Group Consensus of Linear Multi-agent Systems under Nonnegative Directed Graphs
11 Group Consensus of Linear Multi-agent Systems underNonnegative Directed Graphs
Zhongchang Liu,
Member, IEEE,
Wing Shing Wong,
Fellow, IEEE
Abstract —Group consensus implies reaching multiple convergencegroups where agents belonging to the same cluster converge. Thispaper focuses on linear multi-agent systems under nonnegative directedgraphs. A new necessary and sufficient condition for ensuring groupconsensus is derived, which requires the spanning forest of the underlyingdirected graph and that of its quotient graph induced with respect to aclustering partition to contain equal minimum number of directed trees.This condition is further shown to be equivalent to containing clusterspanning trees, a commonly used topology for the underlying graph in theliterature. Under a designed controller gain, lower bound of the overallcoupling strength for achieving group consensus is specified. Moreover,the pattern of the multiple synchronous states formed by all clusters ischaracterized by setting the overall coupling strength be large enough.
Index Terms —Group consensus; coupled linear systems; directedspanning trees; graph topology
I. I
NTRODUCTION
Multi-agent systems (MASs), formed by a network of locallycoupled dynamic agent systems, have been continuingly attractingresearch attentions, and have found wide applications in multi-robotsystems, sensor networks, smart grids, social networks, and so on[1]. While prevalent works concentrate on reaching global consen-sus/synchronization for all agents, recently there arises increasinginterest in the problem of group consensus (or cluster consensus,group/cluster synchronization), where coupled systems converge tomultiple synchronous groups instead of one. Researches on groupconsensus are mainly motivated from multi-modal opinion dynamicsin social networks [2] and clustering of oscillatory networks [3], [4],and have potential applications in building power grids and generatingmultiple coupled formations [5].For the problem of reaching global consensus, extensive studieshave been carried out, yielding comprehensive understandings aboutthe underlying connection structures of the agents in terms ofgraph topologies and the control algorithms subject to various agentdynamics [1], [6]–[10]. In contrast, the mechanisms for achievinggroup consensus are not fully understood yet in the literature. Earlyworks such as [11]–[13] presented sufficient algebraic conditionson the graph Laplacian for achieving a prescribed group consensuspattern. Therein, a common assumption on inter-cluster links is thecoexistence of balanced positive and negative weights, which has theeffect of dismissing any group that has achieved consensus internally.With this assumption, subsequent works such as [14]–[19] designeddistributed control algorithms to cope with different types of agentdynamics. Apart from control algorithms, this stream of studiesmainly rely on two conditions for ensuring cluster consensus: (a)the underlying topology of each cluster should contain a spanning *This work is supported by the National Natural Science Foundationof China (61703445), Natural Science Foundation of Liaoning Province(20180540064), Innovation Support Program for Dalian High-level Tal-ents (2019RQ057), Dalian Science and Technology Innovation Fund(2019J12GX040), and the Hong Kong Innovation and Technology Fund(ITS/066/17FP) under the HKUST-MIT Research Alliance Consortium.Z. Liu is with College of Marine Electrical Engineering, Dalian MaritimeUniversity, Dalian, 116026, P. R. China (Email: [email protected]).W. S. Wong is with Department of Information Engineering, The Chi-nese University of Hong Kong, Shatin, N.T., Hong Kong (Email: [email protected]). tree, and (b) the intra-cluster couplings should be strong enough.On the other hand, for MASs that have all edge weights beingnonnegative, the in-degrees of all nodes in the same cluster from anyother cluster should be equal (i.e., the so-called inter-cluster commoninfluence condition) so as to maintain the group consensus manifoldsinvariant [20]. Under this framework, the underlying topology thatcontains cluster spanning trees was proved to be necessary andsufficient for bidirectionally connected chaotic oscillators in [20],and for unweighted undirected/balanced network of discrete-timesingle integrators in [23]. For general nonnegative digraphs, thistopology was taken as a sufficient condition when enforcing clusterconsensus for agents described by single integrators in discrete time[21] and in continuous time under time-varying topologies [22], andfor agents with generic linear dynamics [24], [25]. In [24], [25], theauthors also showed its necessity when the structure of inter-clusterconnections does not contain any cycle. However, the necessityfor general nonnegative digraphs remains unconfirmed to the bestknowledge of the authors. Other relevant studies focused on thegroup consensus patterns that may emerge in undirected networks orunweighted digraphs from perspectives including group theory [26]–[28] and equitable partitions of graphs [29], [30] without specifyingthe topology of the underlying network. In summary, althoughremarkable results have been reported, there still lacks a unifiedknowledge about the necessary features of nonnegatively weightedunderlying digraphs for ensuring group consensus. In addition, thecharacteristics of the synchronized states in the clusters are rarelyspecified except for MASs with individual dynamics described bysingle integrators [30].Focusing on generic linear MASs under nonnegatively weighteddigraphs, this paper first constructs the quotient graph of the under-lying graph with respect to some given node partition that satisfiesthe inter-cluster common influence condition. The quotient graphcharacterizes the inter-cluster structure, and its Laplacian is shownto be decomposable from the Laplacian of the full underlyinggraph by using similarity transformations. Further invoking existingtheory about m -reducible Laplacians ( [31]) results in a necessaryand sufficient graph topology for ensuring group consensus, whichrequires the spanning forest of the underlying graph and that ofits quotient graph to contain equal minimum number of directedtrees. This condition is shown to be equivalent to containing clusterspanning trees for the underlying graph, and has the distinctive featureof being verifiable without looking into the connection details insideany cluster. Under a designed controller gain for individual linearsystems, the lower bound of the overall coupling strength that canensure group consensus is also specified. Finally, the synchronizedstates in the clusters are characterized, which can exhibit a patternsimilar to that in [30] when the overall coupling strength is largerthan a second bound, while may not when the coupling strength liesin between the group consensus lower bound and the latter bound. Notation : For a set S , its cardinality is denoted by |S| . I n isthe identity matrix of dimension n . n = [1 , , . . . , T ∈ R n . blockdiag { M , . . . , M n } represents the block diagonal matrix con-structed by matrices M , . . . , M n . The symbol “ ⊗ ” stands for theKronecker product. For a square matrix M , its spectrum is denoted a r X i v : . [ ee ss . S Y ] F e b by σ ( M ) , and the real part of its eigenvalue is denoted by Reλ ( M ) .II. P ROBLEM S TATEMENT
Consider a multi-agent system (MAS) consisting of L agentsindexed by the set I = { , . . . , L } . The individual dynamics ofeach agent is described by the following generic linear system model ˙ x l ( t ) = Ax l + Bu l ( t ) , l ∈ I , (1)where x l ( t ) ∈ R n is the state of agent l with initial value x l (0) , u l ( t ) ∈ R n u is the control input, A ∈ R n × n , B ∈ R n × n u , and ( A, B ) is a stabilizable pair.These L agents belong to N distinct clusters denoted by the indexsets C i , i = 1 , . . . , N . Assume without loss of generality that eachcluster C i contains l i ≥ agents ( (cid:80) Ni =1 l i = L ), and the indices arearranged such that C = { , . . . , l } , . . . , C i = { ρ i + 1 , . . . , ρ i + l i } , . . . , C N = { ρ N + 1 , . . . , ρ N + l N } , where ρ = 0 and ρ i = (cid:80) i − j =1 l j , i = 2 , . . . , N . Hence, the set C = {C , . . . , C N } is anontrivial partition of the index set I , and is called a clustering ofthe above multi-agent system. Two distinct agents, l and k in I , aresaid to belong to the same cluster C i if l ∈ C i and k ∈ C i . A. The Group Consensus Problem
In this study, the agents are supposed to be linearly coupled throughtheir control inputs: u l ( t ) = δK (cid:88) k ∈I w lk [ x k ( t ) − x l ( t )] , l ∈ C i , i = 1 , . . . , N (2)where δ > is the overall coupling strength used to compensatefor the underlying topology, K is the controller gain matrix to bedetermined, and w lk ≥ is the weight of the link from agent k toagent l . Then the closed-loop equations for (1) are described by ˙ x l ( t ) = Ax l − δBK L (cid:88) k =1 (cid:96) lk x k ( t ) , l ∈ C i , i = 1 , . . . , N (3)where (cid:96) ll = (cid:80) k ∈I w lk and (cid:96) lk = − w lk for any k (cid:54) = l . Concatenatingvariables in x ( t ) = [ x T ( t ) , . . . , x TL ( t )] T ∈ R nL , we can write (3)into the following compact form: ˙ x ( t ) = ( I L ⊗ A − δ L ⊗ BK ) x ( t ) (4)where L = [ (cid:96) lk ] . Definition 1:
The multi-agent system in (4) achieves group con-sensus with respect to (w.r.t.) the clustering C if for any x l (0) ∈ R n , l ∈ I , lim t →∞ (cid:107) x l ( t ) − x k ( t ) (cid:107) = 0 , ∀ k, l ∈ C i , i = 1 , . . . , N .Note that group consensus problems do not require the con-sensus states in different clusters to be distinct [12], [17], whichare equivalent to the definition of intra-cluster consensus in clusterconsensus problems [20]–[22]. Considering that group consensus isthe prerequisite of reaching cluster consensus and state separationsfor different clusters can be enforced by some extra techniques asused in [21], [22], this study focuses on the fundamental problem ofreaching group consensus only.It is trivial to see that the group consensus problem can besolved if the MAS can achieve global consensus for their states, i.e., lim t →∞ (cid:107) x l ( t ) − x k ( t ) (cid:107) = 0 , ∀ k, l ∈ I . However, global consensusis only a special case of group consensus. The goals of this paper areto reveal general graph topologies that can ensure group consensusfor the MAS (4), and to further shed some light on the patterns ofthe achieved multiple consensus states.
213 4 G G G G
213 4 G G G G G G G G G G
512 6 G G G G G G G G G G bc c L = ¡ b b ¡ c c ¡ c ¡ c + 1 L = ¡ b b ¡ c c ¡ c ¡ c + 1
213 4 G G G G
11 11 3 21 2 121 1214 5 G G G G
11 1 L = ¡ ¡ ¡ ¡ L = ¡ ¡ ¡ ¡ L F L R (a) A graph G partitioned into threesubgraphs G , G , G .
213 4 G G G G
213 4 G G G G
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512 6 G G G G G G G G G G bc c L = ¡ b b ¡ c c ¡ c ¡ c + 1 L = ¡ b b ¡ c c ¡ c ¡ c + 1
213 4 G G G G
11 1
121 1214 5 G G G G
11 1 L = ¡ ¡ ¡ ¡ L = ¡ ¡ ¡ ¡ L F L R (b) The quotient graph G inducedfrom G w.r.t. the partition in (a).Fig. 1. Interaction graph and its quotient graph. B. Useful Graph Theory
A directed graph (digraph) G = ( V , E , A ) is associated with theMAS (1) such that each agent is considered as a node in the nodeset V , while connections among agents correspond to directed edgesin E ⊂ V × V . The adjacency matrix A = [ w lk ] is defined suchthat w lk > if there is a directed edge from agent k to agent l ,and w lk = 0 , otherwise. The in-degree of a node l is the quantity (cid:80) k ∈I w lk . The Laplacian matrix of G is L = [ (cid:96) lk ] with each entry (cid:96) lk being defined in (3). The digraph G is weakly connected if thegraph derived via replacing all directed edges of G with undirectededges is connected. A directed spanning tree of G is a directed treethat contains all the nodes through directed paths in G . A directedspanning forest of G is a digraph consisting of one or more directedtrees that together contain all the nodes of G , but no two of whichhave a node in common. G is said to contain cluster spanning trees w.r.t. the clustering C if for each cluster C i , i = 1 , . . . , N , there existsa node in V which can reach all nodes with indices in C i throughdirected paths in G . Note that the paths used to span a cluster ofnodes may contain nodes belonging to other clusters, and of coursecan also follow inter-cluster edges.It is well-known that a strongly connected graph has an irreducibleLaplacian matrix. For a general graph topology, we invoke from [31]the results of an m -reducible Laplacian matrix in the following. Lemma 1 ( [31]):
Let M ∈ R N × N be a reducible Laplacian matrixof a nonnegative digraph. The following statements are equivalent forany m ∈ { , . . . , N } :(a) M is m -reducible.(b) The zero eigenvalue of M has multiplicity m , and all the othereigenvalues have positive real parts.(c) m is the minimum number of directed trees which together spanthe digraph.By this lemma, a -reducible Laplacian matrix corresponds toa graph that contains a directed spanning tree but is not stronglyconnected. C. Assumptions
Corresponding to the clustering C = {C , . . . , C N } , let the sub-graph of G , denoted by G i , contain all the nodes with indices in C i and the edges connecting them directly (all inter-cluster links areexcluded from G i , and see Fig. 1(a) for an illustration). Then, theLaplacian matrix L of G can be partitioned into the following block-matrix form: L = L L · · · L N L L · · · L N ... ... . . . ... L N L N · · · L NN , (5)where each diagonal block L ii ∈ R l i × l i specifies intra-cluster interactions, and each off-diagonal block L ij ∈ R l i × l j with i (cid:54) = j , i, j = 1 , . . . , N specifies inter-cluster interactions from nodes incluster C j to nodes in C i .To ensure group consensus, the Laplacian L is assumed to satisfythe following condition. Assumption 1:
Every block L ij of the Laplacian L defined in (5)has a constant row sum β ij , i.e. L ij l j = β ij l j , for i, j = 1 , . . . , N .An equivalent description of the above condition is that (cid:80) k ∈C j w lk = (cid:80) k ∈C j w l (cid:48) k for any l (cid:54) = l (cid:48) in C i , and i (cid:54) = j , i.e.,the in-degrees of all nodes in a cluster with respect to another clusterare equivalent. Any clustering C that renders the graph Laplacian L satisfying Assumption 1 is also called an almost equitable partition(AEP) of G [29], [30]. It has been shown in the literature (such as[20], [21], [29]) that Assumption 1 is necessary for the group consen-sus manifold { x ( t ) ∈ R nL : x l ( t ) = x k ( t ) , ∀ l, k ∈ C i , i = 1 . . . , N } to be invariant. An intuitive reasoning is that under this conditiondifferent agents in the same cluster will receive equivalent influencefrom another cluster in the group consensus manifold. Hence, As-sumption 1 is also called the inter-cluster common influence conditionin [20]–[22].In the following presentation, two more basic assumptions aremade to exclude trivial cases. One is that G is at least weaklyconnected (i.e. contains no isolated component) so as to exclude theapparently infeasible graph topologies where agents belonging to thesame cluster happen to reside in different isolated components of thenetwork. The other assumption is that the system matrix A has atleast one eigenvalue located in the closed right half-plane so as toavoid reaching trivial global consensus all the time.III. A CHIEVING G ROUP C ONSENSUS
This section will establish the conditions for ensuring groupconsensus for MAS (4) by bridging the connectivity of G , whichdescribes inter-agent connections, with the connectivity of its inducedquotient graph G w.r.t. C , which describes inter-cluster interactions.The nomenclature of quotient graph follows from [29], in whichthe definition relies on the characteristic matrix that describes thelocalization of each node in each cluster. In the following, we givean intuitive definition of this graph through construction. Definition 2:
Given a graph G and its partition {G , . . . , G N } w.r.t.the clustering C , the quotient graph G induced from G is constructedthrough the following steps:1) collapsing each subgraph G i into a single node with index i ;2) defining a directed edge from node i to node j in G if and onlyif there exists at least one directed edge in G pointing from anode in G i to a node in G j ;3) defining the weight of an edge from node j to node i in G as α ij = 1 l i (cid:88) l ∈C i (cid:88) k ∈C j w lk , i, j = 1 , , . . . , N. (6)Under Assumption 1, each edge weight of G will reduce to α ij = (cid:80) k ∈C j w lk for any l ∈ C i (See Fig. 1(b) for an example.). Also, eachconstant row sum β ij defined in Assumption 1 can be computed by β ij = − α ij for i (cid:54) = j , and β ii = (cid:80) Nj =1 α ij . It follows that underAssumption 1 the Laplacian of G is defined as follows: L G = [ β ij ] i,j =1 ,...,N . (7) A. Necessary And Sufficient Graph Topologies
For each i ∈ { , . . . , N } , define e l ( t ) = x l ( t ) − x ρ i +1 ( t ) as thestate difference between the agent ρ i + 1 in cluster C i and any other agent l ∈ C i \ { ρ i + 1 } . It follows from (3) that ˙ e l ( t ) = Ae l − δBK L (cid:88) k =1 ( (cid:96) lk − (cid:96) ρ i +1 ,k ) x k ( t ) , = Ae l − δBK N (cid:88) j =1 (cid:88) k ∈C j ( (cid:96) lk − (cid:96) ρ i +1 ,k )[ x k ( t ) − x ρ j +1 ( t )]= Ae l − δBK N (cid:88) j =1 (cid:88) k ∈C j ( (cid:96) lk − (cid:96) ρ i +1 ,k ) e k ( t ) (8)where the second equality is valid since for any j ∈ { , . . . , N } and any x ρ j +1 ∈ R n , (cid:80) k ∈C j ( (cid:96) lk − (cid:96) ρ i +1 ,k ) x ρ j +1 = ( (cid:80) k ∈C j (cid:96) lk − (cid:80) k ∈C j (cid:96) ρ i +1 ,k ) x ρ j +1 = 0 due to Assumption 1. Stacking the statedifference vectors e l ( t ) , l ∈ C i \ { ρ i + 1 } , i = 1 , . . . , N in e ( t ) = [ e Tρ +2 ( t ) , . . . , e Tρ + l ( t ) , · · · , e Tρ N +2 ( t ) , . . . , e Tρ N + l N ( t )] T , one can get from (8) that ˙ e ( t ) = ( I L − N ⊗ A − δ ˆ L ⊗ BK ) e ( t ) , (9)where ˆ L ∈ R ( L − N ) × ( L − N ) is in the following block-matrix form ˆ L = [ ˆ L ij ] i,j =1 ,...,N , (10)with each block ˆ L ij ∈ R ( l i − × ( l j − being defined by ˆ L ij = ˜ L ij − l i − γ Tij , i, j = 1 , . . . , N, (11)where γ ij = [ (cid:96) ρ i +1 ,ρ j +2 , · · · , (cid:96) ρ i +1 ,ρ j + l j ] T ∈ R l j − , (12) ˜ L ij = (cid:96) ρ i +2 ,ρ j +2 · · · (cid:96) ρ i +2 ,ρ j + l j ... . . . ... (cid:96) ρ i + l i ,ρ j +2 · · · (cid:96) ρ i + l i ,ρ j + l j ∈ R ( l i − × ( l j − . (13)It is clear from (9) that group consensus can be achieved for anyinitial state x (0) ∈ R nL if and only if I L − N ⊗ A − δ ˆ L ⊗ BK isHurwitz, i.e., the state motions transversal to the group consensusmanifold are stable. By using properties of Kronecker products, thestability of I L − N ⊗ A − δ ˆ L⊗ BK can be ensured by the stabilities of A − δλ l ( ˆ L ) BK for all λ l ( ˆ L ) ∈ σ ( ˆ L ) . It follows that ˆ L plays a keyrole in rendering group consensus. The following lemma specifies itsstability by means of graph topologies. Lemma 2:
Under Assumption 1, all eigenvalues of ˆ L have positivereal parts if and only if the spanning forest of G and that of thequotient graph G contains equal minimum number of directed trees. Proof:
For i = 1 , . . . , N , define S i = (cid:20) l i − I l i − (cid:21) ∈ R l i × l i with inverse S − i = (cid:20) − l i − I l i − (cid:21) . By Assumption 1, one getsthat S − i L ij S j = (cid:20) β ij γ ij L ij (cid:21) , where β ij is the entry of L G definedin (7), and γ ij and ˆ L ij are defined in (12) and (11), respectively. Let S = blockdiag { S , . . . , S N } . Then, one has the following S − L S = β γ · · · β N γ N L · · · L N ... ... . . . ... ... β N γ N · · · β NN γ NN L N · · · L NN . Permutating the columns and rows of S − L S , one can get thefollowing block upper-triangular matrix (cid:20) L G [ γ ij ] i,j =1 ,...,N ( L − N ) × N ˆ L (cid:21) . (14) It follows that the matrix ˆ L is nonsingular with all eigenvalues havingpositive real parts if and only if the two Laplacians L and L G haveequal number of zero eigenvalues. Further using Lemma 1 (b) and(c) yields the conclusion of this lemma.As a stabilizable pair ( A, B ) , for any Q > there is a positivedefinite P > satisfying the following algebraic Riccati equation P A + A T P − P BB T P = − Q. (15)Using the above, we can derive the main result in the following. Theorem 1:
Under Assumption 1, the MAS (4) can achieve groupconsensus w.r.t. C for any initial state x (0) ∈ R nL if and only if L G and L have equal number of zero eigenvalues, or equivalently, thespanning forest of G and that of its quotient graph G contain equalminimum number of directed trees. Proof:
For the sufficiency part, if the conditions in Theorem 1hold, then
Reλ l ( ˆ L ) > for each λ l ( ˆ L ) ∈ σ ( ˆ L ) by Lemma 2. Thenone can choose δ ≥ / min l Reλ l ( ˆ L ) , (16)and let K = B T P where P is defined in (15), such that for each λ l ( ˆ L ) ∈ σ ( ˆ L ) , there holds ( A − δλ l ( ˆ L ) BK ) ∗ P + P ( A − δλ l ( ˆ L ) BK )= A T P + P A − δReλ l ( ˆ L ) P BB T P = − Q − (2 δReλ l ( ˆ L ) − P BB T P ≤ − Q. (17)Hence, A − δλ l ( ˆ L ) BK is Hurwitz for each l = 1 , . . . , L − N whichimplies I L − N ⊗ A − δ ˆ L ⊗ BK is Hurwitz.On the other hand, the violation of the condition in Theorem1 implies by Lemma 2 that the matrix ˆ L has at least one zeroeigenvalue, i.e., there exists at least one l ∗ ∈ { , . . . , L − N } suchthat λ l ∗ ( ˆ L ) = 0 . It turns out that A − δλ l ∗ ( ˆ L ) BK = A is notHurwitz, which implies that group consensus cannot be guaranteedfor all initial states.As seen in the proof of Theorem 1, the positivity of quantity min l Reλ l ( ˆ L ) determines the feasibility of a graph topology forensuring group consensus, while its value determines the overallcoupling strength demanded. The role of this quantity in groupconsensus problems is comparable with that of the minimum realpart of nonzero eigenvalues of the Laplacian L (i.e., min l (cid:54) =1 Reλ l ( L ) where λ ( L ) = 0 ) in global consensus problems [8], [9].The conditions presented in Theorem 1 offer quite a straightfor-ward method to verify group consensusability of an MAS by com-paring properties of the full underlying graph and its quotient graph.As mentioned in the Introduction, previous studies of group/clusterconsensus problems such as [20]–[23] rely on the condition ofcontaining cluster spanning trees w.r.t. a clustering for G . In thefollowing, it is interesting to show in Theorem 2 that this conditionis actually equivalent to that in Theorem 1 after in-depth inspectionson the relations between G and its quotient graph G . B. An Alternative Condition
Subsequent presentation needs the following definitions [30], [32].For any node i in G , a set R ( i ) is a reachable set of i if it contains i and all nodes j that can be reached starting from i via a directedpath in G . A set R p is called a reach if R p = R ( i ) for some i and there is no j such that R ( i ) ⊂ R ( j ) , and the node i is calleda root of this reach. Suppose R p , p = 1 , . . . , m , are the reachesthat together cover all nodes of G . It is clear that if G contains m reaches, then its Laplacian L G is m -reducible. For each reach R p ,the set V p = R p \ ∪ q (cid:54) = p R q is called the exclusive part of R p , and theset F p = R p \ V p denotes the common part of R p . Let F = ∪ mp =1 F p be the union of the common parts. Then, there exists a labeling ofnodes of G such that its Laplacian can be written into the followinglower-triangular form [30]. L G = V . . . V m F · · · F m F (18)where each V p is a Laplacian matrix associated with V p , F is asquare matrix associated with F , and F p ’s are matrices of compatibledimensions.Now we can present the following lemma and theorem which willbe used in the remaining parts of this paper. Their proofs can befound in Appendix A. Lemma 3:
Suppose the Laplacian L of G satisfies Assumption 1,and the associated L G takes the form (18) for some ≤ m < N .Then the graph G contains cluster spanning trees w.r.t. C if and onlyif for each set of subgraphs {G i | i ∈ V p } , p = 1 , . . . , m , the nodestherein can be spanned by a directed spanning tree in G . Theorem 2:
Suppose the Laplacian L of G satisfies Assumption1. The spanning forest of G and that of the quotient graph G w.r.t. C have equal minimum number of directed trees if and only if G contains cluster spanning trees w.r.t. C .A direct combination of Theorem 1 and Theorem 2 leads to thefollowing alternative of Theorem 1. Theorem 3:
Under Assumption 1, the multi-agent system (4) canachieve group consensus w.r.t. C for any initial state x (0) ∈ R nL ifand only if G contains cluster spanning trees w.r.t. C . Remark 1:
In [21], [22], containing cluster spanning trees for adirected underlying graph is found to be a sufficient graph conditionfor achieving group consensus. Its necessity is partly proved by theauthors in [24] for the special case that the quotient graph G isacyclic, which can benefit from the tree-like structure. This paperfurther consolidated this condition as a necessary and sufficient onefor general nonnegative digraphs through comprehensive proofs. Incomparison to checking cluster spanning trees, the new derived graphtopological condition in Theorem 1 is easier to check since thecoupling details inside the clusters are not involved. Moreover, thecondition of comparing the number of zero eigenvalues of the twoLaplacians is also a straightforward algebraic criterion.IV. S YNCHRONIZED S TATES IN C LUSTERS
As is know, if the underlying topology G contains a directed span-ning tree and the inter-agent couplings are strong enough, the MAS(4) can achieve global consensus with x ( t ) → ( L ν T ⊗ e At ) x (0) where ν ∈ R L is the left eigenvector of L such that ν T L = 0 and ν T L = 1 [9]. In this paper, we are interested to see the synchronizedstates in different clusters when the underlying graph of an MASshould be spanned by multiple trees together. To this end, we assumethe Laplacian L G is in the form of (18) for some < m < N . Then,the corresponding Laplacian L of digraph G can be written in thefollowing form: L = L . . . m L m +1 , · · · L m +1 ,m L F (19)where each L p , p = 1 , . . . , m , is the Laplacian associated with nodesin ¯ C p = ∪ i ∈ V p C i , and L F is a square matrix associated with nodesin F = ∪ i ∈ F C i . Lemma 4: If G contains cluster spanning trees, then each L p for p = 1 , . . . , m contains exactly one zero eigenvalue, and the matrix L F is nonsingular with all eigenvalues having positive real parts . Proof:
The first half part follows immediately from Lemma 3and Lemma 1. By Theorem 2, L has m zero eigenvalues totally, thesame number with L G . Therefore, L F must be nonsingular.Denote R = ∪ mp =1 ¯ C p , and rewrite (19) as follows L = (cid:20) L R L FR L F (cid:21) (20)where L R = blockdiag { L , . . . , L m } , and L FR =[ L m +1 , , . . . , L m +1 ,m ] . Similarly, the state vector x ( t ) is alsorepresented as follows x ( t ) = [ x T R ( t ) , x T F ( t )] T . (21)Then, we can derive the final states in each cluster when the overallcoupling strength δ is large enough. Theorem 4:
Under Assumption 1, if the underlying graph G ofthe MAS (4) contains cluster spanning trees w.r.t. C , by selecting K = B T P and δ ≥ λ , λ := min { Reλ l ( L ) : λ l ( L ) (cid:54) = 0 , ∀ l ∈ I} , (22)the state x ( t ) will asymptotically approach the following ( L ν T ⊗ e At ) x (0) , if L is irreducible or -reducible (23a) (cid:20) Ξ ⊗ e At −L − F L FR Ξ ⊗ e At (cid:21) x (0) , if L has the form (19) (23b)as t → ∞ , where Ξ = blockdiag { µ ν T , . . . , µ m ν Tm } with µ p and ν p satisfying ν Tp µ p = 1 being the right and left eigenvector of L p associated with the zero eigenvalue, respectively. Proof:
We only need to prove the case that L is reducible andtakes the form of (19) for some m > . Let n p = | ¯ C p | = (cid:80) i ∈ V p l i be the number of nodes in ¯ C p , and let t = 0 , t p = (cid:80) p − q =1 n q for p =1 , . . . , m . Denote ¯ x p = [ x Tt p +1 , . . . , x Tt p + n p ] T for p = 1 , . . . , m . Itfollows from (4) and (19) that ˙¯ x p ( t ) = ( I ⊗ A − δ L p ⊗ BK )¯ x p ( t ) , p = 1 , . . . , m. (24)If G contains cluster spanning trees w.r.t. C , by Lemma 4, L p hasone zero eigenvalue λ ( L p ) = 0 , and other eigenvalues satisfy min l p (cid:54) =1 Reλ l p ( L p ) ≥ λ . Further using the inequality in (22) yieldsthat δ ≥ / min l p (cid:54) =1 Reλ l p ( L p ) . Hence, one can use methods in[9] to get that for p = 1 , . . . , m , ¯ x p ( t ) → ( µ ν T ⊗ e At )¯ x p (0) , as t → ∞ . (25)It follows that x R ( t ) → (Ξ ⊗ e At ) x R (0) , as t → ∞ . (26)To derive the state of x F ( t ) when t → ∞ , we define the followingtwo variables following (20) and (21): (cid:20) ζξ (cid:21) = ( L ⊗ I n ) x = (cid:20) L R L FR L F (cid:21) (cid:20) x R x F (cid:21) . (27)By (26) and using the fact L R Ξ = 0 , one has that ζ ( t ) = ( L R ⊗ I n ) x R → ( L R ⊗ I n )(Ξ ⊗ e At ) x R (0) = ( L R Ξ ⊗ e At ) x R (0)= 0 , as t → ∞ . (28) Using (4) and (27), we have that (cid:20) ˙ ζ ˙ ξ (cid:21) = ( L ⊗ I n ) ˙ x = ( L ⊗ I n )( I L ⊗ A − δ L ⊗ BK ) x ( t )= ( I L ⊗ A − δ L ⊗ BK ) (cid:20) ζξ (cid:21) (29)It follows from (20) and (21) that ˙ ξ = ( I ⊗ A − δ L F ⊗ BK ) ξ − ( δ L FR ⊗ BK ) ζ. (30)By Lemma 4, there holds Reλ l ( L F ) > , ∀ l . It follows that λ ≤ min l Reλ l ( L F ) , which combining (22) implies that δ ≥ / min l Reλ l ( L F ) . Then, through a similar algebra as in (17), onecan get that A − δλ l ( L F ) BK = A − δλ l ( L F ) BB T P is Hurwitzfor each λ l ( L F ) ∈ σ ( L F ) . That is, I ⊗ A − δ L F ⊗ BK is Hurwitz.Next, solving (30) with (28), one can obtain that ξ ( t ) approacheszero asymptotically. Since ξ = L FR x R + L F x F by (27), it thenfollows from (26) that x F ( t ) → − ( L − F L FR ⊗ I n ) x R ( t ) → − ( L − F L FR Ξ ⊗ e At ) x R (0) , as t → ∞ . (31)Combining (26) and (31) yields the state of x ( t ) in (23b) when L takes the form (19). This completes the proof. Remark 2:
As seen from (23), all clusters C i ’s with i ∈ V p eventually achieve a common synchronous state for p = 1 , . . . , m .For clusters in F , note from (31) that their states x F ( t ) eventuallyenter into the convex hull of x R ( t ) . To see this, by [ L FR L F ] L = L FR |R| + L F |F| = , one has −L − F L FR |R| = |F| where −L FR is a nonnegative matrix and L − F is also a nonnegative matrixsince L F is a nonsingular M -matrix [33]. Hence, −L − F L FR is rowstochastic. This pattern is consistent with that of the MAS with pointmodel (i.e., A = 0 , B = 1 , δ = 1 ) and unweighted digraph [30]. Notethat (23) contains the minimum number of distinct synchronous statesthat can persist, in the sense that no synchronous states in (23) willbe merged by further increasing the overall coupling strength δ . Onthe other side, if δ is decreased such that Reλ ( ˆ L ) ≤ δ < λ ,the generic MAS can achieve group consensus by Theorem 1 butthere is no guarantee for x F ( t ) to enter the convex hull of x R ( t ) due to insufficient inter-cluster coupling strengths compared with theunstable modes of the system matrix A . This differs from MASswith simple integrator models whose final states are irrelevant withthe overall coupling strength [30]. A. Simulation Example
To illustrate the synchronized states, we present a simulationexample for an MAS consisting of agents that belong to clusters C = { , } , C = { , } , C = { , } , C = { , } , C = { , } .The underlying graph G is given in Fig. 2, which contains clusterspanning trees w.r.t. the clustering C = {C , C , C , C , C } , and itsLaplacian L satisfies Assumption 1. The dynamics of the agents aredescribed by harmonic oscillators, that is, for i = 1 , . . . , , (cid:26) ˙ x l ( t ) = x l ( t ) (32a) ˙ x l ( t ) = − x l ( t ) + u l ( t ) . (32b)Selecting Q = I , and solving the algebraic Riccati equation (15), weobtain the controller gain K = B T P = [0 . , . .It is computed that min Reλ ( ˆ L ) = 1 . > and λ = 0 . . Hence,we first set δ = 1 / (2 λ ) = 2 . according to (22) in Theorem 4. Withrandomly generated initial states, the simulated trajectories of the agents are shown in Fig. 3, in which the states of agents form threegroups in such a way that clusters C , C , and C merge into onestate, while C and C each achieves a distinct synchronous state.Note also that the synchronized states of C lie in between the states of clusters C and C when t is large enough. Next, we use a smallervalue for δ by setting δ = 1 / min 2 Reλ ( ˆ L ) = 0 . according to(16). Simulation results in Fig. 4 show that five groups of distinctstates are formed eventually complying with the partition C , i.e., thesuccess of achieving group consensus, but no evident relations canbe observed for the synchronized states in different clusters.
213 4 G G G G
213 4 G G G G G G G G G G
512 6 G G G G G G
11 213 4 G G G G bc c L = ¡ b b ¡ c c ¡ c ¡ c + 1 L = ¡ b b ¡ c c ¡ c ¡ c + 1
213 4 G G G G
11 11 3 21 2 121 1214 5 G G G G
11 1 L = ¡ ¡ ¡ ¡ L = ¡ ¡ ¡ ¡ L F L R
12 34 567 8 9 10 1 2 34 5 G G G G G G G G G G GG GG
213 4 G G G G
11 15 112 34 567 8 9 10 1 2 34 5 C C C C C C C C C C Fig. 2. The graph G (on the left) and its quotient graph G (on the right). G consists of five clusters of nodes with all intra-cluster edge weights equal to and all inter-cluster edge weights equal to . . t (sec) -10010 (a)Trajectories of state component x l ( t ) ; l = 1 ; : : : ; agent 1agent 2agent 3agent 4agent 5agent 6agent 7agent 8agent 9agent 10 t (sec) -10010 (b)Trajectories of state component x l ( t ) ; l = 1 ; : : : ; Fig. 3. The 10 agents achieve group consensus and form 3 distinct syn-chronous states when δ = 1 / λ . t (sec) -10010 (a)Trajectories of state component x l ( t ) ; l = 1 ; : : : ; agent 1agent 2agent 3agent 4agent 5agent 6agent 7agent 8agent 9agent 10 t (sec) -10010 (b)Trajectories of state component x l ( t ) ; l = 1 ; : : : ; Fig. 4. The 10 agents achieve group consensus and form 5 distinct syn-chronous states when δ = 1 / min 2 Reλ ( ˆ L ) < / λ . V. C
ONCLUSIONS
We have investigated the group consensus problem for genericlinear multi-agent systems under nonnegative directed graphs. Withthe aid of m -reducible Laplacian and its decomposed form, wederive a necessary and sufficient condition in terms of the topologiesof the underlying graph and its quotient graph. This condition isshown to be equivalent to an existing condition commonly usedfor undirected graph, and hence a unified understanding of thegraph topologies for ensuring group consensus is established. Weare also able to characterize the synchronized states in differentclusters when the overall coupling of the underlying graph is strongenough, thanks to the identical individual linear system models. It isshown by both theoretical analysis and simulation examples that thecoupling strengths for achieving group consensus and for realizingthe minimum number of distinct synchrony states could be different.Generally, the number of merged states is an outcome of the interplayof the agents’ individual dynamics, the underlying graph topology andthe overall coupling strength δ as shown for coupled oscillators [26]–[29], and thus is hard if not impossible to allow an accurate or explicitspecification. The work in [30] has reported relevant conclusions forMASs with point models and unweighted underlying digraphs byanalysing the structure of the underlying graph Laplacian. Extensionto complex system dynamics could be a challenging task that needsfurther investigation. Another future work is to tailor the conditionsderived for static topologies in this paper to MASs with dynamicallychanging topologies. A PPENDIX
AIn order to prove Lemma 3 and Theorem 2, we need the followingpreliminary results.
Lemma 5: If G contains a directed spanning tree (is stronglyconnected), then G also contains one (is strongly connected). Lemma 6:
Under Assumption 1, if G has a directed spanning treeand its root node is associated with a subgraph G i of G that has adirected spanning tree, then G has a directed spanning tree. Proof:
For the spanning tree of G , suppose without loss ofgenerality that its root node is associated with subgraph G in G . Notethat each directed link of the spanning tree of G is associated withinter-cluster links in G pointing from one subgraph to another. Hence,every subgraph G i , i (cid:54) = 1 is pointed by inter-cluster links originatingfrom some other subgraph. Moreover, every node in each G i , i (cid:54) = 1 ispointed by at least one inter-cluster link due to Assumption 1. Hence,there exists a path from the subgraph G to all nodes outside G viainter-cluster links that are associated with the links of the spanningtree of G . Note that this path can be an extension of a path in thespanning tree of G . It follows that G contains a directed spanningtree with its root being the root of the spanning tree of G . Lemma 7:
Under Assumption 1, if G is strongly connected andthere exists a subgraph G i of G whose nodes can be spanned by adirected tree in G , then G contains a directed spanning tree. Proof:
The strong connectivity of G implies that every node ineach subgraph G j , j ∈ { , , . . . , N } of G is pointed by inter-clusterlinks originating from at least one other subgraph G j (cid:48) , j (cid:48) (cid:54) = j . Usingsimilar arguments as those in the proof of Lemma 6, one sees thatthe directed tree that spans G i can be expanded to reach all nodes in G through inter-cluster links. A. Proof of Lemma 3Proof:
The necessity part follows from Lemma 6 by using thedefinitions of cluster spanning trees and the set V p . For the sufficiencypart, denote by T p for p = 1 , . . . , m the directed spanning tree thatcontains all nodes in {G i | i ∈ V p } . Note that V p shares the same root node with the reach R p . The if part of this lemma implies thatthe subgraph G i associated with this root node of R p is spanned bya component of the directed tree T p . It follows from Lemma 6 thatany cluster of nodes C i with i ∈ R p can be spanned by a directedtree (which contains T p ). The proof is completed when noting thatthe reaches R , . . . , R m contain the labels of all clusters. B. Proof of Theorem 2Proof:
Suppose the minimum number of directed trees whichtogether span G is m . Then the proof of this theorem is convertedto showing the equivalence of the following two statements:(a) G contains cluster spanning trees w.r.t. C .(b) the minimum number of directed trees which together span G is m .For m = 1 , this equivalence has been established by combing Lemma5 to Lemma 7. For < m < N , considering the subset of subgraphsin {G i | i ∈ ∪ mp =1 V p } , one needs at least m directed trees in order tospan all of the nodes therein (at least one directed spanning tree foreach set of subgraphs {G i | i ∈ V p } ).(a) ⇒ (b): By the necessity part of Lemma 3 and its proof, m isa feasible number of directed spanning trees that together span G .Hence, statement (b) holds.(b) ⇒ (a): If (a) does not hold, then according to Lemma 3 thereexists a p ∗ ∈ { , . . . , m } such that the nodes of {G i | i ∈ V p ∗ } cannotbe spanned by any single tree. It follows that more than m directedtrees are needed in order to span all of the nodes in {G i | i ∈ ∪ mp =1 V p } ,i.e., the negation of statement (b) is true. Hence, (b) ⇒ (a) holds.R EFERENCES [1] F. Chen and W. Ren, “On the control of multi-agent systems: A survey,”
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