Structural Controllability of a Consensus Network with Multiple Leaders
aa r X i v : . [ c s . S Y ] M a r Structural Controllability of a Consensus Network withMultiple Leaders
Milad Kazemi M., Mohsen Zamani, and Zhiyong Chen,
Senior Member, IEEE
Abstract —This paper examines the structural controllability for agroup of agents, called followers , connected to each other based on theconsensus law under commands of multiple leaders , which are agents withsuperior capabilities, over a fixed communication topology. It is provedthat the graph-theoretic sufficient and necessary condition for the set offollowers to be structurally controllable under the leaders’ commands isleader-follower connectivity of the associated graph topology. This shrinksto graph connectivity for the case of solo leader. In the approach, weexplicitly put into account the dependence among the entries of the systemmatrices for a consensus network using the linear parameterizationtechnique introduced in [1].
Index Terms —structural controllability, multi-agent systems.
I. I
NTRODUCTION
In recent years, due to the importance of analyzing the complexsystems, the notion of structural controllability has been retakeninto consideration. Defined as controllability of systems for almostevery parameter values, structural controllability has a wide range ofapplications from robotics [2] to biological systems [3]. Lin [4] firstintroduced the structural controllability for single input linear time-invariant (LTI) dynamical systems. He provided a graph-theoreticrepresentation that guarantees the structural controllability for LTIsystems, i.e. controllability for almost every parameter values. Thenew notion of controllability that Lin introduced encouraged otherresearchers to investigate the interaction among systems’ parameters.The authors of [5] presented an algebraic representation of Lin’stheorem and also extended the theorem to scrutinize the structuralcontrollability for multi-input LTI systems. The aforementionedstudies dealt with dynamical systems in which each of the nodesrepresents a first-order dynamical system. The structural controlla-bility of multi-input/multi-output (MIMO) high-order systems wasinvestigated in [6], [7], and [8]. In most cases, one may needto examine beyond the fact that whether a system is structurallycontrollable or not. For instance, when dealing with uncontrollablesystems, declaring the maximum controllable subspace enables us toknow our ability to control the system (see e.g. [9], [10], [11], and[12]).Moreover, in applications such as systems biology, the choice ofinput nodes (driver nodes) is so broad that selecting a proper set ofnodes to ensure the controllability becomes a crucial problem in cellreprogramming or in cancer treatment (see e.g. [13], [14], [15], and[16]). The reference [13] determined the minimum number of drivernodes to guarantee the structural controllability of LTI systems. Inthere, the authors provided a polynomial algorithm to determine thedriver nodes.These studies have a common assumption that the nonzero entriesof the pair ( A, B ) are independent from each other with free choices.Despite the wide application of this theory, it cannot analyze asystem with the same scalar values appearing in more than one place M. Kazemi M. is with the Department of Mechanical Engineering, Amir-kabir University of Technology (Tehran Polytechnic), Tehran, [email protected]. Zamani and Z. Chen are with the School of Electrical Engineering andComputing, University of Newcastle, NSW 2308, [email protected], [email protected] in the pair ( A, B ) , which is indeed the case for many dynamicalsystems, i.e., entries of the pair ( A, B ) cannot be assigned arbitrarily.To overcome this dilemma, one can represent the system by alinearly parameterized model. The authors of [1] extended the notionof structural controllability to linearly parameterized systems withbinary assumptions. This extension has a great application in theareas such as cooperative control of multi-agent systems where thereexists inherent dependence between the entries of the state and inputmatrices that capture the network topology.In this paper, we examine the structural controllability for a groupof agents equipped with a consensus law. The notion of controllabilityfor such a setup was first proposed by [17]. This reference exploitedcontrollability to examine the possibility that a group of intercon-nected agents through consensus law can be steered to any desiredconfiguration under the command of a single leader. Several necessaryand sufficient algebraic conditions for controllability of multi-agentsystems based on eigenvectors of the associated Laplacian graphwere introduced in [17]. The problem was then developed furtherin [18], [19], [20], and [21]. For instance, in [19], it was concludedthat devoid of eigenvalue sharing between the Laplacian matrixassociated with the follower set only and the Laplacian matrixcorresponding to the whole topology is both necessary and sufficientfor controllability. The reference [21] provided a necessary onlycondition for the controllability of followers under multiple leaders.The graph-theoretic representation of these results was introduced in[22]. Moreover, the structural controllability of multi-agent systemswith a switching topology and the structural controllability of higherorder multi-agent systems were studied in [23] and [24], respectively.Another sphere of research relies on the behavior of the system beforeand after establishing link or agent removal. Robustness of structuralcontrollability against node and link removal was investigated in [25]and [26].In this paper, we exploit the linear parameterization techniqueto deal with the dependance among the entries of the pair ( A, B ) when analyzing the structural controllability of interconnected linearsystems. The reference [22] addressed the same problem for the caseof solo leader. Even though the results reported in [22] are correct,the authors neglected the above-mentioned inherent dependence inthe main proof stated there. Moreover, the authors in [23] addressedthe structural controllability for a group of interconnected agentswith multiple leaders under a switching topology and provided thesufficient and necessary condition. Similar to [22], in this referencethere exists an implicit assumption about independence of entriesappearing in the A and B matrices. There seems no clear clue tofix the flaw of the proofs within the same framework. Therefore,we aim to provide an alternative rigorous proof using the linearparameterization technique recently developed in [1].The rest of the paper is organized as follows. The terminology andconcepts used in this paper are defined in Section II. The problemformulation is given in Section III. We study the case where thereexists only one leader among agents in Section IV. Then the resultsof this section are exploited in Section V to examine the multipleleaders case. Finally, Section VI concludes the paper. II. PRELIMINARIES
A. Structural Controllability
Roughly speaking, the concept of controllability, as a paramountproperty of control systems, examines the capability of a system tosteer from any initial state to some desired final value within its entireconfiguration space under a proper control law. The answer to thisexamination is given by controllability tests like the Kalman’s rankcondition [27] or Popov-Belevitch-Hautus (PBH) controllability test[28]. However, for many dynamical systems, the system’s parametersare not precisely known and in some cases, the existence or absenceof system parameters is the only accessible information. In additionto this, some systems have time-variant parameters, and it is com-putationally hard to determine the controllability of these systemsduring the whole process.In order to overcome these challanges, Lin in his seminal paper [4]introduced the notion of structural controllability. The Lin’s theoremprovided a test for checking the controllability of structured LTIsystems, which are LTI systems whose entries of A and B , i.e.,the state and input matrices are either zero or independently freeparameters. As defined in [4], the pair of matrices ( A, B ) witheach entry either being a zero value or an arbitrarily chosen scalarnot depending on other entries, is structurally controllable if thereexists a real controllable pair, say ( ¯ A, ¯ B ) , with the same structure ofzero entries as ( A, B ) . Consequently, the system is concluded to becontrollable for almost every parameter values.The result in [4] is insightful but the concept of structural control-lability introduced in [4] does not apply to systems with the inherentdependence among the entries of A and B . One should note that itis ubiquitous in many practical scenarios like in biological systems,where one explores gene-gene interaction, to have some of theinterconnecting links be related to each other. To accommodate thesesystems, one needs to modify the structural controllability definitionin [4] and propose new test tools. One way is to represent the systemin the linearly parameterized form [29]. This approach is a convenientmethod to analyze LTI systems with parameter repetition in their stateand/or input matrices. We breifly review linear parameterization ofstructured systems in the next subsection. B. Linear Parameterization
Consider the LTI system given as ˙ x = A ( w ) x + B ( w ) u, (1)where A ( w ) ∈ R n × n and B ( w ) ∈ R n × m are functions of an ar-bitrarily selected vector w = (cid:2) w w . . . w σ (cid:3) ⊤ . Suppose thematrices ( A, B ) have p nonzero entries. The definition in [4] onlyapplies to the case that these p entries are exactly represented by w with p = σ .For the more general scenario with σ ≤ p , the matrix pair ( A, B ) can be linearly parameterized as A n × n ( w ) = P k ∈ q c k w k r k , B n × m ( w ) = P k ∈ q c k w k r k , (2)where q = { , . . . , σ } , c k ∈ R n , r k ∈ R × n , and r k ∈ R × m .We provide the following example for further explanation of linearparameterization. Example 2.1:
Consider the following equation ˙ v ˙ v ˙ v = − w − w − w w w w − w w − w v v v + w u. (3)The above LTI system attains the pair ( A, B ) which is a function of (cid:2) w w w (cid:3) ⊤ , and its associated linear parameterization can berepresented as c = , r = (cid:2) − (cid:3) , r = 1 ,c = − , r = (cid:2) − (cid:3) , r = 0 ,c = − , r = (cid:2) − (cid:3) , r = 0 . (4)It is obvious that the vectors c , c , and c are linearly independentof each other and σ = 3 .The pair ( A ( w ) , B ( w )) is called structurally controllable if thereexists a parameter vector w ∈ R σ for which the pair ( A ( w ) , B ( w )) is controllable [1]. We adapt the same definition in this paper.It is worhtwhile noting that, for p = σ , the definitions ofstructural controllability in [4] and [1] are identical and the structuralcontrollability of the system in (1) can be studied by the results in[4]. The structural controllability of the system (1) for σ = p isalso explored in [29] from an algebraic point of view. Despite thewell-approved algebraic structural controllability conditions in [29],in most cases, the graph-theoretic perspective provides more insightsregarding hidden relations that undergo between system’s parameters.In the next subsection, we give a short review on some graph theoryconcepts exploited in this paper. C. Graph Notation
The reference [4] exploited weighted-digraphs to represent dynam-ical systems. This graph representation not only shows the existenceof directed interactions, or links, between the entries of A and B , butalso reveals the strength of those links. This way of demonstratingdynamical systems enabled the author of [4] to introduce graph-theoretic descriptions for structural controllability of single input LTIsystems. In this paper, we deploy the flow graph representation tostudy the dynamical system (1) from graph-theoretic point of view.Consider the weighted graph G with its node set V = { v , v , . . . , v N V } , edge set E = { e , e , . . . , e N E } ,and weight set W corresponded to each link W = { ( e , w ) , ( e , w ) , . . . , ( e N E , w N E ) } . Let N V and N E bethe number of the nodes and the edges, respectively. Then graphrepresentation of the dynamical system (1), which is called the flowgraph denoted by F G , is a digraph. It includes n + m vertices V = { v , v , . . . , v n + m } , where the input nodes take the last m indices, i.e., v n +1 , . . . , v n + m . Moreover, if the { i, j } entry of thematrix [ A, B ] is nonzero, there exists a link from v j to v i . Twonodes are called neighbors if there exists an edge that correspondsthese two nodes, and if all of the nodes are neighbors to each other,the graph is called a complete graph . A path is a set of edges that v v v v w w w w w − w − w − w − w − w Fig. 1. The flow graph of the system defined in (3). connect a set of distinct nodes. The digraph is called connected provided that there exists a bidirectional path between every twodifferent vertices.The flow graph F G has a spanning forest rooted at vertices v n +1 , v n +2 , . . . , v n + m if for any other node of the graph, say v j ∈ { v , v , . . . , v n } , there exists a path from one of the root nodes v i ∈ { v n +1 , v n +2 , . . . , v n + m } to v j .Finally, the Laplacian matrix associated with the graph G is definedas L ( i,j ) = P i = j w ij i = j, − w ij i ∈ neighborhood of j, otherwise. (5)For the sake of clear explanation, we represent the flow graph ofthe system (3) in Fig. 1. Remark 2.1:
It is noteworthy that the vectors c i and r i havegraph-theoretic implications in the system’s associated flow graph.On one hand, the nonzero entry of c i , say j , captures an ingoingedge to node v j with weight w i in its associated flow graph. Onthe other hand, in r i a nonzero entry expresses an outgoing edgefrom node v j in the associated flow graph and j is the index of thatnonzero entry. This is further demonstrated in the following example. Example 2.2:
Consider the following system ˙ x = w w w w x. (6)One can verify that this system can be linearly parameterized withvectors r i , r i and c i related to weight w and w as c = , r = (cid:2) (cid:3) , r = 0 ,c = , r = (cid:2) (cid:3) , r = 0 . (7)The flow graph of this system is represented in Fig. 2.As stated in Remark 2.1, each of the nodes v and v in thecorresponding flow graph (the indices of nonzero entries of c ) hasan ingoing edge with weight w . Moreover, the nonzero entry of r (the first entry) determines the outgoing edge from node v withinthe associated flow graph. This system has two outgoing edges from v (one to node v and the other one to node v ) with weight w . Itis worth noting that the vectors c and r both have a nonzero firstentry. This means that the node v has a self loop with weight w . v v v w w w w Fig. 2. The flow graph of the system defined in (6).
D. Matrix-Algebraic Terminology
In this subsection, we state some notions and results that help usin establishing the main result of the paper.The generic rank denoted by g-rank [ · ] of linearly parameterizedmatrix M ( w ) = (cid:2) A ( w ) B ( w ) (cid:3) = X k ∈ q c k w k r k , where r k = (cid:2) r k r k (cid:3) , is the maximum rank of M ( w ) for allpossible values of w . Furthermore, the pair ( A, B ) is irreducible ifthere exists no permutation matrix Q such that QAQ − = (cid:20) A A A (cid:21) , QB = (cid:20) B (cid:21) , (8)where A ∈ R h × h , A ∈ R ( n − h ) × h , A ∈ R ( n − h ) × ( n − h ) , B ∈ R ( n − h ) × m , and ≤ h ≤ n . It then becomes evident that the systemis structurally controllable if the pair ( A, B ) is irreducible and itsassociated g-rank is equal to the number of states, i.e., n .The irreducibility of the system has a graph-theoretic implicationwhich is stated in the following proposition. Proposition 1: [30] The pair ( A, B ) is irreducible if and only ifthe assocaited flow graph F G has a spanning forest rooted at v n +1 , . . . , v n + m . Remark 2.2:
Proposition 1 was initially developed to address thematrix pairs satisfying the unitary assumption which means that eachweight appears only in one entry of the matrix pair ( A ( w ) , B ( w )) ,i.e., σ = l ; however, the same proof applies to the case in which σ ≤ l without any modification [1].III. P ROBLEM F ORMULATION
Our goal in this paper is to investigate structural controllability fora group of interconnected systems with fixed topology of no self-loops. We consider a group of N interconnected agents and focuson the leader-follower framework, where there exist l agents withsuperior capabilities and access to external commands which we referto as leaders, while the remainder of agents take the follower role.Without loss of generality, the last l agents are considered as leadersmanipulated by some external input, and the remaining N − l agentsare controlled by the consensus law.Each follower can be modeled as a point mass exerted by anexternal load as ˙ x i = − X j ∈N i w ij ( x i − x j ) , (9)where N i is the set that captures the neighbors of the agent i , and w ij = 0 is weight of the edge from j to i . For the sake of simplicity in the notation we suppose that there exists a bijective mapping betweentwo sets { w ij i ∈ N j , i < j } and { w , w , . . . , w α } . In the restof this paper, we exploit w k instead of w ij . Example 3.1:
The system (3) can be rewritten as ˙ v ˙ v ˙ v = − w ( v − u ) + w ( v − v ) + w ( v − v ) w ( v − v ) w ( v − v ) , (10)so, the graph topology for the consensus network, called the com-munication topology, is shown in Fig. 3, while the flow graph is inFig. 1.The leaders do not follow the law in (9), and are controlledexclusively by some external input expressed as ˙ x j = u ⋆j , (11)where j defines the index number of leader vertices j ∈ { N − l + 1 , . . . , N } . The aggregated dynamical model of the wholeinterconnected system can be obtained as [22] ˙¯ x = (cid:20) A ( N − l ) × ( N − l ) B ( N − l ) × l l × ( N − l ) l × l (cid:21) ¯ x + (cid:20) ( N − l ) × u ⋆l × (cid:21) . (12)In the set of equations in (12) the leaders’ positions can be seenas inputs to autonomous dynamics captured by followers only. Thenthe part of above dynamics only associated with followers can besimplified as ˙ x = Ax + Bu, (13)where A = − L ff , where L ff is the part of Laplacian matrixassociated with followers only. And, the matrix B only captures theinteractions between followers and leaders. Our task in paper is toexplore the controllability of the system (13) under the commandsof multiple leaders and establish a graph-theoretic condition whichis both sufficient and necessary for guaranteeing structural controlla-bility.To explore the controllability of multi-agent systems, in the fol-lowing sections, we first consider the case of single leader and derivethe necessary and sufficient condition for this setup. We then extendthe theorem to the case with more than one leader.IV. S TRUCTURAL C ONTROLLABILITY OF MULTI - AGENT S YSTEMS WITH SINGLE LEADER
In this section, a sufficient and necessary condition for structuralcontrollability of a group of agents under a solo leader with fixedcommunication topology is introduced. To this end, let us firstconsider an edge with the weight w k that connects two vertices v i and v j within the flow graph F G that captures the interactions betweenentries of A and B matrices in (13). Without loss of generality,we assume that i < j . Then these two vectors c k ∈ R n × , and r k ∈ R × n have zero entries except their i and j entries i.e. c ( i ) k = − , c ( j ) k = 1 ,r ( i ) k = 1 , r ( j ) k = − . (14)Let us now introduce the set s = { i , . . . , i s }⊂ q where s is thecardinality of set s . Then the matrices C s , R s , and W s can be definedas C s , (cid:2) c i c i . . . c i s (cid:3) ,R s , h r ⊤ i r ⊤ i . . . r ⊤ i s i ⊤ ,W s , diag (cid:2) w i w i . . . w i s (cid:3) . (15) v v v v w w w Fig. 3. The communication topology associated with the system in (3).
Example 4.1:
Consider the graph topology of system representedin (3) under the leadership of node v . This system has, as discussedin Example 2.1, three independent parameters i.e. σ = 3 . Thus, forthis case, we have c = c v v = , r = r v v = (cid:2) − (cid:3) ,c = c v v = − , r = r v v = (cid:2) − (cid:3) ,c = c v v = − , r = r v v = (cid:2) − (cid:3) . (16)Hence, if we consider s = q the C s , R s and W s can be written asfollows C s = − −
10 1 00 0 1 ,R s = − − − ,W s = w w
00 0 w . (17)We need to introduce the notion of transfer matrix for establishingthe main result of this paper. The transfer matrix of { ( c i , r i , r i ) | i ∈ q } , denoted by T , is a block matrix defined as T i,j = (cid:26) r i c j i, j ∈ q r i i ∈ q , j = σ + 1 . (18)We refer to the graph associated with the transfer matrix T as transfergraph denoted by T . This is a directed graph with σ + 1 vertices γ , . . . , γ σ , γ σ +1 and an edge from node γ j to γ i whenever T i,j isnonzero. Example 4.2:
Consider the system (3), if s = { } ⊂ q , C s , R s ,and W s are C s = ,R s = (cid:2) − (cid:3) ,W s = (cid:2) w (cid:3) . (19) γ γ γ γ − − − − − Fig. 4. The transfer graph of the system defined in (3).
Similarly, C q − s , R q − s , and W q − s are C q − s = − −
11 00 1 ,R q − s = (cid:20) − − (cid:21) ,W q − s = (cid:20) w w (cid:21) . (20)For the system (3) the transfer matrix T can be respresented as T = − − − − − . (21)The transfer graph of the above transfer matrix is shown in Fig. 4. Remark 4.1:
As defined in (18), the entry T i,j with i, j ∈ q isobtained by inner product of two vectors r i and c j and this means r i c j = Σ nk =1 r ( k ) i c ( k ) j = r (1) i c (1) j + . . . + r ( n ) i c ( n ) j where r ( k ) i and c ( k ) j represent the k th entry of r i and c j , accordingly. Each of theseterms has a graph representation. The terms r ( k ) i and c ( k ) j seek foran outgoing edge from node v k with weight w i , and another ingoingedge to node v k with weight w j , respectively.Theorem 1 represents a graph-theoretic sufficient and necessarycondition that guarantees the structural controllability among inter-connected agents with fixed topology under single leader. Before westate this result, we first need to introduce some results which enableus to establish the main theorem of this section. Proposition 1 andLemma 4.1 provide results in order to link the irreducibility of thesystem (13) to characteristics of its associated transfer graph T , whichis used in the proof of Theorem 1.We first introduce the following lemma which is inspired by theresult in [1]. In [1] the authors established a connection betweenthe irreducibility property of a linear parameterized representationand the structure of its associated transfer graph when just and values appear within the corresponding c i , r i and r i vectors. Thisassumption does not hold in our case; thus, we need to extend theproof initially stated in [1]. Lemma 4.1:
If the pair ( A, B ) for the system (13) is irreducible,then the associated transfer graph T has a spanning tree rooted at γ σ +1 .In order to prove the above lemma, we follow the approach of [1].To this end, we first introduce the concept of line graph . The linegraph associated with the directed graph F G is also a directed graph L G that represents the adjacencies between the edges of F G . Eachedge of the original graph F G is presented by a node in its associatedline graph L G . Thus, the number of edges in F G is equal to thenumber of veritices in the corresponding L G . It is worth mentioningthat two edges with the same start and end nodes but different weightsin F G , are captured as different nodes in L G . In order to construct L G , one needs parameters associated with each edge in the flow graph F G ,namely start node, end node, and the corresponding weight. Then, thenode ijk is connected to the node jj ′ k ′ in the line graph if thereexist two edges in the flow graph; one from the node v i to the node v j with weight w k and the other one from the node v j to v j ′ withweight w k ′ Proof of Lemma 4.1
Suppose that the relation between entries of ( A, B ) is capturedby the flow graph F G and includes σ independent parameters. Letus consider the equivalent class H o = (cid:8) ijk ∈ V L k = o (cid:9) with V L being the node set of L G and o ∈ { , . . . , σ } . Hence, a quotientgraph like b L can be inaugurated in such way that it has σ nodescorresponding to its independent weights. This quotient graph has anedge between two nodes, if there exists at least one edge betweenthe two sets of nodes in the line graph corresponding to these twonodes. In other words, the quotient graph has an edge from node k to node k ′ , if there exists a node v j such that two edges exist in theflow graph: one from an arbitrary node v i to node v j with weight w k and the other one from v j to an arbitrary node v j ′ with weight w k ′ .Now, we can focus on the transfer matrix. Based on (18), thetransfer graph T has an edge from node γ k to γ k ′ if r k c k ′ = 0 .Based on (14), we know that for all k ’s at most two entries of c k and r k are nonzero. In particular, the entry of c k has a higherindex than that of − . The reverse holds for r k . Therefore, if onecalculates r k c k ′ = Σ nj =1 r ( j ) k c ( j ) k ′ , there exists at most two nonzerosummand say r ( j ) k c ( j ) k ′ + r ( j ) k c ( j ) k ′ which also have the same sign.Hence, one nonzero summands guarantees the existence of an edgefrom γ k to γ k ′ in the corresponding transfer graph. This is analogousto having an edge from an arbitrary node v i to v j ( j ∈ { j , j } ) withweight w k and another edge from v j to an arbitrary node v j ′ withweight w k ′ . Let us now introduce the induced subgraph b T which isobtained from T by deleting the node γ σ +1 and its associated edges.Now based on above-mentioned definitions, we can concludethat b L and b T are isomorphic with the bijection that maps vertex i in b L to vertex γ i in b T . If the pair ( A, B ) is irreducible, byProposition 1, F G has a spanning forest rooted at v n +1 , v n +2 , . . . ,and v n + m . Thus, the associated b L has a spanning forest rooted at { i i ∈ q , r i = 0 } that capture the weights of edges correspond-ing to nodes v n +1 , v n +2 , . . . , v n + m in the original flow graph.Consequently, b T has a spanning forest rooted at γ i s where i s arethe indices of roots of spanning forest for the quotient graph. Sincethere exists an edge from γ σ +1 to each of such γ i ’s in the transfergraph T , it has a spanning tree rooted at γ σ +1 .The following theorem states the necessary and sufficient conditionfor structural controllability of linear parameterized systems. Proposition 2: [1] A linearly parameterized matrix pair ( A, B ) isstructurally controllable if and only if min s ⊂ q ( rank C s + rank R q − s ) = n (22)and T has a spanning tree rooted at γ σ +1 .We can now present the main result of this section. This theoremstates that the connectivity of the system is both the necessary andthe sufficient condition for guaranteeing the structural controllabilityof the system (13) when there exists only one leader in the network. Theorem 1:
Consider the system (13) under the fixed communica-tion topology G and a single leader, l = 1 . This system is structurallycontrollable if and only if G is connected. Proof of Sufficiency
Given the topology is connected , there exists at least N − edgesamong these nodes. We first assume that the connected graph hasexactly N − edges. In this case, the parameters c i ’s correspondingto these N − edges are independent of each other, because each oneof them establishes a link between two different vertices. The sameholds for (cid:2) r i r i (cid:3) vectors. In the worst case scenario, thereexists only one link from the leader to one of the followers. Thus, aconnected topology consists of at least N − σ independent c i vectors. A similar argument can be applied for counting the numberof independent (cid:2) r i r i (cid:3) vectors. Thus as the columns of thematrix C s are linearly independent of each other, one can observe thatrank C s = f . By exploiting the same argument, we can conclude thatrank R q − s = σ − f , where σ and f are the cardinality of the sets q and s , respectively. We now invoke the Proposition 2. It is easy to see thatits first part is satisfied i.e. ( min s ⊂ q ( rank C s + rank R q − s ) = N − ).Now suppose that the number of weights exceeds the number ofstates, i.e., we have more than N − edges, it is obvious that the c i vectors are not independent of each other anymore. The sameholds for (cid:2) r i r i (cid:3) vectors. Let us introduce a subgraph of thistopology which is connected, has N − edges, and contains nosimple cycles. Such a subgraph, as already established, satisfies therank condition in (22) and one can easily conclude that the sameshould hold for the original graph.On the other hand, if we have a connected topology with singleleader, it is easy to see that there exists a spanning tree rootedat leader’s node. Moreover, Proposition 1 declares that the irre-ducibility is equivalent to existence of a spanning forest rooted at v n +1 , . . . , v n + m . Due to the fact that the system has only oneleader, the notion of spanning forest can be considered analogousto the notion of spanning tree. Hence, the system is irreducible.Furthermore, based on Lemma 4.1, we can conclude that T has aspanning tree rooted at γ σ +1 . Hence, the connectivity of the topologyguarantees the existence of a spanning tree for transfer graph. Proof of Necessity
We use proof by contradiction to establish this part. Suppose thatthe system was structurally controllable, while it was not connected.Then the system could be represented as ˙ x = (cid:20) L d × d d × d d × d L d × d (cid:21) x + (cid:20) b d + d − (cid:21) u. (23)The above system can be considered as two separated subsystems:the connected topology which includes the leader and the rest of thetopology. Based on this definition, d represents the number of nodesin the connected topology that includes the leader and the remaining d nodes are considered as a different subsystem. According to Kalman’s theorem the controllability matrix for the system (23) canbe obtained as (cid:2) ¯ B ¯ A ¯ B . . . ¯ A n ¯ B (cid:3) = b ⋆ . . . ⋆ ⋆ ⋆ ... ... . .. ... ⋆ . . . ⋆ d × d × . . . d × , (24)where the ⋆ captures a zero or a nonzero entry. Consequently, therank of the controllability matrix is equal or less than d . Also notethat the controllability matrix includes a zero matrix of dimension d by n . This contradicts with the earlier assumption.V. S TRUCTURAL CONTROLLABILITY OF MULTI - AGENT SYSTEMSUNDER MULTIPLE LEADERS
The previous section introduced the necessary and sufficient condi-tions for the structural controllability of interconnected agents undera solo leader. This result enables us to investigate the structuralcontrollability of multi-agent systems with more than one leader.To this end, we first need to present the notion of leader-followerconnectivity . Definition 5.1:
The graph representation of a set of connectedagents is called leader-follower connected if there exists at least aleader in each of the associated subgraphs which are totally separatedfrom each other.
Remark 5.1:
There exists an analogy between the two notion ofleader-follower connectivity and having a spanning forest rooted atleaders’ vertices. Based on the definition, if the graph has a spanningforest rooted at some special vertices, e.g., leaders, there exist atleast a path between each node of the graph, except leaders, to oneof the leaders’ nodes. This property guarantees the existence of atleast one leader in every totally separated subgraphs which coincideswith Definition 5.1.Before presenting the main results, it is beneficial to review someinformation about the whole interconnected system. Provided themulti-agent system has l leaders, the system matrices A and B are ofdimensions ( N − l ) × ( N − l ) and ( N − l ) × l , accordingly. Moreover,as it is mentioned before, there exists at least one path to each nodefrom one of the leaders in a leader-follower connected topology.Thus, one can easily conclude that there exist at least N − l pathbetween leaders and followers. The following theorem establishesthat the leader-follower connectivity of the topology associated withthe graph is the necessary and sufficient condition for the wholeinterconnected system to be structurally controllable under multipleleaders. Theorem 2:
Consider the system (13) under the communicationtopology G with multiple leaders, i.e., l > . This system isstructurally controllable if and only if the system is leader-followerconnected. Proof of Sufficiency
The goal is to prove the system is structurally controllable providedthat it is leader-follower connected. As mentioned before, a leader-follower connected system with N nodes and l leaders has at least N − l edges. Based on the results derived in single leader case,if we have exactly N − l edges, the parameters c i s are linearlyindependent of each other for every i ∈ q , where q is the setof algebraic independent parameters. The vectors (cid:2) r i r i (cid:3) areindependent of each other as well. Now, if the system is leader-follower connected, again all c i s are linearly independent of eachother. Hence, it is easy to see that for every s ⊂ q the matrices C s and R q − s are full rank, namely rank C s = f and rank R q − s = σ − f . Given that we have N − l independent w k to assign, we can concludethat min s ⊂ q ( rank C s + rank R q − s ) = σ = N − l . The latter isequal to number of system states. Besides, as we stated beforefor the solo leader case, if we have more than N − l edges, it ispossible to introduce a subgraph with N − l edges which satisfies therank condition in (22). On the other hand, the term leader-followerconnectivity suggests that the system has a spanning forest rootedat the leaders vertices. Due to Proposition 1, this means that thecorresponding graph of the system is irreducible. Therefore, basedon Lemma 4.1, we can conclude that the transfer graph T has aspanning tree rooted at γ σ +1 . Based on these two results, the systemis structurally controllable. Proof of Necessity
We use the proof by contradiction to establish the sufficiency part.We assume that the system was not leader-follower connected while itwas structurally controllable. Without the loss of generality, we con-sider that the system consists of two subsystems which are completelyseparated from each other. One of the subsystems is leader-followerconnected and includes all the leaders. This subsystem has N nodes.The remaining N nodes can be seen as a second subsystem. If wecompute the Kalman’s controllability matrix for this system, it iseasy to show that the controllability rank is equal or less than N and the system is not controllable. This contradicts with the initialassumption and the proof is finished.VI. C ONCLUSION
In this paper, the structural controllability of multi-agent systemsunder multiple leaders with fixed topology was scrutinized. Thenecessary and sufficient condition of structural controllability ofmulti-agent systems for the both cases of single and multiple leaderswas developed with the help of the linear parameterization technique.We established that the connectivity of graph topology, in the singleleader situation, and the leader-follower connectivity of the associatedgraph, in the multi leader case, stand not only as the necessarycondition but also as the sufficient condition. Some possible futureresearch directions include investigation of structural controllabilitycondition for switching and linear time-variant topologies.R
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