Distributed Cooperative Control System Algorithms Simulations and Enhancements

Transcription

1 Po Wu and P.J. Antsakls, Dstrbuted Cooperatve Control System Algorthms: Smulatons and Enhancements, ISIS Techncal Report, Unversty of Notre Dame, ISIS , Aprl ( Dstrbuted Cooperatve Control System Algorthms Smulatons and Enhancements Po Wu and Panos J. Antsakls Department of Electrcal Engneerng Unversty of Notre Dame Notre Dame, IN Emal: {pwu1,

2 Abstract In ths paper, we descrbe and smulate a number of algorthms created to address problems n dstrbuted control systems. Based on tools from matrx theory, algebrac graph theory and control theory, a bref ntroducton s provded on consensus, rendezvous, and flockng protocols, and Dubns vehcle model. Recent results are then shown by smulatng and enhancng varous mult-agent dynamc system algorthms. Index Terms Cooperatve systems, dstrbuted control, mult-agent system

3 ISIS TECHNICAL REPORT ISIS APRIL I. INTRODUCTION Dstrbuted control systems refer to control systems n whch the controller elements are not centralzed but are dstrbuted throughout the system wth each component sub-system controlled by one or more controllers. The entre system of controllers s connected by networks for communcaton and montorng. In such mult-agent systems, varous control algorthms to acheve dfferent purposes are consdered, such as the atttude algnment n multple spacecraft settng [3][28], formaton control of unmanned ar vehcles [1], and flockng [2]. See also [27]. Consensus problems under dfferent nformaton constrants have been addressed by many researchers [16]. Jadbabae et al. [3] focus on coordnaton under undrected graphs. Ren et al. [4] extend the result to the drected graph case. Average consensus problem s solved over balanced drected networks n [5]. The speed of consensus can be ncreased by weght optmzaton [20], or by va random rewrng [6] snce the algebrac connectvty of a regular network can be greatly enlarged. The robustness to changes n network topology due to lnk/node falures, tme-delays, and performance guarantees s analyzed n [7]. Asynchronous consensus has been studed n [8][17][18][19]. Rendezvous problems have been ntroduced n [9], n whch all agents are homogeneous and memoryless. It s shown n [10] that synchronous and asynchronous rendezvous s acheved wth the ntal graph connected. The crcumcenter algorthm s desgned [11] to set target ponts as crcumcenters. A related algorthm, n whch connectvty constrants are not mposed, s proposed n [12]. Flockng, whch means convergence to a common velocty vector and stablzaton of nter-agent dstances, s guaranteed as long as the poston and velocty graphs reman connected at all tmes [13]. [14] provdes a stablty result for the case where the topology of agent nterconnectons changes n a completely arbtrary manner. The splt/rejon maneuver and squeezng maneuver are performed wth obstacle avodance. Takng nto account nherent knematc lmtatons of automobles, Dubns vehcle s ntroduced n [21]. A model of Dubns vehcle s a controllable wheeled robot wth a constrant on the turnng angle along a gven route n two dmensons. [22] gave the shortest paths jonng two arbtrary confguratons usng Optmal Control Theory.

4 ISIS TECHNICAL REPORT ISIS APRIL Rendezvous of Dubns vehcles s dscussed n [25]. Smulaton and enhancement of dstrbuted control algorthms mentoned above provdes llustratve examples of how the dynamc system evolves. The dscrete event smulator s mplemented n Java as a class wth headng, poston, velocty and other varables, and wth some Java methods for neghborhood updatng, movng and status dsplayng. Communcaton s accomplshed va message exchangng mechansm takng nto account nose. The behavor of every agent s llustrated as the 2-D poston and trajectores. The paper s organzed as follows. In Secton II, we descrbe recent results of consensus protocols. In Secton III, rendezvous problem s shown. In Secton IV, flockng problems are dscussed. Secton V contans the smulaton results and Secton IV contans concludng remarks and future drectons. II. CONSENSUS PROTOCOLS AND STABILITY THEOREMS A. Defnton and Notatons To descrbe the relatonshps between multple agents, we have a dgraph G to model the nteracton topology. If agent j can receve nformaton from agent, then graph nodes v and v j correspond to agent and j, and a drected edge e j represents a undrectonal nformaton exchange lnk from v to v j, that s, agent j can receve nformaton from agent. The nteracton graph represents the communcaton pattern at certan dscrete tme. Let G = { V, E, A} be a weghted dgraph (or drect graph) of order n wth the set of nodes V = { v1, v2,..., v n }, set of edges E V V, and a weghted adjacency matrx A = [ a j ] wth nonnegatve adjacency elements a j. The node ndces belong to a fnte ndex set I = {1, 2,..., n }. A drected edge of G s denoted by e = ( v, v ), where e j E does not mply e j E edges of the graph are postve,.e., a j > 0 f and only f e j j j. The adjacency elements correspondng to the E. Moreover, we assume a 0 for all I. The set of neghbors of node v s the set of all nodes whch communcate to v, denoted by N = { v V :( v, v ) E. j j }

5 ISIS TECHNICAL REPORT ISIS APRIL A graph G s called strongly connected f there s a drected path from v to v j and v j to v between any par of dstnct vertces v and v j. Vertex v s sad to be lnked to vertex v j across a tme nterval f there exsts a drected path from v to v j n the unon of nteracton graphs n that nterval. A drected tree s a drected graph where every node except the root has exactly one parent. A spannng tree of a drected graph s a tree formed by graph edges that connect all the vertces of the graph. Let 1 denote a n 1column vector wth all entres equal to 1. Let M ( R) represent the set of all n n real matrces. A matrx F M ( R) s nonnegatve, F 0, f all ts entres n n are nonnegatve, and t s rreducble f and only f 1 ( I + F) n > 0. Furthermore, f all ts row sums are +1, F s sad to be a (row) stochastc, whle doubly stochastc f t s both row stochastc and column stochastc. The nteracton graph s tme-dependent snce the nformaton flow among agents may be dynamcally changng. Let G= { G1, G2,..., G M } denote the set of all possble nteracton graphs defned for a group of agents. Note that the cardnalty of G s fnte. The unon of a collecton of graphs { G, G,... G }, each wth vertex set V, s a graph G wth vertex set V G, j = 1,2,..., m. j 1 2 m and edge set equal to the unon of the edge sets of B. Consensus protocols and general stablty theorems Consder the followng synchronous dscrete-tme consensus protocol [4], [7] n 1 x( t + 1) = Aj( t) xj( t ) (1) A () t j= 1 n j= 1 where t {0,1, 2,...} s the dscrete-tme ndex, A () t > 0f nformaton flows from j j v j to v at tme t. The magntude of A () t represents possbly tme-varyng relatve j confdence of agent n the nformaton state of agent j at tme t or the relatve relabltes of nformaton exchange lnks between them. Rewrte (1) n a compact form x( t+ 1) = F( t) x( t) (2) Aj () t where x = [ x1,..., x n ], F = Fj =. An mmedate observaton s that the matrx F n A () t j= 1 j s a nonnegatve stochastc matrx, whch has an egenvalue at 1 wth the correspondng

6 ISIS TECHNICAL REPORT ISIS APRIL egenvalue vector equal to 1. Lemma 1 ([7]): Let G be a dgraph wth n nodes and maxmum degree. Then wth parameter Δ= max ( ) a j j and ε (0,1/ Δ ], F satsfes the followng propertes: ) F s a row stochastc nonnegatve matrx wth a trval egenvalue of 1; ) All egenvalues of F are n a unt crcle; ) If G s a balanced graph, then F s a doubly stochastc matrx; v) If G s strongly connected and 0< ε < 1/ Δ, then F s defned as a prmtve matrx. Wth the connecton between the graph G and the matrx F, we have the stablzaton condton of consensus,.e. the convergence of all agents n fxed topology. Theorem 1 ([7]): Consder a network of n agents wth topology G applyng the consensus algorthm (2). Suppose G s a strongly connected dgraph, then ) A consensus s asymptotcally reached for all ntal states; ) The consensus value s α = (0) wth 1 ; wx ) If the dgraph s balanced (or P s doubly stochastc), an average-consensus s asymptotcally reached and x = ( x (0))/ n. ss w = In swtchng topology, we have the followng results. Theorem 2 ([3]): Let x(0) be fxed and let G be a swtchng sgnal for whch there exsts an nfnte sequence of contguous, nonempty, bounded, tme-ntervals, [t, t +1 ), startng at t 0 = 0, wth the property that across each such nterval, the n agents are lnked together. Then lm x( t) = x 1 (3) t where x ss s a number dependng only on x(0) and G. Theorem 3 ([4]): Let G be a swtchng nteracton graph. The dscrete update scheme (2) acheves consensus asymptotcally f there exsts an nfnte sequence of unformly bounded, nonoverlappng tme ntervals [t, t +1 ), = 1, 2,, startng at t 0 = 0, wth the property that each nterval [t, t +1 ) s unformly bounded and the unon of the graphs across each nterval [t, t +1 ) has a spannng tree. Furthermore, f the unon of the graphs after some fnte tme does not have a spannng tree, then consensus cannot be acheved asymptotcally. The algorthms appled above have a relatve slow convergence speed. It s shown n [6] that small-world network can make the algebrac connectvty more than 1000 ss

7 ISIS TECHNICAL REPORT ISIS APRIL tmes greater than a regular network, whch means that small-world networks go through a spectral phase transton phenomenon achevng ultrafast and more robust consensus. We wll show that n the smulaton. Here, the algebrac connectvty, whch determned the speed of convergence, s defned as: T x Lx λ2 ( L) = mn T 2 (4) 1 x= 0, x 0 x where L s the graph Laplacan defned as L = D A, D j = a j, j j C. Delay n communcaton If communcaton delay exsts, the asynchronous contnuous tme consensus protocol can be descrbed as x () t = a [ x ( t τ ()) t x ( t τ ())] t (5) j j Vj N Assume that the orgnal graph leads to consensus. Introducng delay nto the protocol may affect the performance of the whole dstrbuted control system or even undermne fnal consensus. If delay τ s bounded, global asymptotcal consensus s stll achevable. The followng table shows delay results both n contnuous tme (CT) and dscrete tme (DT). No. Results CT/DT Value Tme 1 π τ Tme- 2λ n CT Unform nvarant Th. 10 n [5] 2 π τ Non- Tme- 2λn CT unform nvarant Th. 5 n[24] τ () t ΔΔ, I Δ 4 Th. 6 n [24] 3 τ () t < 2λ n In [25] 0 t ( k) B 1 j 1 5 Proposton 1 n [26] CT CT DT Nonunform Unform Nonunform Table 1. A Categorzaton of extng consensus delay results Tmevarant Tmevarant Tmevarant

8 ISIS TECHNICAL REPORT ISIS APRIL D. Optmzaton of edge weghts Assume the topology of the graph s known,.e. G = { V, E, A} be a weghted dgraph, then the weghts of graph Laplacan can be optmzed so that maxmum consensus rate s acheved. In [20], the optmzaton problem becomes: maxmze λ ( L) 2 T T subject to ALu (,..., u) A λi 0, I A L( u1,... ur ) A 0 1 r But even when ths LIM problem has a soluton, the system s no longer dstrbuted controlled. In order to desgn a dstrbuted algorthm, we notce from smulatons that generally lnk losses happen when neghbors are at the edge of detecton range. Thus ncreasng weghts correspondng to dstance wll prevent lnk droppng. III. FLOCKING Flockng s ntroduced n [2], wth three flockng rules of Reynolds: ) Flock Centerng: attempt to stay close to nearby flockmates; ) Collson Avodance: avod collsons wth nearby flockmates; ) Velocty Matchng: attempt to match velocty wth nearby flockmates. As stated n [13], the protocol can be expressed as: r = (6a) v v = u (6b) r and v are poston and speed of agent. Agent s steered va ts acceleraton nput u whch conssts of two components, u = α + a, = 1,...,N Component α ams at algnng the velocty vectors of all the agents, whch s smlar n consensus protocols. Component a s a vector n the drecton of the negated gradent of an artfcal potental functon, V, and s used for collson avodance and coheson n the group. Defnton 1 (Potental functon) Potental V j s a dfferentable, nonnegatve, functon of the dstance r j between agents and j, such that 1) V j ( r j ) + as r j 0, 2) V j attans ts unque mnmum when agents and j are located at a desred dstance, and d d r 3) V = 0, f r j > R j j Theorem 5 ([13]) Consder a system of N moble agents wth dynamcs (8), each

9 ISIS TECHNICAL REPORT ISIS APRIL steered by control law wth potental functon. Let both the poston and velocty graphs be tme-varyng, but always connected. Then all parwse velocty dfferences converge asymptotcally to zero, collsons between the agents are avoded, and the system approaches a local extremum of the sum of all agent potentals. Whle [13] solves the rule of algnment and collson avodance, t does not deal wth fragment of agents and obstacle avodance. In [14] these requrements are taken nto account by addng a vrtual leader and settng potental functons of obstacles. A vrtual leader or movng rendezvous pont represents a group objectve. If the vrtual leader moves along a straght lne wth a desred velocty, based on modfed expresson of u = α + a + f (q,p,q r,p r ) A secondary objectve of an agent s to track the vrtual leader. Despte the smlartes between certan terms n these protocols, the collectve behavor would change drastcally and never lead to fragmentaton. IV. RENDEZVOUS ALGORITHMS In rendezvous algorthms, multple agents n the network also have the sensng, computaton, communcaton, and moton control capabltes. Synchronous rendezvous can be performed as stop-and-go maneuvers [10]. A stop-and-go maneuver takes place wthn a tme nterval consstng of two consecutve sub-ntervals. When stopped,.e. n sensng perod, agents are statonary and calculatng where to go. Then agents try to move from ther current postons to ther next way-ponts and agan come to rest. Here we wll not consder collson avodance. Proper way-ponts or target ponts for each agent are of nterest. In fact, target ponts are not unque. Agent 's kth way-pont s the pont to whch agent s to move to at tme t k. Thus f () x t and ( ) agent 's kth movng protocol s: x t denotes the poston of agent after and before movng, then x ( t ) = x ( t ) + u ( x ( t ) x ( t ), x ( t ) x ( t ),..., x ( t ) x ( t )) (7) k k 1 m( tk ) 1 k k 2 k k m ( t k k k) And the postons of agent s relatve neghbors are z x ( t ) x ( t ), j {1, 2,..., m}. u m s defned as { z, z,..., z } D 1 2 m m 1 2 m M 1 2 m 1 2 j j k k u ( z, z,..., z ) D C( z, z,..., z ) 0, z, z,..., zm, (8) m

10 ISIS TECHNICAL REPORT ISIS APRIL and u ( m z,,..., ) 1 z2 z m a corner of 0, z1, z2,..., z m (9) Where C(z) denote the closed dsk of dameter r centered at the pont z/2, and C(z 1, z m ) = C(z j ), j=1,2,,m. Then we have the protocol of rendezvous: Theorem 4 ([10]) Let u 0 = 0 and for each m {1,2,,n-1}, let u m be any contnuous functon satsfyng (6) and (7). For each set of ntal agent postons x 1 (0); x 2 (0); ; x n (0), each agent's poston x (t) converges to a unque pont p R 2 such that for each, j {1,2,,n}, ether p = p j or p - p j > r. Moreover, f agents and j are regstered neghbors at any tme t, then p = p j. A most common target pont would be the centrod of all relatve neghbors, whch would be smulated later. Another target pont would be the crcumcenter [11]. Each agent performs the followng tasks: ) t detects ts neghbors accordng to G; ) t computes the crcumcenter of the pont set comprsed of ts neghbors and of tself; ) t moves toward ths crcumcenter whle mantanng connectvty wth ts neghbors, whch s proved n [11] and wll be demonstrated n the smulaton. Below, n secton VI, we have smulatons of all common target ponts mentoned above. We have also enhanced the performance of rendezvous by addng a relay protocol. It can be shown that nformaton between agents wll flow more quckly wth relay, whch s smlar to enlargng the area of sensng for every agent. V. DUBINS VEHICLES In [22], the dynamc model of the vehcle can be descrbed as (10) where M = (x, y) are the coordnates of the reference pont of the vehcle wth respect to a reference frame, θ s the headng angle wth respect to the frame x-axs. The confguraton of the vehcle s v andω are the lnear and angular veloctes of the vehcle. Wthout loss of generalty, up to a tme axs rescalng, we assume that v(t) = V. V s a constant. The turnng radus

11 ISIS TECHNICAL REPORT ISIS APRIL of the vehcle s lower bounded by a constant value R > 0, whch results n an upper bound on the vehcle s angular veloctyω. For ths model we consder the problem of determnng a path of mnmal length for reachng tangentally a rectlnear route wth a specfed drecton of moton. We denote by T a target rectlnear path n the plane, wth a prescrbed drecton of moton determned by the angle α wth respect to the x-axs (see Fgure 2). We consder the optmal control problem: (11) It s subject to (1) and (2), wth X(0)= (M 0, θ 0 ) and such that, at the unspecfed termnal tme T, M(T) T andθ(t) =α. By applyng Pontryagn s Maxmum Prncple, the optmzaton problem s solvable. After puttng constrants on turnng rate of agents, the characterstcs of Dubns Vehcle wll affect all dstrbuted control algorthms dscussed above, especally rendezvous. Intutvely, knematc constrants such as bounded turnng rate wll restrct the moblty of agents and decrease the performance of dstrbuted control systems, whch s approved by smulatons. VI. SIMULATION The smulaton s mplemented n Java wth the Jprowler tool from Vanderblt Unversty [15]. Jprowler s a probablstc smulator for prototypng and analyzng communcaton protocols of ad-hoc wreless networks. The smulator supports pluggable rado models and multple applcaton modules. The smulaton has added agents the ablty to move n two dmensons, under steady speed varyng ther headngs. The moblty of the agents creates a dynamc topology of nodes connectvty graph. The behavor of agents could be expressed as a sequence of dscrete events. 3-D dstrbuton and movement s avalable, but Java s not effectve n dsplayng 3-D objectves. When characterzng agent movements, two events are crucal n descrbng the behavor of agents, the move event and update event. The move event calculates new postons of each agent by acqurng the orgnal poston, the velocty and the headng. The update event collects nformaton from neghbors, and then apples dfferent protocols to fgure out new velocty and headngs, preparng for the next move event.

12 ISIS TECHNICAL REPORT ISIS APRIL A. Consensus After settng the adjacent matrces wth weghts when ntatng the connecton status of all movng agents, the consensus of 6 agents under fxed dgraph s shown n Fg. 1. Wth strongly connected balanced dgraph or spannng tree, consensus s acheved under nearest neghbor rules. It s plotted that the agent n the dark sde of the lne s dentfed as a neghbor of the agent n the lght sde. Randomly weghted consensus under nearest neghborhood rule s demonstrated n Fg. 2, where fragments are almost nevtable snce the detect radus s less than 2 grds. The darkness of the lnes between agents mples the weght of edges. In Fg. 3, Small-world algorthm s also smulated. Even under a relatve low rewrng rate (<0.01) and wth strong nose (>20%), consensus can be acheved much faster than n nearest neghborhood rules, no matter the graph s weghted or not. Smulatons wth delayed communcaton have smlar performance as shown n Fg. 2, but the consensus value vares when we have random delays for each agents. A straghtforward explanaton of consensus varatons can be deduced from [26], where the asynchronous system has an enlarged graph Laplacan. Although the orgnal graph Laplacan s consstent, dfferent delays of communcaton lnks wll cause the enlarged matrx to vary. Consensus s stll feasble f the delay s bounded, but the consensus value changes every tme when we run the smulaton. One major reason of lnk droppng s that the consensus protocol does not ensure agents always move closer to each other. Consensus s a dstrbuted feedback law whch compensates n some extent the ncreased dstance between agents. But when one agent s at the edge of detecton range of another agent, very lkely they wll lose ther communcaton lnks. Our adaptve weght optmzaton adds weghts to longer lnks proportonally. Ths tme varyng optmzaton wll change consensus value (f acheved) whle mantanng network connectvty.

13 ISIS TECHNICAL REPORT ISIS APRIL Fg. 1. Consensus under fxed topology (6 agents) Fg. 2. Weghted consensus under nearest neghborhood rule (20 agents, fragment)

14 ISIS TECHNICAL REPORT ISIS APRIL Fg. 3. Consensus wth small world algorthm (20 agents) B. Rendezvous Trajectores of all moble agents are plotted to llustrate the process of rendezvous. The dscrete synchronous rendezvous problem has been smulated, but the mplementaton s slghtly dfferent from the stop and go maneuver mentoned n [8]. Agents wll not wat untl all the agents are ready for the next move. For certan ntal condtons, ths protocol wll cause connectvty loss. The result n Fg. 4 shows that rendezvous s straght-forward f the poston graph s connected, but lnk falure caused by lmted sensng radus of agents may render decomposton. Fg. 5 shows by applyng the crcumcenter algorthm n [11], the movement of agents s no longer smooth, but connectvty s well mantaned. Computatonal complexty of the crcumcenter algorthm s defntely larger than rendezvous to neghborhood centrod, thus crcumcenter algorthm s more computatonally costly after scalng. By usng the relay protocol, Fg. 6 s smlar to the fgure of rendezvous to centrod when sensng radus s doubled. It shows performance of rendezvous can be enhanced by addng communcaton cost.

15 ISIS TECHNICAL REPORT ISIS APRIL Fg. 4. Rendezvous to the centrod of neghbors (50 agents) Fg. 5. Rendezvous to the crcumcenter (20 agents)

16 ISIS TECHNICAL REPORT ISIS APRIL Fg. 6. Rendezvous to the centrod wth relay protocol (50 agents) Fgure 7. Rendezvous of Dubns vehcles (10 agents)

17 ISIS TECHNICAL REPORT ISIS APRIL Fg.7 s the smulatons of rendezvous wth the constrants of Dubns vehcle. When turnng rate s bounded, no sudden turnng s allowed. Thus rendezvous s more restrcted and the trajectores of agents are smoother n the fgure. C. Flockng In flockng algorthms, a potental functon s appled to avod collson. In smulaton to the algorthm n [13], takng nto account collson avodance only, fragment s almost nevtable n Fg. 8. Fg. 8. Flockng wth collson avodance (100 agents) Fg. 9 shows algorthm 2 n [14]. A vrtual leader s movng around the gray obstacle. All agents are attemptng to follow t, thus gatherng nto the same drecton. The potental feld and the radus of sensng needs to be carefully adjusted.

18 ISIS TECHNICAL REPORT ISIS APRIL Fg. 9. Flockng wth vrtual leader (100 agents) VII. CONCLUSION In ths paper, a number of recent results n dstrbuted control algorthms were summarzed. We demonstrated some examples of mult-agent dynamc systems. Smulatons of consensus, rendezvous and flockng are nstrumental n studyng varous protocols under dfferent nformaton constrants. Dubns vehcle s a more realstc model to smulate agent knematcs. Future work wll be focused on further understandng asynchronous protocols. For example, wth more constrants such as communcaton delay and lmted agent moblty, whether there are any necessary or suffcent condtons to make these protocols stll feasble. Smulaton software mplementng all above algorthms s avalable from the frst author (pwu1@nd.edu).

19 ISIS TECHNICAL REPORT ISIS APRIL APPENDIX Consensus Rendezvous Flockng/ swarm Algorthms Consensus acheved n DT / CT (dwell tme needed) under nearest neghbor rules, undrected and jontly connected graphs n every perod requred Average consensus acheved for fxed and varant topology: strongly connected balanced dgraph, also a suffcent condton of tme-delays for convergence Consensus acheved n DT / CT (dwell tme needed) under varant graph: jontly spannng tree n every perod Asynchronous consensus acheved, but drecton unspecfed Small world algorthm (random rewrng networks) The algebrac connectvty s more than 1000 tmes greater than regular networks. Rendezvous acheved under a synchronous stop and go maneuver f the ntal graph s connected. Specfed target ponts needed, such as the centrod of neghbors, or the center of the smallest crcle contanng convex hull Asynchronous rendezvous acheved f the drected graph characterzng regstered neghbors s strongly connected. Crcumcenter algorthm n arbtrary dmenson, robust to lnk falures gven strongly connected graph Rendezvous of Dubns Vehcles Flockng acheved under tme-varyng but always connected poston and velocty graphs, collson avodance ensured, no dwell tme needed Flockng satsfyng Reynolds three heurstc rules acheved, takng account obstacle avodance. Group objectve s necessary by settng movng rendezvous ponts to prevent fragments. Authors Jadbabae, A. Je Ln and Morse, A.S. [2], 2003 Olfat-Saber, R.; Murray, R.M. [3], 2004 We Ren; Beard, R.W. [4], 2005 Le Fang; Panos J. Antsakls, Anastass Tzmas [19], 2005 Olfat-Saber, R [5], 2005 J. Ln, A. S. Morse [8], 2003 J. Ln, A. S. Morse [8], 2003 J. Corte s, S. Martı nez, and F. Bullo [9], 2004 Amt Bhata, Emlo Frazzol [23], 2008 H Tanner, A Jadbabae, GJ Pappas [6], 2005 Olfat-Saber, R. [7], 2006 Table 2. A Categorzaton of some recent dstrbuted control algorthms

20 ISIS TECHNICAL REPORT ISIS APRIL REFERENCES [1] J. A. Fax and R. M. Murray, Informaton flow and cooperatve control of vehcle formatons, IEEE Trans Automatc Control, vol. 49, no. 9, pp , Sep [2] C.W. Reynolds, Flocks, herds, and schools: a dstrbuted behavoral model, n Computer Graphcs (ACM SIGGRAPH 87 Conf. Proc.), vol. 21, July 1987, pp [3] A. Jadbabae, L. Je, A.S. Morse, Coordnaton of Groups of Moble Autonomous Agents Usng Nearest Neghbor Rules, Automatc Control, IEEE Transactons on Vol 48, No 6, June 2003 Page(s): [4] W. Ren, R.W. Beard, Consensus seekng n mult agent systems under dynamcally changng nteracton topologes, Automatc Control, IEEE Transactons on Volume 50, No 5, May 2005 Page(s): [5] R. Olfat-Saber, and R. M. Murray, Consensus problems n networks of agents wth swtchng topology and tme-delays, Automatc Control, IEEE Transactons on Vol 49, No 9, Sept Page(s): [6] R. Olfat-Saber, Ultrafast consensus n small-world networks, Amercan Control Conference, Proceedngs of the 2005 [7] R. Olfat-Saber, J.A. Fax, and R.M. Murray, Consensus and Cooperaton n Networked Mult-Agent Systems, Proceedngs of the IEEE, Volume 95, No 1, Jan Page(s): [8] L. Fang; P.J. Antsakls, "Informaton consensus of asynchronous dscrete-tme mult-agent systems," Amercan Control Conference, Proceedngs of the 2005, vol., no., pp vol. 3, 8-10 June 2005 [9] H. Ando, Y. Oasa, I. Suzuk, and M. Yamashta, Dstrbuted memoryless pont convergence algorthm for moble robots wth lmted vsblty, IEEE Trans. Robot. Automat. vol. 15, no. 5, pp , Oct [10] J. Ln, A. S. Morse, and B. D.O. Anderson, The Mult-agent Rendezvous Problem, n Proc. 42nd IEEE Conf. Decson and Control, Dec [11] J. Corte s, S. Martı nez, and F. Bullo, Robust rendezvous for moble autonomous agents va proxmty graphs n arbtrary dmensons, IEEE Trans. Automatc Control, vol. 51, no. 8, pp , Aug

21 ISIS TECHNICAL REPORT ISIS APRIL [12] Z. Ln, M. Broucke, and B. Francs, Local control strateges for groups of moble autonomous agents, IEEE Trans. Autom. Control, vol. 49, no. 4, pp , Apr [13] H.Tanner, A.Jadbabae, G.J.Pappas, Flockng n fxed and swtchng networks, - IEEE Conference on Decson and Control, 2005 [14] R. Olfat-Saber, Flockng for Mult-Agent Dynamc Systems Algorthms and Theory, Automatc Control, IEEE Transactons on vol.51, no.3, March 2006 pp [15] [16] R.Olfat-Saber, J.A.Fax, R.M.Murray, "Consensus and Cooperaton n Networked Mult-Agent Systems", Proc. of the IEEE, vol.95, no.1, pp , Jan [17] L. Fang, P. J. Antsakls, "Asynchronous Consensus Protocols Usng Nonlnear Paracontractons Theory," IEEE Trans on Auto. Control, 2008 [18] L. Fang, P. J. Antsakls, "On Communcaton Requrements for Mult-agent Consensus Seekng," Networked Embedded Sensng and Control, Proceedngs of Workshop NESC 05 [19] L. Fang, P. J. Antsakls, A. Tzmas, "Asynchronous Consensus Protocols: Prelmnary Results, Smulatons and Open Questons," Proc. IEEE CDC, 2005 [20] P. Yang, R. A. Freeman, K. M. Lynch Optmal Informaton Propagaton n Sensor Networks IEEE Internatonal Conference on Robotcs and Automaton, 2006 [21] L. E. Dubns. On curves of mnmal length wth a constrant on average curvature, and wth prescrbed ntal and termnal postons and tangents. Amercan Journal of Mathematcs, 79(3): , [22] P. Soueres, A. Balluch, A. Bcch, Optmal feedback control for route trackng wth a bouded-curvature vehcle, Proc. of the IEEE Int. Conference on Robotcs and Automaton, San-Franssco, Calforna, USA, May [23] Amt Bhata, Emlo Frazzol. Decentralzed algorthm for mnmum-tme rendezvous of Dubns vehcles Amercan Control Conference. Seattle, USA June 2008, Page(s): [24] P. Blman, G. Ferrar-Trecate, Average consensus problems n networks of agents wth delayed communcatons, Techncal Report, arxv:math/ v2, 2005 [25] J.A.Yorke, Asymptotc stablty for one dmensonal dfferental-delay equatons Journal of Dfferental equatons, 7: , 1970.

22 ISIS TECHNICAL REPORT ISIS APRIL [26] A. Nedch and A. Ozdaglar, "Convergence Rate for Consensus wth Delays," LIDS Techncal Report 2774, MIT, Lab. for Informaton and Decson Systems, 2007 [27] vol. 95, No. 1, Proceedngs of the IEEE [28] Bertsekas, D. P.; Tstskls, J. N.; Comments on Coordnaton of Groups of Moble Autonomous Agents Usng Nearest Neghbor Rules, Automatc Control, IEEE Transactons on Volume 52, Issue 5, May 2007 Page(s):