# Global Synchronization of Delayed Complex Networks with Hybrid Coupling, Control Design of Actuator Saturation, and Stochastic Disturbances with Randomly Occurring Nonlinearities

Authors: M. Syed Ali; M. Usha; S. Saravanan; Quanxin Zhu
Journal: Mathematical Problems in Engineering
 (2019)

---

## Abstract

In this paper, we propose and study a global synchronization of delayed complex networks with hybrid coupling, which is composed of constant coupling, discrete-delay coupling, and unbounded distributed-delay coupling, and its actuator saturation control design is then further investigated. Several effective sufficient conditions of global synchronization are attained based on the Lyapunov function and a linear matrix inequality (LMI), which can be easily computed by the interior-point method. In addition, we established the control design of actuator saturation of the addressed stochastic delayed complex networks. More relaxed conditions by employing the new type of augmented matrices by using multitude Kronecker product terms can be handled, which can be introduced. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed synchronization scheme.


---

## Body

## 1. Introduction

In the past decades, complex networks have been gaining increasing research attention due to their potential applications to many real-world systems in various fields of science and engineering. The studies of complex networks have been an active field of research in many scientific and technical disciplines [1–3]. One of the most significant and interesting phenomena in complex networks is the synchronization of all dynamical nodes or vertices connected by links or edges. Examples of such complex networks can be found everywhere in our daily life, from physical objects such as the Internet, multiagent systems, social networks the world wide web (WWW), and electricity distribution network, telephone cell graphs, food webs, neural networks, electrical power grids, cellular and metabolic networks, scientific citation web, living organisms, etc. Complex network is a large number of highly interconnected fundamental units and therefore exhibit very complicated dynamics [4–12].In real-world situation, time delay is ubiquitous in many physical systems due to the finite switching speed of amplifiers, finite signal propagation time in networks, finite reaction times, memory effects, and so on. Furthermore, the time delay may cause undesirable dynamic behaviors such as oscillation, instability, and poor performance. Therefore, the synchronization problem of complex dynamical networks with time delays has become a topic of both theoretical and practical importance. In this regard, some synchronization criteria for the general complex networks with coupling delays which are classified into two types: delay-independent and delay-dependent conditions are derived (see [10, 12–15]).In real-time systems, the complex networks are often largely affected by the noisy environment. During the communications among the nodes of complex networks, the network-induced delays and external disturbances are simply unavoidable. In particular, the ’random’ phenomena often appear due to connections over communication channels, which includes random communication delay, random measurements, and random packet losses, which have lately attracted a great deal attention in the networked control society. As is widely recognized an extensive elegance of practical systems is influenced by additive nonlinear disturbances which might be caused by environmental circumstances. The randomly occurring nonlinearities, also called stochastic nonlinearities, have recently received some interest in the literature; see [16–18]. For example, in [19], the filtering and control problems for continuous-time systems with stochastic nonlinearities have been investigated. Unfortunately, to the best of the authors’ knowledge, the global synchronization problem for complex networks experiencing randomly occurring nonlinearities with actuator saturation and stochastic disturbances has acquired very little interest for time delays, and the purpose of this paper is therefore to fill in such a gap.Nowadays, in digitalized world, control systems are often controlled by digital controllers that make a limitation on the power of the actuator in practical systems. Thus, the analysis and synthesis of controllers of dynamic systems subject to actuator saturation have attracted a considerable attention among control community [20–22]. As for the stabilization of stochastic systems with actuator saturation, only a few works have been done on this subject. To the best of the author’s knowledge, however, the problems of analysis and design for stochastic systems with both state delay and actuator saturation remain open.Motivated by the above observations, in this paper, we investigate the synchronization problem of delayed complex networks (SDCN) with hybrid coupling, actuator saturation and stochastic nonlinearities, multiple stochastic disturbances, time-varying delays, and continuously distributed delays. By employing the properties of Kronecker product [23] and stochastic analysis techniques [24, 25]. We construct a novel Lyapunov functional and introducing several new lemmas; some sufficient conditions are obtained to analyze the synchronization criteria, which are formulated in the form of linear matrix inequalities. A numerical example is presented to illustrate the effectiveness of the proposed control design method.The novelties of this paper are as follows:(1) The global synchronization of the complex networks with hybrid coupling, actuator saturation, randomly occurring nonlinearities, and continuously distributed delays is considered.(2) Novel Lyapunov functional consisting of Kronecker product is constructed to derive the main results.(3) Various types of inequality lemmas are applied in the derivation process which leads to the feasible criteria.(4) Simulation results are provided to show the effectiveness of the proposed results.The rest of this paper is organized as follows. In Section2, a stochastic complex network model with time delays, stochastic nonlinearities, and multiple stochastic disturbances with hybrid couplings and time delays is proposed, and some preliminaries are briefly outlined. In Section 3, the Lyapunov functional method combined with the LMI technique, so as to ensure that the considered complex networks with multiple stochastic disturbances and stochastic nonlinearities are globally synchronized in the mean square, is proposed.Section4 is devoted to deriving the results on controller design of actuator saturation. In Section 5, a numerical example is provided to show the applicability of the obtained results. The conclusions are finally drawn in Section 6.Notations. Throughout this paper, Rn and Rn×m denote, respectively, the n-dimensional Euclidean space and the set of all n,m real matrices. P>0 means that matrix P is real, symmetric, and positive definite. IN represents the n-dimensional identity matrix, diag⋯ stands for a block-diagonal matrix, and col⋯ denotes a matrix column with blocks given by the matrices in{⋯}. If A is a matrix, the notation λmax(A) means the largest eigenvalue of A. The Kronecker product of matrices Q∈Rm×n and Rmp×nq is denoted as Q⊗R; matrix dimensions, if not explicitly stated, are assumed to be compatible for algebraic operations.


## 2. Problem Formulation and Preliminaries

Consider the following array of delayed complex networks with multiple stochastic disturbances and stochastic nonlinearities consisting ofN coupled nodes of the form(1)dxit=Axit+Dxit-τt+δtfB1xit+1-δtgB2xit+C∫-∞tkt-shxisds+Bφutdt+∑j=1Nwij1Γ1xjtdt+dw1t+∑j=1Nwij2Γ2xjt-τtdt+dw2t+σit,xit,xit-τtdw3t,i=1,2,3…,Nwhere xi(t)=colxi1t,…xint∈Rn is the state vector of the ith node at time t; A,D,B1, B2, B, and C are the known real constant matrices; Γ1,Γ2∈Rn×n represents the inner coupling between the subsystems at time t and t-τ(t), respectively, where τ(t) is time-varying delay and satisfy 0≤τ(t)<τ; W(1)=(wij(1))N×N and W(2)=(wij(2))N×N are the outer-coupling configuration matrices representing the coupling strength and the topological structure of the complex networks; φ(·):Rn→Rm is the vector-valued standard saturation function defined as φ(u)=[φ(u1),φ(u2),…,φ(un)]T, where φuq=Signuqmin⁡u¯q,uq, q=1,2,…n furthermore, σi(·,·,·):R×Rn×Rn→Rn is the noise intensity function vector, and wk(t)(k=1,2,3) are mutually independent scalar Brownian motions defined on (Ω,F,P) with(2)Ewkt=0andEwkt2=dt.Moreover, k(·):[0,+∞)→[0,+∞) is the distributed time-delay kernel; and the discrete time delay τ(t) is a time-varying differentiable function satisfying(3)τ˙t≤μ<1,where μ are constants. Finally f(xi(t))=colf1xi1t,f2xi2t,…,fnxint, g(xi(t))=colg1xi1t,g2xi2t,…,gnxint, and h(xi(t))=colh1xi1t,h2xi2t,…,hnxint are continuous nonlinear functions, and δ(t) is a stochastic variable.Define the following random events for system (1):(4)Event  1:  the  system1experiences  the  nonlinear  function  f·,Event  2:  the  system1experiences  the  nonlinear  functiong·,Lettingδ(t) be a Bernoulli distributed sequence defined by(5)δt=1:if  Event  1  occurs,0:if  Event  2  occurs,it follows that δ(t) satisfies Probδt=1=Eδt=δ0, Prob{δ(t)=0}=1-E{δ(t)}=1-δ0, where the constant δ0∈[0,1] reflects the occurrence probability of the event of the nonlinear functions f(·) and g(·). It is further assumed that the variables δ(t) and wk(t)(k=1,2,3) are mutually independent.The controller is given as(6)ut=KxtAssumption 1 (see [26]).
The outer-coupling configuration matrices of the complex networks satisfy(7)wijq=wijq≥0i≠jwij=-∑j=1,j≠iNwijq,q=1,2;i,j=1,2,…,N.Assumption 2 (see [27]).
Nonlinear functionsf(·) and g(·) are bounded functions satisfying fi(0)=gi(0) and for all x1,x2∈R, x1≠x2 and some fixed constants σi-,σi+,ei-,ei+, the active function satisfies(8)σi-≤fix1-fix2x1-x2≤σi+,ei-≤gix1-gix2x1-x2≤ei+,i=1,2,…,nWe denote (9)Δ1=diag⁡σ1+σ1-,…,σn+σn-,Δ2=diag⁡σ1++σ1-2,…,σn++σn-2,Δ3=diag⁡e1+e1-,…,en+en-,Δ4=diag⁡e1++e1-2,…,en++en-2,Assumption 3 (see [28]).
The distributed time-delay Kernelk(·):[0,+∞)→[0,∞) is continuous and integrable and also satisfies(10)k~=∫0+∞ksds<+∞,k˘=∫0+∞sksds<+∞.Assumption 4 (see [29]).
The noise intensity function vectorσi:R×Rn×Rn→Rn. There exist constant matrices Σ1 and Σ2 of appropriate dimensions such that the inequality(11)σit,u1,v1-σjt,u2,v2Tσit,u1,v1-σjt,u2,v2≤Σ1u1-u22+Σ2v1-v22holds for all i,j=1,2,…,N and u1,v1,u2,v2∈Rn.
Combining with the sign⊗ of Kronecker product, model (1) can be rewritten as(12)dxt=IN⊗A+W1⊗Γ1xt+IN⊗D+W2⊗Γ2xt-τt+δtFIN⊗B1xt+1-δtGIN⊗B2xt+IN⊗C∫-∞tkt-sHxsds+IN⊗Bφutdt+W1⊗Γ1xtdw1t+W2⊗Γ2xt-τtdw2t+σt,xt,xt-τtdw3t,where (13)xt=col⁡x1t,x2t,…,xnt,Fxt=col⁡fx1t,fx2t,…,fxNt,Gxt=col⁡gx1t,gx2t,…,gxNt,Hxt=col⁡hx1t,hx2t,…,hxNt,(14)σt,xt,xt-τt=col⁡σ1t,x1t,x1t-τt,…,σNt,xNt,xNt-τt.Next, we give some useful definition and lemmas.Definition 5.
Model (1) is said to be globally synchronized for any initial conditions Πi0(s)(i=1,2,…,N), if the following holds: (15)limt→∞xit-xjt=0∀i,j=1,2,…,N,in which · stands for the Euclidean norm.Lemma 6 (see [30]).
ForK,F∈Rm×n, if x(t)∈Ψ(F), then φ(Kx(t)) can be represented as (16)φKxt=∑i=12mϕϱUϱK+Uϱ-Fxtwhere ∑i=12mϕϱ=1 and ϕϱ≥0.Lemma 7 (see [31]).
For any constant matrixR∈Rn×n, RT=R>0, scalar ρ>0, and vector function ϖ:[0,ρ]→Rn, one has (17)ρ∫0ρϖTsRϖsds≥∫0ρϖsdsTR∫0ρϖsds.Lemma 8 (see [27]).
For any diagonal matricesJ≥0, L≥0, it follows that(18)B1xit-B1xjtfB1xit-fB1xjtT-JΔ1JΔ2∗-Jxit-xitf(xit-fxjt+B1xit-τt-B1xjt-τtfB1xit-τt-fB1xjt-τtT-LΔ1LΔ2∗-L×B1xit-τt-B1xjt-τtfB1xit-τt-fB1xjt-τt≥0.It is equivalent to the following equation:(19)B1xit-B1xjtfB1xit-fB1xjtT-JΔ1JΔ2∗-JB1xit-B1xjtfxit-fxit+B1xit-τt-B1xjt-τtf(xit-τtτ-fxjt-τtT-LΔ1LΔ2∗-L×xit-τ-xjt-τfB1xit-τt-fB1xjt-τt≥0Lemma 9 (see [32]).
Let⊗ denote the notation of Kronecker product. Then, the following relationships hold: (20)1αA⊗B=A⊗αB,2A+B⊗C=A⊗C+B⊗C,3A⊗BC⊗D=AC⊗BD,Lemma 10 (see [33]).
Lete=(1,1,…,1)T, EN=eeT, and U=NIN-EN, P∈Rn×n, x=(x1T,x2T,…,xNT)T, and y=(y1T,y2T,…,yNT)T, with xk,yk∈Rn,(k=1,2,…,N), then (21)xTU⊗Py=∑1≤i<j≤NNxi-xjTPyi-yj.Lemma 11 (see [34]).
LetH, S be n×n any real matrix, e=(1,1,…,1)T, EN=eeT, U=NIN-EN, x=(x1T,x2T,…,xNT)T, y=(y1T,y2T,…,yNT)T with xk,yk∈Rn(k=1,2,…,N), and f(·) and F(·) are functions and defined in system (2). Then for any vectors x and y with appropriate dimensions, the following matrix inequality holds: (22)xTU⊗HFIN⊗Sy=∑1≤i<j≤NNxi-xjTHfSyi-fSyj


## 3. Synchronization Stability Criteria of Complex Networks

In this section, we deal with globally synchronization problem of the complex network (1) or (12) with stochastic nonlinearities.Rewrite system (12) as(23)dxt=IN⊗A+W1⊗Γ1xt+IN⊗D+W2⊗Γ2xt-τt+δ0FIN⊗B1xt+1-δ0GIN⊗B2xt+δt-δ0FIN⊗B1xt-GIN⊗B2xt+IN⊗C∫-∞tkt-sHxsds+IN⊗BUϱK+BUϱ-Fxtdt+W1⊗Γ1xtdw1t+W2⊗Γ2xt-τtdw2t+σt,xt,xt-τtdw3t.Define(24)yt=Aηt,where A=[(IN⊗(A+(BUϱK+BUϱ-F))+W(1)⊗Γ1), 0Nn×rNn, (IN⊗D+W(2)⊗Γ2),0Nn×rNn,δ0F(IN⊗B1), (1-δ0)G(IN⊗B2),IN⊗C,0Nn×rNn], η(t)=colx(t),x(t-τ(t)),F(B1x(t)),G(B2x(t)), ∫-∞tk(t-s)H(x(s))ds,y(t).Then network (4) can be expressed as(25)dxt=yt+δt-δ0Bηtdt+σtdwt,where B=[0Nn×(r+3)Nn,F(IN⊗B1),-G(IN⊗B2),0Nn×2Nn], dw(t)=coldw1t,dw2t,dw3t, σ(t)=[(W(1)⊗Γ1)x(t),(W(2)⊗Γ2)x(t-τ(t)),σ(t,x(t),x(t-τ(t)))].Theorem 12.
For given scalarsτ and μ, the complex networks (23) are said to be globally synchronized if there exist positive definite matrices Pz(z=1,2,3,4,5,6), and positive diagonal matrices J1,J2,L1,L2, such that the following linear matrix inequalities hold for all 1≤i<j≤N:(26)P1<λI(27)Θij=Πi,j9×9<0,where (28)Π1,1=P1A+BUϱK+BUϱ-F+A+BUϱK+BUϱ-FTP1-Nwij1P1Γ1+Γ1TP1-Nwij1,1Γ1TP1Γ1+P2+τP5-B1TJ1Δ1B1-B2TJ2Δ1B2+λΣ1TΣ1,Π1,2=2P1D-Nwij2P1Γ2,Π1,4=B1TJ1Δ2+2δ0P1,Π1,5=B2TJ2Δ2,Π1,7=21-δ0P1,Π1,9=2P1C,Π2,2=-Nwij2,2Γ2TP1Γ2+λΣ2TΣ2-1-μP2-B1TL1Δ1B1-B2TL2Δ1B2,Π2,6=B1TL1Δ2,Π2,7=B2TL2Δ2,Π3,3=k~P6,Π4,4=-J1+P3,Π5,5=-J2+P4,Π6,6=-L1-1-μP3,Π7,7=-L2-1-μP4,Π8,8=-1τP5,Π9,9=1k~P6.Proof.
Choose the Lyapunov functional candidate:(29)Vt=∑l=16Vlt,where (30)V1t=xTtU⊗P1xt,V2t=∫t-τttxTsU⊗P2xsds,V3t=∫t-τttFTIN⊗B1xsU⊗P3FIN⊗B1xsds,V4t=∫t-τttGTIN⊗B2xsU⊗P4GIN⊗B2xsds,V5t=∫t-τtt∫θtxTsdsU⊗P5xsdsdθ,V6t=∫0∞ksds∫t-stHTxθU⊗P6Hxθdθ.Let L be the weak infinitesimal generator of random process {(t,xt),t≥0}. Then, we can obtain [25](31)LV1t,xt=2xTtU⊗P1yt+xTtW1⊗Γ1TU⊗P1W1⊗Γ1xt+xTt-τtW2⊗Γ2TU⊗P1W2⊗Γ2xt-τtσTt,xt,xt-τtU⊗P1σt,xt,xt-τt+2δt-δ0xTtU⊗P1Bηt,(32)=∑1≤i<j≤Nxit-xjtT2P1yit-yjt-Nwij1,1Γ1TP1Γ1xit-xjt-xit-τt-xjt-τtTNwij2,2Γ2TP1Γ2xit-τt-xjt-τt+xit-xjtTP1BUϱK+BUϱ-Fxit-xjt+σit,xit,xit-τt-σjt,xjt,xjt-τtTP1×σit,xit,xit-τt-σjt,xjt,xjt-τt,=∑1≤i<j≤Nxit-xjtTP1A+ATP1+P1BUϱK+BUϱ-F+BUϱK+BUϱ-FTP1-Nwij1P1Γ1+Γ1TP1-Nwij1,1Γ1TP1Γ1xit-xjt+2P1D-Nwij2P1Γ2xit-τt-xjt-τt+2δ0P1B1fxit-fxjt+21-δ0PB2gxit-gxjt+2P1C∫-∞tkt-shxis-hxjsds-xit-τt-xjt-τtTNwij2,2Γ2TP1Γ2xit-τt-xjt-τt+σit,xit,xit-τt-σjt,xjt,xjt-τtTP1σit,xit,xit-τt-σjt,xjt,xjt-τt,(33)LV2t,xt=xTtU⊗P2xt-1-μxTt-τtU⊗P2xt-τt,=∑i=1N-1∑j=i+1Nxit-xjtTP2xit-xjt-1-μxit-τt-xjt-τtTP2xit-τt-xjt-τt,(34)LV3t=FTIN⊗B1xtU⊗P3FIN⊗B1xt-1-μFTIN⊗B1xt-τtU⊗P3FIN⊗B1xt-τt,=∑i=1N-1∑j=i+1NfB1xit-fB1xitTP3fB1xit-fB1xit-1-μfB1xit-τt-fB1xit-τtTP3fB1xit-τt-fB1xit-τt,(35)LV4t=GTIN⊗B2xtU⊗P4GIN⊗B2xt-1-μGTIN⊗B2xt-τtU⊗P4GIN⊗B2xt-τt,=∑i=1N-1∑j=i+1NgB2xit-gB2xitTP4gB2xit-gB2xit-1-μgB2xit-τt-gB2xit-τtTP4gB2xit-τt-gB2xit-τt,(36)LV5t=τxTtU⊗P5xt-∫t-τttxTsU⊗P5xsds,≤τxTtU⊗P5xt-1τ∫t-τttxsdsTU⊗P5∫t-τttxsds=∑i=1N-1∑j=i+1Nτxit-xjtTP5xit-xjt-1τ∫t-τttxis-xjsdsTP5∫t-τttxis-xjsds,(37)LV6t=∫0∞ksHTxtU⊗P6Hxtds-∫0∞ksHTxt-sU⊗P6Hxt-sds=k~HTxtU⊗P6Hxt-∫0∞ksHTxt-sU⊗P2Hxt-sds,=∑i=1N-1∑j=i+1Nk~hxit-hxjtTP6hxit-hxjt-∫-∞tkt-shxis-hxjsTP6hxis-hxjsds,≤∑i=1N-1∑j=i+1Nk~hxit-hxjtTP6hxit-hxjt-1k~∫-∞tkt-shxis-hxjsdsT×P6∫-∞tkt-shxis-hxjsds.According to Assumption 4 and condition (26), it is clear that (38)σit,xit,xit-τt-σjt,xjt,xjt-τtTP1σit,xit,xit-τt-σjt,xjt,xjt-τt≤λ[xit-xjtTΣ1TΣ1xit-xjt+xit-τt-xjt-τtTΣ2TΣ2xit-τt-xjt-τt,1≤i<j≤N.From Assumption 2, by the methods in Lemma 8, for any n×n positive diagonal matrices J1,L1,J2,L2, one has(39)xit-xjtfB1xit-fB1xjtT-B1TJ1Δ1B1B1TJ1Δ2∗-J1xit-xjtfB1xit-fB1xjt+xit-τt-xjt-τtfB1xit-τt-fB1xjt-τtT-B1TL1Δ1B1B1TL1Δ2∗-L1×xit-τt-xjt-τtfB1xit-τt-fB1xjt-τt≥0,(40)xit-xjtgB2xit-gB2xjtT-B2TJ2Δ1B2B2TJ2Δ2∗-J2xit-xjtgB2xit-GB2xjt+xit-τt-xjt-τtgB2xit-τt-gB2xjt-τtT-B2TL2Δ1B2B2TL2Δ2∗-L2×xit-τt-xjt-τtgB2xit-τt-gB2xjt-τt≥0.According to (31)–(40) and Lemma 11, we can obtain (41)LVt≤∑i=1N-1∑j=i+1NξijTtΘijξijt,where (42)
ξ
i
j
t
=
x
i
t
-
x
j
t
T
x
i
t
-
τ
t
-
x
j
t
-
τ
t
T
h
x
i
t
-
h
x
j
t
T
f
B
1
x
i
t
-
f
B
1
x
j
t
T
g
B
2
x
i
t
-
g
B
2
x
j
t
T
f
B
1
x
i
t
-
τ
t
-
f
B
1
x
j
t
-
τ
t
T
g
B
2
x
i
t
-
τ
t
-
g
B
2
x
j
t
-
τ
t
T
∫
t
-
τ
t
t
x
i
s
-
x
j
s
d
s
T
∫
-
∞
t
k
t
-
s
h
x
i
s
-
h
x
j
s
d
s
T
.From Definition 5, we conclude that system (23) has global synchronization. The proof is completed.


## 4. Actuator Saturation and Control Design

In this section, by use of pre- and postmultiplying (27), some matrices and nonlinear matrix inequality (27) can be changed into linear matrix inequality and the final optimization problem can be obtained (43)V0=xT0U⊗P1x0+∫-τ00xTsU⊗P2xs2ds+∫-τ0tFTIN⊗B1xsU⊗P3FIN⊗B1xsds+∫-τ0tGTIN⊗B2xsU⊗P4GIN⊗B2xsds+∫-τ00∫θ0xTsU⊗P5xsdsdθ+∫0∞ksds∫-s0HTxθU⊗P6Hxθdθ,≤λmaxP1x02+∫0-τ00λmaxU⊗P2xs2ds+∫-τ00λmaxU⊗P3FTFIN⊗B12xs2ds+∫-τ00λmaxU⊗P4GTGIN⊗B22xs2ds+∫-τ00∫θ0λmaxU⊗P5xs2dsdθ+∫0∞∫-θ0HTHλmaxU⊗P6xs2dsdθ,≤λmaxP1+τλmaxU⊗P2+τλmaxU⊗P3+τλmaxU⊗P4+12τ2λmaxU⊗P5+12τ2λmaxU⊗P6≤1.Then, the optimization problem can be given as(44)minλmaxP1+τλmaxU⊗P2+τλmaxU⊗P3+τλmaxU⊗P4+12τ2λmaxU⊗P5+12τ2λmaxU⊗P6,(45)minP1,P2,P3,P4,P5,P6,K,Fsuch that inequality (27) holds;(46)Fqxt≤u-q,q=1,2,…n∀xt∈ϕP1.Let(47)X=P1-1,Y=KX,P¯2=XP2X,P¯5=XP5X,J¯1=XJ1X,J¯2=XJ2X,σ¯1=Xσ1XFrom the value|F1(x(t)|≤u¯q, we have(48)-XHq∗-u¯q2<0,q=1,2,…,n,where Hq denotes the qth row of H. Pre- and postmultiplying (27) by diagX,I,I,I,I,I,I,I,I lead to(49)Θij=Πi,j9×9<0,(50)Π1,1=XAT+XTA+YTUqB+BUqY+HTUq-BT+HUq-B-NWij1Γ1X+XΓ1T-Nwij1,1Γ1TXΓ1+P¯2+τP¯5-B1TJ¯1Δ1B1-B2TJ¯2Δ1B2+λΣ¯1TΣ¯1,Π1,2=2D-NWij2Γ2,Π1,4=XB1J1Δ2+2δ0,Π1,5=XB2TJ2Δ2,Π1,7=21-δ0,Π1,9=2C,Π2,2=-Nwij2,2Γ2TP1Γ2+λΣ2TΣ2-1-μP2-B1TL1Δ1B1-B2TL2Δ1B2,Π2,6=B1TL1Δ2,Π2,7=B2TL2Δ2,Π3,3=k~P6,Π4,4=-J1+P3,Π5,5=-J2+P4,Π6,6=-L1-1-μP3,Π7,7=-L2-1-μP4,Π8,8=-1τP5,Π9,9=1k~P6.Therefore, the optimization problem (45) can be rewritten as(51)minΛX,Y,P¯2,P¯5,J¯1,J¯2,Σ¯1such that inequalities (44)-(49) hold, where Λ=λmax(P1)+τλmax(U⊗P2)+τλmax(U⊗P3)+τλmax(U⊗P4)+1/2τ2λmax(U⊗P5)+1/2τ2λmax(U⊗P6) with νmax=1/min(Λ) thenthe controller gain is given asK=YX-1.


## 5. Numerical Example

In this section, we present a numerical example so as to illustrate the advantage and usefulness of our main results.Example 1.
Consider the following complex networks with time delays.
Choose the coupling matricesW(1), W(2) and the linking matrices Γ1, Γ2 as (52)A=-3.20.40.6-2,B1=-1.30.50.3-0.6,B2=-1.30.40.4-1.3C=-1.5-0.30.8-1.3,D=-2.70.90.45-1.7,W1=-2111-2111-2,W2=-3121-2112-3,Γ1=0.7-0.6-0.40.7,Γ2=-0.30.25-0.35-0.4,B=0.10.2.Let the nonlinear function and the noise intensity function vector be given by (53)fxit=0.5xi1t-tanh⁡0.2xi1t+0.2xi2t0.95xi2t-tanh⁡0.75xi2tT,gxit=hxit=0.2xi1t-tanh⁡0.1xi1t0.1xi2tTi=1,2,3,σt,u,v=-0.10.10.2-0.20.1-0.10.2-0.2.Then it is easy verify that k~=0.3,τ=0.9,μ=0.3, δ0=0.56, and (54)Σ1=-0.10.10.1-0.1,Σ2=0.2-0.20.2-0.2,Δ1=-3003,Δ2=1001.We solve the LMIs in Theorem 12. we obtain the feasible solutions as follows: (55)P1=0.15720.01330.01330.1774,P2=6.14211.28561.28566.1421,P3=5.4725-0.0020-0.00205.4863P4=1.05010.17260.17261.0348,P5=4.7825-0.0000-0.00004.7825,P6=0.94870.00130.00130.9510,J1=-0.091700-0.0917,J2=2.8536002.8536,L1=0.2118000.2118,L2=1.0135001.0135.Moreover, the corresponding control gain matrix is obtained; K=0.0785-11.2290.
The initial condition is selected asx0=-0.50.3T. In order to obtain the simulated curve of the system ExT(t)x(t). Simulation results are shown in Figures 1–7. Synchronization of the complex networks with attractor is shown in Figures 1 and 2. Then Figures 3 and 4 state that trajectories are complex networks. It is shown that the trajectory reaches practical randomly nonlinearity. After t=15sec, the trajectory is going to be synchronized, which is the same as the theoretical value presented in the paper. Figure 5 shows the control input with stochastic disturbance. The event time δ(t)=1, its getting 1 occurs, and δ(t)=0, its getting 2 occurs, which is shown in Figures 6 and 7 and it is clear from the figure that once the trajectories reach practical actuator saturation, the interevent time increases; i.e., the control is being updated after long time gaps. There is a maximum time gap between two consecutive control updates after reaching the optimization obtained as νmax=0.7516. The average value of the interevent time is a big multiple of sampling interval.Figure 1
Attractor of the complex networks withx1(t) and x2(t).Figure 2
Attractor of the complex networks withx1(t) and x2(t).Figure 3
State responses of complex networks (23) in Example 1.Figure 4
State responses of complex networks (23) in Example 1.Figure 5
State responses of complex networks (23) in Example 1.Figure 6
The trajectories of the optimization.Figure 7
The trajectories of optimization.


## 6. Conclusion

This paper studies the global synchronization of complex networks with hybrid coupling and stochastic disturbance. By defining appropriate Lyapunov functional and by employing integral inequalities, new delay-dependent globally synchronization criteria have been derived in terms of LMIs. Based on the synchronization criterion and the control design, a sufficient condition is provided to guarantee the globally synchronization of considered system with actuator saturation of randomly nonlinearity. Then method of control gain has also been given. The theoretical results were validated by numerical example and simulations. It should be pointed out that the proposed method in this paper can also be extended to the Markovian jumping complex dynamical systems with parameter uncertainties, partially unknown transition probabilities and random packet loss. This work will be done in the near future.



---

*Source: 9612483-2019-03-06.xml*