Distributed consensus control for linear multi-agent systems with discontinuous observations

Transcription

1 Inernaional Journal of Conrol Vol. 86 No. January Disribued consensus conrol for linear muli-agen sysems wih disconinuous observaions Guanghui Wen a Zhongkui Li b Zhisheng Duan a * and Guanrong Chen ac a Deparmen of Mechanics and Aerospace Engineering College of Engineering Peking Universiy Beijing 87 China; b School of Auomaion Beijing Insiue of Technology Beijing 8 China; c Deparmen of Elecronic Engineering Ciy Universiy of Hong Kong Hong Kong SAR China (Received 5 May ; final version received 5 Augus ) Downloaded by [Peking Universiy] a 7:6 9 December This aricle addresses he disribued consensus problem of linear muli-agen sysems wih disconinuous observaions over a ime-invarian undireced communicaion opology. Under he assumpion ha each agen can only inermienly share is oupus wih he neighbours a class of disribued observer-ype of proocols are designed and uilised o achieve consensus. By using appropriae marix decomposiion i is shown ha consensus in he closed-loop muli-agen sysems under a conneced opology can be convered o he simulaneous asympoic sabiliy of a se of swiching sysems whose dimensions are he same as each agen. From a muliple Lyapunov funcions approach i is proved ha here exiss a proocol o guaranee consensus if he communicaion ime rae is larger han a hreshold value. Furhermore a disribued pinning conrol mehod is employed o solve he consensus problem on an arbirary given opology which needs no be conneced. Paricularly he quesions of wha kind of agens and a leas how many agens should be pinned are addressed. The effeciveness of he analyical resuls is finally verified by numerical simulaions. Keywords: muli-agen sysem; consensus; pinning conrol; disconinuous observaion; lyapunov funcion. Inroducion Recenly disribued cooperaive conrol of muliagen sysems has received considerable aenion from various scienific communiies due o he growing ineres in undersanding he inriguing animal group behaviours such as swarming (Gazi and Passino 3) and flocking (Su Wang and Lin 9; Wen Duan Li and Chen b) and heir poenial applicaions in formaion conrol of saellies (Smih and Hadaegh 5) eaming of muli-roboics (Ren and Sorensen 8) and design of sensor neworks (Yu Chen Wang and Yang 9b). Among he numerous research opics in disribued conrol of muli-agen sysems consensus problem is of paricular imporance which refers o designing an appropriae proocol based only on he local relaive informaion among neighbouring agens o guide all agens o reach an agreemen (see survey papers Olfai-Saber Fax and Murray (7) Ren Beard and Akins (7a) and he references herein). Vicsek Cziro k Ben-Jacob Cohen and Shoche (995) inroduced a simple ye effecive discree-ime model for phase ransiion of a group of auonomous agens and numerically invesigaed he angle consensus behaviour of he muli-agen sysem. By using he ools from he algebraic graph heory a heoreical explanaion of he consensus behavior observed in Vicsek e al. (995) was firs given in Jadbabaie Lin and Morse (3). A general framework of he consensus problem for a nework of agens wih single-inegraor dynamics under a fixed or a swiching opology and possible communicaion ime delays was suggesed and sudied in Olfai-Saber and Murray (4). The consensus condiions derived in Olfai-Saber and Murray (4) were furher relaxed in Ren and Beard (5) by proving ha consensus in muli-agen sysems wih single-inegraor dynamics can be achieved if and only if he ime-varying nework opology conains a direced spanning ree frequenly enough as he nework evolves over ime. Meanwhile he consensus problem for muli-agen sysems wih second- and higher-order dynamics were addressed (Ren and Akins 7; Xie and Wang 7; Hong Chen and Bushnell 8; Wen Duan Yu and Chen d; Ren Moore and Chen 7b; Zhang and Lewis ). Noe ha mos of he above-menioned works are concerned wih he case where he agens are governed by inegraor-ype dynamics. *Corresponding auhor. duanzs@pku.edu.cn ISSN 779 prin/issn online ß 3 Taylor & Francis hp://dx.doi.org/.8/ hp://

2 96 G. Wen e al. Downloaded by [Peking Universiy] a 7:6 9 December However muli-agen sysems wih general linear node dynamics are more popular (Ma and Zhang ; Li Duan Chen and Huang ; Zhang Lewis and Das ; Li Duan and Chen ) which include neworks of agens wih inegraor-ype of dynamics as special cases. In Ma and Zhang () from a saic oupu approach some necessary and sufficien condiions were derived for consensus of muli-agen sysems wih general linear node dynamics under a fixed direced opology. Consensus in muli-agen sysems wih general linear node dynamics under a fixed direced opology was invesigaed wih observerype proocols appropriaely designed in Li e al. () Zhang e al. () Li e al. (). Mos of he above-menioned resuls on he consensus problem in muli-agen sysems wih general linear dynamics are obained based on he assumpion ha informaion is ransmied coninuously among he agens i.e. each agen has o share is sae or oupu informaion wih is neighbours all he ime. However his may no always be he case in realiy. Someimes mobile agens can only communicae wih heir neighbours over some disconneced ime inervals due o for insance emporary sonar equipmen failures or he presence of communicaion obsacles even if he disances among hem are less han he communicaion radius. Ye equipmen failures may be recovered hrough repairing and communicaion obsacles may be bypassed as he sysem evolves in ime. Moivaed by hese facs and based on he works repored in Wen Duan and Chen (a) Wen Duan Li and Chen (c) he consensus problem for muli-agen sysems wih general linear node dynamics based on inermien observaions is sudied in his aricle where he ideal assumpion ha agens could ransmi heir oupu informaion o heir neighbours a all imes is removed. For convenience of analysis he communicaion opology among he agens is assumed o be undireced. For achieving consensus a new class of consensus proocols are proposed and analysed. By using ools from swiching sysems heory i is heoreically shown ha consensus in a closed-loop muli-agen sysem wih a conneced opology can be ensured if he communicaion rae is larger han a hreshold value. The analyical expression of he hreshold value is also explicily given. By using a disribued pinning-based conrol mehod he resuls are hen exended o consensus in muli-agen sysems wih an arbirary opology which needs no be conneced. In paricular he quesions of wha kind of agens and a leas how many agens should be pinned for achieving consensus are addressed and answered. Numerical examples are finally given o verify he heoreical analysis. The res of his aricle is organised as follows. Some preliminaries and he model formulaion are presened in Secion. Consensus problem for muli-agen sysems wih disconinuous observaions under a ime-invarian conneced opology is sudied in Secion 3. Some exensions are given in Secion 4. In Secion 5 several numerical simulaions are provided for illusraion. Conclusions are finally drawn in Secion 6. Throughou his aricle le N and R be he se of naural and real numbers respecively and R nn be he ses of n n real marices. Le I n (O n ) be he n n ideniy (zero) marices and n ( n ) be he n-dimensional column vecor wih all enries equal o one (zero). Marices if no explicily saed are assumed o have compaible dimensions. The marix inequaliy A 4 B means ha boh A and B are square symmeric marices and ha A B is posiive-definie. diag{a a... a N } represens a diagonal marix wih a i i ¼... N being is diagonal elemens. Noaions and k k represen he Kronecker produc and he Euclidian norm respecively.. Preliminaries and formulaion of he model In his secion some preliminaries and he model formulaion for consensus in muli-agen sysems wih disconinuous observaions are inroduced.. Preliminaries An undireced graph G is a pair of (V E) where V¼{... N} is a node se and E{(i j ) i j V}is an edge se in which an edge is represened by an unordered pair of disinc nodes. Two nodes i j are adjacen or neighboring if (i j) is an edge of graph G i.e. (i j) E. A pah on G from node i o node i s is a sequence of ordered edges of he form (i k i kþ ) k ¼... s. An undireced graph is conneced if here exiss a pah beween every pair of disinc nodes oherwise is disconneced. Only simple graphs are considered here i.e. muliple edges and self-loops are forbidden in G. A conneced subgraph of G which is maximal is called a componen of G. The adjacency marix A¼[a ij ] NN of a graph G is defined by a ii ¼ for i ¼... N and a ij ¼ a ji 4 for (i j) E bu oherwise. The Laplacian marix L¼[l ij ] NN is defined as l ij ¼ a ij i 6¼ j and l ii ¼ P N k¼ a ik for i ¼... N. For an undireced graph G boh is adjacency marix and Laplacian marix are symmeric. The following lemmas can be found in graph heory exbooks (e.g. Godsil and Royle ).

3 Inernaional Journal of Conrol 97 Downloaded by [Peking Universiy] a 7:6 9 December Lemma : Suppose ha an undireced graph G has m componens. Then here exis a permuaion marix W of order N such ha 3 e L O O O e L W T O LW ¼ O 5 ðþ O O e Lmm where e L R q q e L R q q... e L mm R q mq m are P symmeric marices wih zero row sums wih m j¼ q j ¼ N and q q m N. Furhermore R(L) ¼ N m where R(L) represens he rank of L. Remark : Any undireced graph G conains a leas one componen and a mos N componens. Thus according o Lemma one has R(L) N. Lemma : Consider a conneced undireced graph G. Then is a simple eigenvalue of is Laplacian marix L and all he oher eigenvalues of L are posiive real numbers.. Formulaion of he model Consider a nework of idenical agens wih linear or linearised dynamics described by _x i ðþ ¼Ax i ðþþbu i ðþ ðþ y i ðþ ¼Cx i ðþ i ¼... N where x i () R n is he sae u i () R m is he conrol inpu y i () R p is he measured oupu and A B and C are consan real marices wih compaible dimensions. The communicaion opology among he N agens is represened by an undireced graph G consising of he node se V¼{... N} and he edge se E{(i j) i j V}. An exising edge (i j) (i 6¼ j) means ha agens i and j can obain informaion from each oher. In some real siuaions agens may only sense he oupus of heir neighbours over some disconneced ime inervals due o he unreliabiliy of communicaion channels failure of physical devices ec. Moivaed by his observaion he following disribued observer-ype of consensus proocol wih disconinuous dynamic oupu measuremens is proposed for each agen i: _v i ðþ ¼Av i ðþþbu i ðþþcf XN j¼ a ij Cðvi ðþ v j ðþþ ðy i ðþ y j ðþþ u i ðþ ¼Kv i ðþ ½k! k! þ Þ _v i ðþ ¼Av i ðþ u i ðþ ¼ ½k! þ ðk þ Þ!Þ k N ð3þ where v i () R n is he sae of he observer embedded in agen i c 4 is he coupling srengh F R np and K R mn are feedback marices A¼[a ij ] NN is adjacency marix of graph G and! 4 4. Le i () ¼ (x i () T v i () T ) T. Then i follows from () and (3) ha _ i ðþ ¼A i ðþþc PN l ij H j ðþ ½k! k! þ Þ j¼ ð4þ _ i ðþ ¼A i ðþ ½k! þ ðk þ Þ!Þ k N where A ¼ A BK A ¼ A O þbk O O H ¼ FC FC and L¼[l ij ] NN is he Laplacian marix of graph G. Remark : The consensus problem for muli-agen sysems wih general linear node dynamics based on inermien relaive sae informaion has been sudied recenly in Wen e al. (a) using he sae informaion of agens which are hard or impossible o obain in pracice. In conras he presen proocol (3) depends only on he relaive oupu informaion of neighboring agens. Definiion : The consensus problem of muli-agen sysem () is solved by proocol (3) if for any iniial condiions he saes of sysem (4) saisfy lim iðþ j ðþ ¼! 8i j ¼... N: ð5þ Remark 3: From Definiion consensus in muliagen sysem () is solved by proocol (3) if and only if boh he saes of agens and he proocols embedded in agens asympoically approach he same values respecively. Finally he following lemmas (referring o Boyd Ghaoui Ferion and Balakrishnan 994) are inroduced. Lemma 3: The maximum real pars of eigenvalues of a marix A R nn is less han where 4 if and only if here exiss a marix P ¼ P T 4 such ha A T P þ PA þ P 5 : Lemma 4: For marices A B C and D of appropriae dimensions one has () (A) B ¼ A (B) ¼ (A B) 8 R; () (A þ B) C ¼ A C þ B C; (3) (A B)(C D) ¼ (AC) (BD).

4 98 G. Wen e al. Downloaded by [Peking Universiy] a 7:6 9 December 3. Main resuls In his secion he mains resuls of his aricle are described and proved. 3. Consensus in muli-agen sysems under a conneced opology In his subsecion disribued consensus in muli-agen sysem () under an undireced conneced opology is addressed. Assumpion : The undireced communicaion opology G is conneced. Under Assumpion i follows from Lemma ha ¼ (/N ) N is he lef eigenvecor of he Laplacian marix L associaed wih he zero eigenvalue. Le s() P ¼ (s () T s () T... s N () T ) T where s i ðþ ¼ i ðþ N N j¼ jðþ i ¼... N. Then sðþ ¼ I N N T In ðþ ð6þ where () ¼ ( () T () T... N () T ) T. I is easy o verify ha s() ¼ if and only if () ¼ () ¼¼ N () for all. I hen follows from (3) and (6) ha _sðþ ¼ðI N A þ clhþsðþ ½k! k! þ Þ _sðþ ¼ðI N A ÞsðÞ ½k! þ ðk þ Þ!Þ ð7þ where A ¼ A BK þbk O O H ¼ FC FC A ¼ A O and L¼[l ij ] NN is he Laplacian marix of graph G. Le Y R N(N ) Y R (N )N T R NN and a diagonal marix D R (N )(N ) be such ha T ¼ ð N Y Þ T ¼ T T LT ¼ ¼ Y T N N D! ð8þ where he diagonal enries of D are he non-zero eigenvalues of Laplacian marix L. By using he linear ransformaion ðþ ¼ T I n sðþ ð9þ wih () ¼ ( () T () T... N () T ) T i follows from Lemma 4 and (7) ha _ðþ¼ði N A þ c HÞðÞ ½k! k! þ Þ _ðþ¼ði N A ÞðÞ ½k! þ ðk þ Þ!Þ k N ðþ where A A H are defined in (7) and is given in (8). Noe ha () for all. Thus he consensus problem for muli-agen sysem () is solved by proocol (3) if and only if i () i ¼ 3... N converge asympoically o zero which is in urn equivalen o ha he following N sysems _z i ðþ ¼ðI N A þ c i HÞz i ðþ ½k! k! þ Þ _z i ðþ ¼ðI N A Þz i ðþ ½k! þ ðk þ Þ!Þ k N ðþ are simulaneously asympoically sable where i i ¼ 3... N are he non-zero eigenvalues of he Laplacian marix L. Taking linear ransformaion i ðþ ¼ I I z O I i ðþ i ¼ 3... N ðþ and using () one obains _ i ðþ ¼T i i ðþ ½k! k! þ Þ _ i ðþ ¼A i ðþ ½k! þ ðk þ Þ!Þ k N ð3þ where T i ¼ A þ c ifc O A ¼ A O and i i ¼ 3... N are he non-zero eigenvalues of he Laplacian marix L. From he above analysis one has convered he consensus problem for muli-agen sysem () wih proocol (3) o he simulaneously asympoical sabiliy problem of N swiching sysems given by (3) which have he same low dimension as a single agen. One can see ha he effecs of he communicaion opology on he consensus are characerised by he non-zero eigenvalues of he Laplacian marix L. Nex an algorihm is presened o consruc proocol (3). Algorihm : Under Assumpion and he condiion ha (A B C) is boh sabilisable and deecable he consensus proocol (3) can be designed as follows. () Choose 4 and selec feedback gain marix K by Ackermann s formula such ha he real pars of he poles of A þ BK are smaller han. () Solve he following linear marix inequaliy (LMI): A T Q þ QA C T C þ Q 5 ð4þ o obain one soluion Q 4. Then le he feedback gain marix be F ¼ Q C T.

5 Downloaded by [Peking Universiy] a 7:6 9 December (3) Take he coupling srengh c / where is he second smalles eigenvalue of he Laplacian marix L. Lemma 5: The maximum real pars of eigenvalues of marices T i are less han where T i ¼ A þ c ifc O F ¼ Q C T Q is a soluion of LMI (4) and i i ¼ 3... N are he non-zero eigenvalues of L. Proof: Similarly o he proof of Proposiion in Li e al. () his lemma can be proved. For he readers convenience a skeched proof is given as follows. Le Q 4 be a soluion of LMI (4). Take F ¼ Q C T. Then one ges ða þ c i FCÞQ þ Q ða þ c i FCÞ T þ Q ¼ AQ þ Q A T c i Q C T CQ þ Q AQ þ Q A T Q C T CQ þ Q ð5þ where he las inequaliy of (5) is derived by using he fac c / wih being he second smalles eigenvalue of he Laplacian marix L. Pre- and pos-muliplying (5) by Q and is ranspose according o (4) yields ða þ c i FCÞQ þ Q ða þ c i FCÞ T þ Q 5 : ð6þ I hus follows from Lemma 3 and (6) ha he maximum real pars of he eigenvalues of marices T i are less han where T i ¼ A þ c ifc O : Lemma 6: There exis some W i 4 such ha Inernaional Journal of Conrol 99 Proof: P Wih ¼ max n j¼... n i¼ a ij he lemma can be proved by using he Gersˇgorin disc heorem (Huang 984). The following heorem is one main resul of his aricler. Theorem : For he muli-agen sysem () under Assumpion and moreover (A B C) is boh sabilisable and deecable he proocol (3) consruced by Algorihm solves he consensus problem if he communicaion ime rae /! 4 /( þ ) þ ln /[( þ )!] where 5 5 ¼ max i¼3... N { max (W i )/ min (P) max (P)/ min (W i )} 4 and marices W i and P are he posiive-definie soluions of LMIs (7) and (8) respecively. Proof: Consruc he following muliple Lyapunov funcion candidae for he ih swiching sysem of (3): ( VðÞ ¼ iðþ T W i i ðþ ½k! k! þ Þ i ðþ T P i ðþ ½k! þ ðk þ Þ!Þ ð9þ where he posiive-definie marices W i and P are he soluions of (7) and (8) respecively k N i ¼ 3... N. For [k! k! þ ) and an arbirarily given k N aking he ime derivaive of V() along he rajecories of sysem (3) gives _VðÞ 5 VðÞ ½k! k! þ Þ ðþ where is defined in Lemma 6. Similarly one has _VðÞ 5 VðÞ ½k! þ ðk þ Þ!Þ: ðþ Noe ha he swiching sysems (3) swiches a ¼ k! and ¼ k! þ k N. Based on he above analysis one obains Vð!Þ 5 e ð! Þ VðÞ T T i W i þ W i T i þ W i 5 ð7þ where 5 T i ¼ A þ c ifc O i ¼ 3... N. Proof: I follows direcly from Lemmas 3 and 5. Lemma 7: There exiss a 4 such ha for all 4 he following LMI A T 4 P þ PA 4 P 5 where A 4 ¼ A O has a soluion P 4. ð8þ 5 e þð! Þ VðÞ ¼ e VðÞ where ¼ (! ) ln. From he condiion /! 4 /( þ ) þ ln/[( þ )!] one has 4. By recursion for any posiive ineger k one ges Vðk!Þ 5 VðÞe k : ðþ For any 4 here exiss an r N such ha r! 5 (r þ )!. Then one has VðÞ 5 Vðr!Þe! 5 VðÞe rþ! 5 e! VðÞe ½=ð!þÞŠ

6 G. Wen e al. Downloaded by [Peking Universiy] a 7:6 9 December i.e. VðÞ 5 e for all where ¼ e! V() and ¼ /(! þ ). This indicaes ha he saes of agens exponenially converge o he same. Remark 4: In Theorem i is assumed ha he agens can obain he inermien relaive oupus of is neighbours periodically. However one may exend he resuls o he consensus in linear muli-agen sysems wih aperiodically inermien oupu communicaions by using he presen approach. Similarly o he proof of Theorem some corresponding heoreical resuls can be derived which are omied here for breviy. Remark 5: To successfully consruc proocol (3) according o Algorihm i should be firsly shown ha boh seps () and () of Algorihm are feasible for some 4. Noe ha here always exis some posiive scalar such ha seps () and () are feasible if (A B C) is sabilisable and deecable. I is also worh noing ha boh seps () and () are feasible for any given 4 if he marix riple (A B C) is conrollable and observable. Remark 6: In he conex of muli-agen sysems wih inermien observaions one ineresing and imporan issue is wha he minimum admissible communicaion rae is for achieving consensus for a given opology G wih fixed parameers and. However LMIs (7) and (8) are solved independenly which may inroduce conservaiveness in deermining he admissible communicaion rae o saisfy he consensus condiion. From he proof of Theorem one can see ha he minimum admissible communicaion rae under a given communicaion opology wih fixed parameers and can be obained by minimising in Theorem. Acually he minimum can be obained by solving he following opimisaion problem: () Minimise i subjec o: W i 4 P 4 W i P 5 i W i T T i W i þ W i T i þ W i 5 A T 4 P þ PA 4 P 5 where T i ¼ A þ c ifc O c i FC AþBK A 4 ¼ A O and i i ¼ 3... N are he non-zero eigenvalues of Laplacian marix L. () Take min ¼ max i¼3... N { i }. Then min is he minimum value of. Remark 7: In Xia and Cao (9) Cai Liu Xu and Sun (9) and Wang Hao and Zuo () some ineresing sae-based inermien feedback mehods are proposed and used o analyse he synchronisaion behaviours of coupled complex dynamical neworks which is a closely relaed opic wih consensus for muli-agen sysems. However from he perspecive of conrol heory he saes of a dynamical sysem are inernal informaion which is difficul or impossible o simulaneously obain. In conras proocol (3) is designed based only on he relaive oupus of neighboring agens which is more pracical. 3. Exensions In he las subsecion consensus in muli-agen sysems wih a conneced communicaion opology is sudied. Noice ha he communicaion opology may be disconneced in real muli-agen sysems due o exernal disurbances and/or sensing range limiaions. I is hus ineresing and imporan o furher invesigae consensus in muli-agen sysems wih an arbirarily given communicaion opology which may no be conneced. Moivaed by he works repored in Li Wang and Chen (4) Xiang Liu Chen Chen and Yuan (7) and Yu Chen and Lu (9a) and based on he analysis given in he las subsecion a pinningbased disribued conrol mehod is uilised here o guaranee consensus in he muli-agen sysem () under an arbirarily given communicaion opology wih disconinuous observaions. Suppose ha he communicaion opology of he muli-agen sysem () is given by G wih node se {... N}. Lemma implies ha changing he order of he node indexes of G will yield a new graph G wih he Laplacian marix L in he form of (). To realise his ransformaion an algorihm is given below. Algorihm : () Se NS ¼ {... N} and m ¼. () Arbirarily selec a node i from NS and use he deph-firs search algorihm Tarjan (97) o find he componen CC(i) of graph G conaining node i. Le m ¼ m þ and NS ¼ NS\ NSCCðiÞ where NSCC(i) denoes he node se of CC(i) NSCCðiÞ[NSCCðiÞ ¼f... Ng and NSCCðiÞ\NSCCðiÞ ¼;. (3) Check he condiion NS ¼;; if no re-perform sep (); if so go o sep (4). (4) Arrange he m componens of G in a sizedescending order. Then relabel he nodes from he firs componen o he las one so as o obain he graph G.

7 Inernaional Journal of Conrol Downloaded by [Peking Universiy] a 7:6 9 December Noe ha G and G are isomorphic o each oher. And under a given proocol (3) consensus in he closed-loop muli-agen sysem () wih opology G can be achieved if and only if consensus in he closedloop muli-agen sysem () wih opology G can be achieved. I is easy o check ha he Laplacian marix L of G is in he form of (). In he following consensus in muli-agen sysem () wih opology G is discussed. Obviously consensus canno be achieved if G is disconneced. To guaranee consensus in sysem () wih an arbirarily given opology a virual leader labeled N þ is inroduced whose dynamics are given as follows: _x Nþ ðþ ¼Ax Nþ ðþþbu Nþ ðþ y Nþ ðþ ¼Cx Nþ ðþ ð3þ where x Nþ () R n is he sae u Nþ () R m is he conrol inpu and y Nþ () R p is he measured oupu. Here he virual leader plays he role of a command generaor providing a reference sae for he followers o rack. Thus i is assumed ha u Nþ () and v Nþ () i.e. he saes of he virual leader evolves wihou being effeced by he followers and he virual leader is no need o observe he sas or oupus of any followers. I is furhermore assumed ha only a subse of agens called pinned agens have access o he oupus of he virual leader bu inermienly. For noaional convenience le P be he se of pinned agens and G be he augmened graph wih adjacency marix A¼½a ij Š ðnþþðnþþ where a (Nþ)(Nþ) ¼ a i(nþ) and a i(nþ) 4 if and only if i P. The objecive here is o find an appropriae conrol proocol for sysem () o achieve consensus in he sense of k i () Nþ ()k¼ 8i ¼... N where Nþ () ¼ (x Nþ () T v Nþ () T ) T. To do so a pinningbased disribued proocol based on (3) is presened for each follower i as follows: _v i ðþ ¼Av i ðþþbu i ðþþcf XNþ a ij Cðvi ðþ v j ðþþ j¼ ðy i ðþ y j ðþþ u i ðþ ¼Kv i ðþ ½k! k! þ Þ _v i ðþ ¼Av i ðþ u i ðþ ¼ ½k! þ ðk þ Þ!Þ k N ð4þ where v i () R n is he sae of he observer embedded in agen i c 4 is he coupling srengh and F R np and K R mn are he feedback marices. Before moving forward he following lemma is inroduced. Le V(i) be he node se of he ih ( i m) conneced componen of G. Lemma 8: Suppose ha for each i {... m} here exiss a leas one node j i such ha j i {V(i) \P}. Then marix b L 4 where b L¼Lþ L is he Laplacian marix of G and ¼ diag{a (Nþ) a (Nþ)... a N(Nþ) } R NN. Proof: Since for each i {... m} here exiss a leas one node j i such ha j i {V(i) \P} i hus follows from Corollaries and in Ren Beard and McLain (5) ha L has a simple zero eigenvalue and all he oher eigenvalues have posiive real pars where L is he Laplacian marix of he augmened graph G. Taking M ¼ (a (Nþ) a (Nþ)... a N(Nþ) ) T R N one has " L¼ b # L M : ð5þ T where b L¼Lþ. Thus all he eigenvalues of b L have posiive real pars. Since b L is symmeric b L is posiivedefinie. Based on he above analysis an algorihm is given here o consruc proocol (4). Algorihm 3: Suppose ha he communicaion opology G has m componen and ha (A B C) is boh sabilisable and deecable. Then he consensus proocol (4) can be designed as follows. () Using Algorihm o relabel he node indexes of G so as o ge a graph G whose Laplacian marix L has he form of (). Then pin m differen nodes j j... j m where j i V(i) and V(i) is he node se of he ih componen of G i ¼... m. () Choose & 4 and selec a feedback gain marix K by Ackermann s formula such ha he real pars of he poles of A þ BK are smaller han &. (3) Solve he LMI A T Q þ QA C T C þ & Q 5 ð6þ o obain one soluion Q 4. Then ake he feedback gain marix F ¼ Q C T. (4) Take he coupling srengh c / where is he smalles eigenvalue of marix b L. To derive anoher main resul he following lemmas are inroduced for which he proofs are simple herefore omied. Lemma 9: The maximum real pars of he eigenvalues of marices T i are less han & where T i ¼ A þ c ifc O F ¼ Q C T Q is a soluion of LMI (6) and i i ¼... N are he eigenvalues of b L.

8 G. Wen e al. Downloaded by [Peking Universiy] a 7:6 9 December Lemma : There exis some W i 4 such ha T T i W i þ W i T i þ &W i 5 ð7þ where & 5 & T i ¼ A þ c ifc O and i i ¼... N are he eigenvalues of b L. Now i is o presen anoher main resul of his aricle. Theorem : For he muli-agen sysem () wih a communicaion opology G he proocol (4) consruced by Algorihm 3 solves he consensus problem if he communicaion rae /! 4 /(& þ ) þ ln%/ [(& þ )!] where 5 & 5 & % ¼ max i¼... N { max (W i )/ min (P) max (P)/ min (W i )} 4 is given in Lemma 7 and marices P and W i are posiivedefinie soluions of LMIs (8) and (7) respecively. Proof: Take x i ðþ ¼x i ðþ x Nþ ðþ and v i ðþ ¼v i ðþ v Nþ ðþ where i ¼... N. Obviously consensus in () can be achieved if and only if k x i ðþk! and kv i ðþk! for all i ¼... N. Based on he above analysis and according o (4) one ges _ vi ðþ ¼Av i ðþþbu i ðþþcf XN j¼ b lij C v j ðþ x j ðþ u i ðþ ¼Kv i ðþ ½k! k! þ Þ _ vi ðþ ¼Av i ðþ u i ðþ ¼ ½k! þ ðk þ Þ!Þ k N ð8þ where b h L¼ b i l ij. NN Le i ðþ ¼ x i ðþ T v i ðþ T T i ¼... N. Then i follows from () and (3) ha _ i ðþ ¼A i ðþþc PN b lij H j ðþ j¼ ½k! k! þ Þ _ i ðþ ¼A i ðþ ½k! þ ðk þ Þ!Þ k N ð9þ where A ¼ A BK þbk O O H ¼ : FC FC A ¼ A O Similarly o he proof of Theorem he res of he proof can be compleed. Remark 8: The minimum admissible communicaion rae for achieving consensus under a given communicaion opology wih fixed parameers & and can be obained by minimising he parameer % in Theorem. And he minimum % can be obained by solving he following opimisaion problem: () Minimise % i subjec o: W i 4 P 4 W i P 5 % i W i T T i W i þ W i T i þ &W i 5 A T 4 P þ PA 4 P 5 where T i ¼ A þ c ifc O A 4 ¼ A O and i i ¼... N are he eigenvalues of b L. () Take % min ¼ max i¼... N {% i }. Then % min is he minimum value of %. 4. Simulaion examples In his secion wo simulaion examples are provided o verify he heoreical analysis. Example : Take each agen in a muli-agen sysem o be a wo-mass-spring sysem wih a single force inpu (Zhang Lewis and Qu ) whose dynamics are described by () wih x i ðþ x i ðþ k k k x i ðþ ¼B x i3 ðþ A A ¼ m m B A k x i4 ðþ m k m m B ¼ B A ð3þ where m and m are wo masses and k and k are spring consans. Furhermore ake he oupu marix B C C A m ¼. kg m ¼. kg k ¼.4 N/m and k ¼. N/m. Some simple calculaions show ha (A B C) is conrollable and observable. Consider a group of four agens wih undireced communicaion opology G as shown in Figure where he weighs are indicaed on he edges. Obviously G is conneced. An observer-ype of consensus proocol in he form of (3) is designed according o Algorihm. Take! ¼ 4 and ¼.

9 Inernaional Journal of Conrol x i () i= x () x () x 3 () x 4 () Figure. Communicaion opology G in Example Wih communicaion Wihou communicaion Figure 3. Consensus of rajecories of x i () i ¼ 3 4 in Example. Downloaded by [Peking Universiy] a 7:6 9 December 4 8 Figure. Inermien communicaion. Then some calculaions give he feedback gain marices in (3) as K ¼ ( ) and 3:838 :65 :594 :65 3:9534 :6675 F ¼ B :594 :6675 8:96 A : ð3þ :739 :67 :7378 According o Lemmas 6 and 7 one may ake ¼.5 ¼.5. Leing c ¼.75 and solving he opimisaion problem in Remark 6 gives min ¼. Thus he minimum communicaion rae is 79.%. Take ¼ 3. which means ha he communicaion rae is 8% see Figure for illusraion. According o Theorem he saes of all agens in sysem () will converge o he same value. The sae rajecories of he agens are shown in Figures 3 6 respecively. Use vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ux 3 E ðþ ¼ kx i ðþ x 4 ðþk i¼ ime o denoe he consensus errors of sysem () under proocol (3). Figure 7 demonsraes ha consensus is indeed achieved. Example : Consider a group of seven agens wih a communicaion opology G as shown in Figure 8 where he agens 3 belong o he firs componen and agens belong o he second componen. The agen dynamics and he parameers are he same x i () i=34. x i3 () i= x () x () x 3 () x 4 () Figure 4. Consensus of rajecories of x i () i ¼ 3 4 in Example. x 3 () x 3 () x 33 () x 43 () Figure 5. Consensus of rajecories of x i3 () i ¼ 3 4 in Example. as hose in Example. To achieve consensus a virual leader labeled by 8 is inroduced. Then an observerype of consensus proocol in he form of (4) is designed according o Algorihm 3. According o

10 4 G. Wen e al. x i4 () i= x 4 () x 4 () x 34 () x 44 () 5 5 Figure 6. Consensus of rajecories of x i4 () i ¼ 3 4 in Example. x i () i= x () x () x 3 () x 4 () x 5 () x 6 () x 7 () x 8 () Figure 9. Consensus of rajecories of x i () i ¼...8in Example. Downloaded by [Peking Universiy] a 7:6 9 December E () 9 8 Consensus errors E () Figure 7. Trajecories of consensus errors E () in Example sep () of Algorihm 3 agens and 7 are chosen as he pinned agens i.e. P¼{ 7}. Simple calculaions yield ha ¼.5493 where is he smalles eigenvalue of b L given in (5). Take & ¼.5 ¼.5 and! ¼ 4. Selecing c ¼.75 and solving he opimizaion problem in Remark 8 gives % min ¼.7. According o.75 Figure 8. Communicaion opology G in Example. x i () i= Theorem he minimum communicaion rae is 78.96%. Take ¼ 3. which means ha he communicaion rae is 8% see Figure for illusraion. The feedback marices F and K are aken o be he same as hose in Example. Then according o Theorem he saes of he agens in sysem () will approach hose of he virual leader. The sae rajecories of agens are shown in Figures 9 respecively. Use vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ux 7 E ðþ ¼ kx i ðþ x 8 ðþk i¼ x () x () x 3 () x 4 () x 5 () x 6 () x 7 () x 8 () Figure. Consensus of rajecories of x i () i ¼...8 in Example. o denoe he consensus errors of sysem () under proocol (4). Figure 3 demonsraes ha consensus is indeed achieved. 5. Conclusions In his aricle he consensus problem for muli-agen sysems wih general linear node dynamics and

11 Inernaional Journal of Conrol 5 x i3 () i= x 3 () x 3 () x 33 () x 43 () x 53 () x 63 () x 73 () x 83 () been proved ha consensus in he closed-loop muliagen sysems can be guaraneed under a fixed conneced undireced opology if he communicaion rae is larger han a hreshold value. Furhermore consensus for muli-agen sysems wih a disconneced communicaion opology has been sudied from a pinning-based disribued conrol approach. The effeciveness of he heoreical analysis has been verified by numerical simulaions. Fuure work will focus on consensus for muli-agen sysems wih higher-order nonlinear dynamics and disconinuous oupu measuremens. Downloaded by [Peking Universiy] a 7:6 9 December Figure. Consensus of rajecories of x i3 () i ¼...8 in Example. x i4 () i=...8. E () Consensus errors E () x 4 () x 4 () x 34 () x 44 () x 54 () x 64 () x 74 () x 84 () Figure. Consensus of rajecories of x i4 () i ¼...8 in Example. 5 5 Figure 3. Trajecories of consensus errors E () in Example. disconinuous oupu measuremens has been sudied. To achieve consensus a class of observer-ype of proocols have been proposed. By using ools from swiching sysems heory and marix analysis i has Acknowledgemens The auhors sincerely hank he Associae Edior and all he anonymous reviewers for heir valuable commens which helped hem improve he manuscrip of his paper. This work was suppored by he Hong Kong Research Grans Council under he GRF Gran CiyU 4/E he Naional Naure Science Foundaion of China under Grans he China Posdocoral Science Foundaion under Grans and he Informaion Processing and Auomaion Technology Prior Discipline of Zhejiang Province-Open Research Foundaion under Gran 8. References Boyd S. Ghaoui L.E. Ferion E. and Balakrishnan V. (994) Linear Marix Inequaliies in Sysem and Conrol Theory Philadelphia PA: SIAM. Cai S. Liu Z. Xu F. and Sun J. (9) Periodically Inermien Conrolling Complex Dynamical Neworks wih Time-varying Delays o a Desired Orbi Physics Leers A Gazi V. and Passino K.M. (3) Sabiliy Analysis of Swarms IEEE Transacions on Auomaic Conrol Godsil C. and Royle G. () Algebraic Graph Theory New York: Springer-Verlag. Hong Y. Chen G. and Bushnell L. (8) Disribued Observers Design for Leader-following Conrol of Muliagen Neworks Auomaica Huang L. (984) Linear Algebra in Sysem and Conrol Theory Beijing China: Science Press. Jadbabaie A. Lin J. and Morse A.S. (3) Coordinaion of Groups of Mobile Auonomous Agens Using Neares Neighbour Rules IEEE Transacions on Auomaic Conrol Li Z. Duan Z. and Chen G. () Dynamic Consensus of Linear Muli-agen Sysems IET Conrol Theory and Applicaions Li Z. Duan Z. Chen G. and Huang L. () Consensus of Muliagen Sysems and Synchronizaion of Complex Neworks: A Unified Viewpoin IEEE Transacions on Circuis Sysems I: Regular Papers

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