Consensus for Clusters of Agents with Cooperative and Antagonistic Relationships
11 Consensus for Clusters of Agents with Cooperative and AntagonisticRelationships
Giulia De Pasquale and Maria Elena Valcher
Abstract — In this paper we address the consensus problem inthe context of networked agents whose communication graphcan be split into a certain number of clusters in such away that interactions between agents in the same clusters arecooperative, while interactions between agents belonging todifferent clusters are antagonistic. This problem set-up arisesin the context of social networks and opinion dynamics, wherereaching consensus means that the opinions of the agents inthe same cluster converge to the same decision. The consensusproblem is here investigated under the assumption that agentsin the same cluster have the same constant and pre-fixed amountof trust (/distrust) to be distributed among their cooperators(/adversaries). The proposed solution establishes how muchagents in the same group must be conservative about theiropinions in order to converge to a common decision.
I. I
NTRODUCTION
Unmanned air vehicles, sensor networks, opinion formation,mobile robots, and biological systems represent just a fewexamples of the wide variety of contexts where distributedcontrol, and in particular consensus and synchronizationalgorithms, have been largely employed [15], [19], [23],[29]. The existence of such a broad area of applicationhas stimulated a rich literature addressing consensus andsynchronization problems for multi-agent networked systemsunder quite different assumptions on the agents’ description,their processing capabilities, the communication structure,the reliability of the communication network, etc.Most of the literature on consensus has focused on theproblem of leading all the agents to a common decision,namely on ensuring that the agents’ describing variablesasymptotically converge to the same value, by assuming thatthe agents are cooperative. Social networks, however, provideclear evidence that mutual relationships may not always becooperative, and yet the dynamics of opinion forming mayexhibit stable asymptotic patterns. In particular, Altafini [1]has shown that in a multi-agent system with cooperativeand antagonistic relationships, bipartite consensus , namelythe splitting of the agents’ opinions into two groups thatasymptotically converge to two opposite values, is possibleprovided that the communication network is structurallybalanced , namely agents split into two groups such thatintra-group relationships are cooperative and inter-grouprelationships are antagonistic. If this is not the case, then theonly equilibrium asymptotically achievable with a DeGroot’s
G. De Pasquale and M.E. Valcher are with the Dipartimento diIngegneria dell’Informazione Università di Padova, via Gradenigo 6B,35131 Padova, Italy, e-mail: [email protected],[email protected] . type of control law is the zero value. This analysis has beenlater extended from the case of simple integrators to thecase of homogeneous agents described by an arbitrary state-space model [32] (see, also, [2], [7]), and has been in turninvestigated by several other authors under different workingconditions.Group/cluster consensus, namely the situation that occurswhen agents split into an arbitrary number of disjointgroups, and they aim to achieve consensus within eachgroup, independently of the others, represents the naturalgeneralisation of the previous problems. It is interesting toremark that also this generalisation arose in the context ofsocial networks. In fact, some opinion dynamics models [12]highlighted how agents with limited confidence levels mayevolve into different clusters, and the members of each clustereventually reach a common opinion.In particular, in [35] the concept of n -cluster synchroniza-tion is introduced for diffusively coupled networks. Threestrategies to achieve synchronization are presented. Firstthe case of purely collaborative agents with different self-dynamics (informed agents and naive agents) is considered,and it is shown that under some homogeneity condition(on the sum of the weights of the edges connecting anyagent of a cluster with the agents of another cluster), n -cluster synchronization is possible. Then the case whenall the agents are identical is considered, assuming againthat the previous homogeneity constraint holds. It is shownhow synchronization may be achieved by suitably exploitingcommunication delays. An alternative scenario is the onewhen cooperative and antagonistic relationships are possible.In that case, the authors assume that the sum of the weightsof the edges connecting any agent of a cluster with the agentsof another cluster is zero. This means that for every agentthe total weight of the agents with whom it collaborates andthe total weight of the agents with whom it has conflictingrelationships, in another cluster, coincide. Synchronisationis possible if and only if the overall matrix representing themulti-agent system has n zero eigenvalues and all the othereigenvalues have negative real part.It is interesting to notice that most of the literature oncluster synchronisation has in fact adopted the same “in-degree balanced condition" adopted in [35], namely theassumption that interactions within the same cluster arecooperative, while interactions between agents of differentclusters may have both signs, but every agent has a a perfectbalance between collaborative relationships and antagonisticrelationships in every other cluster. This is the case, forinstance, of [16], [25], [26], [27], [28], [34], [37]. a r X i v : . [ ee ss . S Y ] A ug In [26] group consensus is investigated for homogeneousmulti-agent systems each of them described by a stabilizablepair ( A, B ) , under the assumption that the communicationgraph admits an “acyclic partition". Under some additionalconditions it is shown that it is always possible to design astate feedback law so that cluster synchronization is achieved.In [27] group consensus for networked systems is investiga-ted, by assuming that agents are modeled as double integrators.Different set-ups, depending on whether agents’ position andvelocity are coupled according to the same topology or not,are explored. In particular, the presence of a leader in eachcluster is considered. It is shown that group consensus can beachieved if the underlying topology for each cluster satisfiescertain connectivity assumptions and the intra-cluster weightsare sufficiently high. Group consensus for networked multipleintegrators is also considered in [28], by assuming that eachcluster has a spanning tree and by introducing a scaling factorwithin each group to make couplings within each clustersufficiently strong.Group consensus of a network of identical oscillatorsto a family of desired trajectories (one for each group) isinvestigated in [34], by making use of pinning control and byadopting, as in [28], scaling factors (the so-called “couplingstrengths") to enforce the coupling within each group, possiblyin an adaptive way. Periodically intermittent pinning controlis adopted in [16] to achieve cluster syncronization to a familyof trajectories: in addition to the in-degree balanced condition,it is assumed that the communication graph restricted to eachsingle cluster is strongly connected.H ∞ group consensus for networks of agents modeledas single-integrators, affected by model uncertainty andexternal disturbances, is investigated in [25]. Conditions onthe coupling strengths that ensure both the achievement ofconsensus within each group, and an H ∞ performance forthe overall system in terms of disturbance rejection, areprovided. The “in-degree balanced condition" is used alsoin [11], where it is referred to as “inter-cluster commoninfluence". Cluster consensus is here achieved by making useof external adaptive inputs, which are the same for agentsbelonging to the same cluster. Group consensus for networkedsystems with switching topologies and communication delaysis investigated in [37]. For a recent survey about consensus,including all results achieved for group/cluster consensus, see[24].The aim of this paper is to address group consensus in adifferent set-up with respect to the one adopted in the previousreferences, a set-up that represents an extension to the caseof an arbitrary number of clusters of the one adopted in [1]for two groups. Indeed, we assume that interactions betweenagents in the same clusters are cooperative, while interactionsbetween agents belonging to different clusters may onlybe antagonistic. This assumption turns out to be quiterealistic in many applications from the economical, biological,sociological fields (see, e.g., [7], [33]) where activation orinhibition, cooperation or antagonistic interactions must betaken into account. Agents in the same group cooperate withthe aim of reaching a common objective, while they tend to compete with agents belonging to different groups or factions.Sociological models were, in fact, the primary motivationbehind the set-up adopted in [1], and the proposed extensionto an arbitrary number of clusters explored in this paper isin perfect agreement with the common perspective adoptedin the tutorial paper by Proskurnikov and Tempo [23] thatfocuses on the relation between social dynamics and multi-agent systems, in the paper by Cisneros-Velarde and Bulloin [5], and in the milestone paper by Davis [6] where theconcept of clustering balance was introduced.Clearly, in this context the “in-degree balanced assumption"adopted in [16], [26], [27], [28], [35], [37] cannot be enforcedwithout leading to a trivial set-up, since inter-cluster weightscan only be nonpositive. However, we will adopt a similar,but weaker, homogeneity condition (in fact, similar to the oneadopted in the first part of [35] for the case of cooperativeagents) that requires that each agent in a group distributes thesame amount of “trust" to the agents in its own group and“distrust" to the agents belonging to adverse groups. This isequivalent to saying that given two arbitrary (not necessarilydistinct) classes, say i and j , the sum of the weights of theincoming edges from all the agents of class j to an agent ofclass i depends on i and j , and not on the specific agent.More in detail, we assume that the communication graphbetween agents is modeled by an undirected, signed, weighted,connected and clustered graph, and that the agents arepartitioned into k clusters, such that intra-cluster interactionsmay only be nonnegative, while inter-cluster interactions canonly be nonpositive. We investigate under what conditions arevised version of the De Groot’s distributed feedback controllaw, that only requires to modify the weight that each agentbelonging to the same class has to give to its own opinion,can lead the multi-agent system to k - partite consensus . Notethat while the typical approach to group consensus requires toenforce the intra-cluster communication (namely the weightsof all the edges within a class) by suitably increasing the“coupling strengths", in this case we only require that eachagent of a cluster increases its level of stubborness or self-confidence, but this value must be the same for all the agentsin the same group. It is worthwhile remarking that, as in theprevious papers about group consensus, the design of thesecoefficients cannot be obtained in a fully distributed way,since - as it will be shown in the following - the algorithmwe propose requires that each cluster is aware of the choicesmade by the clusters preceding it, with respect to somesuitable ordering. However, once the parameters have beenchosen the control algorithm is completely distributed.This work generalizes the preliminary results presentedin [22] for the case of multi-agent systems partitioned intothree groups. The generalisation is not trivial at all, since itrequires to extend the algorithm described in the proof fromthree steps to an arbitrary number of steps. Moreover, theassumptions under which the k -partite consensus problem isachieved have been generalised and better clarified. Finally, k -partite consensus is also investigated for a special class ofnonlinear models.The rest of the paper is organized as follows. In thefollowing some definitions and basic properties in the context of signed graphs are introduced. Section II formalizes the k -partite consensus problem for a multi-agent network,whose agents are described as simple integrators and whosecommunication graph is split into k clusters. Section IIIprovides some preliminary results about k -partite consensus.Section IV provides a complete solution to this problem, underthe aforementioned homogeneity assumption that imposesthat each agent in a group distributes the same amount of“trust" to the agents in its own group and “distrust" to theagents belonging to adverse groups. As a special case, weaddress the case of a complete graph. In Section VI k -partiteconsensus for a class of nonlinear models is studied. Finally,Section VII concludes the work. Notation . Given k, n ∈ Z , with k ≤ n , the symbol [ k, n ] denotes the integer set { k, k + 1 , . . . , n } . In the sequel, the ( i, j ) -th entry of a matrix A is denoted by [ A ] i,j , while the i -th entry of a vector v by [ v ] i . Following [9], we adopt thefollowing terminology and notation. Given a matrix A withentries [ A ] i,j in R + , we say that A is a nonnegative matrix ,if all its entries are nonnegative, namely [ A ] i,j ≥ for every i, j , and if so we use the notation A ≥ . If all the entries of A are positive, then A is said to be a strictly positive matrix and we adopt the notation A (cid:29) . The same notation holdsfor vectors.A symmetric matrix P ∈ R n × n is positive (semi) definite if x (cid:62) P x > ( x (cid:62) P x ≥ ) for every x ∈ R n , x (cid:54) = 0 , andwhen so we use the notation P (cid:31) ( P (cid:23) ).The notation A = diag { A , . . . , A n } indicates a blockdiagonal matrix whose diagonal blocks are A , . . . , A n . Thesymbols n and n denote the vectors in R n with all entriesequal to and to , respectively. A real square matrix A is Hurwitz if all its eigenvalues lie in the open left complexhalfplane, i.e. for every λ belonging to σ ( A ) , the spectrum of A , we have Re( λ ) < .For n ≥ , an n × n nonzero matrix A is reducible [10],[17] if there exists a permutation matrix Π such that Π (cid:62) A Π = (cid:20) A , A , A , (cid:21) , where A , and A , are square (nonvacuous) matrices,otherwise it is irreducible . A Metzler matrix is a real squarematrix, whose off-diagonal entries are nonnegative. If A is an n × n Metzler matrix, then [31] it exhibits a realdominant (not necessarily simple) eigenvalue, known as
Frobenius eigenvalue and denoted by λ F ( A ) . This meansthat λ F ( A ) > Re( λ ) , ∀ λ ∈ σ ( A ) , λ (cid:54) = λ F ( A ) . If A isMetzler and irreducible, then λ F ( A ) is necessarily simple.An undirected, signed and weighted graph is a triple [18] G = ( V , E , A ) , where V = { , . . . , N } = [1 , N ] is the setof vertices, E ⊆ V × V the set of arcs, and
A ∈ R N × N the adjacency matrix of the weighted graph G . An arc ( j, i ) belongs to E if and only if [ A ] i,j (cid:54) = 0 . As the graph isundirected, ( i, j ) belongs to E if and only if ( j, i ) ∈ E , or,equivalently, A is a symmetric matrix. We assume that thegraph G has no self-loops, i.e., [ A ] i,i = 0 for every i ∈ [1 , N ] ,and arcs in E have either positive or negative weights, namelythe off-diagonal entries of A are either positive or negative. If all the nonzero weights take values in {− , } , we call thegraph unweighted . We say that two vertices i and j are friends ( enemies ) if there is a direct edge with positive (negative)weight connecting them.A sequence j ↔ j ↔ j ↔ · · · ↔ j k ↔ j k +1 isa path of length k connecting j and j k +1 provided that ( j , j ) , ( j , j ) , . . . , ( j k , j k +1 ) ∈ E . A graph is said to be connected if for every pair of distinct vertices i, j ∈ [1 , N ] there is a path connecting j and i . This is equivalent to thefact that the adjacency matrix A is irreducible.The graph G is said to be complete if, for every pair ofnodes ( i, j ) , i (cid:54) = j , i, j ∈ V , there is an edge connecting them,namely ( i, j ) ∈ E . Also, G has a (nontrivial) clustering [6]if it has at least one negative edge and the set of vertices V can be partitioned into say k ≥ disjoint subsets V , . . . , V k such that for every i, j ∈ V p , p ∈ [1 , k ] , [ A ] i,j ≥ , while forevery i ∈ V p , j ∈ V q , p, q ∈ [1 , k ] , p (cid:54) = q , [ A ] i,j ≤ .Two vertices i and j are familiar if they belong to the sameconnected component of the same cluster, namely i, j ∈ V h for some h ∈ [1 , k ] and there exists a path (with all positiveweights) from i to j passing only through vertices of V h .II. k - PARTITE CONSENSUS : P
ROBLEM STATEMENT
We consider a multi-agent system consisting of N agents,each of them described as a continuous-time integrator (see[1], [20], [21], [29], [30]). The overall system dynamics isdescribed as ˙ x ( t ) = u ( t ) , (1)where x ∈ R N and u ∈ R N are the state and input variables,respectively. Assumption 1 on the communication structure. [Connec-tedness and clustering] The communication among the N agents is described by an undirected, signed and weightedcommunication graph G = ( V , E , A ) , where V = [1 , N ] is the set of vertices, E ⊆ V × V is the set of arcs, and A is the adjacency matrix of G that mirrors how agentsinteract. The ( i, j ) -th entry of A , [ A ] i,j , i (cid:54) = j , is nonzeroif and only if the information about the status of the j -thagent is available to the i -th agent. We assume that theinteractions between pairs of agents are symmetric and hence A = A (cid:62) . The interaction between the i -th and the j -th agentsis cooperative if [ A ] i,j > and antagonistic if [ A ] i,j < .Also, [ A ] i,i = 0 for all i ∈ [1 , N ] . We also assume that thegraph G is connected and all the agents are grouped in k ≥ clusters, V i , i ∈ [1 , k ] , with n i = |V i | . The aim of this paper is to propose an extension tothe case of k clusters of the results reported in [1] for structurally balanced graphs , namely graphs with two clusters,by proposing conditions under which agents in the samecluster V i , i ∈ [1 , k ] , reach consensus . In other words, weinvestigate conditions ensuring that the state variables of theagents belonging to the same cluster asymptotically convergeto the same value: lim t → + ∞ x k ( t ) = c i , ∀ k ∈ V i , ∀ i ∈ [1 , k ] , independently of their initial values. When dealing with multi-agent systems with cooperativeand antagonistic relationships, one can use the DeGroot’stype distributed feedback control law [1], [29], [36]: u i ( t ) = − (cid:88) j :( j,i ) ∈E | [ A ] i,j | · [ x i ( t ) − sign([ A ] i,j ) x j ( t )] ,i ∈ [1 , N ] , with sign( · ) the sign function, that corresponds,in aggregated form, to u ( t ) = −L x ( t ) , (2)where L is the Laplacian matrix associated with the adjacencymatrix A , defined as [1], [13], [14]: L := C − A , (3)where C is the (diagonal) connectivity matrix, whose diagonalentries are [ C ] ii = (cid:80) h :( h,i ) ∈E | [ A ] ih | , ∀ i ∈ [1 , N ] . In otherwords [ L ] ij = (cid:40)(cid:80) h :( h,i ) ∈E | [ A ] i,h | , if i = j ; − [ A ] i,j , if i (cid:54) = j. (4)As shown in [1], however, this control law leads to anautonomous multi-agent system ˙ x ( t ) = −L x ( t ) , that may achieve a nontrivial consensus only if the under-lying communication graph is structurally balanced. Thisimmediately implies that if the agents can be partitioned into k ≥ clusters, but not into a smaller number of clusters,then the only possible consensus is the one to the zerovalue. This also means that, in the current set-up, a purelydistributed approach in which each agent uses as informationonly the weights it attributes to the information received byits neighbouring agents, whether they are allies or enemies,cannot lead to consensus if not in a trivial form. So, in thispaper we investigate how to modify the distributed controllaw (2), to achieve consensus when the communication graphis connected and signed, but the agents split into k ≥ disjoint groups.For the sake of simplicity, in the following we will assumethat the agents are ordered in such a way that the agentsbelonging to the cluster V are the first n , the agents inthe cluster V are the subsequent n ... and the agents in thecluster V k are the last n k . This assumption entails no loss ofgenerality, since it is always possible to reduce ourselves tothis structure by means of a relabelling of the nodes/agents.Clearly, n + n + · · · + n k = N . Accordingly, the adjacencymatrix of the graph G is block-partitioned as follows A = A , A , . . . A ,k A , A , . . . A ,k ... ... . . . ... A k, A k, . . . A k,k (5)with A i,j ∈ R n i × n j , A i,i = A (cid:62) i,i ≥ , ∀ i ∈ [1 , k ] , A i,j ≤ ∀ i (cid:54) = j , i, j ∈ [1 , k ] , [ A i,i ] (cid:96),(cid:96) = 0 , ∀ i ∈ [1 , k ] , (cid:96) ∈ [1 , n i ] . We Note that this definition is different from the one adopted in most of thereferences cited in the Introduction. consider a distributed control law for the system (1) of thetype u = −M x , (6)where M ∈ R N × N takes the form M = D − A , (7)with A the adjacency matrix of G and D ∈ R N × N a diagonalmatrix that can be partitioned according to the block-partitionof A , namely D = diag {D , D , . . . , D k } , D i ∈ R n i × n i , (8) n i being the cardinality of the i -th cluster. The overall multi-agent system is hence described as ˙ x ( t ) = −M x ( t ) , (9)and the aim of this paper is to investigate if it is possible tochoose the diagonal matrices D i so that all the agents reach k -partite consensus , by this meaning that for every initialcondition x (0) ∈ R N × N (except for a set of zero measure in R N ) all the state variables, associated to agents in the samecluster, converge to the same value, namely lim t → + ∞ x ( t ) = [ c (cid:62) n , c (cid:62) n , . . . , c n (cid:62) n k ] (cid:62) , (10)for suitable c i = c i ( x (0)) ∈ R , i ∈ [1 , k ] , not all of themequal to zero.The diagonal entries of the matrix D are henceforth our designparameters. Each such entry [ D i ] j ∈ R can be seen as thedegree of “stubbornness" of the j -th agent of the i -th cluster.It quantifies how much the j -th individual in the cluster V i isconservative about its opinion. Note that even if the proposedcontrol scheme is not fully distributed, since the agents willnot be able to autonomously decide the level of stubbornessthey have to adopt in order to guarantee that the final targetis achieved, nonetheless the proposed modification of thestandard control law is minimal, since it only requires theagents to modify the weight that each of them gives to itsown opinion. Note that once the diagonal entries of D havebeen set, the remaining control algorithm is implemented ina purely distributed way.III. k - PARTITE CONSENSUS : P
RELIMINARY RESULTS
In order to provide a solution to the k -partite consensusproblem under certain assumptions on the communicationgraph, we first present a simple lemma that provides necessaryand sufficient conditions for k -partite consensus. The resultis elementary and extends the analogous result for consensusof cooperative multi-agent systems. Also, it has similaritieswith Proposition 6 in [37] derived for cooperative networks. Lemma 1:
A multi-agent system (1), whose communica-tion graph G satisfies Assumption 1, adopting the distributedcontrol law (6), and hence described as in (9), with M ∈ R N × N as in (7), A as in (5), D = diag {D , D , . . . , D k } ∈ R N × N and D i ∈ R n i × n i , i ∈ [1 , k ] , diagonal matrices,reaches k -partite consensus if and only if the followingconditions hold:(1) M is a singular positive semidefinite matrix. (2) The kernel of M is spanned by vectors of the type z = [ α (cid:62) n , . . . , α k (cid:62) n k ] (cid:62) , α i ∈ R , i ∈ [1 , k ] . Proof: [Sufficiency] If M is a singular positive semide-finite matrix, then the system ˙ x = −M x is stable (but notasymptotically stable), and for every x (0) ∈ R N lim t → + ∞ x ( t ) = m (cid:88) i =1 b i v i , (11)where m is the dimension of the eigenspace associated withthe (dominant) zero eigenvalue, b i ∈ R are coefficients thatdepend on the initial conditions and v i ∈ R N are the eigenvec-tors associated with the zero eigenvalue (and they can alwaysbe chosen so that they represent a family of orthonormalvectors). By condition (2), each v i is block-partitioned in k blocks, conformably with the clusters’ dimensions, and hence (cid:80) mi =1 b i v i takes the form [ c (cid:62) n , . . . , c k (cid:62) n k ] (cid:62) .[Necessity] If condition (10) holds for (almost) every x (0) ,then must be the dominant eigenvalue of the matrix −M ,and hence, being a symmetric matrix, it follows that M is(singular and) positive semidefinite. Moreover, as condition(10) has to hold for every x (0) that is an eigenvector of −M corresponding to , this implies condition (2). Remark 2:
By referring to the notation adopted withinthe proof of Lemma 1, we can express the steady state valueof the state variable x ( t ) as x ∗ = lim t → + ∞ x ( t ) = m (cid:88) i =1 ( v (cid:62) i x (0)) v i = m (cid:88) i =1 b i v i , (12)where m is the dimension of the eigenspace associated withthe zero eigenvalue and v i are the orthonormal eigenvectorsassociated with . Note that left and right eigenvectorscoincide, because M is a symmetric matrix.We now introduce some additional assumptions on thecommunication graph that will be used in the followinganalysis, and comment on their meaning. Assumption 2 on the communication structure. [Homo-geneity of trust/mistrust] All the agents in a class V i have thesame constant and pre-fixed amount of trust to be distributedamong their cooperators and distrust, specific for each class V j , j (cid:54) = i , to be distributed among the agents in antagonisticclasses. This translates into assuming that the sums of theelements of the rows belonging to the same block assume thesame value, namely for every i, j ∈ [1 , k ] , A i,j n j = c ij n i ,where c ii ≥ and c ij ≤ , ∀ i (cid:54) = j . Note that even if theadjacency matrix is symmetric, c ij may differ from c ji . Example 1:
Consider the undirected, signed, unweighted,connected and clustered communication graph, with k = 3 clusters of cardinality n = 2 , n = 4 , n = 1 , and adjacencymatrix A = − − −
11 0 0 0 − − − − − − − − − − − − − − − − − It is easy to see that this graph satisfies both Assumption1 and Assumption 2, and the parameters c ij are c =1 , c = − , c = − , c = − , c = 2 , c = − , c = − , c = − , c = 0 . Remark 3:
Assumption 2 may be regarded as a gene-ralization of the concept of equitable partition , originallyintroduced in [8] for undirected, unweighted and unsignedgraphs. In an equitably partitioned (unweighted, unsignedand undirected) graph, in fact, all the agents in the samecluster are restricted to have the same number of neighboursin every cluster, i.e. A i,j n j = c ij n i , ∀ i, j ∈ [1 , k ] , andeach c ij is a nonnegative integer number, representing thenumber of unitary entries in each row of A i,j .Moreover, this assumption is similar to the one introducedin the first part of [35] dealing with cooperative multi-agentsystems (see the Introduction), where it was assumed thatthe blocks A ij , i (cid:54) = j, have constant (and nonnegative) rowsums. Assumption 3 on the communication structure. [Closefriendship] There exist k − distinct indices i , i , . . . , i k − ∈ [1 , k ] such that every cluster V h , h ∈ { i , . . . , i k − } , eitherconsists of a single node/agent or for every pair of distinctagents ( i, j ) ∈ V h × V h either one of the following casesapplies:i) ( i, j ) are friends (the edge ( i, j ) belongs to E and it haspositive weight);ii) ( i, j ) are enemies of two (not necessarily distinct)vertices in V i that are familiar to each other. Thismeans that there exist r, s ∈ V i , and belonging to thesame connected component in V i , such that the edges ( r, i ) and ( j, s ) belong to E (and have negative weights).It is worthwhile to better illustrate this graph property.Conditions i) and ii) amount to saying that either the vertices i and j of V h are connected by an edge or there is a pathconnecting them whose intermediate vertices are all in V i .Figure 1 provides a graphical representation of this property.The property holds for V i h and V i k − , but not for V i k , theremaining set.Fig. 1: Graphical representation of Assumption 3.The idea behind this assumption is that if two agents belongto the same clusters V h , h ∈ { i , i , . . . , i k − } , they have aclose relationship: they are either friends or they are enemiesof agents belonging to the same group of friends in V i .From an algebraic point of view, Assumption 3 states thatfor every h ∈ { i , i , . . . , i k − } and for every ( i, j ) , i (cid:54) = j, with i, j ∈ V h either [ A h,h ] i,j > or there exists t ∈ Z + such that [ A h,i A ti ,i A i ,h ] i,j > . As a consequence, forevery diagonal matrix D i such that D i − A i ,i is positivedefinite (see Lemma 10 in the Appendix), and hence ( D i −A i ,i ) − ≥ , we have that [ A h,h + A h,i ( D i −A i ,i ) − A i ,h ] i,j > , ∀ i (cid:54) = j. (13)By referring to the previous Example 1, it is easy to seethat Assumption 3 trivially holds for every choice of i , i ∈ [1 , , i (cid:54) = i . Note that V consists of a single node, while V and V consist of a single connected component. Also,for every choice of i and i , the restriction of the graph tothe clusters V i and V i is a connected graph.IV. k - PARTITE CONSENSUS : P
ROBLEM SOLUTION UNDERTHE HOMOGENEITY CONSTRAINT
We are now in a position to prove that under the ho-mogeneity constraint imposed by Assumption 2 and theclose friendship hypothesis formalised in Assumption 3, wecan always find suitable choices of the diagonal matrices D i , i ∈ [1 , k ] , that lead the multi-agent system, split into k clusters, to k -partite consensus. In particular, we will showthat we can restrict our attention to scalar matrices and henceassume that D i = δ i I n i , for some δ i ∈ R , i ∈ [1 , k ] . Thisamounts to attributing to agents in the same cluster the samelevel of “stubbornness" or “self-confidence", which is specificfor each decision class.
Theorem 4:
Consider the multi-agent system (1), withundirected, signed, weighted and connected communicationgraph G satisfying Assumptions 1, 2 and 3. Assume thatthe agents adopt the distributed control law (6), with M ∈ R N × N described as in (7), A described as in (5), D =diag {D , D , . . . , D k } ∈ R N × N and D i = δ i I n i , i ∈ [1 , k ] . There exist δ i ∈ R , i ∈ [1 , k ] , such that the closed-loop multi-agent system (9) reaches k -partite consensus, namely (10)holds for suitable c i = c i ( x (0)) ∈ R , i ∈ [1 , k ] . Proof:
We assume without loss of generality that i = 1 ,while i h = h +1 for h = 2 , , . . . , k − . In fact, we can alwaysrelabel the clusters, and accordingly permute the blocks of A , so that this condition is satisfied.By Lemma 1, we need to prove that under the theoremassumptions it is always possible to choose the real parameters δ , δ , . . . , δ k so that (1) the matrix M is singular and positivesemidefinite, and (2) its kernel is spanned by vectors taking theform z = [ α (cid:62) n , α (cid:62) n , . . . , α k (cid:62) n k ] (cid:62) , α i ∈ R , i ∈ [1 , k ] .To this end we first address condition (2). By imposing M z = N we obtain the family of equations α i δ i n i = α i c ii n i + k (cid:88) j =1 ,j (cid:54) = i α j c ij n i , i ∈ [1 , k ] , (14)that can be equivalently rewritten as α i δ i = α i c ii + k (cid:88) j =1 ,j (cid:54) = i α j c ij , i ∈ [1 , k ] , and hence in matrix form as ( D − C ) α α ... α k = 0 , (15)where D = diag { δ , δ , . . . , δ k } and C = c c . . . c k c c . . . c k ... ... . . . ... c k c k . . . c kk . So, if we ensure that D − C is a singular matrix, we necessarilyfind a vector w = [ α , α , . . . , α k ] (cid:62) , such that ( D − C ) w = 0 ,and hence [ α (cid:62) n , α (cid:62) n , . . . , α k (cid:62) n k ] (cid:62) is an eigenvector of M associated with the zero eigenvalue. We will later provethat if such a vector exists, we can also ensure that all theeigenvectors of M associated with the zero eigenvalue arenecessarily multiple of it ( is a simple eigenvalue of M ).We now consider condition (1). To impose it, we makeuse of Lemma 9 in the Appendix, by assuming as matrix R the first block of the matrix MM = D − A , −A , . . . −A ,k −A , D − A , . . . −A ,k ... ... . . . ... −A k, −A k, . . . D k − A k,k (16)and then imposing that R is positive definite, namely condition(17) holds: Φ := D − A , = δ I n − A , (cid:31) , (17)and that its Schur complement is positive semidefinite, namelycondition (18) holds.We note that if we assume δ > c ≥ , (19)then Φ n = ( δ I n − A , ) n (cid:29) . By making use ofLemma 10, part i), in the Appendix for D = δ I n , A = A , and z = n , we can claim that Φ = D − A is positivedefinite, i.e., (17) holds.To ensure that (18) holds, we apply again Lemma 9, andimpose condition (20): Φ := D , − A , − A , Φ − A , (cid:31) , (20)as well as condition (21).To address condition (20), we first observe that by Lemma10, part ii), Φ − = ( D − A , ) − is symmetric andnonnegative, and hence so is A , + A , Φ − A , . But thenwe can apply Lemma 10, part i), again, by assuming D = D and A = A , + A , Φ − A , . Indeed, if we impose thefollowing constraint on δ : δ > c + c c δ − c , (22) D − A , − A , ( D − A , ) − A , −A , − A , ( D − A , ) − A , . . . −A ,k − A , ( D − A , ) − A ,k −A , − A , ( D − A , ) − A , D − A , − A , ( D − A , ) − A , . . . −A ,k − A , ( D − A , ) − A ,k ... ... ... ... −A k, − A k, ( D − A , ) − A , −A k, − A k, ( D − A , ) − A , . . . D k − A k,k − A k, ( D − A , ) − A ,k (cid:23) . (18)————————————————————————————————————————————————- H := D − A , − A , ( D − A , ) − A , . . . −A ,k − A , ( D − A , ) − A ,k ... . . . ... −A (cid:62) ,k − A k, ( D − A , ) − A , . . . D k − A k,k − A k, ( D − A , ) − A ,k − −A , − A , ( D − A , ) − A , ... −A k, − A k, ( D − A , ) − A , · Φ − (21) · (cid:2) −A , − A , ( D − A , ) − A , . . . −A ,k − A , ( D − A , ) − A ,k (cid:3) (cid:23) . ————————————————————————————————————————————————-then it is easy to verify that Φ n = ( D − A ) n = ( δ − c ) n − A , Φ − c n = ( δ − c ) n − c ( δ − c ) − c n (cid:29) , where we used the fact that Φ − n = ( D − A ) − n =( δ − c ) − n . Therefore D − A is positive definite, namely(20) holds.Consider, now, the first block of H in (21): Φ := D − A , − A , Φ − A , − [ A , + A , Φ − A , ] · Φ − [ A , + A , Φ − A , ] . We want to prove that for a suitable choice of δ we canensure that Φ is positive definite and impose that its Schurcomplement is positive semidefinite and singular. We observethat from Assumption 3 (see also (13)) and the properties of ( D − A , ) − it follows that A , + A , ( D − A , ) − A , is a nonnegative matrix whose off-diagonal entries are allpositive. On the other hand, by Lemma 11 we can alwayschoose δ > sufficiently large (something that ensures, inparticular, that (22) is met) to guarantee that the entries of Φ − are arbitrarily small, and hence the entries of [ A , + A , ( D −A , ) − A , ]Φ − [ A , + A , ( D −A , ) − A , ] are arbitrarily small. Therefore, the matrix A = − Φ + D ≈A , + A , ( D − A , ) − A , has positive off-diagonalentries. This ensures that − Φ is an irreducible Metzlermatrix.If we now choose δ such that δ > c + c c δ − c + (cid:16) c + c c δ − c (cid:17) ·· (cid:16) δ − c − c c δ − c (cid:17) − (cid:16) c + c c δ − c (cid:17) (23)we ensure that Φ satisfies Φ n (cid:29) . This proves that Φ is positive definite.To generalise the previous reasonings, we need to find acompact way to express each matrix obtained by means of the previous mechanism (based on Lemma 9 and Lemma 10)that consists of recursively imposing that the first block ispositive definite and the opposite of a Metzler matrix, whileits Schur complement is positive semidefinite, and so on. Weintroduce the following notation: M (0) i,j := A i,j M (1) i,j := A i,j + A i, Φ − A ,j = M (0) i,j + M (0) i, Φ − M (0)1 ,j M (2) i,j := A i,j + A i, Φ − A ,j + [ A i, + A i, Φ − A , ]Φ − [ A ,j + A , Φ − A ,j ]= M (0) i,j + M (0) i, Φ − M (0)1 ,j + M (1) i, Φ − M (1)2 ,j = M (1) i,j + M (1) i, Φ − M (1)2 ,j . This allows to equivalently express the previous matrices Φ and Φ given in (17) and (20) as follows: Φ = D − M (0)1 , Φ = D − M (1)2 , . The previous definitions can be generalised thus leading to M ( h ) i,j := M ( h − i,j + M ( h − i,h Φ − h M ( h − h,j , and each block on the upper left corner recursively obtainedthrough this procedure (consider the first block, then take theSchur complement of the first block, and consider the firstblock of the matrix thus obtained...) can be expressed as Φ h := D h − M ( h − h,h , where M ( h − h,h is a Metzler matrix, provided that δ h − hasbeen suitably chosen not only to make Φ h − positive definite,but also sufficiently large so that Φ − h − is arbitrarily small(this may possibly require to further increase the values of δ , . . . , δ h − chosen at the previous stages) and hence all theoff-diagonal entries of M ( h − h,h are positive, since they canbe well approximated by the off-diagonal entries of A h,h + A h, ( D − A , ) − A ,h which are positive, by assumption. Consequently, also − Φ h is an irreducible Metzler matrix, h ∈ [1 , k − . By imposing Φ h n h (cid:29) , we can determinea lower bound on δ h such that Φ h , h ∈ [1 , k − , is positivedefinite. Once we obtain the last Schur complement (whichis also the last “first block") Φ k := D k − M ( k − k,k , we apply to it the same reasoning as before regarding thechoice of δ k − , to ensure that M ( k − k,k is Metzler. Therefore − Φ k is irreducible, Metzler and Hurwitz. By adopting thefollowing recursive procedure, that mimics in the scalar casethe one previously adopted to generate the matrices M ( h ) i,j and Φ h , m (0) i,j := c i,j φ := δ − c , = δ − m (0)1 , m (1) i,j := c i,j + c i, φ − c ,j = m (0) i,j + m (0) i, φ − m (0)1 ,j φ := δ − c , − c , φ − c , = δ − m (1)2 , m (2) i,j := c i,j + c i, φ − c ,j + [ c i, + c i, φ − c , ] φ − [ c ,j + c , φ − c ,j ]= m (0) i,j + m (0) i, φ − m (0)1 ,j + m (1) i, φ − m (1)2 ,j = m (1) i,j + m (1) i, φ − m (1)2 ,j ... m ( k − i,j := m ( k − i,j + m ( k − i,k − φ − k − m ( k − k − ,j ,φ k := δ k − m ( k − k,k , and assuming δ k = m ( k − kk , we can ensure that Φ k n h = 0 . As − Φ k is Metzler andirreducible, and n k is a strictly positive eigenvector of thismatrix corresponding to , then is a simple and dominanteigenvalue of − Φ k [3]. Since the eigenvalues of M are theunion of the eigenvalues of the positive definite matrices in(17) and (20), etc. and of the positive semidefinite and singularmatrix Φ k , that have been obtained from M by repeatedlyapplying the Schur complement formula with respect to thefirst block, then M is positive semidefinite with a simpleeigenvalue in .Now we observe that all the constraints on the δ i , i ∈ [1 , k ] , that we have derived, can be simply obtained fromthe (non symmetric) matrix D − C by imposing that the (1 , -entry of each of the first k − Schur complements,obtained according to the same algorithm that we used todefine the matrices Φ h , h ∈ [1 , k − , are positive, while the k -th one is zero. Indeed, such (1 , -entries just correspondto the coefficients φ , φ , . . . , φ k . But this implies that if wechoose δ i , i ∈ [1 , k ] , according to the previous algorithm,we also ensure that D − C is singular. Therefore D − C hasan eigenvector w = [ α , α , . . . , α k ] (cid:62) , corresponding to ,and hence [ α (cid:62) n , α (cid:62) n , . . . , α k (cid:62) n k ] (cid:62) is an eigenvector of M associated with the zero eigenvalue. Moreover, since we proved that is a simple eigenvalue, all the eigenvectors of M corresponding to have the desired block structure. Remark 5:
By referring to the previous proof and theterminology adopted within, we can deduce for the diagonalmatrix D the following expression: D = diag { m (0)11 I n , m (1)22 I n , . . . , m ( k − kk I n k } + diag { q I n , q I n , . . . , I n k } , where q , q , . . . , q k − are positive real numbers, sufficien-ty large to ensure that the various matrices − Φ h , h =3 , , . . . , k, have nonnegative off-diagonal entries. Example 2:
Consider, again, Example 1. As previouslyremarked, the communication graph satisfies Assumptions 1,2 and 3 for i = 1 and i = 3 (as in the proof). If we applythe algorithm proposed in the proof of the previous theoremwe obtain the constraints δ > , δ > δ − ,δ = 2 δ − (cid:16) − δ − (cid:17) (cid:16) − δ − (cid:17)(cid:104) δ − − δ − (cid:105) − . If we assume δ = 2 then, independently of δ , one gets δ = 2 . It turns out that for every choice of δ > theeigenvector corresponding to the zero eigenvalue of M is z = [ 1 1 | | − (cid:62) . Figure 2 shows the state evolution of the system described asin (9), with adjacency matrix as in Example 1, with randominitial conditions x ( ) taken as realizations of a gaussianvector with mean and variance σ = 4 , i.e. x (0) ∼ N (0 , .The graph shows that tripartite consensus is reached afterabout . units of time with regime values c = − . , c = 0 , c = 1 . .Alternatively, one can choose δ = 3 , δ = 4 and δ = 2 ,and get as dominant eigenvector ¯ z = [ 0 0 | | − (cid:62) . Fig. 2: Graph with homogeneous trust/mistrust weights:tripartite consensus.
Remark 6:
Theorem 4 applies also when the number ofclusters coincides with the number of agents in the network,i.e. each cluster consists of a single node and all nodes areenemies to each other. Indeed, the homogeneity constraint istrivially satisfied with c ij = A ij = [ A ] ij . V. k - PARTITE CONSENSUS FOR MULTI - AGENT SYSTEMSWITH COMPLETE UNWEIGHTED GRAPH
In this subsection we will focus our attention on multi-agent systems with complete, unweighted and undirectedcommunication graphs that are clustered into an arbitrarynumber k of groups. By resorting to a suitable relabelling ofthe agents, we can always assume that the adjacency matrix A is described as follows A = n (cid:62) n − I n − n (cid:62) n . . . − n (cid:62) n k − n (cid:62) n n (cid:62) n − I n . . . − n (cid:62) n k ... ... . . . ... − n k (cid:62) n − n k (cid:62) n . . . n k (cid:62) n k − I n k , n i being the cardinality of the i -th cluster. Also, in this casewe plan to design a distributed control law for the system(1) of the type (6), with M = D − A , and D ∈ R N × N adiagonal matrix, block-partitioned according to the block-partition of A , namely described as in (8), with D i = δ i I n i .Under the previous hypotheses on the adjacency matrix A , Assumptions 1, 2 and 3 are trivially satisfied. So, theexistence of a suitable choice of the coefficients δ i , i ∈ [1 , k ] , that ensures k -partite consensus follows from the previousTheorem 4. On the other hand, the particular structure of A allows to obtain a much simpler proof as well as an explicitexpression of (a possible choice of) the δ i ’s that cannot beobtained in the general homogeneous case. For this reasonwe provide here an independent proof of this result. Theorem 7:
Consider the multi-agent system (1), withundirected, signed, unweighted and complete communicationgraph G split into k clusters, and adjacency matrix describedas above. Assume that the agents adopt the distributed controllaw (6), with M ∈ R N × N described as in (7), D ∈ R N × N described as in (8) and D i = δ i I n i ∈ R n i × n i , for i ∈ [1 , k ] . Then by assuming δ i = 2 n i − , i ∈ [1 , k ] , (24)we can ensure that the closed-loop multi-agent system (9),reaches k -partite consensus. Proof:
By Lemma 1, we need to prove that underthe theorem hypotheses and by assuming the parameters δ i , i ∈ [1 , k ] , as in (24), we can ensure that (1) thematrix M is singular and positive semidefinite, and (2)its kernel is spanned by vectors taking the block form z = [ α (cid:62) n , α (cid:62) n , . . . , α k (cid:62) n k ] (cid:62) , α i ∈ R , i ∈ [1 , k ] .We first verify condition (2). By assuming δ i , i ∈ [1 , k ] , as in(24), and by imposing M z = N , for z described as above,we obtain the family of equations α i n i − k (cid:88) j =1 ,j (cid:54) = i α j n j = 0 , i ∈ [1 , k ] , (25)that can be equivalently rewritten in matrix form as N k α α ... α k = 0 , (26) where N k = n n . . . n k n n . . . n k ... ... . . . ... n n . . . n k . This is clearly a singular matrix and its kernel coincides withthe set of vectors [ α , α , . . . , α k ] (cid:62) , α i ∈ R , i ∈ [1 , k ] ,such that k (cid:88) i =1 α i n i = 0 . (27)This implies that ker M includes all the vectors z = [ α (cid:62) n , α (cid:62) n , . . . , α k (cid:62) n k ] (cid:62) , with α i ∈ R , i ∈ [1 , k ] , satisfying (27). To prove that all the eigenvec-tors of M corresponding to the zero eigenvalue takethe form [ α (cid:62) n , . . . , α k (cid:62) n k ] (cid:62) , α i ∈ R , i ∈ [1 , k ] , let w = (cid:2) w (cid:62) w (cid:62) . . . w (cid:62) k (cid:3) (cid:62) be any eigenvector of M corresponding to . Then condition M w = N implies n i w i = ( (cid:62) n i w i ) n i − k (cid:88) j =1 ,j (cid:54) = i ( (cid:62) n j w j ) n i , i ∈ [1 , k ] , thus ensuring that w i is a scalar multiple of n i for every i ∈ [1 , k ] . Finally, we want to prove that by assuming δ i , i ∈ [1 , k ] , as in (24) we guarantee that M is positive semidefinite andsingular. We first note that under the previous assumptions M can be rewritten as in (28).If we prove that(A) R := 2 n I n − n (cid:62) n is positive definite, and(B) its Schur complement H , given in (29), is positivesemidefinite and singular,then, by Lemma 9, M will be positive semidefinite andsingular.By Lemma 10 part i), we can claim that, since (2 n I n − n (cid:62) n ) n = n n (cid:29) , (A) holds.Now, we observe that, for any vector z = [ α (cid:62) n , α (cid:62) n , . . . , α k (cid:62) n k ] (cid:62) , with α i ∈ R , i ∈ [1 , k ] , α (cid:54) = 0 , satisfying (27), we have n I n − n (cid:62) n ) α n + n α n + · · · + n α k n k , and hence (2 n I n − n (cid:62) n ) − n = − α ( (cid:80) ki =2 α i n i ) n = 1 n n . (30)This allows to verify that the matrix H takes the blockdiagonal form H = diag { n I n − n (cid:62) n , n I n − n (cid:62) n , . . . , n k I n k − n k (cid:62) n k } . Each diagonal block n i I n i − n i (cid:62) n i , i ∈ [2 , k ] , is easilyseen (by a straightforward extension of Lemma 10) to bepositive semidefinite and singular (with as a simple eigen-value). So, we have shown that M is positive semidefiniteand singular and hence condition (B) holds. Consequently, k -partite consensus is asymptotically achieved. M = n I n − n (cid:62) n n (cid:62) n . . . n (cid:62) n k n (cid:62) n n I n − n (cid:62) n . . . n (cid:62) n k ... ... . . . ... n k (cid:62) n n k (cid:62) n . . . n k I n k − n k (cid:62) n k . (28) H = n I n − n (cid:62) n n (cid:62) n . . . n (cid:62) n k n (cid:62) n n I n − n (cid:62) n . . . n (cid:62) n k ... ... . . . ... n k (cid:62) n n k (cid:62) n . . . n k I n k − n k (cid:62) n k − n (cid:62) n n (cid:62) n ... n k (cid:62) n (2 n I n − n (cid:62) n ) − ·· (cid:2) n (cid:62) n n (cid:62) n . . . n (cid:62) n k (cid:3) . (29)————————————————————————————————————————————————- Example 3:
Consider the multi-agent system (9), withcomplete, unweighted communication graph and 5 clustersof size n = 9 , n = 13 , n = 14 , n = 11 , n = 7 . Wehave assumed δ i = 2 n i − , i ∈ [1 , , and that x (0) is arealization of the gaussian random vector with mean andvariance σ = 4 , i.e. x (0) ∼ N (0 , . The system reaches -partite consensus after about . units of time, with regimevalues c = − . , c = 0 . , c = − . , c =0 . , c = 1 . , as illustrated in Fig. 3.Fig. 3: Complete graph: -partite consensus.VI. k - PARTITE CONSENSUS FOR A CLASS OF NONLINEARMODELS
In the following, an extension of the k -partite consensusanalysis to nonlinear systems is proposed. To this aim, byadopting a set-up similar to the one in [1], we consider amulti-agent system described as in (1), with communicationgraph G = ( V , E , A ) satisfying Assumption 1 and subjectedto the feedback law u = f ( x ) , (31)where f : R N → R N is a Lipschitz continuous functionsatisfying f ( ) = . Assumption 4 on the vector field f : We assume for f adistributed additive expression. Specifically, we assume thateach component f i ( x ) , i ∈ [1 , N ] , of the function f dependsonly on the states of the neighbouring agents of the agent i ,namely for every i ∈ [1 , N ] , the function f i ( x ) depends onlyon those entries x j such that ( j, i ) ∈ E , and is expressed as follows f i ( x ) = − (cid:88) j :( j,i ) ∈E (cid:16) [ D ] i ˜ h i ( x i ( t )) − [ A ] i,j ˜ h j ( x j ( t )) (cid:17) , (32)where [ D ] i is a real number, and the nonlinear function ˜ h k ( · ) is the same for all the agents belonging to the same cluster.So, if we assume that the agents are partitioned into k clustersand ordered in such a way that A is described as in (5), thevector x is accordingly partitioned as x = (cid:2) x (cid:62) x (cid:62) . . . x (cid:62) k (cid:3) (cid:62) . with x i ∈ R n i representing the states of the agents belongingto the i -th cluster. The function f can be expressed as theproduct of the matrix M , given in (7), and of a nonlinearfunction h ( x ) : ˙ x = −M h ( x ) , (33)with h ( x ) = [ h ( x ) (cid:62) h ( x ) (cid:62) . . . h k ( x k ) (cid:62) ] (cid:62) , and h i ( x i ) : R n i → R n i , i ∈ [1 , k ] , described as follows h i ( x i ) = [ h i ( x s i +1 ) h i ( x s i +2 ) . . . h i ( x s i + n i )] (cid:62) , (34) s i = (cid:40) , i = 1; (cid:80) j , x a (cid:54) = x b ,h (0) = 0 , (cid:90) x a x b ( h ( z ) − h ( x b )) dz → ∞ as | x a − x b |→ ∞ (cid:111) . The following theorem provides sufficient conditions for anetworked closed-loop system described as in (33) to reach k -partite consensus that extend those given in Theorem 4.Similarly, the extension of Theorem 7 would be possible. Theorem 8:
Consider the multi-agent system (1), withundirected, signed, weighted and connected communicationgraph G satisfying Assumptions 1, 2 and 3, and distributedcontrol law (31) satisfying Assumption 4 and (32). Conse-quently, the multi-agent system is described as in (33), withthe function h ( x ) defined as above, M ∈ R N × N describedas in (7), D ∈ R N × N described as in (8) and D i = δ i I n i ,for i ∈ [1 , k ] . There exist δ i ∈ R , i ∈ [1 , k ] , such that the closed-loop multi-agent system (33) reaches k -partiteconsensus. Proof:
Clearly, the equilibrium points of system (33)are all the vectors x ∗ in R N such that = M h ( x ∗ ) . Wewant to show that it is possible to choose the coefficients δ i , i ∈ [1 , k ] , so that all the equilibrium points of the systemare block partitioned according to the block partitioning of thematrix M , and they are globally simply stable. This ensuresthat the set of all such equilibrium points is the attractor ofevery state trajectory (there cannot be limit cycles and thetrajectories cannot diverge), and hence the multi-agent systemasymptotically reaches k -partite consensus.We have proved (see Theorem 4) that under Assumptions 1,2 and 3 it is possible to choose the coefficients δ , δ . . . δ k ∈ R so that M is a singular positive semidefinite matrix,having as a simple eigenvalue and the correspondingeigenvector takes the form z = [ α (cid:62) n , α (cid:62) n , . . . α k (cid:62) n k ] (cid:62) ,for suitable α i ∈ R , i ∈ [1 , k ] . This implies that theequilibrium points of the system (33) are the vectors x ∗ such that h ( x ∗ ) ∈ (cid:104) z (cid:105) . As the maps h i belong to R ,for every c ∈ R such that c · α i belongs to the imageof the corresponding h i for every i ∈ [1 , k ] , there exist β , β . . . β k ∈ R such that c · [ α (cid:62) n , α (cid:62) n , . . . α k (cid:62) n k ] (cid:62) = h ([ β (cid:62) n , β (cid:62) n , . . . β k (cid:62) n k ] (cid:62) ) .Suppose, without loss of generality, that this is the case for c = 1 , set x ∗ := [ β (cid:62) n , β (cid:62) n , . . . , β k (cid:62) n k ] (cid:62) , and considera suitably modified version of the Lyapunov function V : R N → R adopted in [1]: V ( x ) = k (cid:88) i =1 s i + n i (cid:88) j = s i +1 (cid:90) x j x ∗ j ( h i ( z ) − h i ( x ∗ j )) dz = k (cid:88) i =1 s i + n i (cid:88) j = s i +1 (cid:90) x j β i ( h i ( z ) − α i ) dz, (36)(see (35) for the definition of s i ) for x (cid:54) = x ∗ . Moreover, V ( x ) is radially unbounded and its derivative is ˙ V ( x ) = k (cid:88) i =1 s i + n i (cid:88) j = s i +1 ( h i ( x j ) − h i ( x ∗ j )) ˙ x j = − ( h ( x ) − h ( x ∗ )) (cid:62) M h ( x ) = − h ( x ) (cid:62) M h ( x ) ≤ , where we used the fact that M = M (cid:62) and M h ( x ∗ ) = M z = 0 , and the last inequality holds since M is asingular positive semidefinite matrix. This ensures that everyequilibrium point x ∗ of the system is globally stable and sinceall such equilibrium points have the required block-structure, k -partite consensus is always guaranteed. Example 4:
Consider the multi-agent system (33), withcomplete, unweighted and undirected communication graph, h ( x ( t )) = tanh( x ( t )) , and 4 clusters of size n = 6 , n =9 , n = 11 , n = 7 . We have assumed that x (0) is arealization of the gaussian random vector with mean andvariance σ = 4 , i.e. x (0) ∼ N (0 , and δ i = 2 n i − forevery i ∈ [1 , . The system reaches -partite consensusafter approximately . time units, with regime values c = 0 . , c = − . , c = − . , c = 2 . , asillustrated in Fig. 4. VII. C ONCLUSIONS
In this work we addressed the consensus problem formulti-agent systems with agents split into k groups: agentsbelonging to the same group cooperate, while those belongingto different ones compete. The proposed algorithm representsa modified version of the classical DeGroot’s type ofconsensus algorithm, where the modification pertains howmuch agents in the same group must be conservative abouttheir opinion in order to guarantee that they converge to acommon decision, depending on their initial opinions, namelythey reach k -partite consensus. The degree of stubborness isshared by all the members of the group. We investigated thisproblem under the assumption that agents in the same clusterhave the same amount of trust(/distrust) to be distributedamong their friends(/enemies) with special focus on the caseof complete, signed, unweighted graph for which a simplifiedsolution is proposed. Also, an extension of the k -partiteconsensus problem to a nonlinear set-up has been investigated.Fig. 4: Non linear model with h ( x ( t )) = tanh( x ( t )) andcomplete graph. The graph above shows the evolution of h ( x ( t )) over time. The graph below shows the evolution ofthe state vector x ( t ) .Future research should focus on how k -partite consensusmay be reached even when the homogeneity assumption doesnot hold. In particular, it would be interesting to investigatewhether there is a way to ensure k -partite consensus ina robust way under weaker assumptions on the coopera-tive/antagonistic relationships, for instance assuming that theagents can be partitioned into k groups such that intra-clustersweights are over a certain thresholds and inter-cluster weightsbelow a specific threshold.A PPENDIX
We present here three technical results that are used severaltimes in the paper. Lemma 9: [4] Let M = (cid:20) R SS (cid:62) Q (cid:21) ∈ R n × n , with R ∈ R k × k and Q ∈ R ( n − k ) × ( n − k ) , be a symmetric ma-trix. If R = R (cid:62) is positive definite and its Schur complement Q − S (cid:62) R − S is positive (semi)definite, then M is positive(semi)definite, and σ ( M ) = σ ( R ) ∪ σ ( Q − S (cid:62) R − S ) . Lemma 10:
Let D ∈ R n × n be a diagonal matrix and let A ∈ R n × n be a symmetric Metzler matrix, then:i) D − A is positive definite if and only if there exists astrictly positive vector z ∈ R n such that ( D − A ) z (cid:29) ;ii) If condition i) holds, then ( D − A ) − ≥ and issymmetric. Proof: i) D − A is positive definite if and only if A − D is negative definite, and since A − D is a symmetric matrixthis is equivalent to saying that A − D is Hurwitz. On theother hand, being a Metzler matrix, A − D is Hurwitz ifand only if [9] there exists a strictly positive vector z ∈ R n such that z (cid:62) ( A − D ) (cid:28) . By the symmetry of D − A , thisinequality is equivalent to ( D − A ) z (cid:29) . ii) As A − D is Metzler Hurwitz, then ( A − D ) − is a matrixwith nonpositive entries [3] and hence ( D − A ) − ≥ . Thefact that the inverse of a nonsingular symmetric matrix issymmetric is an elementary algebraic result. Lemma 11:
Given a scalar ε > and matrices A ∈ R n × n , B ∈ R n × m and C ∈ R m × n , with A Metzler andsymmetric, it is always possible to choose a diagonal matrix D ∈ R n × n such that1) ( D − A ) n (cid:29) n ;2) C ( D − A ) − B is a matrix whose entries satisfy | [ C ( D − A ) − B ] i,j | < ε , ∀ i, j ∈ [1 , m ] . Proof:
Keeping in mind the definition of Lapla-cian associated with the nonnegative matrix ¯ A := A − diag { [ A ] , , [ A ] , , . . . , [ A ] n,n } , it turns out that a diagonalmatrix D ∈ R n × n is such that 1) holds if and only if D − A = ∆ + ¯ L , namely D = ∆+ ¯ L + A, where ¯ L is the (symmetric) Laplacianassociated with ¯ A and ∆ is a diagonal matrix, with positivediagonal entries. We assume ∆ = δI n , δ > . We want toprove that it is always possible to choose δ > so that | [ C ( D − A ) − B ] i,j | < ε , for every i, j ∈ [1 , m ] .We note that ( δI n + ¯ L ) − = 1 δ (cid:18) I n + 1 δ ¯ L (cid:19) − = + ∞ (cid:88) t =0 ( − t δ t +1 ¯ L t , and hence C ( D − A ) − B = + ∞ (cid:88) t =0 ( − t δ t +1 C ¯ L t B. Let T be an orthonormal matrix such that T (cid:62) ¯ L T =diag { δ , δ , . . . , δ n } , with δ ≥ δ ≥ · · · ≥ δ n . Since ¯ L ispositive semidefinite (and singular) the δ i ’s are nonnegative and δ n = 0 . Let c i be the i -th column of CT and b (cid:62) i the i -th row of T (cid:62) B , then C ( D − A ) − B = + ∞ (cid:88) t =0 ( − t δ t +1 n − (cid:88) i =1 c i ( δ i ) t b (cid:62) i = 1 δ n − (cid:88) i =1 c i b (cid:62) i (cid:34) + ∞ (cid:88) t =0 (cid:18) − δ i δ (cid:19) t (cid:35) = n − (cid:88) i =1 c i b (cid:62) i δ + δ i , therefore for every i, j ∈ [1 , m ] , | [ C ( D − A ) − B ] i,j | ≤ ( n − · ψδ , where ψ := max i ∈ [1 ,n − h,k ∈ [1 ,m ] | [ c i b (cid:62) i ] h,k | . Therefore by imposing that ( n − · ψδ (cid:28) ε , namely, δ (cid:29) ( n − · ψε , we ensure that 2) holds.R EFERENCES[1] C. Altafini. Consensus problems on networks with antagonisticinteractions.
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