A Graph-Theoretic Characterization of Controllability for Multi-agent Systems Meng Ji and Magnus Egerstedt Abstract In this paper we continue our pursuit of conditions that render a multi-agent networked system controllable. In particular such conditions are sought for networks in which a collection of the agents take on leader roles while the remaining agents execute local consensus-like control laws. Equitable partitions are introduced in order to improve on previous controllability results and the main contribution of this paper is a new necessary condition for controllability. Index Terms Multi-Agent systems Networked system Controllability Graph theory Equitable partitions. I. INTRODUCTION This paper explores the issue of providing a graphtheoretic characterization of the controllability of certain leader-based multi-agent systems. As such it constitutes a continuation of our previous work by considerably improving the necessary condition for controllability. The controllability issue in leader-follower multi-agent systems was first introduced by Tanner in where a necessary and sufficient condition for controllability was given based on the eigenvectors of the graph Laplacian. Although elegant this condition was not graph-theoretic in that controllability could not be directly decided from the graph topology itself. A more topological result was given by Mesbahi and Rahmani in which a sufficient condition for the network to be uncontrollable in the case of one anchored (leader) agent was given. Their result was related to the symmetry and automorphism group of the underlying graph. In this paper we further extend this notion and present a more general condition based on so-called equitable partitions of the underlying graph. Our result thus addresses a scenario where multiple leaders are possible and from an equitable partition point-of-view captures a larger set of graphs. II. NOTATION AND RELIMINARIES In this section we start with some basic notions in graph theory and recall some known results about controllability of multi-agent networks. (We refer the readers to 4 for more details about algebraic graph theory.) Graphs are broadly adopted in the multi-agent literature to encode the interactions in networked systems. An undirected graph G is defined by a set V G = {... n} of nodes and a set E G V G V G of edges. A cell C V G is a subset of the node set. Two nodes i and j are neighbors if (i j) E G and the neighboring relation is indicated with i j while * This work was supported by the U.S. Army Research Office through the grant # 9988. The authors are with the School of Electrical and Computer Engineering Georgia Institute of Technology Atlanta GA {mengjimagnus}@ece.gatech.edu (i) = {j V G : j i} collects all neighbors to the node i. The degree of a node is given by the number of its neighbors and we say that a graph is regular if all nodes have the same degree. A path i i... i L is a finite sequence of nodes such that i k i k k =... L and a graph G is connected if there is a path between any pair of distinct nodes. A subgraph G is said to be induced from the original graph G if it is obtained by deleting a subset of nodes and the edges connected to those nodes. Let A R n n be the ( ) adjacency matrix and B R n m be the ( ) vertex-edge incidence matrix whose orientation is arbitrarily imposed on the graph. The graph Laplacian matrix is defined by L = BB T and it is easy to verify that L = D A where D is the diagonal degree matrix. In the context of multi-agent systems the nodes represent agents and the edges are communication links. In particular an agent i has access to the information pertaining to its neighbors and can use this piece of information to compute its control law. Let x i R d denote the state of node i whose dynamics is described by the single integer ẋ i = u i where u i is the control input. Rendezvous problem is one of the multi-agent control problems that has attracted considerable attentions from many researchers 5 6 7. Some other problems e.g. formation control 8 9 consensus or agreement flocking 4 5 and etc. share the same distributive flavor with the rendezvous problem and thus adopt a similar methodology. In fact a widely adopted control method for multi-agent rendezvous is the Laplacianbased feedback on the form ẋ = Lx where x = x T x T... x T n T denotes the aggregated state vector of the multi-agent system and L = L I d is the augmented Laplacian matrix. Under some circumstances some agents are required to take leader roles while the others follow them. In this paper we use subscript l to denote the affiliation with leaders while f for the followers. For example a follower graph G f is the subgraph induced by the follower set V f. As the leader roles are designated the incidence matrix B can be partitioned as Bf B = B l where B f R n f m and B l R n l m. Here n f n l and m are the cardinalities of the follower group the leader group
and the edge set respectively. As a result the graph Laplacian L is given by Lf l L = fl lfl T L l where L f = B f B T f L l = B l B T l and l fl = B f B T l. As an example Figure shows a leader-follower network with V l = {5 6} and V f = { 4}. This gives B f = and L f = B l = 4 4 l fl = 5 6. Fig.. A leader-follower network with: V f = { 4} and V l = {5 6}. The system we are interested in is the leader-follower system where the followers are governed by the Laplacianbased feedback law given by ẋ f = L f x f + l fl x l () while the leaders s movements are dictated by some exogenous control signals. Under this setup a natural question is as follows: Can the leader(s) move the followers to any desired configuration? From a control theoretic point of view we ask following questions: ) Is this system controllable? ) If not what is the controllability (observability) decomposition? ) What is the controllable subspace? 4) What is the structural significance of the controllable or uncontrollable subspace? Before we answer these questions let us review some basic controllability results: roposition.: Given the system (L f l fl ) the following statements are equivalent: ) The system is completely controllable; ) None of the eigenvectors of L f is in the nullspace of lfl T i.e. v i such that L f v i = λv i for some λ R v i / N (lfl T ) where N ( ) denotes the nullspace; ) The controllability matrix C Lf l fl has full rank; 4) The matrix λi L f l fl has full rank for all λ R. Based on these standard controllability results together with some basic properties of the graph Laplacian we can derive the following lemma. Lemma.: Given a connected graph the system (L f l fl ) is controllable if and only if L and L f do not share any common eigenvalues. roof of Necessity: We can reformulate the lemma as stating that the system is uncontrollable if and only if there exists at least one common eigenvalue between L and L f. First we show the necessity. Suppose the system is uncontrollable. Then from Theorem.() there exists a vector v i R n f such that L f v i = λv i for some λ R with Now since Lf l fl l T fl L l vi l T flv i =. Lf v = i lfl T v i vi = λ λ is also an eigenvalue of L with eigenvector v T i T. III. INTERLACING AND EQUITABLE ARTITIONS Equitable partitions and interlacing theory play an important role for our main results. In this section we introduce some definitions and lemmas needed to support our main results. Definition.: A r-partition π of V(G) with cells C... C r is said to be equitable if each node in C j has the same number of neighbors in C i for all i j. We denote the cardinality of the partition π with r = π. Let b ij be the number of neighbors in C j of a node in C i. The directed graph with the r cells of π as its nodes and b ij edges from the ith to the jth cells of π is called the quotient of G over π and is denoted by G/π. An obvious trivial partition is the n-partition π = {{} {}... {n}}. If a partition contains at least one cell with more than one node we call it a nontrivial equitable partition (NE) and the adjacency matrix of a quotient is given by A(G/π) ij = b ij. Now the equitable partition can be derived from graph automorphisms. For example in the so-called eterson graph shown in Figure (a) one equitable partition π (Figure (b)) is given by the two orbit of the automorphism groups namely the 5 inner vertices and the 5 outer vertices. The adjacency matrix of the quotient is given by A(G/π ) = Since L f is symmetric its left eigenvectors are equal to the right ones. The sufficiency proof will be given after Lemma.8..
The equitable partition can also be introduced by the equal distance partition. Let C V (G) be a given cell and let C i V (G) be the set of vertices at distance i from C. C is said to be completely regular if its distance partition is equitable. For instance every vertex in the eterson graph is completely regular and introduces the partition π as shown in Figure (c). The adjacency matrix of this quotient is given by A(G/π ) =. 5 4 9 (a) 6 8 7 C C Fig.. Example of equitable partitions on (a) the eterson graph G = J(5 ) and the quotients: (b) the NE introduced by the automorphism is π = {C C } C = { 4 5} C = {6 7 8 9 } and (c) the NE introduced by equal-distance partition is π = {C C C } C = {} C = { 5 6} C = { 4 7 8 9 }. The adjacency matrix of the original graph and the quotient are closely related through the so-called interlacing theorem. But first we introduce the following lemma. Definition.: A characteristic vector p i R n of a nontrivial cell C i has s in the positions associated with C i and s elsewhere. A characteristic matrix R n r of a partition π of V (G) is a matrix with the characteristic vectors of the cells as its columns. For example the characteristic matrix of the equitable partition of the graph in Figure (a) is given by =. () Fig.. 4 5 (a) (b) (b) C C (c) 4 The (a) equitable partition and (b) the quotient of a graph. Lemma.: (4 Lemma 9..) Let be the characteristic matrix of an equitable partition π of the graph G and let C  = A(G/π). Then A =  and  = + A where + = ( T ) T is the pseudo-inverse of. As an example the graph in Figure has a nontrivial cell ( ). The adjacency matrix of original graph is A = The adjacency matrix of the quotient is  = + A =. Lemma.4: (4 Lemma 9..) Let G be a graph with adjacency matrix A and let π be a partition of V(G) with characteristic matrix then π is equitable if and only if the column space of is A-invariant. Lemma.5: Given a symmetric matrix A R n n and let S be a subspace of R n. Then S is A-invariant if and only if S is A-invariant. The proof of this well-known fact can for example be found in 6. Remark.6: Let R( ) denote the range space. Suppose V(G) = n C i = n i and π = r then we can find an orthogonal decomposition for R n as R n = R( ) R(Q) () where the matrix Q satisfies R(Q) = R( ) such that its columns together with those of form a basis for R n. Following Lemma.5 R(Q) is also A invariant. Unlike matrix Q is derived from the nullspace of and can be constructed in different ways. One possible choice of such a Q is the n n r matrix with r column blocks Q = Q Q... Q r where Q i R n ni corresponds to C i. Moreover each column sums to zero in the positions associated with C i and has zeros in the other positions. In other words Q i = Q i n (n i ). In Q i the upper and lower parts are zero matrices with appropriate dimensions (possibly empty). One possible choice of Q i would be Q i = Ini T (4) n i (n i ) where R ni is a vector with ones in each position. Based on this method the Q matrix for the equitable
partition in Figure (a) can be given by Q = Q = T and thus Q = T. The other choice of Q matrix is using the orthonormal basis of R( ). We denote this matrix as Q. If we define = ( T ). (5) Note that the invertibility of T follows from the fact that the cells of the partition are nonempty. Moreover it satisfies that T Q = and QT Q = In r. In other words T = Q (6) is a matrix whose columns are defined on an orthonormal basis of R n based on the equitable partition π and and Q have the same column spaces as and Q respectively. Theorem.7: (4 Theorem 9..) If π is an equitable partition of a graph G then the characteristic polynomial of  = A(G/π) divides the characteristic polynomial of A(G). Lemma.8: (4 Theorem 9.5.) Let Φ R n n be a real symmetric matrix and let R R n m be such that R T R = I m. Set Ψ = R T ΦR and let v v... v m be an orthogonal set of eigenvectors for Ψ such that Ψv i = θ i (Ψ)v i where θ i (Ψ) R is an eigenvalue of Ψ. Then ) The eigenvalues of Ψ interlace the eigenvalues of Φ. ) If θ i (Ψ) = θ i (Φ) then there is an eigenvector v of Ψ with eigenvalue θ i (Ψ) such that Rv is an eigenvector of Φ with eigenvalue θ i (Φ). ) If θ i (Ψ) = θ i (Φ) for i = i... l then Rv i is an eigenvector for A with eigenvalue θ i (Φ) for i = i... l. 4) If the interlacing is tight then ΦR = RΨ. Now we are in the position to prove the sufficiency of Lemma. i.e. if L and L f share a common eigenvalue the system (L l fl ) is not completely controllable. roof of Sufficiency of Lemma.: Since L f is a principle sub-matrix of L it can be given by L f = R T LR where R = I nf T R n n f. Following Lemma.8() if L f and L share a common eigenvalue say λ then the corresponding eigenvector satisfies vf v = Rv f = where v is λ s eigenvector of L and v f is that of L f. Moreover we know that Lf l Lv = fl vf vf lfl T = λ L l which gives us l T fl v f = and thus the system is uncontrollable. In fact T is a diagonal matrix with ( T ) ii = C i. Now we have shown that the existence of a common eigenvalue shared by L and L f is a necessary and sufficient condition for the leader-follower network to be uncontrollable. IV. CONTROLLABILITY In this section we will utilize a graph theoretic approach to characterize the necessary condition for a multiple-leader networked system to be controllable. The way we approach this necessary condition is through Lemma.. In what follows we will show first that both L and L f are both similarity to some block diagonal matrices. Then we will furthermore show that under some circumstances the diagonal block matrices resulted from diagonalize L and L f have some diagonal block(s) in common. Lemma 4.: If a graph G has a nontrivial equitable partition (NE) π with characteristic matrix the adjacency matrix A(G) of the graph is similar to a diagonal matrix A Ā = A Q where A is similar to the adjacency matrix  = A(G/π) of the quotient. roof: Let the matrix T = Q be the orthonormal matrix with respect to π as what we have defined in (6). Now let Ā = T T AT = T A T A Q Q T A QT A Q. (7) Since and Q have the same column spaces as and Q respectively they inherit the A invariance property i.e. A = B and A Q = QC. for some matrices B and C. Since their column spaces are orthogonal complements to each other we get and T A Q = T QC = Q T A = Q T B =. In addition let D p = T we get T A = D T A D = D (D T A )D (8) = D ÂD and therefore the first diagonal block is similar to Â. Lemma 4.: Let be the characteristic matrix of a NE in G. R( ) is K-invariant where K is any diagonal block matrix of the form K = diag(k i I ni ) r i=
where k i R n i = C i is the cardinality of the cell and r = π is the cardinality of the partition. Consequently Q T K = where = ( T ) and Q is chosen in such a way that T = Q is a orthonormal matrix. roof: Since =. r = p p... p r where i R ni r is a row block which has s in column i and s elsewhere. On the other hand p i is a characteristic vector representing C i which has s in the positions associated with C i and zeros otherwise. Recall the example given in () = (9) and we can find = while p = T. With a little bit calculation we can find k k K =. = k p k p... k r p r = ˆK k r r where ˆK = diag(k i ) r i= which shows that R( ) is K- invariant. Since R( Q) = R( ) it is K-invariant as well by Lemma.5 and Q T K = Q T ˆK =. By the definition of the equitable partition the subgraph induced by a cell is regular and every node in the same cell has the same number of neighbors outside the cell. Therefore the nodes belonging to the same cell have the same degree and thus by Lemma 4. R( Q) and R( ) are D-invariant where D is the degree matrix given by D = diag(d i I ni ) r i= where d i R denotes the degree of each nodes in cell. Since the graph Laplacian satisfy L(G) = D(G) A(G) Lemma 4. and Lemma 4. together can show that R( Q) and R( ) are L-invariant and thus we have following corollary Corollary 4.: Given the same condition as in Lemma 4. L is similar to a diagonal block matrix L = T T L LT = () L Q where L = T L and L Q = Q T L Q and T = Q defines a orthonormal basis for R n with respect to π. As () defines a similarity transformation it follows that L and L Q carry all the spectrum information of L i.e. they share eigenvalues with L. Now that as we have show in Section II in a leaderfollower network the graph Laplacian can be partitioned as Lf l L = fl lfl T L l according to the leader assigning scheme. Transformations similar to () can be found for L f in the presence of NEs in the follower graph G f. Corollary 4.4: Let G f be a follower graph and let L f be the diagonal sub-matrix of L related to G f. If there is a NE π f in G f and a π in G such that all the nontrivial cells in π f are also cells in π there exists an orthonormal matrix T f such that L f = Tf T Lf L f T f =. () L fq roof: Let f = f (f T f ) where f is the characteristic matrix for π f and let Qf be defined on a orthonormal basis of R( f ). In the above way we have obtained an orthonormal basis for R n f with respect to π f. Moreover L f = D f +A f where A f denotes the adjacency matrix of G f while D f is the degree matrix corresponding to the original graph G. Since all the nontrivial cells in π f are also cells in π D f satisfies the condition in Lemma 4. i.e. nodes from an identical cell in π f have the same degree. Hence from Lemma 4. and Lemma 4. R( ) and R( Q) are L f - invariant and thus L f = Tf T Lf L f T f = () L fq where T f = f Qf L f = T f L f f and L fq = Q T f L f Q f. Again the diagonal blocks of L f share all the spectrum information with L f. Now we are in the position to prove our main result. Theorem 4.5: Given a connected graph G and the induced follower graph G f the system (L f l fl ) is not complete controllable if there exist NEs on G and G f say π and π f such that all the nontrivial cells of π are contained in π f i.e. π and π f such that C i = C i π\π f. roof: In Corollary 4. and Corollary 4.4 we have already shown that L and L f are both similar to some diagonal block matrices. Here we want to show the relation ship between these diagonal block matrices.
Assume π π f = {C C... C r }. According to the given condition C i i =... r. Without loss of generality we can index the nodes in such a way that the nontrivial cells comprise the first n nodes 4 such that n = r i= C i n f < n. Since all the nontrivial cells of π are in π f their characteristic matrices have similar structures = and I f = n n I nf n n r n f r f where is a n r matrix that contains the nontrivial part of the characteristic matrices. Since and f are only normalized and f respectively they have the same block structures. Consequently Q and Q f the matrices containing orthonormal basis of R( ) and R( f ) have following structures Q = Q n (n r ) and Q f = Q where Q is a n (n r ) matrix that satisfies Q T =. n f (n r ) As one can observe Qf is different from Q only by n n f rows of zeros. In other words the special structures of Q and Q f gives us the relationship Q f = R T Q where R = I nf T. Now recall the definition of L Q and L Qf from () and () which gives us L Q = Q T L Q = Q T f R T LR Q f = Q T f L f Qf = L fq. () Therefore L f and L share the same eigenvalues associated with L Q and by Lemma. the system is not completely controllable. In the situation described in Theorem 4.5 the system is not completely controllable. This theorem thus gives us a method to identify uncontrollable situations in a leaderfollower system. Intuitively speaking vertices in the same cell of a NE that satisfy the condition in Theorem 4.5 is not distinguishable from the leaders point of view. In other words if the agents belong to the same cell shared by π and π f and they start from the same point it is impossible for the leaders to pull them apart. Thus the controllable subspace can be obtained by collapsing all the nodes in the same cell into a single meta-node. However since the NEs may not be unique as we have seen in the case of the eterson graph more work is required before a complete understanding of this issue is obtained. Note that our sufficient condition for a graph to be uncontrollable immediately produces a necessary condition for a graph to be controllable and this states as a corollary. Corollary 4.6: Given a connected graph G with the induced follower graph G f a necessary for (L f l fl ) to be controllable is that no NEs π and π f on G and G f exist such that π and π f share all nontrivial cells. Corollary 4.7: If G is disconnected a necessary condition for (L f l fl ) to be controllable is that all of its connected components are controllable. V. EXAMLES AND DISCUSSIONS In this part we will show some uncontrollable situations that are identifiable by our method and discuss the relationship among some existing results our result and our ultimate goal. a) Single Leader with Symmetric Followers. In Figure if we choose node 5 as the leader the symmetric pair () in the follower graph renders the network uncontrollable as stated in. The dimension of the controllable subspace is three while there are four nodes in the follower group. This result can also be interpreted by Theorem 4.5 since all the automorphism groups introduce equitable partitions. b) Single Leader with Equal Distance artitions. We have shown in Figure that the eterson graph has two NEs. One is introduced by the automorphism groups and the other (π ) is introduced by the equal distance groups. Based on π if we choose node as the leader the leader-follower group ends up with a controllable subspace with dimension of two. Since there are four orbits 5 in the automorphism groups this dimension can only be interpreted by the twocell equal distance partitions 6. c) Multiple Leaders. The last example is a modified leader graph based on the peterson graph. In Figure 4 we add another node ( ) connected to { 4 7 8 9 } as the second leader in addition to node. In this network there is an equal distance partition with four cells {} { 5 6} { 4 7 8 9 } and {}. In this situation the dimension of the controllable subspace is still two which is consistent with example b). The examples above can be put into a graph described in Figure 5 which shows the relationship among previous theorems our theorem and the ultimate goal of this line of work. This paper provides a tool which can identify a subset of the uncontrollable follower graph represented by II which represents the graphs satisfy the condition described in Theorem 4.5. Since that condition is based on NEs and is able to deal with multiple leaders we can identify a larger set of graphs than the set of graphs in I which has been revealed by some previous works. Our ultimate goal is to achieve a necessary and sufficient condition which can gives us a topological insight of the network. This condition should 4 We introduce n for convenience. It is easy to verify that n r = n r = n f r f 5 They are { 5 6} {7 } {8 9} { 4} 6 They are { 5 6} and { 4 7 8 9 }
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