Synchronization in Biology, Chemistry, and Physics

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1 Prepared by F.L. Lews Updated: Saturday, February 09, 2013 Synhronzaton n Bology, Chemstry, and Physs Colletve Moton n Anmals, Fsh, and Insets Add dsusson. Refer to refs. 1.1 Colletve Moton n Anmal Groups Equaton Chapter 1 Seton 1 The olletve motons of anmal soal groups are among the most beautful sghts n nature. Eah ndvdual has ts own nlnatons and motons, yet the aggregate moton makes the group appear to be a sngle entty wth ts own laws of moton, psyhology, and responses to external events. Floks of brds, herds of anmals, and shools of fsh are aggregate enttes that take on an exstene of ther own due to the olletve moton nstnts of ther ndvdual members. The aggregate turns and wheels to pursue ts objetves suh as seekng food or mgratng, and to avod predators and obstales. Suh synhronzed and responsve moton makes one thnk of horeographed motons n a dane, yet they are a produt not of planned srpts, but of nstantaneous desons and responses by ndvdual members. In ths seton we study the mehansms for suh synhronzed horeograph behavor of groups n terms of nstantaneous desons made by ther members. The materal n ths seton s from Reynolds []. Fgure. A flok of brds wheels n synhrony. Courtesy pbs.org Fgure. A flok of brds appears to be a sngle omposte entty n ts moton and behavors. Courtesy watt-works.om 1

2 Fgure. A flok of geese on mgraton. Courtesy Freepohoto.om. Fgure. A shool of fsh n synhronzed moton. Courtesy gmagazne.om.au Fgure. A shool of fsh respondng as a sngle organsm. Courtesy aquarumprosmn.om Fgure. A herd of bson on the move n searh of food. Courtesy shelledy.mesa.k12.o.us Fgure. Wldebeest herd on mgraton. Courtesy arkve.org Fgure. Wldebeest stampede to avod danger. Courtesy allposters.om 2

3 Fgure. Zebras blend n a herd and t s dffult for predators to dstngush ndvduals.. Courtesy rownandandrews.om Fgure. Synhronzng bas motons n a herd. Courtesy deswalls.om Colletve motons allow the group to aheve what the ndvdual annot. Benefts of aggregate moton nlude defense from predators, soal and matng advantages, and group foragng for food. In mgratng brd floks, the energy requred by ndvdual brds n flyng s redued by remanng n the wngtp vortex upwash of those ahead. Choreographed motons of groups of lons, wolves, and jakals allow more effent pursut of prey. In olletve moton stuatons, the mportant entty beomes the group, not the ndvdual. Suh behavor has been observed n human rowd pan and mob stuatons, where there s pressure to follow the group s olletve lead, not thnk on one s own [ref]. Reynolds Rules of Moton for Indvduals n Colletve Groups To reprodue the olletve moton of an anmal group n omputer anmaton has been a hallenge. It would be mpossble to srpt the moton of eah ndvdual usng planned motons or trajetores. Analyss of groups based on soal behavors s omplex, yet the ndvduals n olletves appear to follow smple rules that make ther moton effent, responsve, and pratally nstantaneous. The umulatve moton of anmal groups an be programmed n omputer anmaton by endowng eah ndvdual wth the same few rules that allow t to respond to the stuatons t enounters. The responses of the ndvduals aumulate to produe the ombned moton of the group. In large groups, eah ndvdual s aware only of the motons of ts mmedate neghbors. The feld of pereptual awareness of the ndvdual hanges for dfferent types of anmal groups and n dfferent moton senaros. In flokng moton, brds are aware only of a few neghbors ahead of them or besde them. In shools of fsh, shok waves transmtted by the water allow ndvduals to be aware nstantaneously of motons of neghbors that they may not even see. In anmal herds, vbratons of the earth allow ndvduals to be aware of the general moton tendenes of others that may not be lose by. The olletve moton of large groups an be aptured by usng a few smple rules governng the behavor of the ndvduals. Indvdual motons n a group are the result of the balane of two opposng behavors: a desre to stay lose to the group, and a desre to avod ollsons wth other ndvduals [34]. Reynolds has aptured the tendenes governng the 3

4 motons of ndvduals through hs three rules. Reynolds Rules: 1. Collson avodane: avod ollsons wth neghbors 2. Veloty mathng, math speed and dreton of moton wth neghbors 3. Flok enterng: stay lose to neghbors Implementng Reynolds Rules n Dynamal Moton Systems Reynolds rules apture very well the olletve moton of anmal groups and an also be used as ontrol shemes for human systems suh as vehle formatons. Consder a set of N agents { N} and let the ommunaton between the agents be modeled by a general dreted graph. There are many mehansms for mplementng Reynolds rules n dynamal systems ontrol. Of mportane s the defnton of an ndvdual s neghborhood. Agents seek to avod ollsons. Therefore, defne a rle of radus whh eah agent seeks to keep lear of other agents. Defne the ollson neghborhood for agent as N { j: rj } where the dstane between nodes and j s r x x ( p p ( q q ( j j j j Defne the nteraton radus and the nteraton neghborhood by N { j: rj }. It s noted that rad, are dfferent for dfferent anmal groups and dfferent vehles. Moreover, for some groups, suh as floks of brds n mgraton, the ollson and nteraton neghborhoods are not rular. The dynams used to smulate the ndvdual group members an be very smple, yet realst results are obtaned. Consder agent moton n 2-D aordng to the dynams x u ( wth states x [ ] T p q R where ( p( t, q( t s the poston of node n the ( x, y -plane. 2 Ths s a smple pont-mass dynams wth veloty ontrol nputs u [ u u ] T R. p q A sutable law for ollson avodane s gven by u j( xj x (1.1.3 jn whh auses agent to turn away from other agents nsde the ollson neghborhood N. A sutable law for flok enterng s gven by u aj( xj x (1.1.4 jn \ N whh auses agent to turn towards other agents nsde the nteraton neghborhood N and outsde the ollson neghborhood N. These protools an be wrtten n terms of the omponents of veloty as up aj ( pj p (1.1.5 jn \ N 4

5 u a ( q q (1.1.6 q j j jn \ N The relaton between the ollson avodane gans j and the flok enterng gans a j determnes dfferent response behavors nsofar as agents prefer to flok or prefer to avod neghbors. The relaton between the rad, s also nstrumental n desgnng dfferent behavors. Alternatve protools for ollson avodane and flok enterng are gven respetvely by ( x j x ( xj x u j j (1.1.7 r x x jn j jn j ( x x ( x x u a a (1.1.8 j j j j jn \ N r j jn \ N x j x whh are normalzed by dvdng by the dstane between agents. Note that the sum s over omponents of a unt vetor. Therefore, these laws presrbe a desred dreton of moton and result n moton of unform veloty. By ontrast, the laws (1.1.3, (1.1.4 gve velotes that are larger f one s further from one s neghbors. These protools guarantee onsensus behavor n terms of ( x, y poston, yet do not nlude veloty ontrol. Smple motons of a group of N agents n the ( x, y -plane an alternatvely be desrbed by the node dynams p Vos (1.1.9 q Vsn where V s the speed and the headng of agent. Veloty mathng an be aheved by usng the loal votng protool to reah headng onsensus aordng to a ( ( j j jn and a seond loal votng protool to math speeds aordng to V a ( V V ( j j jn As a thrd alternatve, one an use the Newton s law agent dynams x v ( v u n n n wth vetor poston x R, speed v R, and aeleraton nput u R. Ths s more realst than (1.1.2 for moton of physal bodes n n dmensons. For moton n the 2-D plane, one would take x [ ] T p q where ( p( t, q( t s the poston of node n the ( x, y -plane. Consder the dstrbuted poston/veloty feedbak at eah node gven by the seond-order loal neghborhood protools u a ( x x a ( v v a ( x x ( v v (1.13 j j j j j j j jn jn jn 5

6 where 0 s a stffness gan and 0 s a dampng gan. Ths s based on loal votng protools n both poston and veloty so that eah node seeks to math all ts neghbors postons as well as ther velotes. Thus, ths protool realzes Reynolds rules for flok enterng and veloty mathng. A protool suh as (1.1.3 ould be added for ollson avodane. Alternatvely, offsets ould be added for desred relatve separaton as n u a x x a v v ( ( j j j j j jn jn n where j R are the desred offsets of node from node j. Ths law ontans both flok enterng and ollson avodane aspets. See the Seton on Hgher-Order Consensus **. Alternatve methods to loal ooperatve protools for mplementng Reynolds rules nlude potental feld approahes. The methods overed n the Chapter on Potental Feld Moton Control **, nludng approahes for obstale avodane and goal seekng, are ntmately related to the tops n ths seton. Connetvty of Anmal Groups and Vehle Formatons The onnetvty of the dreted graphs just dsussed depends on r j the dstanes between nodes and j. As agents move, these dstanes hange and so the neghbors n the ollson avodane regon N and the flokng regon N hange. Ths s alled a dstane graph topology, and the graph has tme-varyng edges. Nevertheless, the dsusson n the Seton on Dynamally Changng Interaton Topologes ** reveals that the group wll reah onsensus f the edges vary n suh a way that the unon of graphs over eah of an nfnte set of tme ntervals has a spannng tree. However, there s no guarantee that the edges wll not splt so that the graph s no longer strongly onneted or has a spannng tree. For nstane, on reahng an obstale a flok of brds often splts nto two groups, and reforms nto a sngle flok on the other sde. Many papers have been wrtten about defnng potental felds or moton protools that guarantee the group does not lose onnetvty [Le Guo, et.]. 1.2 Leadershp n Anmal Groups on the Move Equaton Seton (Next We have seen that nformaton an be transferred loally between ndvdual neghborng members of anmal groups, yet result n olletve synhronzed motons of the whole group. Loal moton ontrol protools are based on a few smple rules that are followed by all ndvduals. However, n many stuatons, the whole group must move towards some goal, suh as along mgratory routes or towards food soures. In these ases, only a few nformed ndvduals may have pertnent nformaton about the requred dretons of moton. In ths seton we wll study how a small perentage of nformed ndvduals an have a global mpat that results n ontrol of the whole group towards desred goals. The materal n ths seton s from [Couzn 2005]. Some spees have evolved spealzed mehansms for onveyng nformaton about loaton and dreton of goals. One example s the waggle dane of the honeybee that reruts 6

7 hve members to vst food soures. However, n many groups suh as shools of fsh, brd floks, and mammal herds on the move evdene supports the dea that the desons made by all ndvduals follow smple loal protools based on nearest neghbors smlar to Reynolds Rules. Mehansms of nformaton transfer n groups nvolve questons suh as how nformaton about requred moton dretons, orgnally held by only a few nformed ndvduals, an propagate through an entre group by smple mehansms that are the same for every ndvdual. It s not lear how ndvduals reognze leaders who are nformed, or even f ths s requred, or how onflt s resolved when several nformed ndvduals dffer n ther moton preferenes. Protools for Leadershp by a Small Perentage of Informed Indvduals The setup used n [Couzn 2005] s as follows. Consder a set of N agents { N} and let the ommunaton between the agents be modeled by a general dreted graph. The edges of the graph orrespond to agents that are lose together n some sense dependng on the spef anmal group onsdered, as dsussed n the prevous seton. Eah agent has a poston vetor, a dreton vetor, and a speed of moton. One dynams that aptures ths for moton n the ( x, y -plane s the dsrete-tme dynams p( k1 p( k V( k Tos ( k (1.2.1 q( k1 q( k V( k Tsn ( k where V s the speed of agent, ts headng, and x [ ] T p q ts poston. The tme ndex s k and the samplng perod s T. The dreton vetor of agent s v ( k V( kos ( k V( ksn ( k T (1.2.2 Eah agent turns away from others n a ollson avodane regon N aordng to ( x j( k x( k ( k1 j (1.2.3 jn x ( ( j k x k whh auses agent to turn away from other agents nsde the ollson neghborhood. If agents are not deteted wthn the ollson avodane regon, then eah agent wll turn towards and algn wth ts neghbors n an nteraton regon N aordng to ( x j( k x( k vj( k ( k1 j j (1.2.4 \ ( ( jn \ ( N x j k x k jn N v j k Ths s onverted to a unt vetor by ˆ ( 1 ( 1 k k ( 1 (1.2.5 k A perentage p of the group onssts of nformed ndvduals, eah of whom have a preferred desred dreton of moton gven by a unt referene dreton vetor r. The headng ontrol protools are therefore taken as ˆ ( k 1 gr ( k 1 (1.2.6 ˆ ( k 1 g r 7

8 wth g 0 a leader weghtng gan. If g 0, agent s not nformed, exhbts no preferred dreton of travel, and merely algns wth hs neghbors. If g 0, agent s an nformed ndvdual and, as g nreases, agent exhbts a stronger desre to move n ts referene dreton r. 1.3 Human Crowd behavor Durng Pan Stuatons Equaton Seton (Next Vsek paper 1.4 Moton n Physal Partle Systems Equaton Seton (Next Vsek 1.5 Kuramoto Osllators Equaton Seton (Next N Kuramoto osllators have dynams K sn( (1.5.1 N j j wth osllaton frequeny and ouplng gan K [Strogatz 2000]. The osllators are sad to synhronze f ( t ( t 0 ast, j. j Kuramoto took the osllaton frequenes as dstrbuted aordng to a probablty densty funton. He showed that there s a ouplng gan K below whh the osllators reman noherent,.e. do not synhronze. Above ths gan the noherent state beomes unstable, and the osllators splt nto two groups- those wth osllaton frequenes lose to the mean synhronze to a mean frequeny, whle the others drft relatve to ths group. In [Chopra and Spong CDC 2005], synhronzaton s studed usng the Lyapunov 1 T funton V 2, wth [ 1 ] T K 2 N. It s shown that V N os( j( j. It s shown that above some ouplng gan, and f the ntal phase dfferenes are n a ertan ompat 1 set, the osllators onverge to the mean frequeny. N, j 8

9 Equaton Seton (Next 1.6 Synhronzaton of Eletr Power Systems In ths seton we apply the onepts of synhronzaton of oupled systems to dstrbuted eletr power systems ontrol. Consder a power system wth N generators. The dynams of eah node are gven by the flux deay mehanal and generator eletral equatons, whh have the bas form [Ortega 2005] 0 D Pm E EjYjsn( j (1.6.1 M jn E ae u b E os( j j jn and the turbne and valve dynams P m P m d XE ( X E hx E g ( P R 0 The power flow terms are summatons over the neghborhood of node, whh onssts of all generators onneted va transmsson lnes to node. We have assumed the lossless ase for smplty. These equatons have both a feld extaton ontrol u and a power ontrol nput P. The power system transent stablty problem ours at a fast tme sale, and s onerned wth seletng u for fault reovery. For ths problem the mehanal power P m (whh vares more slowly s generally assumed onstant, so only the three flux equatons are onsdered. Classal tehnques desgn AVR and PSS to damp nterarea osllatons. Lnear desgn methods are often used. Soos [2002] used optmal ontrol tehnques. Jang and Dorsey [1994] used Lyapunov methods to desgn loal adaptve ontrollers. In Wang [1997] a nonlnear feedbak lnearzaton method was used. Neural network learnng ontrol was used n Cheng [1997]. Pourboghrat [2004] uses sldng mode methods. Ortega [2005] used a nonlnear passvty based approah for general lossy systems. Partal dfferental equatons were solved for the ontrols, whh have the form u k1e k2 b Ej os( j kjj (1.6.3 jn where k j are nonlnear funtons. On the other hand, the load frequeny ontrol (LFC problem generally gnores the fast exter dynams (thrd equaton and uses lnear methods to desgn the power ontrol nput P. ACE s often used. The lterature s rh. Wang [1994] used a robust adaptve ontrol method based on soluton of a loal deoupled Rat equaton. Geromel [1985] uses output feedbak desgn to get deentralzed loal LFC ontrollers. Venayagamoorthy [2003, 2004] has used nonlnear neuroontrollers, based spefally on DHP, for power system stablty ontrol n a multarea system. The ontrollers only requred loal measurements. He has aheved superor results and mplemented hs ontroller on mroalternators at Unv. of Natal n Durban. Reently proposed ontrollers based on FACTS allow the ontrol of real and/or reatve power flow n AC transmsson lnes by modfyng the reatane. L [2006] desgned a FACTS ontroller usng feedbak lnearzaton. Mshra [2006] desgned a NN FACTS ontroller. Ray jn 9

10 and Venayagamoorthy [preprnt 2006] has desgned a GCSC FACTS optmal ontroller based on HDP. Reurrent NN are used to dentfy the power system dynams. Synhronous, renewable, and CHP generators have dfferent dynams, yet they share the bas swng equaton for rotatng generators, gven for the -th generator as (assumng the lossless ase ( D Pm E G Pe D Pm EG E EjYjsn( j M M jn E ae u b E os( (1.6.5 j j jn wth frequeny, mehanal power P m, eletral power P e, and admttane Y j apturng the networked nteronneton struture between generators. A frequeny equlbrum s haraterzed by frequeny synhronzaton j,, j and balaned power flow 2 Q( Pm Pe EG 0,. In [Dorfler 2010] t was shown that these nteronneted systems are generalzed Kuramoto osllators [Kuramoto 1984], and ondtons for synhronzaton to a ommon frequeny are gven. 10

11 Referenes F. Dorfler and F. Bullo, Transent stablty analyss n power networks and synhronzaton of non-unform Kuramoto osllators, Pro. Ameran Control Conferene, Baltmore, MD, pp , June Y. Kuramoto, Chemal Osllatons, Waves, and Turbulene, Sprnger, Berln, S.H. Strogatz, From Kuramoto to Crawford: explorng the onset of synhronzaton n populatons of oupled osllators, Physa D, vol. 143, pp. 1-20,