields(20)e˙it=−JϕxifixiΦ˜i+JφyigiyiΨ˜i−α1⋅sigm1/n1ei−β1⋅sigp1/q1ei.
Consider the following Lyapunov function:(21)Vt=12∑i=1NeiTteit+12∑i=1NΦ˜iTΦ˜i+12∑i=1NΨ˜iTΨ˜i.
Then, the time derivative ofVt along (20) with the adaptive tuning laws (18) and (19) can be calculated as(22)V˙t=∑i=1NeiTte˙it+∑i=1NΦ˜iTΦ˜˙i+∑i=1NΨ˜iTΨ˜˙i=∑i=1NeiTt−JϕxifixiΦ˜i+JφyigiyiΨ˜i−α1⋅sigm1/n1ei−β1⋅sigp1/q1ei+∑i=1NΦ˜iTJϕxifixiTeit−α1⋅sigm1/n1Φ˜i−β1⋅sigp1/q1Φ˜i+∑i=1NΨ˜iT−JφyigiyiTeit−α1⋅sigm1/n1Ψ˜i−β1⋅sigp1/q1Ψ˜i=−α1∑i=1NeiTt⋅sigm1/n1ei−α1∑i=1NΦ˜iT⋅sigm1/n1Φ˜i−α1∑i=1NΨ˜iT⋅sigm1/n1Ψ˜i−β1∑i=1NeiTt⋅sigp1/q1ei−β1∑i=1NΦ˜iT⋅sigp1/q1Φ˜i−β1∑i=1NΨ˜iT⋅sigp1/q1Ψ˜i=−α1∑i=1N∑j=1reijtm1+n1/n1−α1∑i=1N∑j=1pΦ˜ijm1+n1/n1−α1∑i=1N∑j=1qΨ˜ijm1+n1/n1−β1∑i=1N∑j=1reijtp1+q1/q1−β1∑i=1N∑j=1pΦ˜ijp1+q1/q1−β1∑i=1N∑j=1qΨ˜ijp1+q1/q1.
Applying the Lemma1 yields(23)V˙t≤−2m1+n1/2n1α1N1n1−m1/2n112∑i=1N∑j=1reijt2+12∑i=1N∑j=1pΦ˜i2+12∑i=1N∑j=1qΨ˜i2m1+n1/2n1−2p1+q1/2q1β112∑i=1N∑j=1reijt2+12∑i=1N∑j=1pΦ˜ij2+12∑i=1N∑j=1qΨ˜ij2p1+q1/2q1=−2m1+n1/2n1α1N1n1−m1/2n112∑i=1NeiTteit+12∑i=1NΦ˜iTΦ˜i+12∑i=1NΨ˜iTΨ˜im1+n1/2n1−2p1+q1/2q1β112∑i=1NeiTteit+12∑i=1NΦ˜iTΦ˜i+12∑i=1NΨ˜iTΨ˜ip1+q1/2q1=−2m1+n1/2n1α1N1n1−m1/2n1Vtm1+n1/2n1−2p1+q1/2q1β1Vtp1+q1/2q1.
According to Lemma2, error system (20) is fixed-time stable with the setting time(24)T1=12m1+n1/2n1α1N1n1−m1/2n12n1m1−n1+12p1+q1/2q1β12q1q1−p1.
The proof is completed.For driving system (9) and response system (12) with unknown parameters, the proposed control scheme in Theorem 1 can be extended to design a new adaptive controller.Theorem 2.
Under Assumption1, for driving system (9) and response system (12) with unknown parameters, the controller ut is designed as(25)uit=−Giyi−∑j=1Naijyjt−giyiΨ^i+Jφ−1yiJϕxiFixi+∑j=1Naijxjt+fixiΦ^i+α2⋅sigm2/n2ei+β2⋅sigp2/q2ei+γt⋅sigp2/q2ei,where Φ^i, Ψ^i denote the estimations of Φi and Ψi, α2>0,β2>0 and m2,n2,p2,q2 are all positive odd integers satisfying m2>n2,p2<q2. The following adaptive update laws are proposed to estimate the unknown parameters Φ^i and Ψ^i:(26)Φ^˙i=JϕxifixiTeit−α2⋅sigm2/n2Φ˜i−β2⋅sigp2/q2Φ˜i−γt⋅sigp2/q2Φ˜i,(27)Ψ^˙i=−JφyigiyiTeit−α2⋅sigm2/n2Ψ˜i−β2⋅sigp2/q2Ψ˜i−γt⋅sigp2/q2Ψ˜i,where Φ˜i=Φ^i−Φi, Ψ˜i=Ψ^i−Ψi. And the adaptive updated law γt is designed as(28)γ˙t=∑i=1NeiTt⋅sigp2/q2ei+∑i=1NΦ˜iT⋅sigp2/q2Φ˜i+∑i=1NΨ˜iT⋅sigp2/q2Ψ˜i−α2⋅signγtγtm2/n2−β2⋅signγtγtp2/q2.Then, systems (9) and (12) are fixed-time synchronized with the setting time(29)T2=12m2+n2/2n2α2N2n2−m2/2n22n2m2−n2+12p2+q2/2q2β22q2q2−p2,where N2=Nr+p+q+1.Proof.
Substituting controller (25) into error system (14) yields(30)e˙it=−JϕxifixiΦ˜i+JφyigiyiΨ˜i−α2⋅sigm2/n2ei−β2⋅sigp2/q2ei−γt⋅sigp2/q2ei.
Consider the following positive definite function:(31)Vt=12∑i=1NeiTteit+12∑i=1NΦ˜iTΦ˜i+12∑i=1NΨ˜iTΨ˜i+12γ2t.
Then, the time derivative ofVt along (30) with the adaptive tuning laws (26)–(28) can be calculated as(32)V˙t=∑i=1NeiTte˙it+∑i=1NΦ˜iTΦ˜˙i+∑i=1NΨ˜iTΨ˜˙i+γtγ˙t=∑i=1NeiTt−JϕxifixiΦ˜i+JφyigiyiΨ˜i−α2⋅sigm2/n2ei−β2⋅sigp2/q2ei−γt⋅sigp2/q2ei+∑i=1NΦ˜iTJϕxifixiTeit−α2⋅sigm2/n2Φ˜i−β2⋅sigp2/q2Φ˜i−γt⋅sigp2/q2Φ˜i+∑i=1NΨ˜iT−JφyigiyiTeit−α2⋅sigm2/n2Ψ˜i−β2⋅sigp2/q2Ψ˜i−γt⋅sigp2/q2Ψ˜i+γt∑i=1NeiTt⋅sigp2/q2ei+∑i=1NΦ˜iT⋅sigp2/q2Φ˜i+∑i=1NΨ˜iT⋅sigp2/q2Ψ˜i−α2⋅signγtγtm2/n2−β2⋅signγtγtp2/q2=−α2∑i=1NeiTt⋅sigm2/n2ei−α2∑i=1NΦ˜iT⋅sigm2/n2Φ˜i−α2∑i=1NΨ˜iT⋅sigm2/n2Ψ˜i−α2γt⋅signγtγtm2/n2−β2∑i=1NeiTt⋅sigp2/q2ei−β2∑i=1NΦ˜iT⋅sigp2/q2Φ˜i−β2∑i=1NΨ˜iT⋅sigp2/q2Ψ˜i−β2γt⋅signγtγtp2/q2=−α2∑i=1N∑j=1reijtm2+n2/n2−α2∑i=1N∑j=1pΦ˜ijm2+n2/n2−α2∑i=1N∑j=1pΨ˜ijm2+n2/n2−α2γt⋅signγtγtm2/n2−β2∑i=1N∑j=1reijtp2+q2/q2−β2∑i=1N∑j=1pΦ˜ijp2+q2/q2−β2∑i=1N∑j=1qΨ˜ijp2+q2/q2−β2γt⋅signγtγtp2/q2.
Applying Lemma1 yields(33)V˙t≤−2m2+n2/2n2α2N2n2−m2/2n212∑i=1N∑j=1reijt2+12∑i=1N∑j=1pΦ˜ij2+12∑i=1N∑j=1qΨ˜ij2+12γ2tm2+n2/2n2−2p2+q2/2q2β212∑i=1N∑j=1reijt2+12∑i=1N∑j=1pΦ˜ij2+12∑i=1N∑j=1qΨ˜ij2+12γ2tp2+q2/2q2=−2m2+n2/2n2α2N2n2−m2/2n212∑i=1NeiTteit+12∑i=1NΦ˜iTΦ˜i+12∑i=1NΨ˜iTΨ˜i+12γ2tm2+n2/2n2−2p2+q2/2q2β212∑i=1NeiTteit+12∑i=1NΦ˜iTΦ˜i+12∑i=1NΨ˜iTΨ˜i+12γ2tp2+q2/2q2=−2m2+n2/2n2α2N2n2−m2/2n2Vtm2+n2/2n2−2p2+q2/2q2β2Vtp2+q2/2q2.
According to Lemma2, error system (30) is also fixed-time stable with the setting time(34)T2=12m2+n2/2n2α2N2n2−m2/2n22n2m2−n2+12p2+q2/2q2β22q2q2−p2.
The proof is completed.Remark 4.
In this paper, we provide two adaptive control strategies to ensure the fixed-time synchronization of CNS. Compared with the adaptive controller (17) of Theorem 1, controller (25) of Theorem 2 includes an additional control term γt; such a control strategy can guarantee the fixed-time synchronization which is easier to be achieved, but the whole settling time will be increased.Remark 5.
The proposed controllers (17) of Theorem 1 and (25) of Theorem 2 can make the driving system (9) reach the response system (12) in the predefined time.Remark 6.
Note that the dynamic behavior of node in the error system (14) is discontinuous, which is different from the continuous dynamics of the traditional networks. In this paper, by designing the novel discontinuous feedback control schemes with sig function, chattering will occur. In order to weaken the chattering effect, by using new adaptive dynamic parameter estimations, a unified control framework is designed to guarantee the fixed-time synchronization of the discontinuous network driving system (9) reach response system (12).Remark 7.
The fixed-time impulsive synchronization control for network systems is inferred as follows: for the first stage, only impulsive control is exerted to make the synchronization error converge to one prescribed region. Then, for the second stage, only fixed-time control is exerted to realize the synchronization error converge to zero. Furthermore, it is worth noting that, in many practical applications, the dynamic behavior of CNS is often unknown. Thus, it is significant to develop an effective adaptive synchronization scheme for network systems via impulsive control and fixed-time control; more theoretical and practical aspects can be explored in the future work.


## 4. Numerical Simulation

In this section, a numerical example is presented to show the effectiveness of the proposed schemes obtained in the previous section. We consider the CNS with 10 nodes and choose node dynamics following different dimensional Lorenz and Chen hyperchaotic systems:(35)x˙i1tx˙i2tx˙i3t=0−xi2−xi1xi3xi1xi2︸Fixit+xi2−xi1000xi1000−xi3︸fixitΦi1Φi2Φi3︸Φi+∑j=110aijxj1t∑j=110aijxj2t∑j=110aijxj3t,i=1,2,…,10,(36)y˙i1ty˙i2ty˙i3ty˙i4t=yi4−yi1yi3yi1yi2−yi2yi3︸Gixit+yi2−yi100000yi1yi200000−yi300000yi4︸gixitΨi1Ψi2Ψi3Ψi4Ψi5︸Ψi+∑j=110aijyj1t∑j=110aijyj2t∑j=110aijyj3t∑j=110aijyj4t∑j=110aijyj5t+ui1tui2tui3tui4tui5t︸uit,i=1,2,…,10.As can be seen in Figure1, the weight configuration matrix A is given by(37)A=−40011000110−41101010001−51010110111−60111001000−31001001111−71110000101−30010111010−61110101101−50100000110−3.Figure 1
A simple complex dynamical network systems with ten nodes.The associated parameters are chosen asΦi1=10,Φi2=28,Φi3=8/3 and Ψi1=35,Ψi2=7,Ψi3=12,Ψi4=3,Ψi5=0.5,i=1,2,…,10 such that system (35) and system (36) are chaotic. According to the designing strategy of Theorems 1 and 2, select (35) as the driving system and (36) as the controlled system n<m.Assume thatϕxi=xi1xi2,xi1−xi3T and φyi=0.5yi1−0.5yi2,0.5yi3−0.5yi4T,i=1,2,…,10. The Jacobian matrices of functions ϕxi and φyi are obtained, respectively, as follows:(38)Jϕxi=xi2xi1010−1,Jφyi=0.5−0.500000.5−0.5,and the right inverse matrix of Jφyi is as follows:(39)Jφ−1yi=10−10010−1.Compared with the controller in [53], the designed controllers can show better convergence performance in this article. Based on the controller design (27) and the error dynamics (28) in [53], we obtain(40)e˙i1t=−k⋅signei1ei1α−p⋅ei1,e˙i2t=−k⋅signei2ei2α−p⋅ei2,where k=0.3,p=3,α=3/5. Moreover, the initial values of the state variable for systems (35) and (36) are given by xi10,xi20,xi30T=1+0.05i,−0.05i,0.05iT, yi10,yi20,yi30,yi40T=1+0.05i,−1−0.05i,−0.05i,0.05iT,i=1,2,…,10, and the settling time can be estimated by T=ln1+p/ket01−α/2/1−αp=2.1384. However, the initial values of the state variable for systems (35) and (36) are chosen as xi10,xi20,xi30T=1+is,−is,isT,yi10,yi20,yi30yi40T=1+is,−1−is,−is,isT,i=1,2,…,10, where s is chosen randomly on the interval 0,0.1; by computation, the max settling time can be estimated by Tmax=2.5903, and the trajectories of errors are shown in Figure 2.Figure 2
The synchronization errors between systems (35) and (36) by controller (27) in [53]. (a) ei1ti=1,2,…,10. (b) ei2ti=1,2,…,10.
(a)(b)Based on expressions (26)–(28), by simple computations, we can obtain the adaptive updating laws in following form:(41)Φ^˙i1t=xi2−xi1xi2ei1+ei2−α2⋅signΦ˜i1Φ˜i1m2/n2−β2⋅signΦ˜i1Φ˜i1p2/q2−γt⋅signΦ˜i1Φ˜i1p2/q2,Φ^˙i2t=xi12ei1−α2⋅signΦ˜i2Φ˜i2m2/n2−β2⋅signΦ˜i2Φ˜i2p2/q2−γt⋅signΦ˜i2Φ˜i2p2/q2,Φ^˙i3t=xi3ei2−α2⋅signΦ˜i3Φ˜i3m2/n2−β2⋅signΦ˜i3Φ˜i3p2/q2−γt⋅signΦ˜i3Φ˜i3p2/q2,Ψ^˙i1t=−0.5yi2−yi1ei1−α2⋅signΨ˜i1Ψ˜i1m2/n2−β2⋅signΨ˜i1Ψ˜i1p2/q2−γt⋅signΨ˜i1Ψ˜i1p2/q2,Ψ^˙i2t=0.5yi1ei1−α2⋅signΨ˜i2Ψ˜i2m2/n2−β2⋅signΨ˜i2Ψ˜i2p2/q2−γt⋅signΨ˜i2Ψ˜i2p2/q2,Ψ^˙i3t=0.5yi2ei1−α2⋅signΨ˜i3Ψ˜i3m2/n2−β2⋅signΨ˜i3Ψ˜i3p2/q2−γt⋅signΨ˜i3Ψ˜i3p2/q2,Ψ^˙i4t=0.5yi3ei2−α2⋅signΨ˜i4Ψ˜i4m2/n2−β2⋅signΨ˜i4Ψ˜i4p2/q2−γt⋅signΨ˜i4Ψ˜i4p2/q2,Ψ^˙i5t=0.5yi4ei2−α2⋅signΨ˜i5Ψ˜i5m2/n2−β2⋅signΨ˜i5Ψ˜i5p2/q2−γt⋅signΨ˜i5Ψ˜i5p2/q2,γ˙t=∑i=110∑j=12eijtp2+q2/q2+∑i=110∑j=13Φ˜ijp2+q2/q2+∑i=110∑j=15Ψ˜ijp2+q2/q2−α2⋅signγtγtm2/n2−β2⋅signγt|γt|p2/q2.According to (30), the error dynamics are derived as follows:(42)e˙i1t=−xi2xi2−xi1Φ˜i1−xi12Φ˜i2+0.5yi2−yi1Ψ˜i1−0.5yi1Ψ˜i2−0.5yi2Ψ˜i3−α2⋅signei1ei1m2/n2−β2⋅signei1ei1p2/q2−γt⋅signei1ei1p2/q2,e˙i2t=−xi2−xi1Φ˜i1−xi3Φ˜i3−0.5yi3Ψ˜i4−0.5yi4Ψ˜i5−α2⋅signei2ei2m2/n2−β2⋅signei2ei2p2/q2−γt⋅signei2ei2p2/q2,where the parameters are given as m2=7,n2=5,p2=9,q2=11, α2=20,andβ2=20. The initial values of the state variable for systems (35) and (36) are xi10,xi20,xi30T=1+is,−is,isT,yi10,yi20,yi30,yi40T=1+is,−1−is,−is,isT,i=1,2,…,10, where s is chosen randomly on the interval 0,0.1. Let the initial values of estimations be Φ^i0=Φ^i10,Φ^i20,Φ^i30=1,1,1 and Ψ^i0=Ψ^i10,Ψ^i20,Ψ^i30,Ψ^i40,Ψ^i50=2,2,2,2,2, and the initial value of γt is given as γ0=2.First, according to Theorem1, systems (35) and (36) can realize fixed-time synchronization under controller (17), and settling time T1=2.3283 by computation. The trajectories of error states eit are shown in Figure 3. The trajectories of estimations Φ^it and Ψ^it are depicted in Figures 4(a) and 4(b). Secondly, according to Theorem 2, systems (35) and (36) can realize fixed-time synchronization under controller (25), and the settling time T2=2.4471 by computation. Then, the trajectories of error states eit are shown in Figure 5. The trajectories of estimations Φ^it and Ψ^it are depicted in Figures 6(a) and 6(b). Moreover, the trajectory of γt is illustrated in Figure 7. Finally, as shown in Figures 3 and 4, we can find that the chaotic oscillation has been suppressed completely. Moreover, one can see from Figures 3 and 4 that systems (35) and (36) achieve synchronization within the fixed settling time T1 and T2, respectively.Figure 3
The synchronization errors between systems (35) and (36) by controller (17). (a) ei1ti=1,2,…,10. (b) ei2ti=1,2,…,10.
(a)(b)Figure 4
The synchronization errors between systems (35) and (36) by controller (25). (a)ei1ti=1,2,…,10. (b)ei2ti=1,2,…,10.
(a)(b)Figure 5
The estimations of unknown parameters for systems (35) and (36) by controller (17). (a) Φ^iti=1,2,…,10. (b) Ψ^iti=1,2,…,10.
(a)(b)Figure 6
The estimations of unknown parameters for systems (35) and (36) by controller (25). (a) Φ^iti=1,2,…,10. (b) Ψ^iti=1,2,…,10.
(a)(b)Figure 7
The trajectory of adaptive updated lawγt by controller (25).Compared with the simulation results (see Figure 2 in the work of [53]), our adaptive law with simulation results (Figures 3 and 4) is faster in convergence. From Figures 5 and 6, we can see that the estimations of unknown parameters converge to the true values rapidly. It can be concluded that with the proposed adaptive control approach, better synchronization performances are obtained compared with the previous results.


## 5. Conclusions

This paper has addressed the fixed-time synchronization of different dimensional complex network driving and response systems with unknown parameters. Two suitable controllers and adaptive update laws are established to guarantee the synchronization realized in a given fixed time, respectively. Sufficient conditions have been obtained to guarantee the fixed-time synchronization for the driving and response systems. Meanwhile, our results can be easily extended to the cases of projective synchronization and complete synchronization. Finally, the effectiveness of theoretical analysis has been verified by a numerical simulation.



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