Control of Multi-Agent Systems with Event-Triggered Cloud Access

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1 Control of Multi-Agent Systems with Event-Triggered Cloud Aess Antonio Adaldo, Davide Liuzza, Dimos V. Dimarogonas, and Karl H. Johansson Abstrat This paper investigates a multi-agent formation ontrol problem with event-triggered ontrol updates and additive disturbanes. The agents ommuniate only by exhanging information in a loud repository. The ommuniation with the loud is onsidered a shared and limited resoure, and therefore it is used intermittently and asynhronously by the agents. The proposed approah takes advantage of having a shared asynhronous loud support while guaranteeing a redued number of ommuniation. More in detail, eah agent shedules its own sequene of loud aesses in order to ahieve a oordinated network goal. A ontrol law is given with a riterion for sheduling the ontrol updates reursively. The losed loop sheme is proven to be effetive in ahieving the ontrol objetive and a numerial simulation orroborates the theoretial results. I. ITRODUCTIO The study of networked ontrol systems CS) is motivated by the fat that, nowadays, heterogeneous and geographially distributed devies an be onneted with heap and reliable wireless tehnologies. Speifially, onsensus algorithms have been investigated [1], [2] and tailored for platooning and formation ontrol [3], [4]. On the other hand, several reent papers onsider the possibility of distributed wireless sensors and atuators in CS, devoting the researh effort in oordinating the data pakets and guaranteeing desired performanes [5], [6]. Motivated by the need of saving hardware and software resoures and reduing the transmitted data, event-triggered and self-triggered ontrol strategies have been introdued [7] [9], and later extended to multi-agent oordination [10] [12]. These strategies do not require a fixed sampling period for the feedbak loop, but the ontrol input is updated only when a speifi ondition related to the stability or to some ontrol performane is violated. In the urrent paper a novel event-sheduled loud aess approah is introdued to solve the problem of formation ontrol for a fleet of systems modeled with simple integrator dynamis. We onsider a setup where eah agent proesses information loally. However, all the agents use the same ommuniation hannel and database hosted in the loud, whih are both shared resoures. The loud keeps a redued entralized amount of information and an be aessed by the agents in an asynhronous way under a publish subsribe paradigm [13], [14]. Basially, the agents an intermittently read/write information on the loud in an asynhronous way, while being in an idle mode no ommuniation and no The authors are with the ACCESS Linneaus Center and Shool of Eletrial Engineering, Royal Institute of Tehnology, Stokholm, Sweden. This work was partly supported by the Swedish Researh Counil and the Knut oh Alie Wallenberg Foundation. omputing) between any two onseutive aesses. Suh ontrol infrastruture appears partiularly onvenient when ommuniation onstraints severely limit the possibility of a diret exhange of information among agents. The use of shared resoures hosted in the loud is widely studied in omputer siene, where problems suh as loud aess management, resoure alloations ontrol and ontent deliver are studied [15], [16], while reently the sheduling of a ommon omputational resoure for ontrol system arhitetures has also been onsidered [17], [18]. As motivating example, we onsider the problem of waypoint generation for ontrol formation of a fleet of autonomous underwater vehiles AUVs) [19] [21]. For this kind of agents the ommuniation problems are indeed partiularly severe. Speifially, underwater ommuniation is ahieved by means of expensive and power hungry aousti modems and it is onsiderably limited both in range and bandwidth [19]. Furthermore, the GPS signal is interdited when the vehile is underwater and aurate inertial platforms are expensive. Aousti positioning by means of baselines is also diffiult to be adopted in wide open sea senarios. For this reason, in low ost appliations the AUVs are supposed to now and then surfae to get their exat feedbak position by GPS in order to ompensate the effet of oean urrents and external disturbanes. This kind of senario is onsidered in [22], [23] where, taking advantage of periodi surfaing, wireless ommuniation is used. However, the drawbak is that all AUVs have to surfae at the same instant in order to ommuniate with a leader and reeive the next waypoint in terms of time and position). Furthermore, the marine urrent disturbane is supposed to be the same for all the agents, thus resulting in onservative results when agents are far from eah other, possibly experiening different sea onditions. The main ontribution if this paper is to introdue an asynhronous loud aess for the ontrol system and we exploit a shared database to oordinate a formation of AUVs on the horizontal plane. Speifially, eah vehile surfaes asynhronously with respet to the others, gets its urrent GPS position and a foreast on the maritime onditions related to its region. Using this information, the AUV omputes its new ontrol input and its next surfaing instant. Then, it uploads this new information on the loud and starts a new underwater navigation segment without being able to ommuniate or to get GPS until its next surfaing. ote that, between any two onseutive surfaing intstants, the others ould possibly surfae an arbitrary number of times, thus hanging their ontrol inputs without the AUVs underwater being able to detet suh hange. However, despite the pres-

2 ene of asynhronous and outdated information, we prove that adopting a suitable sheduling rule the fleet onverges to the desired formation keeping a residual error within a given bound. The rest of this paper is organized as follows. In Setion II some notations and bakground onepts are introdued. In Setion III the mathematial model of the multi-agent system is presented, the ontrol objetive is defined and the proposed ontrol algorithm is outlined. In Setion IV suffiient onditions for ahieving the ontrol objetive are identified and in Setion V it is shown how these onditions are attained by sheduling the ontrol updates in an opportunist way. Setion VI shows the appliation of the proposed algorithm to a simulated vehile formation problem. Setion VII onludes the paper with some possible future developments. A. otation II. PRELIMIARIES For any n, 1 n denotes the vetor in R n whose entries are all equal to one, while I n denotes the identity matrix of order n. Operator used on a vetor denotes the Eulidean norm, while if used on a matrix denotes the orresponding indued norm. Operator denotes the Kroneker produt. For the definition and properties of this operator see for example [24]. B. Graph Theory For the purposes of this paper a graph is a tuple G = V,E ) made up of a set V = 1,...,} of nodes and a set E of edges onneting distint nodes. The edge onneting nodes i and j is denoted as i, j) or j,i) indifferently. For eah node i V, the set V i = j V : i, j) E } of the nodes that are onneted to node i by an edge is alled the neighborhood of node i and the nodes j V i are alled neighbors of node i. The number of neighbors d i of a node i is alled the degree of that node. A path between nodes i and j is defined as a sequene i,k 1,...,k m, j of nodes suh that any two onseutive nodes in the sequene are onneted by an edge. A graph is said to be onneted if all possible pairs of nodes are onneted by a path. The matrix A = } a i j suh that 1 if i, j) E a i j = 0 otherwise is alled adjaeny matrix of the graph, the matrix D = diagd 1,...,d } is alled degree matrix of the graph and finally the matrix L = D A is alled Laplaian matrix of the graph. The Laplaian matrix is symmetri and positive semidefinite and 1 is an eigenvetor with zero eigenvalue. Morevoer, the zero eigenvalue has multipliity one if and only if the graph is onneted [1]. Therefore in a onneted graph the eigenvalues of the Laplaian an be denoted as 0 = λ 1 < λ 2... λ. A graph is typially used to desribe networked multi-agent systems: eah node in the graph represents one agent in the network and an edge between nodes i and j represents a possible interation between the orresponding agents. III. SETUP DESCRIPTIO Consider a set of dynamial agents desribed by ẋ i t) = u i t) + ω i t), i = 1,...,, 1) with t 0 and x i t),u i t),ω i t) R n. Here x i t) is the state of agent i, u i t) is the ontrol input applied to it and ω i t) is a disturbane ating on it. Denoting we an rewrite 1) as xt) :=[x 1 t),...,x t) ], ut) :=[u 1 t),...,u t) ], ωt) :=[ω 1 t),...,ω t) ], ẋt) = ut) + ωt). 2) We onsider the rendezvous problem, i.e., the problem of driving the states of the agents lose to eah other in the state spae. More preisely, let xt) be the average of the agents states, xt) := 1 i=1 x i t), let e i t) be the mismath between the state of agent i and the average state, e i t) := xt) x i t), i = 1,...,, and ollet the signals e i t) into the stak vetor et) := [e 1 t),...,e t) ]. Definition 1: We say that the multi-agent system 1) ahieves pratial onsensus if lim sup et) ε, 3) t + where ε > 0 is a given positive onstant. We assume that the agents annot diretly ommuniate with eah other, but have aess to a shared database and a measurement system hosted on a loud. The ommuniation hannel between the agents and the loud is onsidered a shared resoure with limited throughput apaity, and therefore it must be used at disrete time instants and asynhronously by different agents. We onsider pieewise onstant ontrol signals with event-triggered updates. The time instants when agent i updates its ontrol input are denoted as t i,k, with k and we set t i,0 = 0. amely, we have u i t) = u i t i,k ) for t [t i,k,t i,k+1 ). For onveniene, we introdue the funtions l j t) orresponding to the latest update of u j ) before time t [11], l j t) = max t j,k : t j,k t } k ote in partiular that l i t i,k ) = k. It is assumed that when agent i onnets to the loud, say at time t = t i,k = t i,li t), it reeives a measurement of its urrent state xt i,k ), some information about the other agents stored in the database, and some estimate of the disturbanes to whih it is subjet.

3 i t i,li t) t 1,l1 t) t 2,l2 t)... t,l t) x i t i,li t)) x 1 t 1,l1 t)) x 2 t 2,l2 t))... x t,l t)) u i t i,li t)) u 1 t 1,l1 t)) u 2 t 2,l2 t))... u t,l t)) γ i,li t) γ 1,l1 t) γ 2,l2 t)... γ,l t) ρ i,li t) ρ 1,l1 t) ρ 2,l2 t)... ρ,l t) t i,li t)+1 t 1,l1 t)+1 t 2,l2 t)+1... t,l t)+1 TABLE I DATA STORED I THE SHARED DATABASE AT A GEERIC TIME t. Remark 2: The ontrol input u i t i,k ) an be omputed by agent i at time i by using only information downloaded from the loud at time t i,k. For the purposes of the forthoming analysis, onsider also the following signals, z i t) := a i j x j t) x i t)), 7) and the mismathes ũ i t) between a ontrol signal u i t) and the orresponding z i t), ũ i t) := u i t) z i t). 8) The estimate of the disturbanes is given in the form of two oeffiients γ i,k and ρ i,k suh that γ i,k t [t i,k,t i,k+1 ), ω i t) ˆω i,k t) := γ i,k + ρ i,k t t i,k+1 ) t t i,k+1. 4) Remark 1: Disturbane estimates 4) are different for different agents and for different update times of the same agent, taking into aount that disturbanes an vary both in time and spae within the operating region. The ase of a known global upper bound on the disturbanes an still be derived as a partiular ase, by setting γ i,k = γ and ρ i,k = 0 for all i = 1,..., and all k. This partiular model of disturbane estimation is inspired by our motivating example, desribed at the end of this setion. Agent i uses all suh information to ompute its new ontrol input u i t i,k ), and the time t i,k+1 of the next update. Before losing the onnetion to the loud, agent i uploads the values of t i,k, xt i,k ), ut i,k ), γ i,k, ρ i,k, and t i,k+1 on the shared database, so that they an be used later by other agents. Suh values may replae the orresponding old ones uploaded by agent i at the time of the previous onnetion, so that the dimension of the database does not inrease over time. Table I shows the data stored in the shared database at a generi time t. The ontrol signals are obtained as linear diffusive feedbak from other agents in the network. The topology of the interations is desribed by a graph G where eah node represents an agent and the edge i, j) represents a feedbak interation between agents i and j. amely, we set u i t) = a i j ˆx j t i,k ) x i t i,k ) ), t [t i,k,t i,k+1 ), 5) where A = a i j } is the adjaeny matrix of G, is a positive salar gain and ˆx j t i,k ) is an estimate of the state of agent j available at time t i,k. The estimate ˆx j t i,k ) are obtained by using the data available in the loud at time t i,k. amely, 1) is onsidered for agent j under null disturbanes, and it is integrated in the interval [t j,l j t i,k ),t), with t t j,l j t i,k )+1, yielding ˆx j t) = x j t j,l j t)) + u j t j,l j t))t t j,l j t)). 6) The setup proposed above is suitable to desribe a formation ontrol problem for a network of autonomous vehiles under strit ommuniation onstraints. The motivating example of the paper is the problem of a waypoint generation algorithm for a two-dimensional formation of AUVs. Eah agent represents a vehile, and the state of agent i is x i t) = p i t) b i R 2, where p i t) is the horizontal waypoint trajetory of vehile i, i.e. we do not are about the vertial oordinate, and b i is a onstant offset term with respet to the average point of the fleet, so it desribes the position assigned to vehile i within the formation. Sine radio ommuniation is interdited underwater no GPS and no relative information exhange an our), and sine we have assumed that the AUVs are not equipped with expensive sonar modems, the vehiles are ompletely isolated during the navigation, but that they an surfae at disrete time instants to exhange information with a remote repository hosted on a loud. The disturbanes inluded in the model aount for the marine urrents influening the motion of the vehiles. The position measurements may be obtained by GPS and the foreasts 4) on the marine urrents may be omputed from a MAFOR bulletin obtained from a wireless weather station. In fat, foreasts beome more onservative for more distant times in the future, a harateristi whih is embedded into model 4). The proposed ontrol algorithm is summarized below. The algorithm is initialized by setting, for all i V, t i,0 = 0, ˆx i 0) = x i 0) and u i 0) = a i j x j 0) x i 0)). Eah agent i V at eah update time t i,k performs the following operations. 1. Agent i onnets to the loud at time t = t i,k, as sheduled at time t i,k Agent i reeives the measurement of its urrent state x i t i,k ) and uploads it on the shared database. 3. From the database, agent i downloads t j,l j t i,k ), x j t j,l j t i,k )), u j t j,l j t i,k )), t j,l j t i,k )+1, γ j,l j t i,k ) and ρ j,l j t i,k ) for eah j V i. 4. Using x i t), t j,l j t i,k ), x j t j,l j t i,k )) and u j t j,l j t i,k )) for j V i, agent i omputes its new ontrol input u i t i,k ) aording to 5).

4 5. Agent i uploads its new ontrol input u i t i,k ) on the loud. 6. Agent i omputes the parameters γ i,k,ρ i,k by elaborating available information on the disturbanes, and uploads them on the loud. 7. Using γ i,k, t j,l j t i,k ), x j t j,l j t i,k )), u j t j,l j t i,k )), t j,l j t i,k )+1, γ j,l j t i,k ) and ρ j,l j t i,k ) for j V, agent i shedules the time t i,k+1 of its next update and uploads it on the loud. An appropriate sheduling rule will be given later in the paper. 8. Agent i disonnets from the loud and will be unable to ommuniate until its next update at time t i,k+1. When the partiular problem of AUVs oordination is onsidered, t i,k is the k-th surfaing instant for the i-th vehile, and step 8 orresponds to the underwater navigation segment between the surfaing times t i,k and t i,k+1. IV. PRACTICAL COSESUS In this setion our bounded onvergene result is derived. Later on, this result will be related to the sheduling of the ontrol updates. The following assumptions are needed. Assumption 1: The graph G that desribes the feedbak interations is onneted, and its Laplaian has eigenvalues 0 = λ 1 < λ 2... λ. Assumption 2: The disturbanes ω i t) ating on eah agent i = 1,..., are uniformly bounded by ω i t) Ω. Remark 3: Assumption 2 is not related to the estimation model 4). The upper bound Ω is not used in the disturbane estimation nor in sheduling and omputing the ontrol updates, and it is only introdued to haraterize the onvergene radius ε in 3). Assumption 3: There exists a threshold funtion σt) = σ 0 + σ 1 e λ σ t, with positive onstants σ 0, σ 1 and λ σ suh that at any time instant t 0 it holds that ũ i t) σt). 9) Remark 4: Assumption 3 an be fulfilled by sheduling the ontrol updates opportunely. This is shown in Setion V. Theorem 1: Consider the multi-agent system 1) under ontrols 5). Suppose Assumptions 1 to 3 hold. Then pratial onsensus is ahieved with ε = λ σ0 λ2 2 + Ω ) 10) Proof: Consider the following Lyapunov andidate funtion for the error stak vetor et) 1, V t) = et) L 2 I n )et), 11) where L is the Laplaian of the graph G that desribes the agents interations. Denoting zt) = [z 1 t),...,z t) ] 1 When a planar formation problem is onsidered, n = 2 we have zt) = L I n )xt) = L I n )et). 12) Sine L is symmetri we an write et) L 2 I n )et) =et) L I n ) 2 et) Hene, 11) an be rewritten as =L I n )et)) L I n )et)) =zt) zt) = zt) 2. V t) = zt). Consider now the dynamis of this andidate funtion along the system trajetories. Using 2) and 12) we an write żt) = L I n )ẋt) = L I n )ut) + ωt)). 13) ow denote so that we have ũt) = [ũ 1 t),...,ũ t) ], ut) = ũt) + zt), whih substituted into 13) yields Consequently, we have żt) = L I n )ũt) + zt) + ωt)). V t) = d dt zt) = zt) żt) zt) = zt) L I n )ũt) + zt) + ωt)) zt) 14) By the properties of the Kroneker produt and the Eulidean norm we have zt) L I n )zt) λ 2 zt) 2 zt) L I n )ũt) zt) λ ũt) zt) L I n )ωt) zt) λ ωt) Substituting these inequalities into 14) yields V t) λ 2 V t) + λ ũt) + λ ωt). 15) By Assumption 3 we have ũt) σt), whih substituted into 15) yields V t) λ 2 V t) + λ σt) + λ ωt). 16) Aounting for Assumptions 1 and 2, 16) implies that lim supv t) λ σ0 + Ω ) 17) t λ 2 ow observe that, from the Rayleigh-Ritz theorem [24], we have V t) = L I n )et) λ 2 et),

5 or equivalently et) V t) λ 2. 18) Therefore, using 17) into 18), and taking the limit for t +, we have lim sup et) λ ) σt) ωt) t + λ2 2 lim sup + t + λ σ0 λ2 2 + Ω ). Remark 5: The onvergene radius 10) an be arbitrarily redued by inreasing the ontrol gain, reduing the threshold σ 0 or onsidering a better onneted network, whih orresponds to a smaller ratio λ /λ 2 2. V. SCHEDULIG COTROL UPDATES In this setion we present our main result. Speifially, we give a riterion for reursive sheduling of the ontrol updates t i,k suh that Assumption 3 for Theorem 1 holds, and so pratial onsensus is ahieved. Denote ẑ i t) := a i j ˆx j t) ˆx i t)), 19) ẑt) :=[ẑ 1 t),...,ẑ t) ], α :=qσ 0, 20) βt) :=1 q)σ 0 + σ 1 e λ σ t, 21) where 0 < q < 1. Let us introdue the following funtions: ˆΩ i,k t) := ˆω i,k τ)dτ t i,k γ i,k t t i,k ) t [t i,k,t i,k+1 ), = γ i,k t t i,k ) ρ i,kt t i,k+1 ) 2 t t i,k+1, B i,k t) := ˆω j,l j t i,k )t) 2, uti,k ) R i, j,k t) :=max + σt i,k ), λ σt λ j,l 2 jti,k)+1) + B )} i,kt), S i,k t) := d i u i t i,k )t t i,k ) + j:t<t j,l j t i,k )+1 j: t t j,l j t i,k )+1 u j t j,l j t i,k ))t t i,k ) j: t t t j,l j t i,k )+1 j,l j t i,k )+1 ) u j t j,l j t i,k )) t j,l j ti,k )+1 t i,k R i, j,k τ) + στ)dτ, where d i is the degree of node i in the graph G. ow realling 20) and 21), onsider the following salars: T i, j,k =inf τ>t i,k : ˆΩ j,l j t i,k )τ) α }, 2d max T i,k =inf τ>t i,k : S i,k τ) βτ) }, where we denoted d := maxd 1,...,d }. With this notation, the ontrol updates are reursively sheduled as t i,k+1 = Ti, j,k,t i,k}. 22) min j V i i} The sheduling rule 22) introdues a degree of freedom in the hoie of the next onnetion t i,k+1 to the loud. Remark 6: The time instant t i,k+1 an be omputed by agent i at time t i,k by only using information available on the loud at time t i,k. In partiular, the values u i t i,k ) and u j t j,l j t i,k )) are diretly available, fr. Table I. Together with Remark 2, this implies that no entralized omputation is required to implement the proposed ontrol algorithm. The loud is only used as a data repository and it does not need to proess information. All the neessary omputing an be done by the agents in a deentralized way aessing the loud asynhronously. Remark 7: ote that ˆΩ i,k t i,k ) = 0 and S i,k t i,k ) = 0 for any k and for any i V. Morevoer, ˆΩt) and S i,k t) are ontinuous in t with upper-bounded slope. Sine α/d max is a positive onstant and βτ) is lower-bounded by a positive onstant, this implies that times T i, j,k and T i,k annot be infinitely lose to t i,k. Consequently the inter-update times t i,k+1 t i,k are lower-bounded by some positive onstant and the updates do not present aumulation points. Theorem 2: Consider the multi-agent system 1) under ontrols 5). Suppose Assumptions 1 and 2 hold and let the ontrol updates t i,k be sheduled aording to 22). Then pratial onsensus is ahieved with ε as in 10). Proof: We are going to prove that if the ontrol updates are sheduled aording to 22), then 9) holds for all the agents i V and at all the time instants t 0. Then we will obtain the thesis from Theorem 1. Sine t i,0 = 0 for all i V, we have ũ i 0) = 0 < σ0), and therefore at time zero 9) holds for all the agents. ow suppose by ontradition that at a finite time t some agent i attains ũ i t) > σt) for the first time, while ũ j τ) στ) for all τ [0,t) and all j V. Denote also k = l i t), i.e., let t i,k be the latest update for agent i before t. Adding and subtrating ẑ i t) on the right-hand side of 8) we obtain ũ i t) = ẑ i t i,k ) ẑ i t) + ẑ i t) z i t) ). Taking the norm of both sides and applying the triangular inequality yields ũ i t) ẑ i t i,k ) ẑ i t) + ẑ i t) z i t). 23) By the ontradiiton hypothesis we have ũ i t) > σt), therefore 23) implies σt) < ẑi t i,k ) ẑ i t) + ẑi t) z i t). 24)

6 First onsider the term ẑ i t) z i t). We have ẑ i t) z i t) = and onsequently a i j ˆx j t) ˆx i t) x j t)+x i t)), ẑ i t) z i t) d i ˆx i t) x i t) + ˆx j t) x j t). j V i 25) Consider now the terms ˆx j t) x j t). Integrating 1) in [t j,l j t i,k ),t), with t < t j,l j t i,k )+1, we have x j t) =x j t j,l j t i,k )) + u j t j,l j t i,k ))t t j,l j t i,k )) + t j,l j t i,k ) ω j τ)dτ. 26) On the other hand, ˆx j t) an be omputed as in 6). Therefore, using 6) and 26), we have ˆx j t) x j t) ω j τ)dτ ω j τ) dτ, t j,l j t i,k ) t j,l j t i,k ) whih by 4) also implies ˆx j t) x j t) ˆΩ j,l j t i,k )t). The same reasoning an be arried out for the term ˆx i t) x i t), yielding ˆx i t) x i t) ˆΩ i,k t). Substituting the two previous inequalities into 25) yields ẑ i t) z i t) d i ˆΩ i,k t) + j V i ˆΩ j,l j t i,k )t). Sine 22) is applied, we have ˆΩ i,k t) 2d α max, and onsequently α α ẑ i t) z i t) d i + d i α. 27) 2d max 2d max Consider now the term ẑt i,k ) ẑ i t) in 23). Realling 7) and noting that ˆx i t i,k ) = x i t i,k ), beause at time t i,k vehile i reeives the exat measurement of its state, we have ẑ i t i,k ) ẑ i t) = a i j ˆx j t i,k ) x i t i,k ) ˆx j t) + ˆx i t) ). 28) Fousing on the term ˆx i t) x i t i,k ), by 6) applied for j = i, we have ˆx i t) x i t i,k ) = u i t i,k )t t i,k ). 29) Similar reasoning an be applied to the terms ˆx j t i,k ) ˆx j t). However, sine the ontrol updates are asynhronous, u j τ) may be updated one or multiple times during the time interval [t i,k,t i,k+1 ). amely, in the time interval [t i,k,t j,l j t i,k )+1), u j has value u j t j,l j t i,k )), whih is available in the loud at time t i,k, but the possible future values assumed by u j τ) for τ t j,l j t i,k )+1 are unknown at time t i,k. Hene we an write ˆx j t) ˆx j t i,k ) = u j τ)dτ t i,k u j t j,l j t i,k ))t t i,k ) u j t j,l j t i,k )) t t j,l j t i,k )+1, = t j,l j t i,k )+1 t i,k ) + u j τ)dτ t > t j,l j t i,k )+1. t j,l j t i,k )+1 30) Substituting 29) and 30) into 28), taking norms of both sides, and applying the triangular inequality yields ẑi t i,k ) ẑ i t) di u i t i,k )t t i,k ) + j:t<t j,l j t i,k )+1 j:t t j,l j t i,k )+1 ut j,l j t i,k ))t t i,k ) j:t t t j,l j t i,k )+1 j,l j t i,k )+1 ) ut j,l j t i,k )) t j,l j ti,k )+1 t i,k u j τ) dτ. 31) Consider now an agent j that updates its ontrol at least one before time t, and fous on the term u j τ) with τ [t j,l j t i,k )+1,t). Sine 9) holds for all the agents until time t, we an write u j τ) z j τ) + ũ j τ) zτ) + στ)). 32) Morevoer, in τ [t j,l j t i,k )+1,t), sine 9) holds, the state of the system onverges to the region desribed by 17). Therefore, taking into aount that ωτ) B i,k τ) and στ) σt i,k ), we an write zτ) max zti,k ), λ σti,k) + B )} i,kτ). 33) λ 2 Also, sine t i,k τ, 9) holds for all the agents at time t i,k, and we an write zt i,k ) ut i,k ) + σt i,k ). 34) Using 33) and 34) into 32) we an write u j τ) R i, j,k τ) + στ) ), whih substituted into 31) yields ẑ i t i,k ) ẑ i t) S i,k t). ow sine 22) is applied, we have S i,k t) βt), and onsequently ẑi t i,k ) ẑ i t) βt). 35) ow substituting 27) and 35) into 24), we have σt) < α + βt) This is a ontradition, sine α and βt) are defined so that σt) = α + βt). We an onlude that 9) holds for all the agents i at all times t 0.

7 ow sine 9) holds for all the agents uniformly, Assumptions 1 to 3 hold, and Theorem 1 an be applied. Hene pratial onsensus is attained with radius 10). Remark 8: Sine Theorem 2 only requires that t i,k+1 T i, j,k for all j V and t i,k+1 T, the sheduling rule 22) may be relaxed to t i,k < t i,k+1 i,k min Ti, j,k,t i,k}. 36) j V i i} This gives eah agent a degree of freedom in the sheduling of the next onnetion to the loud. Suh degree of freedom may be exploited to avoid loud ongestion due to multiple ontemporary aesses. In fat, if 36) is enfored, agent i is free to hoose t i,k+1 in a given interval, and sine it is aware of the subsequent update times t j,l j t i,k )+1 of all the other agents j i, it an shedule t i,k+1 so that it does not oinide with any of these instants. Remark 9: An alternative upper bound to 34) is zti,k ) ẑti,k ) + α. Fig. 1. Trend of the first onsensus variable x 1) i = p 1) i b 1) i, for eah agent i = 1,...,5 during the simulation. Therefore, funtion R i, j,k t) an also be designed as uti,k ) R i, j,k t) := max min + σt i,k ), ẑti,k ) + α }, λ σt λ j,l 2 jti,k)+1) + B )} i,kt), In this ase, when sheduling the update time t i,k+1, agent i needs to ompute ẑti,k ). If the topology of the onnetions among the vehiles is known, ẑti,k ) an be omputed by using 19) with t = t i,k. VI. UMERICAL SIMULATIOS In order to orroborate the theoretial results, we applied the proposed ontrol algorithm to a formation problem on a simulated network made up of = 5 planar vehiles. The topology of the onnetions is desribed by a omplete graph, so that every agent reeives feedbak from every other agent. The desired formation is desribed by the offsets [ 25, 25), 25,25), 0,0), 25, 25), 25,25)]. The simulation takes plae the time span [0, 50]. The agents are spawned in initial positions randomly extrated in a square of 200 by 200. We pik a ontrol gain = 0.01 and a threshold funtion σt) = e 0.05t. A different value of the additive disturbane is hosen for eah agent, randomly extrated in the range 1.0, 1.0) on both oordinates. At eah hundredth of seond this value is hanged with probability , by randomly extrating a new value from the same range. To model the foreast that the agents reeive about suh disturbanes, at eah update of a vehile i we assign to γ i,k a value of r γ ), where r γ is randomly extrated in 0.0,1.0), while we assign value zero to ρ i,k. With these hoies we have that the norm of the disturbanes is always below the estimate that the agents reeive. Figure 1 illustrates the onvergene of the first position variable for eah vehile during the simulation. Fig. 2. Paths p i t) exeuted by eah vehile i = 1,...,5 during the simulation upper) and detail lower). Figure 2 shows the two-dimensional paths. Finally, Table II shows the update times t i,k in the time span [0,50] for eah agent i = 1,...,5. VII. COCLUSIOS A loud-based ontrol algorithm has been proposed for pratial onsensus of a network of agents with integrator

8 k t 1,k t 2,k t 3,k t 4,k t 5,k TABLE II UPDATE TIMES I THE TIME SPA [0,50]. dynamis under event-triggered updates and additive disturbanes. Suffiient onditions for onvergene have been identified in terms of the network topology and of the sheduling of the ontrol updates. The proposed approah ombines the benefits of event/self-triggered ontrol shemes with the advantage of having a shared asynhronous loud support. Speifially, eah agent shedules its own sequene of loud aesses in order to ahieve a oordinated network formation. The setup is partiularly onvenient for those appliations where diret ommuniation among agents is not always feasible, suh as formation ontrol for AUVs. For this problem, the ontrol algorithm overomes the limitation of having a pre-assigned trajetory for the whole fleet as well as the synhronization of the surfaing of all the agents [20], [23]. Future work will further develop the approah of the paper onsidering different sheduling laws for the loud aesses as well as other ontrol objetives, e.g. leader follower ontrol. Furthermore, more omplex agent dynamis and more omplex models for the foreast on the disturbanes will be studied. REFERECES [1] R. Olfati-Saber, J. A. Fax, and R. M. Murray. Consensus and ooperation in networked multi-agent systems. Proeedings of the IEEE, 951): , [2] W. Ren, R. W. Beard, and E. M. Atkins. A survey of onsensus problems in multi-agent oordination. In Amerian Control Conferene, Portland, Oregon, USA, [3] M. Arak. Passivity as a design tool for group oordination. IEEE Transations on Automati Control, 528): , [4] Dimos V. Dimarogonas and Kostas J. Kyriakopoulos. A onnetion between formation infeasibility and veloity alignment in kinemati multi-agent systems. Automatia, 4410): , [5] D. esi and A. R. Teel. Input-output stability properties of networked ontrol systems. 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