IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 6, JUNE 22 553 Disribued Coaime Corol wih Muliple Dyamic Leaders for Double-Iegraor Dyamics Usig Oly Posiio Measuremes Jiazhe Li, Wei Re, Member, IEEE, ad Shegyua Xu Absrac This oe sudies he disribued coaime corol problem for a group of auoomous vehicles modeled by double-iegraor dyamics wih muliple dyamic leaders. The objecive is o drive he followers io he covex hull spaed by he dyamic leaders uder he cosrais ha he velociies ad he acceleraios of boh he leaders ad he followers are o available, he leaders are eighbors of oly a subse of he followers, ad he followers have oly local ieracio. Two coaime corol algorihms via oly posiio measuremes of he ages are proposed. Theoreical aalysis shows ha he followers will move io he covex hull spaed by he dyamic leaders if he ework opology amog he followers is udireced, for each follower here exiss a leas oe leader ha has a direced pah o he follower, ad he parameers i he algorihm are properly chose. Numerical resuls are provided o illusrae he heoreical resuls. Idex Terms Coaime corol, disribued corol, double-iegraor dyamics, muli-age sysems. I. INTRODUCTION The disribued muli-vehicle cooperaive corol has received icreasig aeio from researchers i differe areas. This is due o is broad applicaios ad is advaages such as low cos, high adapiviy, ad easy maieace, compared wih is ceralized couerpar. The leaderless cosesus problem is a fudameal problem i disribued muli-vehicle cooperaive corol. The objecive is o reach a agreeme o cerai quaiies of ieres amog he vehicles hrough local ieracio. Recely, sigifica progress has bee made i he leaderless cosesus problem (See [] [3] ad refereces herei). A more challegig problem i disribued muli-vehicle cooperaive corol is he coordiaed rackig problem, where here exiss a sigle or muliple dyamic leaders. I he sigle-leader case, he objecive is o drive he saes of he followers o approach he sae of he dyamic leader. This problem ad is varias were ivesigaed i [4] [7], [6]. I he muli-leader case, he objecive is o drive he saes of he followers io he covex hull spaed by hose of he dyamic leaders, also called he coaime corol problem. The coaime corol problem has may applicaios i pracice. For example, suppose ha a group of robos are o move from oe place o aoher, bu oly a subse of hem has he abiliy o deec he hazardous obsacles. This subse of robos ca be desiged as leaders. The oher robos Mauscrip received Ocober 4, 2; revised April 8, 2 ad April 2, 2; acceped Augus 29, 2. Dae of publicaio November 3, 2; dae of curre versio May 23, 22. This work was suppored by Naioal Sciece Foudaio uder gra ECCS-2393, he Naioal Naural Sciece Foudaio of Chia uder Gras 67443, 647, 69422, ad 626, ad he Qig La Projec. Recommeded by Associae Edior Y. Hog. J. Li is wih he School of Auomaio, Najig Uiversiy of Sciece ad Techology, Najig 294, Chia ad was also wih he Deparme of Elecrical ad Compuer Egieerig, Uah Sae Uiversiy, Loga, UT 84322 USA (e-mail: jiazheli983@yahoo.com.c). W. Re is wih he Deparme of Elecrical Egieerig, Uiversiy of Califoria, Riverside, CA 9252 USA (e-mail: re@ee.ucr.edu). S. Xu is wih he School of Auomaio, Najig Uiversiy of Sciece ad Techology, Najig 294, Chia (e-mail: syxu2@yahoo.com.c). Digial Objec Ideifier.9/TAC.2.27468 ca be desigaed as followers. For he followers, oe way o reach he arge area safely is o say i he movig safe area formed by he leaders. I [8], a sop-ad-go sraegy was proposed for vehicles modeled by sigle-iegraor kiemaics uder a fixed udireced ework opology. I [9], he parial differeial equaio heory was exploied ad a hybrid corol schemes was proposed for he leaders. A exesio o a swichig direced ework opology was give i [], where he Lyapuov-based approach was used. I [], he se ipu-o-sae sabiliy ad he se iegral ipu-o-sae sabiliy problems were cosidered for muli-age sysems wih muliple leaders, where all he followers had oliear eighbor-based coordiaio rules. Noe ha [8] [] cosider he muli-age sysems wih sigle-iegraor dyamics. I [2], he followers were assumed o have double-iegraor dyamics, bu he dyamics of he leaders were sigle iegraors. I [3], boh he leaders ad he followers have double-iegraor dyamics. However, he algorihms proposed i [3] require he velociy measuremes o be available. I realiy, i is more difficul o obai velociy ad acceleraio measuremes ha posiio measuremes. We are hece moivaed o desig disribued coaime corol algorihms for auoomous vehicles wih double-iegraor dyamics i he presece of muliple dyamic leaders usig oly posiio measuremes. The case where here exiss a sigle dyamic leader ca be reaed as a special case of muliple dyamic leaders. Whe he absolue posiio measuremes of he vehicles are available, we propose a disribued fiie-ime coaime corol algorihm. We show ha he followers are drive io he covex hull spaed by he dyamic leaders i fiie ime if he ework opology amog he followers is udireced, for each follower here exiss a leas oe leader ha has direced pah o he follower, ad he parameers i he algorihm are properly chose. Whe he absolue posiio measuremes of he vehicles are o available, we propose a disribued adapive coaime corol algorihm usig he relaive posiio measuremes. We show ha he followers will ulimaely move io he covex hull spaed by he dyamic leaders uder similar codiios o he case where he absolue posiio measuremes are available. The salie feaures of he algorihms proposed i his oe are as follows. Firs, boh algorihms ca solve he disribued coaime corol problem wih muliple dyamic leaders for vehicles wih double-iegraor dyamics while removig he requireme o he velociy measuremes. Secod, he firs algorihm guaraees fiie-ime covergece wihou he requireme ha he velociies of he leaders are ideical. Third, i he secod algorihm, he boud o he acceleraios of he leaders is o required o be kow. Fourh, he parameers i he secod algorihm are o required o saisfy ay codiio relaed o he ework opology. I coras, exisig algorihms for vehicles wih double-iegraor dyamics i [3] require he velociy measuremes, ca guaraee fiie-ime covergece oly whe he velociies of he leaders are ideical, require he boud o he acceleraios of he leaders o be kow, ad require parameers i he algorihm o saisfy a cerai codiio relaed o he ework opology whe he acceleraios of he leaders are o ideical. A prelimiary versio of he work has appeared i [5]. T Noaios: Defie p p [; ; ] 2. Give a p ad 2, defie sig() vecor [; ; p] T 2 [sg()jj ; ; sg( p)j pj ] T ad sg() [sg(); ; sg( p )] T, where sg() is he sigum fucio. We use diag(; ; p ) o deoe he diagoal marix of all ; ; p ;(l) o deoe he lh eleme of, ad I p o deoe he p by p ideical marix. 8-9286/$26. 2 IEEE
554 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 6, JUNE 22 II. BACKGROUND AND PRELIMINARIES Cosider a group of s vehicles. We use a graph G (V; E) o deoe he ework opology amog vehicles o s, where V f; ;g is he ode se ad EV2Vis he edge se. A direced edge (j; i) 2Eif vehicle i ca access iformaio from vehicle j bu o ecessarily vice versa. A udireced edge (j; i) 2Eif vehicle i ad vehicle j ca access iformaio from each oher. Here we assume ha (i; i) 62 E. The eighbor se N i of vehicle i is defied as N i fjj(j; i) 2Eg. Suppose ha vehicles o have a leas oe eighbor ad vehicles o have o eighbor. We call vehicles o he followers ad vehicles o he leaders. A graph is udireced if (i; j) 2Eimplies ha (j; i) 2E. We assume ha he graph associaed wih he followers are udireced ad furher assume ha a ij aji, i; j ; ;. A direced pah is a sequece of direced edges of he form (i ;i 2), (i 2;i 3);..., where i j 2V. A udireced pah is defied aalogously. The adjacecy marix A d [aij ] 2 ()2() is defied as a ij > if (j; i) 2Ead a ij oherwise. I is easy o see ha a ij, i ; ;, j ; ; because he leaders have o eighbors. The Laplacia marix L [l ij] 2 ()2() is defied as l ii ;j6i a ij ad l ij a ij, i 6 j. Noe ha L ca be rewrie as L L L 2 s2 s2s : () Suppose ha all he vehicles have double-iegraor dyamics give by _x i () v i (); _v i () u i (); i ; ; s (2) where x i (), v i () ad u i () 2 m are, respecively, he posiio, velociy ad corol ipu associaed wih he ih vehicle. Suppose ha all leaders corol ipus have bee a priori chose as u i () f i (), i ; ; s, where f i() specify he leaders acceleraios ad hece he dyamic covex hull formed by he leaders. I his oe, we focus o he coroller desig for he followers. We have he followig defiiio. Defiiio 2.: Le C p. The se C is said o be covex if for ay x ad y i C, he poi ( )x y is i C for ay 2 [; ]. The covex hull of a se of pois X fx ; ;x qg is he miimal covex se coaiig all pois i X. We use Co(X) o deoe he covex hull of X. Le () Cofx (); ;x ()g ad 7() Cofv (); ;v ()g. The objecive of he disribued coaime corol problem is o desig u i () for all he followers such ha he followers move io he covex hull spaed by he dyamic leaders, i.e., if y()2() kx i () y()k! ad if y()27() kv i() y()k!, i ; ;,as!. Before movig o, we eed he followig assumpios ad lemmas. Assumpio 2.2: kv i()k, kf i()k ad k f _ i()k, i ; ; s, are all bouded. Assumpio 2.3: For each follower, here exiss a leas oe leader ha has a direced pah o he follower. Lemma 2.: [4] Uder Assumpio 2.3, L defied i () is symmeric posiive defiie. Noe from Lemma 2. ha L is iverible. Le x L () [x T (); ;x T ()] T, v L () [v(); T ;v()] T T, ad x d () [x T d(); ;x T d()] T (L L 2 I m)xl(), where x di () 2 m. I follows ha x d () (L L 2 I m )v L () ad x d () (L L 2 I m )_v L (). Lemma 2.2: Uder Assumpio 2.3, if y()2() kx di ()y()k ad if y()27() k _x di () y()k, i ; ;, for all. Proof: Uder Assumpio 2.3, by Lemma 4 i [4] we kow ha each ery of L L 2 is oegaive ad each row sum of L L 2 is equal o oe, which implies ha if y()2() kx di () y()k ad if y()27() k _x di () y()k, i ; ;. Noe from Lemma 2.2 ha x di () ad _x di () belog o, respecively, he covex hull formed by he posiios ad velociies of he leaders. If for each follower x i() ca be drive o xdi () ad v i() ca be drive o _x di (), he he coaime corol problem is solved. Therefore, x di () ad _x di () ca be regarded as, respecively, he desired posiio ad velociy of he ih follower i he covex hull formed by he leaders. Because kv i ()k ad kf i ()k, i ; ; s, are bouded (see Assumpio 2.2), i follows ha k _x d ()k ad kx d ()k are also bouded. We hece assume ha k _x d ()k a ad kx d ()k b. III. MAIN RESULTS A. Coaime Corol Usig Absolue Posiio Measuremes I his secio, we assume ha he absolue posiio measuremes of he vehicles are available bu he velociy ad acceleraio measuremes are o available. We propose he followig coaime corol algorihm: u i () sg z i ()sig [x i () ^x i ()] (3a) _z i()zi()ksig fz i()g _z i () k 2 sg fz i ()g (3b) sg z i ()sig [x i () ^x i ()] (3c) _^x i() k3sg i ; ; a ij [^xi() ^xj ()] j a ij [^xi()xj()] ; (3d) where ^x i (), z i () z i () [x i () ^x i ()], for i ; ;, k, k 2, k 3, ad are posiive cosa scalars, ad a ij, i ; ;, j ; ; s, is he (i; j)h ery of he adjacecy marix A d. Throughou his oe, he soluios o he closed-loop sysems are udersood i he Filippov sese [9]. Remark 3.: I (3d), ^x i () is used o esimae x di (), like ^x fi () i [4]. As will be show i Lemma 3., usig (3d), ^x i () will coverge o x di () i fiie ime. Wihou loss of geeraliy, le ^x i() xdi () for T. Therefore, whe T, ^x i () i (3a), (3b) ad (3c) ca be replaced wih x di (). Coroller (3a) (3c) is moivaed by coroller (8) () i [8] wih a lile modificaio. I (3b) ad (3c), z i() ad z i () are adoped o esimae, respecively, x i ()x di () ad v i () _x di ().Ifz i () coverges o v i () _x di () i fiie ime, u i () i (3a) ca he drive x i() o xdi () ad v i() o _xdi () i fiie ime. Before movig o, we eed he followig lemmas. Lemma 3.: Usig (3d), k^x i() xdi ()k!, i ; ;,i fiie ime if k 3 > a. Proof: Le ^xi () ^x i () x di (), i ; ;, ^x() Followig a similar proof o ha of Theorem 2 i [4], we ca ge ha k^x i() xdi ()k!, i ; ;, i fiie ime if k 3 > a. Lemma 3.2: [7] Cosider he sysem [ ^x T (); ; ^x T ()] T ad V (2) ^x()(l I m ) ^x(). _x () x 2() k sig [x ()] ; _x 2() k 2sg [x ()] (; x) where x ();x 2 () 2, k, k 2 are cosa posiive scalars ad (; x) is a bouded perurbaio wih x [x () x 2 ()] T. Suppose ha here
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 6, JUNE 22 555 exiss a symmeric posiive-defiie marix P such ha he liear marix iequaliy A T P PA " 2 C T C PBB T P< (4) is saisfied, where A 2k 2, B k, 2 C [], ad " is a posiive cosa scalar. The x () ad x 2() will coverge o zero i fiie ime for all bouded perurbaios saisfyig j(; x)j ". Lemma 3.3: [8] Cosider he sysem x() f (; x) K(; x)sgf _x()sig[x()] 2 g, where x() 2, jf (; x)j D, K m K(; x) K M, ad,, D, K m ad K M are cosa posiive scalars. The, x() ad _x() will coverge o zero i fiie ime if > K m(d ( 2 2)). Theorem 3.: Uder Assumpio 2.3, usig (3) for (2), if y()2() kx i () y()k! ad if y()27() kv i () y()k! i fiie ime if > b ( 2 2), k 3 > a ad here exis k >, k 2 > ad a symmeric posiive-defiie marix P such ha (4) is saisfied, where b. I paricular, kx i () x di ()k! ad kv i() _x di ()k!, i ; ;, i fiie ime. Proof: Noe from Lemma 3. ha here exiss a T > such ha ^x i() x di (), i ; ;, for all T. We ex show ha x i(), v i(), ^x i(), z i() ad z i(), i ; ;, will o diverge o ifiiy for all 2 [;T ]. Because from (3a) ku i ()k, i is easy o see ha x i() ad v i() are bouded for all 2 [;T ]. Because from (3d) k _^x i ()k k 3, i follows ha ^x i () is bouded for all 2 [;T ], which implies ha x i () ^x i () is bouded for all 2 [;T ]. Because from (3c) k _z i()k k 2, i follows ha z i () is bouded for all 2 [;T ]. From (3b) we have ha _z i () z i () [v i () _^x i ()] k sig[z i ()] 2. Because z i (), v i () ad _^x i() are bouded, we assume ha kz i() [v i() _^x i()]k < for all 2 [;T ]. Suppose ha jz i(l) ( )j > ( 2 k 2 ) a a cerai 2 [;T ].Ifz i(l) ( ) > ( 2 k 2 ), he i follows ha: _z i(l) ( )z i(l) ( ) v i(l) ( ) _^x i(l) ( ). k z i(l) ( ) < k z i(l) ( ) < : If z i(l) ( ) < ( 2 k 2 ), he i follows ha: _z i(l) ( )z i(l) () v i(l) ( ) _^x i(l) ( ) k z i(l) ( ) > k z i(l) ( ) > :. Therefore, because z i(l) ()j is bouded, z i(l) () will o diverge o ifiiy for all 2 [;T ], which implies ha z i(l) () will o diverge o ifiiy for all 2 [;T ]. Thus x di () ca be used o replace ^x i () for T. For T, because ^x i() x di (), i follows from (2) ad (3) ha: _~z i() ~z i() k sig [~z i()] ; _~z i() k 2sg [~z i()] x di () where ~z i() z i() ~x i(), ~z i() z i() ~x _ i(), ~x i() x i () x di (). If here exiss a symmeric posiive defiie marix P such ha (4) is saisfied, where b, i follows from Lemma 3.2 ha here exiss T 2 > T such ha ~z i() ad ~z i() for all T 2, which implies ha z i () ~x i () ad z i () ~x _ i () for all T 2. I follows from a similar saeme o he above ha x i (), v i(), z i(), z i() are all bouded for all 2 [T ;T 2]. Thus ~x _ i() ca be used o replace z i () for T 2. For T 2, because z i () ~x _ i (), he closed-loop sysem of (2) usig (3a) becomes ~x i() sgf ~x _ i()sig[~x i()] 2 gx di (). Because > b ( 2 2), i follows from Lemma 3.3 ha here exiss T 3 >T 2 such ha ~x i () ad ~x _ i () for all T 3, which implies ha kx i () x di ()k ad kv i () _x di ()k will coverge o zero i fiie ime. I follows from Lemma 2.2 ha if y()2() kx i() y()k! ad if y()27() kv i () y()k! i fiie ime. Nex we show how o choose he gais k ad k 2 i (3) such ha here exiss a symmeric posiive-defiie marix P such ha (4) is saisfied, where b. Lemma 3.4: Give a cosa ">, here exiss a symmeric-posiive defiie marix P such ha (4) is saisfied if k 2 >"ad k 2 2 k2 2 " 2 <k <k 2 2 k2 2 " 2. Proof: Le P z 2. I is easy o see ha P is symmeric 2 posiive defiie if ad oly if z>4. For a give cosa ">, we have ha A T P PA " 2 C T C PBB T P k z 4k 2 4" 2 k k 2 2 z 2 k k 2 2 z 2 : Suppose ha k 2 > " ad k 2 2 k 2 2 " 2 < k < k 2 2 k 2 2 " 2. I is easy o check ha here exiss z > 4 such ha k z 4k 2 4" 2 (k k 2 (2)z 2) 2 <, which implies ha A T P PA " 2 C T C PBB T P<. B. Coaime Corol Usig Relaive Posiio Measuremes I his secio, we assume ha he relaive posiio measuremes of he vehicles are available bu he velociy ad acceleraio measuremes are o available. We propose he followig algorihm: u i () D i ()sg k _^v i () D i ()sg k i ; ; a ij [x i ()x j ()] a ij [x i ()x j ()] k 2^v i () (5a) a ij [x i ()x j ()] a ij [x i()x j()] k 2^v i(); (5b) where ^v i () for i ; ;, D i () diag[d i (); ;d im ()] wih d il () a ij x i(l) () x j(l) () a ij x i(l) ( ) x j(l) ( ) d (6) for l ; ;m, ad k ad k 2 are cosa posiive scalars. Remark 3.2: Because he velociies of he followers are o available, we use ^v i() o esimae v i() for i ; ;. Sice oly relaive posiio measuremes are available, i is difficul o esimae v i () accuraely. Therefore, we le _^v i () _v i () for i ; ;, so ha v i() ^v i() v i() ^v i() for i ; ;. I he followig aalysis we will show ha his is eough o guaraee ha he followers move io he covex hull formed by he leaders. Defie i () a ij [x i () x j ()], i () j a ij [k 2 v j () f j ()] ad v i () i ; ;. Also defie 9() ^v i () v i (), [ T (); ; T ()] T,
556 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 6, JUNE 22 8() [ T (); ; T ()] T ad v() [v T (); ; v T ()] T. Because kv i ()k, kf i ()k ad k f _ i ()k, i ; ; s, are all bouded, i is easy o see ha 8() ad 8() _ are also bouded. Lemma 3.5: Uder Assumpio 2.3, cosider he fucio V () V 2 9( ) 9( _ T ) 2 k 3 sg [9( )] L I m 8( )k 2v() d Fig.. Nework opology associaed wih vehicles o 7. Here i deoes vehicle i, i ; ; 7. where k 3 is a cosa posiive scalar ad V 2 k 3 9 T ()sg[9()] 9 T ()(L I m )8()k 2 9 T ()v(). V () if k 3 saisfies k 3 > max L I m 8() _ 8() ; L I m 8() k 2 kv()k : (7) Proof: See he Appedix. Theorem 3.2: Uder Assumpio 2.3, usig (5) for (2), if y()2() kx i () y()k! ad if y()27() kv i () y()k! as!if k > ad k 2 >. I paricular, kx i() x di ()k! ad kv i () _x di ()k!, i ; ;,as!. Proof: From (2) ad (5) we kow ha _v i () _^v i () _v i (), i ; ;. I follows ha v i() v i(), i ; ;. Equaio (5a) ca be rewrie as u i () D i ()sg[ i ()] k i () k 2^v i (), i ; ;. I follows ha: i() a ij [x i() x j()] a ij fd i()sg [ i()] k i()k 2^v i()g j a ij fd j()sg [ j ()] k j()k 2^v j()g a ijf j() a ij fd i ()sg [ i ()] k i ()g k 2 a ij [v i () v i ()] k 2 a ij fd j ()sg [ j ()] k j ()g a ij v j () k 2 a ij v j () i (): (8) Noe ha (8) ca be rewrie i a vecor form as 9() (L I m )D()sg [9()] k (L I m )9() k 29() _ k2 (L I m )v() 8() (9) where D() is a block diagoal marix of all D i (), i ; ;. Cosider he followig Lyapuov fucio cadidae: V () 2 9() _ 9() T L I m 9() _ 9() V () 2 9T () k I m(k 2) L I m 9() 2 [D()mk3 m] T [D() mk 3 m] Fig. 2. Trajecories of vehicles o 7 usig (3). The circles deoe he leaders while he squares deoe he followers. where k 3 is a cosa saisfyig (7). Uder Assumpio 2.3, i follows from Lemma 2. ha L is symmeric posiive defiie, which meas ha L is also symmeric posiive defiie. Because k 3 saisfies (7), i follows from Lemma 3.5 ha V (). Because k > ad k 2 >, we have ha k I m (k 2 )(L I m ) is symmeric posiive defiie. Therefore, V () is symmeric posiive defiie wih respec o 9(), 9() _ ad D()m k 3 m. For i ; ;, l ; ;m, from (6) we have ha _ d il () I follows ha: a ij v i(l) () v j(l) () 2 sg 2 sg a ij x i(l) () x j(l) () a ij x i(l) () x j(l) () a ij x i(l) () x j(l) () : [D() m k 3 m ] T m i l m [d il () k 3 ] _D() m [d il () k 3 ] _ d il () i l 2 a ij x i(l) () x j(l) () 2 sg a ij v i(l) () v j(l) () a ij x i(l) () x j(l) () 9() 9() _ T [D() k 3 I m ] sg [9()] :
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 6, JUNE 22 557 Fig. 3. Trajecories of x () x () ad v () _x () usig (3), i ; 2; 3. (a) x () x (); v () _x (). Takig he derivaive of V (), we have ha _V () 9() _ 9() T L I m 2 (L I m)d()sg [9()] k(l I m)9() (k 2 ) _ 9() k 2 (L I m )v() 8() 9() 9() _ T 2 k 3 sg [9()] L I m 8() k 2 v() 9 T () k I m (k 2 ) L I m 9() _ 9() 9() _ T [D() k 3 I m ] sg [9()] k 9() T 9() (k 2 ) 9() _ T 2 L I m _ 9(): () Because k > ad k 2 >, we have ha _ V () is egaive semidefiie. I follows ha V () is bouded, which implies ha 9(), _ 9() ad D() are all bouded. Because v() ad 8() are also bouded, i follows from (9) ha 9() is bouded. From () we have ha V () 2k 9() T _ 9() 2(k2 ) _ 9() T L I m 9(): Therefore, V () is bouded. By Barbala s Lemma we have ha V _ ()! as!, which implies ha 9()! ad _9()! as!. Le x F () [x T (); ;x T ()] T, ad v F () [v T (); ;v T ()] T, we have ha (L I m )x F ()(L 2 I m)xl()! ad (LI m)vf ()(L2I m)vl()! as!. I follows ha kx F ()x d ()k! ad kv F () _x d ()k! as!. I follows from Lemma 2.2 ha if y()2() kx i() y()k! ad if y()27() kv i() y()k! as!. IV. NUMERICAL SIMULATIONS This secio gives simulaio resuls o illusrae he heoreical resuls i Secio III. Cosider a group of hree followers ad four leaders i he 2-D space. We assume ha x 4 () [; si(2)] T, x 5 () [:8 :5; si(2)] T, x 6() [:8 :5; si(3) :5] T ad x 7 () [; si(3) :5] T. The ework opology associaed wih Fig. 4. Trajecories of vehicles o 7 usig (5). The circles deoe he leaders while he squares deoe he followers. he seve ages is show by Fig.. We le a ij if (j; i) 2 E ad a ij oherwise. The iiial posiios ad velociies of he followers are chose as x () [3; 2] T, v () [:6; :3] T, x 2 () [6; 2] T, v 2 () [:8; :] T, x 3 () [4; 4] T ad v 3 () [:; :] T. For he algorihm (3), we choose 4,, k 4, k 2 2ad k 3 3. I ca be see ha he codiios i Theorem 3. are saisfied. Fig. 2 shows he rajecories of vehicles o 7 usig (3). I ca be see ha vehicles o 3 move io he covex hull spaed by vehicles 4 o 7. Fig. 3 shows he differece bewee x i () ad x di () ad he differece bewee v i () ad _x di (), i ; 2; 3. For he algorihm (5), we choose k 5ad k 2 3. Fig. 4 shows he rajecories of vehicles o 7 usig (5). I ca be see ha vehicles o 3 move io he covex hull spaed by vehicles 4 o 7. Fig. 5 shows he differeces bewee x i() ad x di () ad he differece bewee v i() ad _xdi (), i ; 2; 3. V. CONCLUSION I his oe, he coaime corol problem has bee ivesigaed for muliple auoomous vehicles wih double-iegraor dyamics i he presece of muliple dyamic leaders. Two disribued coaime corol algorihms have bee derived uder differe cosrais. Differe from he relaed resuls i he lieraure, he proposed algorihms use oly he posiio measuremes of he leaders ad he followers. Therefore, hey ca be realized more easily.
558 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 6, JUNE 22 Fig. 5. Trajecories of x () x () ad v () _x () usig (5), i ; 2; 3. APPENDIX Proof of Lemma 3.5: Because k 3 > k(l I m )8()k k 2kv()k, we have ha _9 T ( ) k 3 sg [9( )] L I m 8( )k2v() d L I m 8( )k 2 v() d9( ) k 3 _ 9 T ( )sg [9( )] d 9 T 3 ( ) k sg [9( )] L I m 8( )k 2 v() j 9( )d L I m 8( )k2v() 9 T 3 () k sg [9()] L I m 8() k2v() V 2 V 2 9 T ( )(L I m ) _ 8( )d 9 T ( ) L I m _ 8( )d: () Because k 3 > k(l I m )[8() _ 8()]k k 2 kv()k, i he follows ha: V () V 2 V 2 9 T ( ) k 3 sg [9( )] L I m 8( ) k 2v()g d 9 T ( ) L I m _ 8( )d 9 T ( ) k 3 sg [9( )] L I m 8( ) _ 8( ) k 2v() d : (2) REFERENCES [] W. Re ad R. W. Beard, Disribued Cosesus Muli-vehicle Cooperaive Corol. Lodo, U.K.: Spriger-Verlag, 28. [2] M. Cao, A. S. Morse, ad B. D. O. Aderso, Agreeig asychroously, IEEE Tras. Auom. Corol, vol. 53, o. 8, pp. 826 838, Sep. 28. [3] F. Xiao, L. Wag, J. Che, ad Y. Gao, Fiie-ime formaio corol for muli-age sysems, Auomaica, vol. 45, o., pp. 265 26, 29. [4] Y. Hog, G. Che, ad L. Bushell, Disribued observers desig for leader-followig corol of muli-age eworks, Auomaica, vol. 44, o. 3, pp. 846 85, 28. [5] W. Re, Muli-vehicle cosesus wih a ime-varyig referece sae, Sys. Corol Le., vol. 56, o. 7 8, pp. 474 483, 27. [6] H. Su, X. Wag, ad Z. Li, Flockig of muli-ages wih a virual leader, IEEE Tras. Auom. Corol, vol. 54, o. 2, pp. 293 37, Feb. 29. [7] Y. Cao, W. Re, ad Z. Meg, Deceralized fiie-ime slidig mode esimaors ad heir applicaios deceralized fiie-ime formaio rackig, Sys. Corol Le., vol. 59, o. 9, pp. 522 529, 2. [8] M. Ji, G. Ferrari-Trecae, M. Egersed, ad A. Buffa, Coaime corol mobile eworks, IEEE Tras. Auom. Corol, vol. 53, o. 8, pp. 972 975, Sep. 28. [9] G. Ferrari-Trecae, A. Buffa, ad M. Gai, Aalysis of coordiaio muli-age sysems hrough parial differeial equaios, IEEE Tras. Auom. Corol, vol. 5, o. 6, pp. 58 63, Ju. 26. [] Y. Cao ad W. Re, Coaime corol wih muliple saioary or dyamic leaders uder a direced ieracio graph, i Proc. IEEE Cof. Decisio Corol, Shaghai, Chia, Dec. 29, pp. 34 39. [] G. Shi ad Y. Hog, Se rackig of muli-age sysems wih variable opologies guided by movig muliple leaders, i Proc. IEEE Cof. Decisio Corol, Alaa, GA, Dec. 2, pp. 2245 225. [2] Y. Lou ad Y. Hog, Muli-leader se coordiaio of muli-age sysems wih radom swichig opologies, i Proc. IEEE Cof. Decisio Corol, Alaa, GA, Dec. 2, pp. 382 3825. [3] Y. Cao, D. Suar, W. Re, ad Z. Meg, Disribued coaime corol for muliple auoomous vehicles wih double-iegraor dyamics: Algorihms ad experimes, IEEE Tras. Corol Sys. Techol., vol. 9, o. 4, pp. 929 938, Jul. 2. [4] Z. Meg, W. Re, ad Z. You, Disribued fiie-ime aiude coaime corol for muliple rigid bodies, Auomaica, vol. 46, o. 2, pp. 292 299, 2. [5] J. Li, W. Re, ad S. Xu, Disribued coordiaed rackig wih muliple dyamic leaders for double-iegraor ages usig oly posiio measuremes, i Proc. Amer. Corol Cof., Sa Fracisco, CA, Jul. 2, pp. 292 297. [6] Y. Su, P. C. Mller, ad C. Zheg, A simple oliear observer for a class of ucerai mechaical sysems, IEEE Tras. Auom. Corol, vol. 52, o. 7, pp. 34 345, Jul. 27.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 6, JUNE 22 559 [7] A. Dávila, J. A. Moreo, ad L. Fridma, Opimal Lyapuov fuio selecio for reachig ime esimaio of super wisig algorihm, i Proc. IEEE Cof. Decisio Corol, Shaghai, Chia, Dec. 29, pp. 845 84. [8] A. Leva, Priciples of 2-slidig mode desig, Auomaica, vol. 43, o. 4, pp. 576 586, 27. [9] A. F. Filippov, Differeial Equaios Wih Discoiuous Righhad Sides. Norwell, MA: Kluwer, 988. Exeded Coroller Syhesis for Coiuous Descripor Sysems Yu Feg, Mohamed Yagoubi, ad Philippe Chevrel, Member, IEEE Absrac This echical oe preses a complee soluio o he osadard H oupu feedback corol problem for coiuous descripor sysems where usable ad oproper weighig fucios are used. I such a problem, he desired coroller has o saisfy wo codiios simulaeously: (i) he closed-loop is admissible ad has a miimum H orm, (ii) oly he ieral sabiliy of a par of he closed-loop is sough. The codiio of he exisece of such a coroller is deduced. A explici characerizaio of he opimal soluio is also formulaed, based o wo geeralized algebraic Riccai equaios (GAREs) ad wo geeralized Sylveser equaios. A umerical example is icluded o illusrae he validiy of he proposed resuls. Idex Terms Comprehesive admissibiliy, descripor sysems, H orm, usable ad oproper weighs. I. INTRODUCTION Descripor (Sigular, Implici) Sysems Have Bee Aracig The Aeio Of May Researchers Over Rece Decades Due To Their Capaciy To Preserve The Srucure Of Physical Sysems Ad To Describe Saic Cosrais Ad Impulsive Behaviors. A Number Of Corol Issues Have Bee Successfully Exeded To Descripor Sysems Ad The Relaed Resuls Have Bee Repored, For Isace I [] [3] Ad The Refereces Therei. The sadard H 2 oupu feedback corol problem for descripor sysems was ivesigaed i [4], ad he opimal coroller was characerized based o wo GAREs. Laer, he auhors proposed a explici formulaio of all opimal corollers for he full iformaio ad he sae-feedback cases i [5]. They showed ha, i coras wih he sae-space case, he usual gai marix defied as a affie fucio of he GARE soluio ca be o-opimal. I boh papers, sufficie codiios abou he solvabiliy of GAREs were give. However, o he bes of he auhors kowledge, soluios o he osadard H 2 oupu feedback corol problem for descripor sysems, where usable ad oproper weighs are cosidered i he overall feedback model, have o ye bee sudied i he lieraure. I fac, he H 2 corol problem requires he defiiio of a sadard model, which is ecessarily based o he physical model of he sysem, he models of dis- Mauscrip received July 9, 2; revised Jauary, 2 ad Jue 7, 2; acceped Ocober 7, 2. Dae of publicaio November 4, 2; dae of curre versio May 23, 22. Recommeded by Associae Edior T. Zhou. The auhors are wih he Isiu de Recherche e Commuicaios e Cyberéique de Naes (IRCCyN) UMR CNRS 6597, Naes B.P. 93, Frace ad also wih he Ecole des Mies de Naes (EMN), Naes 4437, Frace (e-mail: mohamed.yagoubi@mies-aes.fr). Digial Objec Ideifier.9/TAC.2.274689 urbaces ad referece sigals ogeher wih he corol objecives. I his coex (as for may corol problems), i is ofe desirable o ake usable, eve oproper, weighig filers o mee he desig specificaios [6], [7]. These choices geerally resul i a osadard desig problem for plas havig usabilizable (udeecable) fiie dyamics, or eve ucorollable (uobservable) impulsive elemes due o he weighs ivolved. These udesirable elemes ca of course be reaed, for example, by sligh perurbaio o reder he problem sadard [8]. This approach is, however, vulerable o he roubles relaed o lighly-damped poles ad may lead o higher order ad o sricly proper corollers. Moreover, he mehodology of filer absorpio [7], [9] ad he heory of quasi-sabilizig soluios of Riccai equaios [], [] have also bee proposed for solvig hese osadard problems. I addiio, he auhors have equally reaed his problem for descripor sysems wih he presece of usable weighs via sae feedback i [2]. Moreover, i is worh oig ha he well-kow regulaio problems, see [3] [5] ad he refereces herei, ca also be hadled by he use of usable weighig filers. The mai coribuio of his echical oe is a ivesigaio of he exeded H 2 oupu feedback corol problem for coiuous descripor sysems. Sysems ad heir weighs are all described wihi he descripor framework. Hece, i is possible o ake io accou o oly usable weighs, bu oproper weighs as well. This case resuls i osadard H 2 corol problems for which he sadard soluio procedures fail. I he curre echical oe, he exisece of a soluio o his exeded problem (he exeded erm idicaes here ha he desirable coroller ca ad mus sabilize a par of he geeralized closed-loop) is characerized i erms of wo GAREs ogeher wih wo geeralized Sylveser equaios. This echical oe is orgaized as follows. Secio II recalls some basic oaios of descripor sysems ad formulaes he exeded H 2 corol problem. The, based o wo geeralized Sylveser equaios, quasi-admissible soluios o he GAREs are deduced i Secio III. Secio IV characerizes explicily he opimal H 2 oupu feedback corollers. Fially, a umerical example is give i Secio V o illusrae he proposed resuls. Noaio: The superscrips > ad 3 represe he raspose ad complex cojugae raspose, respecively. The oaios F l (; ) ad sad for he lower liear fracioal rasformaio ad Kroecker produc, respecively. RH deoes he se of all proper raioal sable rasfer marices. Moreover, he colum vecor Col(P ) deoes a ordered sack of he colums of he marix P from lef o righ sarig wih he firs colum. II. PROBLEM FORMULATION A. Prelimiaries Cosider he followig coiuous descripor sysem: E _x() Ax() Bu(); y() Cx() Du() where x 2, y 2 p ad u 2 m are he descripor variable, measureme ad corol ipu vecor, respecively. The marix E 2 2 may be sigular, i.e. rak(e) r. The descripor sysem () is said o be regular if de(se A) is o ideically ull. If he descripor sysem is regular, he i has a uique soluio for ay iiial codiio ad ay coiuous ipu fucio [6], [7]. I is said o be impulse-free if deg(de(se A)) rak(e). I is said o be sable if all he roos of de(se A) have egaive real pars. If he descripor sysem is regular, impulse-free ad sable, he i is admissible. I addiio, he descripor sysem () () 8-9286/$26. 2 IEEE