Cluster Synchronization and Controllability of ... - Semantic Scholar
Cluster Synchronization and Controllability of Complex Multi-Agent Networks Weiguo Xia, Ming Cao Faculty of Mathematics and Natural Sciences, ITM, University of Groningen, The Netherlands
Abstract— This paper discloses the similarities between the condition for realizing cluster synchronization and that for uncontrollability in diffusively coupled multi-agent networks, both of which are built upon the characteristics of the networks’ topologies. We first generalize the notions of equitable partitions and almost equitable partitions to make them applicable to directed, weighted graphs. Consequently, we are enabled to characterize the controllable subspace of a given diffusively coupled multi-agent system using graph theoretic ideas. After comparing the condition to realize cluster synchronization and the condition for the network to be controllable, we conclude that those diffusively coupled multi-agent networks that are not controllable usually realize cluster synchronization asymptotically. Simulation results are provided to illustrate the theoretical results.
I. I NTRODUCTION Recently, the study on distributed coordination of multiagent networks has attracted significant attention from researchers in several disciplines, including theoretical biology, theoretical sociology, economics, statistical physics, computer science, network science as well as electrical engineering [1], [2]. The focus of the study is to understand how in complex networks, rich collective behaviors emerge from simple, local, sometimes self-organized agent interactions. We identify two research fields that are extremely active in the past few years, but until recently have been developing more or less independently. One is the study on complex networks, and in particular on synchronization phenomena [1], [3] that have been widely observed in natural and man-made systems. More recently, there is an emerging trend to study cluster synchronization behavior, in which the agents in a connected complex network evolve into several subgroups and only the agents in the same subgroup synchronize with one another as time goes to infinity [4], [5], [6]. In [6], three different cluster synchronization mechanisms have been examined and some necessary and sufficient conditions have been provided to guarantee the stability of the system. The other is the study on cooperative control of multiagent systems [7], [8], and in particular on “controllability” of multi-agent systems [9]. Here, controllability is a classical notion in control theory and a dynamical system is said to be controllable if under suitable control actions as the system’s inputs, the system’s state can be driven from any initial values
to any desired final values within finite time [10]. And some latest work has proposed methods to use the partitioning of the topology of a given network to bound the network’s controllable subspaces [11]. There are of course differences between the cluster synchronization problem for complex networks and the controllability problem for multi-agent systems. For example, cluster synchronization, or synchronization in general, is concerned with a complex network’s collective asymptotic behavior when time approaches infinity; in comparison, controllability is concerned with a system’s dynamic behavior within finite time. This partly explains why the two problems are studied independently in the past. However, we claim in this paper that the methodologies to study the two share striking similarities and by attacking the two problems together, deeper insight can be gained into both of the two topics. Towards this end, we first define generalized equitable partitions and almost equitable partitions for general directed weighted graphs. Then we are able to provide an upper bound for the controllable subspace for a general diffusively coupled multi-agent system. We point out the close relationship between the generalized almost equitable partition of a graph and the constant-row-sums property of the block sub-matrices of the graph’s Laplacian matrix. Furthermore, we show that those diffusively coupled multi-agent networks that are not controllable are in general easier to realize cluster synchronization. The rest of the paper is organized as follows. In Section II we review the cluster synchronization problem for complex networks. In Section III, the controllability problem for multiagent systems is introduced first and then we show the common features between the two problems by looking into several graph partitioning properties of the network topologies described by directed weighted graphs. In the end, a simulation example is shown in Section IV. II. C LUSTER SYNCHRONIZATION We first introduce the cluster synchronization problem for complex networks. Consider the following complex network that consists of 𝑁 coupled agents, whose dynamics are described by 𝑥˙ 𝑖 (𝑡) = 𝑓 (𝑡, 𝑥𝑖 (𝑡)) + 𝑐
978-1-4673-0219-7/12/$31.00 ©2012 IEEE
𝑎𝑖𝑗 Γ(𝑥𝑗 (𝑡) − 𝑥𝑖 (𝑡)),
(1)
𝑗=1
0 This
work was supported in part by grants from the Dutch Organization for Scientific Research (NWO), the Dutch Technology Foundation (STW) and the European Union Seventh Framework Programme [FP7/2007-2013] under grant agreement 257462 HYCON2 Network of Excellence.
𝑁 ∑
where 𝑥𝑖 ∈ IR𝑚 denotes the state of agent 𝑖, 𝑖 = 1, . . . , 𝑁 , 𝑓 : IR+ × IR𝑚 → IR𝑚 is continuous, 𝑐 > 0 is the coupling
165
strength, 𝑎𝑖𝑗 ≥ 0 is the coefficient for the coupling from agent 𝑗 to agent 𝑖 for 𝑖 ∕= 𝑗, 𝑖, 𝑗 = 1, . . . , 𝑁 , 𝑎𝑖𝑖 = 0, and the diagonal matrix Γ = diag{𝛾1 , . . . , 𝛾𝑚 } denotes the inner coupling with 𝛾𝑘 ≥ 0 for 𝑘 = 1, . . . , 𝑚. Directed weighted graphs can be conveniently used to describe the couplings between agents. Define the graph 𝔾 associated with system (1) to be the directed weighted graph with the node set 𝒱 = {1, 2, . . . , 𝑁 } and the edge set ℰ ⊂ {(𝑖, 𝑗) : 𝑖, 𝑗 ∈ 𝒱} for which (𝑖, 𝑗) is an edge of 𝔾 if and only if 𝑖 ∕= 𝑗 and 𝑎𝑗𝑖 ∕= 0, and the weight associated with (𝑖, 𝑗) is 𝑎𝑗𝑖 . Note that since 𝑎𝑖𝑖 = 0, 𝔾 contains no self-edges, i.e., (𝑖, 𝑖) ∕∈ ℰ. The adjacency matrix 𝐴 = (𝑎𝑖𝑗 )𝑁 ×𝑁 of 𝔾 is the 𝑁 × 𝑁 matrix whose elements are the weights 𝑎𝑖𝑗 . Let 𝑑𝑖 = ∑𝑁 𝑗=1,𝑗∕=𝑖 𝑎𝑖𝑗 , for 𝑖 = 1, . . . , 𝑁 and 𝐷 = diag{𝑑1 , . . . , 𝑑𝑁 }. Then the Laplacian matrix of 𝔾 is defined by 𝐿 = 𝐷 − 𝐴. Thus system (1) can also be written as 𝑥˙ 𝑖 (𝑡) = 𝑓 (𝑡, 𝑥𝑖 (𝑡)) − 𝑐
𝑁 ∑
𝑙𝑖𝑗 Γ𝑥𝑗 (𝑡),
𝑗 = 1, . . . , 𝑁,
(2)
𝑗=1
where 𝑙𝑖𝑗 is the 𝑖𝑗-th element of 𝐿. A directed path in 𝔾 is a sequence of distinct nodes 𝑖1 , . . . , 𝑖𝑘 such that (𝑖𝑠 , 𝑖𝑠+1 ) ∈ ℰ for 𝑠 = 1, . . . , 𝑘−1. 𝔾 is said to be strongly connected if there is a directed path from every node to every other node in 𝔾. > 1, is a partition We say that {𝐶1 , 𝐶2 , . . . , 𝐶𝑛 }, 𝑛∩ ∪𝑛 of the set 𝒱 = {1, 2, . . . , 𝑁 } if 𝐶𝑖 ∕= ∅, 𝐶𝑖 𝐶𝑗 = ∅ and 𝑖=1 𝐶𝑖 = 𝒱. We use 𝜋 = {𝐶1 , 𝐶2 , . . . , 𝐶𝑛 } to denote the partition, and we call 𝐶𝑖 ’s the cells and 𝑛 the size of the partition. Now we are ready to define what we mean by cluster synchronization. Definition 1: [6] System (2) is said to realize strict 𝑛cluster synchronization with the partition {𝐶1 , 𝐶2 , . . . , 𝐶𝑛 } if lim𝑡→∞ ∣∣𝑥𝑘 (𝑡) − 𝑥𝑙 (𝑡)∣∣ = 0 for all 𝑘, 𝑙 ∈ 𝐶𝑖 , 𝑖 = 1, . . . , 𝑛, and lim sup𝑡→∞ ∣∣𝑥𝑖 (𝑡) − 𝑥𝑗 (𝑡)∣∣ > 0 for all 𝑘 ∈ 𝐶𝑖 , 𝑙 ∈ 𝐶𝑗 , 𝑖, 𝑗 = 1, . . . , 𝑛 and 𝑖 ∕= 𝑗. If one does not care about the asymptotic differences between the states of the agents in different cells, a weaker notion of cluster synchronization can be defined. System (2) is said to realize cluster synchronization with the partition {𝐶1 , 𝐶2 , . . . , 𝐶𝑛 } if lim𝑡→∞ ∣∣𝑥𝑘 (𝑡) − 𝑥𝑙 (𝑡)∣∣ = 0 for all 𝑘, 𝑙 ∈ 𝐶𝑖 , 𝑖 = 1, . . . , 𝑛. For a partition 𝜋 = {𝐶1 , 𝐶2 , . . . , 𝐶𝑛 } of the graph 𝔾 with ∣𝐶𝑖 ∣ = 𝑙𝑖 , where ∣𝐶𝑖 ∣ is the cardinality of 𝐶𝑖 , we can always relabel the nodes such that the first 𝑙1 nodes lie in 𝐶1 , the next 𝑙2 nodes lie in 𝐶2 , and so on. Then we can write the adjacency matrix 𝐴 and the Laplacian matrix 𝐿 in the following block matrix forms according to the partition ⎤ ⎡ 𝐴11 𝐴12 ⋅ ⋅ ⋅ 𝐴1𝑛 ⎢ 𝐴21 𝐴22 ⋅ ⋅ ⋅ 𝐴2𝑛 ⎥ ⎥ ⎢ 𝐴=⎢ . (3) .. .. ⎥ , .. ⎣ .. . . . ⎦ ⎡ ⎢ ⎢ 𝐿=⎢ ⎣
𝐴𝑛1 𝐿11 𝐿21 .. . 𝐿𝑛1
𝐴𝑛2 ⋅ ⋅ ⋅ 𝐴𝑛𝑛 ⎤ 𝐿12 ⋅ ⋅ ⋅ 𝐿1𝑛 𝐿22 ⋅ ⋅ ⋅ 𝐿2𝑛 ⎥ ⎥ .. .. ⎥ , .. . . . ⎦ 𝐿𝑛2 ⋅ ⋅ ⋅ 𝐿𝑛𝑛
where 𝐴𝑖𝑗 , 𝐿𝑖𝑗 ∈ IR𝑙𝑖 ×𝑙𝑗 , 1 ≤ 𝑖, 𝑗 ≤ 𝑛. Let Ω be a subset of IR𝑚 , 𝑃 = diag{𝑝1 , . . . , 𝑝𝑚 } be a positive-definite diagonal matrix, and Δ = diag{𝛿1 , . . . , 𝛿𝑚 } be a diagonal matrix. QUAD(Δ, 𝑃, Ω) denotes a class of continuous functions 𝑓 (𝑡, 𝑥) : IR+ × IR𝑚 → IR𝑚 satisfying (𝑥 − 𝑦)𝑇 𝑃 (𝑓 (𝑡, 𝑥) − 𝑓 (𝑡, 𝑦) − Δ(𝑥 − 𝑦)) ≤ −𝜖(𝑥 − 𝑦)𝑇 (𝑥 − 𝑦) (5) for some 𝜖 > 0, all 𝑥, 𝑦 ∈ Ω, and all 𝑡 ≥ 0. Now we restate the result on cluster synchronization in [4] as follows. Proposition 1: Let {𝐶1 , 𝐶2 , . . . , 𝐶𝑛 } be a partition of the set 𝒱 and accordingly the Laplacian matrix is in the form (4). Let 𝑃 = diag{𝑝1 , . . . , 𝑝𝑚 } be a positive-definite diagonal matrix, Δ = diag{𝛿1 , . . . , 𝛿𝑚 } be a diagonal matrix, and 𝐼𝑁 ∈ IR𝑁 be the identity matrix. Then system (2) realizes cluster synchronization with the partition {𝐶1 , 𝐶2 , . . . , 𝐶𝑛 } when the following three conditions are satisfied (i) the block matrices 𝐿𝑖𝑗 have constant row sums for 1 ≤ 𝑖, 𝑗 ≤ 𝑛; (ii) 𝑓 (𝑡, 𝑥) ∈ QUAD(Δ, 𝑃, IR𝑛 ); (iii) 𝑧 𝑇 ( 12 𝑐𝛾𝑗 (𝐴 + 𝐴𝑇 ) + 𝛿𝑗 𝐼𝑁 )𝑧 ≤ 0, for all 𝑧 ∈ 𝒲𝑛 , and = {𝑥 = [𝑥𝑇1 , 𝑥𝑇2 , . . . , 𝑥𝑇𝑁 ]𝑇 : 𝑥𝑖 ∈ 𝑗 = 1, . . . , 𝑚, where 𝒲𝑛 ∑ 𝑚 IR , 𝑖 = 1, . . . , 𝑁, and 𝑙∈𝐶𝑘 𝑥𝑙 = 0, 𝑘 = 1, . . . , 𝑛} is a subspace of IR𝑁 𝑚 . In the next section, we introduce the controllability problem for multi-agent systems, and show some relationships between the conditions for realizing cluster synchronization and those for the uncontrollability of a multi-agent system. III. C ONTROLLABILITY OF MULTI - AGENT SYSTEMS A. Problem formulation We consider a multi-agent system consisting of 𝑁 agents and as in Section II we use 𝒱 = {1, . . . , 𝑁 } to denote the set of indices of all the agents. Let 𝑥𝑖 ∈ IR, 𝑖 ∈ 𝒱, denote the state of agent 𝑖. We assign the roles of the leaders and followers to the agents and use 𝒱𝐿 , 𝒱𝐹 ⊂ 𝒱 to denote the sets of indices of the leaders and followers, respectively. Assume that there are altogether 0 < 𝑠 = ∣𝒱𝐿 ∣ < 𝑛 control inputs 𝑢𝑖 ∈ IR, 1 ≤ 𝑖 ≤ 𝑠 and each leader is influenced by only one input. For a leader 𝑖 ∈ 𝒱𝐿 , let [𝑖] ∈ {1, . . . , 𝑠} denote the index of the control input acting on it. Then the dynamics of the leaders are determined by 𝑥˙ 𝑖 =
𝑁 ∑
𝑎𝑖𝑗 (𝑥𝑗 − 𝑥𝑖 ) + 𝑢[𝑖] ,
𝑖 ∈ 𝒱𝐿 ,
(6)
𝑗=1
where 𝑎𝑖𝑗 ≥ 0 is the coefficient for the coupling from agent 𝑗 to agent 𝑖 for 𝑗 ∕= 𝑖, 𝑗 = 1, . . . , 𝑁 , and 𝑎𝑖𝑖 = 0. The followers’ dynamics are governed by linear diffusive couplings 𝑥˙ 𝑖 =
𝑁 ∑
𝑎𝑖𝑗 (𝑥𝑗 − 𝑥𝑖 ),
𝑖 ∈ 𝒱𝐹 ,
(7)
𝑗=1
(4)
where 𝑎𝑖𝑗 ≥ 0 for 𝑗 ∕= 𝑖, 𝑗 = 1, . . . , 𝑁 , and 𝑎𝑖𝑖 = 0. Δ Δ Let 𝑥 = [𝑥1 , . . . , 𝑥𝑁 ]𝑇 and 𝑢 = [𝑢1 , . . . , 𝑢𝑠 ]𝑇 , and we also use weighted directed graphs to describe the couplings
166
between agents as in Section II. Then (6) and (7) can be written in a compact form 𝑥˙ = −𝐿𝑥 + 𝑀 𝑢,
(8)
where 𝐿 is the Laplacian matrix, and the elements of 𝑀 are defined by { 1 if 𝑗 = [𝑖] 𝑚𝑖𝑗 = 0 otherwise, for 1 ≤ 𝑖 ≤ 𝑁 and 1 ≤ 𝑗 ≤ 𝑠. The controllability problem of system (8) has attracted great attention from the area of systems and control [12], [11]. Denote the controllable subspace of system (8) by 𝒦. Note that 𝒦 is the smallest 𝐿-invariant subspace that contains the subspace spanned by the columns of 𝑀 , denoted by im𝑀 [11]. In order to characterize the controllable subspace, we need some more notions from graph theory.
From the above two definitions, we can see the close relationships between the generalized equitable partitions (resp. almost equitable partitions) of a graph and the constantrow-sums property of block matrices 𝐴𝑖𝑗 (resp. 𝐿𝑖𝑗 ) of the associated adjacency 𝐴 (resp. Laplacian matrix 𝐿). Proposition 2: For a partition 𝜋 of a graph 𝔾, we always label the nodes such that the first 𝑙1 nodes lie in 𝐶1 , the next 𝑙2 nodes lie in 𝐶2 , and so on. A partition 𝜋 is a generalized equitable partition of a graph 𝔾 if and only if the row sums of each block 𝐴𝑖𝑗 of the associated adjacency matrix 𝐴 written in form (3) are equal. A partition 𝜋 is a generalized almost equitable partition if and only if the row sums of each block 𝐿𝑖𝑗 of the associated adjacency matrix 𝐿 written in form (4) are equal.
1
5
B. Controllability through generalized almost equitable partitions
2
Given a partition 𝜋 = {𝐶1 , 𝐶2 , . . . , 𝐶𝑛 } of the node set 𝒱 = {1, . . . , 𝑁 } of a graph 𝔾 = (𝒱, ℰ), the characteristic matrix 𝑃 (𝜋) ∈ IR𝑁 ×𝑛 of the partition is defined by { 1 if 𝑖 ∈ 𝐶𝑗 𝑃𝑖𝑗 (𝜋) = 0 otherwise, for 1 ≤ 𝑖 ≤ 𝑁 and 1 ≤ 𝑗 ≤ 𝑛. First consider the case when the graph 𝔾 = (𝒱, ℰ) is unweighted and undirected as in [12], [11], meaning that for any distinct pair of nodes 𝑖 and 𝑗, if (𝑖, 𝑗) ∈ ℰ, then (𝑗, 𝑖) ∈ ℰ and the weights 𝑎𝑖𝑗 = 𝑎𝑗𝑖 = 1. We say agent 𝑗 is a neighbor of agent 𝑖, if 𝑎𝑖𝑗 = 1. A partition 𝜋 is said to be an equitable partition if each node in 𝐶𝑗 has the same number of neighbors in 𝐶𝑖 for all 1 ≤ 𝑖, 𝑗 ≤ 𝑛. If one only cares about the number of neighbors in adjacent cells, while ignoring the structure inside a cell, one can define the notion of almost equitable partition. A partition 𝜋 is said to be an almost equitable partition if each node in 𝐶𝑗 has the same number of neighbors in 𝐶𝑖 for all 1 ≤ 𝑖, 𝑗 ≤ 𝑛 and 𝑖 ∕= 𝑗. For a given 𝑣 ∈ 𝒱, 𝜋 is said to be an almost equitable partition relative to 𝑣 if it is almost equitable and {𝑣} is one of its cells. However, when we consider general directed weighted graphs, the weights 𝑎𝑖𝑗 can be of any nonnegative values. Thus we cannot employ the notion of the number of neighbors any more. Now we generalize the notions of equitable partitions and almost equitable partitions in a natural way. Definition 2: A partition 𝜋 is said to be a generalized equitable partition if for any 𝑘, 𝑙 ∈ 𝐶𝑖 , 𝑖, 𝑗 = 1, . . . , 𝑛, ∑ ∑ 𝑎𝑘𝑟 = 𝑎𝑙𝑟 . (9) 𝑟∈𝐶𝑗
𝑟∈𝐶𝑗
𝑟∈𝐶𝑗
𝑟∈𝐶𝑗
3
Definition 3: A partition 𝜋 is said to be a generalized almost equitable partition if for any 𝑘, 𝑙 ∈ 𝐶𝑖 , 𝑖 ∕= 𝑗, 𝑖, 𝑗 = 1, . . . , 𝑛, ∑ ∑ 𝑎𝑘𝑟 = 𝑎𝑙𝑟 . (10)
Fig. 1.
4
A directed graph 𝔾.
Example 1. Let the adjacency matrix 𝐴 and the Laplacian matrix 𝐿 associated with a graph 𝔾 shown in Fig. 1 be given by ⎡ ⎤ 0 2 0.1 0.1 0 ⎢ 0.1 0 0 0.2 0 ⎥ ⎢ ⎥ ⎢ 0 0.5 0.1 ⎥ 𝐴 = (𝐴𝑖𝑗 )3×3 = ⎢ 0.5 0 ⎥, ⎣ 0 0.5 0.5 0 0.1 ⎦ 0.1 0.1 0 0 0 ⎡ ⎤ −2.2 2 0.1 0.1 0 ⎢ 0.1 −0.3 0 0.2 0 ⎥ ⎢ ⎥ ⎢ 0 −1.1 0.5 0.1 ⎥ 𝐿 = (𝐿𝑖𝑗 )3×3 = ⎢ 0.5 ⎥. ⎣ 0 0.5 0.5 −1.1 0.1 ⎦ 0.1 0.1 0 0 −0.2 It is easy to see that 𝐿𝑖𝑗 , 𝑖, 𝑗 = 1, 2, 3, have constant row sums, which corresponds to a nontrivial generalized almost equitable partition 𝜋 = {𝐶1 = {1, 2}, 𝐶2 = {3, 4}, 𝐶3 = {5}} of 𝔾. Note that 𝜋 is not a generalized equitable partition since the row sums of 𝐴11 are not equal. For given nodes 𝑣1 , . . . , 𝑣𝑠 ∈ 𝒱, 𝜋 is said to be a generalized almost equitable partition relative to 𝑣1 , . . . , 𝑣𝑠 if it is a generalized almost equitable partition and {𝑣1 }, . . . , {𝑣𝑠 } are its cells. Let Π𝐺𝐸𝑃 , Π𝐺𝐴𝐸𝑃 and Π𝐺𝐴𝐸𝑃 (𝑣1 , . . . , 𝑣𝑠 ) denote the sets of all generalized equitable, generalized almost equitable and generalized almost equitable partitions relative to 𝑣1 , . . . , 𝑣𝑠 , respectively. Moreover, we say that a generalized almost equitable partition relative to 𝑣1 , . . . , 𝑣𝑠 is maximal, ∗ (𝑣1 , . . . , 𝑣𝑠 ) if it has the smallwhich is denoted by 𝜋𝐺𝐴𝐸𝑃 est size, that is, if it contains the fewest possible cells. It can be shown that given a graph 𝔾 and nodes 𝑣1 , . . . , 𝑣𝑠 , ∗ (𝑣1 , . . . , 𝑣𝑠 ) always exists uniquely. 𝜋𝐺𝐴𝐸𝑃 Now we can characterize the generalized almost equitable
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partitions using terms of invariant subspaces of the Laplacian matrix 𝐿 of 𝔾 as follows. Lemma 1: A partition 𝜋 of 𝔾 is a generalized almost equitable partition, if and only if im𝑃 (𝜋) is 𝐿-invariant. An upper bound for the controllable subspace is given in [11] for system (8) with multiple leaders when the graph is undirected and unweighted. We provide an upper bound in terms of generalized almost equitable partitions for the general case when the graph is directed and weighted. Proposition 3: Let 𝜋 ∈ Π𝐺𝐴𝐸𝑃 be such that for each 𝑣 ∈ 𝒱𝐿 , {𝑣} is a cell of 𝜋. Then
evolution of the agents’ states are shown in Fig. 2, from which we can see that system (2) realizes cluster synchronization with respect to the partition 𝜋 ∗ (5).
ei1 (t)
2 0 −2 −4
0
5
10
15
0
5
10
15
0
5
10
15
ei2(t)
2 0 −2
𝒦 ⊆ im𝑃 (𝜋). For system (8) with multiple leaders, we have the following bound. Lemma 2: If 𝒱𝐿 = {𝑣1 , . . . , 𝑣𝑠 }, then
ei3(t)
2 0 −2
t
∗ (𝑣1 , . . . , 𝑣𝑠 ))). dim(im𝑃 (𝜋𝐺𝐴𝐸𝑃
dim(𝒦) ≤ From this lemma, we immediately have the following proposition. Proposition 4: Assume that 𝔾 is strongly connected. System (8) with multiple leaders 𝒱𝐿 = {𝑣1 , . . . , 𝑣𝑠 }, is con∗ (𝑣1 , . . . , 𝑣𝑠 ) is trivial, that is, {𝑖} ∈ trollable only if 𝜋𝐺𝐴𝐸𝑃 ∗ 𝜋𝐺𝐴𝐸𝑃 (𝑣1 , . . . , 𝑣𝑠 ) for all 𝑖 ∈ 𝒱. Theorem 1: Given 𝒱𝐿 = {𝑣1 , . . . , 𝑣𝑠 }. For those graph topologies that are strongly connected and have nontrivial ∗ (𝑣1 , . . . , 𝑣𝑠 ), system (8) is not controllable and system 𝜋𝐺𝐴𝐸𝑃 (2) realizes cluster synchronization provided that the conditions (ii) and (iii) in Proposition 1 are satisfied . So in this section, by comparing the conditions for realizing cluster synchronization and checking controllability, we have gained the insight that those multi-agent networks that are uncontrollable in finite time tend to realize cluster synchronization as time goes to infinity. IV. I LLUSTRATIVE EXAMPLE We take the graph 𝔾 and the associated Laplacian matrix 𝐿 in Example 1 as an example. If we take node 5 as a leader, then the maximal generalized almost equitable partition relative to node 5 is 𝜋 ∗ (5) = {{1, 2}, {3, 4}, {5}}. The system (8) with this Laplacian matrix is uncontrollable since the partition 𝜋 ∗ (5) is nontrivial. Now consider the coupled system (2) with the same Laplacian matrix, where 𝑐 = 1, Γ = 𝐼3 , 𝑓 (𝑡, 𝑦) = −𝑦 + 𝑇 𝑔(𝑦) with 𝑦 = [𝑦1 , 𝑦2 , 𝑦3 ]𝑇 ∈ IR3 , ⎡ 1.25 −3.2 𝑇 = ⎣−3.2 1.1 −3.2 4.4
⎤ −3.2 −4.4⎦ , 1
𝑔(𝑦) = [𝑔1 (𝑦1 ), 𝑔2 (𝑦2 ), 𝑔3 (𝑦3 )]𝑇 , and 𝑔1 (𝑤) = 𝑔2 (𝑤) = 𝑔3 (𝑤) = (∣𝑤 + 1∣ − ∣𝑤 − 1∣)/2. The system 𝑑𝑦 𝑑𝑡 = 𝑓 (𝑡, 𝑦) is a 3-D neural network, which has a double-scrolling chaotic attractor. It is easy to see that 𝑓 (𝑡, 𝑦) is globally Lipschitz continuous. Thus, 𝑓 (𝑡, 𝑦) ∈ QUAD(Δ, 𝐼3 , IR3 ). Assume that the initial values of the agents are chosen randomly from [0.5, 0.5] × [0.5, 0.5] × [0.5, 0.5]. Then the trajectories of the
Fig. 2.
The state trajectories of the five agents.
V. C ONCLUSION We have looked at jointly the cluster synchronization problem for complex networks and the controllability problem for multi-agent systems. Using the notions of generalized graph partitioning, we have provided an upper bound for the controllable subspace of a diffusively coupled multi-agent system. In the future, we are interested in studying more control related problems for complex networks by using the integrated set of tools from network theory, control theory and graph theory. R EFERENCES [1] S. H. Strogatz. SYNC: The emerging science of spontaneous order. Hyperion, New York, 2003. [2] M. E. J. Newman. Networks: An introduction. Oxford University Press, 2010. [3] C. W. Wu and L. O. Chua. Synchronization in an array of linearly coupled dynamical systems. IEEE Transactions on Circuits and Systems I, 42:430–447, 1995. [4] W. Wu and T. Chen. Partial synchronization in linearly and symmetrically coupled ordinary differential systems. Physica D, 238:355–364, 2009. [5] W. Xia and M. Cao. Cluster synchronization algorithms. In Proc. of the 2010 American Control Conference, pages 6513–6518, 2010. [6] W. Xia and M. Cao. Clustering in diffusively coupled networks. Automatica, 47:2395–2405, 2011. [7] A. Jadbabaie, J. Lin, and A. S. Morse. Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Transactions on Automatic Control, 48:985–1001, 2003. [8] V. Kumar, N. E. Leonard, and A. S. Morse. Cooperative Control. Springer-Verlag New York, Inc., 2005. [9] A. Rahmani, M. Ji, M. Mesbahi, and M. Egerstedt. Controllability of multi-agent systems from a graph-theoretic perspective. SIAM Journal on Control and Optimization, 48:162–186, 2009. [10] R. E. Kalman. Mathematical description of linear dynamical systems. Journal of Society of Industrial and Applied Mathematics, 1:152–192, 1963. [11] S. Zhang, M. K. Camlibel, and M. Cao. Controllability of diffusivelycoupled multi-agent systems with general and distance regular coupling topologies. In Proc. of the 50th IEEE Conference on Decision and Control and 2011 European Control Conference, pages 759–764, 2011. [12] M. Mesbahi and M. Egerstedt. Graph Theoretic Methods in Multiagent Networks. Princeton University Press, 2010.
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Abstract— This paper discloses the similarities between the condition for realizing cluster synchronization and that for uncontrollability in diffusively coupled multi-agent networks, both of which are built upon the characteristics of the networks’ topologies. We first generalize the notions of equitable partitions and almost equitable partitions to make them applicable to directed, weighted graphs. Consequently, we are enabled to characterize the controllable subspace of a given diffusively coupled multi-agent system using graph theoretic ideas. After comparing the condition to realize cluster synchronization and the condition for the network to be controllable, we conclude that those diffusively coupled multi-agent networks that are not controllable usually realize cluster synchronization asymptotically. Simulation results are provided to illustrate the theoretical results.
I. I NTRODUCTION Recently, the study on distributed coordination of multiagent networks has attracted significant attention from researchers in several disciplines, including theoretical biology, theoretical sociology, economics, statistical physics, computer science, network science as well as electrical engineering [1], [2]. The focus of the study is to understand how in complex networks, rich collective behaviors emerge from simple, local, sometimes self-organized agent interactions. We identify two research fields that are extremely active in the past few years, but until recently have been developing more or less independently. One is the study on complex networks, and in particular on synchronization phenomena [1], [3] that have been widely observed in natural and man-made systems. More recently, there is an emerging trend to study cluster synchronization behavior, in which the agents in a connected complex network evolve into several subgroups and only the agents in the same subgroup synchronize with one another as time goes to infinity [4], [5], [6]. In [6], three different cluster synchronization mechanisms have been examined and some necessary and sufficient conditions have been provided to guarantee the stability of the system. The other is the study on cooperative control of multiagent systems [7], [8], and in particular on “controllability” of multi-agent systems [9]. Here, controllability is a classical notion in control theory and a dynamical system is said to be controllable if under suitable control actions as the system’s inputs, the system’s state can be driven from any initial values
to any desired final values within finite time [10]. And some latest work has proposed methods to use the partitioning of the topology of a given network to bound the network’s controllable subspaces [11]. There are of course differences between the cluster synchronization problem for complex networks and the controllability problem for multi-agent systems. For example, cluster synchronization, or synchronization in general, is concerned with a complex network’s collective asymptotic behavior when time approaches infinity; in comparison, controllability is concerned with a system’s dynamic behavior within finite time. This partly explains why the two problems are studied independently in the past. However, we claim in this paper that the methodologies to study the two share striking similarities and by attacking the two problems together, deeper insight can be gained into both of the two topics. Towards this end, we first define generalized equitable partitions and almost equitable partitions for general directed weighted graphs. Then we are able to provide an upper bound for the controllable subspace for a general diffusively coupled multi-agent system. We point out the close relationship between the generalized almost equitable partition of a graph and the constant-row-sums property of the block sub-matrices of the graph’s Laplacian matrix. Furthermore, we show that those diffusively coupled multi-agent networks that are not controllable are in general easier to realize cluster synchronization. The rest of the paper is organized as follows. In Section II we review the cluster synchronization problem for complex networks. In Section III, the controllability problem for multiagent systems is introduced first and then we show the common features between the two problems by looking into several graph partitioning properties of the network topologies described by directed weighted graphs. In the end, a simulation example is shown in Section IV. II. C LUSTER SYNCHRONIZATION We first introduce the cluster synchronization problem for complex networks. Consider the following complex network that consists of 𝑁 coupled agents, whose dynamics are described by 𝑥˙ 𝑖 (𝑡) = 𝑓 (𝑡, 𝑥𝑖 (𝑡)) + 𝑐
978-1-4673-0219-7/12/$31.00 ©2012 IEEE
𝑎𝑖𝑗 Γ(𝑥𝑗 (𝑡) − 𝑥𝑖 (𝑡)),
(1)
𝑗=1
0 This
work was supported in part by grants from the Dutch Organization for Scientific Research (NWO), the Dutch Technology Foundation (STW) and the European Union Seventh Framework Programme [FP7/2007-2013] under grant agreement 257462 HYCON2 Network of Excellence.
𝑁 ∑
where 𝑥𝑖 ∈ IR𝑚 denotes the state of agent 𝑖, 𝑖 = 1, . . . , 𝑁 , 𝑓 : IR+ × IR𝑚 → IR𝑚 is continuous, 𝑐 > 0 is the coupling
165
strength, 𝑎𝑖𝑗 ≥ 0 is the coefficient for the coupling from agent 𝑗 to agent 𝑖 for 𝑖 ∕= 𝑗, 𝑖, 𝑗 = 1, . . . , 𝑁 , 𝑎𝑖𝑖 = 0, and the diagonal matrix Γ = diag{𝛾1 , . . . , 𝛾𝑚 } denotes the inner coupling with 𝛾𝑘 ≥ 0 for 𝑘 = 1, . . . , 𝑚. Directed weighted graphs can be conveniently used to describe the couplings between agents. Define the graph 𝔾 associated with system (1) to be the directed weighted graph with the node set 𝒱 = {1, 2, . . . , 𝑁 } and the edge set ℰ ⊂ {(𝑖, 𝑗) : 𝑖, 𝑗 ∈ 𝒱} for which (𝑖, 𝑗) is an edge of 𝔾 if and only if 𝑖 ∕= 𝑗 and 𝑎𝑗𝑖 ∕= 0, and the weight associated with (𝑖, 𝑗) is 𝑎𝑗𝑖 . Note that since 𝑎𝑖𝑖 = 0, 𝔾 contains no self-edges, i.e., (𝑖, 𝑖) ∕∈ ℰ. The adjacency matrix 𝐴 = (𝑎𝑖𝑗 )𝑁 ×𝑁 of 𝔾 is the 𝑁 × 𝑁 matrix whose elements are the weights 𝑎𝑖𝑗 . Let 𝑑𝑖 = ∑𝑁 𝑗=1,𝑗∕=𝑖 𝑎𝑖𝑗 , for 𝑖 = 1, . . . , 𝑁 and 𝐷 = diag{𝑑1 , . . . , 𝑑𝑁 }. Then the Laplacian matrix of 𝔾 is defined by 𝐿 = 𝐷 − 𝐴. Thus system (1) can also be written as 𝑥˙ 𝑖 (𝑡) = 𝑓 (𝑡, 𝑥𝑖 (𝑡)) − 𝑐
𝑁 ∑
𝑙𝑖𝑗 Γ𝑥𝑗 (𝑡),
𝑗 = 1, . . . , 𝑁,
(2)
𝑗=1
where 𝑙𝑖𝑗 is the 𝑖𝑗-th element of 𝐿. A directed path in 𝔾 is a sequence of distinct nodes 𝑖1 , . . . , 𝑖𝑘 such that (𝑖𝑠 , 𝑖𝑠+1 ) ∈ ℰ for 𝑠 = 1, . . . , 𝑘−1. 𝔾 is said to be strongly connected if there is a directed path from every node to every other node in 𝔾. > 1, is a partition We say that {𝐶1 , 𝐶2 , . . . , 𝐶𝑛 }, 𝑛∩ ∪𝑛 of the set 𝒱 = {1, 2, . . . , 𝑁 } if 𝐶𝑖 ∕= ∅, 𝐶𝑖 𝐶𝑗 = ∅ and 𝑖=1 𝐶𝑖 = 𝒱. We use 𝜋 = {𝐶1 , 𝐶2 , . . . , 𝐶𝑛 } to denote the partition, and we call 𝐶𝑖 ’s the cells and 𝑛 the size of the partition. Now we are ready to define what we mean by cluster synchronization. Definition 1: [6] System (2) is said to realize strict 𝑛cluster synchronization with the partition {𝐶1 , 𝐶2 , . . . , 𝐶𝑛 } if lim𝑡→∞ ∣∣𝑥𝑘 (𝑡) − 𝑥𝑙 (𝑡)∣∣ = 0 for all 𝑘, 𝑙 ∈ 𝐶𝑖 , 𝑖 = 1, . . . , 𝑛, and lim sup𝑡→∞ ∣∣𝑥𝑖 (𝑡) − 𝑥𝑗 (𝑡)∣∣ > 0 for all 𝑘 ∈ 𝐶𝑖 , 𝑙 ∈ 𝐶𝑗 , 𝑖, 𝑗 = 1, . . . , 𝑛 and 𝑖 ∕= 𝑗. If one does not care about the asymptotic differences between the states of the agents in different cells, a weaker notion of cluster synchronization can be defined. System (2) is said to realize cluster synchronization with the partition {𝐶1 , 𝐶2 , . . . , 𝐶𝑛 } if lim𝑡→∞ ∣∣𝑥𝑘 (𝑡) − 𝑥𝑙 (𝑡)∣∣ = 0 for all 𝑘, 𝑙 ∈ 𝐶𝑖 , 𝑖 = 1, . . . , 𝑛. For a partition 𝜋 = {𝐶1 , 𝐶2 , . . . , 𝐶𝑛 } of the graph 𝔾 with ∣𝐶𝑖 ∣ = 𝑙𝑖 , where ∣𝐶𝑖 ∣ is the cardinality of 𝐶𝑖 , we can always relabel the nodes such that the first 𝑙1 nodes lie in 𝐶1 , the next 𝑙2 nodes lie in 𝐶2 , and so on. Then we can write the adjacency matrix 𝐴 and the Laplacian matrix 𝐿 in the following block matrix forms according to the partition ⎤ ⎡ 𝐴11 𝐴12 ⋅ ⋅ ⋅ 𝐴1𝑛 ⎢ 𝐴21 𝐴22 ⋅ ⋅ ⋅ 𝐴2𝑛 ⎥ ⎥ ⎢ 𝐴=⎢ . (3) .. .. ⎥ , .. ⎣ .. . . . ⎦ ⎡ ⎢ ⎢ 𝐿=⎢ ⎣
𝐴𝑛1 𝐿11 𝐿21 .. . 𝐿𝑛1
𝐴𝑛2 ⋅ ⋅ ⋅ 𝐴𝑛𝑛 ⎤ 𝐿12 ⋅ ⋅ ⋅ 𝐿1𝑛 𝐿22 ⋅ ⋅ ⋅ 𝐿2𝑛 ⎥ ⎥ .. .. ⎥ , .. . . . ⎦ 𝐿𝑛2 ⋅ ⋅ ⋅ 𝐿𝑛𝑛
where 𝐴𝑖𝑗 , 𝐿𝑖𝑗 ∈ IR𝑙𝑖 ×𝑙𝑗 , 1 ≤ 𝑖, 𝑗 ≤ 𝑛. Let Ω be a subset of IR𝑚 , 𝑃 = diag{𝑝1 , . . . , 𝑝𝑚 } be a positive-definite diagonal matrix, and Δ = diag{𝛿1 , . . . , 𝛿𝑚 } be a diagonal matrix. QUAD(Δ, 𝑃, Ω) denotes a class of continuous functions 𝑓 (𝑡, 𝑥) : IR+ × IR𝑚 → IR𝑚 satisfying (𝑥 − 𝑦)𝑇 𝑃 (𝑓 (𝑡, 𝑥) − 𝑓 (𝑡, 𝑦) − Δ(𝑥 − 𝑦)) ≤ −𝜖(𝑥 − 𝑦)𝑇 (𝑥 − 𝑦) (5) for some 𝜖 > 0, all 𝑥, 𝑦 ∈ Ω, and all 𝑡 ≥ 0. Now we restate the result on cluster synchronization in [4] as follows. Proposition 1: Let {𝐶1 , 𝐶2 , . . . , 𝐶𝑛 } be a partition of the set 𝒱 and accordingly the Laplacian matrix is in the form (4). Let 𝑃 = diag{𝑝1 , . . . , 𝑝𝑚 } be a positive-definite diagonal matrix, Δ = diag{𝛿1 , . . . , 𝛿𝑚 } be a diagonal matrix, and 𝐼𝑁 ∈ IR𝑁 be the identity matrix. Then system (2) realizes cluster synchronization with the partition {𝐶1 , 𝐶2 , . . . , 𝐶𝑛 } when the following three conditions are satisfied (i) the block matrices 𝐿𝑖𝑗 have constant row sums for 1 ≤ 𝑖, 𝑗 ≤ 𝑛; (ii) 𝑓 (𝑡, 𝑥) ∈ QUAD(Δ, 𝑃, IR𝑛 ); (iii) 𝑧 𝑇 ( 12 𝑐𝛾𝑗 (𝐴 + 𝐴𝑇 ) + 𝛿𝑗 𝐼𝑁 )𝑧 ≤ 0, for all 𝑧 ∈ 𝒲𝑛 , and = {𝑥 = [𝑥𝑇1 , 𝑥𝑇2 , . . . , 𝑥𝑇𝑁 ]𝑇 : 𝑥𝑖 ∈ 𝑗 = 1, . . . , 𝑚, where 𝒲𝑛 ∑ 𝑚 IR , 𝑖 = 1, . . . , 𝑁, and 𝑙∈𝐶𝑘 𝑥𝑙 = 0, 𝑘 = 1, . . . , 𝑛} is a subspace of IR𝑁 𝑚 . In the next section, we introduce the controllability problem for multi-agent systems, and show some relationships between the conditions for realizing cluster synchronization and those for the uncontrollability of a multi-agent system. III. C ONTROLLABILITY OF MULTI - AGENT SYSTEMS A. Problem formulation We consider a multi-agent system consisting of 𝑁 agents and as in Section II we use 𝒱 = {1, . . . , 𝑁 } to denote the set of indices of all the agents. Let 𝑥𝑖 ∈ IR, 𝑖 ∈ 𝒱, denote the state of agent 𝑖. We assign the roles of the leaders and followers to the agents and use 𝒱𝐿 , 𝒱𝐹 ⊂ 𝒱 to denote the sets of indices of the leaders and followers, respectively. Assume that there are altogether 0 < 𝑠 = ∣𝒱𝐿 ∣ < 𝑛 control inputs 𝑢𝑖 ∈ IR, 1 ≤ 𝑖 ≤ 𝑠 and each leader is influenced by only one input. For a leader 𝑖 ∈ 𝒱𝐿 , let [𝑖] ∈ {1, . . . , 𝑠} denote the index of the control input acting on it. Then the dynamics of the leaders are determined by 𝑥˙ 𝑖 =
𝑁 ∑
𝑎𝑖𝑗 (𝑥𝑗 − 𝑥𝑖 ) + 𝑢[𝑖] ,
𝑖 ∈ 𝒱𝐿 ,
(6)
𝑗=1
where 𝑎𝑖𝑗 ≥ 0 is the coefficient for the coupling from agent 𝑗 to agent 𝑖 for 𝑗 ∕= 𝑖, 𝑗 = 1, . . . , 𝑁 , and 𝑎𝑖𝑖 = 0. The followers’ dynamics are governed by linear diffusive couplings 𝑥˙ 𝑖 =
𝑁 ∑
𝑎𝑖𝑗 (𝑥𝑗 − 𝑥𝑖 ),
𝑖 ∈ 𝒱𝐹 ,
(7)
𝑗=1
(4)
where 𝑎𝑖𝑗 ≥ 0 for 𝑗 ∕= 𝑖, 𝑗 = 1, . . . , 𝑁 , and 𝑎𝑖𝑖 = 0. Δ Δ Let 𝑥 = [𝑥1 , . . . , 𝑥𝑁 ]𝑇 and 𝑢 = [𝑢1 , . . . , 𝑢𝑠 ]𝑇 , and we also use weighted directed graphs to describe the couplings
166
between agents as in Section II. Then (6) and (7) can be written in a compact form 𝑥˙ = −𝐿𝑥 + 𝑀 𝑢,
(8)
where 𝐿 is the Laplacian matrix, and the elements of 𝑀 are defined by { 1 if 𝑗 = [𝑖] 𝑚𝑖𝑗 = 0 otherwise, for 1 ≤ 𝑖 ≤ 𝑁 and 1 ≤ 𝑗 ≤ 𝑠. The controllability problem of system (8) has attracted great attention from the area of systems and control [12], [11]. Denote the controllable subspace of system (8) by 𝒦. Note that 𝒦 is the smallest 𝐿-invariant subspace that contains the subspace spanned by the columns of 𝑀 , denoted by im𝑀 [11]. In order to characterize the controllable subspace, we need some more notions from graph theory.
From the above two definitions, we can see the close relationships between the generalized equitable partitions (resp. almost equitable partitions) of a graph and the constantrow-sums property of block matrices 𝐴𝑖𝑗 (resp. 𝐿𝑖𝑗 ) of the associated adjacency 𝐴 (resp. Laplacian matrix 𝐿). Proposition 2: For a partition 𝜋 of a graph 𝔾, we always label the nodes such that the first 𝑙1 nodes lie in 𝐶1 , the next 𝑙2 nodes lie in 𝐶2 , and so on. A partition 𝜋 is a generalized equitable partition of a graph 𝔾 if and only if the row sums of each block 𝐴𝑖𝑗 of the associated adjacency matrix 𝐴 written in form (3) are equal. A partition 𝜋 is a generalized almost equitable partition if and only if the row sums of each block 𝐿𝑖𝑗 of the associated adjacency matrix 𝐿 written in form (4) are equal.
1
5
B. Controllability through generalized almost equitable partitions
2
Given a partition 𝜋 = {𝐶1 , 𝐶2 , . . . , 𝐶𝑛 } of the node set 𝒱 = {1, . . . , 𝑁 } of a graph 𝔾 = (𝒱, ℰ), the characteristic matrix 𝑃 (𝜋) ∈ IR𝑁 ×𝑛 of the partition is defined by { 1 if 𝑖 ∈ 𝐶𝑗 𝑃𝑖𝑗 (𝜋) = 0 otherwise, for 1 ≤ 𝑖 ≤ 𝑁 and 1 ≤ 𝑗 ≤ 𝑛. First consider the case when the graph 𝔾 = (𝒱, ℰ) is unweighted and undirected as in [12], [11], meaning that for any distinct pair of nodes 𝑖 and 𝑗, if (𝑖, 𝑗) ∈ ℰ, then (𝑗, 𝑖) ∈ ℰ and the weights 𝑎𝑖𝑗 = 𝑎𝑗𝑖 = 1. We say agent 𝑗 is a neighbor of agent 𝑖, if 𝑎𝑖𝑗 = 1. A partition 𝜋 is said to be an equitable partition if each node in 𝐶𝑗 has the same number of neighbors in 𝐶𝑖 for all 1 ≤ 𝑖, 𝑗 ≤ 𝑛. If one only cares about the number of neighbors in adjacent cells, while ignoring the structure inside a cell, one can define the notion of almost equitable partition. A partition 𝜋 is said to be an almost equitable partition if each node in 𝐶𝑗 has the same number of neighbors in 𝐶𝑖 for all 1 ≤ 𝑖, 𝑗 ≤ 𝑛 and 𝑖 ∕= 𝑗. For a given 𝑣 ∈ 𝒱, 𝜋 is said to be an almost equitable partition relative to 𝑣 if it is almost equitable and {𝑣} is one of its cells. However, when we consider general directed weighted graphs, the weights 𝑎𝑖𝑗 can be of any nonnegative values. Thus we cannot employ the notion of the number of neighbors any more. Now we generalize the notions of equitable partitions and almost equitable partitions in a natural way. Definition 2: A partition 𝜋 is said to be a generalized equitable partition if for any 𝑘, 𝑙 ∈ 𝐶𝑖 , 𝑖, 𝑗 = 1, . . . , 𝑛, ∑ ∑ 𝑎𝑘𝑟 = 𝑎𝑙𝑟 . (9) 𝑟∈𝐶𝑗
𝑟∈𝐶𝑗
𝑟∈𝐶𝑗
𝑟∈𝐶𝑗
3
Definition 3: A partition 𝜋 is said to be a generalized almost equitable partition if for any 𝑘, 𝑙 ∈ 𝐶𝑖 , 𝑖 ∕= 𝑗, 𝑖, 𝑗 = 1, . . . , 𝑛, ∑ ∑ 𝑎𝑘𝑟 = 𝑎𝑙𝑟 . (10)
Fig. 1.
4
A directed graph 𝔾.
Example 1. Let the adjacency matrix 𝐴 and the Laplacian matrix 𝐿 associated with a graph 𝔾 shown in Fig. 1 be given by ⎡ ⎤ 0 2 0.1 0.1 0 ⎢ 0.1 0 0 0.2 0 ⎥ ⎢ ⎥ ⎢ 0 0.5 0.1 ⎥ 𝐴 = (𝐴𝑖𝑗 )3×3 = ⎢ 0.5 0 ⎥, ⎣ 0 0.5 0.5 0 0.1 ⎦ 0.1 0.1 0 0 0 ⎡ ⎤ −2.2 2 0.1 0.1 0 ⎢ 0.1 −0.3 0 0.2 0 ⎥ ⎢ ⎥ ⎢ 0 −1.1 0.5 0.1 ⎥ 𝐿 = (𝐿𝑖𝑗 )3×3 = ⎢ 0.5 ⎥. ⎣ 0 0.5 0.5 −1.1 0.1 ⎦ 0.1 0.1 0 0 −0.2 It is easy to see that 𝐿𝑖𝑗 , 𝑖, 𝑗 = 1, 2, 3, have constant row sums, which corresponds to a nontrivial generalized almost equitable partition 𝜋 = {𝐶1 = {1, 2}, 𝐶2 = {3, 4}, 𝐶3 = {5}} of 𝔾. Note that 𝜋 is not a generalized equitable partition since the row sums of 𝐴11 are not equal. For given nodes 𝑣1 , . . . , 𝑣𝑠 ∈ 𝒱, 𝜋 is said to be a generalized almost equitable partition relative to 𝑣1 , . . . , 𝑣𝑠 if it is a generalized almost equitable partition and {𝑣1 }, . . . , {𝑣𝑠 } are its cells. Let Π𝐺𝐸𝑃 , Π𝐺𝐴𝐸𝑃 and Π𝐺𝐴𝐸𝑃 (𝑣1 , . . . , 𝑣𝑠 ) denote the sets of all generalized equitable, generalized almost equitable and generalized almost equitable partitions relative to 𝑣1 , . . . , 𝑣𝑠 , respectively. Moreover, we say that a generalized almost equitable partition relative to 𝑣1 , . . . , 𝑣𝑠 is maximal, ∗ (𝑣1 , . . . , 𝑣𝑠 ) if it has the smallwhich is denoted by 𝜋𝐺𝐴𝐸𝑃 est size, that is, if it contains the fewest possible cells. It can be shown that given a graph 𝔾 and nodes 𝑣1 , . . . , 𝑣𝑠 , ∗ (𝑣1 , . . . , 𝑣𝑠 ) always exists uniquely. 𝜋𝐺𝐴𝐸𝑃 Now we can characterize the generalized almost equitable
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partitions using terms of invariant subspaces of the Laplacian matrix 𝐿 of 𝔾 as follows. Lemma 1: A partition 𝜋 of 𝔾 is a generalized almost equitable partition, if and only if im𝑃 (𝜋) is 𝐿-invariant. An upper bound for the controllable subspace is given in [11] for system (8) with multiple leaders when the graph is undirected and unweighted. We provide an upper bound in terms of generalized almost equitable partitions for the general case when the graph is directed and weighted. Proposition 3: Let 𝜋 ∈ Π𝐺𝐴𝐸𝑃 be such that for each 𝑣 ∈ 𝒱𝐿 , {𝑣} is a cell of 𝜋. Then
evolution of the agents’ states are shown in Fig. 2, from which we can see that system (2) realizes cluster synchronization with respect to the partition 𝜋 ∗ (5).
ei1 (t)
2 0 −2 −4
0
5
10
15
0
5
10
15
0
5
10
15
ei2(t)
2 0 −2
𝒦 ⊆ im𝑃 (𝜋). For system (8) with multiple leaders, we have the following bound. Lemma 2: If 𝒱𝐿 = {𝑣1 , . . . , 𝑣𝑠 }, then
ei3(t)
2 0 −2
t
∗ (𝑣1 , . . . , 𝑣𝑠 ))). dim(im𝑃 (𝜋𝐺𝐴𝐸𝑃
dim(𝒦) ≤ From this lemma, we immediately have the following proposition. Proposition 4: Assume that 𝔾 is strongly connected. System (8) with multiple leaders 𝒱𝐿 = {𝑣1 , . . . , 𝑣𝑠 }, is con∗ (𝑣1 , . . . , 𝑣𝑠 ) is trivial, that is, {𝑖} ∈ trollable only if 𝜋𝐺𝐴𝐸𝑃 ∗ 𝜋𝐺𝐴𝐸𝑃 (𝑣1 , . . . , 𝑣𝑠 ) for all 𝑖 ∈ 𝒱. Theorem 1: Given 𝒱𝐿 = {𝑣1 , . . . , 𝑣𝑠 }. For those graph topologies that are strongly connected and have nontrivial ∗ (𝑣1 , . . . , 𝑣𝑠 ), system (8) is not controllable and system 𝜋𝐺𝐴𝐸𝑃 (2) realizes cluster synchronization provided that the conditions (ii) and (iii) in Proposition 1 are satisfied . So in this section, by comparing the conditions for realizing cluster synchronization and checking controllability, we have gained the insight that those multi-agent networks that are uncontrollable in finite time tend to realize cluster synchronization as time goes to infinity. IV. I LLUSTRATIVE EXAMPLE We take the graph 𝔾 and the associated Laplacian matrix 𝐿 in Example 1 as an example. If we take node 5 as a leader, then the maximal generalized almost equitable partition relative to node 5 is 𝜋 ∗ (5) = {{1, 2}, {3, 4}, {5}}. The system (8) with this Laplacian matrix is uncontrollable since the partition 𝜋 ∗ (5) is nontrivial. Now consider the coupled system (2) with the same Laplacian matrix, where 𝑐 = 1, Γ = 𝐼3 , 𝑓 (𝑡, 𝑦) = −𝑦 + 𝑇 𝑔(𝑦) with 𝑦 = [𝑦1 , 𝑦2 , 𝑦3 ]𝑇 ∈ IR3 , ⎡ 1.25 −3.2 𝑇 = ⎣−3.2 1.1 −3.2 4.4
⎤ −3.2 −4.4⎦ , 1
𝑔(𝑦) = [𝑔1 (𝑦1 ), 𝑔2 (𝑦2 ), 𝑔3 (𝑦3 )]𝑇 , and 𝑔1 (𝑤) = 𝑔2 (𝑤) = 𝑔3 (𝑤) = (∣𝑤 + 1∣ − ∣𝑤 − 1∣)/2. The system 𝑑𝑦 𝑑𝑡 = 𝑓 (𝑡, 𝑦) is a 3-D neural network, which has a double-scrolling chaotic attractor. It is easy to see that 𝑓 (𝑡, 𝑦) is globally Lipschitz continuous. Thus, 𝑓 (𝑡, 𝑦) ∈ QUAD(Δ, 𝐼3 , IR3 ). Assume that the initial values of the agents are chosen randomly from [0.5, 0.5] × [0.5, 0.5] × [0.5, 0.5]. Then the trajectories of the
Fig. 2.
The state trajectories of the five agents.
V. C ONCLUSION We have looked at jointly the cluster synchronization problem for complex networks and the controllability problem for multi-agent systems. Using the notions of generalized graph partitioning, we have provided an upper bound for the controllable subspace of a diffusively coupled multi-agent system. In the future, we are interested in studying more control related problems for complex networks by using the integrated set of tools from network theory, control theory and graph theory. R EFERENCES [1] S. H. Strogatz. SYNC: The emerging science of spontaneous order. Hyperion, New York, 2003. [2] M. E. J. Newman. Networks: An introduction. Oxford University Press, 2010. [3] C. W. Wu and L. O. Chua. Synchronization in an array of linearly coupled dynamical systems. IEEE Transactions on Circuits and Systems I, 42:430–447, 1995. [4] W. Wu and T. Chen. Partial synchronization in linearly and symmetrically coupled ordinary differential systems. Physica D, 238:355–364, 2009. [5] W. Xia and M. Cao. Cluster synchronization algorithms. In Proc. of the 2010 American Control Conference, pages 6513–6518, 2010. [6] W. Xia and M. Cao. Clustering in diffusively coupled networks. Automatica, 47:2395–2405, 2011. [7] A. Jadbabaie, J. Lin, and A. S. Morse. Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Transactions on Automatic Control, 48:985–1001, 2003. [8] V. Kumar, N. E. Leonard, and A. S. Morse. Cooperative Control. Springer-Verlag New York, Inc., 2005. [9] A. Rahmani, M. Ji, M. Mesbahi, and M. Egerstedt. Controllability of multi-agent systems from a graph-theoretic perspective. SIAM Journal on Control and Optimization, 48:162–186, 2009. [10] R. E. Kalman. Mathematical description of linear dynamical systems. Journal of Society of Industrial and Applied Mathematics, 1:152–192, 1963. [11] S. Zhang, M. K. Camlibel, and M. Cao. Controllability of diffusivelycoupled multi-agent systems with general and distance regular coupling topologies. In Proc. of the 50th IEEE Conference on Decision and Control and 2011 European Control Conference, pages 759–764, 2011. [12] M. Mesbahi and M. Egerstedt. Graph Theoretic Methods in Multiagent Networks. Princeton University Press, 2010.
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