Multi-Agent Trajectory Tracking with Self-Triggered Cloud Access

Transcription

1 2016 IEEE 55h Conference on Decision and Conrol CDC ARIA Resor & Casino December 12-14, 2016, Las Vegas, USA Muli-Agen Traecory Tracking wih Self-Triggered Cloud Access Anonio Adaldo, Davide Liuzza, Dimos V. Dimarogonas, Member, IEEE and Karl H. Johansson, Fellow, IEEE Absrac This paper presens a cloud-suppored conrol algorihm for coordinaed raecory racking of neworked auonomous agens. The moivaing applicaion is he coordinaed conrol of Auonomous Underwaer Vehicles. The conrol obecive is o have he vehicles rack a reference raecory while keeping an assigned formaion. Raher han relying on iner-agen communicaion, which is inerdiced underwaer, coordinaion is achieved by leing he agens inermienly access a shared informaion reposiory hosed on a cloud. An even-based law is proposed o schedule he accesses of each agen o he cloud. We show ha, wih he proposed scheduling of he cloud accesses, he agens achieve he required coordinaion obecive. Numerical simulaions corroborae he heoreical resuls. I. INTRODUCTION Neworked vehicle sysems have araced a noable amoun of research in he pas few decades [1] [3]. In mos applicaions, employing a eam of vehicle agens insead of a single plaform has numerous advanages. For example, a group of agens usually provides robusness wih respec o he failure of a single agen in he group. When sampling a propery in a region of he space, a eam of mobile agens will provide a larger number of samples and increased daa redundancy. Also, cerain asks may be srucurally impossible o perform wih a single agen. However, he use of a flee ineviably brings abou he problem of coordinaing he vehicles. Muli-agen coordinaion is paricularly challenging in he case of Auonomous Underwaer Vehicles AUVs because of heir limied communicaion, sensing and localizaion capabiliies [4], [5]. AUVs have numerous applicaions, including, o name us a few, oceanographic surveys, mine search, inspecion of underwaer srucures, and measuremen of chemical properies in a waer body [6]. Underwaer communicaion may be implemened by means of acousic modems, bu such modems are nooriously expensive, power-hungry and limied in boh radius and bandwidh. Underwaer posiioning is also difficul, since good inerial sensors are very expensive, and acousic posiioning sysems A. Adaldo, D. V. Dimarogonas and K. H. Johansson are wih he deparmen of Auomaic Conrol and ACCESS Linnaeus Cener, School of Elecrical Engineering, KTH Royal Insiue of Technology, Osquldas väg 10, 10044, Sockholm, Sweden; s: {adaldo,dimos,kalle}@kh.se. D. Liuzza is wih Universiy of Sannio in Beneveno, Deparmen of Engineering, Piazza Roma 21, 82100, Beneveno, Ialy; davide.liuzza@unisannio.i. This work has received funding from he European Union Horizon 2020 Research and Innovaion Programme under he Gran Agreemen No , AEOROWORKS, from he Swedish Foundaion for Sraegic Research, from he Swedish Research Council, and from he Knu och Alice Wallenberg foundaion. have a limied range. Moreover, GPS is no available underwaer, and a vehicle has o surface whenever i needs o ge a posiion fix [7]. To deal wih communicaion consrains, even- and self-riggered conrol designs [8] can be applied o neworked muli-agen sysems [9]. In his paper, selfriggered muli-agen conrol is considered in combinaion wih he suppor of a shared informaion reposiory hosed on a cloud. Namely, he cloud is inermienly accessed by he agens according o a self-riggered proocol. Since direc communicaion among he agens is inerdiced when hey are underwaer, he agens only exchange daa hrough he cloud reposiory, which is accessed asynchronously. The moivaing applicaion is a leader-following raecory racking ask for a formaion of AUVs subec o disurbances. In he radiional even- and self-riggered neworked conrol, oneo-one communicaion needs o be esablished a leas when an agen needs o updae is conrol signal. Conversely, wih cloud-based approaches [10] [12], he vehicles exchange informaion wihou opening a communicaion channel beween each oher. Each vehicle simply leaves is informaion on he cloud for he ohers o download laer. A cloudsuppored conrol archiecure for muli-agen coordinaion was proposed by he auhors in [12], where he problem of driving a eam of vehicles o a saic formaion is addressed. In his paper, he approach is furher developed o address muli-agen leader-follower racking problems, under more general nework opologies. Two differen coordinaion obecives, namely pracical and asympoic convergence o a given formaion, are formulaed mahemaically, and graphheoreical resuls are used o show ha he proposed cloudsuppored sraegy achieves he desired obecives, despie only using oudaed informaion. Edge-space analysis [13], [14] is used o address direced nework opologies, hus allowing for leader-follower coordinaion. II. PRELIMINARIES In his paper, denoes he euclidean norm of a vecor or he corresponding induced norm of a marix. Moreover, {A} i,k denoes he enry of A in he i-h row and k-h column, while {A} i denoes he row vecor corresponding o he i-h row of A. The null vecor in R n is denoed as 0 n. The se of he posiive inegers is denoed as N, while N 0 = N {0}. A digraph is a uple, wih = {1,, N} and {, i i,, i }. The elemens of and are called respecively verexes and edges of he graph. A pah from verex o verex i is a sequence of verexes saring wih and ending wih i such ha any wo consecuive verexes in he sequence consiue an edge. A spanning ree is a digraph /16/$ IEEE 2207

2 , wih N 1 edges such ha here exiss a node r wih a pah o any oher node. The node r is called he roo of he spanning ree. A digraph, is said o conain a spanning ree if, is a spanning ree for some subse of. Consider a digraph,, and le he edges be denoed as = {e 1,, e M }. The incidence marix of he digraph is defined as B R N M such ha 1 if e k =, i for some, {B} i,k = 1 if e k = i, for some, 0 oherwise. The in-incidence marix is defined as B R N M such ha {B } i,k = {B} i,k if {B} i,k {0, 1} and {B } i,k = 0 if {B} i,k = 1. For a spanning ree, he edge Laplacian [14] is defined as L e = B B. For a digraph, ha conains a spanning ree, bu is no iself a spanning ree, le =, wih, being a spanning ree. Wihou loss of generaliy, le = {e 1,, e N 1 } and = {e N,, e M }. Pariion he incidence and in-incidence marices accordingly as B = [B, B ] and B = [B, B ] respecively. Then, B has a lef-pseudoinverse B [15], and he reduced edge Laplacian is defined as L r = B B + B B B. 1 For any spanning ree, he edge Laplacian is posiive definie, while for any graph ha conains a spanning ree, bu is no iself a spanning ree, he reduced edge Laplacian is posiive definie [14]. III. PROBLEM SETTING Consider a muli-agen sysem composed of N agens indexed as i {1,, N} =, wih kinemaics described by { x i = u i + d i 0, i, 2 x i 0 = x i,0 i. where x i R n represens he sae of agen i, x i,0 is he iniial sae, u i represens a conrol inpu, and d i represens a disurbance signal. The conrol obecive is o have all he agens follow a desired reference raecory r wihin a given olerance. Such conrol obecive is referred o as pracical consensus o he raecory r, and can be formalized as follows. Definiion III.1. The muli-agen sysem 2 is said o achieve pracical consensus o he reference raecory r wih olerance ε > 0 if lim sup x i r ε for all i. Remark III.1. In erms of our moivaing applicaion, we have n = 2, each agen represens an AUV, and x i + b i R 2 represens a waypoin for vehicle i, where b i is a consan bias vecor. In his way, pracical consensus of x 1,, x N corresponds o convergence of he vehicles o a formaion abou r defined by he bias vecors b 1,, b N. However, he analysis remains valid for any n N. The reference raecory can be measured only by a subse of he agens, which are referred o as he leaders in he muli-agen sysem. In his work, we assume ha he agens canno exchange any direc informaion wih each oher. This models he scenarios where, as in our AUV seup, communicaion among he agens is physically inerdiced. In order o exchange informaion, he agens can only upload and download daa on a shared reposiory hosed on a cloud. Namely, when i is conneced o he cloud, an agen can deposi some informaion, and, a he same ime, download some informaion ha was previously uploaded by some oher agens. On he oher hand, when i is no conneced o he cloud, an agen canno exchange informaion a all. For he purposes of his work, an agen s access o he cloud can be considered an insananeous even, while communicaion proocol problems, such as delays and packe losses, are lef ou of he scope of his work. The cloud is modelled as a shared resource wih limied hroughpu and sorage capaciy, and hanks o he conrol algorihm ha we are going o define, i is accessed only inermienly and asynchronously, and he amoun of daa sored herein does no grow over ime. For each agen i, we define he sequence { i,k } k N0 of he agen s accesses o he cloud. Namely, i,k wih k N denoes he ime when agen i accesses he cloud for he kh ime, while convenionally i,0 = 0 for all i. When an agen i accesses he cloud a ime i,k, i also riggers a measuremen of is curren sae, which we denoe as x i,k : x i,k = x i,k. 3 If agen i is a leader, i also produces a measuremen of he curren value of he reference raecory, which we denoe as r i,k : r i,k = r i,k. 4 The disurbance signals and he reference raecory saisfy he following assumpion. Assumpion III.1. The disurbance signals d i in 2 and he reference raecory r saisfy d i δ i and r δ 0, where δ i = δ i,0 δ i, e λ δ +δ i,, i {0, 1,, N}, and δ i,0, δ i,, λ δ, are known non-negaive consans for all i {0, 1,, N}. Our goal is o propose a conrol sraegy such ha each agen uses he informaion acquired from he cloud o aain pracical consensus as by Definiion III.1. To compleely specify he conrol sraegy, we need o define: he conrol signals for each agen, he informaion ha is uploaded and downloaded by each agen when accessing he cloud, and a law for scheduling he cloud accesses. Firs, we define he conrol signals u i. In he proposed conrol sraegy, each signal u i is piecewise consan, and i is updaed upon agen s i cloud accesses, i.e., u i = u i,k [ i,k, i,k

3 AGENT TIME POSITION CONTROL NEXT 1 1,l1 x 1,l1 u 1,l1 1,l ,l2 x 2,l2 u 2,l2 2,l2 +1 N N,lN x N,lN u N,lN N,lN +1 Table III. Schemaic represenaion of he daa sored in he cloud a a generic ime insan 0. Namely, he conrol signals are compued as follows: u i,k = c p i r i,k x i,k + x i,k x i,k, 6 i where c > 0 is a conrol gain, p i = 1 if i and p i = 0 oherwise, i {i}, and x i,k is an esimae of he sae of agen done by agen i a ime i,k. Such esimae is defined laer in his secion. Nex, we define he informaion uploaded and downloaded by each agen when accessing he cloud. When agen i accesses he cloud a ime i,k, i uploads: he curren ime i,k, he measuremen of is curren sae x i,k, he conrol signal u i,k ha is going o be applied unil he following access, and he scheduled ime i,k+1 of he following access. When hese values are uploaded, hey overwrie hose ha were uploaded by he same agen upon he previous access. In his way, he amoun of daa conained in he cloud remains consan. Namely, for each agen, he cloud conains he informaion ha was uploaded upon ha agen s mos recen access. Denoing as l i he index of he mos recen access of agen i before ime, i.e. l i = max{k N 0 i,k }, a abular represenaion of he daa conained in he cloud a a generic ime is given in Table III. Before uploading is own informaion, agen i downloads and sores he informaion corresponding o he agens i. Such informaion is used by agen i o consruc he esimaes x i,k for i ha are used for compuing he conrol signal 6, and also o schedule is following access o he cloud. Namely, he esimae x i,k is compued as follows: x i,k = x,l i,k + u,l i,k i,k,l i,k. 7 Noe ha such esimae coincides wih he sae ha agen would have a ime i,k if no disurbances were acing on i in he ime inerval [,l i,k, i,k. Finally, le us define he rule for scheduling he agens accesses o he cloud. Each agen schedules is own accesses recursively, according o he following rule: i,k+1 = inf{ > i,k Δ i,k ζ i,k σ i,k ς i }, 8 Δ i,k = δ i,k i τdτ, 9 { } ςq ζ i,k = min, 10 q i q 2c q σ i,k =c i + p i u i,k i,k u,h min{,,h +1} i,k i + i + p i Δ i,k + p i δ 0 τdτ i,k + Δ,h + μ τdτ i,h +1 i >,h +1, 11 ς i =ς i,0 ς i, e λ ς +ς i,, 12 where ς i,0, ς i, and λ ς are given non-negaive consans for all i, h = l i,k, δ i and ρ are defined in Assumpion III.1, and μ τ is a bounded scalar signal o be given laer in he paper. The expression of C i,k emerges from he analysis conduced in he following secion, and herefore, will be clarified laer. Noe however ha funcions 9 12 can be compued locally by agen i a ime i,k by using he informaion acquired from he cloud a ha ime, and knowing q for q i q. The funcions ς i wih i are referred o as hreshold funcions, since ς i a hreshold ha σ i,k mus overcome o rigger he cloud access i,k+1 of agen i. Remark III.2. In erms of our moivaing applicaion, an agen s accesses he cloud correspond o he imes when an AUV comes o he waer surface. A posiion measuremen corresponds o GPS fix ha a vehicle can obain while on he waer surface. On he oher hand, when a vehicle is underwaer, i canno communicae wih oher vehicles or access GPS. Neverheless, i has o find a conrol value and he nex surfacing insan coping wih he fac ha in he fuure oher vehicles will surface and updae heir conrol inpu o a ye unknown value. Remark III.3. The fundamenal difference beween he proposed conrol sraegy and he maoriy of he exising selfriggered coordinaion sraegies for muli-agen sysems is ha, in he proposed sraegy, an agen does no require oher agens o exchange informaion when i needs o updae is conrol signal. Conversely, when an agen needs o updae is conrol signal, i uses he informaion ha is already available in he cloud, i.e., ha was previously uploaded by he oher agens upon heir own access imes. IV. MAIN RESULT In his secion, we show how he muli-agen sysem 2, under he conrol algorihm defined by 5 12, can achieve pracical consensus o he reference raecory as by Definiion III.1. Firs, we need o inroduce a digraph induced by he ses and i wih i ha capures he opology of he informaion exchanges processed hrough he cloud. Definiion IV.1. Consider he muli-agen sysem 2 under he conrol law 6. Le = {0} and = {, i i, i } {0, i i }. We say ha he digraph, is he digraph associaed wih he muli-agen sysem. Moreover, we denoe he edges of he digraph as = {e 1,, e M }. Noe ha, i for some i, if and only if agen i downloads he informaion uploaded by agen 2209

4 , while 0, i if and only if agen i is a leader, i.e., if i receives informaion abou he reference raecory. Therefore, he digraph, represens he opology of he informaion exchanges ha are processed hrough he cloud. The following assumpion ensures ha he informaion abou he reference raecory can reach all he agens in he sysem. Assumpion IV.1. The digraph, associaed wih he muli-agen sysem 2 conains a spanning ree wih roo in he verex 0. Namely, we le, wihou loss of generaliy, =, where = {e 1,, e N 1 }, = {e N,, e M }, and, is a spanning ree wih roo in he verex 0. For each edge e l =, i, wih l {1,, M}, we le y l = x x i if, and y l = r x i if = 0. In oher words, y l is he mismach beween he saes of he wo agens whose indexes appear in he edge e l. Le y = [y 1,, y N 1 ], y = [y N,, y M ], and y = [y, y ]. 13 Le B and B be respecively he incidence and inincidence marices of,, and le hem be pariioned as B = [B, B ] and B = [B, B ] according o how is pariioned ino and. Noe ha, leing x = [r, x 1,, x N ], we have y = B I nx 14 y = B I n x. 15 Under Assumpion IV.1, B has a lef-pseudoinverse B see [15] for furher deails, herefore, from 14 and 15, i follows ha y = B B I n y. 16 Finally, le he reduced edge Laplacian L r of, be defined as in Secion II. If, is iself a spanning ree, le y = y, B = B and L r = L e, where L e is also defined in Secion II. Nex, le us inroduce some signals which shall be used in he convergence analysis. Consider he signals v i = c p i r x i + x x i, 17 i for all i. Noe ha v i can be obained from 6 by subsiuing he measuremens r i,k, x i,k and he esimaes x i,k respecively wih r i, x i and x. Le v = [0 n, v 1,, v N ] so ha we can rewrie 17 compacly as v = cb I n y. 18 Le u i be he mismach beween he acual conrol inpu u i and v i for each i, i.e., u i = u i v i, 19 and le u = [0 n, u 1,, u N ]. We are now in he posiion o sae our firs convergence resul. Theorem IV.1. Consider he muli-agen sysem 2, and le Assumpions III.1 and IV.1 hold. If u i ς i for all [0, and all i, hen here exis α, λ > 0 such ha y η for all [0,, where η =α η 0 e cλ + B e cλ τ δτ + ςτ dτ, 0 20 η 0 = y 0, δ = [δ 0, δ 1,, δ N ] and ς = [0, ς 1,, ς N ]. Proof. Subsiuing 19 ino 2, we have x i = v i + u i + d i. 21 Leing d = [ r, d 1,, d N ], 21 can be rewrien compacly as x = v + u + d. 22 Lef-muliplying boh sides of 22 by B I n, and using 14 and 18 and he properies of he Kronecker produc, we have y = cb B I n y B I n u + d, 23 Subsiuing 13 ino 23, observing ha 16 holds hanks o Assumpion IV.1, and using he properies of he Kronecker produc, we have y = cl r I n y B I n u + d, 24 where L r is he reduced edge Laplacian of,, as defined in 1. The Laplace soluion of 24 reads y = e cl y 0 e cl τ B u + ddτ, 0 25 where we have denoed L = L r I n and B = B I n for breviy. Taking norms of boh sides in 25, and using he riangular inequaliy, he properies of he Kronecker produc, Assumpion III.1, and he hypohesis u i ς i for all [0, and all i, we have y e cl y 0 + B e cl τ δτ + ςτ dτ, 0 26 for all [0,, where δ and ς are defined in he heorem saemen. Since L r is posiive definie, L = L r I n is Hurwiz, and herefore here exis α, λ > 0 such ha e cl α e cλ The proof is concluded by subsiuing 27 ino 26. Remark IV.1. The posiive scalar λ mus be smaller han min{eigl r }, bu can be chosen as close o ha as desired. If L r is diagonalizable, one can choose λ = min{eigl r } and α = V V 1, where L r = V ΛV 1 and Λ is diagonal [16]. Corollary IV.1. Under he same hypoheses as Theorem IV.1, we have u i μ i, for all [0, and all i, where μ i = β i η + ς i,

5 β i = c {B + B B B } i, and η is defined in 20. Proof. Subsiuing 13 ino 18, and using 16, we have v i = c{b + B B B } i I n y. Taking norms of boh sides, and using he Cauchy-Swarz inequaliy, we have v i β i y. 29 From 19, using he riangular inequaliy, we have u i v i + u i. 30 Using 29 and he hypohesis u i ς i ino 30 concludes he proof. The nex sep in our analysis is o show ha he condiion u i ς i holds for all 0 and all i if he conrol algorihm defined by 5 12 is applied wih η given by 20. This is formalized in he following heorem. Theorem IV.2. Consider he muli-agen sysem 2, le Assumpions III.1 and IV.1 hold, and le he sysem be conrolled by he algorihm defined by 5 12, 20 and 28. Then he closed-loop sysem does no exhibi Zeno behavior and u i ς i holds for all 0 and all i. Proof. Subsiuing 6 and 17 ino 19, we have u i =c p i r i,k x i,k + x i,k x i,k i p i r x i x x i i 31 for [ i,k, i,k+1, where x i,k, r i,k and x i,k are defined in 3, 4 and 7 respecively. Reordering he erms in 31, we have u i =c i + p i x i x i,k cp i r r i,k c x x i,k 32 i for [ i,k, i,k+1. Firs consider he erm x i x i,k in 32. Inegraing 2 in [ i,k,, and using 3, 5 and 6, we have x i x i,k = u i,k i,k + i,k d i τdτ. 33 Now consider he erm r r i,k in 32. Using 4, we can wrie r r i,k = i,k rτdτ. 34 Finally, consider he erms x x i,k in 32. For hese erms we need o disinguish wo cases, namely,h +1 and >,h +1. Noice ha he laer case corresponds o he fac ha agen updaes is conrol inpu o a value unknown o agen i. In he firs case, inegraing 2 for agen in [,h,, using 3 and 7, and noing ha u = u,h for [,h,,h +1, we have x x i,k =u,h i,k +,h d τdτ, [,h,,h In he second case, similar observaions lead o x x i,k =u,h,h +1 i,k +,h +1 u τdτ + Subsiuing ino 32 yields u i =c i + p i u i,k i,k u,h min{,,h +1 } i,k i + i + p i d i τ p i rdτ i,k d τdτ,h i i,h +1,h d τdτ, >,h +1.,h u τdτ. 37 Taking norms of boh sides in 37, and using he riangular inequaliy and Assumpion III.1, we have u i c i + p i u i,k i,k u,h min{,,h +1 } i,k i + i + p i Δ i,k + p i δ 0 τdτ i,k + Δ,h + u τ dτ i,h +1 i,h +1, 38 for [ i,k, i,k+1. Nex, suppose by conradicion ha some agen i a some ime [ i,k, i,k+1 aains u i > ς i, while u q ς q for all [0, and all q. Then, using Corollary IV.1, we have u τ β ητ + ς τ [0, i. 39 Evaluaing 38 for =, and using 39, we have u i σ i,k, 40 where σ i,k is defined in 11. By 40, u i > ς i implies σ i,k > ς i. Bu his is a conradicion, since he scheduling rule 8 12 and 20 guaranees ha σ i,k ς i for all [ i,k, i,k+1 and all k N 0. To exclude ha he sysem exhibis Zeno behavior, consider he condiions 8 ha rigger he cloud accesses. From 9, we see ha he riggering condiion Δ i,k ζ i,k requires i,k ς q, 2cδ i,0 q for some q, and 2211

6 herefore, i canno generae Zeno behavior. Nex, consider he condiion σ i,k ς i. Evaluaing 11 for = i,k we have σ i,k i,k = c i Δ,h i,k. 41 Recalling ha i,k [,h,,h +1, and noing ha 8 guaranees Δ,h < ς i 2c i for all [,h,,h +1, from 41 we have σ i,k i,k ς i i,k Differeniaing boh sides of 11, and using 9, he coninuiy of σ i,k and he riangular inequaliy, we have σ i,k σ i,k i,k + i,k s i,k τdτ, 43 where s i,k =c i + p i u i,k + δ i + p i δ 0 + <,h +1 u,h + i δ +,h +1 μ 44 From Corollary IV.1, we have u i,k = u i μ i and u,h = u μ for all i such ha <,h +1. Since μ = β η+ς is upper-bounded, δ δ,0 and ρ ρ 0 for all 0, from 44, we have s i,k c i + p i μ i + δ i,0 + p i ρ 0 + μ + δ,0, 45 where μ denoes he maximum value aained by he funcion μ. Denoing he righ-hand side of 45 as s i,k, and subsiuing 42 and 45 ino 43, we have σ i,k ς i i,k 2 + s i,k i,k 46 From 46, a necessary condiion for having σ i,k ς i is ς i i,k 2 + s i,k i,k ς i. 47 Observing ha ς i = ς i i,k ς i, e λ ς i,k +ς i,, we can rewrie 47 as ς i i,k 2 + s i,k i,k ς i i,k ς i, e λ ς i,k +ς i,. For any ς i i,k > ς i, 0 and any λ ς > 0, he posiive soluions in he unknown τ of he equaion ς i i,k 2 + s i,k τ ς i i,k ς i, e λςτ +ς i, is lower-bounded. Therefore, condiion 47 canno be saisfied for i,k arbirarily small. Consequenly, he riggering condiion ς i,k ς i canno generae Zeno behavior eiher. We can conclude ha he closed-loop sysem defined by 2, 5 12 and 20 does no exhibi Zeno behavior. Remark IV.2. Agen i can compue μ for by 20 and 28, and herefore by only using some neighborhood informaion on he nework opology β for and he iniial condiions y 0. Theorems IV.1 and IV.2 amoun o our main resul, which is formalized as follows. Theorem IV.3. Consider he muli-agen sysem 2, le Assumpions III.1 and IV.1 hold, and le he sysem be conrolled by he algorihm defined by 5 12 and 20. Then he closed-loop sysem does no exhibi Zeno behavior and achieves pracical consensus as by Definiion III.1, wih olerance ε = max i { m i }η, where m i is he number of edges in he shores pah from verex 0 o verex i in he graph,, and η = lim η = α B δ +ς, where δ cλ = [δ 0,, δ 1,,, δ N, ] and ς = [0, ς 1,,, ς N, ]. Proof. From Theorems IV.1 and IV.2, we have y η for all 0, where η is defined by 20. Leing, we have herefore lim sup y η. Finally, observing ha r x i m i y yields he desired resul. V. ASYMPTOTIC CONVERGENCE If he disurbances vanish quickly enough, and he reference raecory converges quickly enough o a fixed poin, hen he proposed algorihm, wih only small adusmens, is capable o drive all he agens o he reference poin asympoically. In his case, he following assumpion is needed. Assumpion V.1. Assumpion III.1 holds wih ρ, δ 1,,, δ N, = 0 and λ δ < c min{eigl r }. wih In his scenario, he hreshold funcions are chosen as ς i = ς i,0 e λ ς, 48 0 < λ ς < λ δ < c min{eigl r }. 49 Noe ha Theorem IV.1 and Corollary IV.1 sill hold. Moreover, solving he inegral in 20, and using Assumpion V.1 and 49, we have η η e λ ς, 50 where η = α y 0 + δ0 + ς0, 51 cλ λ δ cλ λ ς and δ, ς are defined in he saemen of Theorem IV.1. In he following heorem we show ha his version of he proposed algorihm is sill Zeno-free. Theorem V.1. Consider he muli-agen sysem 2, le Assumpions IV.1 and V.1 hold, and le he sysem be conrolled by he algorihm defined by 5 11, 20 and 48, wih λ ς saisfying 49. Then, he closed-loop sysem does no exhibi Zeno behavior and u i ς i holds for all 0 and all i. Proof. Reasoning as in Theorem IV.2, we can show ha he scheduling rule 8 11, 20 and 48 guaranees u i ς i for any 0. To show ha he closed-loop sysem is Zeno free, consider again he condiion 8 ha riggers he cloud accesses. From 9, we can see ha he riggering condiion Δ i,k ζ i,k requires δ q,0 e λ δ i,k 1 e λ δ i,k λ δ 2212

7 ς q,0 2c q e λ ς i,k e λ ς i,k for some q. By 49, he previous inequaliy implies ς q,0 2c q + δ q,0 e λ ς i,k δ q,0 λ δ Therefore, he condiion Δ i,k ζ i,k canno generae Zeno behavior. Nex, consider he condiion σ i,k ς i. Wih similar reasoning as in Theorem IV.2, we can show ha 42 and 43 sill hold. Using Corollary IV.1 and 48 and 50, and recalling ha [ i,k, i,k+1, we have u i,k = u i μ i β i η + ς i,0 e λ ς and u,h = λ δ. u μ β η + ς,0 e λ ς for all i such ha <,h +1. Replacing hese wo inequaliies ino 44, and using 48 50, we have s i,k c i + p i β i η + ς i,0 + δ i,0 + p i ρ β η + ς,0 e λ ς [ i,k, i,k+1, where η is defined in 51. Subsiuing 42 and 52 in 43, we have σ i,k ς i i,k 2 + ξ i,k λ ς e λ ς i,k 1 e λ ς i,k, 53 where ξ i,k denoes he coefficien ha muliplies e λ ς in 52. From 53, a necessary condiion for having σ i,k ς i is ς i i,k 2 + ξ i,k λ ς e λ ς i,k 1 e λ ς i,k ς i. 54 Observing ha ς i = ς i i,k e λ ς i,k and ς i i,k = ς i,0 e λ ς i,k, 54 can be rewrien as ς i,0 2 + ξ i,k ς λ i,0 + ξ i,k e λ ς i,k, ς λ ς which has lower-bounded soluions in he unknown i,k. Hence, he riggering condiion σ i,k ς i canno generae Zeno behavior eiher. We can conclude ha he closed-loop sysem does no exhibi Zeno behavior. Theorems IV.1 and V.1 amoun o our asympoic convergence resul, which is formalized as follows. Theorem V.2. Consider he muli-agen sysem 2, le Assumpions IV.1 and V.1 hold, and le he sysem be conrolled by he algorihm defined by 5 11, 20 and 48, wih λ ς saisfying 49. Then he closed-loop sysem does no exhibi Zeno behavior and lim r x i = 0 for all i. Proof. Reasoning as in Theorem IV.3, we have r x i m i η, where m i is defined in Theorem IV.3. From 50, leing in he previous inequaliy yields he desired convergence resul. VI. NUMERICAL SIMULATIONS In his secion, we presen wo numerical simulaions of he proposed algorihm, which demonsrae respecively pracical consensus and asympoic convergence. We consider a sysem made up of N = 4 agens plus he reference raecory, wih graph opology as in Figure 1. Noe ha for Fig. 1. Topology of he muli-agen sysem used in he simulaions. The node r represens he reference raecory r 1 x 1 1 x1 2 x1 3 x Fig. 2. Resuls of he firs simulaion scenario. Top: firs posiion variable x 1 i for each agen and r 1 for he reference raecory. Boom: error norm x i r for each agen. his opology Assumpion IV.1 is saisfied, and we have λ = min{eigl r } = 0.53, B = 1.90 and α = For he firs simulaion, we le c = 1.8, ς = [0.0, , 0.48, 0.57, 0.67] 5.0 e , δ = [99.9, 1.0, 2.0, 3.0, 4.0] 10 3 for all 0. We le he derivaive of he reference raecory and he disurbances be sinusoidal, namely r = δ 0 [cos2πf 0, sin2πf 0 ], d i = δ i [cos2πf i, sin2πf i ]. Noe ha wih his choice Assumpion III.1 is saisfied. The frequencies are chosen as f i = i T for i {0, 1,, N}, where T is he simulaion ime. The simulaion runs for [0.0, 2.5], wih a fixed sep of 10 4 he physical ime scale can be chosen according o he paricular applicaion. Figures 2 and 4 show he resuls of his simulaion. We have a oal of 305 updaes corresponding o an average iner-updae ime of 0.033, wo order of magniudes larger han he simulaion sep. Hence, he simulaion corroboraes he absence of Zeno behavior. For he second simulaion, we le c = 1.0, ς = [0.0, , 0.48, 0.57, 0.67] 10.0 e 1.0, δ = [99.9, 1.0, 2.0, 3.0, 4.0] 10 3 e 1.1. Noe ha, wih hese choices, Assumpion V.1 and 48 are saisfied, and he derivaive of he reference raecory and he disurbances vanish asympoically. The simulaion runs for [0.0, 10.0], wih he same sep as before. Figures 3 and 4 show he resuls of his simulaion. In his case, we have a oal of 1418 updaes, corresponding o an average inereven ime of , sill wo order of magniudes larger han he simulaion sep. Hence, he simulaion corroboraes 2213

8 i i Fig. 4. Top: updaes in he firs simulaion scenario for each agen. Boom: updaes in he second simulaion scenario for [9.0, 10.0] for each agen r 1 x 1 1 x1 2 x1 3 x Fig. 3. Resuls of he second simulaion scenario. Top: firs posiion variable x 1 i for each agen and r 1 for he reference raecory. Boom: error norm x i r for each agen. he absence of Zeno behavior. VII. CONCLUSIONS AND FUTURE DEVELOPMENTS A cloud-suppored conrol algorihm for leader-follower raecory racking in a nework of mobile agens under disurbances has been proposed. The considered seup allows muli-agen coordinaion in case of inerdiced communicaion among he agens. Specifically, he scenario of conrolling a formaion of AUVs has been considered as a moivaing example. The conrol algorihm overcomes he limiaion of having a pre-assigned raecory for he whole flee as well as synchronizing he surfacing of he agens [7]. Sufficien condiions for boh bounded and asympoic convergence have been idenified, in erms of he opology of he informaion exchanges wih he cloud, and of he scheduling of he conrol updaes. The proposed conrol algorihm is effecive in guaraneeing he overall sabiliy despie each agen receiving oudaed informaion and no knowing oher agens fuure conrol inpus. Fuure work will furher develop he approach of he paper considering more complex agen dynamics and conrol obecives. Also, non-idealiies in he cloud access, such as delays and packe losses, will be aken ino accoun. REFERENCES [1] W. Ren, Disribued Consensus in Muli-vehicle Cooperaive Conrol, IEEE Conrol Sysems Magazine, April [2] R. Olfai-Saber, Flocking for muli-agen dynamic sysems: Algorihms and heory, IEEE Transacions on Auomaic Conrol, vol. 51, no. 3, [3] J. A. Fax and R. M. Murray, Graph laplacians and sabilizaion of vehicle formaions, IEEE Transacions on Auomaic Conrol, vol. 15, no. 1, [4] L. Suers, H. Liu, C. Tilman, and D. J. Brown, Navigaion echnologies for auonomous underwaer vehicles, IEEE Transacions on Sysems, Man and Cyberneics Par C: Applicaions and Reviews, vol. 38, no. 4, [5] L. Paull, S. Saeedi, M. Seo, and H. Li, AUV navigaion and localizaion: A review, IEEE Journal of Oceanic Engineering, vol. 39, no. 1, [6] J. W. Nicholson and A. J. Healey, The Presen Sae of Auonomous Underwaer Vehicle AUV Applicaions and Technologies, Marine Technology Sociey Journal, vol. 42, no. 1, [7] E. Fiorelli, N. E. Leonard, P. Bhaa, D. A. Paley, R. Bachmayer, and D. M. Fraanoni, Muli-AUV conrol and adapive sampling in Monerey Bay, IEEE Journal of Oceanic Engineering, vol. 31, no. 4, pp , [8] W. P. M. H. Heemels, K. H. Johansson, and P. Tabuada, An Inroducion o Even-riggered and Self-riggered Conrol, in IEEE Conference on Decision and Conrol, [9] D. V. Dimarogonas, E. Frazzoli, and K. H. Johansson, Disribued even-riggered conrol for muli-agen sysems, IEEE Transacions on Auomaic Conrol, vol. 57, no. 5, pp , [10] M. T. Hale and M. Egersed, Differenially privae cloud-based muliagen opimizaion wih consrains, in American Conrol Conference, [11] P. Pandey, D. Pompili, and J. Yi, Dynamic Collaboraion beween Neworked Robos and Clouds in Resource-Consrained Environmens, IEEE Transacions on Auomaion Science and Engineering, vol. 12, no. 2, pp , [12] A. Adaldo, D. Liuzza, D. V. Dimarogonas, and K. H. Johansson, Conrol of Muli-Agen Sysems wih Even-Triggered Cloud Access, in European Conrol Conference, [13] D. Zelazo, A. Rahmani, and M. Mesbahi, Agreemen via he edge Laplacian, in IEEE Conference on Decision and Conrol, [14] Z. Zeng, X. Wang, and Z. Zheng, Nonlinear Consensus under Direced Graph via he Edge Laplacian, in Chinese Conrol and Decision Conference, [15] M. Mesbahi and M. Egersed, Graph Theoreic Mehods in Muliagen Neworks. Princeon Univerisiy Press, [16] R. A. Horn and C. R. Johnson, Marix Analysis, 2nd ed. Cambridge Universiy Press,