New Synchronization Stability Criteria for ComplexNetworks with Time-varying Coupling Delays

Zhongyang FeiDepartment of Control Science and EngineeringHarbin Institute of Technology, Harbin, China

Email: feizhongyang@gmai1.com

Abstract-This paper focuses on the problem of synchroniza­tion stability analysis for complex dynamical networks with cou­pling delays, both continuous and discrete cases are considered.With the method of delay partitioning, less conservative criteriaare derived in the form of linear matrix inequalities (LMIs).Numerical examples are provided to show the effectiveness ofthe proposed results.

Index Terms-Complex networks; Synchronization stability;Time-varying delays; Delay partitioning; Linear matrix inequal­ity.

I. INTRODUCTION

Complex networks, which are usually referred to as struc­tures that consist of nodes or vertices connected by links andedges, have the advantage of better modeling some practicalcomplex systems in the real world. Examples of such complexnetworks are around us everywhere, from the Internet, foodwebs, biological neural networks to electrical power grids,citation networks of scientists, cellular and metabolic net­works, etc [10]. A lot of attention has been paid to complexnetworks across many fields of science and engineering. Manyproperties, including dynamical evolution and node diversity,have been widely studied. Till now, much work has been doneto complex networks in the literature [1], [8], [11], [12], [13],[14], [15], [16], [19], [20].

In practical industries, time delays are always unavoidable,which are the main sources of poor performance in a system,even cause oscillation, instability and divergence. So it isnot only a theoretical problem but also a practical one toanalyze and synthesize complex networks with time delays,a lot of work has been done to such systems [9], [17],[18]. As one of the basic problems of complex networkswith time delays, synchronization stability analysis has drawnmuch attention in the literature [3], [4], [5], [6], [10]. In[10], the synchronization phenomena and criteria was studied,and some of the synchronization conditions were derivedfor both delay-independent and delay-dependent asymptoticstabilities in terms of linear matrix inequalities (LMIs). Soon,by carefully treating with the delay parts, a new criterion was

°This work was partially supported by National Natural Science Foundationunder Grant (60504008), The Research Found for the Doctoral Programmeof Higher Education of China (20070213084), Key Laboratory of IntegratedAutomation for the Process Industry (Northeastern University), Ministry ofEducation of China.

1-4244-2386-6/08/$20.00 ©2008 IEEE

Huijun GaoDepartment of Control Science and EngineeringHarbin Institute of Technology, Harbin, China

Email: hjgao@hit.edu.cn

given according to a novel Lyapunov-Krasovskii functional(LKF) which guaranteed the synchronization of the system[5]. Then some simple synchronization criteria for complexdynamical networks were provided based on the concept ofmatrix measure [2]. Though some precursors have obtainedsome achievements in synchronization stability of complexnetworks with time delays, we find that the results are stillleaving much room for improvement. Moreover, to the bestof our knowledge, there are still no available results for thesynchronization stability for complex networks with time­varying delays, which motivate the present study.

In this paper, we revisit the problem of complex networkswith time-varying delay and present some new synchronizationstability conditions, both continuous systems and discretesystems are included. By partitioning the constant delay, weconstruct novel LKFs, and create the new criteria by meansof LMIs, which are readily solvable by standard software. Itis proved to be less conservative with the delay fractioningparts getting thinner. Illustrative examples are also shown todemonstrate the merits of the new criteria.

The rest of the paper is organized as follows. The models,preliminaries and some definitions with the delay-dependentconditions that guarantee the synchronized states to be asymp­totically stable for both continuous-time and discrete-timecomplex networks are presented in Sections 2 and 3. Illus­trative examples are mentioned to show the effectiveness ofthe new criteria in Section 4, and some conclusions are givenin Section 5.

Notation: The notation used in this paper is fairly standard.~n represents the n-dimensional Euclidean space. P > 0 meansthat matrix P is real symmetric and positive definite; Thesuperscripts "T" and "-1" represent matrix transposition andmatrix inverse respectively; In and On denote the identity ma­trix and zero matrix with n-dimensions. Matrices are assumedto be compatible for algebraic operations if their dimensionsare not explicitly stated. The notation sym(A) stands forAT +A.

II. CONTINUOUS-TIME NETWORKS

In this section, we will consider the synchronization stabil­ity for continuous-time complex networks with time varyingdelays. First, we will introduce some preliminaries and defi­nitions, and then present the new criterion.

A. Problem Formulation and Preliminaries

Consider the following continuous-time complex dynamicalnetwork with a time-varying coupling delay:

xi = f (xi)+ cN

∑j=1

Gi jAx j(t− τ(t)), i = 1,2, . . . ,N, (1)

where N is the number of coupled nodes; f : Rn → Rn iscontinuously differentiable, xi = [xi1,xi2, . . . ,xin]

T are the statevariables of node i; the constant c represents the couplingstrength; An×n ∈ Rn×n is the constant inner-coupling matrixof the nodes; τ(t) is a time-varying differentiable function,which satisfies the assumption that

h1 ≤ τ(t)≤ h2, (2)τ(t) < µ, (3)

where 0≤ h1 < h2 and µ are all constants. Note that h1 maynot be equal to 0. G = (Gi j)N×N represents the outer-couplingmatrix of the network, in which Gi j is defined as follows: ifthere exists a connection between node i and node j ( j 6= i),then Gi j = G ji = 1, or else Gi j = G ji = 0 ( j 6= i), and thediagonal elements of matrix G are defined by

Gii =−N

∑j=1, j 6=i

Gi j =−N

∑j=1, j 6=i

G ji, i = 1,2, . . . ,N. (4)

Suppose that network (1) is connected in the sense that thereare no isolated clusters, which means G is an irreduciblematrix.

Definition 1: The delayed dynamical network in (1) is saidto achieve asymptotic synchronization if

x1(t) = x2(t) = . . . = xN(t) = s(t) as t → ∞, (5)

where s(t) is a solution of an isolate node, satisfying s(t) =f (s(t)).

Lemma 1: Consider the delayed dynamical network in (1).The eigenvalues of the outer-coupling matrix G are denotedas

0 = λ 1 ≥ λ 2 ≥ λ 3 ≥ . . .≥ λ N .

If the following N − 1 of n-dimensional linear time-varyingdelayed differential equations are asymptotically stable abouttheir zero solutions:

yk(t) = J(t)yk(t)+ cλ kAyk(t− τ(t)), k = 2,3, . . . ,N, (6)

where J(t) = f (s(t) ∈Rn×n is the Jacobian of f (x(t)) at s(t),then the synchronized states (5) are asymptotically stable.

B. New Criterion for Synchronization Stability

In this subsection, we will show our new result on synchro-nization stability for general complex dynamical networks withtime-varying delays. We first treat the time delay τ(t) into twoparts which is the constant part h1 and the time-varying parth(t), so we have

τ(t) = h1 +h(t),

where h(t) satisfies

0≤ h(t)≤ h2−h1,

then we know h(t) = τ(t) < µ from the definition of τ(t).Then we part the constant delay into m pieces, a LKF

candidate is constructed as follows with such approach

Vk(t) = Vk1(t)+Vk2(t)+Vk3(t)+Vk4(t), (7)

where

Vk1(t) = yTk (t)Pkyk(t),

Vk2(t) =∫ t

t− h1m

ϒT (s)Rk1ϒ(s)ds+

∫ t−h1

t−τ(t)yT

k (s)Rk2yk(s)ds,

Vk3(t) =∫ t

t−h2

yTk (s)Mkyk(s)ds,

Vk4(t) =∫ 0

− h1m

∫ t

t+vyT

k (s)Qk1yTk (s)dsdv

+∫ −h1

−h2

∫ t

t+vyT

k (s)Qk2yk(s)dsdv,

and

ϒk(s) =

yk(s)

yk(s− 1m h1)

...yk(s− m−1

m h1)

.

Then based on the standard Lyapunov method, we present ournew criterion for synchronization stability of the general com-plex dynamical networks with time-varying coupling delays inthe following theorem.

Theorem 2: Consider the complex dynamical networkswith coupling time-varying delay τ(t) in (1), it is said that theasymptotical synchronization in (5) is achieved if there existmatrices Pk > 0, Rk1 > 0, Rk2 > 0, Mk > 0, Qk1 > 0, Qk2 > 0and Sk, k = 2,3, . . . ,N such that for a given integer m≥ 1, thefollowing LMI holds

Θk = W TP PWP +W T

R1R1WR1 +W T

R21R2WR21 +W T

M MWM

+(µ−1)W TR22

R2WR22 +W TQ1

Q1WQ1 +W TQ2

Q2WQ2

+sym(SkWS) < 0, (8)

where

P =[

0 PkPk 0

], R1 =

[Rk1 00 −Rk1

],

M =[

Mk 00 −Mk

], Q1 =

[−Qk1 0

0 Qk1

],

Q2 =

−Qk2 0 00 −Qk2 00 0 Qk2

,

WP =[

In 0n,(m+3)n0n,(m+3)n In

],

WR1 =[

Imn 0mn,4n0mn,n Imn 0mn,3n

],

WR21 =[

0n,mn In 0n,3n],

WR22 =[

0n,(m+1)n In 0n,2n],

WM =[

In 0n,(m+3)n0n,(m+2)n In 0n,n

],

WQ1 =

√mh1

In −√

mh1

In 0n,(m+2)n

0n,(m+3)n

√h1m In

,

WQ2 =√

1h2−h1

0n,(m+1)n In −In 0n,n

0n,mn In −In 0n,2n0n,(m+3)n (h2−h1) In

,

WS =[−J(t) 0n,mn −cλ kA 0n,n In

].

Remark 1: The Theorem 2 showed to be less conservativethan most existing results because of the concept of delayfractioning to the constant delay. Moreover, with the increaseof fractioning number m, the proposed result demonstrates itssuperiority, which will be well illustrated via an example inSection 4.

III. DISCRETE-TIME NETWORKS

In this section, we will provide some results for discrete-time dynamic networks with coupling time-varying delays.Unless otherwise defined, we endorse the same meaning tothe notations used in the above section.

A. Problem Formulation and Preliminaries

Consider the following discrete complex dynamical networkwith time-varying coupling delays:

xi(t +1) = f (xi(t))+cN

∑j=1

Gi jAx j(t−τ(t)), i = 1,2, . . . ,N, (9)

where τ(t) satisfies the assumption that

1≤ h1 ≤ τ(t)≤ h2, (10)

where h1, the minimum delay bound, and h2, maximum delaybound, are all constant positive scalars. Moreover, τ(t) is alsorequired to be a positive integer and supposed to be time-varying in the whole dynamic process.

Definition 2: The discrete-time delayed dynamical networkin (9) is said to achieve asymptotic synchronization if

x1(t) = x2(t) = . . . = xN(t) = s(t) as t → ∞, (11)

where s(t) is a solution of an isolate node, satisfying s(t +1) =f (s(t)).

Lemma 3: Consider the delayed dynamical network (9).The eigenvalues of the outer-coupling matrix G are denotedas

0 = λ 1 ≥ λ 2 ≥ λ 3 ≥ . . .≥ λ N .

If the following N−1 linear time-varying delayed differentialequations are asymptotically stable about their zero solutions:

yk(t +1) = J(t)yk(t)+ cλ kAyk(t− τ), k = 2,3, . . . ,N, (12)

where J(t) = f (s(t))∈Rn×n is the Jacobian of f (x(t)) at s(t),then the synchronized states (11) are asymptotically stable.

B. New Criterion for Synchronization Stability

In this subsection, we mainly consider the stability analysisproblem. More specifically, we divide the time delay τ(t) intotwo parts: the constant delay part h1 and the time-varying parth(t). With dividing the constant part h1 into m parts, we get:

τ(t) = τm+h(t),

where τ and m are all positive integers and h(t) satisfies thefollowing condition

0≤ h(t)≤ h2− τm.

Then we are in the position to give a new criterion fordiscrete complex networks to be synchronization stability. Tothis end, we construct a new LKF as follows

Vk(t) = Vk1(t)+Vk2(t)+Vk3(t)+Vk4(t), (13)

where

Vk1(t) = yTk (t)Pkyk(t),

Vk2(t) =t−1

∑i=t−τ

ϒT (i)Rk1ϒ(i)+

t−1

∑i=t−h2

yTk (i)Rk2yk(i),

Vk3(t) =−τm+1

∑i=−h2+1

t−1

∑j=t−1+i

yTk ( j)Mkyk( j),

Vk4(t) =−1

∑i=−τ

t−1

∑j=t+i

δTk ( j)Qk1δ k( j)

+−τm−1

∑i=−h2

t−1

∑j=t+i

δTk ( j)Qk2δ k( j),

δ k( j) = yk( j +1)− yk ( j) ,

and

ϒk(i) =

yk(i)

yk(i− τ)...

yk(i− τm+ τ)

.

Our new criterion for the discrete complex dynamical networkswith time-varying coupling delays is shown in the followingtheorem.

Theorem 4: Consider the discrete complex networks in (9)with time-varying coupling delay τ (t) . For given positiveintegers τ and m, the network is said to achieve asymp-totic synchronization in (11) if there exist matrices Pk > 0,Rki > 0, i = 1,2, Mk > 0, Qk j > 0, j = 1,2, Uk > 0, Vk > 0,k = 2,3, . . . ,N and Xk, Yk, Zk satisfying

Θk = W TP PWP +W T

R1R1WR1 +W T

R2R2WR2 +W T

M MWM

+τW TQ1

Qk1WQ1 +(h2− τm)W TQ2

Qk2WQ2

+sym(SWS)+ τUk +(h2− τm)Vk < 0,

Πk1 =[

Uk XkXT

k Qk1

]≥ 0, Πk2 =

[Vk YkY T

k Qk2

]≥ 0,

Πk3 =[

Vk ZkZT

k Qk2

]≥ 0, (14)

where

P =[

Pk 00 −Pk

], R1 =

[Rk1 00 −Rk1

],

R2 =[

Rk2 00 −Rk2

], M =

[Mk 00 −Mk

],

S =[

X Y Z],

WP =[

Jk (t) 0n×mn cλ kA 0n×nIn 0n×(m+2)n

],

WR1 =[

Imn 0mn×3n0mn×n Imn 0mn×2n

],

WR2 =[

In 0n×(m+2)n0n×(m+2)n In

],

WM =[ √

(h2− τm+1)In 0n×(m+2)n0n×(m+1)n In 0n×n

],

WQ1 =[

Jk (t)− In 0n×mn cλ kA 0n×n],

WQ2 = WQ1 , WS =

In −In 0n×(m+1)n0n×mn In −In 0n×n

0n×(m+1)n In −In

.

Remark 2: Here we try different methods to treat withcontinuous-time complex networks and discrete-time complexnetworks. With the continuous case, we use Jensen’s inequalityto built Theorem 2, with discrete case, we use the approachmentioned in [7] to get the LMIs in Theorem 4.

IV. ILLUSTRATIVE EXAMPLESIn this section, we will use numerical examples to demon-

strate the advantage of the new criteria proposed in this paper.

A. Continuous-time Networks

Example Consider a continuous-time network with 5 nodes,with each node a simple three-dimensional stable linear systemdescribed by x1

x2x3

=

−x1−2x2−3x3

,

which is asymptotically stable at s(t) = 0, and Jacobian isgiven as J = diag{−1,−2,−3} . In addition, assume thatthe inner-coupling matrix is A=diag{1,1,1} , and the outer-coupling matrix is described by the following irreduciblesymmetric matrix satisfying condition (4):

G =

−2 1 0 0 11 −3 1 1 00 1 −2 1 00 1 1 −3 11 0 0 1 −2

.

The set of the eigenvalue of G is{0,−1.382,−2.382,−3.618,−4.618} .

Set that the coupling strength c = 0.5. Find the allowableupper boundary h2 for fixed lower boundary h1 = 0.8 andconstant µ = 0.05 with different m. Then alter µ to 0.01with h1 unchanged, find the feasible upper boundary h2 withdifferent m. The numerical results are shown in Table 1.

Theorem 1h2 with µ = 0.05

(h1 = 0.8)h2 with µ = 0.01

(h1 = 0.8)m = 1 0.807 0.814m = 2 0.909 0.937m = 3 0.925 0.951m = 4 0.930 0.956m = 5 0.932 0.958

Table 1 The upper delay boundary with different µ

From Table 1, we can easily see that the conservatismdecreases significantly with the fractioning becoming thinner.

B. Discrete-time Networks

Example Consider a discrete-time complex network with 3nodes, in which each node is a simple two-dimensional stablelinear system described by[

x1 (t +1)x2 (t +1)

]=

[0.8 0

0.05 0.9

][x1 (t)x2 (t)

],

which is asymptotically stable at s(t) = 0, and Jacobian is

given as J =[

0.8 00.05 0.9

]. Suppose that the inner-coupling

matrix is A=[

0.08 00.16 0.08

], and the outer-coupling matrix

is described by the following irreducible symmetric matrix:

G =

−1 1 01 −2 10 1 −1

.

The set of the eigenvalue of G is {0,−1,−3} .

With setting the coupling strength c = 0.5, we assume thath1 = h2, which means that the system is with constant timedelays. Then we can get the comparison with the methodmentioned in [5] in Table 2.

Method the constant time delay τ

Gao et al. [5] 27Theorem 1, m = 1 27Theorem 1, m = 2 32Theorem 1, m = 3 33Theorem 1, m = 5 35

Table 2 The maximum delay bound comparison amongdifferent methods

From Table 2, we can easily see that Theorem 4 is muchless conservative, especially when the fractioning number mincreases.

Then we fix the lower bound h1 to find the allowable time-varying delay bound h2 that guarantees the discrete complexnetworks to be asymptotically stable. Detailed comparison isgiven in Table 3.

lower bound h1 fractioning number m upper bound h24 2 148 2 16

12 2 1816 2 2020 2 23

Table 3 Allowable upper bound of h2 for various h1

V. CONCLUSIONSIn this paper, the problem of synchronization stability for

complex networks with time-varying delay has been studied,both continuous and discrete cases are taken into considera-tion. Benefiting from the novel LKF proposed with the ideaof delay fractioning, new delay-dependent sufficient conditionson the synchronization stability for such networks have beenpresented in terms of LMIs based on the standard Lyapunovtheory. In order to show the advantage of derived results,numerical examples are presented to illustrate the reducedconservatism in most existing results.

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