122 IEEE TRANSACTIONS ON CYBERNETICS, VOL. 48, NO. 4, APRIL 218 A Barycenrc Coordnae-Based Approach o Formaon Conrol Under Dreced and Swchng Sensng Graphs Tngru Han, Zhyun Ln, Senor Member, IEEE, Ronghao Zheng, and Mnyue Fu, Fellow, IEEE Absrac Ths paper nvesgaes wo formaon conrol problems for a leader follower nework n 3-D. One s called he formaon marchng conrol problem, he obecve of whch s o seer he agens o manan a arge formaon shape whle movng wh he synchronzed velocy. The oher one s called he formaon roang conrol problem, whose goal s o drve he agens o roae around a common axs wh a arge formaon. For he above wo problems, we consder dreced and swchng sensng opologes whle he communcaon s assumed o be bdreconal and swchng. We develop approaches ulzng barycenrc coordnaes oward hese wo problems. Local conrol laws and graphcal condons are acqured o ensure global convergence n boh scenaros. Index Terms Barycenrc coordnae, formaon conrol, leader follower nework, swchng sensng graph. A. Poson-Based Conrol Each agen s able o sense s poson n a global coordnae sysem, hen s conrolled o reach a arge formaon, whch s prescrbed by he desred poson n he global coordnae sysem. B. Dsplacemen-Based Conrol Each agen s able o sense he relave posons of s neghbors n a global coordnae sysem, hen s conrolled o reach a arge formaon, whch s prescrbed by he desred dsplacemens n he global coordnae sysem. Thus, dsplacemen-based conrol requres ha he axes of he agens local coordnae sysems should have he same orenaons. I. INTRODUCTION FORMATION conrol has araced much neres due o s remendous engneerng applcaons [1]. Recenly, many works have been repored on hs opc such as [2] [4], some of whch focus on swarm and flockng [5] [7], conanmen conrol [8] [1] and achevng a gven paern [11] [16]. In he lgh of he knd of sensed varables for formaon conrol, he reference [17] dvdes he exsng leraure no hree caegores, ha s, poson-, dsplacemen-, and dsancebased conrol, whch are oulned as follows. Manuscrp receved Ocober 28, 216; revsed January 21, 217; acceped March 9, 217. Dae of publcaon March 31, 217; dae of curren verson March 15, 218. Ths work was suppored by he Naonal Naural Scence Foundaon of Chna under Gran 61673344 and Gran 6153335. Ths paper was recommended by Assocae Edor X.-M. Sun. (Correspondng auhor: Zhyun Ln. T. Han s wh he College of Elecrcal Engneerng, Zheang Unversy, Hangzhou 3127, Chna (e-mal: hanngru@zu.edu.cn. Z. Ln s wh he College of Elecrcal Engneerng, Zheang Unversy, Hangzhou 3127, Chna, and also wh he School of Auomaon, Hangzhou Danz Unversy, Hangzhou 3127, Chna (e-mal: lnz@zu.edu.cn. R. Zheng s wh he College of Elecrcal Engneerng, Zheang Unversy, Hangzhou 3127, Chna, and also wh he Zheang Provnce Marne Renewable Energy Elecrcal Equpmen and Sysem Technology Research Laboraory, Zheang Unversy, Hangzhou 3127, Chna (e-mal: rzheng@zu.edu.cn. M. Fu s wh he School of Elecrcal Engneerng and Compuer Scence, Unversy of Newcasle, Callaghan, NSW 238, Ausrala, and also wh Zheang Unversy, Hangzhou 3127, Chna (e-mal: mnyue.fu@newcasle.edu.au. Color versons of one or more of he fgures n hs paper are avalable onlne a hp://eeexplore.eee.org. Dgal Obec Idenfer 1.119/TCYB.217.2684461 C. Dsance-Based Conrol Each agen s able o sense he relave posons of s neghbors n s own local coordnae sysem, hen s conrolled o reach a arge formaon, whch s prescrbed by he desred neragen dsances. Ths ndcaes ha he agens do no need o have a common orenaon of local coordnae sysems. Mos recenly, an approach based on barycenrc coordnaes for formaon shape conrol s nroduced n [18] [2] whch can be execued n local frames. Ths s because he barycenrc coordnae s a geomerc noon characerzng he relave posons of a pon wh respec o oher pons n absence of he global frame [21]. We sum up he barycenrc-coordnaebased conrol as below. D. Barycenrc-Coordnae-Based Conrol Each agen s able o sense he relave posons of s neghbors n s own local coordnae sysem lke he one for dsance-based conrol, and s conrolled o reach a arge formaon, whch s prescrbed by he desred barycenrc coordnaes of every agen wh respec o s neghbors. As a comparson, he barycenrc-coordnae-based conrol requres less advanced sensng capably han poson-based and dsplacemen-based conrol, and needs less neracons among agens han dsance-based conrol, as shown n Fg. 1. Along he recen research based on barycenrc coordnaes, [18] and [19] adop complex barycenrc coordnaes o solve he formaon conrol problem n 2-D. Laer on, he work [2] mplemens real barycenrc coordnaes o sele 2168-2267 c 217 IEEE. Personal use s permed, bu republcaon/redsrbuon requres IEEE permsson. See hp://www.eee.org/publcaons_sandards/publcaons/rghs/ndex.hml for more nformaon.
HAN e al.: BARYCENTRIC COORDINATE-BASED APPROACH TO FORMATION CONTROL 123 Fg. 1. Sensng capably versus neracons. he formaon conrol problem n hgher dmensons. In he exsng works menoned above, [18] and [2] boh assume ha he sensng opology s fxed. However, dscussng swchng opologes becomes more aracve due o he exsence of sensng falure. From hs perspecve, [19] consders swchng opologes, bu only a pecular swchng sgnal s dscussed, ha s, an agen can eher sense all s neghbors or sense none of hem. Besdes, he developed conroller n [19] s no fully dsrbued because he compuaon of some parameers nvolves he knowledge concernng he whole nework. Movaed by hs need, our prevous work [22] sudes he formaon conrol problem over swchng opologes for a leader follower nework under he premse ha he followers le n he convex hull spanned by he leaders n he arge confguraon. The conroller n [22] s fully dsrbued, however, a convexy assumpon s mposed on he desred formaon shape, whch lms he applcably. In hs paper, we am o remove he convexy assumpon n [22] by allowng he neracon weghs o be negave, whch makes he seup general enough o nclude any desred formaon paern. Accordng o he praccal demand n engneerng, wo formaon conrol problems are nvesgaed n hs paper, namely, formaon marchng conrol problem and formaon roang conrol problem. The ask of formaon marchng s o drve he agens o realze any arge formaon whle movng wh he synchronzed velocy. Formaon marchng can be ofen seen n UAVs maneuverng and arge deecng. The goal of formaon roang s o seer he agens o move around a common axs wh any specfed formaon shape. Formaon roang conrol can fnd many useful applcaons such as saelles and spacecrafs flyng around he earh and crcular moble sensor neworks collecng measuremens [23] [25]. References [26], [27] sudy formaon roang conrol problems, where Ln e al. ulzed he Lyapunov-based approach, whch, however, s only workable for undreced and fxed opologes. In hs paper, alhough consderng dreced and swchng scenaros, we can also show ha he formaon roang conrol can be acheved by ulzng he barycenrc-coordnae-based conrol. To overcome he dffcules nduced by swchng opologes for he above wo problems, a communcaon graph s nroduced and an auxlary sae nformaon s exchanged, wh whch a fully dsrbued conrol law s developed for each problem. To show global convergence oward he desred formaon shape under swchng opologes, he dea of perssen excaon s adoped. Wh hs mehod, he graphcal condons are obaned o ensure ha all he agens are able o globally converge o he desred formaon. As he frs aemp o develop he barycenrc-coordnae-based conrol for general swchng opologes, sngle-negraor knemac model for each agen s consdered n hs paper. Wh he purpose of exendng he resuls o he second-order cases or more complex agen dynamcs, one way s o ake advanage of he backseppng phlosophy, ha s o say, desgn he acceleraon npu such ha he velocy sgnal n he second-order case converges o he desgned velocy npu n he paper [2]. The res of hs paper s organzed as follows. We revew some prelmnares abou graph heory n Secon II and wo formaon conrol problems are formulaed n Secon III. We develop a conrol law for he formaon marchng conrol and prove s convergence n Secon IV. Moreover, a conrol law s proposed for he formaon roang conrol and s sably s analyzed n Secon V. Three smulaon examples are presened n Secon VI o valdae our heorecal resuls. Secon VII concludes hs paper and pons ou possble fuure research. Noaon: R represens he se of real numbers. Denoe 1 n he n-dmensonal vecor of ones and I n he deny marx of order n. The symbol s he Kronecker produc. II. PRELIMINARIES A dreced graph s defned by G = (V, E, whch consss of a verex se V of elemens called nodes and a se E V V of ordered pars of nodes called edges. For each node V, denoe N + ={ V : (, E} he se of s n-neghbors whle N ={h V : (, h E} represens he se of s ou-neghbors. An n n Laplacan marx L correspondng o a dreced graph G s defned as w f = and N + L(, = f = and N + w k f = k N + where L(, s he (, -h enry of L and w R\{} s called he wegh on edge (,. For a dreced graph G = (V, E, a node v s sad o be k-reachable from a subse R V f here exss a pah from a node n R o node v afer deleng any k 1 nodes excep node v. A me-varyng graph s defned as G( = (V, E(, represenng a graph wh s edge se changng over me. The unon graph G([ 1, 2 ] s defned by G([ 1, 2 ] = V, E(. [ 1, 2 ] For a me-varyng graph G(, a node v s called only k- reachable from a se R f here exss T > such ha for all, node v s k-reachable from R n he unon graph G([, +T, where T s called he perod.
124 IEEE TRANSACTIONS ON CYBERNETICS, VOL. 48, NO. 4, APRIL 218 For a graph G = (V, E, aspannng k-ree of G rooed a R ={r 1, r 2,...,r k } V s a subgraph of G such ha: 1 every node r R has no n-neghbor; 2 every node v / R has k n-neghbors; 3 every node v / R s k-reachable from R. For a graph G = (V, E, aspannng k-fores of G rooed a R ={r 1, r 2,...,r m } V wh m k s a subgraph of G such ha: 1 every node r R has no n-neghbor; 2 every node v / R has k n-neghbors; 3 every node v / R s k-reachable from R. Remark 1: We would lke o menon ha a spannng k-ree has exacly k roos whle a spannng k-fores may have more han k roos. A confguraon n R 3 of n nodes s defned by her coordnaes n R 3, denoed as p = [p T 1,...,pT n ]T R 3n, where p R 3 for = 1,...,n. Moreover, we say p s generc f he coordnaes p 1,...,p n do no sasfy any nonrval algebrac equaon wh neger coeffcens [28]. A framework n R 3 s a graph G equpped wh a confguraon p, denoed as (G, p. Two frameworks (G, p and (G, q are called smlar f p p = γ A ( q q for all, V where γ> s a scalng facor and A s a unary marx. For a marx A R n n,heassocaed graph s defned as G(A, whch has n nodes labeled by 1,...,n, and an edge (, exss f and only f he (, -h enry of A s nonzero. III. PROBLEM FORMULATION In engneerng applcaons, agens may perform marchng when maneuverng and searchng arge. Moveover, n some scenaros, agens may execue crcular moons such as flyng around a landmark. Thereby, wo specfc problems oward formaon conrol wll be suded n hs paper. One of hem s called he formaon marchng conrol problem, he goal of whch s o seer he agens o form a arge formaon shape whle movng wh he synchronzed velocy. The oher one s called he formaon roang conrol problem, whose obecve s o drve he agens roang around a common axs wh a arge formaon shape. Measuremens beween agens may be undreconal due o he lm of equpped devces such as cameras whch only have a concal feld of vson. In addon, measuremens may fal somemes due o severe envronmen facors. So a more general seup wh dreced and swchng sensng graphs s adoped o model he mulagen sysems n hs paper. We consder a leader follower nework of N agens n he hree dmensons, where here are m leaders labeled from 1 o m and N m followers labeled from m+1 on. Defne a arge confguraon p = [p T l, pt f ]T R 3N for all he agens where p l = [p T 1,...,pT m ]T aggregaes he saes of he leaders whle p f = [p T m+1,...,pt N ]T aggregaes he saes of he followers. Throughou hs paper, a vecor noaon equpped wh he subscrp l s used o represen he aggregaed sae for he leaders and lkewse f s used for he followers. Denoe he global coordnae sysem by g and he agen s local coordnae sysem by.lez R 3 be he 3-D poson of agen n he global frame g. Moreover, le z ( be he value of agen s poson n agen s local frame, and from now on we use he superscrp ( o represen he value n.leo be he orgn of n g, ha s, O ( =. Defne R he roaon marx from o g,.e., z O = R (z ( O (, V. Furhermore, defne R he roaon marx from and,.e., z ( O ( = R (z ( O (. Suppose each agen s governed by he followng dynamcs: ż ( ( = u ( (, = 1,...,N (1 where u ( R3 represens he conrol npu of agen. The leaders are sad o be n a smlar formaon p l f her posons sasfy z ( = γ Ap + c, = 1,...,m (2 where A R 3 3 s a unary marx, c R 3 s a consan vecor mplyng he ranslaon, and γ > s a scalng facor. We assume n hs paper ha he leaders are n a smlar formaon p l and we concenrae on devsng conrollers for he followers. I s assumed ha each agen s provded wh sensors such as cameras and manages o sense he relave posons of s neghbors n s local frame. A me-varyng graph Ḡ( = (V, Ē( s adoped o model he sensng graph, where V = V l V f wh V l ={1,...,m} and V f ={m + 1,...,N}. Tha s o say, (, Ē( ndcaes ha agen has he ably o sense he relave poson of agen n a me. Le N be he se of agen s n-neghbors n graph Ḡ( and lkewse N ( denoes he se of agen s ou-neghbors. Noe ha N f and only f (, Ē(. Moreover, each agen s assumed o equp devces o communcae wh oher agens. However, he devces mplemened for sensng and communcaon may be dfferen. Thus, We adop anoher me-varyng graph H( o model he communcaon graph, where an edge (, ndcaes ha agen can ransm nformaon o agen. Usually, devces suppor wo-way communcaon and communcaon dsance s longer han sensng dsance, whch makes reasonable o have he followng assumpon. Assumpon 1: The communcaon graph H( s bdreconal. Moreover, he communcaon graph H( conans he sensng graph Ḡ( as a subgraph a any me. Moreover, he followng assumpons are also made hroughou hs paper. Assumpon 2: The arge confguraon p = [p T l, pt f ]T s generc. Assumpon 3: Agen has access o he roaon marx R, N. Assumpon 4: The nerval beween any wo swchng nsans sasfes a dwell me condon. Tha s o say, here exss τ D > such ha +1 τ D for all =, 1,... f he sensng graph Ḡ( swches a, 1, 2,...
HAN e al.: BARYCENTRIC COORDINATE-BASED APPROACH TO FORMATION CONTROL 125 Fg. 2. Smple llusrave example for formaon marchng. Fg. 4. Geomerc explanaon for he orhogonal proecon operaor. Fg. 3. Smple llusrave example for formaon roang. A. Formaon Marchng Conrol Problem The obecve s o seer all he agens converge o form a smlar formaon wh a common velocy v r (. Asmple llusrave example for formaon marchng s gven n Fg. 2. Le he moon of he leaders be governed by ż ( ( = v r ( (, = 1,...,m (3 where v r ( ( s he common velocy. Assume ha each agen, = 1,...,N, has access o he velocy v r ( (. Remark 2: For he suaon ha v r ( ( s only known o a subse of he followers, some echnques can be ncluded o make possble (see [22]. For smplcy of analyss, we wre (3 n g as ż ( = v r (, = 1,...,m. (4 We nex gve he precse defnon abou achevng a smlar formaon for he formaon marchng conrol. Defnon 1: A smlar formaon p = [p T l, pt f s sad o be ]T globally unformly asympocally acheved for he formaon marchng conrol problem f for any δ> and for any ε> here exss T > such ha for any and for any z ( sasfyng z ( γ Ap c <δ, = 1,...,N ( + T( z ( γ Ap c v r (τdτ <ε (5 where A, c, and γ are deermned n (2. Then we are able o gve he descrpon for he formaon marchng conrol problem as below. Suppose he leaders are n a smlar formaon p l and governed by (4. Gven Assumpons 1 4 and relave poson measuremens z ( z ( for N, desgn a fully dsrbued conrol law u ( for each follower, and fnd he correspondng graphcal condons, under whch (5 s sasfed. B. Formaon Roang Conrol Problem The obecve s o drve all he agens surround a common axs wh a smlar formaon. We use a un vecor R 3 whch passes a pon ρ R 3 o represen hs axs, denoed by (ρ. Tha s, all he agens fnally move on he crcular orbs around (ρ whle keepng a desred formaon shape. A smple llusrave example for formaon roang s shown n Fg. 3. We assume he synchronzed angular velocy ω = 1for smplcy snce follows he same analyss f ω akes oher values. Moreover, we nroduce a new coordnae sysem for each agen such ha he z-axs of s parallel o. Le and share he common orgn. The roaon marx from o s denoed as R R 3 3, ha s, z ( = R z (, where he superscrp ( represens he value n. Remark 3: Noce ha we do no defne he x-axs and y- axs of explcly, so R can ake dfferen values. However, does no affec he analyss and conclusons. The nex assumpon s requred o acheve formaon roang. Assumpon 5: Each leader manages o sense he relave poson ρ ( z( and each agen has access o he vecor (. Remark 4: For he suaon ha he vecor ( s only known o a subse of he agens, ( can be known o all he agens hrough communcaon f Assumpon 3 holds. Le he moon of he leaders be governed by ( ż ( π ( ( = R 1 R R 2 P ( ρ ( z(, = 1,...,m (6 where he funcon P(x s he orhogonal proecon operaor on he un vecor x defned by P(x = I xx T. Geomercally, P(x s an orhogonal proecon marx ha proecs any vecor ono he orhogonal complmen of x. See Fg. 4 for an llusraon. And R(π/2 s aaned by he roaon marx cos( sn( R( = sn( cos(. 1 Denoe R he roaon marx from o g, ha s, z O = R (z ( O (, V. Then we can wre he dynamcs (6 n he global frame g as ż = R 1 R( π 2 R P( (ρ z, = 1,...,m. (7 Remark 5: Noe ha he dynamcs (7 ndcaes ha every leader moves surroundng (ρ anclockwse (lookng along he negave drecon of wh he angular velocy ω = 1. We nex gve he formal defnon abou achevng he smlar formaon for he formaon roang conrol problem. Defnon 2: A smlar formaon p = [p T l, pt f ]T s sad o be globally unformly asympocally acheved for he formaon roang conrol problem f for any δ> and for any ε> here exss T > such ha for any and for any z ( sasfyng z ( γ Ap c <δ, = 1,...,N, hen holds ha for all + T z ( R 1 R( R (γ Ap + c ρ ρ <ε (8 where A, c, and γ are deermned n (2.
126 IEEE TRANSACTIONS ON CYBERNETICS, VOL. 48, NO. 4, APRIL 218 Now, we can gve he descrpon for he formaon roang conrol problem. Suppose ha he leaders are already n a smlar formaon p l and governed by (6. Gven Assumpons 1 5 and relave poson measuremens z ( z ( for N, desgn a fully dsrbued conrol law u ( for each follower, and fnd he correspondng graphcal condons, under whch (8 s sasfed. IV. FORMATION MARCHING CONTROL In hs secon, we frs focus on desgnng a fully dsrbued conrol law for formaon marchng conrol problem. Then more nsgh behnd he deas of barycenrc-coordnae-based conrol s nroduced. A las, we conduc he convergence analyss for he whole leader follower nework under he proposed conrol law. A. Dsrbued Conrol Law The basc dea o ulze he barycenrc-coordnae-based conrol s o esablsh he barycenrc coordnae represenaon for each agen wh respec o s n-neghbors. A leas four n-neghbors are necessary o have hs represenaon. Thus, f N < 4, hen agen wll no make use of he sensed measuremens and us abandon hem. In erm of hs mechansm, an neracon graph G( = (V, E( s consruced dependng on whch n-neghbors are chosen o nerac. Denoe N he se of agen s n-neghbors n G( and N ( he se of agen s ou-neghbors. In oher words, f N < 4, hen N =. Oherwse, N = N. We develop he followng lnear swchng conrol law for each follower = m + 1,...,N: ξ = 1 2 ξ ( k ( z ( z ( N ż ( = v r ( ( k (ξ + (9 k (R ξ N N ( where ξ s an auxlary sae wh he values for he leaders beng zero,.e., ξ 1 = = ξ m =, and k ( R\{} s he wegh assocaed wh edge (, n G( sasfyng k ((p p =. (1 N Remark 6: By Assumpon 4 and he desgn of k (s we know ha k (s are pecewse consan. Remark 7: Noce ha compung k (s from(1 sdsrbued and can be carred ou by each agen wh known p and p ( N. Moreover, z ( z ( ( N s acqured by onboard sensors and k R ξ ( N ( s aaned hrough communcaon. Thus, he conroller (9 s fully dsrbued, namely, all he parameers can be deermned n a dsrbued way. An llusraon of nformaon flow ncludng sensng and communcaon s dsplayed n Fg. 5, where he sold lnes represen he measuremens obaned by sensors and he dashed lnes sand for he nformaon exchange hrough communcaon. Fg. 5. Illusraon of nformaon flow ncludng sensng and communcaon for (9. Now we defne ζ = R ξ and s obaned ha ζ = R ξ = 1 2 R ξ ( k (R z ( z ( = 1 2 ζ and we also have ż = R ż ( = R v ( r ( = v r ( N N N N k ( ( z z k (R ξ + k (ζ + N ( N ( k (ζ. k (R R ξ Hence, he sysem (9 can be wren n he global frame ζ = 1 2 ζ k ( ( z z, N ż = v r ( k (ζ + (11 k (ζ. N N ( Denoe L( he Laplacan of G( wh he weghs k (s compued by (1, whch has he followng srucure: [ ] m m L( = m (N m. (12 L lf ( L ff ( Accordng o he desgn of k (s, L( sasfes (L( I 3 p = and L(1 =. (13 B. Barycenrc Coordnae-Based Idea In hs secon, we am o provde more nsgh behnd he deas of barycenrc-coordnae-based conrol. To gve clearer explanaon abou he basc dea, we consder a fxed graph G. The barycenrc coordnae s a geomerc noon characerzng he relave poson of a pon wh respec o oher pons. In he hree dmensons, a leas four oher pons are needed for a pon o have a barycenrc coordnae represenaon. Recall (1 ( k p p = N + where p has a barycenrc coordnae represenaon wh respec o s n-neghbors, ha s k p = k p. N + N +
HAN e al.: BARYCENTRIC COORDINATE-BASED APPROACH TO FORMATION CONTROL 127 The basc dea of barycenrc-coordnae-based conrol s ha each follower drves N + k (z z o zero. The followng lemma shows ha (L I 3 z = and de(l ff = ndcae ha he followers reach he arge formaon. Lemma 1: Suppose ha he leaders are n a smlar formaon. If (L I 3 z = and de(l ff =, hen z f = (I N m γ Ap f + 1 N m ( c + v r (τdτ. Proof: Noe ha (L I 3 p = and L1 =. Then we have [ ] (L I 3 (I N γ Ap + 1 N (c + v r (τdτ = (L I 3 (I N γ Ap = (L γ Ap = (I N γ A(L( I 3 p =. Snce de(l ff =, z f solved from (L I 3 z = s unque. Therefore, he concluson follows. The nex lemma provdes he fac ha f he number of he leaders s less han 4, hen L ff mus be sngular. Before ha, we denoe by L(G he se of all Laplacan marces wh nonzero weghs on he edges n G. Isruehaforany marx L L(G, we have L1 =. For a se R V, lel R be he sub-marx consruced from L by removng he rows and columns correspondng o R. Lemma 2: Consder a graph G = (V, E wh n nodes and generc ξ,ζ,η R n. Gven he se R ={r 1,...,r m },fm < 4, hen de(l R = for any L {L L(G : Lξ =, Lζ =, Lη = }. The proof of Lemma 2 s gven n he Appendx. The nex heorem presens he relaonshp beween he nonsngulary of L ff and he opology connecvy of G. Theorem 1: Consder a graph G = (V, E wh n nodes and generc ξ,ζ,η R n. Gven he se R ={r 1,...,r m } wh m 4, he followng saemens are equvalen. 1 Every node n V\R s 4-reachable from R. 2 For almos all 1 L {L L(G : Lξ =, Lζ =, Lη = }, all he prncpal mnors of L R are nonzero. The proof of Theorem 1 s saed n he Appendx. Remark 8: The basc dea of dsplacemen-based conrol s o run consensus on z p for every agen where p s he arge confguraon. Thus, dsplacemen-based conrol asks for he common orenaon of all he local frames due o he presence of p. However, he man dea of barycenrccoordnae-based conrol s o drve N + k (z z o zero, where p s only used o compue k. As a consequence, barycenrc-coordnae-based conrol makes possble o devse coordnae-free conrol laws. Moreover, Lemma 1 shows ha he leaders are able o reconfgure he formaon shape whle he followers can only ac as dummes, whch canno be acheved by dsplacemen-based conrol. Remark 9: Dsance-based conrol ulzes nonlnear local consrans, whch only guaranees local asympoc convergence and s nonrobus wh respec o dsance msmaches [3]. Global convergence based on dsance-based conrol s only avalable for some specal graphs [31] whle 1 The phrase for almos all means for all values excep hose n some proper algebrac varey wh Lebesgue measure zero [29]. barycenrc-coordnae-based conrol makes global convergence possble for more general graphs by usng lnear dsrbued conrol laws. Moreover, barycenrc-coordnaebased conrol requres less edges o acheve a arge formaon shape han dsance-based conrol. Specfcally, for dsance-based conrol, he underlyng graph needs o be globally rgd, whch requres more lnks beween leaders and followers compared wh he graphcal condon presened n Theorem 1. C. Sably Analyss In hs secon, we provde he convergence analyss for he whole sysem under he conroller (11 for he formaon marchng conrol problem. Defne z and ζ he aggregaed vecors of all z s and ζ s, respecvely. Under he dsrbued conrol law (11, he overall closed-loop sysem can be wren as [ ] ([ ] [ ] [ ] ż H( z 1N = I ζ L( U 3 + v ζ r (14 where H( = [ ] Lff T (, U = [ 1 2 I N m The nex heorem provdes graphcal condons o ensure global convergence for he whole sysem. Theorem 2: Under conrol law (11, for almos all chosen weghs, a smlar formaon p = [p T l, pt f can be globally ]T unformly asympocally acheved for he formaon marchng conrol problem f he followng wo condons hold. 1 Every follower s only 4-reachable from V l n he neracon graph G( wh perod T. 2 For any and any follower, here exss [, + T such ha N F N + (, where F s a spannng 4-fores rooed a V l n he unon graph G([, + T and N F represens he se of s n-neghbors n F. Before presenng echncal proof for Theorem 2, we develop he nex lemma and nroduce a lemma summarzed from [32] whch provdes a perssenly excng condon ensurng exponenal sably for swchng sysems. Gven generc ξ,ζ,η R n, denoe (F ={L(, : : L L(F, Lξ =, Lζ =, Lη = } and (G ={L(, : : L L(G, Lξ =, Lζ =, Lη = }, where L(, : represens he -h row of L. Lemma 3: Consder a graph G = (V, E wh n nodes and aser ={r 1,...,r m } where m 4. Suppose ha every node n V\R s 4-reachable from R and F s a spannng 4-fores rooed a R. If for any node holds ha s a subspace sasfyng (F (G, hen for almos all 2 L formed by L(, :, all prncpal mnors of L R are nonzero. We pospone he proof of Lemma 3 o he Appendx. Denoe he se T ={ } of elemens n [,, where here exss a posve such ha for all, T ( =, holds ha. Furhermore, le Ɣ be he se of funcons 2 The phrase for almos all L can be undersood by for almos all k s used o consruc L. ].
128 IEEE TRANSACTIONS ON CYBERNETICS, VOL. 48, NO. 4, APRIL 218 v( defned on [,, where for every v( here exss T such ha: 1 v( and v( are connuous and bounded on [, \T ; 2 v( and v( have fne lms as + and, where T. Lemma 4 [32]: Le V( : R + R n r be a regulaed marx funcon (one-sded lms exs for all R +, and sasfy for some posve δ and α 1, and all R + +δ V(τ 2 dτ<α 1. (15 Suppose also ha he enres of V( le n Ɣ. LeM be a real consan n n marx wh M + M T = I n and le B be a real consan n r marx wh rank r. Then [ ] VB T ẋ = x (16 BV T M s exponenally sable f and only f for some posve δ and α 2, and all R + +δ V(τV T (τdτ α 2 I. (17 Lemma 5 [33]: Consder m marces A 1,...,A m R n n. If rank(a 1 + A 2 + + A m = n, hen he marx A T 1 A 1 + A T 2 A 2 + +A T m A m s posve defne. Proof of Theorem 2: Denoe y = z 1 N v r (τdτ and he sysem (14 becomes [ ] ([ ] [ ] ẏ H( y = I ζ L( U 3. (18 ζ Snce he fac (L( I 3 [ (I N γ Ap + 1 N c ] = (L( γ Ap = (I N γ A(L( I 3 p = we ge ha { y = (I N γ Ap + 1 N c ζ (19 = s an equlbrum pon of sysem (18, whch mples ha { ( z ( = (I N γ Ap + 1 N c + v r (τdτ ζ = s an equlbrum soluon of sysem (14. Recall ha he leaders are n a smlar formaon, ha s o say, y l ( = y l. Apply he coordnae ransformaon e f = y f y f for he followers and we have [ ] ([ ] [ ] ėf L T = ff ( ef ζ f L ff ( 1 2 I I 3. (2 n ζ f Noce ha he sysem (2 can be derved by subsung [ ] ef x =, M = 1 ζ f 2 I n I 3, B = I n I 3, and V = Lff T I 3 no (16 n Lemma 4. In wha follows, we prove ha all he condons descrbed n Lemma 4 are sasfed. Suppose he graph G( swches a, 1, 2,... From Assumpon 4 we know ha +1 τ D for all =, 1, 2... Moreover, we are always able o fnd a τ M >τ D large enough such ha +1 τ M for all =, 1, 2,...If for some nerval [, +1 here s no such a τ M, hen [, +1 can be paroned o sasfy hs condon. Suppose every follower s only 4-reachable from V l. Then here exs T > such ha every follower s 4-reachable from V l n he unon graph G([, + T for all. Snce L ff ( s pecewse consan, s obaned ha L ff ( s regulaed. We choose δ as δ = T + 2τ M. Snce k (s are aken from a fne se, we aan ha L ff ( 2 s unform upper-bounded, whch ndcaes ha here exss a posve α 1 such ha for all, holds ha +δ L T ff (τ 2 dτ<α 1. Therefore, accordng o Lemma 4, remans o show ha here exss a posve α 3 such ha for all, holds ha +δ L T ff (τl ff (τdτ α 3 I. (21 In he followng, we wll prove ha (21 s sasfed. Consderng any, whou loss of generaly, denoe ( l1, l1 +1] and + δ [ l2, l2 +1. We le W := Moreover, we defne and +δ L T ff (τl ff (τdτ. H := L T ( l1 +1L( l1 +1 + +L T ( l2 1L( l2 1 X := L( l1 +1 + +L( l2 1. Thus, for almos all L(, he nequaly L( k (, =, k = l 1 + 1,...,l 2 1 leads o he fac ha X(, =, whch mples ha G(X = G(L( l1 +1 G(L( l2 1. Furhermore, s also known ha G([ l1 +1, l2 = G(L( l1 +1 G(L( l2 1 = G(X. Snce (X I 3 p = and X1 N =, he marx X can be also regarded as a Laplacan marx wh he followng srucure: [ ] where X ff = L ff ( l1 +1 + +L ff ( l2 1. Due o l1 +1 τ M and + δ l2 τ M,wehave l2 l1 +1 T. Hence, every follower s 4-reachable from V l n he unon graph G([ l1 +1, l2. Accordng o he second graphcal condon n Theorem 2, G([ l1 +1, l2 has a spannng 4-fores F and N F N ( for some [ l1 +1, l2. Thus, we can nfer ha for each, hese{x(, :} nduced by all possble Xs sasfes {L(, : : L L(F, (L I 3 p = } {X(, :}. Then follows from Lemma 3 ha: X ff de ( X ff =. (22
HAN e al.: BARYCENTRIC COORDINATE-BASED APPROACH TO FORMATION CONTROL 129 Now we come o look a H whch has he followng form: [ ] where H ff H ff = L T ff ( l 1 +1L ff ( l1 +1 + +L T ff ( l 2 1L ff ( l2 1. By Lemma 5 and he nequaly (22, s ceran ha de ( H ff =. Togeher wh he fac ha H ff s posve sem-defne, can be obaned ha H ff s posve defne. Noe ha he marx W can be wren as W = Lff T ( l 1 L ff ( l1 ( l1 +1 + Lff T ( l 1 +1L ff ( l1 +1( l1 +2 l1 +1 + + Lff T ( l 2 1L ff ( l2 1( l2 l2 1 + Lff T ( l 2 L ff ( l2 ( + δ l2. Snce we have shown ha H ff s posve defne, n mples ha for any vecor v = holds ha v T H ff v >. So can be easly obaned ha for any vecor v =, holds ha v T Wv >. Thus, W s also posve defne. To prove (21 holds, s requred o show ha he smalles egenvalue of W s unform lower bounded on. Noce ha L ff (s are aken from a fne se. Moreover, he number of swches durng [, + δ s no more han δ/τ D, whch ndcaes ha H s also aken from a fne se. Snce for all =, 1, 2,..., holds ha τ D +1 τ M. Moreover, s also known ha l1 +1 τ M and + δ l2 τ M. Hence, we can nfer from he expresson of W ha here exss α 3 > such ha for all W α 3 I. Thus, he sae [e T f,ζt f wll converge o zero exponenally, whch equvalenly ndcaes ha z f converges o z f ]T exponenally. Therefore, he concluson follows. Remark 1: Consderng sac opologes, follows from Theorem 1 ha he graphcal condon ha every follower s 4-reachable from V l s necessary for global convergence under (9. Moreover, we can nfer from Theorem 2 ha hs graphcal condon s also suffcen snce for sac opologes condon (1 mples condon (2. To sum up, 4-reachably s necessary and suffcen for sac opologes o ensure convergence. For swchng opologes, Theorem 2 shows ha he graph does no need o sasfy 4-reachably all he me. I only requres ha he unon graph sasfes he condons frequenly, whch s much mlder. V. FORMATION ROTATING CONTROL In hs secon, we frs develop a fully dsrbued conrol law for he formaon roang conrol problem and hen analyze s sably. A. Dsrbued Conrol Law In hs secon, a fully dsrbued conrol law s proposed for he formaon roang conrol problem. Le η be he esmaon of ρ ( z ( for every agen. Indeed, for he leaders, we have η = ρ (. Then we gve z( he followng esmaon algorhm for each follower : η = ż ( + (z ( z ( + R η η. (23 N Defne x = R η and we oban ha ẋ = R η = R ż ( + ( R z ( z ( + R η η = ż + = ż + N N N ( z z + R R η x ( z z + x x. Thus, he algorhm (23 can be wren n g as ẋ = ż + ( z z + x x. (24 N Lemma 6: Under (24, for each follower, x globally unformly asympocally converges o ρ z f and only f every follower s only reachable from he leaders. Proof: Le y = x (ρ z and s known ha y = for = 1,...,m. Moreover, we aan ha ẏ =ẋ +ż = ( y y, = m + 1,...,N. (25 N Noce ha (25 s he well-known consensus algorhm wh he saes of he leaders beng zero. Wh he same echnque used n he proof of [22, Th. 5.1], we drecly oban ha y unformly asympocally converges o zero f and only f every follower s only reachable from he leaders. By he ransformaon y = x (ρ z we can rewre he conroller (7 as ξ ż = R 1 R( π 2 R P( x, = 1,...,m. (26 Nex, we are able o propose he conroller for he followers: ( ż ( π ( = R 1 R R 2 P ( η k (ξ N + k (R ξ, (27 N ( = R 1 ˆRR ξ 1 2 ξ ( k ( z ( z ( where N 1 ˆR = 1 and k s are desgned n he same way as he conrol law (11.
121 IEEE TRANSACTIONS ON CYBERNETICS, VOL. 48, NO. 4, APRIL 218 Defne ζ = R ξ and we derve ha ζ = R ξ = R R 1 ˆRR R 1 R ξ 1 2 ζ k ( ( z z N = R 1 ˆRR ζ 1 2 ζ N Moreover, we also have ( ż = R ż ( π ( = R R 1 R R 2 P ( k (ζ + N k ( ( z z. η N ( π = R 1 R( R P( x 2 k (ζ + N N ( k (ζ k (ζ. Hence, he conrol law (27 can be wren n he global coordnae sysem by ż = R 1 R( π 2 R P( x k (ζ + N ( k (ζ ζ = R 1 ˆRR ζ 1 2 ζ N N k ( ( z z. (28 B. Sably Analyss In hs secon, we analyze he convergence of he whole sysem under he conroller (28. The nex heorem presens he same condons as he ones n Theorem 2 o guaranee global convergence. Theorem 3: Under conrol law (28, for almos all chosen weghs, a smlar formaon p = [p T l, pt f can be globally ]T unformly asympocally realzed for he formaon roang conrol problem f he followng wo condons hold. 1 Every follower s only 4-reachable from V l n he neracon graph G( wh perod T. 2 For any and any follower, here exss [, + T such ha N F N + (, where F s a spannng 4-fores n he unon graph G([, + T and N F represens he se of s n-neghbors n F. The proof of Theorem 3 requres a lemma concernng cascade sysems. Lemma 7 [34]: Consder he followng nonlnear mevaryng sysem: ẋ 1 = f 1 (, x 1 + g(, xx 2 (29 ẋ 2 = f 2 (, x 2 (3 where x = [x1 T, xt 2 ]T. The sysem (29 and (3 s globally unformly asympocally sable f he followng condons are sasfed. 1 For each r > here exss c(r > such ha f x( < r hen x(;, x( c(r. 2 Sysem ẋ 1 = f 1 (, x 1 s globally unformly asympocally sable. 3 Sysem (3 s globally unformly asympocally sable. Proof of Theorem 3: Le ψ = R 1 R( R (z ρ + ρ. Then for he leaders, we know ha ψ = R 1 Ṙ( R (z ρ + R 1 R( R ż π = R 1 Ṙ( R g(z ρ R 1 R( R( 2 sn( cos( = R 1 cos( sn( sn( cos( ( + R 1 cos( sn( P 1 R P( x ( z ( ρ ( ( x (. Wh he fac ha for he leaders, x = ρ z and he z-axs of s parallel o. Thus, we deduce ha ψ =. Nex we consder he followng: ψ = R 1 Ṙ( R (z ρ + R 1 R( R ż sn( cos( = R 1 cos( sn( ( z ( ρ ( π R 1 R( R( 2 + R 1 R( R R P( (y + ρ z N ( k (ζ N = R 1 R( R( π 2 R P( y + R 1 R( R k (ζ k (ζ. We defne N ( N k (ζ φ = R 1 R( R ζ and π f ( = R 1 R( R( R P( y. 2 For he leaders, we have ψ = and φ =. For he followers, s obaned ha φ = R 1 Ṙ( R ζ + R 1 R( R ζ = R 1 Ṙ( R ζ + R 1 R( R R 1 ˆRR ζ 1 2 ζ k ( ( z z = R 1 N Ṙ( R ζ R 1 R( ˆRR ζ 1 2 φ k ( ( ψ ψ N = 1 2 φ N k ( ( ψ ψ
HAN e al.: BARYCENTRIC COORDINATE-BASED APPROACH TO FORMATION CONTROL 1211 and ψ = N ( k (φ N k (φ + f. Noe ha ψ ( = z ( and φ ( = ζ (. Le { ψ = (I N γ Ap + 1 N c φ = and apply he coordnae ransformaon e f = ψ f ψf. Then we oban [ ] ([ ] ėf L T [ ] [ ] = ff ( ef ff φ f L ff ( 1 2 I I 3 +. (31 n φ f Fg. 6. Targe confguraon. We have shown n Secon IV ha [ ] ([ ] ėf L T ff ( = φ f L ff ( 1 2 I n I 3 [ ef φ f ] s globally unformly asympocally sable. Moreover, from Lemma 6 we also oban ha y globally unformly asympocally converges o zero. Consequenly, on he bass of Lemma 7, o complee he proof only remans o verfy ha he condon (1 n Lemma 7 s sasfed. Le ([ ] L T ff ( C( = L ff ( 1 2 I I 3 n and [ ] [ ] ff ( ef b( =,μ=. φ f For each r >, suppose [ ] μ( < r y f ( hen we know μ( < r and y f ( < r. The soluon of (31 can be expressed by μ( = e C(τdτ μ( + e τ C( d b(τdτ. Take norm on boh sdes and we have μ( e C(τdτ μ( + e τ C( d b(τ dτ. Snce he sysem (2 s exponenally sable, here mus exs C(τdτ posve k and λ such ha e ke λ(, whch leads o he fac ha μ( ke λ( μ( + ke λ( τ b(τ dτ. Noe ha every sae y ( s a convex combnaon of y 1 (,...,y N ( [35, p. 78]. So s vald ha y ( < r, and ogeher wh he fac ha f ( y (, we can nfer ha b( s upper bounded by a funcon c y (r. Hence, we have μ( ke λ( μ( + c y (r ke λ( τ dτ k μ( + k c y(r kr + k c y(r λ λ. Fg. 7. Swchng graph G( ha swches beween wo graphs. Fg. 8. Graphs G 1 and G 2 swch every 1 s. Therefore, here exss c(r such ha [ ] μ( < c(r. y f ( Hence, ψ converges o γ Ap +c globally unformly asympocally. Equvalenly, z globally unformly asympocally converges o R 1 R( R (γ Ap + c ρ + ρ. Hence, he concluson follows. Remark 11: To mee Assumpon 3, one suffcen condon s ha he orenaons of all local coordnae sysems are conssen. However, hs s no necessary. For nsance, wh exra edges for nformaon exchange, Assumpon 3 can be sasfed. Apar from edge (, E, by addng (,, (k,, (k, E, agen manages o sense z z and z k z whle can ge z z and z k z. Then R can be compued by solvng he equaon as below ( z ( z ( = R z ( z ( ( z ( z ( + R z ( k z ( = z ( k z (.
1212 IEEE TRANSACTIONS ON CYBERNETICS, VOL. 48, NO. 4, APRIL 218 Fg. 9. (a Traecores of formaon marchng. (b Error beween he real poson and he arge poson. (c Error beween he rue velocy and he arge velocy. (d Norm of he conrol npu. Fg. 11. Formaon shape can be reconfgured by he leaders. Fg. 1. T = 2. Error beween he real poson and he arge poson for he case VI. SIMULATIONS In hs secon, we presen hree smulaon examples o llusrae our heorecal resuls. We consder a sysem of four leaders whch move n a smlar formaon and oher sx followers. The arge confguraon p s supposed o be he one n Fg. 6, where he leader se s V l ={1,...,4} and he follower se s V f ={5,...,1}. We draw he neracon graph G( used n he smulaons n Fg. 7, and can be checked ha G( sasfes he wo condons n Theorems 2 and 3 by selecng he perod as T = 2. Frs, we carry ou he smulaons usng he conroller (4 for he leaders and (11 for he followers. A smulaon resul s ploed n Fg. 9, where Fg. 9(a shows he raecores of all he agens and he poson error, velocy error, and conroller npu are ploed n Fg. 9(b (d, respecvely. There are glches n he sgnals n Fg. 9(c and (d snce swchng occurs durng hese me nsans. Bu, we can sll see from he resuls ha he agens globally reach he arge smlar formaon for he formaon marchng conrol. Moreover, we change he swchng sgnal n Fg. 7 such ha G 1 and G 2 swch every 1 s, shown n Fg. 8. The poson
HAN e al.: BARYCENTRIC COORDINATE-BASED APPROACH TO FORMATION CONTROL 1213 Fg. 12. (a Traecores of formaon roang. (b Error beween he real poson and he arge poson. (c Error beween he rue velocy and he arge velocy. (d Norm of he conrol npu. error for hs suaon s recorded n Fg. 1, from whch we can see ha larger perod T ndcaes slower convergence. Second, we consder a scenaro ha all he agens have reached he arge smlar formaon for he formaon marchng conrol as shown n Fg. 9(a. Then we mpose exra force on he leaders o change he leaders formaon shape and also use he same conroller (11 for he followers. A smulaon resul s gven n Fg. 11, whch demonsraes ha he formaon shape of he whole group can be reconfgured by he leaders whle he followers only ac as dummes. Thrd, we adop he conrol law (7 for he leaders and (28 for he followers. A smulaon resul s ploed n Fg. 12, where Fg. 12(a shows he raecores of all he agens and he poson error, velocy error, and conroller npu are ploed n Fg. 12(b (d, respecvely. We can see ha a smlar formaon s realzed for he formaon roang conrol. VII. CONCLUSION Ths paper nvesgaes he wo formaon conrol problems for a leader follower nework n he hree dmensons, ha s, formaon marchng conrol problem and formaon roang conrol problem. By exchangng an auxlary sae va communcaon, fully dsrbued conrol laws are developed for boh scenaros. Moreover, he same condons are obaned o guaranee global convergence for boh problems. One fuure sudy s o exend he conrol laws n hs paper o second-order cases or more complex agen dynamcs. One possble approach s o ake advanage of he backseppng phlosophy. Specfcally, desgn he acceleraon npu such ha he velocy sgnal n he second-order case converges o he desred velocy sgnal produced by he proposed conrol laws n hs paper. Anoher fuure sudy s o consder conroller desgn wh smooh veloces even n he presence of opology swchng, whch seems o be neresng and praccal. APPENDIX Proof of Lemma 2: We denoe U = V\R and relabel he nodes n R and U consecuvely. As a resul he marx L s ransformed o he form [ ] L L11 L := 12 L 21 L 22 where L 11 correspond o he nodes n R and L 22 correspond o he nodes n U. Thus, we have [ L21 L 22 ] 1 =, [ L21 L 22 ] Pξ = [ L21 L 22 ] Pζ =, [ L21 L 22 ] Pη =.
1214 IEEE TRANSACTIONS ON CYBERNETICS, VOL. 48, NO. 4, APRIL 218 Due o ha ξ,ζ, and η are generc, we know ha he rows of L 22 le n an (n 4-dmensonal lnear subspace. Snce m < 4, we can nfer ha L 22 s no of full row rank, whch means ha de(l R = for any L {L L(G : Lξ =, Lζ =, Lη = }. We summarze wo resuls n [2] for provng Theorem 1. Lemma 8 [2]: Consder a spannng 4-ree T = (V, E wh n nodes and generc ξ,ζ,η R n. Then for all L {L L(T : Lξ =, Lζ =, Lη = }, all prncpal mnors of L R are nonzero, where R s he roo se. Lemma 9 [2]: Consder a roo se R = {r 1, r 2,...,r k } such ha any node n V\R s k-reachable from R. Then for he Laplacan L of G wh almos all weghs w s: 1 all he prncpal mnors of L R are nonzero; 2 de(m = where M s a sub-marx of L by deleng he rows correspondng o he k roos and any k columns. Proof of Theorem 1: (1 = (2 If (1 holds, we add edges (r k, r 1, (r k, r 2, (r k, r 3, (r k, r 4 for k = 5,...,m and hus consruc a new graph, denoed by G, whch has a spannng 4-ree as s subgraph wh he roo se R ={r 1, r 2, r 3, r 4 }. We denoe hs spannng 4-ree by T and hus by Lemma 8 we know ha all he prncpal mnors of L R are nonzero for any L {L L(T : Lξ =, Lζ =, Lη = }. Noce ha he dfference beween L {L L(G : Lξ =, Lζ =, Lη = } and L {L L(T : L ξ =, L ζ =, L η = } s ha some nonzero weghs n L become zero n L. Accordng o he fac ha eher a polynomal s zero or s no zero almos everywhere, we can nfer ha for almos all L {L L(G : Lξ =, Lζ =, Lη = }, all he prncpal mnors of L R are nonzero. Moreover, snce R R, afer relabelng he nodes properly, s obaned ha for almos all L {L L(G : Lξ =, Lζ =, Lη = }, all he prncpal mnors of L R are nonzero. (2 = (1 Suppose ha here exss a node / R such ha afer deleng hree nodes, whou loss of generaly, say {1, 2, 3}, s no reachable from R. Defne a se U ncludng he nodes ha are no reachable from R afer removng nodes 1, 2, and 3. Denoe Ū = V U {1, 2, 3}. Noe ha Ū canno be empy snce here are m nodes n R. Moreover, s rue ha here s no edge leadng from a node n Ū o a node n U. Afer relabelng he nodes, he marx L becomes he one wh he srucure as L 11 L 12 L 13 L := L 21 L 22 L 31 L 32 L 33 where L 11 correspond o {1, 2, 3}, L 22 correspond o he nodes n U, and L 33 correspond o he nodes n Ū. Thus, s known ha [ L21 L 22 ] 1 =, [ L 21 L 22 [ L21 L 22 ] Pζ =, [ L 21 L 22 ] Pξ = ] Pη =. Snce ξ,ζ, and η are generc, he rows of [L 21 L 22 ] le n an (n Ū 4-dmensonal lnear subspace. Moreover, [L 21 L 22 ] has n Ū 3 rows. Thus, we can nfer ha [L 21 L 22 ]sno of full row rank, whch means de(l R = for any L {L L(G : Lξ =, Lζ =, Lη = }. Proof of Lemma 3: Noe ha each L(, : (F les n a 1-D lnear subspace. 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Saarbrücken, Germany: VDM-Verlag, 28. Zhyun Ln (SM 1 receved he bachelor s degree n elecrcal engneerng from Yanshan Unversy, Qnhuangdao, Chna, n 1998, he maser s degree n elecrcal engneerng from Zheang Unversy, Hangzhou, Chna, n 21, and he Ph.D. degree n elecrcal and compuer engneerng from he Unversy of Torono, Torono, ON, Canada, n 25. He s currenly a Professor wh he School of Auomaon, Hangzhou Danz Unversy, Hangzhou. Precedng o hs poson, he held a Professor poson wh Zheang Unversy from 27 o 216, and a Pos-Docoral Fellow poson wh he Unversy of Torono from 25 o 27. He held a Vsng Professor posons a several unverses ncludng he Ausralan Naonal Unversy, Canberra, ACT, Ausrala, he Unversy of Caglar, Caglar, Ialy, he Unversy of Newcasle, Callaghan, NSW, Ausrala, he Unversy of Technology Sydney, Ulmo, NSW, Ausrala, and Yale Unversy, New Haven, CT, USA. Hs curren research neress nclude dsrbued conrol, esmaon and opmzaon, cooperave conrol of mulagen sysems, hybrd conrol sysem heory, and robocs. Dr. Ln s currenly an Assocae Edor for he IEEE CONTROL SYSTEMS LETTERS, Hybrd Sysems: Nonlnear Analyss, andheinernaonal Journal of Wreless and Moble Neworkng. Ronghao Zheng receved he bachelor s degree n elecrcal engneerng and he maser s degree n conrol heory and conrol engneerng from Zheang Unversy, Hangzhou, Chna, n 27 and 21, and he Ph.D. degree n mechancal and bomedcal engneerng from he Cy Unversy of Hong Kong, Hong Kong, n 214. He s currenly wh he College of Elecrcal Engneerng, Zheang Unversy. Hs curren research neres ncludes dsrbued algorhms and conrol, especally, he coordnaon of neworked moble robo eams wh applcaons n auomaed sysems and secury. Tngru Han receved he bachelor s degree n auomaon from Zheang Unversy, Hangzhou, Chna, n 212, where he s currenly pursung he Ph.D. degree n conrol heory and conrol engneerng wh he College of Elecrcal Engneerng. Hs curren research neress nclude mulagen sysems, neworked conrol, and dsrbued algorhms. Mnyue Fu (F 3 receved he bachelor s degree n elecrcal engneerng from he Unversy of Scence and Technology of Chna, Hefe, Chna, n 1982, and he M.S. and Ph.D. degrees n elecrcal engneerng from he Unversy of Wsconsn-Madson, Madson, WI, USA, n 1983 and 1987, respecvely. From 1987 o 1989, he served as an Asssan Professor wh he Deparmen of Elecrcal and Compuer Engneerng, Wayne Sae Unversy, Dero, MI, USA. He oned he Deparmen of Elecrcal and Compuer Engneerng, Unversy of Newcasle, Callaghan, NSW, Ausrala, n 1989, where he s currenly a Char Professor of Elecrcal Engneerng. He was a Vsng Assocae Professor wh he Unversy of Iowa, Iowa Cy, IA, USA, from 1995 o 1996, a Senor Fellow/Vsng Professor wh Nanyang Technologcal Unversy, Sngapore, n 22, and a Vsng Professor wh Tokyo Unversy, Bunkyo, Tokyo, n 23. He has held a ChangJang Vsng Professorshp wh Shandong Unversy, Jnan, Chna, a Vsng Professorshp wh Souh Chna Unversy of Technology, Guangzhou, Chna, and a Qan-Ren Professorshp wh Zheang Unversy, Hangzhou, Chna. Hs curren research neress nclude conrol sysems, sgnal processng and communcaons, neworked conrol sysems, smar elecrcy neworks, and super-precson posonng conrol sysems. Dr. Fu has been an Assocae Edor for he IEEE TRANSACTIONS ON AUTOMATIC CONTROL, Auomaca, he IEEE TRANSACTIONS ON SIGNAL PROCESSING, andhejournal of Opmzaon and Engneerng.