Distributed Linear Supervisory Control

Transcription

1 3rd IEEE Conference on Decision and Conrol December -7, Los Angeles, California, USA Disribued Linear Supervisory Conrol Ali Khanafer, Tamer Başar, and Daniel Liberzon Absrac In his work, we propose a disribued version of he logic-based supervisory adapive conrol scheme Given a nework of agens whose dynamics conain unknown parameers, he disribued supervisory conrol scheme is used o assis he agens o converge o a cerain se-poin wihou requiring hem o have explici knowledge of ha se-poin Unlike he classical supervisory conrol scheme where a cenralized supervisor makes swiching decisions among he candidae conrollers, in our scheme, each agen is equipped wih a local supervisor ha swiches among he available conrollers The swiching decisions made a a cerain agen depend only on he informaion from is neighboring agens We apply our framework o he disribued averaging problem in he presence of large modeling uncerainy and suppor our findings by simulaions I INTRODUCTION Logic-based swiching supervisory conrol has been proposed as a mehod o overcome limiaions of adapive conrol schemes [] A fundamenal difference beween he wo approaches is ha while adapive conrol requires coninuous uning of parameers, supervisory conrol relies on logicbased swiching among a collecion of candidae conrollers Coninuous uning suffers from well-known issues such as loss of sabilizabiliy In he classical supervisory conrol scheme, a cenralized supervisor esimaes he sae of he plan, and based on he hisory of esimaion errors, i acivaes a cerain candidae conroller For a more deailed sudy of supervisory conrol, see Chaper 6 of [] Supervisory conrol has been used in various problems and applicaions [3] [] In [3], [], he se-poin conrol problem has been sudied using a supervisory conrol framework I has also been uilized in pah-following problems for underacuaed sysems wih large modeling uncerainies [6] Recenly, supervisory conrol has been exended o addresses he problem of sabilizing uncerain sysems wih quanized oupus [] In his work, and moivaed by is aracive properies, we exend he supervisory conrol framework o a disribued seing A disribued version of supervisory conrol can have wide applicaions in sabilizaion and racking problems over neworked sysems in he presence of large modelling uncerainies Saemen of Conribuions The main conribuion of his paper is exending he cenralized supervisory conrol framework o a disribued Research suppored in par by an AFOSR MURI Gran FA and in par by a gran hrough he Informaion Trus Insiue of he Universiy of Illinois The auhors are wih he Coordinaed Science Laboraory, ECE Deparmen, Universiy of Illinois a Urbana-Champaign, USA {khanafe,basar,liberzon}@illinoisedu seing We firs provide a deailed descripion of he main componens in his scheme We prove ha when he se in which he unknown parameers ake values is finie, he swiching sops in finie ime a each node Furher, we provide sufficien condiions for achieving se-poin racking using his framework wihou requiring he individual agens o have explici knowledge of he desired se-poin Finally, we apply his scheme o he disribued averaging problem in he presence of unknown parameers, and we suppor our findings wih simulaions Noaion and Terminology We denoe he i-h row of a marix X R n m by [X] i R m, and he (i, j)-h enry of ha marix by [X] ij R Similarly, we denoe he i-h enry of a vecor x R n by [x] i R We adop he game-heoreic noaion x i o mean he collecion of vecors x j for all j i The ideniy marix is denoed by I, and he all-ones vecor is denoed by A direced graph is a pair G = (V, E), where V is he se of nodes and E V V is he se of edges Given G, we denoe an edge from node i V o node j V by (i, j) When (i, j) E if and only if (j, i) E, we call he graph undireced We call an undireced graph conneced if i conains a pah beween any wo nodes in V We use he words nodes and agens inerchangeably Organizaion In Secion II, we inroduce he sysem model and presen he problem formulaion The main componens of he disribued supervisory conrol scheme are provided in Secion III Secion IV conains he sabiliy analysis of he proposed scheme An applicaion o he disribued averaging problem is presened in Secion V We conclude he paper in Secion VI and provide ideas for fuure work II SYSTEM MODEL Consider a nework wih n nodes, and le x R n be he sae of he nework, where [x] i R is he sae of node i I is possible o exend his seing o he case where he sae of he i-h agen is k i -dimensional, where k + +k n = n; however, in his paper, we resric our aenion o he case where he sae of each node is scalar for simpliciy Le u R n be a vecor consising of he inpus o all he nodes wih [u] i R being a scalar inpu o node i Furher, le y R n be a vecor consising of he oupus of all he nodes wih [y] i R being a scalar oupu of node i Similar o he sae variables, i is possible o allow he nodes o ake muliple inpus and produce muliple oupus, and he resricion o //$3 IEEE 8

2 3 A p = Fig : A pah graph wih 3 nodes and is corresponding A p he single-inpu single-oupu se-up is for purpose of clariy in presenaion The nework is described by a graph whose opology is unknown, ie, he inerconnecions among he n nodes are no known Le P = {,, r} be a finie index se To each p P, we associae a graph G p = (V p, E p ), where V p is he se of verices, and E p V p V p is he se of edges The index p P is unknown o he nodes, and is corresponding graph, G p, describes he acual nework under sudy The graphs G p, p p, are differen possibiliies of wha G p migh be To each graph G p, here corresponds a linear dynamical sysem represened by a riple (A p, B p, C p ), where A p, B p, C p R n n Each riple represens a differen possibiliy of he acual sysem (A p, B p, C p ) ha governs he dynamics of he nework In paricular, we assume ha he nodes operae according o he following linear dynamics: ẋ = A p x + B p u, x() = x, () y = C p x We define he neighborhood of node i in he graph G p as N p (i) = {j V p : (j, i) E p } Noe ha we have no explicily included i in N p (i) o allow for applicaions where node i is no able o measure is own sae, for example In order o capure he underlying nework opology, we mus have ha he sae x i, conrol u i, and measuremen y i of node i can only depend on he saes, conrol inpus, and measuremens of he nodes in N p (i) To his end, we impose he following sparsiy consrain on he marices {A p, B p, C p : p P}: j / N p (i) = [A p ] ij = [B p ] ij = [C p ] ij =, p P () Under his consrain, he marices A p, B p, C p can be seen as an encoding of he opology of he graph G p To demonsrae he sparsiy consrain, consider he 3-node pah graph shown in Fig For his graph, he marix A p mus have he shown srucure, where can be any nonzero real number Furher, in order o be able o design decenralized conrollers, we mus resric he knowledge of node i abou he graph G p In paricular, we assume ha he knowledge of node i abou he opology of G p is only local; his can be capured by resricing he knowledge of node i o he se {[A p ] i, [B p ] i, [C p ] i } Formally, we make he following assumpion Assumpion : The se P is finie, and he se {[A p ] i, [B p ] i, [C p ] i : p P} is known o node i Our goal is o design decenralized conrol inpus [u] i, via an exension of he classical supervisory conrol scheme, in order o rack he following sable linear reference model: ẋ m = A m x m, x m () = x m, (3) y m = C m x m, where x m R n and A m, C m R n n Define he racking error, e T, as follows: e T = y m y The problem we are solving here is no he general racking problem, because here is no exernal reference signal The reason behind inroducing he reference model is moivaed by applicaions where he agens aemp o converge o a cerain se-poin wihou he explici knowledge of ha poin An example of such a scenario is he disribued averaging problem where nodes aemp o compue he average of heir iniial values, x, wihou knowing he value of he average a priori We will apply our framework o he disribued averaging problem in Secion V Moreover, he sandard sabilizaion problem, ie, regulaing he sae x o he origin, is a special case of he problem we are solving and can be achieved by removing he reference model, ie, seing x m In he following secion, we will inroduce he disribued supervisory conrol scheme, and explain he funcions of is main componens in deail III DISTRIBUTED SUPERVISORY CONTROL ARCHITECTURE Fig illusraes he general archiecure of he disribued supervisory conrol scheme In his scheme, each node has access o a bank of candidae conrollers ha ake as inpu he oupus of he nodes in is neighborhood as well as he racking error The Sparse Filers block in he figure emphasizes ha he local dynamics and conrollers of node i can only use informaion from neighboring nodes I should be noed ha here is no cenralized sparse filer implemened, and his block is inroduced for he sake of demonsraion only In his secion, we will precisely explain how he informaion from he neighboring nodes affec he dynamics and conrol inpus of node i Each node has a local supervisor: a dynamical sysem ha akes as inpu he oupus and conrol inpus of he neighboring nodes and produces a swiching signal The swiching signal provided by he supervisor acivaes one of he available conrollers The choice of a given conrol inpu is inended o minimize he racking error We will sudy he supervisor in more deails nex A The Disribued Supervisor We will refer o he collecion of he local supervisors by he disribued supervisor As illusraed in Fig 3, he disribued supervisor has hree main blocks: a muli-esimaor, a monioring signal generaor, and a swiching logic componen As in he cenralized supervisory conrol case, here are cerain properies we require from he individual blocks of he local supervisors which are crucial for achieving racking In paricular, he muli-esimaors mus guaranee ha a leas one esimaion error e p is small This will guaranee ha 9

3 Local Conrollers [u p ] i i Local Supervisor [u] i Sparse Filers Node i u i y i y m [y] i Model Fig : Disribued supervisory conrol archiecure swiching hals in finie ime As for he candidae conrollers, hey mus ensure ha he closed loop sysem is deecable wih respec o he esimaion error The swiching logic mus ensure ha he esimaion error is bounded, while avoiding fas swiching Here, we will work wih a specific choice of hese hree blocks y u Disribued Muli-Esimaor y p p P + e p e n p Monioring Signal Generaor Monioring Signal Generaor µ p Swiching Logic µ n p Fig 3: The disribued supervisor Swiching Logic Disribued Muli-Esimaor and Candidae Conrollers For now, we assume ha he conrol inpu u is given We will explain how o selec he conrol below The disribued muli-esimaor is a collecion of local muli-esimaors ha are implemened a he nodes A node i, he local muliesimaor is a dynamical sysem ha akes as inpu he oupus and conrol inpus of he neighboring nodes, and i produces an esimae [y p ] i, p P A each node, we adop he sandard Luenberger observer o design he muliesimaor Le he marix L p be sparse: j / N p (i) = [L p ] ij = () The esimaor equaions a node i can hen be wrien as [ẋ p ] i = [A p ] ij [x p ] j + [B p ] ij [u] j + [L p ] ij [y p y] j, j N(i) [y p ] i = j N(i) [C p ] ij [x p ] j, wih arbirary iniial values [x p ()] i To wrie he esimaor equaions more compacly, le x p = [[x p ],, [x p ] T and y p = [[y p ],, [y p ] T, for all p P Recalling ha he marices A p, B p, L p, C p are sparse, we can now wrie ẋ p = A p x p + B p u + L p (y p y), x p () = x p, y p = C p x p, where x p = [[x p ()],, [x p () ] T I is imporan o noe ha x p, y p are no sored a any node in he nework, since n hey are cenralized quaniies, and are inroduced merely for noaional simpliciy We define he esimaion error as e p = y p y, p P We denoe he esimaion error a he i-h node by e i p = [y p y] i, p P As for he candidae conrol inpus a node i, we assume hey are linear and given by [u p ] i = [K p ] ij [x p ] j + [F p ] ij [e T ] j, p P, j N(i) where he gain marices K p and F p mus be sparse o guaranee ha he conrollers are decenralized Formally, we have he following consrain on he gain marices: j / N i p = [K p ] ij = [F p ] ij =, p P () Similar o he esimaors, for each p P, we collec he conrol inpus of he nodes ino he vecor u p = [[u p ],, [u p ] T We can hen wrie u p = K p x p + F p e T, p P In general, he number of candidae conrol inpus need no be equal o P = r However, we will assume in his paper, for simpliciy, ha each node has access o r conrollers Monioring Signal Generaors Each node implemens a monioring signal generaor which keeps rack of he hisory of he esimaion errors This allows he swiching decisions (o be explained nex) o be based on he hisory of errors insead of he insananeous esimaion error values The monioring signals can be defined as any norm of he esimaion error Here, we define he monioring signal a he i-h node as he square of he L norm of e i p Formally, we wrie µ i p() = e i p(s) ds (6) I is more convenien for implemenaion purposes o express he monioring signal as an ordinary differenial equaion (ODE): µ i p = e i p, µ i p() =, p P Swiching Logic The swiching logic a each node akes he monioring signals µ i p, p P, as inpus and produces a swiching signal σ i : [, + ) P which deermines he conroller o be applied a each ime insan In paricular, we have [u] i = [u σi ] i, i {,, n} The chosen conroller should correspond o he monioring signal ha has he lowes value However, if we se σ i = min p P µ i p, we run ino he risk of fas swiching, which could be derimenal for he sabiliy of he sysem [] To his end, we will employ hyseresis swiching logic a each node wih hyseresis consan h i > The hyseresis consan is inroduced in order o preven σ i from swiching is value oo quickly A each node, we firs iniialize he swiching signal as follows: σ i () = min p P µi p() 6

4 Le ˆp i () := arg min p P µ i p() The signal σ i swiches is value a ime if µ iˆp i + h i µ i σ i Fig illusraes he hyseresis based logic a node i no i() = min pp µi p() ˆp i = arg min µ i p i =ˆp i pp µ iˆp i + h i apple µ i i yes Fig : Hyseresis based swiching logic IV STABILITY ANALYSIS In his secion, we will obain sufficien condiions for driving he racking error o zero Our approach will consis of wo main seps Firs, we will show ha swiching a all he nodes will hal in finie ime Then, assuming ha he swiching has sopped a all he nodes, we will sudy he deecabiliy properies of he closed-loop sysem In order o prove ha swiching erminaes in finie ime, i is insrumenal o show ha e p converges o zero exponenially fas When p = p, we have ẋ p ẋ = (A p + L p C p )(x p x) To guaranee ha x p converges exponenially fas o x, we need o impose he following condiion Condiion : The marix A p + L p C p is Hurwiz wih L p, C p saisfying () and (), respecively Remark : This condiion can be viewed as a disribued version of deecabiliy for he plan In he case when C p = I, for all p P, his condiion can be saisfied via diagonal dominance Diagonal dominance can be achieved by choosing [L p ] ii < [A p ] ii max [A p ] ij p P Noe ha he maximizaion can be carried ou locally a each node because of Assumpion To guaranee ha L p is sparse, we can selec i o be a diagonal marix Wih such choice of L, he marix A p + L p becomes diagonally dominan wih negaive diagonal enries, and by Gershgorin s circle heorem, i follows ha he marix is Hurwiz Under Condiion, x p converges exponenially fas o x, and consequenly e p = C p (x p x) converges o zero exponenially fas regardless of he applied conrol u We now have he following proposiion, which is an immediae exension of is counerpar in he cenralized archiecure [], [] j i Proposiion : For all i {,, n}, here exiss a ime Ti and an index qi P such ha σ i() = qi, for all T i Moreover, e i q L, for all i {,, n} i Proof: Because e p converges o zero exponenially fas, i follows from (6) ha µ i p is bounded Le K i N be such ha µ i p K i By definiion, µ i p is a nondecreasing funcion, for all p P Hence, each µ i p mus have a limi Because P is finie, here exiss a ime T i such ha eiher µ i p K i or µ i p( ) µ i p( ) < h i for all > T i ; herefore, a mos one more swich can occur for T i This in urn implies ha here exiss a ime Ti such ha σ i () = qi, q i P, for Ti Because µi p is bounded, µi q mus also be bounded i By (6), i hen follows ha e i q L i Noe ha afer he swiching sops, he esimae of node i, qi, migh no mach ha of anoher node j, q j In oher words, he percepion of node i abou he underlying graph will in general be differen han ha of node j This leads o new analysis challenges ha were no presen in he cenralized srucure In order o sudy he sabiliy of he sysem following erminaion of swiching, we firs define ˆx q := [[x q ],, [x q n ] T, q := [q,, q n] T Furher, we need o consruc he following marices: [A q ] [B q  q :=, ˆB ] [C q ] q :=, Ĉq :=, [A q n [B q n [C q n ˆK q := [K q ] [K q n, ˆF q := [F q ] [F q n, ˆL q := [L q ] [L q n Wih hese definiions, we can wrie he conrol law u afer he swiching sops as u = ˆK q ˆx q + ˆF q e T Define x := [x T, ˆx T q ]T Afer he swiching sops, he closed-loop sysem becomes: ẋ = Ax + Dx m ê q = Cx, where [ Ap B A = p ˆFq C p B p ˆKq ( ˆB q ˆFq + ˆL q )C p  q + ˆB q ˆKq + ˆL q Ĉ q [ ] Bp ˆFq C D = m, ˆB q ˆFq C m C = [ C p Ĉ q ] Consider now he marix [ ] Bp ˆFq + L Γ = p ˆB q ˆFq + ˆL, q and noe ha [ Ap + L A Γ C = p C p B p ( ˆK q ˆF q Ĉ q ) L p Ĉ q  q + ˆB q ( ˆK q ˆF q Ĉ q ) ], ] 6

5 Using oupu injecion, we can wrie ẋ = (A Γ C)x + Γ ê q + Dx m (7) To achieve racking, he marix A Γ C mus be Hurwiz Hence, in addiion o Condiion, we need o impose he following condiion Condiion : The marix Âq + ˆB q ( ˆK q ˆF q Ĉ q ) is Hurwiz for all q = [q,, qn] T wih {q,, qn} P, while saisfying () and () Remark : Assume ha B p = C p = I, for all p P, and le us selec [K p ] i = [A p ] i, for all i and p Noe ha such selecion for [K p ] i is made possible by Assumpion In his case, Condiion simplifies o requiring ˆF q o be sparse and Hurwiz This can be achieved by selecing F p = ki, where k R > We are now ready o sae he main resul of his secion Denoe he sae o which he reference model converges by x m Proposiion : Under Condiions and, and assuming ha x m converges asympoically o x m, he sae of he plan x remains bounded, and i asympoically converges o x = (A p + B p ( ˆK q ˆF q Ĉ q )) B p ˆFq C m x m Proof: Under Condiions and, he marix A Γ C is Hurwiz We know from Proposiion ha e i q L i for all i {,, n} Noing ha ê q = [e q,, en q ], n we conclude ha ê q converges o zero as Then, because x m is bounded, we deduce from (7) ha x mus remain bounded Using he fac ha ê q converges o zero, he seady-sae expression follows immediaely from (7) Remark 3: Because he objecive of he conroller is o enable he plan o rack he reference model, we are ineresed in cases where x = x m Assuming ha B p, C p, C m are all equal o he ideniy marix, for all p P, he seadysae expression simplifies o x = (A p + ˆK q ˆF q ) ˆFq x m Hence, by seing K p = A p for all p P, we will have x = x m if and only if p = q, ie, when all he nodes correcly idenify he unknown opology Oherwise, here will be a discrepancy beween x and he reference rajecory x m Noneheless, in cerain scenarios, his discrepancy may be negligible as we will demonsrae in Secion V Finally, we noe ha he muli-esimaors and conrollers we used here are only a specific possibiliy which we adoped o demonsrae he idea behind disribued supervisory conrol One possible variaion is o selec he conrol inpus as u p = K p y p + F p e T, p P By following similar seps o he above, one can show ha, wih his choice of conrollers, he marix ha is required o be Hurwiz in Condiion becomes  q + ˆB q ( ˆK q ˆF q )Ĉq Hence, differen choices of he conrollers will provide differen condiions on he sysem parameers o ensure sabiliy We are currenly invesigaing differen design choices ha would place less resricions on he sysem parameers V APPLICATION: TRACKING CONSENSUS DYNAMICS In his secion, we apply he disribued supervisory conrol scheme o he disribued averaging problem [3], [] in he case where he dynamics of he nodes conain unknown parameers In disribued averaging neworks, he nodes aemp o converge o he average of heir iniial values, x(), by performing local averaging When he dynamics of he nodes conain unknown parameers, adapive conrol echniques have been applied o solve his problem in [] By performing logic-based swiching, our scheme enables convergence o he average wihou requiring coninuous uning of parameers as in he adapive conrol approach In [3], [6], he problem of achieving consensus when he underlying opologies are swiching has been sudied Noe ha he opology in our case is unknown, bu fixed, and he swiching is performed a each node o choose he conroller ha minimizes he racking error To specialize he reference model (3) o he disribued averaging dynamics, we assume ha A m is he negaive of he weighed Laplacian marix of a conneced undireced graph In paricular, we have A m = A T m, A m =, [A m ] ij, [A m ] ij = (i, j) / E, i j, where he weighs [A m ] ij, j i are randomly generaed The conneciviy of he graph corresponding o A m is necessary for he convergence o he average [3] We assume ha here is full sae observaion across he nework; we herefore se C m = I and C p = I, for all p P We also se B p = I, for all p P Because he agens aemp o compue he average of heir iniial values, we se x m = x and x p = x, for all p P We consider a nework of n = agens and se x = [,, ] T The agens will herefore aemp o converge o T x = 3 We le P =, ha is, here are possible opologies, and we se p = The marices {A p } p P are generaed a random, wihou any conneciviy requiremens In order o saisfy Condiion, we pick L p = ki, for all p P, where k R is seleced as explained in Remark In view of Remark, we se K p = A p and F p = I, for all p P, in order o saisfy Condiion We will run wo experimens, where we generae differen {A p } p P, A m marices, each ime while respecing he conneciviy consrain on A m Fig demonsraes he rajecories of he sae of he nework x, he sae of he reference model x m, he swiching signals σ i, and he racking error e T for he firs experimen In his case, all he agens correcly converge o he correc opology G p, and, hence, converge o he average value 3 The racking error herefore converges o zero Fig 6 illusraes he same signals for he second experimen In his case, agen 3 does no selec he correc opology, ie, q3 Noneheless, i converges o 39, and he racking error is very small The remaining nodes all converge o 3 A poenial fuure research direcion is quanifying he racking error in he even where qi p 6

6 x xm 3 σ et 8 6 Fig : All he agens correcly idenify he unknown opology x 6 xm 3 σ et 8 6 Fig 6: One of he agens does no idenify he correc opology VI CONCLUSION We proposed a disribued version of he classical cenralized supervisory conrol scheme Our scheme is based on logic-based swiching among candidae conrollers a each node The swiching decisions performed a each node depend only on informaion from neighboring nodes The goal of he conrollers is o rack a se-poin, wihou requiring he agens o have explici knowledge of his poin The classical sabilizaion or regularizaion problem is a special case of his se-poin racking problem We showed ha swiching sops in finie ime a each node, and we provided sufficien condiions for sabiliy We applied our scheme o he disribued averaging problem when he dynamics of he agens conain unknown parameers Simulaion resuls demonsraed he efficacy of our scheme Fuure work will focus on making Condiion less sric, generalizing he problem o racking of a reference model wih a reference inpu signal, exending he scheme o handle ime-varying unknown parameers, and designing incenive schemes o ensure ha he majoriy of he agens idenify he underlying nework REFERENCES [] J P Hespanha, D Liberzon, and A S Morse, Overcoming he limiaions of adapive conrol by means of logic-based swiching, Sysems & Conrol Leers, vol 9, no, pp 9 6, 3 [] D Liberzon, Swiching in sysems and conrol Springer, 3 [3] A S Morse, Supervisory conrol of families of linear se-poin conrollers par i exac maching, Auomaic Conrol, IEEE Transacions on, vol, no, pp 3 3, 996 [], Supervisory conrol of families of linear se-poin conrollers robusness, Auomaic Conrol, IEEE Transacions on, vol, no, pp, 997 [] J P Hespanha, D Liberzon, and A S Morse, Logic-based swiching conrol of a nonholonomic sysem wih parameric modeling uncerainy, Sysems & Conrol Leers, vol 38, no 3, pp 67 77, 999 [6] A P Aguiar and J P Hespanha, Trajecory-racking and pahfollowing of underacuaed auonomous vehicles wih parameric modeling uncerainy, IEEE Transacions on Auomaic Conrol, vol, no 8, pp , 7 [7] L Vu, D Chaerjee, and D Liberzon, Inpu-o-sae sabiliy of swiched sysems and swiching adapive conrol, Auomaica, vol 3, no, pp , 7 [8] I Al-Shyoukh and J S Shamma, Swiching supervisory conrol using calibraed forecass, IEEE Transacions on Auomaic Conrol, vol, no, pp 7 76, 9 [9] S Baldi, G Baiselli, E Mosca, and P Tesi, Muli-model unfalsified adapive swiching supervisory conrol, Auomaica, vol 6, no, pp 9 9, [] L Vu and D Liberzon, Supervisory conrol of uncerain linear imevarying sysems, IEEE Transacions on Auomaic Conrol, vol 6, no, pp 7, [], Supervisory conrol of uncerain sysems wih quanized informaion, Inernaional Journal of Adapive Conrol and Signal Processing, vol 6, no 8, pp , [] J P Hespanha, Logic-based swiching algorihms in conrol, PhD disseraion, Yale Universiy, 998 [3] R Olfai-Saber and R M Murray, Consensus problems in neworks of agens wih swiching opology and ime-delays, IEEE Trans Auoma Conr, vol 9, no 9, pp 33, [] V D Blondel, J M Hendrickx, A Olshevsky, and J N Tsisiklis, Convergence in muliagen coordinaion, consensus, and flocking, in Proc Join h IEEE Conf Decision and Conrol and European Conrol Conf, December [] J Yao, D J Hill, Z-H Guan, and H O Wang, Synchronizaion of complex dynamical neworks wih swiching opology via adapive conrol, in Decision and Conrol, 6 h IEEE Conference on IEEE, 6, pp 89 8 [6] W Ren and R W Beard, Consensus seeking in muliagen sysems under dynamically changing ineracion opologies, IEEE Transacions on Auomaic Conrol, vol, no, pp 6 66, 63