Consensus of Multi-agent Systems Under Switching Agent Dynamics and Jumping Network Topologies

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1 Inernaonal Journal of Auomaon and Compung 35, Ocober 206, DOI: 0007/s z Consensus of Mul-agen Sysems Under Swchng Agen Dynamcs and Jumpng Nework Topologes Zhen-Hong Yang Yang Song,2 Mn Zheng,2 We-Yan Hou 3 School of Mecharonc Engneerng and Auomaon, Shangha Unversy, Shangha , Chna 2 Shangha Key Laboraory of Power Saon Auomaon Technology, Shangha , Chna 3 School of Informaon, Zhengzhou Unversy, Zhengzhou , Chna Absrac: Consensus of mul-agen sysems s an neresng research opc and has wde applcaons n scence and engneerng The agens consdered n mos exsng sudes on consensus problem are me-nvaran However, n many cases, agen dynamcs ofen show he characersc of swchng durng he process of consensus Ths paper consders consensus problem of general lnear mul-agen sysem under boh swchng agen dynamcs and jumpng nework opologes Whn he proposed mul-agen sysem, he agen dynamc swchng s assumed o be deermnsc, whle he nework opology jumpng s consdered respecvely for wo cases: deermnsc jumpng Case and Markov jumpng Case 2 By applyng he dwell me and he average dwell me echnques, a suffcen consensus and an almos sure consensus condons are provded for hese wo cases, respecvely Fnally, wo numercal examples are presened o demonsrae he heorecal resuls Keywords: Mul-agen sysems, consensus, swchng agen dynamcs, swchng opologes, Markov process Inroducon Mul-agen sysem MAS s a class of dynamc sysems wh a group of auonomous agens Recen years have wnessed an ncreasng aenon on consensus of MASs due o s broad applcaons n scence and engneerng Dverse applcaons of consensus problem nclude cooperave conrol of unmanned ar vehcles, 2], formaon conrol of moble robos 3, 4] and flockng of mulple agens 5, 6] Ingeneral, consensus problem s o desgn dsrbued proocols for dfferen knds of MASs under varous communcaon opologes assumpons o reach expeced performances The nework opologes consdered n he sudes of consensus can be roughly classfed as fxed opology 7 0] and swchng opology 22] Fxed opology s suable for he case ha he communcaon channel s deal whle he nformaon neracon pah among he agens remans unchanged durng he process of consensus A basc mehod based on graph heory was frs nroduced n consensus analyss for sngle- and double-negraor MASs wh fxed opology 7, 8] By desgnng dsrbued conrollers wh Rcca mehod, he consensus of MASs wh general lnear agen was reached n 9, 0] I s shown ha, for he fxed opology case, a necessary condon of consensus s ha Research Arcle Manuscrp receved Augus 27, 204; acceped March 30, 205; publshed onlne July 25, 206 Ths work was suppored by Naonal Naural Scence Foundaon of Chna No , and Shangha Naural Scence Fund No 3ZR46300 Recommended by Edor-n-Chef Huo-Sheng Hu c Insue of Auomaon, Chnese Academy of Scences and Sprnger-Verlag Berln Hedelberg 206 he fxed opology should be conneced In pracce, MASs are ofen subjec o he change of conrol commands or unconrolled exernal facors, eg, nose, lnk falure and nework congeson In such suaons, he nework opology s no fxed bu swches among a se of sub-opologes The swchng among sub-opologes can be caegorzed as conrollable 4] and sochasc 5 2] due o dfferen swchng-drven mechansms When opology swchng s conrollable, he consensus can be acheved by allocang suable dwell me for he sub-opologes wh dfferen jon connecvy properes 4] For sochasc opology swchng case, he models of Bernoull and Markov are ofen appled o descrbe he characerscs of swchng The defnons of consensus under sochasc crcumsances can be specfcally dvded no mean square MS consensus and almos sure AS consensus The MS consensus requres ha he expecaon of he norm of he sae dfference beween any wo agens, e, E x x j ], j should converge o zero asympocally, whle he AS consensus requres ha he sae rajecores of all he agens converge o same value wh he probably one In 5 8], by usng graph heory and Lyapunov mehod, MS consensus was suded respecvely for double-negraor and general lnear MASs wh Bernoull or Markov swchng opology Moreover, nspred by nonnegave marx heory, boh MS and AS consensus problems were addressed for sngle-negraor MAS n 9 20] By her resuls, even hough all he sub-opologes are dsconneced, MS or AS consensus can also be reached f some jon conneced condons are sasfed However, o he bes of our knowledge, here s no jon conneced AS consensus condon repored for general lnear MAS The df-

2 Z H Yang e al / Consensus of Mul-agen Sysems Under Swchng Agen Dynamcs and Jumpng Nework Topologes 439 fculy les n ha general lnear agen dynamcs make he nonnegave marx echnques employed n he prevously menoned works no applcable To deal wh he AS consensus for general lnear MASs, some resrcve assumpon, e, a leas one sub-opology s conneced, s nroduced o make he problem racable n mahemacs 2] Invesgang weaker AS consensus condon for general lnear MAS s an neresng fuure research opc I should be noed ha a common pon of all he aforemenoned leraures s ha he consdered agens are lnear me-nvaran However, he dynamcs of agens n many MASs may vary n some cases For nsance, as dscussed n 22], he dynamcs of an arcraf n low and hgh aack angles can be que dfferen Generally, addonal power oupu from he hrus vecorng nozzles s needed n hgh angle of aack aude, whle s no necessary n low aack angle case Therefore, s worhy o ncorporae such facor of agen dynamcs swchng no he sudes of MAS consensus To he bes of he auhors knowledge, here s no relevan resul repored on hs case Moreover, because he swches exs smulaneously n agen dynamcs and nework opology, and mgh be subjec o dfferen swchng-drve mechansms, hus he consensus analyss wll be more complcaed Ths paper ams a brdgng hs gap We assume ha he agen dynamc swchng s deermnsc and conrolled by a pecewse consan swchng sgnal, whle he nework opology jumpng s consdered respecvely for wo cases: deermnsc jumpng Case and Markov jumpng Case 2 The consensus s hen nvesgaed for such wo cases The man conrbuons of hs arcle can be concluded as hree pars: modelng mehod for MASs wh boh swchng agen dynamcs and jumpng nework opologes s proposed; 2 a suffcen consensus condon s presen for Case by applyng he dwell me and he average dwell me echnques; 3 moreover, by usng he ergodcy of he Markov process, a suffcen almos sure consensus condon s gven for Case 2 The res par of hs paper s organzed as follows Secon 2 gves some basc conceps and a prelmnary resul of graph heory The dealed modelng of an MAS wh swchng agen dynamcs and jumpng nework opologes s gven n Secon 3 A suffcen consensus and almos sure consensus condon are proved n Secon 4 Ths s followed by a numercal example n Secon 5 and a bref concluson n Secon 6 Noaons Le r n n and r n be he se of n n real marces and n-dmenson real column vecors, respecvely I n represens he deny marx n 0 nsann-dmenson column vecor wh all enres equal o 0 Marces and vecors, f no explcly saed, are assumed o have compable dmensons The superscrp T means he ranspose of real marces For a marx A vecor x, le A x denoes s 2-norm A square marx s Hurwz f all of s egenvalues have negave real pars The marx nequaly A> B means ha A B s posve sem- defne The Kronecker produc, denoed by, faclaes he manpulaon of marces by he followng properes: A BC D =AC BD; 2A B T = A T B T 2 Prelmnares In hs secon, some basc conceps and a prelmnary resul of graph heory are nroduced We use a dgraph G =V,E,A o descrbe he neracon opology among agens, where V =, 2,,N} s a fne nonempy se of nodes e, agens, E V V s a se of edges, and A = a j] R N N s a weghed adjacency marx sasfyng ha a j > 0fj, E and a j =0,oherwse Anedgej, ne means ha agen can oban nformaon from agen j Here, we exclude he self-connecon, e,, / E and a =0 Asequence of edges, 2, 2, 3,, k, k wh j, j E, j 2,,k} s called a dreced pah from agen o agen k A dgraph G s sad o be conneced or conan a dreced spannng ree f here s a node such ha here exss a dreced pah from hs node o all he oher nodes n G Oherwse,G s sad o be dsconneced The Laplacan marx L =l j] R N N of he dgraph G s defned as l = j aj, lj = aj Lemma 23] All he egenvalues of L have nonnegave real pars Zero s an egenvalue of L, wh as he rgh egenvecor, referred as λ L = 0 Furhermore, zero s a smple egenvalue of L f and only f G s conneced 3 Problem formulaon Consder an MAS wh N connuous-me agens ẋ = A γ] x + B γ] u where N =, 2,,N}, x R n and u R m are he sae and npu of he agen, respecvely γ S =, 2,,S} s a deermnsc swchng sgnal whch desgnaes dynamcs of all he agens a every nsan Le D, beheswchesofγ haakeplacenhenerval, andτ γ s he average dwell me of γ durng, D Then,, τ γ In hs paper, we assume ha all A j], j S are no Hurwz f A j] s Hurwz, he consensus can be reached by jus seng u =0and A j],b j] s sablzable The swchng nework opology among agens s modeled as a dgraph G σ = V,E σ,a σ, where opology swchng sgnal σ akes values n a fne se P =, 2,,p} whch corresponds o he graph se G,,G p} Ths paper consders wo scenaros of σ: Case σ s deermnsc and conrolled by a pecewse consan swchng sgnal Then, smlar o γ, we can defne N, and τ σ as he swches and average dwell me of σ durng he nerval, respecvely and N, τ σ The nal condons of MAS n Case nclude: he nal sae x 0, N, he nal dynamcs ẋ 0 = A γ0] x 0 + B γ0] u 0 as well as he nal

3 440 Inernaonal Journal of Auomaon and Compung 35, Ocober 206 communcaon graph G σ0,whereγ 0 = γ0 and σ 0 = σ0 Case 2 σ s subjec o a fne-sae Markov process wh saonary ranson probables Prσ + h =j σ =} = λ jh + oh, j where h>0andλ j > 0 s he ranson rae from mode a me o mode j a me + h Leλ = p j=,j λj and defne he ranson rae marx as Λ = λ j] p p The nal condons of MAS n Case2 nclude: he nal sae x 0, he nal dynamcs ẋ 0 = A γ0] x 0 + B γ0] u 0, and he nal probably dsrbuon F = f,f 2,,f p]ofσ, where f = Prσ0 = }, P Markov process σ s assumed o be rreducble and aperodc Therefore, s ergodc and exponenally converges o a unque nvaran dsrbuon π =π,π 2,,π p]whch can be obaned by solvng he equaon 0 = πλ and p = π = For MAS, s clear ha reached consensus s relaed wh he swchng mechansms We now defne consensus noons for MAS n hese wo cases as follows Defnon TheMASssadoreachaconsensus n Case wh deermnsc swchng γ andσ f lm x x j =0,, j N hold for any nal condons; an almos sure consensus n Case 2 wh deermnsc swchng γ and Markov jumpng σ f Pr lm x x j =0} =,, j N hold for any nal condons For MAS, followng conrol law s desgned for agen : u = K γ] j P a j σ xj x, N 2 where K γ] R m n s feedback marx o be desgned and a j σ s he, j-h elemen of adjacency marx A σ of G σ Defnng x = ] x T,x T 2,,x T T N and usng Laplacan marx L σ of G σ,wehave ẋ = I N A γ] L σ B γ] K γ] x 3 Then, nroduce he followng varable ransformaon: where ξ =T I n x T = and s nverse marx s T = 0 Rewrng 3 wh respec o ξ, weoban ξ = I N A γ] TL σ T B γ] K γ] ξ 4 Then, paron T and T as follows ] Y ] T =, T = N W 2 Y 2 where Y R N Denoe ξ = T, ξ T,ξR] T where ξr = x x 2 T, x 2 x 3 T,, x N x N T] T, and ξ = x Noe ha L σ N = 0 and defne l σ as he frs row of L σ, follows ha 7 can be rewren as ξ = A γ] ξ lσw 2B γ] K γ] ξ R 6 ξ R = I N A γ] Y 2L σ W 2 B γ] K γ] ξ R = F γ] σ ξ R Here, we call ξr = F j] σ ξr as he j-h subsysem of 7 Defnon 2 The sysem 7 s sad o be: exponenally sable f here exs ρ > 0 such ha ρ holds for any nal condons; exponenally almos sure sably } f here exs ρ> ln ξ 0 such ha Pr lm R ρ = holds for any nal condons Proposon The consensus almos sure consensus s acheved for he MAS f he sysem gven by 7 s exponenally sable almos sure sable wh respec o ξ R The proof s sraghforward So we om Nex, he swchng sgnals are fxed as γ = γ and σ = σ, ha s, he MAS s wh me-nvaran agen dynamcs and lm ln ξ R fxed opology Clearly, from 7 and Proposon, he MAS can reach a consensus f F γ] σ s Hurwz The followng lemma gves he condon Lemma 2 For MAS wh me-nvaran agen dynamcs A γ],b γ] and fxed opology G σ,hemarxf γ] σ n 7 s Hurwz f and only f G σ s conneced and all marces A γ] λ L σ B γ] K γ] =2,,N are Hurwz, where λ L σ are he nonzero egenvalues of he Laplacan marx L σ assocaed wh G σ Proof Wh he paron of T and T n 5 and noe ha L σ N = 0, one can oban TL σt 0 l σw ] 2 = 0 N Y 2L σw 2 5 7

4 Z H Yang e al / Consensus of Mul-agen Sysems Under Swchng Agen Dynamcs and Jumpng Nework Topologes 44 where l σ s he frs row of L σ I follows ha excep zero egenvalue λ L σ, marx Y 2L σw 2 has he same egenvalues as L σ Then, s possble o fnd a nonsngular marx M σ such ha Y 2L σw 2 = M σλ σm σ,whereλ σ s an upperrangular marx wh he dagonal enres be egenvalues of L σ excludng zero egenvalue λ L σ Therefore I N A γ] Λ σ B γ] K γ] F γ] σ =M σ I n M σ I n Denoe F γ] σ = I N A γ] Λ σ B γ] K γ] Noe ha he elemens of F γ] σ are eher block dagonal or block upper rangular Hence, F γ] σ s Hurwz f and only f he elemens along he dagonal, e, A γ] λ L σ B γ] K γ] =2,,N are Hurwz Snce A γ] s no Hurwz, mples ha all λ L σ 0,=2,,N Thus, by Lemma, G σ s conneced Assumpon The dgraph se G,,G p} conans r conneced graphs, where r s an neger sasfyng r p Whou loss of generaly, le G,,G r} and res be he ses of conneced and dsconneced dgraphs, respecvely Under Assumpon, one can defne λ + = mn ReλL k > 0 k P,λ L k 0 For an arbrary j S, snce A j],b j] s sablzable, here exss a soluon P j] > 0 o he followng Rcca nequaly P j] A j] + A j] T P j] 2λ + P j] B j] B j] T P j] + εi < 0 8 where ε>0 Then, he marx K j] s consruced as K j] = B j] T P j] 9 I s easy o verfy by Lemma 2 ha he proposed conroller 2 wh K j] defned n 9 guaranees ha F j] s Hurwz f G s conneced Then, by Assumpon, one can oban ha f r, F j] s Hurwz, oherwse F j] s no Based on hese facs, we can fnd wo consans α j] > 0 and β j] > 0 sasfyng ha j] ef e αj] βj], r 0 +βj], r < p e αj] The consans α j], β j] can be calculaed by akng advanage of he followng lemma Lemma 3 24] For a square marx F j], δ j] > max k Re λ k F j], α j] > 0 such ha j] ef eαj] +δj], 0 Feasble values for α j] can be obaned as follows: for any fxed symmerc and posve defne marx Q, leh j] > 0 be he soluon of hen F j] δ j] I H j] α j] + H j] F j] λ max =ln λ mn δ j] I T = Q H j] H j] From Lemma 3, s clear ha f F j] selec a δ j] δ j] > 0andβ j] < 0, hen β j] = δ j] 4 Man resuls 2 s Hurwz, one can = δ j],whereasff j] s no, α j] can be calculaed by 2 4 Consensus condon for Case In hs subsecon, boh agen dynamc swchng sgnal γ and opology jumpng sgnal σ are assumed o be pecewse consan and rgh connuous funcons of me Then, by applyng he dwell me and he average dwell me echnques, frs man resul s gven as follows Theorem Suppose Assumpon holds Then, he MAS wh proocol 2 can reach a consensus under deermnsc agen dynamc swchng γ and deermnsc opology jumpng σ f here exs average dwell me τ σ and τ γ such ha T d, 0 T c, 0 q 3 α + + qβ d β c + ρ 0 4 τ γ τ σ +q where ρ>0, α =max j S α j], α =max p α β d = max r+ p, j S β j], β c = mn r, j S β j], T c, 0 and T d, 0 are he oal acvaon mes of conneced and dsconneced dagraphs durng 0,, respecvely Proof If γ =j s nvaran n me nerval,, one can defne Φ j, as he ranson marx of j-h subsysem of 7 Assume ha swchng acons of γ ake k mes n nerval 0, and denoe, 2, k as he swchng nsans sasfyng 0 < < 2 < k < Le Ψ, 0 be he ranson marx of 7, holds ha k Ψ, 0 = Φ γk, k Φ γs s+, s

5 442 Inernaonal Journal of Auomaon and Compung 35, Ocober 206 where 0 = 0 From 0, he norm bound of Ψ, 0 can be esmaed as ln Ψ, 0 ln Φ γ k, k + k ln Φ γs s+, s α γ k] N, k r β γ k] T, k + = = T, k + k =r+ k r = β γ k] β γs] = T k s+, s+ α γs] N s+, s =r+ β γs] T s+, s 5 where N, represens he number of acvaons of he graph G n he nerval, andt, represenshe cumulave resdence me of G n, By rearrangng he erms n 5, we oban ln Ψ, 0 ln Φ γk, k + k ln Φ γs s+, s α γ k] N, k r β γ k] T, k + = = T, k + k =r+ k r = β γ k] β γs] = T k s+, s+ α γs] N s+, s =r+ β γs] T s+, s 6 Defne α =max j S α j], β c =mn r, j S β j] and β d =max r+ p, j S β j] I follows ha ln Ψ, 0 p α N, k + k N s+, s where T c, 0 = T d, 0 = = β ct c, 0 + β d T d, 0 r T, k + = =r+ T, k + r k T s+, s = =r+ k T s+, s 7 are oal acvaon mes of conneced and dsconneced dgraph, respecvely Noe ha he vss for whch σ s = σ + s are couned wce n he summaon of N, k + k N s+,s Precsely k N, k + N s+, s=n, 0 + D, 0 8 where D, 0 s he number of swchng acon of γ occurences when dgraph G s acve durng 0, Subsung 8 no 7, nocng ha p = D, 0 = D, 0, N, 0 = N, 0, and by he defnon of average p = dwell me of γ and σ, follows ha ln Ψ, 0 α N, 0 + D, 0 β ct c, 0+ β d T d, 0 α τ γ + τ σ β ct c, 0 + β d T d, 0 9 where α =max p α From 3 and noce he fac ha T c, 0 + T d, 0 =, leadso T c, 0 +q 20 T d, 0 q 2 +q Snce ξ R =Ψ, 0ξ R0, 20, 2 and condon 4 enal ha ln ξ R ln Ψ, 0 lm lm ρ 22 Remark Snce boh σk andγk are deermnsc swchng sgnals meanng ha hey can be conrolled, so s reasonable o place some resrcons on dwell me and average dwell me of σk andγk ntheoremoreacha consensus Accordng o he same reason, assumng average dwell me consran of γk n he followng Theorem 2 s also reasonable 42 Consensus condon for Case 2 Dfferen from Case, he agen dynamcs swchng γ s also consdered o be deermnsc, whle opology jumpng σ s assumed o be subjec o a sochasc Markov process n hs subsecon Correspondngly, he consensus noon s modfed as almos sure consensus The followng heorem gves he condon Theorem 2 Suppose Assumpon holds Then, he MAS wh proocol 2 can reach an almos sure consensus under deermnsc agen swchng γ and sochasc Markov opology jumpng σ f here exss average dwell me τ γ and a consan ρ>0 such ha η j] > 0, j S 23 α mnη j] + ρ 0 24 τ γ j where η j] = p = απλ + r = βj] π p =r+ βj] π, α and α are defned n Theorem Proof Inroduce he followng ndcaor funcons:, σ = J = 0, σ I j] =, γ =j 0, γ j Le r j],0 = I j] τdτ denoe he fracon of me n 0 he nerval 0,whenj-h subsysem s funconng Le T j], 0 = I j] τj 0 τdτ be he cumulave resdence

6 Z H Yang e al / Consensus of Mul-agen Sysems Under Swchng Agen Dynamcs and Jumpng Nework Topologes 443 me of subsysem j durng 0, when he dgraph G s acve By combnng 6 and 7, we oban ln Ψ, 0 p α N, k + k N s+, s = S r β j] T j], 0 + S β j] T j], 0 j= = j= =r+ Recallng 8 and nocng p = D, 0 = D, 0, hen ln Ψ, 0 αd, 0 + p α N, 0 S r j= = β j] = T j], 0 + S j= =r+ β j] T j], 0 For a Markov process σ}, lee π ] denoe he expecaon wh respec o he unque nvaran dsrbuon π Then from 25], for each mode : E πn, 0] = π π λ 25 E πt j] ]=r j],0 π 26 From he defnon of average dwell me of γ andexplong he ergodc law of large numbers, one obans wh Probably lm ln Ψ, 0 α τ γ p α π λ = S r r j] β j] π + S r j] j= = j= where r j] = lm r j],0 Then, wh Proposon, lm sup ln Ψ, 0 α τ γ =r+ β j] π sasfyng ha s j= rj] = s r j] η j] j= where η j] = p = απλ + r = βj] π p =r+ βj] π Snce ξ R =Ψ, 0ξ R0 From condon 23 and 24, wh probably, lm sup ln ξr lm sup ln Ψ, 0 α mn jη j] ρ τ γ 5 Smulaon In hs secon, we gve wo examples o llusrae he valdy of he resuls Example Consder he MAS wh four agens where each agen has wo knds of dynamcs whch are specfed as ] ] A ] 0 05 =, B ] = 0 05 and A 2] = ], B ] = ] I can be checked ha A ] and A 2] are no Hurwz The nework opology s assumed o be swched beween G and G 2 whcharegvennfgwhheweghoneach edge assumed o be Clearly, G s conneced whle G 2 s no Then, by akng advanage of 8 and 9 wh ε =, feasble K ] and K 2] are obaned as ] K ] = K 2] = I can ndeed be verfed ha he proposed conroller 2 wh he above K ] and K 2] guaranees ha F ], F 2] are Hurwz, whle F 2], F 2] 2 are no, whch s conssen wh Lemma 2 Moreover, suable α j] and β j] defned n 0 for esmang e F j] are calculaed by Lemma 3 as α ] =0982 4, α 2] =7 7, β ] =06, β 2] =04, α ] 2 = 068 0, α 2] 2 = , β ] 2 } = 00, β 2] 2 = 002 Ths leads o α = max α ],α2] = 7 7, α 2 = } max α ] 2,α2] 2 = 068 0, α = maxα,α 2 } = 7 7, } } β c =mn β ],β2] =04 and β d =max β ] 2,β2] 2 = 002 Fg ] Communcaon opologes In smulaon, we choose he nal saes as x 0 =, 23 T, x 20 = 5, 3 T, x 30 = 4, 22 T, x 40 = 05, 2 T Clearly, o make condon 4 n Theorem hold, mus sasfy ha q< βc, and also noe ha β d larger q wll lead o larger τ γ and τ σ, so n hs smulaon, we choose a proper q =02, τ γ = 20, and τ σ = 20 Then, 4 can hold wh ρ< To llusrae sysem behavor, le boh γ andσ be perodc swchng sgnals sasfyng ha γk = 2 kt,kt + 0 kt +0,kT + 40, k =0,, 2, 2 kt 2,kT 2 +6 σk =, k =0,, 2, kt 2 +6,kT where T =50andT 2 = 40 I s easy o check ha > 0 and ρ>0 such ha > T d, 0 T c, 0 q =02 α + + qβ d β c + ρ ρ 0 τ γ τ σ + q Then, by Theorem, consensus can be reached and can ndeed be seen from he smulaon resul n Fg 2

7 444 Inernaonal Journal of Auomaon and Compung 35, Ocober 206 a Frs componen rajecores a Frs componen rajecores b Second componen rajecores b Second componen rajecores Fg 2 Smulaon for Theorem : consensus of four agens Fg 3 Smulaon for Theorem 2: consensus of four agens Example 2 Consder he same MAS as n he frs example, excep ha he opology swchng s drven by a Markov process wh ranson rae marx ] Λ= Then, he assocaed nvaran dsrbuon s π = ] Snce urns ou ha η ] =043 > 0, η 2] =0084 > 0, he frs condon of Theorem 2 s sasfed Accordng o hs heorem, almos sure consensus can α be reached f τ γ 33 To es hs resul, mn η j] ρ he sgnal γk s changed o 2 kt,kt + 0 γk =, k =0,, 2, kt +0,kT + 8 where T = 28 Thus, he condons of Theorem 2 are sasfed and he smulaon resul s shown n Fg 3, whch showsheconsensussachevedasexpeced 6 Conclusons Consensus problems of mul-agen sysems under boh swchng agen dynamcs and jumpng nework opologes have been addressed n hs paper The agen dynamc swchng s assumed o be deermnsc, whle he nework opology jumpng s consdered respecvely for wo cases: deermnsc jumpng and Markov jumpng By applyng he dwell me and he average dwell me echnques, a suffcen consensus and almos sure consensus condon are provded for hese wo cases, respecvely The proof reles on he fac ha hese wo dfferen swches are ndependen Noe ha hs arcle focuses on he dynamcs swchng of each agen as synchronous One fuure drecon s o nvesgae consensus problem of mul-agen sysem wh asynchronous swchng agen dynamcs Anoher fuure research s ha he agen dynamc swchng and nework opology swchng are no ndependen bu neracve References ] Y Eun, H Bang Cooperave conrol of mulple unmanned aeral vehcles usng he poenal feld heory Journal of Arcraf, vol 43, no 6, pp , ] WRen,RWBeard,EMAlknsInformaonconsensus n mulvehcle cooperave conrol: Collecve group behavor hrough local neracon IEEE Conrol Sysems Magazne, vol 27, no 2, pp 7 82, ] JPDesa,JPOsrowsk,VKumarModelngandconrol of formaons of nonholonomc moble robos IEEE Transacons on Robocs and Auomaon, vol 7, no 6, pp , 200 4] H A Clarke, W H Chen Trajecory generaon for auonomous soarng UAS Inernaonal Journal of Auomaon and Compung, vol 9, no 3, pp , 202 5] R Olfa-Saber Flockng for mul-agen dynamc sysems: Algorhms and heory IEEE Transacons on Auomac Conrol, vol 5, no 3, pp , 2006

8 Z H Yang e al / Consensus of Mul-agen Sysems Under Swchng Agen Dynamcs and Jumpng Nework Topologes 445 6] Z Q Wu, Y Wang Dynamc consensus of hgh-order mulagen sysems and s applcaon n he moon conrol of mulple moble robos Inernaonal Journal of Auomaon and Compung, vol 9, no, pp 54 62, 202 7] G Y Mao, S Y Xu, Y Zou Necessary and suffcen condons for mean square consensus under Markov swchng opologes Inernaonal Journal of Sysems Scence, vol 44, no, pp 78 86, 203 7] R Olfa-Saber, J A Fax, R M Murray Consensus and cooperaon n neworked mul-agen sysems Proceedngs of he IEEE, vol 95, no, pp , ] H Y Zhao, J H Park, Y L Zhang, H Shen Dsrbued oupu feedback consensus of dscree-me mul-agen sysems Neurocompung, vol 38, pp 86 9, 204 8] W Ren Consensus sraeges for cooperave conrol of vehcle formaons IET Conrol Theory and Applcaons, vol, no 2, pp , ] Z L, Z Duan, G Chen Dynamc consensus of lnear mulagen sysems IET Conrol Theory and Applcaons, vol5, no, pp 9 28, 20 9] A Tahbaz-Saleh, A Jadbabae Consensus over ergodc saonary graph processes IEEE Transacons on Auomac Conrol, vol 55, no, pp , ] I Mae, J S Baras, C Somaraks Convergence resuls for he lnear consensus problem under Markovan random graphs SIAM Journal on Conrol and Opmzaon, vol 5, no 2, pp , 203 0] K Hengser-Movrc, K Y You, F L Lews, L H Xe Synchronzaon of dscree-me mul-agen sysems on graphs usng Rcca desgn Auomaca, vol 49, no 2, pp , 203 ] W N, D Z Cheng Leader-followng consensus of mulagen sysems under fxed and swchng opologes Sysems & Conrol Leers, vol 59, no 3 4, pp , 200 2] D Vengersev, H Km, J H Seo, H Shm Consensus of oupu-coupled hgh-order lnear mul-agen sysems under deermnsc and Markovan swchng neworks Inernaonal Journal of Sysems Scence, vol 46, no 0, pp , ] B Lu, F Wu, S Km Swchng LPV conrol of an F-6 arcraf va conroller sae rese IEEE Transacons on Conrol Sysems Technology, vol 4, no 2, pp , ] Y F Su, J Huang Two consensus problems for dscreeme mul-agen sysems wh swchng nework opology Auomaca, vol 48, no 9, pp , 202 3] H Km, H Shm, J Back, J H Seo Consensus of oupucoupled lnear mul-agen sysems under fas swchng nework: Averagng approach Auomaca, vol 49, no, pp , 203 4] W Y Xu, J D Cao, W W Yu, J Q Lu Leader-followng consensus of non-lnear mul-agen sysems wh jonly conneced opology IET Conrol Theory and Applcaons, vol 8, no 6, pp , 204 5] Y Zhang, Y P Tan Consenably and proocol desgn of mul-agen sysems wh sochasc swchng opology Auomaca, vol 45, no 5, pp 95 20, ] K Y You, Z K L, L H Xe Consensus condon for lnear mul-agen sysems over randomly swchng opologes Auomaca, vol 49, no 0, pp , ] W Ren, R W Beard Consensus seekng n mulagen sysems under dynamcally changng neracon opologes IEEE Transacons on Auomac Conrol, vol 50, no 5, pp , ] M Tanell, B Pcasso, P Bolzern, P Colaner Almos sure sablzaon of unceran connuous-me Markov jump lnear sysems IEEE Transacons on Auomac Conrol, vol 55, no, pp 95 20, ] P Bolzern, P Colaner, G De Ncolao Almos sure sably of Markov jump lnear sysems wh deermnsc swchng IEEE Transacons on Auomac Conrol, vol 58, no, pp , 203 Zhen-Hong Yang s curren a maser suden a Shangha Unversy, Chna Hs research neress nclude consensus of mul-agen sysem E-mal: yangzhh@shueducn ORCID D:

9 446 Inernaonal Journal of Auomaon and Compung 35, Ocober 206 Yang Song receved B Sc degree n 998 and Ph D degree n 2006, boh from Nanjng Unversy of Scence and Technology, Chna Currenly, he s an assocae research fellow n Shangha Unversy, Chna Hs research neress nclude swched sysem, neworked conrol heory and applcaon E-mal: y song@shueducn Correspondng auhor ORCID D: Mn Zheng receved Ph D degree n 2008 from Souheas Unversy, Chna Currenly, he s an assocae research fellow n Shangha Unversy, Chna Hs research neress nclude me-delay sysems sably analyss and synhess of eleoperaon robo sysem, nework conrol sysem, and nellgen conrol sysem E-mal: zhengmn203@shueducn ORCID D: We-Yan Hou receved he bachelor degree from Xdan unversy n 986, receved he M Sc degree from Tsnghua Unversy, Chna n 998, and hs he Ph D degree from Shangha Unversy, Chna n 2004 He s now a professor n School of Informaon Engneerng, Zhengzhou Unversy, Chna Hs research neress nclude dsrbued nellgen conrol, wreless communcaon, and dgal mage processng E-mal: houwy@zzueducn ORCID D:

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