On consensus in multi-agent systems with linear high-order agents

Transcription

1 Proceedings of the 7th World Congress he International Federation of Automatic Control On consensus in multi-agent systems with linear high-order agents Peter Wieland, Jung-Su Kim, Holger Scheu, and Frank Allgöwer Institute for Systems heory and Automatic Control, University of Stuttgart, Germany Abstract: Consensus of a group of agents in a multi-agent system is considered All agents are modeled by identical linear nth order dynamical systems and the interconnection topology between the agents is modeled as a directed weighted graph We provide answers to the questions whether the group converges to consensus and what consensus value it eventually reaches Furthermore, we give a necessary and sufficient condition for convergence to consensus in the double integrator case and propose an LMI-based design for group consensus in the general case An example is used to illustrate the results Keywords: Multi-agent systems; Consensus; Directed graph; LMI INRODUCION Recently, the consensus problem among multi-agent systems MAS has received a lot of attention in the literature he interest in this problem is mainly motivated by the huge variety of applications in various areas, eg unmanned aerialvehicles, mobile robots, satellites, formation control, and sensor networks, to name only a few For a nice overview of recent results on the topic, see Ren et al 27; Olfati-Saber et al 27; anner et al 27 and the references therein Numerous results have been proposed on consensus for MASs, most of which consider agents that are modeled by single or double integrators However, in some applications, agents of higher dynamical order are required if consensus of more than two variables is aimed at While the consensus problem for agents modeled as integrator chains of length greater than two was reported quite recently Ren et al, 26, only little attention has been paid to the inherent instability of integrator chains of length greater than one o overcome this problem and for full generality, it is interesting to investigate the consensus problem considering agents modeled by general linear time-invariant LI single-input systems It is well known that the interconnection topology among agents plays a pivotal role in reaching group consensus in a MAS Graphs are commonly employed in order to model the interconnection topology in a MAS cf de Gennaro and Jadbabaie 26; Ren et al 26; Ren and Atkins 25; Olfati-Saber 26 In the most general setup, directed and weighted graphs are used A graph can be completelycharacterizedbyits LaplacianmatrixL, hence, it is important to fully understand the properties of this matrix, most notably its eigenstructure which is crucial for the consensus problem In the past, research was mainly focused on the analysis of MAS consensus: Given the interconnection topology and some consensus algorithm, the question is answered, whether the states of all agents converge to some common value, the consensus For agents modeled as single integrator with arbitrary interconnection topology or double integrator with an interconnection topology admitting a real Laplacian spectrum, connectedness of the MAS is necessary and sufficient for convergence to consensus In the general case however, it was reported recently Ren and Atkins, 25 that connectedness of the MAS is only a necessary condition for convergence to consensus Whether or not a sufficiently connected MAS actually converges to consensus depends on the gains in the consensus algorithm herefore, it is required to develop a systematic method to determine the gains or to find tight bounds on the gains in the consensus algorithm Consideringthe status of recentconsensus research, firstly, this paper presents conditions for consensus of MASs with agents modeled as general LI systems hese conditions explicitly depend on the gains in the consensus algorithm, the parameters of the agent model, and the interconnection topology o obtain these conditions, we derive the characteristic equation of the closed-loop consisting of all agents and a given consensus algorithm he characteristic equation enables us to derive conditions on the gains and to unveil the dynamic evolution of the consensus state Secondly, an implication of the left-eigenvector of the Laplacian matrix corresponding to the zero eigenvalue is elucidated: Non-zero elements in the left-eigenvector correspond to agents that can be chosen as the root of a spanning tree in the graph describing the interconnection topology of the MAS his implies that if the ith component in the left-eigenvector is non-zero, the ith agent has some influence on the consensus value his influence is quantified by the components of the left-eigenvector hirdly, a systematic way to choose the gains in the consensus algorithm is proposed for agents modeled as double integrators and general LI systems In the case of double integrator agents, we present exact convergence bounds that are tighter than previously proposed bounds For agents modeled as general LI systems, an LMI based design method for the consensus algorithm is proposed Unlike in previous results, in our design methods a desired convergence rate can be specified he remainder of this paper is organized as follows: We start by presenting some basic facts from graph theory in Section 2 Section 3 exposes some results on analysis of consensus algorithms for agents modeled by LI systems followed by the presentation of design methods for the double integrator and the general LI system case in /8/$2 28 IFAC / KR-755

2 Section 4 he results are illustrated on an example in Section 5 before Section 6 concludes the paper 2 Notation 2 PRELIMINARIES hroughout this paper, we write and for the all ones and all zero vector of appropriate dimension he ith unit vector in R n is denoted e i he imaginary unit is written as j := 22 Basic graph theory o make the paper self-contained, we startby an overview of concepts from graph theory We consider weighted and directed graphs given as G = {V, E, W } where V = {,, N} is a nonempty finite vertex set with each vertex i representing one agent and the number of vertices denoted as V = N, E V V is a set of ordered pairs of vertices called edges satisfying i, j E if and only if there is a link from vertex i called the parent to vertex j called the child, ie information flows from agent i to agent j, W : V V [, is a mapping that assigns positive weights to the edges of G and satisfies Wi, j if and only if i, j E he graph G is undirected if Wi, j = Wj, i for all i, j =,,N, it is balanced if N j= Wi, j = N j= Wj, i for i =,,N We assume that there are no selfloops, ie i, i E, i =,,N A directed path p in G is a finite sequence of vertices p = {i, i 2,, i l } satisfying i k, i k+ E for k =,,l Vertex i is an ancestor of vertex j and vertex j is a descendant of vertex i if there is a directed path from i to j We define the sets P i V as the set of parents of i, C i V as the set of children of i, A i V as the set of ancestors of i, and D i V as the set of descendants of i A directed tree is a directed graph where every vertex except for one distinct vertex, called the root of the tree, has exactly one parent Define the restriction W Ẽ : V V [, of the mapping W : V V [, to a subset { Ẽ E as Wi, j if i, j Ẽ, W Ẽi, j := if i, j Ẽ he graph G is said to contain a spanning tree if there exists a subset Ẽ E such that the graph G = {V, Ẽ, W Ẽ } is a directed tree A graph G is strongly connected if any vertex in V is the root of a spanning tree of G he graph G can be completely characterized by its Laplacian matrix L = D A where D = [d ij ] is the diagonal matrix of the vertex in-degrees, ie N d ii = Wj, i, d ij = if i j, j= and A = [a ij ] is the adjacency matrix defined as a ii =, a ij = Wj, i if i j As all row-sums of L vanish, the Laplacian matrix has always at least one zero eigenvalue with corresponding right-eigenvector he properties of the Laplacian matrix L of an undirected graph are very well understood Godsil and Royle, 24; Fiedler, 973 In particular, the connectedness of a graph can be expressed in terms of the second smallest eigenvalue λ 2 L of the Laplacian, which equals the algebraic connectivity x Lx ag := min x = x x x of an undirected graph Adding edges to an undirected graph, ag cannot decrease In a general directed graph, however, ag cannot be related to graph connectivity without further knowledge of the topology Instead, it was shownrecently Ren et al, 24 that the eigenvalue with the second smallest real part can be used to characterize connectedness of a directed graph Unfortunately, the second smallest real part may decrease when new edges are added to the graph he following facts are used throughout the paper: Fact he zero eigenvalue of L is simple and all the other eigenvalues have positive real part if and only if G contains a spanning tree cf Ren et al 24 Fact 2 If G is balanced, the algebraic connectivity satisfies ag Reλ 2 L, where λ 2 L is the eigenvalue of L with the second smallest real part, and ag > if and only if G is connected cf Wu 25 In the remainder of this paper, we denote the eigenvalues of L as λ L,, λ N L with Reλ i L Reλ i+ L, i =,,N and λ L = Let p be a left-eigenvector of L corresponding to the eigenvalue λ L = and satisfying p = If G is balanced, then p = N If G contains a spanning tree, p is uniquely defined because the zero eigenvalue is simple 3 CONSENSUS ANALYSIS We consider a MAS of N linear agents with dynamics ξ n i +α ξ i + +α ξ i = u i, i =,,N, where ξ k i denotes the kth derivative of ξ i and u i is an input Every controllable single-input linear system can be written as 3 he consensus dynamics In this section we analyze consensus in the group of agents defined by he group of agents is said to reach consensus if the following requirement is satisfied: I ξ k i t ξ k j t as t for all i, j =,,N, i j and all k =,,n In many applications, an additional requirement is that II ξ k i t < for all t >, i =,,N, and k =,,n A consensus algorithm frequently found in literature is given as N u i = β k γ i Wj, iξ k j ξ k i 2 j= for i =,,N, where β k >, k =,,n and γ i >, i =,, N are design parameters to be determined, while the weights Wj, i, i, j =,,N are assumed to be given by the interconnection topology he values γ i, i =,,N can be interpreted as positive scalings of the weights of the edges incident to vertex i Algorithm 2 uses only state information of agents j satisfying j P i to 542

3 determine the input of agent i, ie algorithm 2 is local in nature Using the definitions ξ u γ ξ =, u =, Γ = diag, ξ N u N γ N algorithm 2 can be written in matrix form as u = β k ΓLξ k 3 Note that L := ΓL is a Laplacian matrix corresponding to the graph G = {V, E, W } with Wi, j := γ j Wi, j If G is strongly connected, γ i, i =,,N can be chosen such that G is balanced heorem 3 Given the MAS with algorithm 3, Requirement I is satisfied if and only if the polynomial N s n + α k + β k λ j ΓLs k 4 j=2 } {{ } =: p j s is Hurwitz In that case all agents converge to a consensus state, ie ξ k i t ζ k t, i =,,N, k =,,n for t and the consensus state ζt evolves according to ζ n + α ζ + + α ζ = 5 with initial condition given as the weighted average ζ k = p ξ k, k =,,n where p = Γ p p Γ o prove heorem 3, we need the following Lemma, the proof of which is omitted due to space limitations: Lemma 4 Given an N N matrix L and constants α k, β k, k =,,n, det s n I + s k α k I + β k L = N i= s n + s k α k + β k λ i L Proof he closed loop consisting of N agents described by and the consensus algorithm 3 reads ξ n = α k I + β k ΓLξ k Define ζt = p ξt, η i t = ξ i t ξ t, i = 2,,N, and η = η 2,, η N, ie ζη = p ξ + ξ = Sξ, I }{{}}{{} =: S =: S 2 with S = S + S 2 and p 2 p N S = = + I p 2 p N }{{}}{{} =: 2 =: Note that SΓL = S 2 ΓL 2 Consequently, we obtain the transformed dynamics ζ n = α k ζ k, 6 η n = α k I + β k I ΓL η I k 7 Requirement I is satisfied if and only if η k t, k =,, for t in which case ξ k i t ζ k t, i =,,N, k =,,n and ζt evolves according to 5 It remains to verify that the assertions given in the theorem are satisfied if and only if the η-dynamics 7 is stable, ie the characteristic polynomial of the η-dynamics is given by 4 o that end, let K k = α k I β k ΓL, k =,,N and define x = ξ, ξ,,ξ to write the closed loop dynamics in state-space form as I ẋ = x I I K K n 2 K }{{} =: M he characteristic equation of the dynamic matrix M can be computed using the Schur complement as detsi M = s N K k det si s k = det s n I s k K k = Using Lemma 4 and the fact that the first eigenvalue of ΓL is λ ΓL =, the characteristic polynomial can be rewritten as N s n + α k s k p j s 8 } {{ } =: p s j=2 where p s corresponds to the ζ-dynamics 6 and hence the remaining part, which equals to 4, corresponds to the η-dynamics 7 Corollary 5 Given the MAS with algorithm 3, Requirement II is satisfiedif andonlyif 5 is stable, ie if the dynamics of the individual agent given by with input u i is stable Proof By heorem 3, ξ k i t ζ k t, i =,,N, k =,,n, ie ξ k i t < iff ζ k t < Remark 6 Polynomial 4 has real coefficients even if L has eigenvalues with non-zero imaginary part: If λ i+ L and λ i L are complex conjugate eigenvalues of L, then p i+ s p i s has real coefficients Remark 7 By heorem 3, the consensus state ζt evolves independently of the interconnection topology given by L but the initial condition does depend on the interconnection topology 32 he consensus state It was shown in heorem 3 that the initial value of the consensus state depends on the left-eigenvector p of L 543

4 corresponding to the eigenvalue zero More precisely, agent i s initial state has some influence on the consensus state if and only if p e i he theorem given below enlightens this fact from an algebraic graph theory point of view o that end, we need the following definitions and lemma a direct corollary to the Perron-Frobenius heorem: A matrix A is said to be non-negative positive, denoted A A >, if all its elements are non-negative positive Equivalently, a vector x is said to be nonnegative positive, denoted x x >, if all its components are non-negative positive A matrix A is said to be stochastic if A = ; the spectral radius of a stochastic matrix A is ρa = he graph induced by an N N matrix A = [a ij ] is the graph GA on N vertices that contains an edge from vertex j to vertex i if and only if a ij A is irreducible if and only if GA is strongly connected Lemma 8 adapted from Horn and Johnson 985 If A is an irreducible stochastic matrix, then λ = = ρa is a simple eigenvalue, A = and q > : q A = q heorem 9 Given a graph G with Laplacian matrix L and a vector p satisfying p L = and p = Assume G contains a spanning tree hen p and p e i > if and only if the ith vertex of G can be chosen as the root of a spanning tree in G Proof Denote the set of roots of spanning trees in G as V = {i V {i} D i = V} V with N = V > and define V = V \ V with N = V here exists no directed path from any element of V to any element of V hus, without loss of generality, we assume L L = L L where L is a N N Laplacian matrix corresponding to the connections between the elements of V Let D be the matrix of vertex in-degrees and A the adjacency matrix such that L = D A ; L corresponds to a strongly connected graph G, hence D exists and A is irreducible Define L = D L = I D A Let à = D A which is again an irreducible matrix From L = we deduce à =, hence à is stochastic By Lemma 8, there is a vector q > such that q à = q, which implies q L = It follows that q = D q > and q L = Hence q = q, is a left-eigenvector of L with q e i > if i V, q e i = if i V, and q L = Finally p = q q satisfies p e i > if and only if i V p e i = if i V and p is the unique vector satisfying p L = and p = heorem 9 conveys that the consensus state is influenced by the initial state of agent i iff there is a directed path from vertex i to every other vertex in the graph Intuitively, as all agents converge to the consensus state, it can only depend on information that is available to all agents at least indirectly Observe that p i ζ k = p ξ k i V = γ i ξ k i p i i V γ i is a weighted average of the initial states of agents corresponding to the vertices in V where the weights can be adjusted arbitrarily by appropriately choosing the values γ i, i V he remaining values γ i, i V do not affect the consensus state but may affect the eigenvalues of L 4 CONSENSUS DESIGN So far we reduced the problem of checking convergence to consensus to investigate whether a given polynomial is Hurwitz he coefficients of the polynomial depend on the interconnection topology, the single agent dynamics, and the consensus algorithm, namely the values β k, k =,, In this section, we derive methods to determine the gains β k, k =,,n in the consensus algorithm, such that convergence to consensus is guaranteed In the sequel we assume γ i =, i =,,N, ie Γ = I for simplicity If γ i for some i, all results remain valid substituting L by L 4 Double integrator agents We first consider the case of double-integrator dynamics ξ i = u i, i =,,N 9 which received a lot of attention in consensus literature de GennaroandJadbabaie, 26; Olfati-Saber, 26; Ren and Atkins, 25 Applying the consensus algorithm u i = β Wj, iξ j ξ i + β Wj, i ξ j ξ i, we know that the consensus state grows unbounded since 9 is unstable For problems where this unboundedness is acceptable, the following theorem gives a necessary and sufficient condition for convergence to consensus heorem Assume the group of agents defined by 9 with consensus algorithm and interconnection topology G = {V, E, W } he agents reach consensus if and only if G contains a spanning tree, β >, β >, and Imλi L β 2 β > max i=2,,n Reλ i L λ i L Proof Using heorem 3 we need to check that N s 2 + β λ j Ls + β λ j L j=2 } {{ } =: p j s 2 2 is Hurwitz if and only if β >, β >, holds, and G contains a spanning tree Polynomial 2 is Hurwitz if and only if all real factors of 2 are Hurwitz In the first part of the proof, we shall show that 2 is not Hurwitz if G does not contain a spanning tree In the second part of the proof we show that, under the assumption that G contains a spanning tree, 2 is Hurwitz if and only if β >, β >, and holds Part : If G does not contain a spanning tree, λ 2 L =, thus p 2 s = s 2 is not Hurwitz Part 2: In the sequel, we assume that G contains a spanning tree, thus Reλ i L >, i = 2,,N Assume Imλ i L = for some i hen the correpsonding p i s is Hurwitz if and only if β λ i L > and β λ i L > which is equivalent to β > and β > If Imλ i+ L for some i +, i such that λ i+ L and λ i L are complex conjugate In that case p i+ s p i s has real coefficients Let λ i± L = σ i ± jω i, then p i+ s p i s = s 4 s 3 s 2 s 2β σ i 2β σ i + β 2 σ2 i + ω2 i 2β β σ 2 i + ω2 i β 2 σ2 i + ω2 i 544

5 which is Hurwitz if and only if β >, β >, and β 2 > 2 ωi 3 β σ i σ i + jω i For ω i =, condition 3 reduces to β 2/β > which is satisfied for any β >, β > Consequently, 3 is satisfied for i = 2,,N iff is satisfied Hence, if G contains a spanning tree, 2 is Hurwitz iff β >, β > and hold Corollary Assume the group of agents as in heorem and G balanced If β > and < β β 2 ag, then ξ k i t ζ k t, i =,,N, k =,,n Proof Corollary is a direct consequence of heorem and Fact 2 Remark 2 Choosing β =, condition of heorem reduces to β > max i=2,,n Reλ i L Imλ i L λ i L As this bound on β is necessary and sufficient for consensus, it needs to be tighter than the sufficient bound 2 β > max i=2,,n λ i L cos π 2 ta Reλ il Imλ il of heorem IV2 in Ren and Atkins 25 his fact can be seen observing that 2 λ i L cos π 2 ta Reλ il Imλ = 2 Reλ i L il In fact, given some r >, if β >, β = rβ >, and β 2 = β β r max 2 4 i=2,,n Reλ i L it can be observed after some computation that all roots s i of polynomial 2 satisfy Res i β β = r < 42 General LI single-input systems For the general case of agents modeled as LI system with consensus algorithm 2, an LMI-based method to find values β k, k =,, that guarantee convergence to consensus is given in the following theorem heorem 3 Assume the group of agents defined by with consensus algorithm 3 using Γ = I and interconnection topology G containing a spanning tree Define M = α α n 2 α If, for some ν, there exist a matrix Q R N N, Q = Q and a vector κ R N such that the LMIs C + Reλ i LC R + Imλ i LC I 5 with QM C = + M Q + 2νQ QM, + M Q + 2νQ en κ C R = + κe n e n κ + κe, n κe C I = n e n κ e n κ κe n Im d d max µ µ 2 µ 3 L 2 d, σ L 2 d max, σ min σ σ min d max d 2d max 2d Re Fig Sets containing the non-zero eigenvalues of L are satisfied for i = 2,, N, the agents reach consensus choosing β = β,,β = Q κ, and the roots s j of polynomial 4 satisfy Res j < ν, j =,,nn Proof Assume λ i L = σ i + jω i, i = 2, N he characteristic polynomial of M σi e ẇ = M w n ω i e n β ω i e n σ i e n }{{}}{{} β w =: M =: B }{{} =: K is given by p 2 i s if λ i = σ i ω i = or p i+ s p i s if λ i± = σ i ± jω i Hence, if there exist a constant ν and a matrix P = P such that M P + P M K B P P B K 2ν P, 6 polynomials p 2 i s as well as p i + s p i s are Hurwitz with roots s j satisfying Res j < ν, j =,,2n With P P = P, Q = P, κ = Qβ, LMI condition 6 reduces to 5 he design proposed in heorem 3 contains some sources of conservatism due to the fixed block-diagonal structure of P and most notably because the same P is used for all λ i L, i = 2,,N Allowing different matrices P i for different eigenvalues λ i L would result in a nonlinear matrix inequality due to the terms P i e n β Exploiting convexity, the LMI conditions of heorem 3 can be relaxed to obtain conditions that are easier to check Denote L L = conv{λ 2 L,, λ N L} where conv is the convex hull Observe that Cλ = C + ReλC R + ImλC i is affine in λ and Cλ if and only if Cλ where λ denotes the complex conjugate of λ Hence, given values µ i C, i =,, q such that L L conv{µ,, µ q, µ,,µ q }, Cµ i, i =,,q is a sufficient condition for LMI 5 being satisfied for i = 2,,N By Gershgorin s disk theorem see Horn and Johnson 985 we know that λ i L Bd max := {λ C : λ d max d max } for i =,,N where d max = max i j P i Wj, i is the maximum vertex in-degree Hence, given any σ σ min := Reλ 2 L and any d d max, one obtains L L L 2 d, σ := {λ C : Reλ σ} Bd Let µ = σ + jσ 2d σ σ, µ 2 = d σ 2d σ + jd, and µ 3 = 2d + jd to obtain a simple polygon containing L 2 d, σ, namely conv{µ, µ 2, µ 3, µ, µ 2, µ 3 } L 2 d, σ L L Figure illustrates the sets L 2 d, σ, L 2 d max, σ min, and the values µ i, i =,,3 A sufficient condition for LMI 5 being satisfied for i = 2, N is 545

6 ζt, ξit ζt ξ t ξ it 9 8 Fig 2 Graph G of an imploding star with a directed circle given by Cµ i, i =,,3 If the interconnection topology is balanced, one may choose σ = ag 7 5 EXAMPLE As an example we consider again individual agents modeled as 9 We assume the interconnection topology given by Figure 2 with all edge weights equal to one All vertices on the circle can be chosen as the root of a spanning tree he spectrum of the Laplacian L of G reads { λl, 2 3 ± j, ± j 3, ± j, ± j 3, ± j }, 2, In accordance with heorem 9, one obtains the vector p =, 2,, 2, ie all nodes on the circle have the same influence on the consensus state while the node in the center does not influence the consensus state As graph G is not balanced, the algebraic connectivity ag is meaningless in this example in fact ag < Conditions and 4 evaluate to β 2 > β 2 696, = β β 2 β r > , respectively Using the LMI of heorem 3 with ν = 2, one obtains β 9649 and β 2455 as a solution with r = β β 374 and β r 6552 > 493 Hence, we know that Res i r and Res i ν by Remark 2 and heorem 3, respectively; here r > ν, ie Remark 2 gives a better estimate of the convergence rate than heorem 3 In fact, max i Res i 3745 which is very close to r Figure 3 shows simulation results for some initial conditions satisfying p ξ = ζ = and p ξ = ζ = All agents converge to consensus exponentially with convergence rate larger than r he consensus state evolves according to ζt = ζ + ζt = t, ζt = ζ = as expected 6 SUMMARY his paper was concerned with the consensus problem for MASs with agents modeled as general LI systems Motivated by the fact that there exist only few results on the problem, the presentpapergeneralizedseveralexisting results on consensus with agents modeled as a single or double integrator he characteristic equation of the whole MAS was derivedexplicitly, whichwas thenused to obtain a condition on the consensus algorithm ensuring convergence In addition, a meaningful interpretation for the role of the left-eigenvector of the Laplacian corresponding to its 6 5 ζt, ξit 3 7 ζt ξ t ξ it t Fig 3 Simulation results for topology given in Figure 2 zero eigenvalue was given Namely this eigenvector quantifies the influence of each agent on the consensus state As far as design is concerned, a systematic method was proposed to choose the gains in the consensus algorithm such that the MAS reaches consensus asymptotically with prespecified convergence rate 5 5 REFERENCES Maria Carmela de Gennaro and Ali Jadbabaie Decentralized control of connectivity for multi-agent systems In Proc 45th IEEE Conf Decision and Control, pages , 26 Miroslav Fiedler Algebraic connectivity of graphs Czechoslovak mathematical journal, 232:298 35, 973 Chris Godsil and Gordon Royle Algebraic Graph heory, volume 27of Graduateextsin Mathematics Springer, 24 Roger A Horn and Charles R Johnson Matrix Analysis Cambridge University Press, 985 Reza Olfati-Saber Flocking for multi-agent dynamic systems: Algorithms andtheory IEEE rans Automat Contr, 53:4 42, 26 Reza Olfati-Saber, J Alexander Fax, and Richard M Murray Consensus and cooperationin networkedmultiagent systems Proc IEEE, 95:25 233, 27 Wei Ren and Ella Atkins Second-order consensus protocols in multiple vehicle systems with local interactions In Proc of theaiaa Guidance, Navigation, and Control Conference and Exhibit, 25 Wei Ren, Randal W Beard, and imothy W McLain Coordination variables and consensus building in multiple vehicle systems In V Kumar, N Leonard, and A S Morse, editors, Cooperative Control, A Post-Workshop Volume 23 Block Island Workshop on Cooperative Control, volume 39 of LNCIS, pages 7 88 Springer, 24 Wei Ren, Kevin Moore, and YangQuan Chen High-order consensus algorithms in cooperative vehicle systems In Proceedings of the 26 IEEE International Conference on Networking, Sensing and Control ICNSC, pages , 26 Wei Ren, Randal W Beard, and Ella M Atkins Information consensus in multivehicle cooperative control IEEE Control Systems Magazine, 272:7 82, 27 Herbert G anner, Ali Jadbabaie, and Georges J Pappas Flocking in fixed and switching networks IEEE rans Automat Contr, 525: , 27 Chai Wah Wu Algebraic connectivity of directed graphs Linear and Multilinear Algebra, 533:23 223,