Research Article Adaptive Synchronization of Complex Dynamical Networks with State Predictor

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1 Appled Mahemacs Volume 3, Arcle ID 39437, 8 pages hp://dxdoorg/55/3/39437 Research Arcle Adapve Synchronzaon of Complex Dynamcal eworks wh Sae Predcor Yunao Sh, Bo Lu, and Xao Han Key Laboraory of Beng for Feld-Bus Technology & Auomaon, orh Chna Unversy of Technology, Beng 44, Chna College of Scence, orh Chna Unversy of Technology, Beng 44, Chna Correspondence should be addressed o Bo Lu; bolu@ncueducn Receved July 3; Acceped 7 Augus 3 Academc Edor: Mchael Chen Copyrgh 3 Yunao Sh e al Ths s an open access arcle dsrbued under he Creave Commons Arbuon Lcense, whch perms unresrced use, dsrbuon, and reproducon n any medum, provded he orgnal work s properly ced Ths paper addresses he adapve synchronzaon of complex dynamcal neworks wh nonlnear dynamcs Based on he Lyapunov mehod, s shown ha he nework can synchronze o he synchronous sae by nroducng local adapve sraegy o he couplng srenghs Moreover, s also proved ha he convergence speed of complex dynamcal neworks can be ncreased va desgnng a sae predcor Fnally, some numercal smulaons are worked ou o llusrae he analycal resuls Inroducon In recen years, he synchronzaon problem of complex neworks s a ho opc n many areas, ncludng mahemacs, physcs, bology, compuer, and arfcal nellgence 37 A complex dynamcal nework s composed of a large number of nodes o be seered o arrve synchronzaon However, s so hard and mpraccal o conrol all nodes o synchronze n he large complex nework In order o solve hs problem, he pnnng conrol s nroduced, whch can decrease he number of conrollers for synchronzaon of he complex neworkspnnngconrolschemesaneffecvewayo conrol dynamcal neworks o a desred sae The dea of he pnnng conrol s o seer a small fracon nodes, and all nodes of he complex nework can acheve conrol arge va localzed feedback of hose nodes A lo of ousandng works abou he synchronzaon problem of complex neworks are presened n he recen references 37 In,Wang and Chen used specfcally and randomly pnnng conrol sraeges for scale-free chaoc dynamcal neworks De Lells e al 9 consdered he synchronzaon of complex neworks hrough local adapve couplng The auhors proposed a decenralzed adapve pnnng conrol scheme for synchronzaon of undreced neworksusngalocaladapvesraegyobohcouplng srenghs and feedback gans In 4, Yu e al nvesgaed he synchronzaon va pnnng conrol on general complex neworks In 5, 6, second-order consensus problem for mulagen sysems was nvesgaed by usng pnnng conrol mehod The convergence speed of complex dynamcal neworks s a sgnfcan ssue Through lmed communcaon, each agencanforecashefuuresaesofsneghborsandself; as well as he new conrol law can be consruced by he predced saes Wh hs sraegy, he complex dynamcal nework can evolve more quckly o equlbrum Movaed by, hs paper nvesgaes he synchronzaon va desgnng a sae predcor I s proved ha all nodes wll asympocally synchronze o he gven homogeneous saonary sae usng he adapve sraegy o he couplng srenghs desgned, f he complex dynamcal nework s conneced and a leas one node s nformed Inroducng he sae predcor for formaon algorhm n mulagen sysems, he smulaon resulsshowhausnghesaepredcornmulagen sysems can mprove he speed of he sysem o complee he desred ask The res of hs paper s desgned as follows Secon gves a model of he complex dynamcal nework Some prelmnares are nroduced o solve he adapve synchronzaon Secon 3 provdes he heorecal analyss of adapve synchronzaon of he complex dynamcal nework Furhermore, some more dealed analyses are presened n

2 Appled Mahemacs hs secon Secon 4 gves some smulaons o llusrae our heorecal resuls Concluson s fnally summarzed n Secon 5 Prelmnares and Problem Saemen Consder a complex dynamcal nework descrbed by x () =f(x ()) a c () (x () x ()) +γ a c () ( x p () x p ()), () where x () = (x (), x (),,x n ()) T R n, ( =,,,) represens he sae vecor of he node a me ; s he neghbor se of node ; f( ) R n s connuously dfferenable; a s he couplng wegh beween any wo agens, where a and a =; c () denoes he couplng srenghs beween nodes and node ;heweghedcouplng confguraon marx of he sysem s defned as u u u u u u U=u = d R, () u u u wh u = a c = =, = a c and u = a c for = Desgn he sae predcor for he conrol law as X p = LX, (3) where X p =( x p, xp,, xp ) and γ s he mpac facor of he sae predcor Under sae predcor (3), nework ()canberewrenas x () =f(x ()) a c () (x () x ()) γ( a a k c () c k () (x x k ) a a p c () c p () (x x p )) Defnon ework (4) s sad o acheve synchronzaon f (4) lm x () x () =, =,,, (5) where he homogeneous sae sasfes x () =f(x (),) = (6) The adapve conrol a node s desgned as c () = a a p c p () x () x() T x () x () + a a k c k () x k () x () T x k () x (), where c () Inhefollowng,somenecessaryassumponsandlemmas are saed Assumpon The couplng srenghs of he nework are bounded: (7) c () c (8) Assumpon 3 (see 5) The vecor feld f :R n R n ( =,,,) n nework (4) sasfes he Lpschz condon; here exss a posve consan ρ>,such ha f (x) f(y) ρ x y (9) Assumpon 4 The weghs sasfy he balance condon: a c () = a c (), () Lemma 5 (see 9) For any vecors x, y R n and posve defne marx G R n n,hefollowngmarxnequalyholds: x T y x T Gx+y T G y () Lemma 6 (see 6) Supposng ha a and b are vecors, hen for any posve-defne marx E,he followng nequaly holds: a T b nf E> {at Ea + b T E b} () Lemma 7 (see 38) The marx A of an undreced graph G s rreducble f and only f he undreced graph s conneced 3 Man Resuls In hs secon, we wll gve dealed analyss of he adapve synchronzaon of he nework wh he sae predcor By usng he Lyapunov funcon approach, adapve synchronzaon condons of such work are obaned Theorem 8 Consderng nework (4) wh nodes seered by adapve conrol (7), under Assumpons 4, hen all nodes

3 Appled Mahemacs 3 wll asympocally synchronze o he gven homogeneous saonary sae; ha s, lm x () x () = (3) Proof Le x () x () x() Consruchefollowng Lyapunov funcon: where V () =V () +V (), (4) +γ a c () a p c p () x T () x () = γ = = ρ a c () = + γ a c () a p c p () x T () x p () = a c () a k c k () x T () x () a c () V () = = x T () x (), ( m) c () V () =, = (5) +γ = +γ a c () a p c p () x T () x () a c () a k c k () x T () x k () wherem >s suffcenly large By he defnon of he marx U, soseehau s symmerc and rreducble By Lemmas 5 7, dfferenang V (),wecanhave V () = = x T () f (x ()) f(x ()) = x T () a c () ( x () x ()) γ x T () +γ x T () ρ = + a c () a k c k () ( x () x k ()) a c () a p c p () ( x () x p ()) x T () x () a c () x T () x () = = γ a c () x T () x () = +γ = a c () a k c k () x T () x () a c () a k c k () x T () x k () γ = a c () a p c p () x T () x p () ρ a c () γ = x T () x () + = x T () x () + = x T () x () + γ = + γ = + γ = a c () a k c k () a c () +γ a c () a p c p () p a c () +γ a c () a p c p () p a c () a k c k () x T () x () a c () a k c k () x T k () x k () a c () a p c p () x T () x () + γ a c () a p c p () x T p () x p () =

4 4 Appled Mahemacs = ρ a c () γ = x T () x () = + x T () x () +γ = γ = a c () a k c k () a c () +γ a c () a p c p () p a c () a k c k () x T () x k () a c () a p c p () x T () x p () ρ a c () γ = x T () x () + = x T () x () + = x T () x () + γ = + γ = + γ = + γ = a c () a k c k () a c () +γ a c () a p c p () p a c () +γ a c () a p c p () p = ρ a c () γ = a c () a k c k () x T () x () a c () a k c k () x T k () x k () a c () a p c p () x T () x () a c () a p c p () x T p () x p () + a c () a c () a k c k () + γ + γ + γ x T () x () = a c () a p c p () a c () a k c k () a c () a p c p () + a c () +γ a c () a p c p () p x T () x () + γ a c () a k c k () x T k () x k () = + γ a c () a p c p () x T p () x p () = = ρ = a c () γ a c () a k c k () +γ a c () a p c p () x T () x () + = a c () +γ a c () a p c p () p x T () x () + γ a c () a k c k () x T k () x k () = + γ a c () a p c p () x T p () x p () = (6) Therefore,

5 Appled Mahemacs 5 V () ρ l γl +γ a c () l = x () x T () xt () d x () ρ l γl +γ a c () l x () = l + γl + x T () xt () l + γl x () x () d l + x () γl γ a c () l = x + x T () xt () γ a c () l () x () = d x () γ a c () l = γ a c () l = x + x T () xt () γ a c () l () x () = d x () γ a c () l = x = x T () xt () ρ+γ(w +V ) () x () d, ρ+γ(w +V ) x () (7) where l = u = =, = a c (), W = = a c ()l, V = = a c ()l,,=,,, Dfferenang V (),wege V () = ( m) c () c () = ( m) c () = { { a a p c p () x () x () T p { x () x () + a a k c k () x k () x() T

6 6 Appled Mahemacs = ( m) x k () x () } } } = + ( m) = = x T () xt () a c () a p c p () x T () x () a c () a k c k () x T k () x k () ( m)(w +V ) d ( m)(w +V ) x () x () x () (8) Combnng V () and V (), and snce m>s suffcenly large, hen we can have V () = V () + V () ρ+(γ+ m)(w +V ) x T () xt () d ρ+(γ+ m)(w +V ) (9) x () x () <, x () where l, W, V are defned as before So, These complee he proof lm x () x () = () Theorem 9 ework (4) solves an agreemen problem faser han he same nework whou he sae predcor Proof The proof s smlar o ha of Theorem n 33 4 Smulaons In hs secon, we gve he numercal smulaons o llusrae he analycal resuls Consder a small nework wh he undreced opology descrbed as he followng symmerc marx: A= , () where all nodes of he pagebreak nework and her synchronous goal wll obey he same nonlnear dynamcs descrbed as he Lorenz sysem: x =(x x ) f (x ()) =f(x,x,x 3 { )= x = 8x x x 3 x { x 3 =x x 8 { 3 x3, () as shown n Fgure Fgure descrbes he convergence of he sae errors on hex-axs, y-axs, and z-axs, respecvely From hs fgure, we can see ha all nodes of he above nework can synchronze o he synchronous sae gradually Wh he same nal sae and he same nonlnear dynamcs, nfluenced by he same adapve sraegy, can be easy o fnd as n Fgure 3 ha he nework wh a sae predcor can also synchronze o he synchronous sae and be faser han he nework whou a sae predcor 5 Concluson In hs paper, we have nvesgaed he adapve synchronzaon of complex dynamcal neworks wh nonlnear dynamcs By nroducng local decenralzed adapve sraeges o hecouplngsrenghs,wehaveprovedhaheneworkwh a sae predcor can synchronze o he synchronous sae

7 Appled Mahemacs 7 y-axs x z x z x y x y x x x x z-axs x-axs Fgure : The nonlnear dynamcs of he nework Traecores convergence (z-axs) Traecores convergence (x-axs) Traecores convergence (y-axs) Fgure : The raecores of four agens n he dynamcal nework whou a sae predcor x z x z x y x y x x x x Traecores convergence (x-axs) Traecores convergence (y-axs) Traecores convergence (z-axs) Fgure 3: The raecores of four agens n he dynamcal nework wh a sae predcor when γ = Acknowledgmens Ths work was suppored n par by he aonal aural Scence Foundaon of Chna under Gran no 63449, no 6746, he Beng aural Scence Foundaon Program 43, Scence and Technology Developmen Plan Proec of Beng Educaon Commsson (no KM39), and Fundng Proec for Academc Human Resources Developmen n Insuons of Hgher Learnng Under he Jursdcon of Beng Muncpaly (PHR855) References X F Wang and G Chen, Pnnng conrol of scale-free dynamcal neworks, Physca A,vol3,no3-4,pp5 53, P De Lells, M d Bernardo, and F Garofalo, Synchronzaon of complex neworks hrough local adapve couplng, Chaos, vol 8, no 3, Arcle ID 37, p 8, 8 3HSu,ZRong,MChen,XWang,GChen,andHWang, Decenralzed adapve pnnng conrol for cluser synchronzaon of complex dynamcal neworks, EEE Transacons on Cybernecs,vol43,pp ,3 4WYu,GChen,JLü, and J Kurhs, Synchronzaon va pnnng conrol on general complex neworks, SIAM Journal on Conrol and Opmzaon, vol 5, no, pp , 3 5 BLu,XWang,HSu,YGao,andLWang, Adapvesecondorder consensus of mul-agen sysems wh heerogeneous nonlnear dynamcs and mevaryng delays, eurocompung, vol8,pp89 3,3 6 H Su and W Zhang, Second-order consensus of mulple agens wh couplng delay, Communcaons n Theorecal Physcs,vol5,pp 9,9 7 BLu,WHu,JZhang,andHSu, Conrollablyofdscreeme mul-agen sysems wh mulple leaders on fxed neworks, Communcaons n Theorecal Physcs, vol 58, pp , 8 H Su, M Chen, J Lam, and Z Ln, Sem-global leaderfollowng consensus of lnear mul-agen sysems wh npu sauraon va low gan feedback, IEEE Transacons on Crcus and Sysems,vol6,pp88 889,3 9 GWen,ZDuan,WYu,andGChen, Consensusofmulagen sysems wh nonlnear dynamcs and sampled-daa nformaon: a delayed-npu approach, Inernaonal Robus and onlnear Conrol,vol3,no6,pp6 69,3 WYu,PDeLells,GChen,MdBernardo,andJKurhs, Dsrbued adapve conrol of synchronzaon n complex neworks, IEEE Transacons on Auomac Conrol,vol57, no 8, pp 53 58, H Su, M Chen, X Wang, H Wang, and V Valeyev, Adapve cluser synchronzaon of coupled harmonc oscllaors wh mulple leaders, IET Conrol Theory and Applcaons, vol 7, pp , 3 M Wang, H Su, M Zhao, M Chen, and H Wang, Flockng of mulple auonomous agens wh preserved nework connecvy and heerogeneous nonlnear dynamcs, eurocompung, vol 5, pp 69 77, 3 3 W Zhang, H Su, and J Wang, Compuaon of upper bounds for he soluon of connuous algebrac rcca equaons, Crcus,SysemsandSgnalProcessng, vol 3, pp , 3

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