The Laplacian Pencil and Its Application to Consensus Theory

The Laplacian Pencil and its Application to Consensus Theory

Fabio Morbidi a

aNeCS team, Inria Grenoble Rhˆone-Alpes, 655 Avenue de l’Europe, 38334 Montbonnot Saint Martin, France.

Abstract

In this paper we present a generalization of the classical continuous-time consensus algorithm, obtained by replacing the graph Laplacian with a ﬁrst-order matrix polynomial in the real variable s. The stability properties of the resulting consensus protocol are studied in terms of scalar parameter s for some relevant families of undirected and directed graphs, and for general regular networks by leveraging the spectral theory of matrix pencils. In this context, the original notion of Laplacian pencil is introduced. Finally, by interpreting variable s as the control input of a bilinear system, a graph-theoretic controllability analysis of the proposed protocol is performed. Extensive numerical simulations and examples illustrate our theoretical ﬁndings.

Key words: Multi-agent systems; Consensus algorithm; Cooperative control; Autonomous vehicles; Sensor networks

1 Introduction the classical graph Laplacian is retrieved). The stability properties of the new protocol depend on the value In the last decade we have experienced an impetuous of s and are as variegated as those of the deformed growth in multi-agent systems research [4,23,24], as nu- consensus algorithm, thus making it suitable for a wide merous dedicated sessions in international conferences range of mobile multi-robot applications. For example, and special issues in control and robotics journals have for formation control, target enclosing or coverage path witnessed. Distributed control, and especially consensus planning (cf. Sect. 6), and notably for human-swarm problems, have had a prominent role in this stream of interaction [22] where the vehicles in a team are coor- research [29,34]. Consensus theory originated from the dinated at high level by a supervisor via parameter s. work of Tsitsiklis [39], Jadbabaie et al. [19] and Olfati- In line with this last applicative example, in this paper Saber et al. [30], where the consensus problem was we are not interested in decentralized mechanisms for formulated for the first time in system-theoretic terms. selecting the variable s. Starting from these seminal contributions, numerous ex- For directed graphs, it is shown that the proposed con- tensions to the prototypal consensus protocol have been sensus protocol exhibits complex grouping behaviors proposed in the literature: among the most relevant are even for elementary network topologies. the cases of time-varying network topology [27], of net- Note that while in [25] the spectral theory of quadratic works with delayed [30] or quantized noisy communi- eigenvalue problems was crucial for the stability analysis of the deformed consensus protocol, its counterpart cation and link failure [10,20], of random networks [37], in this paper is represented by the more widely-studied of finite-time [40] and Riemannian consensus [35]. spectral theory of matrix pencils (see, e.g. [7, 11, 15]): In this paper, we continue the line of research inau- this theory is instrumental in characterizing the prop- gurated by [25], where a new continuous-time consen- erties of the new algorithm for general regular graphs sus algorithm, termed “deformed consensus protocol”, and motivated us to introduce the original notion of was introduced and its stability properties studied Laplacian pencil. Note that our notion differs from that in terms of a scalar parameter s. Following a similar recently proposed in [38] to describe undirected multi- paradigm, we propose here a simpler consensus proto- commodity flow networks and involving a symmetric col where the standard Laplacian matrix is replaced by normalized Laplacian. Lp(s)=D − sA, a first-order matrix polynomial in the An interesting feature of the proposed consensus pro- real variable s,whereD and A are the degree and adja- tocol is that it can be interpreted as a bilinear system cency matrix of the communication graph, respectively. with scalar input s, and its controllability properties In this way, the state of each node is updated using a completely characterized in graph-theoretical terms. combination of the states of its neighbors, which are In particular, the commutator of matrices D and A scaled by the parameter s (note, however, that for s =1 turns out to play a crucial role in the controllability analysis. In this respect, as also pointed out in [33] for Correspondingauthor,Tel.:+330322825902 the controlled agreement dynamics, the symmetry of a Email address: [email protected] (Fabio network has an important impact on the controllability Morbidi). of the new consensus protocol. Note that differently from the recent literature on clus- Note that the Laplacian L is a symmetric positive ter synchronization or group consensus (see [41,42] and semidefinite matrix. the references therein), in this paper the inter-agent couplings are not diffusive, unless s is equal to unity Definition 3 (κ-regular graph) A graph G is called (in fact, if s = 1 the row sums of the dynamic ma- κ-regular when every node has precisely κ neighbors. trix of our system are not zero, in general). This also distinguishes our work from [8] where a matrix pencil Lemma 1 G is a κ-regular graph if and only if matrices formalism is used to study the synchronization of sym- D and A commute, i.e. DA = AD. metrically and diffusively coupled networks. Finally, G κ κ although the proposed consensus algorithm may admit Proof: [Necessity ] If is -regular, then D = In and we trivially have κ A = A κ. [Sufficiency] DA = AD a bipartite consensus solution as the protocol studied d a d a i, j ∈{,...,n} d in [1] for networks with antagonistic interactions, our implies that i ij = j ij , 1 ,where i denotes the degree of node i. But this equality is true only analysis in this paper is not uniquely restricted to such d d ∀ i, j G κ a clustering phenomenon. if i = j , i.e. only if is -regular. Beside the aforementioned practical applications, we believe that the study of the protocol proposed in this Lemma 2 Let G be a κ-regular graph. Then λi(L)= work may help to shed new light on known results, and κ − λi(A), i ∈{1,...,n}. to gain a deeper insight into consensus theory by adopting a different perspective. Definition 4 (Bipartite graph) A graph G is called The rest of the paper is organized as follows. Sect. 2 bipartite if its node set V can be divided into two disjoint presents some preliminaries on algebraic graph theory sets V1 and V2, such that every edge connects a node in and matrix pencils. The stability properties of the new V1 to one in V2.IfG is connected, its bipartition {V1,V2} consensus protocol with undirected graphs are studied is unique. in Sect. 3. Sect. 4 is devoted to the study of the nonlinear controllability properties, and in Sect. 5 the sta- Definition 5 (Signless Laplacian matrix Q [5]) bility analysis is extended to directed graphs. Finally, The signless Laplacian matrix of graph G is defined as in Sect. 6, the theory is illustrated with numerical sim- Q = D + A. ulations, and in Sect. 7 the main contributions of the paper are summarized and possible directions for future Note that as L, the signless Laplacian Q is a symmet- research are outlined. ric positive semidefinite matrix, but it is not necessarily Notation: Throughout this paper, k [−1, 1, −1, 1, singular. ...,(−1)n−1, (−1)n]T ∈ IR n, 1 denotes a column vector of n ones, Jm×n the m × n ones matrix, In the n × n Property 1 (Spectral properties of Q [5, 14]) identity matrix, λi(A)thei-th eigenvalue of matrix A, (1) Let G be a κ-regular graph. Then pL(λ)= ∗ T n ( · ) and ( · ) the conjugate transpose and transpose of (−1) pQ(2κ − λ) where pL(λ) denotes the charac- a matrix, respectively, diag(b) a diagonal matrix with teristic polynomial of the Laplacian L. the elements of vector b put on its main diagonal, ⊗ the If G is a bipartite graph, then pL(λ)=pQ(λ). Kronecker product, ∅ the empty set, |·|the cardinality · (2) The least eigenvalue of Q of a connected graph is of a set, the floor function which maps a real number equal to 0 if and only if the graph is bipartite. In this to the largest previous integer, deg[p(λ)] the degree of λ case, 0 is a simple eigenvalue and the corresponding the polynomial with real coefficients p( ), and and eigenvector is v =[vk] where vi =1, i ∈ V1 and matrix inequalities in the positive definite and positive vj = −1, j ∈ V2,being{V1,V2} the bipartition of G. semidefinite sense, respectively. (3) In any graph, the multiplicity of the eigenvalue 0 2 Preliminaries of Q is equal to the number of bipartite connected components of G. In this section, we review some basic notions of algebraic In a sense, the least eigenvalue of Q can be interpreted graph theory and summarize some terminology regard- G ing matrix pencils for later reference. as a measure of how close is to being a bipartite graph. A similar idea has been utilized for the second-smallest 2.1 Algebraic graph theory eigenvalue of L, which is known as the algebraic connec- tivity of G [3]. For this reason, the least eigenvalue of Q Let G =(V, E) be a graph (or network) where is called the algebraic bipartiteness of graph G in [9] . V = {1,...,n} is the set of nodes, and E is the set of Let D =(V, E)beadirected graph (or digraph,for edges [12]. All graphs in this paper are finite, with no short) where V = {1,...,n} is the set of nodes, and self-loops and multiple edges. E ⊆ V × V is the set of edges. In the case of directed a Definition 1 (Adjacency matrix A) The adjacency graphs, we can define the adjacency matrix as ij =1 j, i ∈ E a matrix A =[aij ] of graph G is an n × n matrix defined as if ( ) and ij = 0 otherwise, and the degree main in in aij =1if {i, j}∈E and aij =0otherwise. trix as D =diag(d1 ,...,dn ), where di denotes the in-degree of node i with i ∈{1,...,n}. With these def- Definition 2 (Laplacian matrix L) The Laplacian initions in hand, the in-degree Laplacian L(D) and in- matrix of graph G is an n×n matrix defined as L = D−A degree signless Laplacian Q(D)ofD, can be defined as where D = diag(A1) isthedegreematrix. in the undirected case.

2 Definition 6 (Bipartite digraph) A digraph D = 3 Problem formulation (V, E) is called bipartite if its node set V can be divided into two disjoint sets V1 and V2, such that E ∩ V × V ∅ E ∩ V × V ∅ It is well-known [30], that if the static undirected com- ( 1 1)= and ( 2 2)= . munication graph G is connected, each component of the T n state vector x [x1,...,xn] ∈ IR of the linear time- Definition 7 (Rooted out-branching [24]) A digraph invariant system, D =(V, E) is a rooted out-branching, if: x˙ (t)=−Lx(t), (1) (1) It does not contain a directed cycle, (2) It has a node vR (root) such that for every other asymptotically converges to the average of the initial node v ∈ V , there is a directed path from vR to v. 1 n states x1(0),...,xn(0), limt→∞ xi(t)= n i =1 xi(0) 1 T T = n x0 1,wherex0 [x1(0), ..., xn(0)] , i.e., average 2.2 Matrix pencils consensus is achieved. Let us now consider the following generalization of the Laplacian L. The following definitions are drawn from [7, Sect. 4.5]. Definition 11 The parametric Laplacian of the graph Definition 8 (Matrix pencil) Let H, K ∈ Cm×n. G is an n × n matrix defined as, The matrix polynomial H − λK in the indeterminate λ, is called a matrix pencil. Lp(s)=D − sA, (2)

Definition 9 (Regular and singular pencils) Let where s is a real parameter. , ∈ Cn×n − λ H K .Ifdet(H K) is not identically zero s for all values of λ,thepencilH − λK is called regular. Note that Lp( ) is a symmetric matrix (but not positive Otherwise is called singular.WhenH − λK is regular, semidefinite as L, in general), and that similarly to the λ − λ deformed Laplacian studied in [25], we have Lp(1) = L p( ) det(H K) is called the characteristic poly- − nomial of H − λK and the eigenvalues of H − λK are and Lp( 1) = Q. Inspired by (1), in this paper we study definedtobe: the stability properties of the following linear system, t − s t , • The roots of p(λ)=0, x˙ ( )= Lp( ) x( ) (3) •∞ n − λ λ

• The pencil −H + λK is regular, In order to state our ﬁrst result, note that Lp(s)in(2) • ∩ { } can be interpreted as a matrix pencil in the indetermi- ker(H) ker(K)= 0 , nate s, and to reinforce this fact, we will refer to it as • All the eigenvalues of −H + λK are real. the Laplacian pencil.

3 Property 2 (Properties of the Laplacian pencil) from the solution of (3), which can be written as, • n The Laplacian pencil is a regular pencil, cf. Def. 9 T (in fact, the degree matrix D is always nonsingular). x(t)= exp(−λi(Lp(s)) t)(ui (s) x0) ui(s). (4) • By Prop. 1, if A is nonsingular, all eigenvalues of the i=1 Laplacian pencil are finite and the same as the eigen- −1 −1 values of DA or A D. On the other hand, if A is where we have used the spectral decomposition of T singular, the Laplacian pencil admits an infinite eigen- Lp(s)=U(s) Λ(s) U (s), being U(s)=[u1(s), u2(s), value with algebraic multiplicity n − deg[det(Lp(s))]. ..., s • un( )] the matrix consisting of normalized and Since D is positive definite and A is symmetric, mutually orthogonal eigenvectors of Lp(s)andΛ(s)= by Prop. 2 the negated Laplacian pencil has all diag(λ1(Lp(s)),...,λn(Lp(s))). real eigenvalues. On the other hand, if G is a bipartite graph and s = −1, The next theorem shows an interesting connection be- we have the following result [43, Th. 5]. tween the stability properties of system (3) and the spectral properties of the Laplacian pencil for regular graphs. Proposition 3 Consider the system x˙ = −Lp(−1) x. If the graph G is connected and bipartite with bipartition Theorem 1 Let the graph G be κ-regular, then: {V1,V2}, we have that,

1 (1) The smallest and second smallest eigenvalue in mod- lim xi(t)=− lim xj (t)= xi(0)− xj (0) , t→∞ t→∞ n ulus of the negated Laplacian pencil are the values i ∈ V j ∈ V of s for which system (3) is marginally stable. Sys- i ∈ V1 j ∈ V2 1 2 tem (3) is asymptotically stable for all s between xi t − xj t these two eigenvalues. and the quantity i ∈ V1 ( ) j ∈ V2 ( ) is time (2) If the smallest (or analogously, second smallest) invariant. eigenvalue in modulus of the negated Laplacian pencil is simple (i.e. its algebraic multiplicity is one), Remark 2 This behavior is referred to as bipartite con- and the associated unit-norm eigenvector is z = sensus in [1] and cluster anticonsensus in [43]: in fact the z/z, we have that, states of the nodes in V1 and V2 asymptotically assume the same absolute value but have opposite signs. T lim x(t)=(z z ) x0. t→∞ In order to illustrate Th. 1 and Prop. 3, in the next section we study the stability properties of system (3) In the proof of Theorem 1 below, we will make use of the for some special families of graphs. Motzkin-Taussky theorem [21, Ch. II, §2.5]. Note that although not strictly necessary in view of Lemma 2, we 3.2 Illustrative examples believe that this result is insightful and it may serve as a basis, in future research, for more general statements. In this section we completely characterize the stability properties of system (3) for the path graph Pn,cycle Lemma 3 (Motzkin-Taussky) If Z(s)=V + s W n×n graph Cn, Petersen graph J(5, 2, 0), and complete graph with V, W ∈ C , is diagonalizable for every complex K s s n (the reader is referred to [12] for a precise deﬁnition of number , then the eigenvalues of Z( ) are of the form these graphs). Note that Cn, J(5, 2, 0) and Kn are regular λi sλi i ∈{ ,...,n} (V)+ (W), 1 . graphs,andthatPn and Cn (with n even) are bipartite graphs. In order to prove our ﬁrst result (Prop. 4), we Proof of Theorem 1: Since D and A are symmetric ma- need the following lemma [15, Th. 8.5.1]. trices and commute being G κ-regular (recall Lemma 1), the negated Laplacian pencil is diagonalizable for every Lemma 4 (Sturm sequence property) Consider real s (cf. [2, Fact 8.17.1] and [18, Sect. 4.5]), and then the following n × n symmetric tridiagonal matrix, by Lemma 3, we have that λi(−Lp(s)), i ∈{1,...,n}, are linear functions of parameter s.Moreover,since ⎡ ⎤ a1 b1 ⎢ b a b ⎥ λ − s −κ< ,i∈{ ,...,n}, ⎢ 1 2 2 ⎥ i( Lp( )) s =0 = 0 1 ⎢ . ⎥ ⎢ b a .. ⎥ , T = ⎢ 2 3 ⎥ ⎣ .. .. ⎦ and since the eigenvalues of the negated Laplacian . . bn−1 pencil are all real by the third item of Property 2, bn−1 an λi(−Lp(s)) < 0 ∀ i and system (3) is asymptotically stable only for those values of s which are between the (r) smallest and second smallest eigenvalue in modulus of where bj =0 , ∀ j ∈{1,...,n − 1}.LetT denote the negated Laplacian pencil. On the other hand, sys- the leading r × r principal submatrix of T.Then, tem (3) is marginally stable exactly for s equal to these the number of negative eigenvalues of T is equal to eigenvalues. The second item of the statement follows the number of sign changes in the Sturm sequence

4 1, det(T(1)), det(T(2)), ...,det(T(n)). The result is still to n in the natural order, moving counterclockwise), we valid if zero determinants are encountered along the have that: way, as long as we deﬁne a “sign change” to mean a transition from + or 0 to −,orfrom− or 0 to +,but • If n is even: not from + or − to 0. · For |s| < 1, system (3) is asymptotically stable. · For |s| > 1, system (3) is unstable. Proposition 4 (Path graph Pn) For the path graph · For s = −1, system (3) is marginally stable. In this Pn with n ≥ 2 nodes (we number the nodes from 1 to n case, the states associated to n/2 nodes asymptot- 1 T in the natural order from left to right), we have that: ically converge to n x0 k and the states associated n/ • |s| < to the other 2 nodes asymptotically converge to For 1, system (3) is asymptotically stable. − 1 T k • For |s| > 1, system (3) is unstable. n x0 . • For s = −1, system (3) is marginally stable. In this • If n is odd, let case, it is possible to identify two groups of n/2 nodes (if n is even), or one group of n/2 nodes and one μ 1 . of n/2 +1nodes (if n is odd). The states associated n +1 to the nodes in one group asymptotically converge to cos π 1 T n n x0 k and the states associated to the nodes in the − 1 T k other group converge to n x0 . Then: · s ∈ μ, − s For ( 1), system (3) is asymptotically stable. Proof: In this case, Lp( ) is a symmetric tridiagonal · For s<μor s>1, system (3) is unstable. matrix, ⎡ ⎤ · For s = μ, system (3) is marginally stable, and the −1 s components of the state vector x asymptotically con- ⎢ ⎥ verge, in general, to n diﬀerent values. ⎢ s −2 s ⎥ ⎢ ⎥ − s ⎢ .. ⎥ . Lp( )=⎢ . ⎥ Proof: In this case −Lp(s) is a circulant matrix [6], ⎣ s −2 s ⎦ −Lp(s)=circ[−2,s,0, ..., 0,s]. It is well-known that s −1 circulant matrices are diagonalizable by the Fourier matrix and hence their eigenvalues can be computed in closed form. In fact, the eigenvalues of a general n×n circulant − p s The Sturm sequence of L ( ) is given by, matrix C =circ[c1,c2,..., cn], are given by:

2 i−1 for n =2, 1, −1, 1 − s λi(C)=ρC(ω ),i∈{1,..., n}, (6) for n =3, 1, −1, 2 − s2, 2s2 − 2 √ 2πj/n 2 2 4 2 where ω e , j= −1, and the polynomial ρC(ξ)= for n =4, 1, −1, 2 − s , 3s − 4,s − 5s +4 n−1 2 cnξ + ...+ c3ξ + c2ξ + c1 is called the circulant’s . . . . representer [6, Th. 3.2.2]. By applying this formula to matrix −Lp(s), we have that Therefore by Lemma 4, if |s| < 1, s =0 , all the eigenvalues of −Lp(s) are strictly negative and system (3) is 2 π (i − 1) λi(−Lp(s)) = 2 cos s − 2,i∈{1,..., n}. asymptotically stable. On the other hand, the Sturm se- n quence of Lp(s), is given by, (7) From a systematic study of (7) in terms of variable s,the for n =2, 1, 1, 1 − s2 ﬁrst two items of each bullet in the statement are imme- 2 2 diately proved. As far as the marginally-stable behavior for n =3, 1, 1, 2 − s , 2 − 2s is concerned, for n even (5) holds true, and for n odd: n , , , − s2, − s2,s4 − s2 for =4 1 1 2 4 3 5 +4 . . 2 2 π n/2·0 . . lim exp(−Lp(μ) t) x0 = circ cos , . . t →∞ n n |s| > s π n/ · π n/ · from which we deduce that for 1, Lp( ) has a neg- 2 2 1 , 2 2 2 , ative eigenvalue, and hence system (3) is unstable. We cos n cos n also note that −Lp(0) = −D, and hence the system is s s − π n/ · n − asymptotically stable for =0. Finally, for = 1,we ..., 2 2 ( 1) . have that: cos x0 n 1 T lim exp(−Lp(−1) t) x0 = kk x0. (5) J , , t →∞ n Proposition 6 (Petersen graph (5 2 0)) For the Petersen graph, we have that: Proposition 5 (Cycle graph Cn) For the cycle graph • For s ∈ (−3/2, 1), system (3) is asymptotically stable. Cn with n>2 nodes (we number the nodes from 1 • For s>1 or s<−3/2, system (3) is unstable.

5 • For s = −3/2, system (3) is marginally stable, and Lemma 5 (Controllability rank condition [17]) the components of the state vector x asymptotically Consider the nonlinear system, converge, in general, to ten different values. x˙ = f(x)+g(x) u, (8) Proof: Since the eigenvalues of the adjacency matrix of ∈ n u ∈ the Petersen graph are 3, 1 and −2 with multiplicity 1, 5 where x IR , IR are the state vector and scalar con- and 4, respectively [3, Sect. 1.4.5], then, trol input, respectively, and f(x) and g(x) are analytical vector fields. System (8) is weakly controllable at x0 if and only if rank (C )=n at x0,wherethen × n control- λ1(−Lp(s)) = 3 s − 3, lability matrix C is defined as, λ2(−Lp(s)) = ... = λ6(−Lp(s)) = s − 3, 2 n−1 C g, adf g, adf g, ...,adf g , (9) λ7(−Lp(s)) = ... = λ10(−Lp(s)) = −2 s − 3, where adf g =[f, g] (the Lie bracket of f and g)and , −1 ≥ from which the first two items of the statements fol- adf g =[f adf g], 2. low. For the third item, by numbering the nodes from 1 to 5 and from 6 to 10 in the natural order moving Note now that system (3) can be rewritten as: counterclockwise along the external and internal “ring” − s, of the graph, respectively (cf. [12, Fig. 1.8]), we have x˙ = Dx+ Ax (10) 3 1 1 that limt →∞exp(−Lp(− ) t) x0 = (N + J10×10) x0, 2 3 5 which is a bilinear system and has the form (8) with where, f(x)=−Dx, g(x)=Axand u = s. The controllability circ[1, −1, 0, 0, −1] −I5 matrix in (9) reduces in this case to: N = , 2 2 −I5 circ[1, 0, −1, −1, 0] C = Ax, Γx, (DΓ − ΓD)x, (D Γ + ΓD − 2DΓD)x, (D3Γ − ΓD3 − 3D2ΓD +3DΓD2)x, (D4Γ + ΓD4 and circ[ · ] denotes a circulant matrix (see the proof of Prop. 5). −4D3ΓD − 4DΓD3 +6D2ΓD2)x,... , (11) Proposition 7 (Complete graph Kn) For the com- where Γ DA − AD is the commutator of matri- plete graph Kn with n>2 nodes, we have that: ces D and A. From (11), we immediately deduce that • For s ∈ (−(n − 1), 1), system (3) is asymptotically if Γ = 0, system (10) is weakly controllable nowhere. stable. In particular, by recalling Lemma 1, we obtain the fol- • For s<−(n − 1) or s>1, system (3) is unstable. lowing result. • s − n − For = ( 1), system (3) is marginally stable and Proposition 8 If the graph G is κ-regular, system (10) the components of the state vector x asymptotically is weakly controllable nowhere. converge, in general, to n different values. The next proposition shows that path graphs Pn and Proof: Since the eigenvalues of the adjacency matrix of complete bipartite graphs Km,n are also critical for con- n − − the complete graph are 1 and 1 with multiplicity 1 trollability. and n − 1, respectively [3, Sect. 1.4.1], then, Proposition 9 If G∈{Pn,Km,n}, system (10) is λ1(−Lp(s)) = (n − 1) s − (n − 1), weakly controllable nowhere.

λ2(−Lp(s)) = ... = λn(−Lp(s)) = − s − (n − 1), Proof: If G = Pn,for ≥ 1 we have that:

3 5 2−1 from which we obtain the ﬁrst two items of the state- adf g =adf g =adf g = ... =adf g, ment. For the third item, it is suﬃcient to observe that 2 4 6 2 1 ... . lim exp(−Lp(−(n − 1)) t) x0 =(In − Jn×n) x0. adf g =adf g =adf g = =adf g t →∞ n

Instead, with G = Km,n, we have that Γ = 0 if m = n, 4 Controllability analysis and Γ = 0 but

In this section we change our viewpoint and by inter- 1 3 1 2−1 adf g = adf g = ...= ad g, preting variable s as a control input to system (3), we (n − m)2 (n − m)2(−1) f study its controllability properties. In other words, by 2 1 4 1 2 adopting a graph-theoretic formalism, we are here inter- adf g = adf g = ...= adf g, ested in determining to which points we can steer trajec- (n − m)2 (n − m)2(−1) tories from vector x0 at the initial time. The following Lemma (see [17, Ths. 2.2, 2.6]) is useful for our forth- if m = n with ≥ 1. Therefore, in both cases the control- coming analysis. lability matrix C cannot be full rank.

6 Prop. 8 and Prop. 9 indicate that all the families of 2 graphs considered in Sect. 3.2, render system (10) weakly controllable nowhere (in other words, there do not exist initial vectors x0 from which we can reach any desired point in the state space of system (10)). Actually, as 4 also observed in [33], graphs that exhibit very specialized 1 5 structures are the most problematic from a controllability viewpoint: vice versa, there are better chances to steer the state vector of system (10) towards any desired end point with networks not presenting strong symmetries (cf. Example 1 below). 3 Fig. 1. Graph considered in Example 1. Remark 3 (On network controllability literature) Note that the controllability analysis conducted in this where q x2 + x3. By computing the determi- section differs significantly from that recently per- nant of C , we can see that system (10) is weakly formed in [33] for the controlled agreement dynamics controllable everywhere except in the submanifold (see also [28, 31, 44] and the references therein). In fact, 5 2 2 M = x ∈ IR x1x4x5(x − x )=0 . differently from [33], system (10) is nonlinear and we 2 3 do not inject control signals at leader nodes, the only input being the scalar parameter s and all nodes being 5 Extension to directed graphs identical in our formulation. In this section we assume that the communication graph If system (10) is weakly controllable at the point x0, is directed and contains a rooted out-branching, and par- then the control input s(t)thatsteersthestatefromx0 alleling Sect. 3 we will study the stability properties of at the initial time t0 to a desired end point xd at the the system, t − D s t , final time tf , can be computed, for instance, by solving x˙ ( )= Lp( ( )) x( ) (14) the following optimal control problem: in terms of the real parameter s,where“D(s)” indicates that the parametric Laplacian Lp is now relative to a 1 T min (x(tf ) − xd) Ξ (x(tf ) − xd)+ directed topology. Note that similarly to Sect. 3, we have s(t) 2 that Lp(D(1)) = L(D)andLp(D(−1)) = Q(D). It is tf D 1 T t t rs2 t dt well known [24], that if the digraph contains a rooted (x ( ) Θx( )+ ( )) out-branching, the state trajectory of system, 2 t0 t − D t , s.t. x˙ (t)=−Dx(t)+Ax(t) s(t), x(t0)=x0, x˙ ( )= L( ) x( ) (15)

T T T satisﬁes limt→∞ x(t)=(u1v ) x0 where u1 and v1, where Ξ = Ξ 0, Θ = Θ 0, r>0, are given 1 are, respectively, the right and left eigenvectors associ- weights. From the Euler-Lagrange necessary conditions ated with zero eigenvalue of L(D), normalized such that for optimality, we have that the optimal control is the T bilinear state-costate feedback, u1 v1 = 1. Moreover, we have that (15) achieves average consensus for every initial state x0 if and only if D is weakly connected and balanced (i.e., for every node, s t −1 T t η t ,t∈ t ,t , ( )= r x ( ) A ( ) [ 0 f ] (12) the in-degree and out-degree are equal). Because of the space constraints, we will restrict our sta- and the state-costate equations, after eliminating s(t) bility analysis of system (14) to two families of balanced and dropping the time index, are: digraphs which exhibit intriguing grouping behaviors.

Proposition 10 (Directed cycle Dn) For the directed 1 T x˙ =−Dx− Axx Aη, x(t0)=x0, cycle graph Dn with n>2 nodes (we number the nodes r from 1 to n in the natural order, moving counterclock- η − η 1 η T η, η t t − . wise), we have that: ˙ = Θx+ D + r A x A ( f )=Ξ(x( f ) xd) (13) • For s =1and for n>2, average consensus is achieved. • If n is even: Example 1 Consider the graph G in Fig. 1. The con- · For |s| < 1, system (14) is asymptotically stable. trollability matrix C of system (10) is, in this case: · For |s| > 1, system (14) is unstable. · For s = −1, system (14) is marginally stable. In this ⎡ ⎤ n/ q −qq−qq case, the states associated to 2 nodes asymptoti- 1 T k ⎢ ⎥ cally converge to n x0 and the states associated to ⎢ x1 + x3 + x4 x1 x1 x1 x1 ⎥ 1 T ⎢ ⎥ the other n/2 nodes converge to − n x0 k. C = ⎢ x1 + x2 + x4 x1 x1 x1 x1 ⎥ . ⎣ ⎦ • If n is odd: q + x5 2x5 4x5 8x5 16x5 · System (14) is asymptotically stable for s ∈ x4 −2x4 4x4 −8x4 16x4 (ϑ(n), 1),where,

7 1 H ϑ(n)= . being ( ) the Heaviside step function: n(n − 2) + 1 cos π n 0, if <0, H( ) · For s<ϑ(n) or s>1, system (14) is unstable. 1, if ≥ 0. · For s = ϑ(n), system (14) is marginally stable. At steady-state, we have that the i-th component of • For s<σ(n) or s>1, system (14) is unstable. the state vector of system (14) obeys, • For s =1, average consensus is achieved. • For s = σ(n), system (14) is marginally stable: xi(t)=A sin 2πf(n) t + φi(n)+φ◦ ,i∈{1,...,n}, · For n ∈{3, 6, 9}, we can identify three groups of nodes ({1}, {2}, {3} for n =3, {1, 4}, {2, 5}, {3, 6} n { , , } { , , } { , , } n where A and φ◦ are positive constants, the frequency for =6,and 1 4 7 , 2 5 8 , 3 6 9 for =9), whose associated states asymptotically converge to 1 n(n − 2) + 1 the same constant value. f(n)= tan π , · For n =5, the components of the state vector of sys- 2π n tem (14) experience steady-state quasi-periodic oscillations. and the phase · For n =12, we can identify three groups of nodes, {1, 4, 7, 10}, {2, 5, 8, 11}, {3, 6, 9, 12}, whose associ- π i − φ n 2 ( 1) ,i∈{ ,...,n}. ated states experience steady-state periodic oscilla- i( )= n n − 1 tions with frequency 1/π. n ( 2) + 1 π · n/∈{ , , , , } tan n For 3 5 6 9 12 , the components of the state vector of system (14) experience steady-state periodic oscillations with frequency: Proof: Since the degree matrix is equal to the identity 1 π(δ(n) − 1)(2n − 3) matrix for the directed cycle, the parametric Lapla- fo(n)= tan . cian Lp(Dn(s)) coincides with the deformed Lapla- π n cian Δ(Dn(s)) (cf. [25]). Therefore, the proof of this proposition, which relies, again, on formula (6) being Proof: Similarly to Prop. 5 and Prop. 10, −Lp(D(s)) − D s Lp( n( )) a circulant matrix, is identical to that of is a circulant matrix in this case, −Lp(D(s)) = Prop. 12 in [25]. circ[−2, 0, ..., 0,s,s], and its eigenvalues can be computed in closed-form using formula (6), Note that the directed cycle is a bipartite digraph for n D ≡ D even (cf. Prop. 5). The directed cycle n n(1) has λ − D s s 2π(i−1)(n−1) a connection set {1} (cf. [12, p. 9]), i.e. a directed edge i( Lp( ( ))) = exp n j i i n i ∈{ ,...,n} connects node to node +1mod ,for 1 . 2π(i−1)(n−2) Let us now generalize this notion and consider the di- +exp n j − 2,i∈{1,...,n}, { , } rected graph with connection set 1 2 : this means that (16) directed edges connect node i to node i +1andi +2, √ 1 where j= −1. The ﬁrst two items of the statement mod n. We will refer to this digraph as Dn(1, 2). easily follow from a systematic study of (16) in terms of parameter s.Inparticular,fors =1, Lp(D(s)) reduces Proposition 11 (Digraph Dn(1, 2)) For the digraph to the standard Laplacian and being Dn(1, 2) strongly Dn(1, 2) with n>2 nodes (we number the nodes from 1 n connected and balanced, average consensus is achieved. to in the natural order, moving counterclockwise), we s σ n have that: On the other hand, if = ( ): • For n ∈{3, 6, 9}, we have that: • System (14) is asymptotically stable for s ∈ (σ(n), 1), where, lim exp − Lp(D(σ(n))) t x0 = t →∞ 2 1 σ(n)= , circ 2, −1, −1,...,2, −1, −1 x0. 2π(δ(n)−1)(n−1) 2π(δ(n)−1)(n−2) n cos n +cos n • For n =5, the eigenvalues of −Lp(D(σ(n))) and are {−10, ±j2tan(7π/5), ±j2tan(21π/5)}.The ∞ two pairs of purely-imaginary eigenvalues induce 1 δ(n)=2H(n−3)+ [H(n−5−7 )+H(n−9−7 )], oscillations with frequencies f1 = π tan(7π/5), 1 =0 f2 = π tan(21π/5), respectively (cf. [36, p. 134]). (3−tan2 θ)tanθ By using the identities tan(3θ)= 2 and 1 Note that along the same line we can consider connection √ 1−3tan θ √ sets of the form {1, 2, 3}, {1, 2, 3, 4},...,{1, 2,...n− 1}: tan(7π/5) = 5+2 5,weobtainf1/f2 =2+ 5 however, this extension goes beyond the scope of this paper which is an irrational number. This means (cf. [36, and it is left as a subject of future research. Sect. 8.6]), that the components of the state vector

8 / π/ − . s 1 1 1 cos (16 5) 1 2361 (recall Prop. 10). For = ϑ(5), stable steady-state oscillations arise. • For the digraph in Fig. 2(b), system (14) is asymptotically stable for s ∈ (−1.4818, 1).Fors = −1.4818, 2 5 2 5 which has been computed on a trial-and-error basis, we experience stable steady-state oscillations. • For the digraph in Fig. 2(c), system (14) is asymptotically stable for s ∈ (−1.2253, 1).Fors = −1.2253 (the second smallest real eigenvalue in modulus of the 3 4 3 4 negated Laplacian pencil), the system is marginally (a) (b) stable and the components of the state vector asymp- 1 totically converge, in general, to ﬁve diﬀerent values.

6 Numerical simulations 2 5 In order to illustrate the theory presented in Sect. 3 and Sect. 5, extensive numerical simulations have been car- ried out. Consider a team of n vehicles modeled as single integrators, p˙ i(t)= νi(t), i ∈{1,...,n},wherepi(t)= T 2 2 [pix(t),piy(t)] ∈ IR and νi(t) ∈ IR denote respec- 3 4 tively the position and the input of agent i at time t.Let (c) the control input of vehicle i be of the form, Fig. 2. Example 2: variations on the directed cycle D5. νi(t)= s pj (t) − pi(t) , (17) of system (14) experience steady-state quasi-periodic j ∈N(i) oscillations. • For n =12, −Lp(D(σ(n))) has a pair of purely- where N (i) denotes the set of nodes adjacent to node i in imaginary eigenvalues, ±j2, which induce periodic os- the communication graph. Then, the collective dynam- cillations of period π, it has a pair of zero eigenvalues ics of the group of agents adopting control (17), can be and all the other eigenvalues have negative real parts. writtenincompactformasp˙ (t)=−(Lp(s) ⊗ I2) p(t) • n/∈{, , , } − p D σ n T T T 2n For 3 6 9 12 , L ( ( ( ))) has a pair of where p =[p1 ,..., pn ] ∈ IR . purely-imaginary eigenvalues, In the sequel, we will implicitly assume that parameter s is transmitted in real-time to all the agents by a (human) π δ n − n − ± ( ( ) 1)(2 3) , supervisor. In the next two sections, we will examine j2tan n separately the case of undirected and directed graphs. which induce periodic oscillations of period 1 fo(n)= 6.1 Undirected topology π(δ(n)−1)(2n−3) π tan n . All the other eigenvalues, n instead, have negative real parts. Fig. 3(a) shows the trajectory of = 6 vehicles imple- menting the control law (17), when the communication graph is the cycle C6 (for easiness of reference, the ini- The behavior of system (14) with the communication D , D , s σ n tial positions of the agents are marked with circles). In digraphs 12(1 2) and 5(1 2) for = ( ), is shown order to make the agents rendezvous at the same point in Sect. 6.2. while avoiding the two gray obstacles, we set s(t)=−1 for t ∈ [0, 40) s, and s(t)=1fort ∈ [40, 80] s. The time Remark 4 Note that differently from Sect. 3, since x y D s evolution of the -, -coordinates of the vehicles is re- Lp( ( )) is nonsymmetric, it can also admit complex- ported in Fig. 3(b). As it is evident in the figures, the conjugate eigenvalues and the components of the state vehicles first cluster in two groups and then converge to vector of system (14) may experience stable steady-state the same point among the obstacles (recall Prop. 5). oscillations, as we have seen in Prop. 10 and Prop. 11 above. It is worth pointing out here that this is not possible, instead, for protocol (15) which does not admit 6.2 Directed topology periodic solutions. Fig. 5(a) shows the behavior of system, We conclude this section with an illustrative example which shows the sensitivity of behavior of system (14) p˙ (t)=−(Lp(D(s)) ⊗ I2) p(t), (18) to the topology of the communication digraph. with the communication digraph D12(1, 2) (see Fig. 4(a)). Example 2 The initial positions of the 12 agents are marked with circles in Fig. 5(a), and in the simulation we set s(t)= • For the digraph in Fig. 2(a), i.e. D5, system (14) is σ(12) = −2fort ∈ [0, 30) s and s(t)=0fort ∈ asymptotically stable for s ∈ (ϑ(5), 1),whereϑ(5) = [30, 60] s. The time evolution of the x-, y-coordinates

9 p 2.5 A3 1 s switches 1x p2x 2 0.5 p3x p 0 4x [m] 1.5 p5x A1 * −0.5 p6x 1 A5 −1 0.5 0 10 20 30 40 50 60 70 80

y [m] time [s] 0 s switches A6 2 p1y −0.5 p2y A2 1 p3y −1 p4y * [m] 0 p5y −1.5 −1 A4 p6y −2 −2

−1 −0.5 0 0.5 1 0 10 20 30 40 50 60 70 80 x [m] time [s] (a) (b)

Fig. 3. Simulation results: undirected graph C6. (a) The 6 agents rendezvous at the same point while avoiding the two obstacles (gray rectangles): this is made possible by switching s from −1 to 1 (the initial positions are marked with circles and the ﬁnal position with a diamond); (b) Time evolution of the x-, y-coordinates of the agents (top and bottom, respectively). of the 12 vehicles is reported in Fig. 5(b). From an in- where the objective of the latter task is to ensure that spection of Fig. 5(a) and Fig. 5(b), we see that in the at least one agent eventually moves to within a given ﬁrst half of the simulation the agents split into three distance from any point in the target environment. groups {1, 4, 7, 10}, {2, 5, 8, 11}, {3, 6, 9, 12},andthe vehicles in the same group move, equally spaced, along a common closed trajectory with frequency 1/π.Inthe 7 Conclusions and future work second half of the simulation, when s = 0, the agents converge instead to the origin (recall Prop. 11). Fig. 6 shows In this paper we have presented a generalization of the the behavior of system (18) when the communication di- classical continuous-time consensus protocol, and we graph is D5(1, 2) (see Fig. 4(b)) and s = σ(5) = −4at have analyzed its graph-dependent stability properties all times. In particular, Figs. 6(a)-(c) report the quasi- in terms of a real parameter s. We leveraged the spectral periodic trajectories of the 5 agents at times t =3s, theory of matrix pencils to establish explicit stability t =15s,andt = 40 s, respectively (cf. Prop. 11). Again, conditions for regular graphs: this motivated us to in- the initial positions of the agents are marked with cir- troduce the notion of Laplacian pencil. By interpreting cles. Note that if we imagine the agents tracing out tra- parameter s as a control input, the graph-theoretic jectories on a torus, then each trajectory is dense on it, controllability properties of the proposed consensus pro- i.e. each trajectory comes arbitrarily close to any given tocol have been analyzed, showing the important role point on the torus. played by the symmetry of the communication network. Finally, it is worth pointing out that the two control in- The theory has been illustrated with the help of worked stances described in this section can be regarded as prim- examples and numerical simulations. itives for more sophisticated tasks, such as, mobile tar- There are a number of potential venues for further de- gets’ enclosing [16] and path coverage [32], respectively, velopment. First of all, we aim at studying the properties of the proposed protocol when the communication network is not static but changes over time, as conse- 1 1 quence, for example, of displacement of mobile nodes. 2 12 Second, we are going to investigate whether the parametric Laplacian and the deformed Laplacian studied 3 11 in [25], represent two instances of a more general family 2 5 of N-order Laplacian matrix polynomials in the real 4 10 variable s. This line of investigation is supported by the existing spectral theory of nonlinear eigenvalue problems [7, 13]. Third, we would like to explain the pecu- 5 9 liar grouping behaviors observed with directed graphs within a more general conceptual framework: in par- 6 8 ticular, we aim at extending the path-coverage mecha- 7 3 4 (a) (b) nism induced by quasi-periodic oscillations described in Sect. 6.2, to generic directed topologies. Finally, work Fig. 4. (a) Digraph D12(1, 2) considered in Fig. 5; (b) Digraph is underway (cf. [26]) to generalize our results to agents D5(1, 2) considered in Fig. 6. with more involved dynamical models.

10 s switches 2 A3 A6 2 p2x,p5x,p8x,p11x 1.5 1 A5 [m] 0 1 p3x,p6x,p9x,p12x −1 p1x,p4x,p7x,p10x

0.5 A8 A9 0 10 20 30 40 50 60 A12 time [s] y [m] 0 A10 2 −0.5 A1 1 p2y,p5y,p8y,p11y A11

−1 A7 [m] 0 A2 p3y,p6y,p9y,p12y p1y,p4y,p7y,p10y −1.5 A4 −1

−1.5 −1 −0.5 0 0.5 1 1.5 2 0 10 20 30 40 50 60 x [m] time [s] (a) (b) Fig. 5. Simulation results: directed graph D12(1, 2). (a) Trajectory of the 12 agents (the initial positions are marked with circles and the ﬁnal position with a diamond); (b) Time evolution of the x-, y-coordinates of the agents (top and bottom, respectively: note that the same color convention as in (a) has been adopted for the agents).

2 2 2 A2 1.5 1.5 1.5 1 1 1 A1 0.5 0.5 0.5 0 0 0 y [m] y [m] y [m] −0.5 −0.5 −0.5 A5 −1 −1 −1 A4 A3 −1.5 −1.5 −1.5 −2 −2 −2

−2 −1 0 1 2 −2 −1 0 1 2 −2 −1 0 1 2 x [m] x [m] x [m] (a) (b) (c) Fig. 6. Simulation results: directed graph D5(1, 2). Trajectory of the 5 agents for s = σ(5) = −4: (a) t = 3 s (the initial positions are marked with circles); (b) t =15s;(c)t =40s.

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