Output Group Consensus for Heterogeneous Lnear Mut-Agent Systems Communcatng over Swtchng Topoogy Jahu Qn Qchao Ma We Xng Zheng Department of Automaton Unversty of Scence and Technoogy of Chna Hefe 37 P R Chna E-ma: jhqn@ustceducn mqc4@maustceducn Schoo of Computng Engneerng and Mathematcs Western Sydney Unversty Sydney NSW 75 Austraa E-ma: wzheng@westernsydneyeduau Abstract: In ths paper we am to nvestgate the output group consensus probem for a swtchng network of heterogeneous near systems such that outputs of the agents synchronze wth each other n every custer From the nterna mode prncpe perspectve a necessary condton s frst derved n terms of the system dynamcs Wth ths necessary condton a dynamc controer s then desgned to sove the output group consensus probem n two steps namey consensus of reference generators and output trackng for each agent It s obtaned that f every possbe underyng topoogy of each custer contans a drected spannng tree then group consensus for the reference generators can be reazed wth suffcenty strong ntra-custer coupng strength An approprate controer s then desgned to force the output of each agent to track the output of the reference generators A smuaton exampe s gven at ast to vadate our theoretca fndngs Key Words: Output group consensus swtchng topoogy heterogeneous systems nterna mode prncpe Introducton Recenty nspred by the observaton that n the engneerng word a mut-agent system may consst of agents governng by dfferent system dynamcs especay wth dfferent dmensons the output consensus probem for heterogeneous mut-agent systems has drawn attenton of many researchers [4 3] As eaborated n [4 3] n achevng output synchronzaton there necessary exsts a reference generator whch generates the nontrva synchronzed trajectory In vew of ths to reaze the output consensus a- mong mutpe heterogeneous agents a dynamc controer s broady adopted Intutvey n eader-foowng consensus the dynamc controer drves the output of the foower systems to that of the exo-system [] Whe n eaderess consensus the dynamc controer generates the reference trajectory functon as exo-system for each system to track [4 3] Rea-word systems are usuay composed of severa nteractng custers of couped agents [] Then a more genera consensus probem termed group/custer consensus probem whch consders mutpe custers under genera coupng topoogy nvovng possbe negatve coupngs has aso receved growng attentons recenty [3 7 4 5] It frequenty arses when agents wthn the same custer are cooperatve whe the agents from dfferent custers are repusve and/or cooperatve [7] In the group consensus probem the nvarance of consensus manfod a subspace n whch the states of agents wthn the same custer are dentca does not hod wth just dffusve coupngs For ths nvarance probem of neary couped nonnear systems [ 7] provde a necessary and suffcent common nter-custer coupng condton whch refers to the scenaro that the coupngs each agent n the same custer receves from any other custer sum up equay to guar- Ths work was supported n part by the Natona Natura Scence Foundaton of Chna under Grant 647369 the Youth Innovaton Promoton Assocaton of Chnese Academy of Scences and the Austraan Research Counc under Grant DP4986 antee the nvarance of the group consensus manfod Under ths condton ntensve research concernng group consensus probem for homogeneous mut-agent systems has been conducted [3 7] however for cooperatve networks Specfcay assumng that the coupngs each agent receves from any other custer sum up to zero whch naturay nvove repusve coupngs and s termed n-degree baanced condton group consensus for homogeneous mut-agent systems s nvestgated n [8 9 4] To date few research works have concentrated on group/custer consensus for heterogeneous mut-agent systems except for [6] where output custer consensus for heterogeneous near systems s studed under n-degree baanced condton based on the nterna mode prncpe Motvated by the above dscusson we am to further address the output group consensus probem for heterogeneous near mut-agent systems The contrbutons of ths paper are as foows The genera common nter-custer coupng condton whch aows for repusve coupngs between a- gents from dfferent custers s nvestgated From the vewpont of nterna mode prncpe a necessary condton concernng system dynamcs s derved A dstnct feature of such condton compared wth that n [3] es n that the nfuence brought by common nter-custer coupngs s expcty nvoved 3 We consder a genera framework such that the near mut-agent systems communcate over swtchng network topoogy n the presence of repusve coupngs 4 To address the consensus probem over swtchng topoogy a mutpe Lyapunov functon approach s apped and structura condtons wth respect to network structure and ntracuster coupng strength are provded An nterestng and consstent concuson wth those eaborated n [8 9] s fnay made that f each custer contans a drected spannng tree at each nstant and moreover the ntra-custer coupng strength s strong enough then the reference generators are synchronzed wthn each custer The remander of the paper s arranged nto fve sectons In Secton we ntroduce reevant graph notons and formu-
ate the probem The man resuts concernng output group consensus are presented n Sectons 3 and 4 foowed by an ustratve exampe n Secton 5 The paper s fnay wrapped up wth concudng remarks n Secton 6 Notatons: Let I n be the dentty matrx and n n the zero matrx n R n n dag{a a q } denotes the dagona matrx wth a beng the -th dagona eement The spectrum of a square matrx A denoted by σa s the set of a egenvaues of A The magnary axs s denoted by jr Premnary Graph Notons The nteracton topoogy s represented by a drected graph G = V E A of order N wth a fnte nonempty set of nodes V = { N} a set of edges E V V and a weghted adjacency matrx A = [ ] a j R N N where a j s the weght aso caed coupng strength n ths work of the drected edge j satsfyng a j f j s an edge of G and a j = otherwse Moreover assume a = for a V to avod sef-oops Note that a j for nter-custer coupngs can be ether postve or negatve correspondng respectvey to the cooperatve and compettve nteractons The Lapacan matrx L of G = V E A s defned as L = dag{ N } A where = N j= a j = N [] A drected path s a sequence of edges n a drected graph of the form 3 q q A dgraph has a drected spannng tree f there exsts at east one node caed the root havng a drected path to every other node The nteracton graph G s swtchng among fnte gven dgraphs Gven an nfnte sequence of consecutve nonoverappng tme nterva [t k t k+ k N wth t = t k+ t k > τ where τ s caed the dwe tme and across whch the nteracton topoogy s fxed The tme sequence t t s caed the swtchng sequence at whch the nteracton topoogy changes Let G σt be the nteracton graph at tme t wth σt : [ + { p} Hence t s assumed that Gt swtches among p dfferent nteracton graphs { G G p} System Mode and Probem of Interest Consder a group of N agents governed by heterogeneous near system dynamcs: ẋ t = A x t + B u t y t = C x t = N where x = [x xn ] T R n u R m and y R p are respectvey the state nput and output of agent A R n n B R n m C R p n In what foows at no oss of generaty assume that the N nodes each representng a heterogeneous near system are dvded nto q q > dsjont custers namey V V q such that q = V = V and the number of nodes n a custer say V s N q These N nodes can be abeed n such a way that they are ndexed as j= N j + j= N j where N = e V = { j= N j + j= N j} Let ī denote the subscrpt of the custer whch node beongs to e V ī and G be the underyng topoogy of custer V = q e VG = V For ater use defne κ = κ = p= N p Output Group Consensus Probem: Desgn approprate contro aws u for = N as foows ζ = F ζ + O y + G e ζ + ey a u = K x t + K ζ t + G e ζ + ey b where ζ R m e ζ = N j= aσt j ζ j ζ e y = N j= aσt j y j y such that for any nta states of the heterogeneous system there hods m t y t y j t = ī = j j = N In ths paper our am s to ntroduce approprate controers and derve suffcent condtons to sove the above output group consensus probem To ths purpose a prerequste requrement s that the group consensus manfod S = { [x T xt N ]T : C x = C j x j ī = j } shoud be nvarant for heterogeneous system couped through a and b As eaborated n [] and [7] a necessary and suffcent condton for S to be nvarant through neary couped ordnary dfferenta equaton s that the common nter-custer coupng condton s satsfed e a σt j j V = d σt k V k k = q k 3 where d σt k s a constant rreevant to the choce of and j n custers V k and V respectvey Ths means for agents wthn the same custer the sums of the weght of the ncomng coupngs from any of the other custer are the same Throughout ths paper ths common nter-custer condton s adopted Under the common nter-custer coupng condton the Lapacan matrx L σt = [ σt j ] N N of G σt s wrtten as foows L σt + D σt L σt q L σt = L σt q L σt qq + D σt q where L σt k coupngs from custer V k to custer V D σt k = q k specfes the nter-custer = d σt I N = represents q j= j dσt j I N for = q Note that L σt the Lapacan matrx of G σt 3 Interna Mode Prncpe In ths secton we frst present a premnary resut whch s the fundamenta ngredent of our man resut Ths premnary resut extends that presented n [3] for compete consensus to the case where group consensus s taken nto consderaton Insertng a and b nto yeds ẋ t = A x + B K x + B K ζ + B G e ζ + ey 4a ζ t = F ζ + O y + G e ζ + ey 4b whch can be equvaenty transformed nto the foowng compact form { ˆx = Â ˆx + ˆBe ˆx 5a ŷ = Ĉ ˆx 5b
where ˆx = [ ˆx T ˆxT N ]T ˆx = [x T ζt ]T  ˆB and Ĉ are bock dagona matrces wth each bock beng [ ] [ ] A + B  = K B K B G ˆB O C F = Ĉ G = [ C ] In terms of group consensus for 5a and 5b the foowng resut extends the nterna mode prncpe proposed n [3] For the sake of brevty the network topoogy s assumed to be fxed n the next theorem e we use j nstead of σt j Theorem Consder N near state-space modes couped through dynamc controers a and b If y y j and ζ ζ j as t for ī = j j = N then there exst a scaar m matrces S R m m and R R p m = q where σs C + and S R s observabe and matrces Π R n m Γ R p m and Λ R m m such that A Π + B Γ + j Λ j = Π S 6a C Π = R j= 6b Furthermore there exsts z R m such that m y t R e S t z = 7 t V = q Proof Snce y y j and ζ ζ j as t system 5a has an attractve nvarant subspace M where C j x j = C x and ζ j = ζ for j V = q Reca that the common nter-custer coupng condton s mposed on M one hence has ˆx =  ˆB L I p+m C ˆx 8 where C = dag{ C C q } Foowng the anayss n [3] assume at no oss of generaty that M contans no exponentay stabe modes contans ony modes that are observabe at the output and 3 s non-trva wth dmenson m In vew of such assumpton there exsts a matrx S R m m such that  ˆB L I p+m C Φ = ΦS Partton Φ nto One then has  [Π T ΣT ]T ˆB Φ = [Π T ΣT ΠT N ΣT N ]T j C j Π j + Σ j = [Π T Σ T ]T S j= ths competng the frst part of the proof wth Γ = K Π + K Σ and Λ j = G C j Π j + Σ j Next we w prove that C Π = R V Snce y = y j one has C Π = C j Π j j V Then there exsts some matrx R such that C Π = R V for = q By the fact that modes n M are observabe at the output one has S R s observabe for = q It s aways [ possbe ] to fnd a transformaton matrx T = [Φ Σ] such S that T ÂT = where σs C H + and H s Hurwtz Remark The couped term N j= jλ j n 6a arses from the empoyment of e ζ and e y n 4a If the atter two terms e ζ and e y are removed n controer desgn then 6a reduces to the form obtaned n [3] 4 Output Group Consensus Usng Reatve Controer State In ths secton we w sove the output group consensus probem by desgnng approprate contro aws u = N To resove ths group consensus probem we frst ntroduce an assumpton whch s requred such that the probem s feasbe based on the necessary condton proposed n the precedng secton Smar condtons can aso be found n [5 3] Assumpton For each V = q there exst compatbe matrces Γ Π and Ψ such that A Π + B Γ = Π S C Π = R 9 B Ψ = Π where S R m m R R p m σs jr Remark The frst two condtons n Assumpton are broady used n the exstng terature concernng consensus of heterogeneous near systems [] Intutvey these two condtons requre that a the system matrces contan a common egen-space whch s refected by the matrx S The states of the agents n each custer are fnay controed nto the common egen-space The thrd condton s made to factate the controer desgn such that the nfuence from common nter-custer coupngs can be compensated Inspred by Remark we consder the foowng reference generator for each agent V = q ζ t = S ζ t where ζ R m These types of generators produce trajectores for agents n each custer to track To synchronze the above reference generators n each custer we consder the foowng dstrbuted contro protoco for agent V = q ζ t = S ζ t + j= H ī c j a σt j ζ j t ζ t where c j = c ī = j = otherwse c j = ; H ī s to be desgned ater In the seque we ca c the ntra-custer coupng strength whch s used to refect strong versus weak coupngs To track the trajectores of the reference generators for each agent the foowng feedback controer s expoted x t = A x t + B u t + H ỹ t y t a u t = K x t + K ζ t + Ψ j= c j σt j ζ j b for V = q where a s a Luenberger observer The feedback controer b uses the nformaton from the Luenberger observer and the reference generators
The controer of the above form can be seen as extenson of those proposed n [3] In ths paper we choose K as K and K = K Π + Γ where K s to be determned Next we w prove that output group consensus for heterogeneous system can be acheved va a and b In what foows we frst prove that the reference generators couped as n can reaze group consensus exponentay fast Then by dynamc controer a b we w show that the output of each agent tracks that of ts correspondng reference generator thereby eadng to output group consensus 4 Group Consensus for Couped Reference Generators In ths subsecton we w show that the couped reference generators acheve group consensus n an exponenta manner To ths purpose we transform the group consensus probem nto a stabty probem by ntroducng proper error varabes Then the stabty anayss s performed wth respect to the error system dynamcs Frst we ntroduce some notatons Let Lσt k k = q k be the sub-matrx of the foowng matrx [ ] L σt k dk k = N Lσt k [ ] N where = dag{ q } = N I N Suppose that the underyng topoogy of each custer contans a drected spannng tree durng each nterva [t k t k+ k = a postve defnte matrx say σt k correspondng to Lσt k exsts such that Lσt k T σt k σt k L σt k < Let N σt k = dag{c σt k L σt k + Lσt k T σt k c q σt k q L σt k qq + L σt kt qq σt k q } N σt k = Lσt k T σtk + σt k Lσt k N σtk where σtk = dag{ σt k σt k q } Now we are ready to present our resut Lemma Suppose that the common nter-custer coupng condton 3 s satsfed If for each t k = the underyng graph G σt k of each custer contans a drected s- pannng tree and ntra-custer coupng strength c = q satsfes c > γλ max P + λ max PS + S T P φλ mn P ηλ mn P 3 then group consensus for the couped generators can be acheved exponentay fast wth γ > satsfyng the foowng nequates { PS + S T P c η + φ PP γp nκ γτ < where P s symmetrc postve defnte κ satsfes λ max σt k+ P < κλ mn σt k P for t k = φ and η > are defned such that λ mn N σt k φλ max σt k σt k L σt k + Lσt k T σt k η σt k = q Proof See Appendx for the detaed proof Remark 3 In the statement of Lemma the varabes φ κ and η are we defned n ght of that there are fnte dfferent topooges for the communcaton network to take 4 Output Group Consensus va Dynamc Controer Based on the resut estabshed n the precedng subsecton n what foows we sha prove that the output group consensus s asymptotcay acheved for heterogeneous systems Especay we w show that each heterogeneous near system tracks the trajectory of the correspondng reference generator Theorem Under Assumpton suppose that the condtons stated n Lemma hod Then the output group consensus for mut-agent system can be acheved through dynamc controer a and b where K and H are desgned such that A + B K and A + H C are Hurwtz for = N Proof Defne error varabes ɛ = x Π ζ and ν = x x One has ɛ =A + B K ɛ B K ν ν =A + H C ν for = N Snce A + H s Hurwtz ν tends to zero as tme approaches nfnty at an exponenta rate Reca that K s desgned such that A + B K s Hurwtz t s therefore concuded that ɛ reaches zero exponentay fast for = N Next we w verfy output consensus Bearng the above concuson n mnd t can be obtaned that x Π ζ and x x exponentay fast as tme tends to nfnty Recang the fact that y = C x by Assumpton one has y C Π ζ = R ζ exponentay fast Snce ζ ζ j ī = j = accordng to Lemma y y j ī = j = the resut s vad 5 An Iustratve Exampe In ths secton we present an exampe to ustrate our theoretca fndngs Exampe : Consder the fve near systems movng on the pane wth two possbe network topooges shown n Fg The communcaton graph swtches from graph a to graph b perodcay wth a perod T = s Two custers are consdered n ths exampe such that V = { } V = {3 4 5} The systems dynamcs are chosen such that 3 A = B = C = 3 5 A = B = C 9 = 3 7 4 A 3 = B 3 3 3 = C 3 = 5 5 6 A 4 = B 4 = C 4 = [ ] [ ] 3 5 A 5 = B 4 5 = C +3 5 = 5 [ ] 3 3
e t y t y t y t y t a b 5 5 3 3 4 3 3 4 - -3-3 - Fg : Two dfferent nteracton topooges among fve heterogeneous agents Underyng topoogy of each custer contans a drected spannng tree The nter-custer coupngs satsfy n-degree baanced condton wth d = d = S = By computaton one has [ ] 3 R = [ ] S = 4 4 [ ] 3 5 R 3 3 = [ ] The nta vaues of the agents are randomy chosen from nterva [ ] [ ] R The output trajectores of the fve agents are shown n Fg Whe the output trackng error s depcted n Fg 3 It s observed obvousy that group consensus s acheved asymptotcay - custer custer - 4 6 8 4 6 8 Tme 3 custer custer - 4 6 8 4 6 8 Tme a D pot of the output trajectores of the fve agents - custer T=s T=5s T=s - -5 5 5 5 y t - custer T=s T=s T=5s - -6-4 - 4 6 8 4 y t b D pot of the output trajectores of the fve agents Fg : The output trajectores of y T = [y y ] = 5 It s observed that group consensus s asymptotcay acheved 6 Concuson 8 custer custer In ths paper we have nvestgated the output group consensus contro for a network of heterogeneous near systems communcatng over swtchng topoogy A necessary condton has been derved n terms of the system dynamcs wth whch a dynamc controer s then desgned to sove the output group consensus probem n two separatve steps: consensus of reference generators; and output trackng for each agent An approprate controer s then proposed n order to track the output of the reference generators to reaze group consensus The smuaton has vadated the effectveness of our theoretca fndngs Future works nvove further dscussons on how ntercuster coupngs nfuence the evouton of the reference generators and reaxaton of the assumpton mposed on system dynamcs 6 4 - -4 4 6 8 4 6 8 Tme Fg 3: The trackng error trajectores of e = y R ī ζ = 5 It s cear that the trackng error asymptotcay vanshes
7 Appendx: Proof of Lemma In ths proof we consder a genera system dynamcs for every V ζ t = S ζ t + j= H ī c j a σt j ζ j t ζ t 5 Pre-mutpyng the above system wth I m and notng that = I N for = q one then arrves at δ =dag { I N S I Nq S q } δ [ Lσt I m ] dag { IN H I Nq H q } δ 6 where δ = [ζ κ + ζ κ + T ζ κ + ζ κ T ] T δ = [ δ T δ T q ] T Lσt takes the foowng form L σt = c Lσt + D σt Lσt q L σt q c Lσt q qq + D σt q 7 σt k P δ t [t k t k+ Evdenty group consensus s asymptotcay acheved f system 6 s asymptotcay stabe Now consder the foowng Lyapunov functon canddate for the error system dynamcs 6: Vt = q = δt Choosng H ī = P for V and takng the dervatve of Vt aong the trajectory of error system dynamcs 6 gves Vt = = = δ T δ T σt k P S δ [ c σt k L σt k + σt k δ T = j= j σt k L σt k j P P j δ j ] D σt k P P δ Let us frst consder the ast two terms n the dervatve of Vt Defne y t = I N P δ then one has δ T = j= j + = δ T σt k =y T [ N σt k I m ] y σt k L σt k j P P j δ j D σt k φy T σt k I m y = φ P P δ = Therefore one obtans the foowng nequaty V = γ δ T = δ T σt k P P δ 8 σt k [ P S + S T P c η + φ P ] δ δ T where t s assumed that σt k P δ = γv 9 Hence Vt e γt t k Vt k for t [t k t k+ For any t > t s aways possbe to fnd an nteger s such that t s t < t s+ Therefore one has Vt exp γt ts Vt s exp γt ts exp nκ γt s t s Vt s s exp γt t s + nκ γt j t j Vt If nκ γτ < hods then one has δt as t + exponentay fast Ths competes the proof References [] V N Beykh I V Beykh and E Mosekde Custer synchronzaton modes n an ensembe of couped chaotc oscators Phys Rev E 633: 366-366-4 [] C Gods and G Doye Agebrac Graph Theory New York: Sprnger [3] Y Han W Lu and T Chen Achevng custer consensus n contnuous-tme networks of mut-agents wth ntercuster non-dentca nputs IEEE Trans Autom Contro 63: 793 798 5 [4] A Isdor L Marcon and G Casade Robust output synchronzaton of a network of heterogeneous nonnear agents va nonnear reguaton theory IEEE Trans Autom Contro 59: 68 69 4 [5] M Ja Enhancng synchronzabty of dffusvey couped dynamca networks: a survey IEEE Trans Neura Netw Learn Syst 47: 9 3 [6] Z Lu and W S Wong Output custer synchronzaton of heterogeneous near mut-agent systems n Proc 54th IEEE Conf Decson Contro Osaka Japan 5 pp 853 858 [7] W Lu B Lu and T Chen Custer synchronzaton n networks of couped nondentca dynamca systems Chaos : 3 [8] J Qn H Gao and W X Zheng Exponenta synchronzaton of compex networks of near systems and nonnear oscators: a unfed anayss IEEE Trans Neura Netw Learn Syst 63: 5 5 5 [9] J Qn and C Yu Custer consensus contro of generc near mut-agent systems under drected topoogy wth acycc partton Automatca 499: 898 95 3 [] J Qn C Yu and B D O Anderson On eaderess and eader-foowng consensus for nteractng custers of doubentegrator mut-agent systems Automatca 74: 4 6 [] Y Su and J Huang Cooperatve output reguaton of near mut-agent systems IEEE Trans Autom Contro 574: 6 66 [] A T Wnfree The Geometry of Boogca Tme Sprnger- Verag New York 98 [3] P Weand R Sepuchre and F Agöwer An nterna mode prncpe s necessary and suffcent for output synchronzaton Automatca 475: 68 74 [4] J Yu and L Wang Group consensus n mut-agent systems wth swtchng topooges and communcaton deays Syst Contro Lett 596: 34 348 [5] H Hu W Yu G Wen Q Xuan and J Cao Reverse group consensus of mut-agent systems n the cooperatoncompetton network IEEE Trans Crcut Syst-II: Exp Bref 63: 36 47 6 j= P S + S T P c η + φ P < γp