Research Article Control of Synchronization and Stability ...downloads.hindawi.com/journals/mpe/2015/271759.pdfBy Lyapunov stability theorem, the decentralized controllers with similar

Research ArticleControl of Synchronization and Stability for NonlinearComplex Dynamical Networks with Different DimensionalSimilar Nodes and Coupling Time-Varying Delay

Luo Yi-ping Luo Xin Deng Fei and Hu Jun-qiang

Hunan Institute of Engineering Xiangtan Hunan 411104 China

Correspondence should be addressed to Luo Xin 97646021qqcom

Received 10 January 2015 Revised 25 March 2015 Accepted 30 March 2015

Academic Editor Xinggang Yan

Copyright copy 2015 Luo Yi-ping et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper discusses the stability and synchronization for the nonlinear coupled complex networks with different dimensionalnodes and the external coupling satisfies the condition of dissipationThe definition of synchronization of the complex dynamicalnetworks is proposed as the manifold By Lyapunov stability theorem the decentralized controllers with similar parameters aredesigned to synchronize such dynamical networks asymptotically in which the characteristics are variable delayed Finally anumerical example is given to illustrate the effectiveness of the designed method

1 Introduction

Extensively existing in various phenomena of all kinds ofareas in the world such as social network and World WideWeb (WWW) complex dynamical networks have receivedmore and more attention in recent years [1 2] A lotof researchers have analyzed coupling complex dynamicalnetworks and got abundant results [3ndash7] Representativephenomenon in complex dynamical networks is synchro-nization among all dynamical nodes So in the past few yearssynchronization is the interesting subject for researchers [8ndash10] As a result widespread and varied criteria for stabilityand synchronization in dynamical network have been derived[11ndash14] For instance [13] discussed the adaptive pinningsynchronization in complex networks with nondelay andvariable delay coupling Reference [14] designed controllerswith synchronization conditions to achieve the synchroniza-tion of nonlinear coupled dynamic complex networks withunanimous delay Anyway the existing works on this topicnormally focus on the dynamical complex networkwith samedimensions of nodes

However attributes of individual nodemight be differentSuch as the Super Smart Grid every user as a node possessessimilar but discrepant equipment and every kind of equip-ment of user is one dimension of node By using different

dimensions of node to express different attributes of user wecan define the Super Smart Grid as complex networks withdifferent dimensional similar nodes

Though [15] mentioned that one synchronization schemeis applicable to the complex networks in which the nodescontain different dimensions both the nodes and couplingof nodes are linear In reality of engineering the nonlinearcoupling might be more complicated than the linear systemRecently [16] also researched coupled complex dynamicalnetworks with different dimensions nodes which discussedthe asymptotic synchronization of this network by decentral-ized dynamical compensation controllers without mention-ing time delay As we all know time delay widely exists in var-ious phenomena of nature engineering networks biologicalsystem and human social activities Generally speaking timedelay is inevitable because the information spread througha complex network that is characterized by limited speed oflong distance signal transmission Furthermore in some realsituations delay is varied which is called time-varying delay

All of the above discussions are to construct the nonlinearcomplex dynamical networks with different dimensionalsimilar nodes and coupling time-varying delay In this paperwe assumed that the dimension of individual node in thiscoupled complex dynamical network is different Since nodeshave some similar behaviors and state connections we define

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 271759 6 pageshttpdxdoiorg1011552015271759

2 Mathematical Problems in Engineering

the synchronization of the dynamical networks as the man-ifold Considering the time-varying delay in this networkthe decentralized control strategies are designed to achievethe stability and synchronization asymptotically for similarnodes of complex dynamical networks In the end numericalexamples are given to demonstrate the effectiveness of ourproposed results Finally Section 5 presents conclusions

2 Formulation of the Problemand Preliminaries

Here we consider a complex dynamical network which hassimilar nodes with different dimensions and coupling time-varying delay

119894= 119864

119894119909

119894

+

119873

sum

119895=1119895 =119894

119888

119894119895(ℎ

119894119895(119909

119895 (119905 minus 120591 (119905))) minus ℎ

119894119894(119909

119894 (119905 minus 120591 (119905))))

+ 119865

119894119906

119894

(1)

where the 119894th node satisfies 119894 = 1 2 119873 119909

119894isin 119877

119899119894 and119909

119894= (119909

1198941 119909

1198942 119909

119894119899119894)

119879 and 119906

119894isin 119877

119898times119899119894 is the control inputof node 119894 Respectively 119864

119894isin 119877

119899119894times119899119894 119865

119894isin 119877

119899119894times119898 both areknown as constant matrices The sufficiently smooth ℎ

119894119895are

the nonlinear vector fields defined as 119877

119899119895rarr 119877

119899119894(119894 119895 =

1 2 119873) 119888

119894119895isin 119877 are the outer coupling configuration

parameters 119888

119894119895shows the coupling strength and topology of

the dynamical network and satisfies 119888

119894119894= minus sum

119873

119895=1119895 =119894119888

119894119895 And

119888

119894119895= 0 if there is a connection between node 119894 and node

119895 (119894 = 119895) otherwise 119888

119894119895= 0 The diagonal elements of matrix

are defined as 119862 = (119888

119894119895)

119873times119873

Assumption 1 Consider network (1) of 119873 different dimen-sional nodes There exist 119873 matrixes 119878

119894isin 119877

1198991times119899119894(119878

119894= 0)

119861

119894isin 119877

119898times119899119894 and 119906

119894= 119861

119894119909

119894+ 119860

119894V119894conforming to

119878

119894(119864

119894+ 119865

119894119861

119894) = (119864

1+ 119865

1119860

1) 119878

119894

119878

119894119865

119894= 119865

1

(2)

where 119894 = 1 119873

Assumption 2 Under Assumption 1 conditions there exist119867

119894isin 119877

1198991times1198991 and 119894 = 1 119873 satisfying

119878

119894ℎ

119894119895(119909

119894 (119905 minus 120591 (119905))) = 119867

119894119878

119895119909

119895 (119905 minus 120591 (119905)) (3)

Remark 3 Notice every node of the complex dynamical net-work has its own dimensions and there is no certain connec-tion for the dimension among different nodes Assumption 1shows 119873 matrixes 119864

119894+ 119865

119894119861

119894have some same eigenvalues

It further means that the state of different nodes containssimilar behaviors

Lemma 4 (see [17]) For any vectors 119909 119910 isin 119877

119899times119899 and positivedefinite matrix 119882 isin 119877

119899times119899 the following matrix inequity holds2119909

119879119910 le 119909

119879119882119909 + 119910

119879119882

minus1119910

3 Synchronization for Complex Network byDecentralized Controllers

In this section based on Assumptions 1ndash2 and Lemma 4 atfirst we propose the definition of synchronization manifoldand then synthesise the decentralized dynamical compensa-tion controllers to synchronize the complex network asymp-totically

Definition 5 A complex dynamical network is said to achievethe asymptotical synchronization if

119878

1119909

1= 119878

2119909

2= sdot sdot sdot = 119878

119894119909

119894= 119904 (119905) as 119905 997888rarr infin (4)

where 119904(119905) isin 119877

1198991 is a solution of targeted state satisfying

119904 (119905) = 119865

1119904 (119905) (5)

For our synchronization scheme we define the error vectorsas

119890

119894= 119878

119894119909

119894minus 119904 (119905) (6)

From (5) and (6) Assumptions 1 and 2 the dynamical errorequation is given as

119890 = 119878

119894

119894minus 119904 (119905) = 119878

119894[

[

119864

119894119909

119894+ 119865

119894119906

119894

+

119873

sum

119895=1119895 =119894

119888

119894119895(ℎ

119894119895(119909

119895 (119905 minus 120591 (119905))) minus ℎ

119894119894(119909

119894 (119905 minus 120591 (119905))))

]

]

minus 119864

1119904 (119905) = 119878

119894[

[

(119864

119894+ 119865

119894119861

119894) 119909

119894+ 119865

119894119860

119894V119894

+

119873

sum

119895=1

119888

119894119895ℎ

119894119895(119909

119895 (119905 minus 120591 (119905)))

]

]

minus 119864

1119904 (119905) minus

119873

sum

119895=1

119888

119894119895119867

119894119904 (119905)

= (119864

119894+ 119865

1119861

1) 119878

119894119909

119894+ 119865

1119860

1V119894

+

119873

sum

119895=1

119888

119894119895119867

119894119878

119895119909

119895 (119905 minus 120591 (119905)) minus 119864

1119904 (119905)

minus

119873

sum

119895=1

119888

119894119895119867

119894119904 (119905 minus 120591 (119905)) = 119864

1119890

119894+ 119865

1119861

1119878

119894119909

119894+ 119865

1119860

1V1

+

119873

sum

119895=1

119888

119894119895119867

119894119890

119895 (119905 minus 120591 (119905))

(7)

where 119867

119894 isin 119877

1198991times1198991(119894 = 1 119873) are norm-bounded

which means there is a constant 119872 satisfying

1003817

1003817

1003817

1003817

119867

119894

1003817

1003817

1003817

1003817

le 119872 (8)

Mathematical Problems in Engineering 3

Theorem 6 In this section one decentralized dynamical com-pensation controller is designed to achieve our synchronizationas mentioned before which is

119906

119894= 119861

119894119909

119894+ 119860

119894V119894

V119894

= 119889

119894119860

minus1

1119865

minus1

1119877119890

119894minus 119860

minus1

1119861

1119878

119894119909

119894

(9)

where 119894 = 1 119873 constant 119896

119894gt 0 119865

minus1

1119877is a right inverse

matrix for 119865

1and satisfies 119865

1119865

minus1

1119877= 119868

1198991 which generally means

119865

1has full row rankHere from (9) error dynamical systems (7) become the

following form

119890

119894= 119864

1119890

119894+ 119889

119894119890

119894+

119873

sum

119895=1

119888

119894119895119867

119894119890

119895 (119905 minus 120591 (119905))

(119894 = 1 119873)

(10)

Select 119888 = max1lt119894lt119873

|119888

119894119895| and 120575 = max eig((119864

119879

1+ 119864

1)2) If there

exists a constant 120574 satisfying

(120575 + 119889

119894+

119888119872119873

2

+ 120574) lt 0 (11)

120574 gt

119888119872119873

2

gt 0(12)

then the error dynamical systems will achieve asymptoticalsynchronization

Proof First we select the following Lyapunov function can-didate

119881 (119905) =

119873

sum

119894=1

119890

119879

119894119890

119894

2

(13)

The derivative of 119881(119905) along the trajectories with errordynamical systems is

119881 (119905) =

119873

sum

119894=1

(

119890

119879

119894119890

119894

2

+

119890

119879

119894119890

119894

2

)

=

119873

sum

119894=1

[

119890

119879

119894(119865

119879

1+ 119865

1) 119890

119894

2

+ 119889

119894119890

119879

119894119890

119894]

+

119873

sum

119894=1

119873

sum

119895=1

119888

119894119895119890

119879

119894119867

119894119890

119895 (119905 minus 120591 (119905))

le (120575 + 119889) 119890

119879

119894119890

119894+

119873

sum

119894=1

119873

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119888

119894119895

1003816

1003816

1003816

1003816

1003816

1003817

1003817

1003817

1003817

119867

119894

1003817

1003817

1003817

1003817

1003816

1003816

1003816

1003816

1003816

119890

119879

119894119890

119895 (119905 minus 120591 (119905))

1003816

1003816

1003816

1003816

1003816

(14)

where 119889 = max1lt119894lt119873

|119889

119894| By Lemma 4 (11) (12) and the

parameters we set before we derive the following inequation

119881 (119905) le (120575 + 119889 +

119888119872119873

2

)

119873

sum

119894=1

119890

119879

119894119890

119894

+

119888119872119873

2

119873

sum

119895=1

119890

119879

119895(119905 minus 120591 (119905)) 119890

119895 (119905 minus 120591 (119905))

le minus120574

119873

sum

119894=1

119890

119879

119894119890

119894

+

119888119872119873

2

119873

sum

119895=1

119890

119879

119895(119905 minus 120591 (119905)) 119890

119895 (119905 minus 120591 (119905))

(15)

If there exists a function 119892(120576) satisfying (12) where

119892 (120576) = 120576 minus 120574 +

119888119872119873

2

exp (120576120591) (16)

we derive the following results

119892 (0) lt 0

119892 (infin) gt 0

119892 (120576) gt 0

(17)

Then from 0 to infin we can find 120576 gt 0 to suit

119892 (120576) = 120576 minus 120574 +

119888119872119873

2

exp (120576120591) = 0(18)

Set

119872

0= supminus120591le119904le0

119881 (119904)

119882 (119905) = exp (120576120591) 119881 (119905) 119905 ge 0

(19)

119876 (119905) = 119882 (119905) minus ℎ119872

0 ℎ gt 1 (20)

Hence based on Lyapunov lemma it is available to make thestate of dynamics systems achieve synchronization asymp-totically under manifold (4) if we can prove 119876(119905) lt 0 119905 isin

(minus120591 infin)Firstly because of 119876(119905) = 119882(119905) minus ℎ119872

0 then 119876(119905) lt

0 119905 isin (minus120591 0) Secondly the target is to prove the followinginequation

119876 (119905) lt 0 119905 isin (0 infin) (21)

If there exists 119905

0isin (0 infin) which satisfies

119876 (119905

0) = 0

119876 (119905

0) gt 0

(22)

119876 (119905) lt 0 minus120591 le 119905 le 119905

0 (23)

4 Mathematical Problems in Engineering

the time derivative of 119876(119905

0) is

119876 (119905

0) = 120576119882 (119905

0) + exp (120576119905

0)

119881 (119905

0)

le 120576119882 (119905

0) minus 120574 exp (120576119905

0) 119881 (119905

0)

+

119888119872119873

2

exp (120576119905

0) 119881 (119905

0minus 120591 (119905))

le (120576 minus 120574) 119882 (119905

0)

+

119888119872119873

2

exp (120576120591) 119882 (119905

0minus 120591 (119905))

lt (120576 minus 120574 +

119888119872119873

2

exp (120576120591)) ℎ119872

0

(24)

It is clear that there is a contradiction between (22) and (24)so (19) is positive and according to hypothesis (19) we setℎ rarr 1 then

119881 (119905) lt exp (minus120576119905) 119872

0 (25)

The proving for Theorem 6 ends

Remark 7 Compared with other similar published resultsthis paper discussed the complex dynamical networks withtime-varying delay with designing different controllers Bydefining a new function 119881 we find a new way to proverelevant problems and get some results for complex dynam-ical networks with time-varying delay Under the researchconditions of similar literatures having no time delay it isavailable to accept that the method in this paper can simplifythe process and save cost of control while we discuss one kindof complex dynamical network without uncertain parts

4 Numerical Examples

The following dynamical error equation illustrates the theo-retical effectiveness of our proposed synchronization themesderived in Section 3 To put it simply we choose a ten-node network in which the first node has 2 dimensions thesecond node has 3 dimensions and the rest of them have 4dimensions Consider

119890

119894= 119864

1119890

119894+ 119889

119894119890

119894+

119873

sum

119895=1

119888

119894119895119867

119894119890

119895 (119905 minus 120591 (119905))

(119894 = 1 119873)

(26)

where we choose the upper bound about time-varying delayas 120591 = 01 and without loss of generality the parameters fordynamical error equation are chosen as follows

119864

1= (

minus2 5

0 3

)

119864

2= (

minus2 2 3

0 1 2

0 0 1

)

119864

3= sdot sdot sdot = 119864

10= (

minus2 2 3 5

0 1 2 minus1

0 0 1 0

0 0 0 rand + 2

)

119861

1= (

1 0

0 1

)

119861

2= (

1 minus1 minus2

0 1 minus1

)

119861

119894= (

1 minus1 minus2 minus3

0 1 minus1 1

)

119860

119895= (

minus1 0

3 2

)

(27)

where 119894 = 3 4 10 119895 = 1 2 10 Consider

119878

1= (

1 0

0 1

)

119878

2= (

1 3 0

0 1 0

)

119878

3= sdot sdot sdot = 119878

10= (

1 3 0 0

0 1 0 0

)

(28)

Consider the connection type for dynamical network so wechoose the outer coupling matrix as

119862

=

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

minus1 0 0 0 1 0 0 0 0 0

1 minus3 minus1 1 0 0 1 0 0 1

0 1 minus5 1 1 1 1 minus1 0 1

1 1 1 minus8 1 1 0 1 1 1

0 0 1 1 minus3 0 0 1 0 0

0 1 1 1 0 minus6 1 1 1 0

0 1 0 0 0 1 minus3 0 1 0

0 0 0 1 1 0 0 minus2 0 0

0 0 0 0 0 0 1 0 minus1 0

0 1 0 0 0 0 0 0 0 minus1

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)10times10

(29)

Mathematical Problems in Engineering 5

0 05 1 15 2 25 3

0

2

4

6

8

10

12

minus2

minus4

minus6

e

i1

(t)

t

Figure 1 State of error for the first dimension of nodes

0 05 1 15 2 25 3

0

2

4

6

8

10

12

minus2

e

i2

(t)

t

Figure 2 State of error for the second dimension of nodes

In this example initial conditions of each node are chosen as

119909

0

1= [1 2]

119909

0

2= [3 4 5]

119909

0

3= [minus05 2 1 4]

119909

0

4= [1 12 0 1]

119909

0

5= [minus5 4 3 5]

119909

0

6= [2 minus4 2 1]

119909

0

7= [minus2 3 minus1 1]

119909

0

8= [3 1 1 0]

119909

0

9= [0 5 3 2]

119909

0

10= [2 1 minus05 2]

(30)

Besides according to Theorem 6 and the assumptions weproposed before 119867

119894 lt 1 = 119872 and 120575 = 4 Also we are able

to find out 119889

119894lt minus10 after calculating these parameters The

results are shown in Figures 1 and 2 which exhibit that thedynamical networks achieve synchronization asymptotically

5 Conclusions

In this paper for the coupling time-varying delay com-plex networks with different dimensional similar nodes thedecentralized controllers are designed to synchronize suchnetworks According to the results of numerical example justif the nodes contain similar behaviors by Lyapunov stabilitytheorem we can find decentralized controllers with similarparameters to verify that our stability and synchronizationcontrol theme in this paper is effective

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by Natural Science Foundation ofChina under Grant no 11372107 and no 61174211

References

[1] H Liu J Chen J-A Lu andMCao ldquoGeneralized synchroniza-tion in complex dynamical networks via adaptive couplingsrdquoPhysica A vol 389 no 8 pp 1759ndash1770 2010

[2] X F Wang and G Chen ldquoComplex networks small-worldscale-free and beyondrdquo IEEE Circuits and Systems Magazinevol 3 no 1 pp 6ndash20 2003

[3] X Wang and J Z Huang ldquoEditorial uncertainty in learningfrom big datardquo Fuzzy Sets and Systems vol 258 pp 1ndash4 2015

[4] C Vitolo Y Elkhatib D Reusser C J Macleod and WBuytaert ldquoWeb technologies for environmental Big DatardquoEnvironmental Modelling amp Software vol 63 pp 185ndash198 2015

[5] W J Mallon ldquoBig datardquo Journal of Shoulder and Elbow Surgeryvol 22 no 9 p 1153 2013

[6] K Kambatla G Kollias V Kumar andAGrama ldquoTrends in bigdata analyticsrdquo Journal of Parallel and Distributed Computingvol 74 no 7 pp 2561ndash2573 2014

[7] C Zheng M Sun Y Tao and L Tian ldquoAdaptive-impulsivecontrol for generalized projective synchronization between twocomplex networkswith time delayrdquo inProceedings of the ChineseControl and Decision Conference (CCDC rsquo10) pp 3574ndash3578Xuzhou China May 2010

[8] H R Karimi ldquoRobust synchronization and fault detection ofuncertain master-slave systems withmixed time-varying delaysand nonlinear perturbationsrdquo International Journal of ControlAutomation and Systems vol 9 no 4 pp 671ndash680 2011

[9] D H Ji J H Park W J Yoo S C Won and S M LeeldquoSynchronization criterion for Lurrsquoe type complex dynamicalnetworks with time-varying delayrdquo Physics Letters A vol 374no 10 pp 1218ndash1227 2010

6 Mathematical Problems in Engineering

[10] N Li Y Zhang J Hu and Z Nie ldquoSynchronization for generalcomplex dynamical networks with sampled-datardquo Neurocom-puting vol 74 no 5 pp 805ndash811 2011

[11] T Chen X Liu and W Lu ldquoPinning complex networks by asingle controllerrdquo IEEE Transactions on Circuits and Systems IRegular Papers vol 54 no 6 pp 1317ndash1326 2007

[12] X F Wang and G Chen ldquoPinning control of scale-freedynamical networksrdquo Physica A Statistical Mechanics and ItsApplications vol 310 no 3-4 pp 521ndash531 2002

[13] Y Liang X Wang and J Eustace ldquoAdaptive synchronization incomplex networks with non-delay and variable delay couplingsvia pinning controlrdquo Neurocomputing vol 123 pp 292ndash2982014

[14] B Zhuo ldquoPinning synchronization of a class of complexdynamical network with doupling delayrdquo International Journalof Nonlinear Science vol 9 no 2 pp 207ndash212 2010

[15] Y H Wang Y Q Fan Q Y Wang and Y Zhang ldquoStabilizationand synchronization of complex dynamical networks withdifferent dynamics of nodes via decentralized controllersrdquo IEEETransactions on Circuits and Systems I Regular Papers vol 59no 8 pp 1786ndash1795 2012

[16] L Zhang Y Wang and Q Wang ldquoSynchronization for non-linearly coupled complex dynamical networks with differentdimensional nodesrdquo in Proceedings of the 26th Chinese Controland Decision Conference (CCDC rsquo14) pp 3632ndash3637 IEEEChangsha China May- June 2014

[17] JWu and L Jiao ldquoSynchronization in complex delayed dynami-cal networks with nonsymmetric couplingrdquo Physica A vol 386no 1 pp 513ndash530 2007

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