# Finite-Time Nonfragile Synchronization of Stochastic Complex Dynamical Networks with Semi-Markov Switching Outer Coupling

Authors: Rathinasamy Sakthivel; Ramalingam Sakthivel; Boomipalagan Kaviarasan; Chao Wang; Yong-Ki Ma
Journal: Complexity
 (2018)

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## Abstract

The problem of robust nonfragile synchronization is investigated in this paper for a class of complex dynamical networks subject to semi-Markov jumping outer coupling, time-varying coupling delay, randomly occurring gain variation, and stochastic noise over a desired finite-time interval. In particular, the network topology is assumed to follow a semi-Markov process such that it may switch from one to another at different instants. In this paper, the random gain variation is represented by a stochastic variable that is assumed to satisfy the Bernoulli distribution with white sequences. Based on these hypotheses and the Lyapunov-Krasovskii stability theory, a new finite-time stochastic synchronization criterion is established for the considered network in terms of linear matrix inequalities. Moreover, the control design parameters that guarantee the required criterion are computed by solving a set of linear matrix inequality constraints. An illustrative example is finally given to show the effectiveness and advantages of the developed analytical results.


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## Body

## 1. Introduction

During the past twenty years, the investigation of complex dynamical networks (CDNs) that consist of a huge number of interacting dynamical nodes has received a great deal of attention from various science and engineering areas, such as social networks, ecological prey-predator networks, protein networks, power grids, and ecosystems [1, 2]. It should be mentioned that the analysis of dynamical behaviors of CDNs has become a hot research topic in recent years. Among many dynamical behaviors, synchronization phenomenon is the most important behavior and several interesting and efficient methodologies have been developed in the literature to solve the synchronization problem of various kinds of CDNs; for instance, see [3–7]. In [4], the problem of inner synchronization of a CDN has been investigated by considering two different types of guaranteed cost dynamic feedback controllers, where the control gains corresponding to two feedback controllers have different dimensions subject to the topological structure of the CDN. In [5], the problem of outer synchronization between two hybrid-coupled delayed dynamical networks has been discussed by using the aperiodically adaptive intermittent pinning control, where a simple and elegant pinned-node selection scheme is proposed to achieve the required result. It is worth mentioning that the CDN representing real-time systems is generally affected by external noise factors or stochastic disturbances [8]. Thus, the consideration of external noise factors in the study of synchronization of CDNs is of great importance in the viewpoints of both theoretical and practical. By taking this fact into account, in recent years, research communities have eagerly investigated the problems of synchronization of CDNs with external stochastic disturbances, see for example [9–12]. Sakthivel et al. [10] presented some sufficient conditions that ensure the synchronization and solve the state estimation problem of discrete-time stochastic complex networks in the presence of uncertain inner couplings, where the interval matrix approach is employed to characterize the uncertainties encountered in the inner coupling terms. Li et al. [11] developed a new synchronization criterion for a class of discrete-time stochastic complex networks subject to partial mixed impulsive effects, where by using the Lyapunov stability theory and the variation of parameters formula, the required criterion is obtained.It should be pointed out that the interconnection topology among the nodes of CDNs plays a significant role in the study of synchronization problem. In the existing literature, there have been two reported kinds of interconnection topologies, which are constant/fixed topology [13] and time-varying topology [14]. In [14], it has been illustrated that the time-varying interconnection topology is more general than the fixed one. Nevertheless, in most of the real networks, the connectivity of the network topology might be unfixed or randomly changing due to new creation or link failures. In order to tackle these types of issues, it is more appropriate to model the CDNs with randomly switching topologies that are governed by a Markov process. Based on this scenario and following the seminal works reported in [15, 16], some interesting and significant results about synchronization of CDNs with Markov jump topologies have been discussed; see [17–19]. Specifically, in [18], the problem of nonfragile synchronization for a class of discrete-time complex networks subject to Markov jumping switching topology has been investigated in a unified framework that includes the nonfragile H∞ synchronization control and nonfragilel2–l∞ synchronization control problem as its special cases. However, it is worth pointing out that the aforementioned papers have considered constant transition rates in the Markov process. It should be mentioned that the Markov process might consist of time-varying transition rates when modeling practical systems. Such kind of process is known as semi-Markov process and few interesting results regarding semi-Markov jump systems have been addressed in recent literature [20–23]. Interestingly, in [21], by employing the supplementary variable technique and plant transformation, the state estimation and sliding mode control problems have been investigated for semi-Markovian jump systems in the presence of mismatched uncertainties. Apparently, semi-Markov jump systems are comparatively more general than the traditional Markov jump systems [24]. Following the aforementioned seminal works, the concept of semi-Markov process has further been employed in the network topology to obtain the synchronization criteria for CDNs (see [25, 26] and the references therein). However, only very few results about the synchronization of CDNs with semi-Markov jump topology have been reported in the literature, which stimulates us to do this present work.It is worth mentioning that most of the available controller design approaches have a predominant assumption that the designed controller can be implemented accurately. But in some real situations, such an assumption is not always true as the controllers are often very sensitive or fragile to their parameters’ variations. Furthermore, it should be noted that a small perturbation in controllers may lead to undesirable oscillatory behavior or even instability [27]. Hence, it is desirable as well as necessary to ensure the insensitivity of the controller to certain parameter perturbations without loss of the robust stability and thus, the investigation of nonfragile or resilient controller design that has been capable of tolerating some level of controller parameter gain variations has been enormously increased in recent years [28–31]. To mention a few, in [28], a robust resilient control problem of discrete-time Markov jump nonlinear systems has been solved by employing the linear matrix inequality and stochastic analysis techniques; in [29], based on the dissipative theory and the event-triggered sampling scheme, the nonfragile control design problem for a class of network-based singular systems with input time-varying delay and external disturbances has been addressed. Therefore, it is reasonable to consider the nonfragile control design in the study of synchronization of CDNs. It is noteworthy that only few research papers regarding the nonfragile control design for achieving the synchronization of CDNs have been published; see [32, 33]. On the other hand, it is worth mentioning that most of the existing results based on the classical control theory dealt with the asymptotic property of control system trajectories over an infinite-time interval and did not possess any restriction to the system states. But in many practical problems, it is required that the described system state does not exceed a certain bound during a fixed finite-time interval [34]. According to this fact, a great number of interesting results on finite-time control design have been proposed for the synchronization of various CDNs; for instance, see [35–38]. To the best of our knowledge, however, the problem of robust nonfragile synchronization has not been fully investigated for a class of CDNs over a prescribed finite-time interval.Motivated by the above analysis, in this paper, we focus on the finite-time nonfragile synchronization problem is investigated for a class of CDNs subject to semi-Markov jump topology and stochastic noises. More precisely, a new delay-dependent sufficient condition under which the considered CDNs are synchronized to the target network within a given finite-time interval is developed in terms of linear matrix inequalities by utilizing the Lyapunov stability theory and the stochastic analysis techniques. Subsequently, based on the developed condition, a design algorithm of the proposed nonfragile state feedback controller that can ensure the finite-time stochastic synchronization of the addressed network is presented. Eventually, a numerical example is shown to illustrate the effectiveness of the proposed theoretical results.The rest of this paper is organized as follows: in Section2, the problem formulation of the network model under study and the preliminaries required to obtain the main results are given. The finite-time stochastic synchronization criterion for the considered network model is presented in Section 3. A numerical example and its simulations are provided in Section 4. Conclusion of this paper is given in Section 5.


## 2. Problem Formulation and Preliminaries

In this paper, we consider a class of complex dynamical networks (CDNs) with semi-Markov jump outer coupling and stochastic noise, which consists ofN identical nodes and is defined over the Wiener process probability space (Ω,F,P), where Ω is the sample space, F is the algebra of events and P is the probability measure defined on F. Such a network model can be described in the following form:(1)dxit=Axit+ft,xit+λ∑j=1NTijσtΛxjt-τt+uitdt+ρt,xit,xit-τtdwt,xit=ϕit,∀t∈-τ2,0,i=1,2,…,N,where xit∈Rn denotes the state vector of the ith node; A is a known real constant matrix with suitable dimension; f(·,·)∈Rn represents a nonlinear vector-valued function; the constant λ>0 is the coupling strength of network; Tijσt are the elements of the outer coupling matrix Tσt which describes the network topological structure and is assumed to follow a semi-Markov process σt which to be defined later. In particular, Tijσt is defined as follows: if there exists a connection between node i and node j, then Tijσt>0; otherwise, Tijσt=0. Further, the diagonal elements of the outer coupling matrix are given as follows: Tiiσt=-∑j=1,j≠iN‍Tijσt; Λ represents the inner coupling matrix and is a positive diagonal matrix with appropriate dimension; uit∈Rn is the control input of the ith node which to be defined later; the function ρ(·,·,·):R×Rn×Rn→Rn is the noise intensity vector-valued function; wt is a 1-dimensional Brownian motion defined on the probability space Ω,F,P with Ewt=0, Ew2t=1 and Ewswt=0 for s≠t, where E is the mathematical expectation; τ(t) is the time-varying delay function satisfying 0≤τ1≤τ(t)≤τ2<∞ and τ˙t≤μ<1, where τ1, τ2, and μ are known scalars; and ϕit denotes the initial value of the ith node’s state and is assumed to be a continuous vector-valued function.Now, let us define the semi-Markov jump process of the outer coupling matrix. The processσt,t≥0 is a continuous-time homogeneous semi-Markov process with right continuous trajectories and takes values in a finite set S=1,2,…,N. More precisely, σt is associated with the transition probability matrix Π=πpqhN×N which is given by the following transition rates:(2)Probσt+h=q∣σt=p=πpqhh+oh,ifp≠q,1+πpphh+oh,ifp=q,whereh>0 is the sojourn time, limh→0⁡oh/h⁡=0 and πpqh≥0 for p≠q is the transition rate from mode p at time t to mode q at time t+h and πpph=-∑q=1,q≠pN‍πpqh. For notational simplicity, we hereafter denote the semi-Markov process parameter σt by p. For example, Tσt is denoted by Tp.To synchronize all theN identical nodes in the network (1) to a common value, let us define the synchronization error vector as eit=xit-st, where st∈Rn is the state vector of the unforced isolated node that can be expressed as dst=Ast+ft,stdt and is assumed to be noise-free, that is, ρt,st,st-τt=0. Based on this error vector, we now choose a robust state feedback controller to achieve the synchronization of network (1), which is insensitive to the uncertain perturbations or gain fluctuations and of the form:(3)uit=Kp+αtΔKpteit,i=1,2,…,N,where Kp is the feedback controller gain matrix that is to be determined in the forthcoming section, ΔKpt is a time-varying matrix representing the controller gain fluctuations, and αt is a stochastic variable describing the randomly occurring controller gain fluctuations. It is here assumed that ΔKpt takes the form ΔKpt=MpΓptNp, where Mp and Np are known real constant matrices and Γpt is an unknown time-varying matrix satisfying ΓpTtΓpt≤I. Further, it is assumed that the stochastic variable αt obeys the Bernoulli distribution with the following probability rules: (i) Probαt=1=Eαt=α¯ and (ii) Probαt=0=1-Eαt=1-α¯, where α¯∈0,1.Then, by using (1) and (3), the closed-loop form of the error system can be obtained as follows:(4)deit=A+Kp+αtΔKpteit+gt,eit+λ∑j=1NTijpΛejt-τtdt+ρ^t,eit,eit-τtdwt,p∈S,i=1,2,…,N,where gt,eit=ft,xit-ft,st and ρ^t,eit,eit-τt=ρt,xit,xit-τt-ρt,st,st-τt. By using the Kronecker product properties and mathematical manipulations, the error system (4) can be written in the following compact form: (5)det=IN⊗A+IN⊗Kp+α¯IN⊗MpΓptNp+αt-α¯IN⊗MpΓptNpet+Gt,et+λTp⊗Λet-τtdt+ρ^t,et,et-τtdwt,where et=e1Tt,e2Tt,…,eNTtT, Gt,et=gTt,e1t,gTt,e2t,…,gTt,eNtT, and ρ^t,et,et-τt=ρ^t,e1t,e1t-τt,ρ^t,e2t,e2t-τt,…,ρ^t,eNt,eNt-τtT.In order to develop the main results, the following assumptions and definition are required.Assumption 1.
For the nonlinear functionf(·,·), there exists a known real constant matrix G such that ft,xit≤Gxi(t) for any xit∈Rn.Assumption 2.
The noise intensity functionρ(·,·,·):R+×Rn×Rn→Rn is uniformly Lipschitz continuous in terms of the following inequality of trace inner product: traceρTt,xit,xit-τtρt,xit,xit-τt≤ηixiTtxit+ζixiTt-τtxit-τt, where ηi and ζi i=1,2,…,N are known nonnegative constants.Assumption 3.
For eachp∈S, all the real parts of eigenvalues of Tσt are negative except an eigenvalue 0 with multiplicity 1, which means that the reverse of the graph generated by the matrix Tσt contains a rooted spanning directed tree for every p∈S.Definition 4 (see [34]).
The considered network (1) is said to be stochastically synchronized in finite-time with respect to c1,c2,τ2,T∗,Zp if there exist positive definite matrix Zpp∈S and positive constants T∗, c1, c2 with c2>c1 such that the following condition holds: (6)ExTt0IN⊗Zpxt0≤c1⟹ExTtIN⊗Zpxt<c2,t0∈-τ2,0,t∈0,T∗.


## 3. Main Results

Based on the Lyapunov-Krasovskii stability theory, this section aims to develop a new set of delay-dependent sufficient conditions that can guarantee the stochastic synchronization of the considered network model (1) over a finite-time interval. Moreover, based on these conditions, a design of the robust nonfragile state feedback control (3) for the network model under consideration is provided in terms of linear matrix inequalities (LMIs).Theorem 5.
Consider the network model (1) with Assumptions 1–3. For given positive scalars c1, c2, T∗, δ, μ, λ, α¯∈0,1, τ1, τ2, symmetric matrix Zpp∈S, and diagonal matrices R3, R4, the considered network (1) is stochastically synchronized in finite-time under the nonfragile controller (3), if there exist symmetric matrices Pp>0(p∈S), Ql>0l=1,2,3, Rv>0v=1,2 and positive scalars κpp∈S, ϵ1 such that the following matrix inequalities hold:(7)Pp-κpIN<0,p∈S,(8)Ω¯ph=Ωpijh7×7ϵ1ϑpυpT∗-ϵ1IN0∗∗-ϵ1IN<0,(9)c1eδT∗λ1+τ1λ2+τ2λ3+τ2λ4+τ13λ5+τ123λ6-λ7c2<0,where(10)Ωp11h=IN⊗PpIN⊗A+IN⊗ATIN⊗PpT+IN⊗PpIN⊗Kp+IN⊗KpTIN⊗PpT+IN⊗Q1+IN⊗Q2+IN⊗Q3+τ12IN⊗R1+κpIN⊗R3+IN⊗GIN⊗GT+∑q=1NπpqhIN⊗Pq,Ωp13=IN⊗PpλTp⊗Λ,Ωp15=IN⊗Pp,Ωp22=-IN⊗Q1-τ122IN⊗R2,Ωp33=-1-μIN⊗Q2+κpIN⊗R4,Ωp44=-IN⊗Q3,Ωp55=-IN⊗IN,Ωp66=-IN⊗R1,Ωp77=-IN⊗R2,ϑp=α¯IN⊗PpIN⊗Mp00⋯0︸6T,υp=IN⊗Np00⋯0︸6,λ1=max⁡λmaxIN⊗P~p,p∈S,λ2=max⁡λmaxIN⊗Q~1p,p∈S,λ3=max⁡λmaxIN⊗Q~2p,p∈S,λ4=max⁡λmaxIN⊗Q~3p,p∈S,λ5=max⁡λmaxIN⊗R~1p,p∈S,λ6=max⁡λmaxIN⊗R~2p,p∈S,λ7=min⁡λminIN⊗P~p,p∈S,τ12=τ2-τ1,and the rest of elements of Ωpijh are zero.Proof.
To develop the finite-time stochastic synchronization criterion for the network model (1), it is enough to establish the finite-time stochastic stability criterion for the closed-loop error system (5). For this purpose, we select the Lyapunov-Krasovskii functional as follows:(11)Vet,t,p=∑l=13Vlet,t,p,where(12)V1et,t,p=eTtIN⊗Ppet,V2et,t,p=∫t-τ1teTsIN⊗Q1esds+∫t-τttesIN⊗Q2esds+∫t-τ2teTsIN⊗Q3esds,V3et,t,p=τ1∫-τ10∫t+θteTsIN⊗R1esdsdθ+τ12∫-τ2-τ1∫t+θt-τ1eTsIN⊗R2esdsdθ,with Pp>0, Qll=1,2,3, Rvv=1,2 and τ12=τ2-τ1.
Based on Ito’s differential formula [9], the stochastic derivative of Vet,t,p can be obtained as(13)dVet,t,p=£Vet,t,p+Veet,t,pρt,et,et-τtdwt,where £Vet,t,p=£V1et,t,p+£V2et,t,p+£V3et,t,p and Veet,t,p=∂V(et,t,p)/∂e.
Now, by calculating the time derivative ofV(et,t,p) along the solution trajectories of the error system (5), we can get(14)£V1et,t,p=2eTtIN⊗PpIN⊗A+IN⊗Kp+αtIN⊗MpΓptNp+αt-α¯×IN⊗MpΓptNpet+λTp⊗Λet-τt+Gt,et+∑q=1NπpqheTtIN⊗Pqet+trace⁡ρTt,et,et-τtIN⊗Ppρt,et,et-τt,(15)£V2et,t,p≤eTtIN⊗Q1+IN⊗Q2+IN⊗Q3et-eTt-τ1IN⊗Q1et-τ1-1-μeTt-τtIN⊗Q2et-τt-eTt-τ2IN⊗Q3et-τ2,(16)£V3et,t,p=τ12eTtIN⊗R1et+τ122eTt-τ1IN⊗R2et-τ1-τ1∫t-τ1teTsIN⊗R1esds-τ12∫t-τ2t-τ1eTsIN⊗R2esds.Further, by applying Jensen’s single integral inequality [6] to the integral terms in (16), we can get the following inequalities:(17)-τ1∫t-τ1teTsIN⊗R1esds≤-∫t-τ1teTsdsIN⊗R1∫t-τ1tesds,-τ12∫t-τ2t-τ1eTsIN⊗R2esds≤-∫t-τ2t-τ1eTsdsIN⊗R2∫t-τ2t-τ1esds.On the other hand, it follows from Assumption 2 and condition (7) that(18)trace⁡ρTt,et,et-τtIN⊗Ppρt,et,et-τt≤κptrace⁡ρTt,et,et-τtρt,et,et-τt≤κpeTtIN⊗R3et+eTt-τtIN⊗R4et-τt,where κp(p∈S) are positive scalars and R3, R4 are known constant matrices.
Moreover, according to Assumption1, we can obtain the following inequality:(19)eTtIN⊗GIN⊗GTet-FTt,etFt,et>0.
Then, by combining (13)–(19) and taking mathematical expectation, it can be obtained that(20)EdVet,t,pdt≤E£Vet,p,t≤EξTtΩpijh7×7+ϑpΓptυp+ϑpΓptυpTξt,where (21)ξt=eTteTt-τ1eTt-τteTt-τ2FTt,et∫t-τ1teTsds∫t-τ2t-τ1eTsdsTand the elements of Ωpijh, ϑp, and υp are defined in the theorem statement. Moreover, based on Lemma 2 in [6], for any positive scalar ϵ1, the right-hand side of (20) can equivalently be written as(22)Ωpijh7×7+ϑpΓptυp+ϑpΓptυpT≤Ωpijh7×7+ϵ1ϑpϑpT+ϵ1-1υpTυp.Based on the Schur complement, it is noted that (22) is equivalent to the left-hand side of (8). Thus, it can be observed that E£Vet,t,p<0 if the LMIs (7) and (8) hold. Furthermore, if there exists a constant δ>0, it yields that E£Vet,t,p<δEVet,t,p. From which, it can be obtained that Ee-δtVet,t,p<EVe0,0,p0, where p0=σ0. Next, define the following new parameters: P~p=Zp-1/2PpZp-1/2, Q~1p=Zp-1/2Q1Zp-1/2, Q~2p=Zp-1/2Q2Zp-1/2, Q~3p=Zp-1/2Q3Zp-1/2, R~1p=Zp-1/2R1Zp-1/2, R~2p=Zp-1/2R2Zp-1/2. Then, it follows from condition (11) and 0≤t≤T∗ that(23)EVet,t,p<eδtVe0,0,p0≤eδteT0IN⊗Ppe0+∫-τ10eTsIN⊗Q1esds+∫-τ00eTsIN⊗Q2esds+∫-τ20eTsIN⊗Q3esds+τ12∫-τ10eTsIN⊗R1esds+τ122∫-τ20eTsIN⊗R2esds=eδT∗max⁡λmaxIN⊗P~p+τ1max⁡λmaxIN⊗Q~1p+τ2max⁡λmaxIN⊗Q~2p+τ2max⁡λmaxIN⊗Q~3p+τ13max⁡λmaxIN⊗R~1p+τ123max⁡λmaxIN⊗R~2p×EeTsIN⊗Zpes≤c1eδT∗λ1+τ1λ2+τ2λ3+τ2λ4+τ13λ5+τ123λ6,where λaa=1,2,…,6 are defined in the theorem statement and c1 is given in Definition 4.
On the other hand, from (11), we can have(24)EVet,t,p≥EeTtIN⊗Ppet=min⁡λminIN⊗P~pEeTtIN⊗Zpet≥λ7EeTtIN⊗Zpet.
Now, by combining the inequalities (23) and (24), we can get(25)EeTtIN⊗Zpet≤c1λ1+τ1λ2+τ2λ3+τ2λ4+τ13λ5+τ123λ6e-δT∗λ7<c2.It is clear to see that the inequality (25) is the same as that in (9) which is the desired condition. Hence, it can be concluded that the closed-loop error system (5) is finite-time stochastically stable which means that the considered network model (1) is stochastically synchronized within a prescribed finite-time interval. Thus, the proof of this theorem is completed.Theorem 6.
Consider the network model (1) with Assumptions 1–3. For given positive scalars c1, c2, T∗, δ, μ, λ, α¯∈0,1, τ1, τ2, symmetric matrix Zpp∈S and diagonal matrices R3, R4, the considered network (1) is stochastically synchronized in finite-time under the non-fragile state feedback controller (3), if there exist symmetric matrices Xp>0p∈S, Q^lp>0l=1,2,3, R^vp>0v=1,2, any matrices Ypp∈S with appropriate dimensions and positive scalars κ^p, p∈S, r, all=1,2,3, bvv=1,2, ϵ1 such that the following matrix inequalities hold:(26)Xp-κ^pIN>0,p∈S,(27)Ω¯^ph=Ω^pijh12×12<0,(28)rZp-1<Xp<Zp-1,0<Q^lp<alZp-1,0<R^vp<bvZp-1,(29)c1τ1a1+τ2a2+τ2a3+τ13b1+τ123b2-eδT∗c2c1∗-r<0,where (30)Ω^p11h=IN⊗AIN⊗Xp+IN⊗XpIN⊗AT+IN⊗Yp+IN⊗YpT+IN⊗Q^1p+IN⊗Q^2p+IN⊗Q^3p+τ12IN⊗R^1p+πpphIN⊗Xp,Ω^p13=λTp⊗ΛIN⊗Xp,Ω^p15=IN⊗IN,Ω^p18=α¯IN⊗Mp,Ω^p19=IN⊗XpIN⊗NpT,Ω^p110=IN⊗XpIN⊗R3,Ω^p111=IN⊗XpIN⊗G,Ω^p112=πp1hXpT,…,πpp-1hXpT,πpp+1hXpT,…,πpNhXpT,Ω^p22=-IN⊗Q^1p-τ122IN⊗R^2p,Ω^p33=-1-μIN⊗Q^2p,Ω^p310=IN⊗XpIN⊗R4,Ω^p44=-IN⊗Q^3p,Ω^p55=-IN⊗IN,Ω^p66=-IN⊗R^1p,Ω^p77=-IN⊗R^2p,Ω^p88=-ϵ1IN⊗IN,Ω^p99=-ϵ1IN⊗IN,Ω^p1010=-κ^pIN⊗IN,Ω^p1111=-IN⊗IN,Ω^p1212=-diag⁡X1,…,Xp-1,Xp+1,…,XN,κ^p=-1κp,and the remaining parameters of Ω^pijh are zero. Moreover, if the obtained LMIs are feasible, then the desired state feedback controller gain matrices in (3) are computed by Kp=YpXp-1.Proof.
LetXp=Pp-1 and pre- and postmultiply the matrix Ω¯ph by diagIN⊗Xp,…,IN⊗Xp︸4,IN⊗IN,IN⊗Xp,IN⊗Xp,IN⊗IN,IN⊗IN. Now, we introduce the following new variables: XpQlXp=Q^lpl=1,2,3, XpRvXp=R^vpv=1,2, and Yp=KpXp.
Moreover, from Theorem5, it is noticed that λ7<IN⊗Zp-1/2PpZp-1/2<λ1, 0<IN⊗Zp-1/2Q1Zp-1/2<λ2, 0<IN⊗Zp-1/2Q2Zp-1/2<λ3, 0<IN⊗Zp-1/2Q3Zp-1/2<λ4, 0<IN⊗Zp-1/2R1Zp-1/2<λ5 and 0<IN⊗Zp-1/2R2Zp-1/2<λ6. According to the congruence transformation, these relations can be changed into λ1-1IN⊗Zp-1<IN⊗Xp<λ7-1IN⊗Zp-1, 0<IN⊗Q^lp<λ7-2λl+1IN⊗Zp-1(l=1,2,3) and 0<IN⊗R^vp<λ7-2λv+4IN⊗Zp-1(v=1,2). Now, if we set λ7=1, λ1-1=r, λl+1≤all=1,2,3 and λv+4≤bvv=1,2, then the constraints in (28) can easily be deduced. Moreover, the conditions in (26), (27), and (29) can be obtained from (7), (8), and (9), respectively, which are the desired conditions. Hence, the proof is completed.Remark 7.
It should be mentioned that the constraints in (27) are cannot be solved directly via MATLAB LMI control toolbox due to the existence of the time-varying terms ∑q=1N‍πpqh. To overcome this difficulty, the transition rates πpqh are assumed to be bounded and satisfy πpq-≤πpqh≤πpq+, since they are partially measurable in practice which is mentioned in [26]. Moreover, in this case, the following assumptions are made as in [26]:(31)πpqh=∑k=1Nγkπpq,k,∑k=1Nγk=1,γk≥0,πpq,k=πpq-+k-1πpq+-πpq-N-1,p≠q,q∈S,πpq+-k-1πpq+-πpq-N-1,p=q,q∈S.Now, we are able to present the sufficient conditions guaranteeing the stochastic synchronization of the considered network (1) over a finite-time interval in terms of LMIs in which all the elements are either constants or constant matrices according to Remark 7. Thus, we have the following theorem.Theorem 8.
For given positive scalarsc1, c2, T∗, δ, λ, μ, α¯∈0,1, τ1, τ2, symmetric matrix Zpp∈S and diagonal matrices R3, R4, the considered network (1) with Assumptions 1–3 is stochastically synchronized in finite-time under the non-fragile controller (3), if there exist symmetric matrices Xp>0p∈S, Q^lp>0l=1,2,3, R^vp>0v=1,2, any matrices Ypp∈S with appropriate dimensions and positive scalars κ^pp∈S, r, all=1,2,3, bvv=1,2, ϵ1 such that the following matrix inequality, (26), (28), and (29) hold:(32)Ω^pij,k12×12<0,k=1,2,…,N,where (33)Ω^p11,k=IN⊗AIN⊗Xp+IN⊗XpIN⊗AT+IN⊗Yp+IN⊗YpT+IN⊗Q^1p+IN⊗Q^2p+IN⊗Q^3p+τ12IN⊗R^1p+πpp,kIN⊗Xp,Ω^p112,k=πp1,kXpT,…,πpp-1,kXpT,πpp+1,kXpT,…,πpN,kXpT,and the remaining elements of Ω^pij,k are the same as those defined in Theorem 6. Further, the nonfragile state feedback controller gain matrices in (3) are calculated by Kp=YpXp-1.Proof.
Based on Remark7, the time-varying element πpqh may take values in the interval πpq-,πpq+. Then, by using (31) and following the similar lines in the proof of Theorem 6, it is easy to obtain the inequality (32) which completes the proof.Remark 9.
It should be noted that, so far in the literature, several control approaches have been proposed for the synchronization problem of several CDNs [3–6], wherein the interconnection topology among the nodes are assumed to be fixed. However, in practice, this assumption is practically difficult or even impossible. However, yet now, there were no results reported in the existing literature for the synchronization analysis of stochastic CDNs with switching topology. According to this fact, in this paper, finite-time synchronization problem of stochastic CDNs with switching topology is investigated. Furthermore, due to random behavior in the dynamics of stochastic CDNs, it is very difficult to determine the exact fixed control value. Therefore, in this paper, the feedback control gain is considered with uncertain terms, which is more significant to reflect the realistic scenarios.


## 4. An Illustrative Example

This section provides an illustrative example to verify the developed theoretical results in the previous section. For the sake of simplicity, consider a class of CDNs in the form of (1) with five identical nodes and the state vector of each node being three-dimensional, that is, N=5 and n=3.Let us select the network matrix and the nonlinear function as(34)A=1-1.40-0.921.2-0.502-1.2,ft,xit=0.5exp⁡-0.01ttanh⁡xi1t0.5exp⁡-0.01ttanh⁡xi2t0.5exp⁡-0.01ttanh⁡xi3t.It is clear to see that ft,xit satisfies Assumption 1 with G=diag{0.5,0.5,0.5}. In this example, we consider the semi-Markov jump topology with two modes, whose connectivity graph is shown in Figure 1. The inner coupling matrix is assumed to be Λ=diag{0.2,0.5,0.7} and the coupling strength is chosen as λ=0.5. The time-varying delay is taken as τ(t)=0.5+0.5sin(t) from which it can be obtained that τ1=0, τ2=1 and μ=0.5.Figure 1
Connected topology of the considered CDNs.
(a)
(b)Based on Figure1, the jumping coupling configuration matrices T(σ(t)) for σ(t)=1,2 can be expressed as (35)T1=000001-100001-100100-100001-1,T2=-110000000010-100001-1000101.Moreover, the elements π12h and π21h of transition rate matrix are assumed to lie in the intervals 0.1,2 and 0.8,1.7, respectively. So, in light of (31), the transition rates π12h and π21h can be represented as π12h=∑n=12‍γnπ12,k and π21h=∑n=12‍γnπ21,k, respectively, with π12,1=0.1, π12,2=2, π21,1=0.8, and π21,2=1.7. The stochastic variable representing the controller gain fluctuations is chosen as α(t)=0.25+0.25sin(t). Furthermore, the uncertain parameter matrices in the control gain are taken as (36)M1=0.010.010.020.020.020.010.020.010.10,M2=0.010.020.020.020.030.010.020.010.01,N1=0.010.020.030.010.030.010.020.020.01,N2=0.020.010.020.010.010.010.020.010.02and Γpt=sint. The rest of parameters involved in the simulation are set to be c1=1, c2=2, δ=0.01, and T∗=1. Then, by solving the LMIs (26), (28), and (29) in Theorem 6 along with (32) in Theorem 8 with the aid of MATLAB LMI control toolbox, we can get a set of feasible solutions from which the nonfragile state feedback control gain matrices can be obtained as follows:(37)K1=-17.7856172.5392-19.7897-155.4332-30.329066.867117.6228-66.6954-40.4541,K2=-19.2269-2.5815-51.26294.3104-29.1095-64.493239.984957.5098-35.7536.Here, our aim is to design the nonfragile state feedback controller such that the considered network (1) is robustly synchronized with the target network within a desired finite-time interval. For the simulation purposes, we set the initial conditions for the states of the nodes and the isolated node as follows: x10=101110T, x20=91110T, x30=11109T, x4(0)=10910T, x5(0)=101110T, and s(0)=989T. The noise intensity function is taken as ρt,xit,xit-τt=0.1sin⁡xi1t0.1sin⁡xi2t-τt0.1sinxi3t-τtT.Based on these values, simulations are drawn in Figures2–12. Specifically, the state responses of the first, second, and third nodes together with the isolated node are plotted in Figures 2, 3, and 4, respectively, wherein the dotted line represents the isolated node and the dashed lines denote the five identical nodes. It can easily be observed from these figures that the states of the nodes are exactly synchronized with the states of the isolated node within short period which shows the efficiency of the proposed nonfragile control strategy. Moreover, the corresponding error state responses and the control response curves are given in Figures 5–7 and Figures 8–10, respectively. Further, Figure 11 shows the jumping mode of the semi-Markov switching topology. In addition, to realize the finite-time synchronization, the time evolution of eiTtZpeiti=1,2,3,4,5,p∈S is depicted in Figure 12. It can be seen from Figure 12 that the states of the error system do not exceed the prescribed threshold c2=2, which means that the synchronization of considered network (1) is achieved within a given finite-time interval. Thus, it can be concluded from the simulations that the designed nonfragile control algorithm effectively works even in the presence of stochastic noise and time-varying coupling delay.Figure 2
State responses of first node.Figure 3
State responses of second node.Figure 4
State responses of third node.Figure 5
Error state responses of first node.Figure 6
Error state responses of second node.Figure 7
Error state responses of third node.Figure 8
Control input of first node.Figure 9
Control input of second node.Figure 10
Control input of third node.Figure 11
Jumping mode.Figure 12
Evolution ofeiTtZpeit.


## 5. Conclusion

In this paper, we have studied the robust finite-time nonfragile synchronization problem for a class of CDNs with semi-Markov jump outer coupling, time-varying coupling delay, randomly occurring gain variation and stochastic noise. In particular, we have considered the semi-Markov switching topology to obtain the synchronization criterion. Moreover, we have introduced a stochastic variable satisfying the Bernoulli distribution to represent the random gain variations in the controller design. By employing the Lyapunov-Krasovskii stability theory and some stochastic analysis techniques, we then have developed a new finite-time stochastic synchronization criterion for the considered network in terms of linear matrix inequalities and have presented a design algorithm for the proposed nonfragile state feedback controller to a solution of the obtained set of linear matrix inequalities. At last, we have provided a numerical example to verify the obtained theoretical results. In addition, it should be pointed out that one of the future research topics would be to investigate the problem of finite-time mixedH∞ and passivity synchronization of stochastic singular CDNs with semi-Markov switching outer coupling delay and actuator saturation.



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