Decentralized Event-Triggered Cooperative Control with Limited Communication

Transcription

1 Decentralzed Event-Trggered Cooperatve Control wth Lmted Communcaton Abstract Ths note studes event-trggered control of Mult-Agent Systems (MAS) wth frst order ntegrator dynamcs. It extends prevous work on event-trggered consensus by consderng lmted communcaton capabltes through strct peer-to-peer non-contnuous nformaton exchange. The approach provdes both a decentralzed control law and a decentralzed communcaton polcy. Communcaton events reure no global nformaton and are based only on local state errors; agents do not reure a global samplng perod or synchronous broadcastng as n sampled-data approaches. The proposed decentralzed event-trggered control technue guarantees that the nter-event tmes for each agent are strctly postve. Fnally, the deas n ths note are used to consder the practcal scenaro where agents are able to exchange only uantzed measurements of ther states. Keywords: event-trggered control; consensus; uantzaton; mult-agent systems.. Introducton An ncreasng nterest n controllng large scale dynamcal systems composed of several to many autonomous moble agents exsts n dfferent academc, commercal, and mltary areas. Ths thrust s related to the large number of applcatons n whch a group of coordnated agents s potentally able to outperform a sngle or a number of systems operatng ndependently (Ren, Beard, and Atkns 007). An mportant problem n Mult-Agent Systems (MAS) s to desgn and mplement decentralzed algorthms for control and communcaton of agents. It s well understood that each agent should be able to determne ts own control laws ndependently and based only on local nformaton. Ths has been an mportant research topc (Ren, Beard, and Atkns 007; Moreau 004; J and Egerstedt 007; Tanner, Jadbabae and Pappas 003). These papers consder agents wth contnuous-tme dynamcs and t s assumed that agents can have contnuous access to the states of ther neghbors. In many applcatons the agents transmt ther relevant varables such as

2 poston, velocty, headng, etc. to a subset of the agents not contnuously but at dscrete ponts n tme. It s mportant to dscern how freuently the agents should establsh communcaton n order to preserve propertes of smlar control algorthms that assume contnuous nformaton exchange. The sample-data approach s commonly used to estmate the samplng perods (Can and Ren 009; Can and Ren 00; Hayakawa, Matsuzawa, and Hara 006; Lu, Xe, and Wang 00; Qn and Gao 0). An mportant drawback of perodc transmsson s that t reures synchronzaton between the agents, that s, all agents need to transmt ther nformaton at the same tme nstants and, n some cases, t reures a conservatve samplng perod for worst case stuatons. In event-trggered broadcastng (Astrom and Bernhardson 00; Astrom 008; Tabuada 007; Wang and Lemon 008; Wang and Lemon 0; Donkers and Heemels 00; Garca and Antsakls 03; Anta and Tabuada 00) a subsystem sends ts local state to the network only when t s necessary, that s, only when a measure of the local subsystem state error s above a specfed threshold. Event-trggered control schemes offer a new pont of vew, wth respect to conventonal tme-drven strateges, on how nformaton could be sampled for control purposes. The semnal work (Astrom and Bernhardson 00) provded an nterestng comparson between conventonal tme drven samplng and the new event-drven samplng, emphaszng the practcal advantages of the latter. Tabuada (007) presented a trggerng condton based on norms of the state and the state error ex( t ) x( t), that s, the last measured state mnus the current state of the system, where the k measurement receved at the controller node s held constant untl a new measurement arrves. When ths happens, the error s set eual to zero and starts growng untl t trggers a new measurement update. The use of event-trggered control strateges n networked systems (Dmarogonas, Frazzol, and Johansson 0; Dmarogonas and Johansson 009; Dmarogonas and Frazzol 009; Yu and

3 Antsakls 0; Garca and Antsakls 0; Sun and El-Farra 0; Seyboth, Dmarogonas, and Johansson 03) provdes a more robust and effcent use of network bandwdth. Its mplementaton n MAS also provdes a hghly decentralzed way to schedule transmsson nstants whch does not reure synchronzaton compared to perodc sampled-data approaches. The work n the present paper s smlar to (Dmarogonas, Frazzol, and Johansson 0; Dmarogonas and Johansson 009; Dmarogonas and Frazzol 009) where the consensus problem wth sngle ntegrator dynamcs, event-based communcaton, and connected and undrected graphs was consdered. The man advantage of our approach compared to these papers s that we consder both the reducton of actuaton and communcaton updates whle they only focus on reducton of update nstants,.e. they stll assume that contnuous communcaton exsts among agents n order to calculate the error thresholds. Snce contnuous access to the states of neghbors s typcally not possble we extend the work n (Dmarogonas, Frazzol, and Johansson 0) to consder the exchange of nformaton among agents at dscrete tme nstants whch are, n general, non-perodc and based on local events. The present paper also provdes an mportant extenson to consder the case where the agents are able to transmt only a uantzed verson of ts measured state. Smlar work (Seyboth, Dmarogonas, and Johansson 03) uses a dfferent threshold that does not reure contnuous access to the states of neghbors. The approach n ths note preserves the decentralzed nature of the event computatons compared to (Seyboth, Dmarogonas, and Johansson 03) where an estmate of the second egenvalue of the Laplacan matrx (L) s used to trgger communcaton events. The communcaton polcy descrbed n the present paper s decentralzed n the sense that each agent computes ts transmsson nstants based on local nformaton. We provde asymptotc convergence to the ntal average usng the new threshold that consders only the last receved states of the neghbors. The polcy ensures strctly postve nter-event tmes. For the case when

4 uantzed measurements are used we are able to show convergence to a bounded regon around the ntal average; ths bound s proportonal to the uantzaton parameter. An extended scheme s also proposed n order to guarantee strctly postve nter-event tmes n the presence of uantzaton. The remander of ths document s organzed as follows: Secton addresses the event-trggered control strategy that consders lmted knowledge of states of neghbors. Secton 3 presents smlar results usng uantzed measurements. Secton 4 provdes llustratve examples and conclusons are gven n Secton 5.. Decentralzed Consensus We consder a set of n agents that are modeled as a sngle ntegrator: x u,... n. () where x s the state and u s the control nput assocated to agent. Snce contnuous measurements from neghbors are not avalable to each agent, then the control nput s obtaned usng the last measurements receved from each neghbor N as follows: u() t u( tk, t ) ( ( ) ( )),... k x tk x tk n () N where x( t k ) represents the last measurement transmtted by agent at ts update tme tk and N s the set of neghbors of agent. Smlarly, x t represents the last measurements receved from ( ) k neghbor at the correspondng tme t k. In general, the update ntervals are nonperodc and the update nstants for each agent are dfferent from those of other agents,.e. t and are not necessarly eual. The events are also computed based only on local nformaton, that s, events are desgned based on nformaton that s avalable to each agent. We propose the followng threshold: k t k

5 a( a N ) N e () t z ( tk, tk ) (3) where e() t x ( t ) x () t k represents the novel nformaton wth respect to the last transmtted measurement, 0 a (/ N ), 0, N s the cardnalty of, and N z( tk, t ) ( ( ) ( )). k x tk x tk (4) N In ths paper we use the notaton ( t, t ) to represents pecewse constant varables that are updated k at tmes t, when the local agent transmts an update, and also at all tmes t for k k N, when the agent receves an update from any of ts neghbors. At each node the updates of the pecewse constant versons of the states x( t k ) and x ( t ) k are as follows. When an event s trggered at tme t t k the local agent updates ts local pecewse constant verson of ts state usng the current measurement x () t,.e. x ( t ) ths measurement to ts neghbors. x t and transmts k () On the other hand, when the local agent receves an update from any of ts neghbors N at correspondng tmes t t k contanng a current measurement x () t, the local agent uses ths measurement to update ts pecewse constant verson of the state x, that s, x ( t ) x t. Note k () that () and (4) are functons of x( t k ) and all neghbors states x ( t ) for k N are updated at all correspondng tme nstants t and t. k k, therefore, they E. (3) s smlar to the threshold n [8]; however, the threshold n (Dmarogonas, Frazzol, and Johansson 0) s based on the contnuous varable z () t whch s gven by: z() t ( x() t x ()). t (5) N

6 It s clear that z () t s a functon of the contnuous measurements of local agent x () t, and t s also a functon of the contnuous measurements of all neghbors x () t. It s evdent that the local agent s not able to desgn ths threshold snce contnuous measurements from neghbors are not avalable. In ths work we try to reduce both the actuaton updates and the communcaton updates, whle the authors of (Dmarogonas, Frazzol, and Johansson 0) only consdered the reducton of actuaton updates assumng that the agents can have access to the contnuous states of ther neghbors. Therefore the threshold n (Dmarogonas, Frazzol, and Johansson 0) cannot be used n the present paper. When an event s trggered by agent we have e( t ) x ( t ) x ( t) x ( t ) x ( t ) 0 because k k k k t t k s an event tme for agent. We also have that a( a N ) N e () t z ( tk, tk ) (6) holds for any value of z ( t, t ). Note that the trggerng condton (3) guarantees that (6) s k satsfed. Let xt (... t ) [ x ( t )... x ( t )] T k kn k n kn represent the vector contanng the latest broadcasted updates by each agent n the network, that s, ths vector s a functon of all update tmes t k for = n. Assume that nput and communcaton delays are neglgble. The next result shows convergence for a group of agents usng the new threshold (3) under control (). Theorem. Consder a group of agents x u for = n, wth control nputs gven by () and wth event-based updates gven by (3). Assume that the communcaton graph s connected and undrected. Then all agents asymptotcally stablze to ther ntal average. Proof. Consder the ISS Lyapunov functon V (/ ) x T Lx. We have that

7 V x() t Lx () t x() t LLx( t... t ) x( t... t ) e() t LLx( t... t ) T T T T k kn k kn k kn z ( t, t ) ( e( t) e ( t)) z ( t, t ) k k N where x() t [ x ()... t x ()] t T and et () [ e ()... t e()] t T. By usng the neualty n n a a xy x y, for a>0, we have: a a k k k k k k a a N V z ( t, t ) N z ( t, t ) e ( t ) z ( t, t ) e ( t ). (7) By symmetry of the undrected communcaton graph and usng (6), we have: k a k V a N z ( t, t ) N e ( t) an z( t, t ) (8) whch mples V 0 for 0 a (/ N ) and 0. Because V 0, V 0 mples that V has a fnte lmt and V 0 as t. We have: Snce an 0lmV a N z ( t, t ) 0. (9) t 0 and (, ) 0 then k k k z t t an z( t, t ) 0 for = n. k k Thus, from (9), we have z ( t, t ) 0 as t for = n. In vew of (6) and (), when k k z ( t, t ) 0 then all errors e( t ) reset and reman eual to zero, that s, snce z ( t, t ) 0 as k t for = n, then we have that lm e ( t) 0 for = n. We can also wrte t k T T T T V x() t Lx () t x() t LL( x() t e()) t z() t z() t z() t Le() t 0. (0) Smlarly, Because lm e ( t) 0 t 0 lm lm ( ) T T V z t z( t) z( t) Le( t ). () t t for = n, t follows from (0) and () that

8 T lm zt ( ) zt ( ) lm z( t) 0 () t that s l m z ( t) 0 for = n. Recall the defnton of z () t n (5), we have t t lm ( x () t x ()) t 0 t N for = n whch can be wrtten n vector form as lm Lx( t) 0. (3) t When the nteracton graph s connected the Laplacan L has a smple zero egenvalue wth the assocated egenvector. Therefore n n lm x ( t) lm x ( t),,... n. (4) t t For undrected graphs t can be shown that the ntal average remans constant. Defne the average N x() t x () t, we have the followng: N N N xt () x () t ( x ( t) x ( t)) ( e ( t) e ( t)) 0 (5) N N and N xt () x(0) x(0), then the ntal average remans constant. The authors of (Dmarogonas, Frazzol, and Johansson 0) were able to show that at any gven tme there exsts at least one agent n the network for whch ts nter-event tme s strctly postve. In ths note we show that the nter-event tmes, not for at least one, but for all agents, are always strctly postve. Corollary. Consder a group of agents x u, = n, wth control nputs gven by () and wth updates (3). Assume that the communcaton graph s connected. Then the nter-event tmes for each agent = n, are strctly postve.

9 Proof. Consder the evoluton of the term e () t over the nterval t [ tk, tk ) when e () t s contnuous: d e () t e () t e () t e () t z ( t, t k ) () (, ) () (, ) k e t z t t k k ae t z t t a k k (6) dt and consder the dfferental euaton: (, ) (7) a z a tk t wth ntal condton ( t ) e ( t ) 0. Then, we have: k k t at ( ) a t k k k e () t () t e z () d, t [ t, t ). (8) A lower bound for the nter-event tmes of agent s obtaned by fndng the mnmum tme t such as () t z ( t, t ), where k k a( a N ) N 0. We analyze two cases here, the frst case s when z ( t, t ) 0 at the last update nstant. In k t k ths case, from (8), () t takes a fnte tme t>0 to grow from zero to z ( t, t ) snce k z ( t, t ) 0. The second case s when z ( t, t ) 0. In ths case () t 0for t[ t, t ), k k k t k t and we have that (6) holds, therefore agent does not generate any event durng that tme nterval. When agent receves an update from ts neghbors then z ( t, t ) 0 and the frst case holds,.e. k the error takes a fnte tme t>0 to grow from zero to z ( t, t ). k Remark. The selecton of threshold (3) s ntutve because t really s a functon of local nformaton and t s also related to how fast the error wll grow at any gven tme and trgger the next event. In fact, (), (3), and (8) tell us a clear pcture of the communcaton pattern. Because z ( t ) k s used n (), t determnes how fast the correspondng agent moves wth respect to ts prevous transmtted value and a proportonal threshold s used for the same agent as seen n (3). If

10 z ( t, t ) 0 for some agent at some update nstant k t then e () t x () t z ( t, t ) 0, ths k k means that the agent wll not move, the error remans eual to zero, and the current x () t remans eual to the last update x ( t ). It s clear that there s no need to send addtonal updates f the k current nformaton has not changed and no events should be trggered. Ths s the man reason that the error s compared usng strctly greater than n (3) nstead of eual as n (Dmarogonas, Frazzol, and Johansson 0). The man beneft s that we are able to lower bound the nter-event tmes, not for at least one agent, but for all of them. Recent work (Seyboth, Dmarogonas, and Johansson 03) proposes a dfferent threshold that does not reure contnuous access to the states of neghbors. The threshold s a functon of tme and other tunng parameters. The approach n ths note preserves the decentralzed nature of the soluton compared to (Seyboth, Dmarogonas, and Johansson 03) snce one of the tunng parameters depends on global nformaton,.e. on the second egenvalue of L. Algorthms for estmaton of the second egenvalue of the Laplacan have been presented n (Aragues, et.al. 0), (Franceschell, et.al. 009), and (Yang, et.al. 00). The algorthm n (Aragues et.al. 0) s especally practcal for mplementaton n the event-based approach n (Seyboth, Dmarogonas, and Johansson 03) snce the estmate of the second egenvalue always remans smaller than the true second egenvalue of L. Ths s a condton on the tunng parameter stated n (Seyboth, Dmarogonas, and Johansson 03) for convergence of the consensus algorthm. 3. Decentralzed Consensus wth Quantzaton It was assumed n the last secton that the sensor s able to m easure the state of the system wth nfnte precson. In realty, however, the measured varables have to be uantzed n order to be represented by a fnte number of bts to be used n processor operatons and to be transmtted over

11 a dgtal communcaton channel. In ths secton we study the effects of sgnal uantzaton on the convergence of the event-trggered control approach prevously descrbed n ths paper. We defne a unform uantzer as a functon : V such that: ( ) f ( ),( ) ( ) (9) where represents the uantzaton step and V {...,,0,,,...}. The above uantzer represents an nfnte rate, unform, passve uantzer wth bounded uantzaton error. The only varables that are avalable to compute the control nputs and the state errors for each agent are the uantzed states of the agents. The control nputs are now gven by: u () ( ( )) ( ( )),... t x tk x tk n (0) N and the uantzed state error s defned as follows: e () t ( x( t )) x ( ( t)),... n. () k Theorem 3. Consder a group of agents x u for = n, and each agent transmts t uantzed output x ( ( t k )) to ts neghbors at some tme nstants t k. The control nputs are gven by (0) and the event-based updates are trggered when ( e ( t)) a( a) N M ( t, t ) () k k s satsfed, where 0 a,, M (, ) ( ( )) ( ( )) tk t k x tk x t N. Assume that the communcaton graph s connected and undrected. Then all agents asymptotcally stablze to a bounded regon around ther ntal average gven by: lm x ( t) x(0),... n. (3) t

12 N N and the average remans constant,.e. xt () x () t x (0). Proof. Consder the canddate Lyapunov functon V V where x V ( ) d wth 0 V x ( ( t)) x ( t) x ( ( t)) u( t), for =,,n. Note that V 0 snce the seres nterconnecton of a sngle ntegrator and a passve memoryless functon s los sless (Khall 00). We have V u () t ( x ()) t ( x ( t )) ( x ( t )) ( x ( t )) e () t k k k N x ( ( tk )) x ( ( )) ( ( )) ( ( )) ( ( )) ( ) tk x tk x tk x tk e t N (4) and consder the followng relaton: N x ( ( t (5) k )) x ( ( )) ( ( )). tk x t N Then we have: V ( x( t )) ( x ( t )) e ( t) ( x( t )) ( x ( t )) k k N N M( tk, t ) ( ( )) (, ) a e t M tk t N a ( a) a M ( t, t ) N ( e ( t)) k k (6) where the neualty a a xy x y, for a>0, has been used to obtan the second lne n (6). By usng the threshold () we can guarantee that ( e ( t)) a( a) N M ( t, t ) k k (7) holds. Then we obtan V ( ) M ( t, t ) (8) ( a) whch mples V 0 for 0a and 0. k

13 Because V 0, V 0 mples that V has a fnte lmt and V 0 as t. We have the followng: 0 lm V ( ) M ( t, t ) 0. (9) t ( a) We also have that 0 and (, ) 0 k k M t t = n; conseuently M ( t, t ) 0 for k = n. From (9) we can see that ( ) M ( t, t ) 0 as t k k whch means that M ( tk, tk ) 0 n. Snce as t for = for = n. By defnton M( tk, t ) ( x ( t )) x ( ( tk )) k k uadratc terms. Then we have that 0 we have M ( t, t ) 0 as t k, whch conssts of a summaton of lm x ( ( t )) lm ( x ( t )),,... n. (30) t N In vew of (0), (7), and (30) all errors k k t lm e ( t) 0 for = n, whch s euval ent to t lm x ( ( tk )) lm x ( ( t)),... n (3) t e therefore, t follows from (30) and (3) that () t reset and reman eual to zero, that s, we have that t lm x ( ( t)) lm x ( ( t)),,... n. (3) t t It can be shown that for undrected graphs, and usng uantzaton of the states n ths case, the ntal average remans constant. Defne the average N x() t x () t, we have the followng: ( x ( t)) xt () x () t x ( ( t)) e () t e () t 0 (33) N N N N and xt () x(0) x (0). N N

14 Let lm ( x ( t)), from (3) and (33) we have that t x (0). (34) The last statement can be shown by contradcton. Assume that x(0) then, from (9), we have that ether x () t x(0) for = n, or x () t x(0) for = n, and n both cases (33) does not hold and we have a contradcton. From (34) and (9) we obtan (3) and the proof s complete. It s mportant to note that nter-event tmes are not lower bounded when usng uantzaton. It s stll possble to defne solutons for ths type of traectores n the sense of Krasowsk, as t s shown n (Ceragol, De Perss, and Frasca 0), by ntroducng deal sldng modes for traectores that contan accumulaton ponts. On the other hand, the computatons assocated wth the event trggered communcaton polcy reure contnuous sensng, uantzng, and computng and comparng errors and tme-varyng thresholds. In practce, all these operatons can be performed locally by each agent s processor unt freuently but not contnuously. Ths mplementaton dsassocates traectores from deal sldng modes and creates a chatterng effect. In order to prevent the undesred chatterng effect that may be present when a system transmts updates very freuently n the boundary of a uantzaton level and ts assocated Zeno behavor we * ntroduce a mnmum update nterval 0. The mnmum update nterval s useful not only for avodng Zeno behavor but also for a practcal mplementaton of a non-contnuous sensng and uantzng scheme. In the followng we relax the assumpton that the errors need to be calculated * contnuously; nstead, we ntroduce a samplng tme T, 0 T whch allows for a practcal mplementaton of the event trggered approach. Note that the samplng tme T s only used to check the error perodcally but communcaton between agents s stll event based snce at every samplng

15 tme each agent decdes f transmsson of nformaton s needed based on the sze of the current error. In selectng * we want to ensure that the error e( t) x( tk ) x ( ) t remans bounded n a desred regon e () t for the tme nterval t t t. Ths means that f an update s * [ k, ] k trggered by the -th agent at tme t k then we have: * e () t ( x ( t )) ( x ()) t, t[ t, t ] k k k (35) Let us consder n ths case a fxed threshold. An event s trggered when: e () t p (36) where p s an nteger snce e ( t) vares n ncrements of. In general, asymptotc convergence of the uantzed outputs s not acheved n ths case, but convergence to a bounded regon around the ntal average can be shown by evaluatng the dfference of the states of any one agent and the remanng agents n the network. Choose, wthout loss of generalty, an agent and re-label t as x c and the remanng agents as x x n-. Let A and L represent the adacency and the Laplacan matrces assocated wth the communcaton graph correspondng to the remanng agents. Corollary 4. Consder a group of agents x u for = n, and each agent transmts ts uantzed output x ( ( t )) to ts neghbors at some tme nstants t. The control nputs are gven by k k (0) and the event-based updates are trggered accordng to (36). Consder a sampled noncontnuous event mplementaton. Assume that the orgnal (before choosng an agent) communcaton graph s connected and undrected. Then all agents stablze to a bounded regon around ther ntal average and the followng s satsfed: when T s desgned to satsfy lm x[ ] x [ ] p G ( L dag{ A}) (37) c c

16 T N * mn mn /,, (38),..., n where A c s a row vector contanng the entres, other than the cc, a entry, n the c-th row of the T orgnal adacency matrx A and G [ A c L ]. Proof. Frst, gven control nputs (0), trggerng condton (36), and for any confguraton wth fnte ntal condtons and for fxed and connected topologes (such as the ones consdered here) there always exsts a fnte and postve constant S such that error e( t) x ( t ) x ( t) as follows: k x S. Consder the behavor of the Solvng (39) wth ntal condton ( t ) 0 we obtan d e () t e () t x () t. S (39) dt k e t t S * c (40) * * ( k ) ( ) k for 0<c<. Then * c 0 s a lower bound on the n S ter-event tmes that the -th agent uses to * broadcast ts measurements. Addtonally, usng the mnmum update nterval estmated by (40), we guarantee that x ( (t * )) ( x ( t )). Also note that the ntal average usng k k uantzaton remans constant as t was shown n Theorem 3. Defne [ ] x [ ] [ ], whch represents the dfference between the chosen agent and any x c other remanng agent at the T-dscretzed tme nstants ndexed by. We have the followng: [ ( [ ]) ( [ ]) [ ] ] x[ ] T xk x k x c N [ ] [ ] [ ] [ ] T e [ ] [ ] e [ ] x [ ] x [ ] c c N (4)

17 for = n-, where T t and [ ] x[ ] ( x[ ]). It s clear that n ths case the event k k tmes t take place at some of the dscrete tme nstants labeled as. k k Euaton (4) can be wrtten n a compact form: [ ] Q[ ] TG( [ ] e [ ]) X c [ ] (4) where QInTLT dag{ Ac}, Xc[] ( xc[ ] xc[]) n, [] [[]... []] T T n, [] [ c[], []... n []], e [] [ e [], e []... e []] T. The response of (4) to ntal condton [0] s gven by: c n The norm of [ ] satsfes: l [ ] Q [0 ] Q TG([ l] e [ l]) Q l X [ l ]. (43) l l c l [ ] Q [0] ( p ) T G Q T Q l0 l0 l. (44) Snce the orgnal communcaton graph s connected then agent x c has drected paths to all followers and 0T mn/ N then, by lemma 8.3 n (Ren and Cao 0), Q has all ts,..., n egenvalues wthn the unt crcle and lm Q 0. Addtonally, from Lemma.6 and Lemma.8 n (Ren and Cao 0), we have that l lm Q ( In Q) and l 0 p G L dag A c lm [ ] ( { }) (45) whch s euvalent to (37). Remark. Any agent n the network can be selected as x c resultng n dfferent expressons n (37) accordng to the remanng agents communcaton graph. The mnmum of these expressons holds as a bound n (37). Snce the ntal average s constant the agents converge around the ntal average.

18 Remark 3. Threshold (36) s constant once we choose a uantzaton parameter. Ths threshold choce makes sense because the error () vares n ncrements of. Addtonally, from (37), the regon of convergence can be reduced by choosng a smaller, by tradng off samplng-nter-event tme. 4. Examples Example. Consder eght agents exchangng uantzed measurements of postons usng a neglgble samplng tme for computng the error. The uantzaton parameter s =0.5. Smulaton results are shown n Fg.. Ths fgure shows that the non-uantzed states of the agents converge to a regon around the ntal average whch s In ths example all the uantzed states reach a common value and the bound (3) s satsfed. In addton, snce the uantzed states reach a common value, the agents do not move and no addtonal events are generated after approxmately 3 seconds as t can be seen n the center and rght plots of Fg., where the broadcastng perods for each agent are shown. States of agents 5 Broadcastng perods (-4) 4.5 Broadcastng perods (5-8) Tme (sec) Tme (sec) Tme (sec) Fg.. Quantzed consensus. Left: states of eght agents. Center: Broadcastng perods agents -4, agent (), agent (), agent 3(x), agent 4(+). Rght: Broadcastng perods agents 5-8, agent 5(), agent 6(), agent 7(x), agent 8(+).

19 States of agents Broadcastng perods (-4) Broadcastng perods (5-8) Tme (sec) Tme (sec) Tme (sec) Fg.. Quantzed consensus wth strctly postve nter-event tmes. Left: states of eght agents. Center: Broadcastng perods agents -4, agent (), agent (), agent 3(x), age nt 4(+). Rght: Broadcastng perods agents 5-8, agent 5(), agent 6(), agent 7(x), agent 8(+). Example. We consder the same system as n Example but the dfference s that we ntroduce a mnmum nter-event tme eual to 0.3 seconds whch also serves as a samplng nterval for calculatng the error. We select p= and =0.5. Smulaton results are shown n Fg.. In ths case the average of the non-uantzed states also remans constant over tme, The uantzed values do not reach a common value although the dfference of any par of them remans bounded. The agents keep sendng updates when they reach ths regon but usng event tmes eual or greater than the mnmum update nterval whch s 0.3 seconds as t can be observed n the center and rght plots of Fg.. The agents converge to a bounded regon around the ntal average eual to 4.85 and the bound (37) s satsfed. The mnmum theoretcal bound for ths example s eual 7 whch s conservatve snce, from Fg. and after transent response, the dfference between any two agents s less than Conclusons

20 Decentralzed control and broadcastng laws for consensus were presented n ths note. The man advantage of ths formulaton compared to smlar work s that we were able to reduce both actuaton and transmsson updates; contnuous montorng of states of neghbors s no longer needed. Asymptotc convergence to ntal average was shown. We offered an mportant extenson to consder transmsson of uantzed measurements. In order to avod Zeno behavor n ths case we ntroduced a strctly postve mnmum update nterval and convergence to a bounded regon around the ntal average was obtaned. References Anta, A., and Tabuada, P. (00), To sample or not to sample: Self-trggered control for nonlnear systems, IEEE Transactons on Automatc Control, 55, Aragues, R., Sh, G., Dmarogonas, D.V., Sagues, C., and Johansson K.H. (0), Dstrbuted algebrac connectvty estmaton for adaptve event-trggered consensus, n Proceedngs of the Amercan Control Conference, pp Astrom, K.J., and Bernhardson, B.M., (00), Comparson of Remann and Lebesgue samplng for frst order stochastc systems, n Proceedngs of the 4st IEEE Conference on Decson and Control, pp Astrom, K.J. (008) Event based control, n A. Astolf and L. Marcon, (eds.), Analyss and Desgn of Nonlnear Control Systems, pp Sprnger-Verlag, Berln. Cao, Y., and Ren, W. (009), Sample-data formaton control under dynamc drected nteracton, n Proceedngs of the Amercan Control Conference, pp Cao, Y., and Ren, W. (00), Mult-vehcle coordnaton for double ntegrator dynamcs under fxed undrected/drected nteracton n a sampled-data settng, Internatonal Journal of Robust and Nonlnear Control, 0, Ceragol, F., De Perss, C., and Frasca, P (0), Dscontnutes and hysteress n uantzed average consensus, Automatca, 47, Dmarogonas, D.V., Frazzol, E., and Johansson, K.H. (0), Dstrbuted event-trggered control for mult-agent systems, IEEE Transactons on Automatc Control, 57, Dmarogonas D.V., and Johansson, K.H. (009), Event-trggered control for mult-agent systems, n Proceedngs of Jont 48 th IEEE Conf. Decson and Control and 8 th Chnese Control Conference, pp Dmarogonas, D.V., and Frazzol, E. (009), Dstrbuted event-trggered control strateges for mult-agent systems, n Proceedngs of the 47 th Allerton Conference on Communcatons, Control, and Computng, pp Donkers, M.C.F., and Heemels, W.P.M.H. (00), Output-based event-trggered control wth guaranteed Lnfty-gan and mproved event-trggerng, n Proceedngs of the 49th IEEE Conf. Decson and Control, pp Fr anceschell, M., Gasparr, A., Gua, A., and Seatzu, C. (009), Decentralzed Laplacan egenvalues estmaton for th th networked mult-agent systems, n Proceedngs of Jont 48 IEEE Conf. Decson and Control and 8 Chnese Control Conference, pp Garca, E., and Antsakls, P.J. (03), Model-based event-trggered control for systems wth uantzaton and tmevaryng network delays, IEEE Transactons on Automatc Control, 58, Ga rca, E., and Antsakls, P.J. (0), Decentralzed model-based event-trggered control of networked systems, n Proceedngs of the Amercan Control Conference, pp

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