Journal of Xiamen University (Natural Science)

Transcription

1 () Joual of Xiame Uivesiy (Naual Sciece) Vol. 48 No. 4 J ul. 2009, 3 (, 36005) :,,.,,,.,.,. : ;;; : TP 393 :A : (2009) ,( dyamic age s). ( muliage sysems),, [ ], [2 ], [3 ].,,,.., [4-5 ].,. [ 6-0 ]..,,,,,. [0 - ],..., 3 : :( A050002) edu. c. A T A. A : 0 ( A ; 0) A ( ). I gi 0 0,I = {,2,, }. G = ( v,, A), v = { v,, v},α v v,a = [ aij ], aij 0. I. eij = ( vi, v j ).,eij,aij > 0. i, j I,aii = 0, aij = 0. G,aij = aji, A. G (Laplacia maix) L = [ lij ],i j, lij = - aij, lii = a ij. G i j, gi T L = 0 [8 ]. G,.,0 = < 2 [8 = ] max.,,, [0 ].,.,. ( i l) ( ) = ui, i I () ( i l) ( ) R i ( ) l, (0) i ( )

2 494 () 2009 = i ( ), < 0, i ( ) = 0, ui, l [ ]., [ ] ui = - gij k ij [ l- k = 0 k ( ( i k) ( ) - ( j k) ( ) ) ] (2) kij > 0, 0 = k > 0,,, l - (posiive gai). j i, gi, gij = 0, gii = 0., Πi, j I, aij = gij k ij. l = l = 2,() (2). [ 8 ]., [0 - ]., ui = - gij k ij [ l - k = 0 k ( ( i k) ( - ij ( ) ) - ( j k) ( - ij ( ) ) ) ] (3) l = [8-9 ]., l = 2. l = 2, (), (3) i ( ) = - aij { 0 [ i ( - ij ( ) ) - j ( - ij ( ) ) ] + [g i ( - ij ( ) ) - g j ( - ij ( ) ) ]}, i I (4), aij. 0 =, = (4), ( ) = - L k( - k ( ) ) - L kg( - k ( ) ) (5) ( ) = [ ( ), 2 ( ),, ( ) ] T,( - k ( ) ) = [ ( - k ( ) ) ], 2 ( - k ( ) ) ],, ( - k ( ) ) ] T, g( - k ( ) ) = [ g ( - k ( ) ), g 2 ( - k ( ) ),, g ( - k ( ) ) ] T, ( - ) / 2,, 2,,, k ( ) { ij ( ) :i,,, },L k = [ lkij ] - aij, j i, k ( ) = ij ( ) lki j = 0, j i, k ( ) ij ( ) (6) lkij, j = i,l k gi T L K = 0 T [9 ]. (5) gi :gi T ( ) = 0.,, gi T g( ) =, R,= Ave (g( ) ) = Ave (g(0) ). g( ) [8 ] g i ( ) =+ g i ( ), i I (7) g i ( ) R, g i = i ( ) = 0. (7) g 0 i (s) ds = 0 (+ g i (s) ) ds, i I (8) (8) i ( ) = + i ( ) - i (0) + i (0), i I (9) c( ) =, i ( ) = i ( ) - i (0) + i (0), i I, gc( ) =,g i ( ) = g i ( ), i I. c( ), gc ( ), i ( ) g i ( ) (7) (9) g i ( ) = gc( ) + g i ( ) (0) i ( ) = c( ) + i ( ), i I () (0), (0) () ij ( ) i ( ) = i ( ) (2) g i ( - ij ( ) ) = gc( - ij ( ) ) + g i ( - ij ( ) ) (3) i ( - ij ( ) ) = c( - ij ( ) ) + i ( - ij ( ) ), i I (4) i ( ) = - aij { 0 [ i ( - ij ( ) ) - c( - ij ( ) ) + c( - ij ( ) ) - j ( - ij ( ) ) ] + (4) [g i ( - ij ( ) ) - gc( - ij ( ) ) + gc( - ij ( ) ) - g j ( - ij ( ) ) ]}, i I (5) (2), (3), (4) (5) i ( ) = - aij { 0 [ i ( - ij ( ) ) - j ( - ij ( ) ) ] + [g i ( - ij ( ) ) - g j ( - ij ( ) ) ]}, i I (6) g i ( ) = g i ( ), i I, (6) gx ( ) = A 0 x ( ) g L k x ( - k ( ) ) (7) x ( ) = [ ( ),, ( ), + ( ),, 2 ( ) ] T, gx ( ) x ( ), g Koecke, A 0 = 0 I 0 0, x ( - k ( ) ) = [ ( - k ( ) ),, ( - k ( ) ), + ( - k ( ) ),, 2 ( - k ( ) ) ] T. (7), (4)., : 0,,, 0 k ( ) hk (8),(4) gx ( ) = A 0 x ( ) g L ks x ( - k ( ) ),

3 4 : 495 s = ( ) l0 (9) L ks, L ks = L s. L s Gs = ( v, s, A s).. l0 Z. ( ) : R l0,.,. [9 ] ( diffeeia2 ble) x ( ) R W = W T : 0, : h - k [ x ( ) - x ( - k ( ) ) ] T W [ x ( ) - x ( - k ( ) ) ] -( k ) k ( ) (8). gx T ( s) W gx (s) ds, > 0 (20) 2 [6 ], ( ) gi p T ( 0) + gi p T g( 0) g( ) gi p T g (0), p - L p T gi =, (0) g (0) (4), : G, hk > 0, Qk = Q T k : 0,,, 2 3 ( + ) = ( + ) ( + ) ω ( + ) ( + ) = A 0 T + A 0 + ( j + ) = - A j - A 0 T ( ; 0 (2) ( hk A 0 T Qk A 0 - h - k Q k), A j = 0 0 gl j, j, ( i + ) ( j + ) = A i T ( ( i + ) ( i + ) = A i T ( hkq k) A j + h - j Q j, hkq k) A j, i < j, hkq k) A i - h - i Q i, i, (4), ( ) gi p T (0) + gip T g(0),g ( ) gi p T g(0). Lyap uov V ( ) = x T ( ) x ( ) + - h k (s - + hk) gx T (s) Qk gx ( s) ds (22) (7), (22), gv ( ) = gx T ( ) x ( ) + x T ( ) gx ( ) + h kgx T ( ) Q kgx ( ) - x T ( ) [ A T 0 h - + A 0 + A 0 T ( hkq k) A 0 + k Q k ] x ( ) - i = A 0 T ( hkq k) A i ] x ( - i ( ) ) - x T ( - i ( ) ) [ A i T + h - k Q i + i = A T i A T i ( hkq k) A 0 ] x ( ) + hj Q j h - k y T ( )y ( ). y ( ) = [ x T ( ) - h k gx T ( s) Q kgx ( s) ds x T ( ) [ ( A i + h - k Q i) + x T ( - i ( ) ) i = A k x ( - k ( ) ) + x T ( - k ( ) ) Qk x ( - k ( ) ) = x T ( - ( ) ) x T ( - 2 ( ) ) x T ( - ( ) ) ] T,(2). (2), gv ( ) < 0, (4),2,. hk,(2). Malab.,, (2).. (2),,.,,. 0, 0 i, j I ij ( ) = 0,(7) gx ( ) = A 0 x ( ) + B x ( - 0 ) (23) B = gl, x ( ), A 0, L 2 G, ij ( ), i, j I

4 496 () ij ( ) < max = mi acaw0, (24) w0 w0 = , G 2 L, (4), ( ) gi p T ( 0) + gi p T g (0),g ( ) gi p T g(0). (23),(23). (23) x ( s) = (s I2 - A 0 - e - 0 s B ) - x (0) (25) H (s) = s I2 - A 0 - e - 0 s B, H (s) s = 0 (ope L H P). de ( H ( s) ) = de (s I2 - A 0 - e - 0 s B ) = de s I e - - I e - 0 s L 0 s L s I + de (s 2 I + s e - 0 s L + e - 0 s L ), Z( s) = s 2 I + s e - 0 s L + e - 0 s L, H (s) Z( s). L G, L 0 = < 2 = max ( L ). i, i =,, L i, i Z ( s) = 0 s 2 + s e - 0 s i + e - 0 s i = 0, i =,, (26) i =,s = 0. i = 2,,, (26) + s+ i s 2 (s) = s+ s 2 e - 0 s = 0 (27) = e - 0 s,, (23) i. s = j w ( s ) ( j w ) = + jw - w 2 e - jw 0, A ( w) = + 2 w 2 w 2, ( w) = - + acaw - 0 w (28) 0 >,( j w), ( s), ( j w).. (23) i 0 =,( j w), (23). 0 <,( j w). ( j w) ( s) w x ( j w), ( w x ) = - + acaw x - 0 w x = - (2 k + ), k = 0,, (29) A ( w x ) = + 2 w 2 x w 2 x < (30) (30) - w 4 x w 2 x + 2 < 0 w x > w0 = (29) 0 = k+ acaw x w x (32) w x - [2 kacaw x - d 0 = d w x w 2 x w x + 2 w 2 ] x (3) (32), k = 0,, (33) k = 0,, w x > 0, d 0 < 0. d w x 0 = acaw0.. w0 2 2,. (24),,. 2. 2,. { Gs s = ( ) l0 }, : 3 (8) s l0 Gs, hk > 0, Qk = Q T k : 0,,2,,

5 4 : 497 s = s 2s 3s ( + ) s 3 22s 23s 3 ( + ) s s ω 3 ( + ) s ( + ) ( + ) s s = A 0 T + A 0 + ( j + ) s = - A js - A 0 T ( ; 0, s l0 ( hka 0 T Qk A 0 - h - k Q k), j, A js = 0 0 gl js, ( i + ) ( j + ) s = A is T ( ( i + ) ( i + ) s = A is T ( hkq k) A js + h - j Q j, hkq k) A js, i < j, hkq k) A is - h - i (34) Q i, i, (4), ( ) gi p T (0) + gi p T g(0),g ( ) gi P T g(0). (9). (22) (9) L yap uov, (22),gV s ( ) = y T ( ) s y ( ), s l0. (34),gV s ( ) < 0, s l0. (4),2,. 3 6,6 6. ( 0) = ( ) T,g (0) = ( ) T =. 2, max = ,3 ij ( ) = s ij ( ) = s,i, j I,. 2, 3, (24) max,,. max,. Fig. A weighed udieced gaph 4..,,.,,. 2 ( ij ( ) = s) (a) ; (b) Fig. 2 Tajecoies of he ages ( ij ( ) = s)

6 498 () ( ij ( ) = s) (a) ; (b) Fig. 3 The ajecoies of he ages ( ij ( ) = s) : [ ] Olfai 2 Sabe R, Muay R M. Disibued coopeaive cool of muliple vehicle fomaios usig sucual po2 eial f ucios [ C ]/ / Poceedigs of he 5h IFAC Wold Cogess. Baceloa : Spai,2002 : - 6. [2 ] ToeJ, Tu Y. Flocks, heds, ad schools : a quaiaive heoy of flockig [J ]. Physical Review E,998,59 (4) : [3 ] Pagaii F,Doyle J,Low S. Scalable laws fo sable e2 wok cogesio cool[j ]. Poceedigs of he 40h IEEE Cofeece o Decisio ad Cool,200, : [4 ] Lych N A. Disibued algoihms [ M ]. Moga Kauf2 ma :Moga Kauf ma Publishe,997. [ 5 ] De Goo M H. Reachig a cosesus[j ]. Ameica Sais2 ical Associaio,974,69 (345) :8-2. [6 ] Kashyap A,Basa T,Sika R. Quaized cosesus[j ]. Auomaica,2007,43 (7) : [7 ] Re W,Bead R W. Cosesus seekig i muliage sys2 ems ude dyamically chagig ieacio opologies [J ]. IEEE Tasacios o Auomaic Cool, 2005, 50 (5) : [8 ] Olfai2Sabe R,Muay R M. Cosesus poblems i e2 woks of age s wih swichig opology ad ime2delays [J ]. IEEE Tasacios o Auomaic Cool, 2004, 49 (9) : [9 ] Su Y G, Wag L, Xie G M. Aveage cosesus i e2 woks of dyamic age s wih swichig opologies ad muliple ime2vayig delays[j ]. Sysems & Cool Le2 es,2007,9 (5) : [0 ] Re W, Akis E M. Secod2ode cosesus poocols i muliple vehicle sysems wih local ieacios[ C]/ / Poc A IAA Guidace,Navigaio ad Cool Cof. Sa Facisco,CA,USA,2005 : - 3. [ ] Re W. Secod2ode cosesus algoihm wih exe2 sios o swichig opologies ad efeece models[ C]/ / Ameica Cool Cofeece : Cosesus Algoihm of he Secod2ode Ne wok wih Muliple Time2vayig Delays L Β Sheg2yag,SUN Hog2fei 3 ( School of Ifomaio Sciece ad Techology,Xiame Uivesiy,Xiame 36005,Chia) Absac : This pape discusses he secod2ode cosesus algoihm fo he ewok wih muliple ime2vayig delays. The sudied poocol is affeced o oly by he ime delays bu also by he swichig opologies of a ewok. The sudy shows ha he ewok ca achieve cosesus if ceai codiios ae saisfied. The codiios ae ifeed by ioducig a special L yapuove f ucio. Fu2 hemoe,we aai a commo uppe boud ime2delays by discussig he poblem i f equecy domai. This implies ha he ewok ca achieve cosesus,whe he ime delays geeaed by age commuicaio ae lowe ha he uppe boud ime2delay. Moeove, he commo uppe boud ime2delay is depede o he paamees of he ewok which ca educe i s cosevaism. Fially,we al2 so discuss he case of he swichig opologies. I his case,he ewok ca also achieve cosesus. Key wods : ime2delay ; liea maix iequaliy ; muliage sysem ; swichig opology