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1 Systems & Cotrol Letters 59 (010) Cotets lsts avalable at SceceDrect Systems & Cotrol Letters joural homepage: Surroudg cotrol cooperatve aget etworks Fe Che, We Re, Yogca Cao Departmet of Electrcal ad Computer Egeerg, Utah State Uversty, UT, USA a r t c l e f o a b s t r a c t Artcle hstory: Receved Aprl 010 Receved revsed form 8 July 010 Accepted 18 August 010 Avalable ole October 010 Keywords: Multaget systems Cooperatve cotrol Surroudg cotrol Decetralzed estmato Laplaca matrx I ths paper, two surroudg cotrol problems are proposed, where a team of followers s used to surroud a team of leaders. The problems are solved uder a decetralzed estmato-ad-cotrol framework. Usg tools from algebrac graph theory ad dyamcal systems theory, t s show that the two teams, volvg a team of leaders ad a team of followers, preserve some desred covergece propertes, eve f the geometrc ceter of the leaders ca oly be obtaed from estmators. A smulato example s preseted to verfy the valdty of the derved results. 010 Elsever B.V. All rghts reserved. 1. Itroducto The exploso computato ad commucato capabltes has made t possble to coordate large umbers of autoomous vehcles commucatg through a etwork to perform a varety of challegg tasks, whch are beyod the ablty of a sgle vehcle. Ths gves rse to a ew research area, cooperatve cotrol of multaget systems. Recet years have wtessed a tesve ad growg terest ths area. I partcular, cosesus [1 5], formato cotrol [6], swarmg [7], flockg problems [8], ad sychrozato [9] have receved sgfcat atteto. For a multaget system, regardless of the cooperatve task t performs, a graph s a atural choce to descrbe the formato flows of the system, ad graph coectvty s reported to be crtcal to esure system stablty. The combato of algebrac graph theory ad dyamcal systems theory has revved a broad terest the aalyss of multaget systems. I partcular, some elegat results o the dstrbuto of the egevalues of Laplaca matrces have bee derved, ad have bee used related areas, lke sychrozato of chaotc oscllators [10,11]. I ths paper, motvated by the prevous research o multaget coordato, we propose a ew cooperatve cotrol problem. We start descrbg ths problem by a example. Ths paper was ot preseted at ay IFAC meetg. Correspodg author. Tel.: ; fax: E-mal addresses: fche@lve.com (F. Che), we.re@usu.edu (W. Re), yogca.cao@aggemal.usu.edu (Y. Cao). Suppose a team of umaed groud vehcles (UGVs) s set to detect ad establsh a corrdor through a hostle terra. 1 To protect the UGVs from potetal threats, aother team of armed robotc vehcles (ARVs) s dspatched to provde groud coverage for the UGVs. That s, ARVs must surroud the UGVs. What kd of algorthms ca the ARVs use to acheve ths purpose? Ca they stll be decetralzed? I ths paper, ths problem, referred to as a surroudg cotrol problem, s formulated uder a leader follower framework, where a team of followers s used to surroud a team of leaders. To desg cotrollers for the followers, the geometrc ceter of the leaders should be kow. Geerally, ths pece of formato caot be obtaed drectly sce each follower mght have access to oly a subset of the leaders. Therefore, a decetralzed estmator s costructed at each follower to estmate the geometrc ceter of the leaders. The, the estmated ceter s used to desg cotrollers for the followers, whch results a system wth coupled estmato ad cotrol. The framework of multaget coordato by smultaeous decetralzed estmato ad cotrol s proposed [13]. I ths paper, we frst assume that each follower has the kowledge of the geometrc ceter of the leaders, ad propose a cotrol law to acheve some desred covergece propertes. We the costruct a dstrbuted estmator ad show that whe usg the estmated formato, the resultg system stll has the desred covergece propertes f the cotrol parameters are chose approprately. 1 Ths example s motvated by [1] /$ see frot matter 010 Elsever B.V. All rghts reserved. do: /j.syscole

2 F. Che et al. / Systems & Cotrol Letters 59 (010) The surroudg cotrol problem ca be cosdered a verse cotamet cotrol problem, where a group of agets are drve to be cotaed a partcular area specfed by aother group durg ther trasportato. The cotamet cotrol problem was proposed ad studed [14] uder a udrected etwork topology ad was exteded to a drected etwork topology [15] ad to corporate swarmg behavor [16]. As show [14,15], a decetralzed cosesus-lke protocol ca be used to solve the cotamet cotrol problem. However, for the surroudg cotrol problem, the stuato s qute dfferet. To solve ths problem, some global formato, e.g., the geometrc ceter of the leaders, eeds to be kow. To keep the decetralzed ature of the cotroller, a estmator s used the cotroller, whch brgs some dffcultes stablty aalyss of the resultg system. The surroudg cotrol problem s also closely related to the target eclosg problem [17,18], where there s oly oe leader volved. I [17], the task of capturg a movg object s dvded to two problems: the eclosure problem ad the graspg problem. It s assumed that each robot ca recogze up to two robots as ts eghbors ad the eghborhood s defed by the agles of the robots. A smlar problem s vestgated [18] by usg the cyclc pursut strategy. Smple feedback cotrol laws are desged to acheve the desred global behavor. It s assumed that the posto formato of the leader s avalable to all the agets. The problem of steerg a group of ucycles to form a collectve uform crcular moto aroud a fxed target wth equal agular dstaces s studed [19]. The problem s dvded to two subproblems. Oe s to acheve a coordated moto f the agets are close to the target, whle the other s to avgate the agets closer to the target f they are far away from the target. I [0], the authors study a model of self-propelled partcles that move at a costat speed o the surface of a sphere. Le group represetato s used to detfy crcular formatos. Shape cotrol laws are proposed to solate the crcular formatos of the partcles arraged symmetrc patters. Note that the surroudg cotrol problem, multple leaders eed to be surrouded, whch s more dffcult tha the sgle leader case, especally whe oly local formato ca be used. I addto, we cosder more geeral commucato topologes, rather tha a specal topology such as the cyclc pursut topology. Fally, we do ot assume that the posto formato of the leaders s avalable to all the followers. The rest of the paper s orgazed as follows: Secto, the otato ad termology used throughout ths paper are troduced. The surroudg cotrol problem ad the balaced surroudg cotrol problem are formulated Secto 3. They are solved Sectos 4 ad 5, respectvely. A smulato example s gve Secto 6. Fally, Secto 7 summarzes the ma coclusos.. Mathematcal prelmares Let R d deote the d-dmesoal Eucldea space. The detty matrx s deoted by I, 0 s the vector wth all zeros, ad 1 s the vector wth all oes. Uless otherwse stated, the orm used throughout ths paper s the Eucldea orm. For a set S, S deotes the umber of elemets S. For x R d ad S R d, defe x S f x y. y S A udrected graph of order s deoted by G (V, E) comprsg a set V {1,,..., } of odes ad a set E {(, j) j} of edges. If there s a edge from ode to ode j, the ode ad ode j are eghbors of each other. The set of all eghbors of ode s deoted by N {j V j}. A path a graph G s a sequece of odes such that from each of ts odes there s a edge to the ext ode the sequece. A graph s called coected f every par of dstct odes the graph ca be coected through some path. For a graph G = (V, E), let A = [a j ] be the adjacecy matrx where a j = 1 f (, j) E ad a j = 0 otherwse. I addto, let D dag(d 1, d,..., d ) be the degree matrx wth d = j N a j, where d s called the degree of ode. The Laplaca matrx of the graph G s gve by L D A. Here L s symmetrc postve sem-defte, ad L has a smple zero egevalue f ad oly f the graph G s coected [1]. I ths paper, the egevalues of a Laplaca matrx L s reumbered the followg way: 0 = λ m (L) λ 1 (L) λ (L) λ (L) λ max (L). 3. Problem descrpto Suppose that there are a umber of agets whch behave ether as leaders or followers. The set of the leaders s deoted by V L, whle the set of the followers s deoted by V F. Both V L ad V F are assumed to be oempty. I partcular, let N > 0 be the umber of the leaders, ad > 0 be the umber of the followers. Let x R d deote the posto of aget that moves R d. I ths paper, the leaders are assumed to be statoary. I addto, assume that the followers obey to the sgle-tegrator dyamcs,.e., ẋ = u, V F. (1) If the cotrol put u makes the followers surroud the leaders evetually,.e., lm x j(t) co(v F ) = 0 () for all j V L, where co(v F ) deotes the covex hull formed by the postos of the followers, the we say that a surroudg cotrol problem s solved. If a addtoal codto that the fal cofgurato of all the followers forms a regular polytope cetered at the geometrc ceter of the leaders s mposed, we say that a balaced surroudg cotrol problem s solved. Of course, a balaced surroudg cotrol problem s more dffcult tha a surroudg cotrol problem because t requres a addtoal codto o the fal cofgurato of the followers. We assume that each follower s equpped wth a sesg devce ad a commucato devce. Let the sesg radus of each follower be r. A udrected graph G s (V F, E s ) s used to descrbe the sesg relatoshps betwee the followers, where E s {(, j) x x j < r}. Let N s {j V F (, j) E s }, whch s the set of all followers who are wth the sesg radus of follower. A udrected graph G c (V F, E c ) s used to descrbe the commucato relatoshps betwee the followers, where E c {(, j) followers, j ca commucate wth each other}. I addto, let N c {j V F (, j) E c }, whch s the set of all the followers who ca commucate wth follower. The posto formato exchage betwee the leaders ad the followers s acheved va commucato. A udrected graph Ḡ (V L V F, Ē) s used to descrbe the formato flows betwee the leaders ad the followers, where (, j) Ē for V L ad j V F f ad oly f posto formato ca be commucated betwee F leader ad follower j. For leader, let N deote the set of the L followers who ca commucate wth leader. For follower, let N deote the set of the leaders who ca commucate wth follower. Before desgg the cotrollers, four assumptos are order. Assumpto 1. The umber of the followers satsfes d + 1, where d s the dmeso of the movg space. Remark 1. It ca be verfed that Assumpto 1 s also ecessary for the purpose of surroudg cotrol. For example, whe d = 1, f = d, the there s oly oe follower, whch caot surroud multple leaders.

3 706 F. Che et al. / Systems & Cotrol Letters 59 (010) Assumpto. Itally, there are o d followers wth postos x p, = 1,..., d, p V F, such that d x = a j x pj j=1 for all V F where a j 0 ad d j=1 a j = 1. Remark. Whe d = 1, Assumpto meas that the set of the tal postos of all the followers should ot be a sgleto. Whe d =, Assumpto meas that, tally, the followers should ot be o a le. Assumpto 3. Itally, the followers are placed wth a crcle wth radus r. Assumpto 4. The graph G c s assumed to be fxed ad coected. I the graph Ḡ, a leader ca commucate wth at least oe follower tally. 4. Surroudg cotrol problem I ths secto, the surroudg cotrol problem s studed. I the followg, we frst focus o the oe-dmesoal case (d = 1), ad the exted the results to the hgher-dmesoal case (d > 1) x s avalable I ths subsecto, t s assumed that x 1 j V L x j s avalable to all the followers. Ths assumpto holds whe the postos of all the leaders ca be obtaed by ay follower. If x s avalable to all the followers, the the cotrol law s desged for each follower as u = (x x j ) j N s + κ x + ξ sg [x (0) x j (0)] x (3) j N s (0) where, κ, ad ξ are postve costats, ad sg( ) s the sgum fucto. Remark 3. I the cotrol put, the frst term s a repulsve force betwee follower ad ts eghbors. Ths term s used to elarge the covex hull co(v F ) formed by the followers. The secod term s a attractve force, whch s used to drve the followers to x + ξ sg j N s(0)[x (0) x j (0)]. It s frst show that the algorthm (3) has the ablty of collso avodace. Lemma 1. Suppose that Assumptos 1 3 hold for the system (1) wth the cotrol put (3). If x (0) > x j (0),, j V F, the x (t) > x j (t) for all t 0. Proof. For otatoal coveece, defe ξ ξ sg{ j N s [x (0) x j (0)]}. Because Assumpto 1 holds, the umber of the followers satsfes. From Assumpto 3, we kow that the sesg graph G s (0) s a complete graph tally. I addto, due to Assumpto, we kow that f x (0) > x j (0), the ξ > ξ j. Suppose that, at tme t 1, x j s stll less tha but suffcetly close to x such that N s \ {j} = N s j \ {} ad x x j < m,j VF,ξ ξ j ξ ξ j. The, the followg perod of tme whose legth s supposed to be T, oe has ẋ (t) ẋ j (t) = (x x k ) + κ ( x + ξ x ) k N s (x j x k ) κ ( x + ξ j x j ). k N s j Because N s \ {j} = N s j \ {} ad x > x j, oe kows that ẋ (t) ẋ j (t) κ ((ξ ξ j ) (x x j )). Sce ξ ξ j m k,l VF,ξ k ξ l ξ k ξ l ad x x j < m k,l VF,ξ k ξ l ξ k ξ l, oe has ẋ (t) ẋ j (t) > 0. Therefore, at tme t 1 + T, x (t 1 + T) x j (t 1 + T) = x (t 1 ) x j (t 1 ) + t1 +T t 1 > x (t 1 ) x j (t 1 ), (ẋ (τ) ẋ j (τ))dτ whch dcates that the dstace betwee x ad x j wll become larger. Thus x j wll ever catch up wth x,.e. x j (t) < x (t) for all t 0. Remark 4. Ths lemma dcates that f there s o collso tally, the collso avodace betwee followers s guarateed the followg tme terval. Remark 5. It ca be draw from ths lemma that f x q (0) = max j VF x j (0) ad x p (0) = m j VF x j (0), the x q (t) = max j VF x j (t) ad x p (t) = m j VF x j (t) for all t 0. I the followg, the equlbrum pots of the system (1) s aalyzed. Lemma. Suppose that Assumptos 1 3 hold. If the rght had sde of (3) equals to 0 for all V F, the [ x ξ, x + ξ] [m j VF x j, max j VF x j ]. Proof. If [ x ξ, x + ξ] [m j VF x j, max j VF x j ] does ot hold, the ether x ξ < m j VF x j or x + ξ > max j VF x j. Suppose x ξ < m j VF x j. Due to Lemma 1, the set M { V F, x = m j VF x j } s oempty ad M V F. For p M, oe has ẋ p = (x p x j ) + κ x + ξ sg j N s p k N s p(0) [x p (0) x k (0)] xp. (4) Sce x p = m j VF x j, oe kows that (x p x j ) 0. (5) j N s p Because ξ sg k Np(0) s (x p(0) x k (0)) = ξ, oe kows that x + ξ sg [x p (0) x k (0)] x p = x ξ x p < 0. (6) k N s p(0) The, (4) (6) yeld ẋ p < 0, whch s a cotradcto. Smlarly, t ca be proved that x + ξ > max j VF x j wll also lead to a cotradcto. The ext lemma shows that u = 0 wll be acheved asymptotcally. Lemma 3. For the system (1) wth the cotrol put (3), f ( 1), the lm u = 0 for all V F. κ > Proof. Let x [x 1,..., x ] T. Defe a Lyapuov fucto caddate V = 1 ẋt ẋ. (7)

4 F. Che et al. / Systems & Cotrol Letters 59 (010) Because ẍ = u = (ẋ ẋ j ) κ ẋ, oe has j N s ẍ = L s (t)ẋ κ ẋ, (8) wth L s (t) the Laplaca matrx of the graph G s at tme t, whch yelds V = ẋ T ẍ = ẋ T ( L s (t)ẋ κ ẋ) = ẋ T ( L s (t) κ I)ẋ { λ max [L s (t)] κ } ẋ. From Gershgor Dsc Theorem, t ca be proved that λ max (L s (t)) ( 1). If κ > ( 1), the V < 0 for ẋ 0, whch leads to the result that lm ẋ (t) = 0 for all V F. Remark 6. The codto κ > ( 1) ca be replaced wth λ max (L s (t)) < κ. Because λ max (L s (t)) s determed by the topology of the graph G s, a terestg problem s to vestgate the relatoshps betwee the topology of G s ad the dyamcs of the system (1) wth the cotrol put (3) whe ad κ are both fxed. Ths problem becomes eve more terestg whe the graph G s s geerated by some complex etwork models, e.g., small-world model [] or scale-free model [3]. The cotrol of complex etworks has receved tesve atteto from varous dscples, see, for example, [4] ad [5]. By usg Lemmas ad 3, oe ca prove the followg result. Theorem 1. Suppose that Assumptos 1 3 hold for the system (1) wth the cotrol put (3). If κ > ( 1) ad ξ max VL x x, the the surroudg cotrol problem s solved asymptotcally. Proof. From Lemma 3, oe kows that lm u = 0 for all V F. Because Assumptos 1 3 hold, we kow that Lemma holds. The, by Lemma, oe has [ x ξ, x + ξ] [m x j, max x j ] as t. Because ξ max VL x x, oe kows that x [ x ξ, x + ξ] [m x j, max x j ] for all V L as t, whch dcates lm x co(v F ) = 0 for all V L. 4.. x s ot avalable I ths subsecto, t s ot assumed that x s avalable to all the followers. Therefore, a decetralzed estmator eeds to be costructed at each follower to estmate x. The followers exchage ther estmates va the commucato graph G c. The estmator s gve as follows: ẏ = κ j N c (y j y ), (9) where y s follower s estmate of x, κ > 0 s a costat, ad the tal codto of y satsfes 1 y (0) = N j NL (0) N F j (0) x j(0) (10) F for all V F. Recall from Secto 3 that for leader j, N j deotes the set of the followers who ca commucate wth leader j, whle for L follower, N deotes the set of the leaders who ca commucate wth follower. Remark 7. Note that to mplemet the estmator, the absolute postos of the leaders at the tal tme are requred. The ext lemma shows that the estmator (9) s globally asymptotcally stable. Lemma 4. Suppose that Assumpto 4 holds. Usg the estmator (9) wth the tal codtos (10), oe has lm y x = 0, V F. Proof. Defe e y x, the ė = ẏ = κ j N c Let (e j e ). e [e 1,..., e ] T (11) ad V(e) 1 et e. The oe has V = e T ė = κe T (L c e). (1) Due to Assumpto 4, we kow that tally, a leader ca commucate wth at least a follower. Therefore, t ca be verfed that e1 = 0, whch yelds that V κλ (L c ) e. (13) Sce the graph G c s fxed ad coected, oe has λ (L c ) > 0 [1], whch leads to V < 0 for e 0. The, oe ca desg the followg ew cotrol law u = y + ξ sg [x (0) x k (0)] x, (14) k N s (0) where y s gve by (9) ad (10). Although the estmator s globally asymptotcally stable, whe x s replaced wth the estmator (9) wth the tal codtos (10), t s ot clear whether the resultg system of (1) usg (14) s stll stable. Therefore, stablty aalyss s eeded. Before that, a lemma s order. Lemma 5. Suppose that Assumptos 1 3 hold for the system (1) wth the cotrol put (14). Defe M { V F x = m j VF x j (0)} ad M { V F x = max j VF x j (0)}. Let p M ad q M. If u = 0 for all V F, the [y p ξ, y q + ξ] [m x j, max x j ]. Proof. The proof s smlar to that of Lemma, ad hece omtted here. Theorem. Suppose that Assumptos 1 4 hold for the system (1) wth the cotrol put (14). If κ <, ad ξ max VL x x, the the surroudg cotrol problem s solved asymptotcally. Proof. Defe V 1 ẋt ẋ + 1 aet e (15) where e s defed (11) ad a > (max V d c F ) wth d c beg the λ (L c ) degree of ode G c. It ca be obtaed that ẍ = u = κ (y j y ) ẋ, j N c

5 708 F. Che et al. / Systems & Cotrol Letters 59 (010) whch ca be rewrtte a matrx form as ẍ = κ(l c y) ẋ. The oe has V = ẋ T ẍ + ae T ė = ẋ T ẋ + κẋ T (L c y) + ae T ė. (16) By usg the fact that L c 1 = 0, we kow that L c y = L c (e+ x1) = L c e, whch yelds that κẋ T (L c y) = κẋ T (L c e). Accordg to Cauchy Schwartz equalty, oe has ẋ T L c e ẋ T L c e ẋ L c e 1 (ẋt ẋ + (L c e) T (L c e)) = 1 (ẋt ẋ + e T L c e). By usg the Gershgor Dsc Theorem, λ max (L c ) < max VF d. Because a egevalue λ of L c correspods to a egevalue λ of L c, oe kows that λ max (L c ) 4(max V F d ), whch yelds ẋ T L c e 1 ẋt ẋ + (max V F d ) e T e. (17) Usg (13) ad (17), (16) ca be further reduced to V ẋ T ẋ + 1 κẋt ẋ + κ(max d c ) e T e aκλ (L c ) e V F ẋ T 1 + κ ẋ + κ(max d c ) aκλ (L c ) e. (18) V F Because κ < ad a > (max V d c F ), oe has V λ (L c ) < 0 wheever ẋ 0 or e 0, whch yelds lm ẋ = 0. By usg Assumptos 1 3, we kow that Lemma 5 holds, whch yelds [y p ξ, y q + ξ] [m x j, max x j ], as t. I addto, by usg Assumpto 4, we kow Lemma 4 hold. The oe ca coclude [ x ξ, x + ξ] [m x j, max x j ], (19) as t. Because ξ max VL x x, oe kows that x [ x ξ, x + ξ] [m x j, max x j ] for all V L as t, whch dcates lm x co(v F ) = 0 for all V L. Remark 8. Whe x s ot avalable, we caot guaratee collso avodace due to the fact that the followers have dfferet estmates of x. However, the followers ca be equpped wth local collso avodace capabltes ad mechasms. For example, soars or frared sesors ca be stalled o the followers such that collso avodace s guarateed whe the followers are close to each other or close to a leader. Remark 9. To solve the surroudg cotrol problem, we eed some global formato,.e., x, max VL x x, ad. The geometrc ceter of the leaders x s obtaed by usg a decetralzed estmator. Therefore, max VL x x ca also be calculated a decetralzed way. Moreover, the umber of the followers ca be obtaed by usg some decetralzed commucato protocols computer scece. I other words, the cotrol laws that we desged are decetralzed. The results Theorem ca be geeralzed to the case of hgher dmesos, provded that some tal codtos are met. Theorem 3. For the case of hgher dmesos,.e., d > 1, suppose that Assumptos 1 4 hold for the system (1) wth the cotrol put (14), ad tally co(v F ) s a hyperrectagle, whch s the geeralzato of a rectagle for hgher dmesos. If κ < ad ξ max VL x x, the the surroudg cotrol problem s solved asymptotcally. Proof. The proof s smlar to that of Theorem. 5. Balaced surroudg cotrol problem I ths secto, the balaced surroudg cotrol problem s cosdered. That s, () holds ad all followers coverge to a cofgurato that forms a regular polytope wth a geometrc ceter at x. Here, we assume that the geometrc ceter of the leaders s avalable to all the followers. If t s ot avalable, the estmator preseted the last secto ca be used, ad the stablty aalyss ca be doe uder the estmato-ad-cotrol framework as preseted the last secto. We hece assume that all agets share a commo coordate system cetered at x. I ths secto, we oly cosder the two-dmesoal case, but the derved results ca be exteded to the case of a hgher dmeso. To smplfy the aalyss, the polar coordate system s used, where r ad θ are, respectvely, the radus ad agle of follower. Suppose that ṙ = η (0) θ = ω. (1) I the ew coordate system, we desg η ad ω. Note that, oce η ad ω are desged, the cotrollers u for the orgal system (1) ca be specfed as follows: [ ] η cos θ u = r ω s θ. η s θ + r ω cos θ We frst desg ω. Defe θ j m{(θ θ j ) mod π, (θ j θ ) mod π}, whch measures the dstace betwee θ ad θ j. If θ j < π, there are two possble cases: ether 0 θ θ j < π or π π < θ θ j π. We use a udrected graph G θ (V F, E θ ) to deote the agle relatoshps betwee the followers, where E θ {(, j) θ j < π }. The set of the eghbors of follower G θ s deoted by N θ. The requremet that the followers form a regular polytope dcates that the agles of the followers should dstrbute a balaced way, whch s defed as follows. Defto 1 (Balaced Dstrbuto). The agle vector θ [θ 1,..., θ ] T s sad to be a balaced dstrbuto f ad oly f θ j mod π = 0 ad θ θ j for all j, V F, ad j V F.

6 F. Che et al. / Systems & Cotrol Letters 59 (010) a 1 (θ θ j ) + a (θ θ j ), 0 θ 4 θ j < π, 1 π r j a π 1 + a π 4, θ θ j π π, 1 a 3 ((π) (θ θ j ) ) + a 4 ((π) (θ θ j ) ), π π < θ θ j π, where a 1 = a ( π ), a = ((π) (π π ) ) ( π )4 a 4, a 3 = a 4 ((π) (π π ) ), ad a 4 < 0. Box I. To desg ω, we frst defe a potetal fucto r j. The ω s desged such that the agles of the followers chage alog the egatve gradet of j N θ r j. The fucto r j should have the followg propertes: () t s a repulsve force whe 0 θ θ j < π, ad () t becomes a attractve force whe π π < θ θ j π. I ths way, a balaced dstrbuto wll be acheved (see Box I). Lemma 6. The followg propertes hold for the fucto r j : 1. r j 0, r. j < 0 for 0 < θ (θ θ j ) θ j < π ad r j > 0 for (θ θ j ) π π < θ θ j < π, 3. r j s cotuously dfferetable. Proof. Ths ca be show by some smple calculatos, ad s hece omtted. The, the cotrol law ω s defed as ω = j N θ r j θ, V F. () To mplemet (), we eed the followg assumpto. Assumpto 5. The edge set of G θ s a subset of the edge set of the commucato graph G c,.e., E θ E c. Ths assumpto ca be acheved by choosg approprate tal values of θ(0). For example, we ca choose θ(0) such that G θ (0) s a subgraph of a lear cyclc pursut graph [6]. The let E c be the lear pursut graph, we ca have E θ (t) E c for all t 0. Therefore, the followers ca exchage ther formato through the commucato graph G c. Because of Assumptos 1 ad 5, we kow that the cotroller () ca be mplemeted. Lemma 7. Suppose that Assumptos 1 ad 5 hold. The set Θ {θ 0 < θ θ j < π} s postvely varat for the system (1) usg (). Proof. Defe V 1 V F,j V F, j r j. (3) It ca be verfed that (1) usg () becomes θ = V θ whch yelds V 0. By otg that θ θ j = 0 or θ θ j = π lead to V = ad V(t) V(0) <, (4) oe ca coclude that Θ s postvely varat. Theorem 4. Suppose that Assumptos 1 ad 5 hold. For the system (1) usg () wth θ(0) Θ, θ wll acheve a balaced dstrbuto asymptotcally. Proof. By usg Lemma 7 ad LaSalle s varat prcple [7], we kow that the system wll coverge to the largest varat set S {θ V(θ) = 0}. Because V = V θ = θ, t follows that V = 0 mples that θ = 0. Therefore, t s straghtforward to obta that lm θ(t) = 0. That s, the agles of all the followers wll stop chagg evetually. Reumber the followers such that θ p1 θ p θ p. For otatoal coveece, defe θ p+1 = θ p1 ad θ p0 = θ p. It ca be verfed that θ p,p +1 = π. (5) V F If θ s ot a balaced dstrbuto, oe kows that the set { V F θ p,p +1 π } s ot empty. I addto, t ca be proved that s ot empty where { V F θ p1,p < π }. Ths s show by cotradcto. Suppose s empty, the for all p, oe has θ p1,p π, whch dcates that p 1 for all p. By repeatg the above statemets, oe ca obta = V F, whch yelds θ p,p +1 π for all V F. Because θ s ot balaced, oe kows that there s at least oe follower V F such that θ p,p +1 > π, whch leads to V F θ p,p +1 > π. (6) Ths cotradcts wth (5). Therefore, oe ca coclude that s ot empty. For p, oe has θ p = = j p,θ p,j< π j p,θ p,j< π r j θ p r j (θ p θ j ) (θ p θ j ). If j {p 1, p,..., p 1 }, oe kows 0 < θ p θ j < π, whch r yelds j (θ (θ p θ j ) p θ j ) < 0 by usg Lemma 6. Smlarly, t ca be proved that r j (θ p θ j ) (θ p θ j ) < 0 holds also for j {p 1, p,..., p 1 }. The, oe ca coclude θ p θ = 0. > 0, whch cotradcts wth

7 710 F. Che et al. / Systems & Cotrol Letters 59 (010) (a) t = 0. (b) t = 0.5. (c) t = 1.5. (d) t = 5. Fg. 1. Surroudg cotrol of moble etworks. The leaders are deoted by, whle the followers are represeted by. We ext desg η. The ma purpose of η s to drve all agets to a crcle cetered at x ad wth a radus ξ. The cotroller for η s of the followg form: [ ] η = βsg r ξ r s (θ θ) r cos(θ θ), (7) where β > 0 s a costat, ad r ad θ are, respectvely, the radus ad agle of the geometrc ceter of the leaders. From Assumptos 1 ad 5, we kow that Theorem 4 holds. Therefore, θ wll acheve a balaced dstrbuto asymptotcally. I the followg, t s show that θ s bouded for all V F. If 0 θ θ j < π, let r j(r a ) = V(0), where V(0) s the value of the fucto V, defed by (3), at tme 0. I addto, oe has that r j ( θ θ j ) V(t) V(0) = r j (r a ), whch yelds that r a θ θ j < π. It thus follows that r j π π π θ 4a a r a r a. (8) If π π < θ θ j π, let r j (r b ) = V(0). The oe has that π π < θ θ j r b, whch yelds that r j θ 4a 4 r b (π) r b (π) π π r b π π. (9) Defe r max max π 4a a r a π π r a, 4a 4 r b[(π) r ] b (π) π π oe has that r b π π, (30) θ ( 1)r max. (31) Theorem 5. Suppose that Assumptos 1 ad 5 hold. For the system (1) usg () ad (0) usg (7), f β > r( 1)r max ( r(ξ r ) 1 + 1), ξ > max( r, max VL x x / cos(π/)), ad θ(0) Θ, the the balaced surroudg cotrol problem s solved asymptotcally. Here r max s a costat defed by (30). Proof. Defe V = 1 [ r ξ r s (θ θ) r cos(θ θ)]. The oe has V r ξ r s (θ θ) r cos(θ θ) r θ [ r(ξ r ) 1 + 1] β.

8 F. Che et al. / Systems & Cotrol Letters 59 (010) Therefore, C( x, ξ) wll be the crcumcrcle of the regular polytope formed by all the followers, where C( x, ξ) deotes the crcle cetered at x ad wth the radus ξ. Because the radus of the crcle of the regular polytope s r = ξ cos(π/). Sce ξ max VL x x / cos(π/), oe kows that x B( x, r ) co(v F ), V L, (33) where B( x, r ) {x R x x r }. 6. Smulatos Fg.. The estmates of the geometrc ceter of the leaders. Because β > r( 1)r max ( r(ξ r ) 1 + 1), oe kows V < 0. Therefore, the system wll coverge to lm r (t) = ξ r s (θ θ) r cos(θ θ). (3) Thus, as t, (r r cos(θ θ)) ξ r s (θ θ), whch yelds r + r r r cos(θ θ) ξ. Ths further leads to r cos θ r r cos θ cos θ + r cos θ + s θ r r s θ s θ + r s θ ξ. The oe has r (r cos θ r cos θ) + (r s θ r s θ) ξ. I ths secto, a smulato example as show Fg. 1 s frst preseted to llustrate Theorem 3. Ths example cludes four leaders ad four followers. The cotrol parameters are specfed as follows: κ = 1, r = 8, ad ξ = max Vf x x. The edge set E c of the commucato graph G c s defed to be E c {(, + 1) = 1,..., V f 1}. Itally, each leader ca commucate wth a follower, ad the covex hull of the followers s set to be a rectagle, where the covex hull s formed by the dashed les (Fg. 1(a)). Whe the followers are movg, the rectagle mght ot be kept because the followers have dfferet estmates of the geometrc ceter of the leaders x (Fg. 1(b) ad (c)). However, as the estmates coverge to x, the rectagle s recovered ad the followers ca surroud the leaders evetually (Fg. 1(d)). Fg. shows the estmates of the geometrc ceter of the leaders, whch s [0, 0] ths case. The dashed les deote the followers estmates of the x dmeso of the geometrc ceter, whle the sold les represet the estmates of the y dmeso of the geometrc ceter. (a) t = 0. (b) t = 5. (c) t = 15. (d) t = 100. Fg. 3. Balaced surroudg cotrol of moble etworks.

9 71 F. Che et al. / Systems & Cotrol Letters 59 (010) I the followg, we preset aother example as show Fg. 3 to llustrate Theorem 5. Ths example cludes 4 leaders ad 6 followers. Itally, the agles of the followers are geerated such that 0 < θ θ j < π for all j,, j V F (See Fg. 3(a)). Note that ths example, co(v F ) s ot requred to be a hyperrectagle tally. The cotrol parameters are chose to satsfy β > r( 1)r max ( r(ξ r ) 1 + 1) ad ξ > max( r, max VL x x / cos(π/)), ad the commucato graph G c s geerated to satsfy Assumpto 5. The covex hull of the followers s dcated by dashed les. The geometrc ceter of the leaders s deoted by a crcle. As ca be see from Fg. 3, ot oly ca the followers surroud the leaders, but also the fal cofgurato of the followers forms a regular polytope cetered at the geometrc ceter of the leaders. It ca be show that the dstaces, deoted by dotted les, from the followers to the geometrc ceter are of the same values fally. 7. Coclusos I ths paper, a ew cooperatve cotrol problem, surroudg cotrol, s proposed, ad partally solved. To solve ths problem, some global formato s eeded,.e., the geometrc ceter of the leaders. To ths am, a decetralzed estmator s costructed to estmate the geometrc ceter, ad s used the cotroller. By usg tools from graph theory ad dyamcal systems theory, we show that our cotrollers guaratee that the followers ca surroud the leaders evetually. Ths paper focuses o the statoary leader case ad a fxed commucato graph. The model we proposed s of course very smple, but t serves as a atural startg pot for the study of more complcated models, for stace, whe the commucato graph s tme varyg. Aother future drecto s to exted the curret paper to cosder movg leaders. The most challegg part the exteso mght be to desg a estmator that ca track the geometrc ceter of the movg leaders. I addto, the stablty aalyss could also be more volved. As stated the troducto, the algorthm proposed ths paper may ultmately fd applcatos real lfe. But before ths ca happe, some compellg ssues such as the effects of tme delays ad dsturbaces should be satsfactorly addressed. Ackowledgemet Ths work was supported by the Natoal Scece Foudato uder Grat ECCS Refereces [1] A. Jadbabae, J. L, A.S. Morse, Coordato of groups of moble autoomous agets usg earest eghbor rules, IEEE Trasactos o Automatc Cotrol 48 (6) (003) [] R. Olfat-Saber, J.A. Fax, R.M. Murray, Cosesus ad cooperato etworked mult-aget systems, Proceedgs of the IEEE 95 (1) (007) [3] W. Re, R.W. Beard, E.M. Atks, Iformato cosesus multvehcle cooperatve cotrol, IEEE Cotrol Systems Magaze 7 () (007) [4] M. Cao, A.S. Morse, B.D.O. Aderso, Reachg a cosesus a dyamcally chagg evromet: a graphcal approach, SIAM Joural o Cotrol ad Optmzato 47 () (008) [5] G. Xe, L. Wag, Cosesus cotrol for a class of etworks of dyamc agets, Iteratoal Joural of Robust ad Nolear Cotrol 17 (10 11) (007) [6] Z. 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