Neurocomputing 331 (2019) 346–355

Contents lists available at ScienceDirect

Neurocomputing

journal homepage: www.elsevier.com/locate/neucom

Synchronization analysis of network systems applying sampled-data

controller with time-delay via the Bessel–Legendre inequality

Hongru Ren

a , b , Junlin Xiong

b , Renquan Lu

a , ∗, Yuanqing Wu

a

a Guangdong Provincial Key Laboratory of Intelligent Decision and Cooperative Control, Guangdong University of Technology, Guangzhou 510 0 06,

Guangdong, China b Department of Automation, University of Science and Technology of China, Hefei 230026, Anhui, China

a r t i c l e i n f o

Article history:

Received 11 October 2018

Revised 13 November 2018

Accepted 17 November 2018

Available online 22 November 2018

Communicated by Bo Shen

Keywords:

Integral inequality

Network systems

Sampled data control

Synchronization

Time delay

a b s t r a c t

In this paper, taking advantage of aperiodic sampled data control technique, the synchronization issue of

network systems is brought into focus. During the transmission of sampled data, time-varying delays are

concerned. Applying input delay method, the sampled data network systems can be rebuilt via continu-

ous systems with another new delay term in the distributed controller. Unlike the utilization of Jensen

and Wirtinger-based inequalities in most literature, the Bessel–Legendre integral inequality is introduced,

which can relax conservative further. The characteristics of this integral inequality are adequately merged

with the establishment of augmented Lyapunov functional. Two sufficient conditions for synchronizability

of network systems are established. In the end, a simulation example is illustrated to verify the efficacy

and advantage of the designed approach.

© 2018 Elsevier B.V. All rights reserved.

g

i

a

[

i

a

c

H

p

s

m

T

u

r

s

o

i

s

t

c

l

1. Introduction

For the past few years, a large amount of problems of network

systems have been researched extensively [1–8] . For example, in

[9] , based on time variant measurement topology, the issue of dis-

tributed estimation for cars formation was addressed. In [10] , the

synchronization problem of network systems with general linear

dynamics was solved by a distributed event-triggered strategy. In

particular, the synchronization problem becomes a hotspot topic

in network systems investigation owing to its widespread practical

applications. For instance, formation control, cooperative control

of unmanned aircrafts and underwater vehicles, communication

among wireless sensor networks and flocking of mobile vehicles

are some typical issues in military and civilian area which the

synchronization problems are involved. In [11] , using a complex

Laplacian-based technique, the fundamental formation control

problem was investigated which required the network systems to

achieve an explicit but arbitrary formation shape. In [12] , for the

synchronization tracking, an iterative learning control scheme was

proposed, which can be utilized in the network systems with a

fixed communication topology. The results were not only suitable

for the homogeneous network systems, but extended to hetero-

∗ Corresponding author.

E-mail addresses: rhr_5207@163.com (H. Ren), xiong77@ustc.edu.cn (J. Xiong),

lurenquan2012@163.com (R. Lu), yqwuzju@163.com (Y. Wu).

s

s

[

d

https://doi.org/10.1016/j.neucom.2018.11.061

0925-2312/© 2018 Elsevier B.V. All rights reserved.

eneous ones also. Afterwards, with regard to the synchronization

ssue of high-order nonlinear network systems, a novel distributed

daptive iterative learning control strategy was further designed in

13] . In addition, the distributed synchronization protocol design

ssue for network systems with oriented communication graphs

nd ordinary linear dynamics was discussed in [14] .

For network systems, researchers poured attention into syn-

hronization problem utilizing continuous-time control strategies.

owever, continuous-time control strategy sometimes is not ap-

ropriate due to some inherent characteristics of the networked

cenarios. In order to address such issue, sampled-data control

ethod should be a better choice rather than continuous-time one.

he plant of sampled-data system is continuous-time, while the

pdate mode of control law is discrete-time [15–20] . Although pe-

iodic sampling approach was widely used under some circum-

tances, it is necessary to employ aperiodic sampling pattern in

rder to saving energy further. Some crucial problems were stud-

ed in literature with the aim of analysing aperiodic sampled-data

ystems. In [21] , stability analysis of the acylic sampled data sys-

ems was concerned for application to embedded and networked

ontrol. In [22] , the synchronization of network systems with non-

inear dynamics applying aperiodic sampled data controllers was

tudied. In [23] , the continuous time systems with time-varying

ampling intervals were rebuilt as discrete time-varying systems. In

24] , the initial sampled data systems were reconstructed by time

elay systems employing the input delay approaches. Adopting

H. Ren, J. Xiong and R. Lu et al. / Neurocomputing 331 (2019) 346–355 347

s

b

f

f

m

a

p

o

m

a

l

r

d

s

s

o

c

n

i

t

n

l

c

e

m

p

l

c

c

l

o

n

d

t

t

t

w

s

i

a

i

I

[

e

s

e

p

t

s

t

s

l

b

t

t

a

f

a

e

w

t

u

d

t

w

t

s

c

E

c

M

A

i

i

c

z

d

a

c

p

2

2

k

l

o

E

w

a

e

p

a

e

e

m

s

t

L{

W ∑

r

a

t

x

F

t

c

v

s

c

d

s

d

D

ampled data control laws, a refresh method to evaluate the sta-

ility of continuous linear systems was proposed in [25] . Derived

rom the discrete time Lyapunov theorem, the above approach af-

ords simpler stability conditions of the continuous time system

odels. In [26] , utilizing Lyapunov functionals with discontinuity

t the impulse moments, exponential stability for time-varying im-

ulsive systems including nonlinear dynamics was established.

Owing to the fact that control signals are transmitted from

ne system to another in a wireless communication network in

ost situations, nonlinear dynamics and transmission delays usu-

lly arise inevitably during control process [27–33] . The prob-

ems of time delay for network systems are widely studied by

esearchers. In [34] , the cooperative output regulation issues for

iscrete time linear network systems suffering time delay were

tudied. The network systems were subject to jointly connected

witching networks. The distributed control law was composed

f a distributed switched observer and a purely decentralized

ontrol law. In [35] , the synchronization problems of time-delay

etwork systems with continuous-time single-integrator dynam-

cs were studied and the smart leader was introduced. In [36] ,

he synchronization issues of a kind of second order continuous

etwork systems with jointly-connected topologies and time de-

ay were investigated. Moreover, a linear neighbor-based proto-

ol suffering time delay was introduced. The imperfect network

nvironment may influence the stability of network systems re-

arkably. As a consequence, the sampled-data control approach is

opular and effective for analyzing network systems with time de-

ay. Among the existing literature, the input delay method is a

ompelling approach to model the sampled-data control inputs via

ontinuous-time functions. By applying this approach, the closed-

oop control systems can be presented as switched systems with

rdered and multiple time-varying delays [37,38] .

More recently, by Lyapunov functional approach, the synchro-

ization analysis of network systems with time variant delays has

rew a lot of attentions. The essential technical steps concerning

his approach are correlated with the design of the functional, and

he choice of proper bounding approaches as well. As is known

o all that the Jensen inequality is one of the bounding methods,

hich has been employed diffusely, despite an unavoidable con-

ervatism. Afterwards, to relax this conservatism, extended Jensen

nequalities were studied, which involved Jensen inequality via the

doption of attached quadratic terms. The alleged Wirtinger-based

nequality and its further extensions were investigated in [39,40] .

n addition, generalized integral inequalities were investigated in

41] , which were derived from Legendre polynomials and Bessel in-

quality. Particularly, if the degree of Legendre polynomials is cho-

en as two, it can be called the second-order Bessel–Legendre in-

quality.

There are many progress and achievements on synchronization

roblems for network systems in the existing literature. However,

he synchronization issue of network systems has not been fully

tudied and some problems are required to be probed. Based on

he discussion mentioned above, this paper considers the issue of

ynchronization for network systems with time-varying delays uti-

izing aperiodic sampled-data control approach. The main contri-

utions are as follows. First, time-varying delays with no restric-

ion on their derivatives are considered in this paper. Furthermore,

he characteristics of the Bessel-Legendre inequality are adequately

dapted to the establishment of augmented Lyapunov-Krasovskii

unctionals. In addition, unlike the utilization of Wirtinger-based

nd Jensen inequalities in most literature, the Bessel-Legendre in-

quality is adopted to address the synchronization issue of net-

ork systems, which is less conservative.

The structure of the paper is arranged as follows. First, sys-

em model, the corresponding problem description and some

seful lemmas are explained by Section 2 . Then in Section 3 ,

erived from the input delay approach and Lyapunov functional,

wo novel sufficient conditions for synchronizability issue of net-

ork systems with time variant delays are provided. Meanwhile,

wo relevant sampled-data synchronization control laws are de-

igned. In the end, the validity of the proposed synchronization

ontrol law is testified by numerical simulation in Section IV.

Notations: Throughout the entire paper, R

n represents the real

uclidean space with dimension n . R

a ×b is utilized to denote the

ollection of all a × b real matrices. M

T is the matrix transpose of

, and M

−1 is the inverse matrix. For any square matrix A , He (A ) = + A

T . The notation ‖ · ‖ means the Euclidean vector norm or the

nduced matrix 2-norm. Q > 0 ( Q ≥ 0) represents that Q is pos-

tive definite (positive semi-definite). The set S n + stands for the

ollection of symmetric positive definite matrices. 0 expresses a

ero matrix, and I N stands for the N -dimensional identity matrix.

iag( ���) represents a block diagonal matrix or diagonal matrix,

nd diag p ( J ) denotes a block diagonal matrix with p blocks of J .

ol { ���} stands for a column vector. P �Q represents the Kronecker

roduct.

. Problem formulation and preliminaries

.1. Network systems

So as to introduce the concept of network systems, simple

nowledge of graph theory is presented first. G = (V , E , A ) is uti-

ized to represent an oriented graph containing the collection

f vertices V = { v 1 , v 2 , . . . , v N } , the collection of oriented edges

⊆V × V , and the adjacency matrix A = [ a i j ] N×N . For convenience,

e represent G = (V , E , A ) as G ( A ) if there is no ambiguity. When

vertex j can take over signals from vertex i , then an oriented

dge e ij ∈ E exists in G ( A ), which can be represented via vertices

air of vertices ( v i , v j ). The element of adjacency matrix A is set as

i j = 1 if and only if an oriented edge ( v i , v j ) consists in G ( A ), or

lse, a i j = 0 .

From vertex v i to v j , an oriented path is a series of ordered

dges in E , indicated by (v i , v c1 ) , (v c1 , v c2 ) , . . . , (v ck , v j ) with inter-

ediate vertices v cp , p = 1 , . . . , k . An oriented graph is strong or

trongly connected if there exists an oriented path between any

wo distinct vertices v i and v j . For graph G ( A ), the corresponding

aplacian matrix L = [ l i j ] N×N is set by

l i j = −a i j , i � = j,

l ii = −∑ N j =1 , j � = i l i j .

(1)

ith regard to a strong oriented graph G ( A ), L is subject to N j=1 l i j = 0 , which is known as the diffusion property, and is ir-

educible as well.

Consider the network systems with N isolated linear systems,

bbreviated as nodes, described by the following dynamic charac-

eristics:

˙ i (t) = Ax i (t) + Bu i (t) , i = 1 , 2 , . . . , N. (2)

or the i th node, x i (t) = [ x i 1 (t ) , . . . , x in (t )] T ∈ R

n stands for the sys-

em state and u i (t) ∈ R

p denotes the designed control law. The

onstant real matrices A and B have proper dimensions.

The objective of the paper is, for network systems (2) , to pro-

ide a distributed controller u i for each node i such that the

ynchronization can be realized. However, to reduce the energy

onsumption of communication, such design task for node i only

epends on the signals from its neighbors and its own at separate

ampling instants instead of successive signals.

Before deriving the main results, it is essential to retrospect one

efinition and two lemmas as follows.

efinition 1 [42] . If the following equations hold,

348 H. Ren, J. Xiong and R. Lu et al. / Neurocomputing 331 (2019) 346–355

t

2

c

L

a

e⎡⎣h

w

L

a⎡⎣h

�

R

L

t

a

c

3

d

u

w

m

Z

p

c

t

t

h

u

·

t

B

t

w

η

lim

→∞

‖ x i (t) − x j (t) ‖ = 0 , ∀ i, j = 1 , . . . , N, (3)

the network systems (2) can be considered to achieve the synchro-

nization for any initial conditions.

Lemma 1 [43] . With regard to an oriented strong graph G , its

Laplacian matrix L meets the diffusive coupling condition (1) with∑ N j=1 l i j = 0 , and is irreducible as well. Moreover, L has an eigenvalue

0 with multiplicity 1, and the relevant normalized left eigenvector

� = [ � 1 , � 2 , . . . , � N ] T ∈ R

N satisfying ∑ N

i =1 � i = 1 , namely, �

T · L = 0 .

In addition, for all i = 1 , 2 , . . . , N, ϱi > 0 is satisfied.

Lemma 2 [44] . With regard to an oriented and strong graph G , let

� = [ � 1 , � 2 , . . . , � N ] T be the unique left normalized eigenvector of

corresponding Laplace matrix L with regard to the 0 eigenvalue, � =diag { � 1 , � 2 , . . . , � N } > 0 , and W N×N = � − � �

T . Accordingly, ∀ P > 0,

the following two equations hold:

x T (t)(W L � P C) g(x (t))

= −1

2

N ∑

i =1

N ∑

j =1 , j � = i � i l i j

(x i (t) − x j (t)

)T P C

[g(x i (t)) − g(x j (t))

],

x T (t)(W � P C) g(x (t))

=

1

2

N ∑

i =1

N ∑

j =1 , j � = i � i � j

(x i (t) − x j (t)

)T P C

[g(x i (t)) − g(x j (t))

].

Remark 1. It can be noticed that g ( x ( t )) stands for any function of

x ( t ). For example, ˙ x (t) and x ( t ) are the most common two alterna-

tives of g ( x ( t )). When nonlinear dynamics exist in network systems,

then g ( x ( t )) can be substituted by a nonlinear function of x ( t ).

2.2. Bessel-Legendre integral inequality

In the first place, let us introduce an inequality which is the

kernel of the theoretical derivation of this paper. It is consistent

with the inequality shown in [45] recently, which is a special cir-

cumstance of the Bessel-Legendre inequality in [41] as well. The

corresponding proving process can be sought out in [45] or in [41] .

Lemma 3. For a differentiable function z in [ a, b] → R

n , and a matrix

M ∈ S n + , the inequality

∫ b

a

˙ z T (v ) M

z (v ) dv ≥ 1

b − a G

T diag (M, 3 M, 5 M) G (4)

holds, where

G =

⎡

⎢ ⎢ ⎢ ⎢ ⎣

z(b) − z(a )

z(b) − z(a ) − 2

b − a

∫ b

a

z(v ) dv

z(b) − z(a ) − 6

b − a

∫ b

a

δa,b (v ) z(v ) dv

⎤

⎥ ⎥ ⎥ ⎥ ⎦

,

δa,b (v ) = 2

(v − a

b − a

)− 1 .

Remark 2. The Wirtinger-based inequality of [39] is included

in the inequality (4) with the aid of the third element of G .

Except as signals ∫ b

a z(v ) dv , z ( a ) and z ( b ), an additional signal∫ b a δa,b (v ) z(v ) dv is introduced to accomplish this improvement.

Compared with the Jensen and Wirtinger-based inequalities, the

utilize of the Bessel–Legendre integral inequality can relax the con-

servatism. In the next section, the characteristics of this integral

inequality are adapted to the establishment of extended Lyapunov

functional adequately. Pursuing the approach developed in [46] ,

more precise conditions are derived in this paper. t

.3. Parameter-dependent matrix inequalities

Applying another construction method, the reciprocally convex

ombination inequality in [47] is shown via the following lemma.

emma 4. With regard to any R ∈ S n + , it is assumed that there is

matrix X ∈ R

n ×n so that [ R X X T R

] ≤ 0 . Accordingly, the following in-

quality

1

βR 0

0

1

1 − βR

⎤

⎦ ≤[R X

X

T R

], ∀ β ∈ (0 , 1) ,

olds.

Next, a variational form of Lemma 4 is presented alternatively,

hich is derived from the classical bounding technique in [48] .

emma 5. With regard to any given M 1 ∈ S n + , M 2 ∈ S

n + , Y 1 ∈ R

2 n ×n

nd Y 2 ∈ R

2 n ×n , the inequality

1

βM 1 0

0

1

1 − βM 2

⎤

⎦ ≤ �M

(β) , ∀ β ∈ (0 , 1) ,

olds, where

M

(β) = He ( Y 1 [ I n 0 n ×n ] + Y 2 [ 0 n ×n I n ] )

− βY 1 M

−1 1 Y T 1 − (1 − β) Y 2 M

−1 2 Y T 2 .

emark 3. It can be noticed that the key distinction between

emmas 4 and 5 is that, the lower bound depends specifically upon

he uncertain argument β in Lemma 5 . Eventually, at the cost of

dditional decision variables, this reliance on β results in a de-

rease of conservatism.

. Sampled data control law

In the paper, the synchronization control protocal with sampled

ata signals for system i is given as follows, i = 1 , . . . , N,

i (t) = αK

N ∑

j =1 , j � = i a i j

[x j (t k − η(t k )) − x i (t k − η(t k ))

],

t ∈ [ t k , t k +1 ) , k ∈ N , (5)

here α > 0 stands for a coupling strength parameter, K is the gain

atrix to be designed. t k represents the updating instant of the

OH. Moreover, it is supposed that the sampled signals have ex-

erienced a time-varying transmission delay η( t ) from sampler to

ontroller which satisfies 0 ≤η1 ≤η( t ) ≤η2 . Hence, data informa-

ion at discrete sampling instant t k − η(t k ) is applied to the con-

roller design task at t k . Then, by employing ZOH, the control signal

olds between two adjacent updating moments t k and t k +1 . The

pdating instants of ZOH are subject to η(t 0 ) = t 0 < t 1 < · · · < t k <

· · < lim t→∞

t k = + ∞ . Suppose that the sampling intervals are al-

erable with an upper bound h b as

(t k +1 − η(t k )) − (t k − η(t k )) = t k +1 − t k ≤ h b , k ∈ N .

y defining h 2 � h b + η2 , one has

k +1 − (t k − η(t k )) = t k +1 − t k + η(t k ) ≤ h 2 , k ∈ N , (6)

here h 2 represents the maximal time lag between the instant t k −(t k ) at which the state information is updated, and the instant

k +1 at which the next sampling is triggered.

H. Ren, J. Xiong and R. Lu et al. / Neurocomputing 331 (2019) 346–355 349

x

S

x

w

i

[

t{

w

R

c

f

b

f

3

o

t

t

G

G

a

ξ

ξ

ξ

ξ

ξ

a

f

δ

δ

3

T

G

a

S

t

t

�

w

�

a

g

g

A

P

a

V

w

V

a

R

i

[

t

s P

Define

(t) =

⎡

⎢ ⎢ ⎣

x 1 (t) x 2 (t)

. . . x N (t)

⎤

⎥ ⎥ ⎦

, x (t k − η(t k )) =

⎡

⎢ ⎢ ⎣

x 1 (t k − η(t k )) x 2 (t k − η(t k ))

. . . x N (t k − η(t k ))

⎤

⎥ ⎥ ⎦

.

ubstituting (5) into (2) leads to

˙ (t) = (I N � A ) x (t) − α(L � BK) x (t k − η(t k )) , (7)

here t ∈ [ t k , t k +1 ) , k ∈ N . Inspired by Fridman et al. [24,37,38] , the

nput delay approach is utilized. Defining h (t) = t − t k + η(t k ) , t ∈ t k , t k +1 ) , it can be found from (6) that h 1 ≤ η(t k ) ≤ h (t) < t k +1 − k + η(t k ) ≤ h 2 , where h 1 = η1 . Then, we rewrite (7) as

˙ x (t) = (I N � A ) x (t) − α(L � BK) x (t − h (t)) , x (t) = φ(t) , −h 2 ≤ t ≤ 0 ,

(8)

here h 1 ≤ h ( t ) ≤ h 2 and h 12 � h 2 − h 1 .

emark 4. It can be observed that ˙ h (t) = 1 for t � = t k . However, no

onstraints are made on the derivative of the time-varying delay

unctions ˙ η(t) . Therefore, η( t ) are fast variant delays which the sta-

ility conditions do not rely on the derivative of the time delay

unctions ˙ η(t) [49] .

.1. Some useful notations

On the basis of Lemma 3 together with Lemma 4 or 5 , two the-

rems are given for the synchronization issue of system (8) with

ime variant delays. For the simplicity of expression, in this sec-

ion we utilize the following notations.

e i =

[0 nN×(i −1) nN I nN 0 nN×(14 −i ) nN

], i = 1 , . . . , 14 ,

w i =

[0 n ×(i −1) n I n 0 n ×(14 −i ) n

], i = 1 , . . . , 14 ,

� = (I N � A ) e 1 − α(L � BK) e 3 ,

i j = Aw 1 + αi j BKw 3 , αi j = α(l i j /� j ) ,

2 (χ ) =

[

χ1 − χ2

χ1 + χ2 − 2 χ5

χ1 − χ2 − 6 χ6

]

, G 3 (χ ) =

[

χ2 − χ3

χ2 + χ3 − 2 χ7

χ2 − χ3 − 6 χ8

]

,

4 (χ ) =

[

χ3 − χ4

χ3 + χ4 − 2 χ9

χ3 − χ4 − 6 χ10

]

, �(χ) =

[G 3 (χ ) G 4 (χ )

],

nd

0 (t) =

⎡

⎢ ⎣

x (t) x (t − h 1 )

x (t − h (t)) x (t − h 2 )

⎤

⎥ ⎦

, ξ1 (t) =

1

h 1

⎡

⎢ ⎣

∫ 0

−h 1

x (t + s ) ds ∫ 0

−h 1

δ1 (s ) x (t + s ) ds

⎤

⎥ ⎦

,

2 (t) =

1

h (t) − h 1

⎡

⎢ ⎣

∫ −h 1

−h (t) x (t + s ) ds ∫ −h 1

−h (t) δ2 (s ) x (t + s ) ds

⎤

⎥ ⎦

,

3 (t) =

1

h 2 − h (t)

⎡

⎢ ⎣

∫ −h (t)

−h 2

x (t + s ) ds ∫ −h (t)

−h 2

δ3 (s ) x (t + s ) ds

⎤

⎥ ⎦

,

4 (t) = (h (t) − h 1 ) ξ2 (t) , ξ5 (t) = (h 2 − h (t)) ξ3 (t) ,

6 (t) =

⎡

⎢ ⎣

∫ −h 1

−h 2

x (t + s ) ds

h 12

∫ −h 1

−h 2

δ4 (s ) x (t + s ) ds

⎤

⎥ ⎦

,

nd looking up the functions δa, b given in Lemma 3 , where the

unctions δi are shown as follows

1 (s ) = 2

s + h 1

h 1

− 1 , δ2 (s ) = 2

s + h (t)

h (t) − h 1

− 1 ,

3 (s ) = 2

s + h 2

h 2 − h (t) − 1 , δ4 (s ) = 2

s + h 2

h 12

− 1 .

.2. “Reciprocally convex” – based result

heorem 1. It is assumed that the oriented graph G is strong.

iven controller gain matrix K, time-varing delay 0 ≤η1 ≤η( t ) ≤η2 ,

nd upper bound of sampling intervals h b > 0, if there exist P ∈

5 n + , S 1 , S 2 , R 1 , R 2 ∈ S n + , M , N ∈ R

14 n ×2 n and a matrix X ∈ R

3 n ×3 n so

hat for any i, j = 1 , . . . , N, and any κ = 1 , 2 , the following inequali-

ies hold:

=

[˜ R 2 X

X

T ˜ R 2

]≤ 0 , �i j (h κ ) = �0 (h κ ) − �T (w )��(w ) < 0 ,

here h 1 = η1 , h 2 = h b + η2 , and for any θ ∈ R ,

0 (θ ) = He

(G

T 1 (θ ) P G 0 + M g 1 (θ ) + N g 2 (θ )

)+

ˆ S

+ T i j

(h

2 1 R 1 + h

2 12 R 2

)i j − G 2 (w ) T ˜ R 1 G 2 (w ) ,

G 1 (θ ) =

[w

T 1 h 1 w

T 5 h 1 w

T 6 w

T 11 + w

T 13

ˆ G

T 1 (θ )

]T ,

ˆ G 1 (θ ) = (h 2 − θ )(w 11 + w 14 ) + (θ − h 1 )(w 12 − w 13 ) ,

G 0 =

[T

i j w

T 1 −w

T 2 w

T 1 +w

T 2 −2 w

T 5 w

T 2 −w

T 4

ˆ G

T 0

]T ,

ˆ G 0 = h 12 (w 2 + w 4 ) − 2(w 11 + w 13 ) ,

ˆ S = diag (S 1 , −S 1 + S 2 , 0 n ×n , −S 2 , 0 10 n ×10 n ) ,

˜ R i = diag (R i , 3 R i , 5 R i ) ,

nd

1 (θ ) = (θ − h 1 )

[w 7

w 8

]−

[w 11

w 12

],

2 (θ ) = (h 2 − θ )

[w 9

w 10

]−

[w 13

w 14

].

fterwards, the network systems (2) can realize the synchronization.

roof. For system (8) , choose the Lyapunov-Krasovskii functional

s

(x (t) , ˙ x (t)

)= V P

(x (t)

)+ V S

(x (t)

)+ V R

(x (t) , ˙ x (t)

), (9)

here

V P

(x (t)

)=

� x T (t)(W � P ) � x (t) ,

V S

(x (t)

)=

∫ t

t−h 1

x T (s )(W � S 1 ) x (s ) ds

+

∫ t−h 1

t−h 2

x T (s )(W � S 2 ) x (s ) ds,

R

(x (t) , ˙ x (t)

)= h 1

∫ 0

−h 1

∫ t

t+ θ˙ x T (s )(W � R 1 ) x (s ) d sd θ

+ h 12

∫ −h 1

−h 2

∫ t

t+ θ˙ x T (s )(W � R 2 ) x (s ) d sd θ,

nd

� x (t) = col{ x (t) , h 1 ξ1 (t) , ξ6 (t) } . �

emark 5. The Lyapunov functionals V S and V R have been utilized

n [47] and [39] already. About the choice of V P , for example in

47] , the usual approach is to select a quadratic term relying on

he momentary system state x ( t ) merely. While it was demon-

trated in [39] that it is necessary to augment V for purpose of

350 H. Ren, J. Xiong and R. Lu et al. / Neurocomputing 331 (2019) 346–355

�

w

G

b

x

T

b

ξ

O

c

l

n

c{T

w

c

ξ

a

G

c

a

h

ϕ

w

fi

g

g

C

f

fully taking advantage of the Wirtinger-based inequality. The aug-

mented form is comprised of ∫ 0 −h 1

x (t + s ) ds and

∫ −h 1 −h 2

x (t + s ) ds,

which can be found apparently in the Wirtinger-based inequality.

Furthermore, considering the Bessel-Legendre inequality shown in

Lemma 3 , which is an augmented form of the Wirtinger-based

inequality, the term V P ought to contain two extra signals∫ 0 −h 1

δ1 (s ) x (t + s ) ds and

∫ −h 1 −h 2

δ4 (s ) x (t + s ) ds as well. Such state ex-

tension in V P was first put forward in [50] .

Along the trajectories of (8) , the target of the following deriva-

tion is to obtain an upper bound of the derivative of Lyapunov

functional V (x (t) , ˙ x (t)

). Therefore, an extended state vector is uti-

lized as

ϕ(t) = col

{

ξ0 (t ) , ξ1 (t ) , ξ2 (t ) , ξ3 (t ) , ξ4 (t ) , ξ5 (t ) }

.

To avoid complexity, the time variable is left out without any pos-

sible ambiguity. Specifically, it means that, for example, h and ϕrepresent h ( t ) and ϕ( t ) in the sequel, respectively.

For t ∈ [ t k , t k +1 ) , by finding the time derivative of functional

V P ( x ( t )) along the trajectory of (8) , we can see that

d

dt V P

(x (t)

)= 2

� x T (t)(W � P ) � x (t) .

Accordingly, it is needed to represent ˙ � x (t) and

� x (t) by the extended

vector ϕ. For one thing, it can be noticed that

˙ x (t) =

� ϕ.

Then, it can be found that

d

dt

∫ 0

−h 1

x (t + s ) d s =

d

d t

∫ t

t−h 1

x (u ) d u = x (t) − x (t − h 1 ) , (10)

is the first component of h 1 ˙ ξ1 (t) . Afterwards, adopting integration

by parts leads to

d

dt

∫ 0

−h 1

s · x (t + s ) ds = s · x (t + s ) | 0 −h 1 −

∫ 0

−h 1

x (t + s ) ds

= h 1 x (t − h 1 ) −∫ 0

−h 1

x (t + s ) ds.

Therefore, the second component of h 1 ˙ ξ1 (t) is given by

d

dt

∫ 0

−h 1

δ1 (s ) x (t + s ) ds

=

d

dt

∫ 0

−h 1

(2 s

h 1

+ 1

)x (t + s ) ds

=

2

h 1

· d

dt

∫ 0

−h 1

s · x (t + s ) ds +

d

dt

∫ 0

−h 1

x (t + s ) ds

= 2 x (t − h 1 ) − 2

h 1

∫ 0

−h 1

x (t + s ) ds + x (t) − x (t − h 1 )

= x (t) + x (t − h 1 ) − 2

h 1

∫ 0

−h 1

x (t + s ) ds. (11)

Consequently, from (10) and (11) , it can be concluded that

h 1 ˙ ξ1 (t) =

[e 1 − e 2

e 1 + e 2 − 2 e 5

]ϕ.

By using similar methods, it can be seen that

˙ ξ6 (t) =

[e 2 − e 4

ˆ G 0 (e )

]ϕ,

where ˆ G 0 (χ ) = h 12 (χ2 + χ4 ) − 2(χ11 + χ13 ) . Hence, it can be sum-

marized that

˙ x (t) =

� G 0 ϕ,

here

�

0 =

[� T e T 1 − e T 2 e T 1 + e T 2 − 2 e T 5 e T 2 − e T 4 ˆ G

T 0 (e )

]T .

For another, let us derive an expression of � x (t) relying on ϕ. To

egin with, it can be seen that

(t) = e 1 ϕ, h 1 ξ1 (t) = h 1

[e 5 e 6

]ϕ.

hen, to express ξ 6 ( t ) depending on the augmented state ϕ, it can

e noticed that

6 (t) =

⎡

⎢ ⎢ ⎣

(∫ −h 1

−h

+

∫ −h

−h 2

)x (t + s ) ds

h 12

(∫ −h 1

−h

+

∫ −h

−h 2

)δ4 (s ) x (t + s ) ds

⎤

⎥ ⎥ ⎦

. (12)

n one hand, from (12) , it can be seen that the first nN elements

an be represented as (e 11 + e 13 ) ϕ. On the other hand, for the

ast nN elements of ξ 6 ( t ), two dissimilar expressions of δ4 ( t ) are

eeded, which are based on δ2 ( t ) and δ3 ( t ), respectively. Then, it

an be found that

h 12 δ4 (s ) = (h − h 1 ) δ2 (s ) + (h 2 − h ) , h 12 δ4 (s ) = (h 2 − h ) δ3 (s ) − (h − h 1 ) .

herefore, it can be seen that

h 12

(∫ −h 1

−h

+

∫ −h

−h 2

)δ4 (s ) x (t + s ) ds

= (h − h 1 )

(∫ −h 1

−h

δ2 (s ) x (t + s ) ds −∫ −h

−h 2

x (t + s ) ds

)

+ (h 2 − h )

(∫ −h 1

−h

x (t + s ) ds +

∫ −h

−h 2

δ3 (s ) x (t + s ) ds

)=

ˆ G 1 (h, e ) ϕ,

here ˆ G 1 (θ, χ) = (h 2 − θ )(χ11 + χ14 ) + (θ − h 1 )(χ12 − χ13 ) . Ac-

ordingly, it can be concluded that

6 (t) =

[e 11 + e 13

ˆ G 1 (h, e )

]ϕ,

nd

� x (t) = G 1 (h, e ) ϕ, where

1 (θ, χ) =

[χ T

1 h 1 χT 5 h 1 χ

T 6 χ T

11 + χ T 13

ˆ G

T 1 (θ, χ)

]T .

Furthermore, recalling the structure of the extended state ϕ, it

an be noticed that the last four elements of ϕ can be regarded

s linear combination of the other elements. That is, ξ4 (t) = (h − 1 ) ξ2 (t) and ξ5 (t) = (h 2 − h (t )) ξ3 (t ) directly lead to

T He

((W � M ) g 1 (h, e ) + (W � N ) g 2 (h, e )

)ϕ = 0 ,

here matrices M , N ∈ R

14 n ×2 n and functions g 1 ( · ), g 2 ( · ) are de-

ned as

1 (θ, χ) = (θ − h 1 )

[χ7

χ8

]−

[χ11

χ12

],

2 (θ, χ) = (h 2 − θ )

[χ9

χ10

]−

[χ13

χ14

].

onsequently, the derivative of V P ( x ( t )) is given as

d

dt V P

(x (t)

)= ϕ

T He

(G

T 1 (h, e )(W � P ) � G 0 + (W � M ) g 1 (h, e )

+ (W � N ) g 2 (h, e ) )ϕ. (13)

By deriving the time derivative of functional V S ( x ( t )), it can be

ound that

d V S

(x (t)

)= ϕ

T � S ϕ, (14)

dt

H. Ren, J. Xiong and R. Lu et al. / Neurocomputing 331 (2019) 346–355 351

w

0

t

=

w

ϒ

ϒ

F

ϒ

ϒ

w

β

ϒ

w

�

C

F

r

w

�

I

i

e

h

V

b

=T

s

T

t

t

S

o

f

�

[w

G

a

g

g

T

w

P

P

w

G

F

S

N

here � S = diag (W � S 1 , W � (−S 1 + S 2 ) , 0 nN×nN , −W � S 2 ,

10 nN×10 nN ) .

To obtain the derivative of functional V R (x (t) , ˙ x (t)

), we can see

hat

d

dt V R

(x (t) , ˙ x (t)

)

˙ x T (t) (

h

2 1 (W � R 1 ) + h

2 12 (W � R 2 )

)˙ x (t) + ϒ1 + ϒ2 ,

here

1 = −h 1

∫ t

t−h 1

˙ x T (u )(W � R 1 ) x (u ) du,

2 = −h 12

(∫ t−h 1

t−h

+

∫ t−h

t−h 2

)˙ x T (u )(W � R 2 ) x (u ) du.

rom Lemma 3 , it can be found that

1 ≤ −ϕ

T G 2 (e ) T � R 1 G 2 (e ) ϕ,

2 ≤ − 1

βϕ

T G 3 (e ) T � R 2 G 3 (e ) ϕ − 1

1 − βϕ

T G 4 (e ) T � R 2 G 4 (e ) ϕ

= −ϕ

T �(e ) T

⎡

⎣

1

β� R 2 0

0

1

1 − β� R 2

⎤

⎦ �(e ) ϕ,

here � R i = diag (W � R i , 3 W � R i , 5 W � R i ) , i = 1 , 2 , and

=

h − h 1 h 12

. Next, it can be noticed from Lemma 4 that

2 ≤ −ϕ

T �(e ) T � ��(e ) ϕ,

here

� =

[� R 2

� X

� X

T � R 2

]≤ 0 .

onsequently, it can be summarized that

d

dt V R

(x (t) , ˙ x (t)

)≤ ϕ

T [ � T (

h

2 1 (W � R 1 ) + h

2 12 (W � R 2 )

)�

− G 2 (e ) T � R 1 G 2 (e ) − �(e ) T � ��(e ) ] ϕ. (15)

inally, putting the previous formulas (13) –(15) together, and

ecalling Lemmas 1 and 2 , then we have

d

dt V

(x (t) , ˙ x (t)

)≤ 1

2

N ∑

i =1

N ∑

j =1 , j � = i � i � j ϕ

T �i j ( h ) ϕ

here

˜ ϕ = ϕ i − ϕ j ,

i j (θ ) = �0 (θ ) − �T (w )��(w ) ,

�0 (θ ) = He

(G

T 1 (θ, w ) P G 0 + M g 1 (θ, w ) + N g 2 (θ, w )

)+

ˆ S

+ T i j

(h

2 1 R 1 + h

2 12 R 2

)i j − G 2 (w ) T ˜ R 1 G 2 (w ) ,

G 0 =

[T

i j w

T 1 −w

T 2 w

T 1 +w

T 2 −2 w

T 5 w

T 2 −w

T 4

ˆ G

T 0 (w )

]T ,

ˆ S = diag (S 1 , −S 1 + S 2 , 0 n ×n , −S 2 , 0 10 n ×10 n ) ,

˜ R i = diag (R i , 3 R i , 5 R i ) ,

� =

[˜ R 2 X

X

T ˜ R 2

]≤ 0 .

t can be noticed the fact that �ij ( h ) is affine respecting h ,

n order that �ij ( h ) is convex. In consequence, the two in-

qualities �ij ( h 1 ) < 0 and �ij ( h 2 ) < 0 indicate �ij ( h ) < 0 for all

∈ [ h 1 , h 2 ]. Afterwards, it can be found that d dt

V (x (t ) , ˙ x (t )

)< 0 and

(x (t) , ˙ x (t)

)≤ V

(x (0) , ˙ x (0)

), which indicates that V

(x (t) , ˙ x (t)

)is

ounded. Hence, � x T (t )(W � P ) � x (t ) is bounded as well and

� i � j λmin (P ) ‖

� x i (t) − �

x j (t) ‖

2

≤ 1

2

N ∑

i =1

N ∑

j =1 , j � = i � i � j

(� x i (t) − �

x j (t) )T

P

(� x i (t) − �

x j (t) )

� x T (t)(W � P ) � x (t) = O (e −�t ) .

herefore, based on Definition 1 , synchronization in network

ystems (2) suffering time variant delay η( t ) can be guaranteed.

heorem 2. It is assumed that the oriented graph G is strong. Given

ime-varing delay 0 ≤η1 ≤η( t ) ≤η2 , upper bound of sampling in-

ervals h b > 0 and parameters ϑg , (g = 1 , 2 , 3) , if there exist P 2 ∈

4 n + , S 1 , S 2 ∈ S n + , M , N ∈ R

14 n ×2 n , X ∈ R

3 n ×3 n , U ∈ R

n ×n and a diag-

nal matrix J ∈ S n + so that ∀ i, j = 1 , 2 , . . . , N and any κ = 1 , 2 , the

ollowing inequalities hold:

¯ =

[R 2 X

X

T R 2

]≤ 0 , (16)

�1 + �2 (h κ ) Z T

Z −δ J

]< 0 , (17)

here h 1 = η1 , h 2 = h b + η2 , and for any θ ∈ R ,

�1 = He

(w

T 1 δ1 (A J w 1 + αi j BUw 3 )

),

�2 (θ ) = He

(G

T 12 (θ ) P 2 G 02 + M g 1 (θ ) + N g 2 (θ )

)+ S

− G

T 2 (w ) R 1 G 2 (w ) − �T (w ) ��(w ) ,

12 (θ ) =

[h 1 w

T 5 h 1 w

T 6 w

T 11 + w

T 13

ˆ G

T 1 (θ )

]T ,

ˆ G 1 (θ ) = (h 2 − θ )(w 11 + w 14 ) + (θ − h 1 )(w 12 − w 13 ) ,

G 02 =

[w

T 1 −w

T 2 w

T 1 +w

T 2 −2 w

T 5 w

T 2 −w

T 4

ˆ G

T 0

]T ,

ˆ G 0 = h 12 (w 2 + w 4 ) − 2(w 11 + w 13 ) ,

S = diag ( S 1 , −S 1 + S 2 , 0 n ×n , −S 2 , 0 10 n ×10 n ) ,

R i = δi +1 diag ( J , 3 J , 5 J ) ,

Z = δ(A J w 1 + αi j BUw 3 ) ,

δ = h

2 1 δ2 + h

2 12 δ3 ,

nd

1 (θ ) = (θ − h 1 )

[w 7

w 8

]−

[w 11

w 12

],

2 (θ ) = (h 2 − θ )

[w 9

w 10

]−

[w 13

w 14

].

hen, the network systems (2) can accomplish the synchronization

ith the control gain K = U J −1 in (5) .

roof. Based on Theorem 1 , we choose P = diag (P 1 , P 2 ) , where

1 ∈ S n + and P 2 ∈ S

4 n + . By selecting P 1 = δ1 J, R 1 = δ2 J and R 2 = δ3 J,

here J ∈ S n + is a diagonal matrix, it can be seen that

T 1 (θ ) P G 0 =

[w

T 1 G

T 12 (θ )

][P 1 0

0 P 2

][

i j

G 02

]

,

= w

T 1 δ1 J(Aw 1 + αi j BKw 3 ) + G

T 12 (θ ) P 2 G 02 .

urthermore, we set

J = J −1 , U = K J , P 2 = diag 4 ( J ) P 2 diag

4 ( J ) , S 1 = J S 1 J ,

¯ 2 = J S 2 J , X = J X J , M = diag

14 ( J ) M diag

2 ( J ) ,

¯ = diag

14 ( J ) N diag

2 ( J ) .

352 H. Ren, J. Xiong and R. Lu et al. / Neurocomputing 331 (2019) 346–355

�

A[f

p

t

g

R

a

a

R

T

Y

i

a

T

w

t

T

q

t

v

T

v

h

S

d

1⎡⎣w

�

a

i

t

t

Pre- and post-multiplying � by diag 6 ( J ) directly leads to (16) . Pre-

and post-multiplying �ij ( h κ ) by diag 14 ( J ) and applying Schur com-

plement, it can be found that (17) holds. �

3.3. Another synchronization result

In the last section, Theorem 1 is obtained based on

Lemmas 3 and 4 . By substituting Lemma 5 for Lemma 4 , an analo-

gous analysis of synchronization can be accomplished alternatively,

which results in the following conditions.

Theorem 3. Assume that the oriented graph G is strong. Given time-

varing delay 0 ≤η1 ≤η( t ) ≤η2 , controller gain K and upper bound of

sampling intervals h b > 0, if there exist P ∈ S 5 n + , S 1 , S 2 , R 1 , R 2 ∈ S

n + ,M , N ∈ R

14 n ×2 n and two matrices Y 1 , Y 2 ∈ R

14 n ×3 n so that ∀ i, j =1 , . . . , N and any κ = 1 , 2 , λ = 1 , 2 , the following inequality holds: [

�0 (h κ ) − He

(Y 2 G 3 (w ) + Y 1 G 4 (w )

)Y λ

Y T λ

− ˜ R 2

]

< 0 ,

where h 1 = η1 , h 2 = h b + η2 , and �0 , ˜ R 2 are given in Theorem 1 .

Accordingly, the network systems (2) can realize the synchronization.

Proof. Some proof steps are ignored due to the fact that the proof

process is similar to the technique in Theorem 1 . Thereinto, the

major difference consists in the utilization of Lemma 4 superseded

by Lemma 5 . From Lemma 4 , it can be found that

ϒ2 ≤ −ϕ

T �(e ) T

⎡

⎣

1

β� R 2 0

0

1

1 − β� R 2

⎤

⎦ �(e ) ϕ

≤ −ϕ

T �(e ) T � �M

(β)�(e ) ϕ,

where

� �M

(β) = He ([

� X 1

� X 2

])− β �

X 1 � R

−1 2

� X

T 1 − (1 − β) � X 2

� R

−1 2

� X

T 2 .

Consequently, we conclude that

d

dt V R

(x (t) , ˙ x (t)

)≤ ϕ

T [ � T

(h

2 1 (W � R 1 ) + h

2 12 (W � R 2 )

)�

− G 2 (e ) T � R 1 G 2 (e ) − �(e ) T � �M

(β)�(e ) ] ϕ.

(18)

Finally, putting the previous formulas (13) –(18) together, and re-

calling Lemmas 1 and 2 , then we have

d

dt V

(x (t) , ˙ x (t)

)≤ 1

2

N ∑

i =1

N ∑

j =1 , j � = i � i � j ϕ

T �i j ( h ) ϕ

where

�i j (θ ) = �0 (θ ) − �T (w )�M

(β)�(w ) ,

�M

(β) = He ([

X 1 X 2

])− βX 1

R

−1 2 X

T 1 − (1 − β) X 2

R

−1 2 X

T 2 .

Then, denoting �( w ) by �, it can be seen that

�i j (h ) = �0 (h ) − He (�T

[X 1 X 2

]�)

+ β�T X 1 R

−1 2 X

T 1 �

+(1 − β)�T X 2 R

−1 2 X

T 2 �.

Denoting �T X 1 = Y 2 and �T X 2 = Y 1 leads to

�i j (h ) = � + βY 2 R

−1 2 Y T 2 + (1 − β) Y 1 R

−1 2 Y T 1

= β(� + Y 2 R

−1 2 Y T 2 ) + (1 − β)(� + Y 1 R

−1 2 Y T 1 ) ,

where

= �0 (h ) − He

([Y 2 Y 1

][ G 3 (w ) G 4 (w )

]).

pplying Schur complement, it can be obtained that

� Y λY T λ

− ˜ R 2

]< 0 ,

or λ = 1 , 2 , can lead to �ij ( h ) < 0. Therefore, referring to the

roof in Theorem 1 and according to Definition 1 , synchroniza-

ion in network systems (2) with time-varying delay η( t ) can be

uaranteed. �

emark 6. To compare the conservatism of Theorems 1 and 3 , we

ssume that the conditions given in Theorem 1 are satisfied. By

pplying Schur complement, it can be noticed that � ≥ 0 leads to

˜ 2 − X

R

−1 2 X

T ≥ 0 , ˜ R 2 − X

T ˜ R

−1 2 X ≥ 0 .

hen, by choosing

T 1 =

˜ R 2 G 4 (w ) + X

T G 3 (w ) , Y T 2 =

˜ R 2 G 3 (w ) + X G 4 (w ) ,

t can be found that

�0 (h κ ) − He

(Y 2 G 3 (w ) + Y 1 G 4 (w )

)+ Y 1 R

−1 2 Y T 1

= �0 (h κ ) − �T (w )

(� +

[˜ R 2 − X

R

−1 2

X

T 0

0 0

])�(w )

≤ �0 (h κ ) − �T (w )��(w ) ,

nd

�0 (h κ ) − He

(Y 2 G 3 (w ) + Y 1 G 4 (w )

)+ Y 2 R

−1 2 Y T 2

= �0 (h κ ) − �T (w )

(� +

[0 0

0

˜ R 2 − X

T ˜ R

−1 2

X

])�(w )

≤ �0 (h κ ) − �T (w )��(w ) .

herefore, the comparison demonstrates that Theorem 3 al-

ays brings about better (or at least the same) outcomes

han Theorem 1 . Generally, the underlying improvement of

heorem 3 over Theorem 1 , at the cost of a visible growth of the

uantity of decision variables, is the revealment of a tradeoff be-

ween the numerical complexity and the reduction of the conser-

atism.

heorem 4. Assume that the oriented graph G is strong. Given time-

aring delay 0 ≤η1 ≤η( t ) ≤η2 , upper bound of sampling intervals

b > 0 and parameters ϑg , (g = 1 , 2 , 3) , if there exist matrices P 2 ∈

4 n + , S 1 , S 2 ∈ S n + , M , N ∈ R

14 n ×2 n , Y 1 , Y 2 ∈ R

14 n ×3 n , U ∈ R

n ×n and a

iagonal matrix J ∈ S n + such that for any i, j = 1 , . . . , N and any κ =

, 2 , λ = 1 , 2 , the following inequality holds:

�1 + �3 (h κ ) Z T Y λZ −δ J 0

Y T λ

0 −R 2

⎤

⎦ < 0 ,

here h 1 = η1 , h 2 = h b + η2 , and for any θ ∈ R ,

¯3 (θ ) = He

(G

T 12 (θ ) P 2 G 02 + M g 1 (θ ) + N g 2 (θ )

)+ S

− G

T 2 (w ) R 1 G 2 (w ) − He

(Y 2 G 3 (w ) + Y 1 G 4 (w )

),

nd other variables or matrices can be found in Theorem 2 . Accord-

ngly, the network systems (2) can realize the synchronization with

he control gain matrix K = U J −1 in (5) .

Proof: Referring to the proof in Theorem 2 , we can find the way

o prove Theorem 4 in like manner.

H. Ren, J. Xiong and R. Lu et al. / Neurocomputing 331 (2019) 346–355 353

Fig. 1. Topology graph G of the network systems.

0 5 10 15 20 25 30 35-2

0

2

0 5 10 15 20 25 30 35-2

0

2

Fig. 2. State trajectories of the network systems.

4

t

a

t

a

A

.

T

G

l

[

b

A

w

x

x

p

s

0

A

s

K

s

t

F

a

F

t

. Numerical example

In Section 4 , a numerical simulation is provided to demonstrate

he availability of the designed approach. The topology graph G

mong the network systems is provided in Fig. 1 . The adjacent ma-

rix A and the Laplacian matrix L of the network systems are given

0 5 10 150

0.2

0.4

0.6

0.8

1

1.2

sam

plin

g in

terv

als

Fig. 3. Aperiodic sampling interv

s

=

⎡

⎢ ⎢ ⎣

0 1 0 0 1

1 0 1 0 0

0 1 0 1 0

0 0 1 0 1

1 0 0 1 0

⎤

⎥ ⎥ ⎦

, L =

⎡

⎢ ⎢ ⎣

2 −1 0 0 −1

−1 2 −1 0 0

0 −1 2 −1 0

0 0 −1 2 −1

−1 0 0 −1 2

⎤⎥⎥⎦

herefore, it can be found that the directed communication graph

of network systems is strong. Correspondingly, the normalized

eft eigenvector to the eigenvalue 0 can be calculated as � =1 / 5 1 / 5 1 / 5 1 / 5 1 / 5] T .

The dynamic characteristics of the i th system can be described

y Eq. (2) corresponding to the following parameter matrices,

=

[0 −1 . 7458

3 . 5539 0

], B =

[−2 . 0458 −1 . 3388

−1 . 5095 1 . 3022

],

here x i (t) =

[x i 1 (t) x i 2 (t)

]T , i = 1 , 2 , 3 , 4 , 5 .

The initial states of the network systems are given as follows:

1 (0) =

[−1 . 5

0 . 3

], x 2 (0) =

[0 . 8

1 . 5

], x 3 (0) =

[−0 . 5

0 . 8

],

4 (0) =

[1 . 5

−0 . 8

], x 5 (0) =

[−0 . 8

−1 . 5

].

Recalling the framework of the sampling controller (5) , we sup-

ose that the coupling strength is α = 0 . 5 . The time-varying delays

atisfy 0.1 ≤η( t ) ≤ 0.2, and upper bound of sampling lags is h b = . 5 . Therefore, it can be seen that h 1 = 0 . 1 and h 2 = h b + η2 = 0 . 7 .

ccording to Theorem 2 , we can find the gain K of the proposed

ampling controller by

=

[−0 . 3391 0 . 1329

−0 . 1630 0 . 4565

].

Filling the gain matrix K and time-varying delays η( t k ) into the

ampling controller (5) , the state trajectories of the network sys-

ems under such sampled-data control law are demonstrated in

ig. 2 which signifies that the network systems can eventually re-

lize synchronization.

The corresponding aperiodic sampling intervals are shown in

ig. 3 . The x -axis value of each stem stands for a sampling time

k , k = 0 , 1 , . . . . The height of every stem represents the time span

20 25 30 35

als with upper bound h b .

354 H. Ren, J. Xiong and R. Lu et al. / Neurocomputing 331 (2019) 346–355

0 5 10 15 20 25 30 35-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

Fig. 4. Sampled-data control signal u i ( t ) for system 1.

0 5 10 15 20 25 30 350

0.5

1

1.5

2

2.5

3

Fig. 5. Synchronization error ε( t ) attenuate exponentially to 0.

0 5 10 15 20 25 30 350

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Fig. 6. Norm of the state trajectories ‖ x i ( t ) ‖ of the network systems.

5

t

a

o

t

S

t

a

e

c

c

c

b

l

i

A

e

R

t

C

P

G

(

R

of the aperiodic sampling interval t k +1 − t k related to the maxi-

mum sampling interval h b . From Fig. 3 , we can observe that the

sampling controller needs less data transmission. Therefore, com-

munication burden and energy consumption can be reduced by

this way intuitively.

In Fig. 4 , the sampling control signals u i ( t ) for (2) are given.

From (5) , it can be found that u i (t) , t ∈ [ t k , t k +1 ) use the informa-

tion of the states x i (t k − η(t k )) with time-varying delays η( t k ). Fur-

thermore, the signals u i (t) , t ∈ [ t k , t k +1 ) keep a constant during the

sampling intervals [ t k , t k +1 ) due to the ZOH.

For the purpose of demonstrating the availability of our de-

signed algorithm further, the synchronization error of each sys-

tem is defined as ε(t) =

√ ∑ 5 j=2 ‖ x j (t) − x 1 (t) ‖ . The synchroniza-

tion error ε( t ) is illustrated in Fig. 5 with a fast convergence rate

which can be seen that the synchronization of network systems is

achieved within a short time.

The responses of the norm of the state trajectories ‖ x i ( t ) ‖ of the

network systems are presented in Fig. 6 , from which we can ob-

serve that the network systems achieve the synchronization with

the sampled-data controller which suffers time-varying delays.

. Conclusions

In the paper, utilizing aperiodic sampled data control approach,

he synchronization issue of network systems suffering time vari-

nt delays has been analyzed. Adopting input delay method, the

riginal sampled data systems have been rebuilt as continuous sys-

ems with novel time variant delay terms in the control signals.

o as to derive the synchronization conditions of network sys-

ems, the Bessel–Legendre inequality has been adopted. The char-

cteristics of this inequality have been adequately adapted to the

stablishment of extended Lyapunov functionals. Two novel suffi-

ient conditions for synchronizability of network systems and the

orresponding controller design methods have been presented. By

ontrast, it has been identified that the latter sufficient condition

rings better (or at least the same) outcomes than the former. At

ast, a numerical simulation has been shown to verify the availabil-

ty and advantage of the designed approach.

cknowledgment

This work was partially supported by the National Natural Sci-

nce Foundation of China ( U1611262 , 61773357 ), the Innovative

esearch Team Program of Guangdong Province Science Founda-

ion (2018B030312006), the Fundamental Research Funds for the

entral Universities (2017FZA5010), the Science and Technology

lanning Project of Guangdong Province (2017B010116006), the

uangdong Natural Science Funds for Distinguished Young Scholar

2018B030306013).

eferences

[1] H. Dong , Z. Wang , S.X. Ding , H. Gao , On H-infinity estimation of randomly oc-curring faults for a class of nonlinear time-varying systems with fading chan-

nels, IEEE Trans. Autom. Control 61 (2) (2016) 479–484 .

[2] R. Lu , P. Shi , H. Su , Z.G. Wu , J. Lu , Synchronization of general chaotic neuralnetworks with nonuniform sampling and packet missing: A switched system

approach, IEEE Trans. Neural Networks Learn. Syst. 29 (3) (2016) 523–533 . [3] H. Dong , H. Nan , Z. Wang , W. Ren , Variance-constrained state estimation for

complex networks with randomly varying topologies, IEEE Trans. Neural Net-works Learn. Syst. 29 (7) (2017) 2757–2768 .

[4] F. Yang , H. Dong , Z. Wang , W. Ren , F.E. Alsaadi , A new approach to non-fragilestate estimation for continuous neural networks with time-delays, Neurocom-

puting 197 (4) (2016) 205–211 .

[5] C. Wen , F. Liu , Observer-based consensus of second-order multi-agent systemswithout velocity measurements, Neurocomputing 179 (2016) 298–306 .

[6] G. Wen , W. Yu , J. Wang , D. Xu , J. Cao , Distributed node-to-node consensus ofmulti-agent systems with time-varying pinning links, Neurocomputing 149 (1)

(2015) 1387–1395 .

H. Ren, J. Xiong and R. Lu et al. / Neurocomputing 331 (2019) 346–355 355

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[7] H.X. Hu , W. Yu , Q. Xuan , L. Yu , G. Xie , Consensus of multi-agent systems inthe cooperation-competition network with inherent nonlinear dynamics, Neu-

rocomputing 158 (2015) 134–143 . [8] L. Ma , Z. Wang , H.-K. Lam , Event-triggered mean-square consensus control

for time-varying stochastic multi-agent system with sensor saturations, IEEETrans. Autom. Control 62 (7) (2017) 3524–3531 .

[9] D. Viegas , P. Batista , P. Oliveira , C. Silvestre , C.L.P. Chen , Distributed state esti-mation for linear multi-agent systems with time-varying measurement topol-

ogy, Automatica 54 (2015) 72–79 .

[10] W. Hu , L. Liu , G. Feng , Consensus of linear multi-agent systems by distributedevent-triggered strategy, IEEE Trans. Cybern 46 (1) (2017) 148–157 .

[11] Z. Lin , L. Wang , Z. Han , M. Fu , Distributed formation control of multi-agentsystems using complex laplacian, IEEE Trans. Autom. Control 59 (7) (2014)

1765–1777 . [12] S. Yang , J.X. Xu , D. Huang , Y. Tan , Optimal iterative learning control design

for multi-agent systems consensus tracking, Syst. Control Lett. 69 (2) (2014)

80–89 . [13] X. Jin , Adaptive iterative learning control for high-order nonlinear multi-agent

systems consensus tracking, Syst. Control Lett. 89 (2016) 16–23 . [14] Z. Li , G. Wen , Z. Duan , W. Ren , Designing fully distributed consensus proto-

cols for linear multi-agent systems with directed graphs, IEEE Trans. Autom.Control 60 (4) (2015) 1152–1157 .

[15] P. Shi , Filtering on sampled-data systems with parametric uncertainty, IEEE

Trans. Autom. Control 43 (7) (1998) 1022–1027 . [16] H. Gao , J. Wu , P. Shi , Robust sampled-data H ∞ control with stochastic sam-

pling, Automatica 45 (7) (2009) 1729–1736 . [17] X. Meng , T. Chen , Optimal sampling and performance comparison of periodic

and event based impulse control, IEEE Trans. Autom. Control 57 (12) (2012)3252–3259 .

[18] S. Liu , T. Li , L. Xie , M. Fu , J.F. Zhang , Continuous-time and sampled-data-based

average consensus with logarithmic quantizers, Automatica 49 (11) (2013)3329–3336 .

[19] Z.W. Liu , Z.H. Guan , X. Shen , G. Feng , Consensus of multi-agent net-works with aperiodic sampled communication via impulsive algorithms us-

ing position-only measurements, IEEE Trans. Autom. Control 57 (10) (2012)2639–2643 .

20] M. Guo , D.V. Dimarogonas , Nonlinear consensus via continuous, sampled, and

aperiodic updates, Int. J. Control 86 (4) (2013) 567–578 . [21] Y. Oishi , H. Fujioka , Stability and stabilization of aperiodic sampled-data con-

trol systems using robust linear matrix inequalities, Automatica 46 (8) (2010)1327–1333 .

22] Y. Wu , H. Su , P. Shi , Z. Shu , Z.G. Wu , Consensus of multiagent systems usingaperiodic sampled-data control, IEEE Trans. Cybern. 46 (9) (2016) 2132–2143 .

23] Y.S. Suh , Stability and stabilization of nonuniform sampling systems, Automat-

ica 44 (12) (2008) 3222–3226 . [24] E. Fridman , A refined input delay approach to sampled-data control, Automat-

ica 46 (2) (2010) 421–427 . 25] A. Seuret , A novel stability analysis of linear systems under asynchronous sam-

plings, Automatica 48 (1) (2012) 177–182 . 26] P. Naghshtabrizi , J.P. Hespanha , A.R. Teel , Exponential stability of impulsive sys-

tems with application to uncertain sampled-data systems, Syst. Control Lett. 57(5) (2008) 378–385 .

[27] K. Liu , E. Fridman , Wirtinger inequality and Lyapunov-based sampled-data sta-

bilization, Automatica 48 (1) (2012) 102–108 . 28] H. Ma, H. Liang, Q. Zhou, C.K. Ahn, Adaptive dynamic surface control design for

uncertain nonlinear strict-feedback systems with unknown control directionand disturbances, IEEE Trans. Syst. Man Cybern. Syst. (2018), doi: 10.1109/TSMC.

2018.2855170 . 29] Y. Zhang , Y.P. Tian , Consensus of data-sampled multi-agent systems with ran-

dom communication delay and packet loss, IEEE Trans. Autom. Control 55 (4)

(2010) 939–943 . 30] L. Ma , Z. Wang , Y. Liu , F.E. Alsaadi , Exponential stabilization of nonlinear

switched systems with distributed time-delay: An average dwell time ap-proach, Eur. J. Control 37 (2017) 34–42 .

[31] H. Ma, Q. Zhou, L. Bai, H. Liang, Observer-based adaptive fuzzy fault-tolerantcontrol for stochastic nonstrict-feedback nonlinear systems with input quan-

tization, IEEE Trans. Syst. Man Cybern. Syst. (2018), doi: 10.1109/TSMC.2018.

2833872 . 32] L. Ma , Z. Wang , Y. Liu , F.E. Alsaadi , A note on guaranteed cost control for non-

linear stochastic systems with input saturation and mixed time-delays, Int. J.Robust Nonlinear Control 27 (18) (2017) 4 4 43–4 456 .

[33] H. Ma, H. Liang, H. Ma, Q. Zhou, Nussbaum gain adaptive backstepping controlof nonlinear strict-feedback systems with unmodeled dynamics and unknown

dead zone, Int. J. Robust Nonlinear Control (2018), doi: 10.1002/rnc.4315 .

34] Y. Yan , J. Huang , Cooperative output regulation of discrete-time linear time-de-lay multi-agent systems under switching network, Neurocomputing 241 (2017)

108–114 . [35] Z. Ma , Z. Liu , Z. Chen , Modified leader-following consensus of time-delay mul-

ti-agent systems via sampled control and smart leader, Int. Journal Control Au-tom. Syst. 15 (6) (2017) 2526–2537 .

36] P. Lin , Y. Jia , Consensus of a class of second-order multi-agent systems with

time-delay and jointly-connected topologies, IEEE Trans. Autom. Control 55 (3)(2010) 778–784 .

[37] K. Liu , E. Fridman , L. Hetel , Stability and L 2 -gain analysis of networked controlsystems under round-robin scheduling: a time-delay approach, Syst. Control

Lett. 61 (5) (2012) 666–675 .

38] K. Liu , E. Fridman , Delay-dependent methods and the first delay interval, Syst.Control Lett. 64 (1) (2014) 57–63 .

39] A. Seuret , F. Gouaisbaut , Wirtinger-based integral inequality: Application totime-delay systems, Automatica 49 (9) (2013) 2860–2866 .

40] H.B. Zeng , Y. He , M. Wu , J. She , Free-matrix-based integral inequality for sta-bility analysis of systems with time-varying delay, IEEE Trans. Autom. Control

60 (10) (2015) 2768–2772 . [41] A. Seuret , F. Gouaisbaut , Hierarchy of LMI conditions for the stability analysis

of time-delay systems, Syst. Control Lett. 81 (2015) 1–7 .

42] W. Ren , R.W. Beard , Consensus seeking in multiagent systems under dynami-cally changing interaction topologies, IEEE Trans. Autom. Control 50 (5) (2005)

655–661 . 43] Q. Ma , J. Lu , H. Xu , Consensus for nonlinear multi-agent systems with sampled

data, Trans. Inst. Meas. Control 36 (5) (2014) 618–626 . 44] J. Lu , D.W. Ho , Globally exponential synchronization and synchronizability for

general dynamical networks, IEEE Trans. Syst. Man Cybern. Part B Cybern. 40

(2) (2010) 350–361 . 45] H.B. Zeng , Y. He , M. Wu , J. She , New results on stability analysis for systems

with discrete distributed delay, Automatica 60 (2015) 189–192 . 46] P.G. Park , W.I. Lee , S.Y. Lee , Auxiliary function-based integral inequalities for

quadratic functions and their applications to time-delay systems, J. Frankl. Inst.352 (4) (2015) 1378–1396 .

[47] P.G. Park , J.W. Ko , C. Jeong , Reciprocally convex approach to stability of systems

with time-varying delays, Automatica 47 (1) (2011) 235–238 . 48] Y.S. Moon , P. Park , W.H. Kwon , Y.S. Lee , Delay-dependent robust stabilization

of uncertain state-delayed systems, Int. J. Control 74 (14) (2001) 1447–1455 . 49] E. Fridman , U. Shaked , Delay-dependent stability and H ∞ control: Constant and

time-varying delays, Int. J. Control 76 (1) (2003) 48–60 . 50] K. Liu , A. Seuret , Y. Xia , Stability analysis of systems with time-varying de-

lays via the second-order Bessel-Legendre inequality, Automatica 76 (2016)

138–142 .

Hongru Ren received the B.E. in Automation from Uni-

versity of Science and Technology of China, Departmentof Automation, Hefei, Anhui, China in 2013, and he is

currently pursuing the Ph.D. degree also in Department

of Automation, University of Science and Technology ofChina, Hefei, Anhui, China. His research interests include

networked control systems and Kalman filtering.

Junlin Xiong received his B.Eng. and M.Sc. degrees from

Northeastern University, China, and his Ph.D. degree fromthe University of Hong Kong, Hong Kong, in 20 0 0, 20 03

and 2007, respectively. From November 2007 to Febru-

ary 2010, he was a research associate at the Univer-sity of New South Wales at the Australian Defence Force

Academy, Australia. In March 2010, he joined the Univer-sity of Science and Technology of China where he is cur-

rently a professor in the Department of Automation. Cur-rently, he is an associate editor for the IET Control Theory

and Application. His current research interests are in the

fields of negative imaginary systems, large-scale systemsand networked control systems.

Renquan Lu received his Ph.D. degree in Control Scienceand Engineering from Zhejiang University, Hangzhou,

China, in 2004. He was supported by the National Sci-ence Fund for Distinguished Young Scientists of China

in 2014, honored as the Distinguished Professor of Pearl

River Scholars Program of Guangdong Province, the Dis-tinguished Professor of Yangtze River Scholars Program

by the Ministry of Education of China in 2015 and 2017,respectively. Currently, he is a professor of the School

of Automation at Guangdong University of Technology,Guangzhou, China. His research interests include complex

systems, networked control systems, and nonlinear sys-

tems.

Yuanqing Wu received the Ph.D. degree with the Depart-

ment of Control Science and Engineering, Zhejiang Uni-versity, Hangzhou, China, in 2016. He is currently with

the Intelligent Information Processing Laboratory, Guang-

dong University of Technology, Guangzhou, China. From2014 to 2015, he was a Research Assistant with the School

of Electrical and Electronic Engineering, Nanyang Tech-nology University, Singapore. His current research inter-

ests include multiagent systems, large-scalar network sys-tems, output regulation theory, sampled-data control, and

event-triggered control.